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Mathematics Department
The Mathematics Department consists of nine full-time faculty, as well as
adjunct and online instructors. It offers transitional, general studies, and curriculum mathematics classes on the main
campus, as distance learning classes, in some high schools, and online. The goal of the department is to provide the highest
quality instruction at all levels by employing current best practices in mathematics education. Faculty members are a dynamic, forward-thinking team who:
Integrate technology into all classes. This includes web-based materials as well as the use of the graphing calculator.
Offer collaborative learning opportunities and use alternative forms of assessment in all classes.
Stay abreast of new trends and update skills through conferences, workshops, self-study, and faculty collaboration.
Monitor progress by assessing student achievement data.
Continuously review and update course materials.
Are actively involved in campus activities such as the Phi Theta Kappa Honor Society and the Technology Challenge Summer Camp.
Several faculty members are ATE (Advanced Technological Education) Exemplary Faculty. |
More About
This Textbook
Overview
Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many courses such as divisors, Euclidean algorithms, modular arithmetic, equivalence relations, and injectivity and surjectivity of functions. The material has been tested by real students over many years so all the essentials are covered. With over 300 exercises to help you test your progress, you'll soon learn how to think like a mathematician.
Editorial Reviews
From the Publisher
"In this book, Houston has created a primer on the fundamental abstract ideas of mathematics; the primary emphasis is on demonstrating the many principles and tactics used in proofs. The material is explained in ways that are comprehensible, which will be a great help for people who seem to hit the wall regarding what to do when confronted with the creation of a proof... In this book, Houston takes a systematic and gentle approach to explaining the ideas of mathematics and how tactics of reasoning can be combined with those ideas to generate what would be considered a convincing proof."
Charles Ashbacher, Journal of Recreational Mathematics |
2WN10 - Scientific Computing
Matlab
The assignments form an essential part of the course. Some of the
exercises require writing computer programs.
It is advised to do the programming in Matlab.
TU/e has a nice
Matlab-website.
The parts you should look at are Chapter 1: Basic elements of Matlab
and Chapter 6: Programming in Matlab.
Course materials
The course is based on the book Iterative methods for Sparse Linear Systems by
Yousef Saad.
The second edition of this book is available from the author's
website as a PDF file with links. A copy of the PDF file is available
here. |
CAS Online Course 2, Second Edition Insurance Accounting. August 31, 2012, for those who purchased the 1st Edition, the original online course will be available and unchanged 2nd Edition: CA2, 2nd Edition, is the current version that became effective on September 1,. 2012. Given a case, evaluate one or more entities' insurable interests. personal risk management needs.
Mathematics Explorations Second Edition Teacher - ActiveMath. y the actual purchaser of the book. w w w .goodyearbooks. *This activity lesson is based on a "Mathematics Detective" article written by David B. The answer given in the answer key for this item was D, 6.17. how long it takes for it to stop? Solutions. 1a. Victim A—Missing Person T b. Victim B—Missing Person Z c. |
Formats
Book Description
Publication Date: March 27, 2009Editorial Reviews
About the Author
Kara Monroe is a graduate of Ball State University with a degree in Mathematics Education. Kara taught high school mathematics in Lakeland, Florida. Kara went on to join Ivy Tech Community College Richmond as the manager of instructional technology. While at Ivy Tech, Kara has earned her MBA from Jones International University and a Ph.D. in Higher Education Leadership from Capella University. Kara serves as the Executive Director of Finance, Facilities, and Information Technology at Ivy Tech Community College Richmond. However, her passion remains with students in the classroom and she seeks every opportunity to teach and develop courses in mathematics, computer information systems, business, and education leadership.
This is a great book for what it is designed for. It does a fabulous job of presenting a lot of material in a concise manner. The explanations are easy to follow, and there is a lot of variety in the techniques used. You will not be able to use this book alone to learn all the math taught in high school, but that is not the intent. The intent is to use it as a supplement for another class or in a tutoring session.
A great addition to a student's library. This "textbook" will provide the student with a resource guide for the math courses they are bound to take during their high school years. And for many students, this will assist them well into their college years. As mentioned in the beginning of the text, "The purpose of this book is to outline the basic skills a student at the middle school level - or a student who may be struggling in high school mathematics - needs and to demonstrate those skills in an easy to understand way." I know it will help my daughter as she begins 8th grade next week!
This book should act as a supplement to your studying. Certain sections are quite thorough, while others only provide quick refreshers. Here is my take on each section:
Essential Math Skills - This is a good, quick introduction to basic math skills, but I don't think that it will be useful for a high school student. Upon entering high school, a student should have mastered such topics as fractions and integers, and if he or she hasn't, then he or she is in trouble.
Algebra I - You are inundated with information in this section, and although the explanations are clear, there are not enough examples.
Algebra II - Dr. Monroe provides plenty of examples, and her step-by-step method of solving them is very clear and helpful. I liked how she details the four ways to solve systems of linear equations.
Geometry - This chapter does not cover geometric proofs, which are fundamental to the subject.
Trigonometry - A section on vectors would have been useful. Also, Dr. Monroe could have provided general instruction on how to use a graphing calculator to graph functions.
Pre-Calculus - I liked the inclusion of word problems, which are sometimes left out in similar study aides. The chapter does lack sections on limits and polar coordinates.
Calculus - This section could have been broadened to include more topics and flesh out certain concepts. It should be used only for a basic review of derivatives and integrals.
Statistics - The section on Probability is very limited, and Probability is an integral part to the study of Statistics.
Pros: Tutor in a book - I have taught and tutored high school and college math students in various topics such as algebra, calculus, statistics, probability, engineering, etc. for the past seven years. During this time I have come across a variety of books. Many of them are popular among students but in my opinion they are just a collection of solved problems and unsolved exercises. Even the solved problems leave students guessing about intermediate steps. No book in my opinion has so far made a sincere effort to help students develop the concepts of mathematics by guiding them in a step - by - step hands on approach. And this is where this book steps in. It solves the problems in elementary steps and in the process explains the concept very coherently. It is a tutor in a book!
Arsenal of solutions - The authors seem to have done research on various types of problems which generally show up on tests and board exams and they have produced a detailed solution mechanism for each problem type. So, this does help students develop an arsenal of solutions to deal with different kinds of problems he/she might encounter on tests. Let's be honest here, at the high school level there are not infinite types of problems. Hence, to a certain extent Dr. Monroe's book has solved the ubiquitous uncertainty about different popular forms in which problems can show up on the exam.
Variety of topics - Even though this is pretty much standard for all books, I like the organization of this book as far as presenting the topics is concerned. Dr. Monroe has tried to build up the students skills in a highly structured manner. She starts with an introduction to some basic math skills and gradually develops it to the level of calculus while systematically traversing algebra, geometry, trigonometry and pre-calculus. This also shows the foresight of the author in trying to help a student build advanced skills gradually and comfortably instead of inundating the student with high flying concepts and making math as dreadful as the rumor is.
Good for teachers - This book should be read by teachers as well. I very much like the writing/teaching style. The writing style of the book can be likened to having a live tutor. The problem with many teachers is not that they do not know the subject matter but that they do not know how to present the matter to a struggling student. This book in my opinion will help teachers learn how to teach more effectively.
Lucid writing style - This book is written in simple English, so it will benefit students who are not the native speakers of English as well.
If you follow the book you will not only know math but you will also develop problem solving skills. I do not have any doubt when I say that you will score in top percentile on the sections discussed in the books when taking your board exams and standardized tests such as the SAT.
Cons: I like the glossary section at the end of the book but I miss the index. I hope the publishers add the index sections in the future editions so that navigating through the book becomes easier.
The author should have included Probability as a separate chapter as it is a very confusing topic in general and as detailed learning as the other topics in this book would have provided great help to students.Read more › |
Elementary Linear Algebra - 4th edition
Summary: The text starts off using vectors and the geometric approach, plus, it features a computational emphasis. The combination helps students grasp the concepts. At the same time, it provides a challenge for mathematics majors |
The major in Mathematics will enable students to: 1)
obtain a background in mathematics (including Analysis,
Algebra and Differential Equations) sufficient for
graduate work; 2) organize and analyze information,
solve problems readily and construct logical arguments;
3) do mathematics, using modern technology when appropriate;
4) appreciate the utility of mathematics.
MPT refers to the Mathematics
Placement Test. This test consists of 55 multiple
choice questions over arithmetic and algebra. Sample
questions may be obtained from the Mathematics-Physics
department or from the Admissions office.
COURSES IN MATHEMATICS (MAT)
013 Intermediate Algebra Topics include real
and complex numbers, factoring, solving linear and non-linear
equations and inequalities in one variable, and applications
of equations. Many course topics will be illustrated
with calculators and/or computers. Does not satisfy
General Education requirements. May not be taken concurrently
with or after completion of the equivalent of MAT 133
(or the equivalent of any higher numbered MAT course)
with a C- or better. (F/S)
104 Mathematics for Elementary and Middle School
Teachers—Prerequisite: Admission in the Teacher
Ed. program or permission. This course prepares prospective
teachers of elementary and middle school mathematics.
An understanding of the basic operations in mathematics
and their grounding in sets is developed. Topics include
sets, numeration, measurement, geometry, logic, number
theory, algebraic reasoning, statistics, and problem
solving. Mathematical learning experiences using manipulatives
are the focus of the laboratory portion of the course.
Three hours lecture, one two-hour laboratory per week.
(F/S)
143 Statistics—Prerequisites: MAT 013 or Math
ACT 21 or greater or MPT 25 or greater or permission.
This course is an algebra-based introduction to descriptive
and inferential statistics. Topics include discrete
and continuous distributions, measures of central tendency
and variability, correlation and regression, and analysis
of variance.
202 Introduction to the History of Mathematics—Prerequisite:
MAT 175 or permission. Designed to give an appreciation
of the historical development of mathematics. The course
includes a survey of various concepts and their developments
from antiquity to present, with emphasis on the people
involved in the growth of mathematics. (S–odd)
303 Number Theory—Prerequisite: MAT 135 or permission.
An introduction to the theory of numbers. A variety
of topics dealing with the properties of integers will
be covered. These include division and factorization
properties of integers, prime numbers and perfect numbers,
number representations, linear diophantine equations,
linear and quadratic congruences, quadratic reciprocity
and others.
323 Modern Geometry—Prerequisite: MAT 135 or
permission. A study of some modern aspects of mathematics,
including logical foundations of geometries and an introduction
to non-Euclidean geometries. Recommended for prospective
secondary teachers of mathematics. (F–odd)
353 Numerical Analysis—Prerequisite: MAT 205
and CSC 133 or permission. Topics include finite difference
formulas, polynomial approximations and interpolation,
numerical differentiation and integration, solutions
of differential equations. The emphasis is on numerical
techniques used in conjunction with digital computers.
Work on the computer will be included. (S–even)
363 Complex Variables—Prerequisite: MAT 135
and MAT 213 or permission. The course will introduce
the subject of complex variables. In particular, we
will study the properties of complex numbers; analytic
and elementary functions; complex integration including
definite, contour, and line integrals; series; residues
and poles.
401 Mathematics Seminar—Prerequisite: permission.
Seminars and discussions of mathematical topics of current
interest to the staff and students. Students will generally
be expected to present one or more seminars. May be
taken as many as three times.
403 Advanced Calculus I—Prerequisite: MAT 135
and MAT 213 or permission. Topics covered include an
advanced treatment of the theory of continuous functions
in n-dimensional space; line, path, surface integrals
and their applications; conditional, absolute, and uniform
convergence of series, and the representation of functions
by power and trigonometric series. (F)
413 Advanced Calculus II—Prerequisite: MAT 403.
A continuation of MAT 403. Topics covered vary but frequently
include differentiation, Riemann-Stieltjes Integral,
sequences of functions, series of functions, special
types of series and functions of several variables.
423 Methods of Teaching Mathematics in the Secondary
School Mathematics teaching and learning in grades
7 through 12. Topics include objectives, teaching materials,
use of technology in mathematics teaching and learning,
curriculum organization and principles of presenting
mathematical content to secondary students. Does not
count toward a major in mathematics. (S–even)
453 Great Ideas in Mathematics—Prerequisite
and/or Corequisite: MAT 403 and MAT 433 or permission.
This course will attempt to tie together the many concepts
already discussed in previous classes as well as to
provide a medium in which a student is encouraged to
assimilate a variety of complex mathematical ideas into
a cohesive and thoughtful presentation. An additional
course aspect is the administering of an exam typifyng
the subjects common to a degree in mathematics at College
of the Ozarks. Writing Intensive. (F)
49V (1-6) Special Problems—Prerequisite: permission.
An opportunity to do directed research under the supervision
of a program faculty member.
COURSES IN PHYSICS (PHY)
104 Fundamentals of Physical Science An introductory
course in the physical sciences designed primarily for
non-science majors. This course covers important topics
in physics and astronomy. Three hours lecture and one
two-hour lab per week. (F/S)
114 Astronomy A course intended for students
with little or no scientific background, covering the
solar system, distances and motions of stars, the light
from stars, stellar evolution, star clusters, galaxies
and cosmology. Sky observation with the college's 8"
reflector is included. It is recommended that PHY 104
and MAT 013 be taken prior to Astronomy. (F)
124 Introduction to Geology A comprehensive
study of physical and historical geology with an emphasis
on geological phenomena occurring in North America.
Some emphasis is placed on the geology of the Ozark
Region. Three hours of lecture and one two-hour lab
period per week. (S)
204 Physics for the Life Sciences—Prerequisite:
MAT 133 and MAT 142 or MAT 135 or permission. A course
intended for Biology majors, Medical Technology and
Pre-Nursing students and other interested students.
This course will cover the basic concepts of physics
(mechanics, heat, light, sound, electricity, atomic
and nuclear physics) and their applications to the life
sciences. (S–odd)
225 General Physics I—Prerequisite: High school
physics or PHY 104 and MAT 175. Covers the principles
of mechanics. For engineering and science majors. Three
lecture periods and one two-hour experimental laboratory
and one two-hour computational laboratory each week.
(F) |
Buy PDF
Standards-Driven Math Vocabulary Ranking Intervention Protocol (VRIP) is a textbook and classroom supplement for students, parents, teachers and administrators who need better options for math intervention classes ranging in difficulty from pre-algebra to geometry. This book is from the official Standards-Driven Series (Standards-Driven and Math VRIP are trademarks of Nathaniel Max Rock). Included are more than 750 middle school and high school math vocabulary words ranked in order from easiest to hardest for maximum standards-driven informed intervention instruction. Three diagnostic (pre & post) tests are included. Concise definitions for all words are included and each is indexed to the standard from which the word came. There is a discussion format for all words. Ranked folded-page study sheets are included for all levels. Author Nathaniel Max Rock is an engineer by training with a Masters Degree in business and is a math teacher. He brings years of life-learning and math-learning experiences to this work which is used as a supplemental text in his math intervention classes. If you are struggling in a "standards-based" math class, then you need this book! 424 pages (E-Book ISBN#0-9749392-4-2 (ISBN13#978-0-9749392-4-7)) |
The following computer-generated description may contain errors and does not represent the quality of the book: These examples are intended to provide a complete course of elementary algebra for classes in which the bookwork is supplied by the teacher. In the choice of the subjects included, and in their arrangement, I have throughout followed the recommendations of the Committed on the Teaching of Mathematics appointed by the Mathematical Association. Among new features may be mentioned: The postponement until after easy simultaneous equations of the long rules for multiplication and division. The postponement until after quadratics of complicated fractions, H.C.F., L.C, M., Square root, and Literal Equations. The early introduction and extensive use of Graphs. The inclusion of some of the applications to Geometry which form such a prominent feature in modem continental text-books. The treatment of fractional indices from a numerical point of view, so as to lead up to the use of four-figure logarithm tables. The stress laid oo numerical checks of all kinds. The large selection of problems, including very easy ones. |
The Identity Matrix
When dealing with matrix computation, it is important to understand the identity matrix. We can think of the identity matrix as the multiplicative identity of square matrices, or the one of square matrices. Any square matrix multiplied by the identity matrix of equal dimensions on the left or the right doesn't change. The identity matrix is used often in proofs, and when computing the inverse of a matrix.
Study Your Way
Easy Help. Fun Teachers.
Expert teachers who know their stuff bring personality & fun to every video. |
Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
Need to learn MATHEMATICA? Problem SOLVED
Take full advantage of all the powerful capabilities of Mathematica with help from this hands-on guide. Filled with examples and step-by-step explanations, Mathematica Demystified takes you from your very first calculation all the way to plotting complex fractals.
Using an intuitive format, this book explains the fundamentals of Mathematica up front. Learn how to define functions, create 2-D graphs of functions, write basic programs, and use modules. You'll move on to 3-D graphics, calculus, polynomial, linear, and differential equations, dynamical systems, and fractals. Hundreds of examples with concise explanations make it easy to understand the material, and end-of-chapter quizzes and a final exam help reinforce learning.
This self-teaching guide offers: A quick way to get up and running on MathematicaCoverage of Mathematica 6 and 7Tips for avoiding and correcting syntax errorsDetails on creating slideshow presentations of your workNo unnecessary technical jargonA time-saving approach to performing better on an exam or at work
Simple enough for a beginner, but challenging enough for an advanced user, Mathematica Demystified is your shortcut to mastering this fully integrated technical computing software.
Synopsis:
The fast and easy way to learn MathematicaAbout the Author
Jim Hoste, Ph.D. (Claremont, CA) is a Professor at Pitzer College, and a member of the Mathematical Sciences Research Institute. Hes been an Associate Editor of the Journal of Knot Theory and Its Ramifications since 1991, and has authored dozens of research publications. |
Syllabus PAT
PAT REVISED SYLLABUS ENGLISH
(As per 10th class Grammar of H.P. Board of School Education) :-
A variety of questions involving the use of particular structures with a context. Text types used include cloze, gap-filling, sentences-completion; sentence-reordering, dialogue-completion and sentencetransformation ( including combining sentences). The Grammar syllabus will include the following areas:
Effects of current and Magnets: Potential, Potential difference, Ohm's law; Series combination of resistors, parallel combination of resistors; Power dissipation due to current; Inter relation between P,V.I and R. Magnetic field, field lines, field due to a current carrying wire, field due to current carrying coil or solenoid; Force on current carrying conductor, Fleming's left hand rule. Electro magnetic induction. Induced potential difference, induced current. Alternating current,; frequency of AC, Advantage of AC over DC. Domestic electric circuits.
Refraction; laws of refraction: Image formed by a convex lens; functioning of a lens in human eye; problems of vision and remedies. Applications of spherical mirrors and lenses. Appreciations of concept of refraction; velocity of light; refractive index twinkling of stars; dispersion of light. Scattering of light.
ALGEBRA
1. Polynomials:- Zeros of a polynomial. Relationship between zeros and coefficients of a polynomial with particular reference to quadratic polynomials, Statement and simple problems on division algorithm for polynomials with real coefficients.
2. Pair of Linear Equations in Two Variables:- Pair of linear equations in two variables; Geometric representation of different possibilities of solutions inconsistency; Algebraic conditions for number of solutions; Solution of pair of linear equations in two variables algebraically by substitution by elimination and by cross multiplication; Simple situational problems must be included; Simple problems on equations reducible to linear equations may be included.
3. Quadratic Equations:- Standard form of a quadratic equation ax2 + bx + c = 0, (a ≠ 0). Solution of the quadratic equations (only real roots) by factorization and by completing the square, i.e by using quadratic formula. Relationship between discriminate and nature of roots. Problems related to day to day activities to be incorporated.
4. Arithmetic Progression:- Motivation for studying AP. Derivation of standard results of finding the nth term and sum of first n terms.
TRIGONOMETRY
1. Trigonometric Ratios:- Trigonometric ratios of an acute angle of a right-angled triangle. Proof of their existence (well defined); motivate the ratios, whichever are defined at 0° & 90°. Values (with proofs) of the trigonometric ratios of 30°, 45° & 60°. Relationships between the ratios.
2. Trigonometric Identities:- Prro and applications of the identity sin2 A + cos2 A = 1. Only simple identities to be given. Trigonometric ratios of complementary angles.
3. Heights and Distances:- Simple and believable problems on heights and distances. Problems should not involve more than two right triangles. Angles of elevation / depression should be only 30°, 45°, 60°.
COORDINATE GEPMETRY
1. Lines (in two dimensions):- Review the concepts of coordinate geometry done earlier including graphs of linear equations. Awareness of geometrical representation of quadratic polynomials. Distance between two points and section formula ( internal). Area of a triangle.
GEOMETRY
1. TRIANGLES:- Definitions, examples, counter examples of similar triangles.
* If a line is drawn parallel to one side of a triangle to intersect to other two sides in distinct points, the other two sides are divided in the same ratio.
* If a line divides two sides of a triangle in the same ratio, the line is parallel to the third side.
* If in two triangles, the corresponding angles are equal, their corresponding sides are proportional and the triangles are similar.
* If the corresponding sides of two triangles are proportional, their corresponding angles are equal and the two triangles are similar.
* If one angles of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the two triangles are similar.
* If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse, the triangles on each side of the perpendicular are similar to the whole triangle and to each other.
* The ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.
* In a right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides.
* In a triangle, if the square of one side is equal to the sum of the squares on the other two sides, the angles opposite to the first side is a right triangle. 2. CIRCLES:
Tangents to a circle motivated by chords drawn from points coming closer and closer and closer to the point.
* The tangent at any point of a circle is perpendicular to the radius through the point of contact.
* The lengths of tangents drawn from an external point to circle are equal. 3. CONSTRUCTIONS:-
Division of a line segment in a given ration (internally). Tangent to a circle from a point outside it. Construction of a triangle similar to a given triangle.
MENSURATION
1. AREAS OF PLANE FIGURES:- Motivate the area of a circle ; area of sectors and segments of a circle. Problems based on areas and perimeter/ circumference of the above said plane figures. (In calculating area of segment of a circle, problems should be restricted to central angle of 60°, 90° & 120° only. Plane figures involving triangles, simple quadrilaterals and circle should be taken).
2. SURFACE AREAS AND VOLUMES:-
(i) Problems on finding surface areas and volumes of combinations of any two of the following cubes, cuboids, spheres, hemispheres and right circular cylinder, cones. Frustum of a cone.
(ii) Problems involving concerting one type of metallic solid into another and other mixed problems. (Problems with combination of not more than two different solids be taken)
STATISTICS
1. STATISTICS:- Mean, median and mode of grouped data (bimodal situation to be avoided). Cumulative frequency graph.
2. PROBABILITY:- Classical definition of probability. Connection with probability as given in Class IX. Simple problems on single events, not using set notation. |
This is a short, focused introduction to MATLAB, a comprehensive software system for mathematical and technical computing. It contains concise explanations of essential MATLAB commands, as well easily understood instructions for using MATLAB's programming features, graphical capabilities, simulation models, and rich desktop interface. Written for MATLAB 7 it can also be used with earlier (and later) versions of MATLAB.This book teaches how to graph functions, solve equations, manipulate images, and much more. It contains explicit instructions for using MATLAB's companion software, Simulink, which allows graphical models to be built for dynamical systems. MATLAB's new "publish" feature is discussed, which allows mathematical computations to be combined with text and graphics, to produce polished, integrated, interactive documents.
MATLAB is an incredibly rich and complicated software system; this book focuses first on the essentials, then develops finer points through numerous examples
Reviews & endorsements
"Major highlights of the book are completely transparent examples of classical yet always intriguing mathematical, statistical, engineering, economics, and physics problems. In addition, the book explains a seamless use with Microsoft Word for integrating MATLAB outputs with documents, reports, presentations, or other on-line processes. Advanced topics with examples include: Monte Carlo simulation, population dynamics, and Linear Programming. Overall, it is an outstanding textbook, and, likewise, should be an integral part of the technical reference shelf for most IT professionals. It is a great resource for wherever MATLAB is available!"
ACM UbiquityBrian R. Hunt, University of Maryland, College Park Brian R. Hunt is an Associate Professor of Mathematics at the University of Maryland.
Ronald L. Lipsman, University of Maryland, College Park Ronald L. Lipsman is a Professor of Mathematics and Associate Dean of the College of Computer, Mathematical and Physical Sciences at the University of Maryland.
Jonathan M. Rosenberg, University of Maryland, College Park Jonathon M. Rosenberg is a Professor of Mathematics at the University of Maryland.
Kevin R. Coombes, University of Texas, M. D. Anderson Cancer Center Kevin R. Coombes is an Associate Professor of Biostatistics and Biomathematics at the M.D. Anderson Cancer Center, University of Texas.
John E. Osborn, University of Maryland, College Park John E. Osborn is a Professor of Mathematics at the University of Maryland.
Garrett J. Stuck, University of Maryland, College Park Garrett J. Stuck is a Professor of Mathematics at the University of Maryland |
The development of geometry from Euclid to Euler to Lobachevsky, Bolyai, Gauss and Riemann is a story that is often broken into parts – axiomatic geometry, non-Euclidean geometry and differential geometry. This poses a problem for undergraduates: Which part is geometry? What is the big picture to which these parts belong? In this introduction to differential geometry, the parts are united with all of their interrelations, motivated by the history of the parallel postulate. Beginning with the ancient sources, the author first explores synthetic methods in Euclidean and non-Euclidean geometry and then introduces differential geometry in its classical formulation, leading to the modern formulation on manifolds such as space-time. The presentation is enlivened by historical diversions such as Huygens's clock and the mathematics of cartography. The intertwined approaches will help undergraduates understand the role of elementary ideas in the more general, differential setting. This thoroughly revised second edition includes numerous new exercises and a new solution key. New topics include Clairaut's relation for geodesics and the use of transformations such as the reflections of the Beltrami disk. less |
Euless Algebra 2Bob These more advanced courses are primarily taught by math and science experts at the University. My background in linear algebra is significant due to my advanced science related to my research endeavors in advanced science systems. Linear Algebra also used differential equations |
Digital signal processing concepts have been successfully taught in the lower division for the past ten years and now have been introduced into an engineering course designed for juniors and seniors in high school. This hands-on curriculum allows students to do real time audio, video, and communication projects using a board similar to a Texas Instruments DSK and a block diagram based development tool. The infinity curriculum, supported with a text book using precalculus mathematics, allows students to gain experience with basic concepts of DSP that they will study later with more rigor if they choose to major in engineering. The course has been adopted by over eighty high schools and has been taught in some high schools for three years. |
I don't blame anyone who didn't have good enough teachers to explain the how and the why of algebra. But throwing it out isn't the solution.
You may never use 2x² - 13x - 24 = (2x + 3)(x - 8) in real life, but understanding the relationship between a dependent variable and its independent variables will keep you from being fooled by pretty graphs the media throws at you. Knowing how exponential decay works will help you to recognize that Al Gore's "hockey stick curve" is something he's just pulling out of thin air. Even something as potentially confusing as integration can help you to understand why the debt crisis isn't going to go |
ISBN13:978-0201347302 ISBN10: 020134730X This edition has also been released as: ISBN13: 978-0201384086 ISBN10: 0201384086
Summary: This...show more. Students using this text will receive solid preparation in mathematics, develop confidence in their math skills and benefit from teaching and learning techniques that really work. For mathematics teachers. ...show less
(Each chapter begins with a ''Preliminary Problem'' and concludes with a ''Hint for Solution to the Preliminary Problem,'' ''Questions from the Classroom,'' ''Chapter Outline,'' ''Chapter Review,'' and a ''Selected Biography.'') (*indicates optional section.)
Integers and the Operations of Addition and Subtraction. Multiplication and Division of Integers. Divisibility. Prime and Composite Numbers. Greatest Common Divisor and Least Common Multiple. *Clock and Modular Arithmetic.
5. Rational Numbers as Fractions.
The Set of Rational Numbers. Addition and Subtraction of Rational Numbers. Multiplication and Division of Rational Numbers. Proportional Reasoning.
Instructor's EditionBook Daddy Fort Wayne, IN
Hardcover Very Good 020134730X Purchase Protected By Our Satisfaction Guarantee |
Precalculus: Mathematics for Calculus
9780840068071
ISBN:
0840068077
Edition: 6 Pub Date: 2011 Publisher: Brooks Cole
Summary: Designed to give students a background in mathematics theory and introduce them to mathematics concepts this textbook is comprehensive without being daunting. Students are introduced to modelling and problem solving and they are given a rigorous workout on what they have learned as they work through the book. It has many graphs that chart mathematical ideas that students can assimilate with ease. It is written in a c...lear and readable style that will aid comprehension and enjoyment. This is just one of the many cheap math textbooks we have available for students to acquire in great condition.
Stewart, James is the author of Precalculus: Mathematics for Calculus, published 2011 under ISBN 9780840068071 and 0840068077. Nine hundred seventy three Precalculus: Mathematics for Calculus textbooks are available for sale on ValoreBooks.com, two hundred nine used from the cheapest price of $89.30, or buy new starting at $155.18 Free Edition: Same as student edition but has free copy markings. Almost new condition. SKU:97808... [more] |
Algebra - 2nd edition
Summary: Algebra, Second Edition, by Michael Artin, discusses concrete topics of algebra in greater detail than most textbooks, preparing readers for the more abstract concepts. This book covers all of the topics that are important to the average mathematician, and are covered in the typical course. Linear algebra is tightly integrated throughout |
,...
Show More, should form a connecting link between arithmetic and the more scientific works to be studied later, and should stimulate a desire to proceed further in mathematics. It is to meet this demand in the spirit described that this book has been prepared. The time for introducing the work depends upon circumstances. In some cases elementary algebra of this nature should be begun in the first year of the high school, while other conditions make it advisable to take it up in the latter part of the grammar-school course. In either event it is desirable that pupils should have some knowledge of algebra before they leave school. For those who are not to, pursue the subject further this book furnishes such algebra as is necessary for the intelligent reading of formulas and the solution of equations found in elementary industrial manuals. Those who continue their school work will find the subject treated in this book in such a way as to stimulate an interest in their later work, and will meet no obsolete forms that must be unlearned before proceeding. In sequence of topics the author has continued the plan adopted in his arithmetics, that of recognizing the value of the various courses of study in use in different parts of the country r |
Parents - information about Math Mammoth products
Both of these series of books present basically the same lessons (they are nearly identical in the lesson content). They also both contain explanations for the concepts, along with problems & exercises.
A comprehensive coverage of math topics in each grade; e.g. a complete curriculum.
Worktexts that are great for review, reinforcement, for filling in gaps, for supplements, etc. Also, if you are not sure about Math Mammoth, you can purchase one of the Blue Series books to try it out.
Includes tests, cumulative reviews, and an additional worksheet maker. It also comes with a bonus software Soft-Pak.
(none of those)
More expensive (because it has more)
cheaper
BOTH include explanations for the concepts right in the student texts. This means there is no separate textbook and workbook, but those are combined into a single text.
BOTH concentrate on conceptual development in a logical progression.
BOTH include plenty of varied practice problems and word problems.
I also offer Review workbooks for grades 1-5, which can be used to review the math concepts of a specific grade level.
The two other Math Mammoth series (the Golden and Green series) are worksheets only, and are meant to be used by people who can explain the math to students (such as teachers, tutors, or math knowledgeable parents). |
Quantitative Reasoning
Core Competencies-Quantitative Reasoning
During the 2010 – 2011 academic year, faculty coordinating College Algebra and Statistics for Everyday Life (SEL) performed an assessment of students' ability in quantitative reasoning and mathematical reasoning. The following five learning outcomes were measures using purposefully designed or identified questions from course exams.
Learning Outcomes: Statistics for Everyday Life
Learning Outcomes: College Algebra
Express and manipulate mathematical information, concepts, and thoughts in verbal, numeric, graphical and symbolic form while solving a variety of problems.
Develop and interpret mathematical models such as formulas, graphs, tables, and schematics, and draw conclusions and/or inferences from them.
Apply appropriate quantitative skills to analyze, evaluate and determine the best feasible solution to real-world problems
Solve multiple-step problems through different (inductive, deductive and symbolic) modes of reasoning
Solve multiple-step problems through different (inductive, deductive and symbolic) modes of reasoning.
All SEL multiple-choice questions in the test-bank were mapped to one or more of the five learning outcomes. For each of the five course exams, the percent correct for each question was recorded by learning outcome. Results from the F2010 semester led to improvements in some test questions for the S2011 course offering.
College Algebra faculty designed open-ended test questions for each of the five learning outcomes and a standard five-level scoring rubric was applied to all mid-term and final exams. A score of 3 or 4 on any individual question was defined as meeting the learning outcome while a score of 0, 1, or 2 was considered unacceptable. The percent of students performing at an acceptable level was recorded and compared to the percent of students receiving a course grade of C- and above, the necessary criteria to enroll in the "next" math course. Results from the F2010 semester led to more rigorous grading policies and an automated homework system necessitating multiple efforts to reach a correct solution. |
Description of Helping Students Understand Algebra Step by Step by Carson Dellosa
Assist students to easily transition from arithmetic to algebra! Teachers can use the Helping Students Understand series as a full unit of study or as a supplement to their curriculum while parents can use this series to help their struggling students grasp algebraic concepts. This book includes step-by-step instructions with examples, practice problems using the concepts, real-life applications, a list of symbols and terms, tips, answer keys, reference, and NCTM Standards and Expectations for 2004. 128 pages.
Product:
Helping Students Understand Algebra Step by Step
Vendor:
Carson Dellosa
Media Type:
Book
Minimum Grade:
7th Grade
Maximum Grade:
7th Grade
Number of Pages:
128
Weight:
0.47 pounds
Length:
11 inches
Width:
8.5 inches
Height:
0.25 inches
Vendor Part Number:
CD-404020
There are currently no reviews for Helping Students Understand Algebra Step by Step. |
Graphing Sine
Elizabeth Kupiecki
(Teacher (K-12))
This was a great introduction to both the unit circle (if students have not seen it before) and the graph of a sine function. The review in the beginning and the practice problems helped prepare students to solve for the y values at points along the unit circle eventually leading to the sine graph. I think there was some room for improvement at the review at the beginning only covered the ratios for the different functions and did not include a review example of how to set up an equation to solve for an unknown side. I have found that setting up these equations is more of a struggle for students compared to just writing trig ratios.
I think that the flow of the lesson was great. It incorporated a lot of appealing visual elements and the flow of it made sense and was easy to follow. I enjoyed the tip on how to change the mode of the calculator from radians to degrees.
I enjoyed how students "built" a table by solving a series of problems. Initially I wondered why decimal values were used instead of exact, but this later made the graph much easier to interpret. Overall, I think this stand-alone lesson presented a difficult concept in a very understandable way.
Technical Remarks:
The powerpoint worked well, all the navigational buttons were fuctional.
However, most of the correct answers said "correct" which would need to be changed before using this with students. Likewise answers where a student needed to change the angle mode said "rad", the answer for using cosine instead of sine said "cos", etc.
Additionally some of the animations seemed a bit too long, especially the appearance of the forward button. I understand this instructionally as you want to make sure students take adequate time on the lesson, but I found myself reading the slide and then having to wait to move forward.
Time spent reviewing site:
40
1 year ago
Katlyn Witt
(Teacher (K-12)) |
This secondrevised andextendededition of the self-contained and unified approach to Bernstein functions and their subclassesbrings together old and establishs new connections. Applications of Bernstein functions in different fields of mathematics (such as probability theory, potential theory, operator theory, integral equations, functional calculi,... more... |
Precalculus Mathematics for Calculus - With CD stu...show moredents develop insight into mathematical ideas. The authors' attention to detail and clarity, as in James Stewart's market-leading Calculus text, is what makes this text the market leader. ...show less
Overview. Angle Measure. Trigonometry of Right Triangles. Discovery Project: Similarity. Trigonometric Functions of Angles. The Law of Sines. The Law of Cosines. Review. Test. Focus on Modeling:Surveying5.255.80 +$3.99 s/h
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Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
WATERBURY PUBLIC SCHOOLS
Moving Forward for Student Success
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
1
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
Background
The Waterbury Public Schools Curriculum Framework for Mathematics builds on the Common Core State Standards for Mathematics. The standards in this framework are the culmination of an extended, broad-based effort to fulfill the
charge issued by the states to create the next generation of pre-kindergarten–12 standards in order to help ensure that all students are college and career ready in mathematics no later than the end of high school.
The Council of Chief State School Officers (CCSSO) and the National Governors Association Center for Best Practice (NGA) began a multi-state standards development initiative in 2009, the two efforts merged. The standards in this
document draw on the most important international models as well as research and input from numerous sources, including state departments of education, scholars, assessment developers, professional organizations, educators from
pre-kindergarten through college, and parents, students, and other members of the public. In their design and content, refined through successive drafts and numerous rounds of feedback, the Standards represent a synthesis of the best
elements of standards-related work to date and an important advance over that previous work. As specified by CCSSO and NGA, the Standards are (1) research and evidence based, (2) aligned with college and work expectations, (3)
rigorous, and (4) internationally benchmarked. A particular standard was included in the document only when the best available evidence indicated that its mastery was essential for college and career readiness in a twenty-first-century,
globally competitive society. The standards are intended to be a living work: as new and better evidence emerges, the standards will be revised accordingly.
Waterbury Public Schools Mathematics Department Statement of Philosophy
Waterbury Public Schools provides a rich and rigorous mathematics curriculum that prepares students for rewarding postsecondary experiences. . All courses are carefully aligned to the Common Core State Standards in Mathematics. A
rich and rigorous mathematics education is about becoming an effective problem solver. This entails evaluating given information, accessing prior knowledge and intertwining these to move toward a potential solution. Attaining such
abilities requires students to become driven, independent, competent and confident in their math abilities. Based on this; the philosophy underscoring the units is that of teaching mathematics for understanding, this philosophy will have
tangible benefits for both students and teachers. For students, mathematics should cease to be seen as a set of disjointed facts and rules. Rather, students should come to view mathematics as an interesting, powerful tool that enables
them to better understand their world. All students should be able to reason mathematically; thus, activities will have multiple levels so that the able student can go into more depth while a student having trouble can still make sense out
of the activity. For teachers, the reward of seeing students excited by mathematical inquiry, a redefined role as guide and facilitator of inquiry, and collaboration with other teachers should result in innovative approaches to instruction,
increased enthusiasm for teaching, and a more positive image with students and society. Students exiting the Waterbury Public Schools Mathematics program will understand and be able to solve non-routine problems in nearly any
mathematical situation they might encounter in their daily lives. In addition, they will have gained powerful heuristics, vis-à-vis the interconnectedness of mathematical ideas, that they can apply to most new problems typically requiring
multiple modes of representation, abstraction, and communication. This knowledge base will serve as a springboard for students to continue in any endeavor they choose, whether it be further mathematical study in high school and
college, technical training in some vocation, or the mere appreciation of mathematical patterns they encounter in their future lives. Furthermore, instruction and assignments are designed to aid students in improving their testing skills.
An additional goal of the Waterbury Public Schools Department is that all students are prepared for the numerous standardized tests that they will encounter as they progress through high school and beyond.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
2
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
Seeing Structure in Expressions (A-SSE) The Real Number System (N-RN) Interpreting Functions (F-IF)
Interpret the structure of expressions Extend the properties of exponents to rational exponents Understand the concept of a function and use function notation
Write expressions in equivalent forms to solve problems Use properties of rational and irrational numbers. Interpret functions that arise in applications in terms of the
context
Analyze functions using different representations
Arithmetic with Polynomials and Rational Expressions (A-APR) Quantities (N-Q)
Perform arithmetic operations on polynomials Reason quantitatively and use units to solve problems Building Functions (F-BF)
Understand the relationship between zeros and factors of
polynomials Build a function that models a relationship between two
Use polynomial identities to solve problems quantities
Rewrite rational expressions Build new functions from existing functions
Creating Equations (A-CED) Linear, Quadratic, and Exponential Models (F-LE)
Create equations that describe numbers or relationships Construct and compare linear, quadratic, and exponential
models and solve problems
Interpret expressions for functions in terms of the situation they
Reasoning with Equations and Inequalities (A-REI) model
Understand solving equations as a process of reasoning and explain
the reasoning Interpreting Categorical and Quantitative Data (S-ID)
Solve equations and inequalities in one variable
Solve systems of equations Summarize, represent, and interpret data on a single count or
Represent and solve equations and inequalities graphically measurement variable
. Summarize, represent, and interpret data on two categorical
and quantitative variables
Interpret linear models
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
3
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
High School - Algebra
Expressions. An expression is a record of a computation with numbers, symbols that represent numbers, arithmetic operations, exponentiation, and, at more advanced levels, the operation of evaluating a function. Conventions about
the use of parentheses and the order of operations assure that each expression is unambiguous. Creating an expression that describes a computation involving a general quantity requires the ability to express the computation in general
terms, abstracting from specific instances.
Reading an expression with comprehension involves analysis of its underlying structure. This may suggest a different but equivalent way of writing the expression that exhibits some different aspect of its meaning. For example,
p + 0.05p can be interpreted as the addition of a 5% tax to a price p. Rewriting p + 0.05p as 1.05p shows that adding a tax is the same as multiplying the price by a constant factor.
Algebraic manipulations are governed by the properties of operations and exponents, and the conventions of algebraic notation. At times, an expression is the result of applying operations to simpler expressions. For example, p + 0.05p
is the sum of the simpler expressions p and 0.05p. Viewing an expression as the result of operation on simpler expressions can sometimes clarify its underlying structure.
A spreadsheet or a computer algebra system (CAS) can be used to experiment with algebraic expressions, perform complicated algebraic manipulations, and understand how algebraic manipulations behave.
Equations and Inequalities. An equation is a statement of equality between two expressions, often viewed as a question asking for which values of the variables the expressions on either side are in fact equal. These values are the
solutions to the equation. An identity, in contrast, is true for all values of the variables; identities are often developed by rewriting an expression in an equivalent form.
The solutions of an equation in one variable form a set of numbers; the solutions of an equation in two variables form a set of ordered pairs of numbers, which can be plotted in the coordinate plane. Two or more equations and/or
inequalities form a system. A solution for such a system must satisfy every equation and inequality in the system.
An equation can often be solved by successively deducing from it one or more simpler equations. For example, one can add the same constant to both sides without changing the solutions, but squaring both sides might lead to
extraneous solutions. Strategic competence in solving includes looking ahead for productive manipulations and anticipating the nature and number of solutions.
Some equations have no solutions in a given number system, but have a solution in a larger system. For example, the solution of x + 1 = 0 is an integer, not a whole number; the solution of 2x + 1 = 0 is a rational number, not an
integer; the solutions of x2 – 2 = 0 are real numbers, not rational numbers; and the solutions of x2 + 2 = 0 are complex numbers, not real numbers.
The same solution techniques used to solve equations can be used to rearrange formulas. For example, the formula for the area of a trapezoid, A = ((b1+b2)/2)h, can be solved for h using the same deductive process.
Inequalities can be solved by reasoning about the properties of inequality. Many, but not all, of the properties of equality continue to hold for inequalities and can be useful in solving them.
Connections to Functions and Modeling. Expressions can define functions, and equivalent expressions define the same function. Asking when two functions have the same value for the same input leads to an equation; graphing
the two functions allows for finding approximate solutions of the equation. Converting a verbal description to an equation, inequality, or system of these is an essential skill in modeling.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
4
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
Explanations and Examples
Mathematical Practice Standards
Mathematically proficient students
should be able to:
HS.MP.1. Make sense of problems High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and
and persevere in solving them. goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try
special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might,
depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can
explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They
check their answers to problems using different methods and continually ask themselves, "Does this make sense?" They can understand the approaches of others to solving complex problems and
identify correspondences between different approaches.
HS.MP.2. Reason abstractly and High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols,
quantitatively. and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the
problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.
HS.MP.3. Construct viable High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of
arguments and critique the reasoning statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions,
of others. communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose.
High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—
explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to
clarify or improve the arguments.
HS.MP.4. Model with mathematics. High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design
problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated situation, realizing
that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts
and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the
results make sense, possibly improving the model if it has not served its purpose.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
5
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
Explanations and Examples
Mathematical Practice Standards
Mathematically proficient students
should be able to:
HS.MP.5. Use appropriate tools High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet,
strategically. a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions
generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that
technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical
resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
HS.MP.6. Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose,
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of
precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
HS.MP.7. Look for and make use of By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an
structure. existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated
things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that
to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose
functions, and transform figures.
HS.MP.8. Look for and express High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2
regularity in repeated reasoning. + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high
school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
6IF 3. Recognize that HS.MP.8. Look for Recursive and explicit rules for sequences are first introduced in the context of Unit 1 Boardworks High Identifying Arithmetic Sequence Series
sequences are functions, and express chemistry. For example, the number of hydrogen atoms in a hydrocarbon is a Patterns School Algebra CC Similarities and
sometimes defined regularity in function of the number of carbon atoms. This relationship may be defined by the PowerPoints Differences At a Snail's Pace
recursively, whose domain repeated reasoning. recursive rule "add two to the previous number of hydrogen atoms" or explicitly as Note Taking
h = 2 + 2c. The function may also be represented in a table or a graph or with F.IF.3 Lesson Recursive Exploring Geometric
is a subset of the integers... Summarizing
concrete models. and Explicit Formulas.doc Sequences
Cooperative
CT.9-12.1.C.3.a.(2) Learning
CAST's Universal Design Exploring Infinite Series
Determine equivalent Nonlinguistic
for Learning Guidelines –
representations of an algebraic Representations
Educator Checklist Generating Recursive
equation or inequality to Vocabulary Sequences to Explore
simplify and solve problems. Lesson: Recursive and Development Exponential Patterns
Explicit Geometric
Formulas Geometric Sequences &
Series
Lesson: Introduction to
Recursion (Has YouTube
Clips so you will have to
download clips and bring
them in if you choose to
utilize this lesson.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
7 1. Write a function HS.MP.1. Make Example; You frequently go to the gym to work out lifting weights. You plan to Unit 1 Boardworks High Identifying Generating Recursive
that describes a relationship sense of problems gradually increase the size of the weights over the next month. You always put two Patterns School Algebra CC Similarities and Sequences to Explore
between two quantities.* and persevere in plates that appear to be the same weight on each side of the bar. The plates are not PowerPoints Differences Exponential Patterns
solving them. labeled but you do know the bar weighs 20 kg. How can we express the total weight Note Taking
a. Determine an explicit you lifted on any day?" Students will assign a variable for the weight of a plate (say Math Tasks: Geometric Sequences And
HS.MP.2. Reason Summarizing
expression, a recursive w) and derive an expression for the total weight lifted, 4w +20 or its equivalent. Discounting Tickets Series
abstractly and Cooperative
process, or steps for Learning
quantitatively
calculation from a context.
Modeling Nonlinguistic Sums of Sequences
HS.MP.4. Model Representations
CT.9-12.1.C.1.a.(1) Identify, Adventures With The Fish
with mathematics.
describe, create and generalize Pond: Population Modeling Vocabulary Study of Functions
numeric, geometric and HS.MP.5. Use Development
statistical patterns with tables,
graphs, words and symbolic
appropriate tools
rules. strategically.
CT.9-12.1.C.2.a.(2) Identify
HS.MP.6. Attend to
precision.
an appropriate symbolic
representation for a function HS.MP.7. Look for
or relation displayed and make use of
graphically or verbally. structure.
HS.MP.8. Look for
and express
regularity in
repeated reasoning.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
8 2. Write arithmetic HS.MP.4. Model Arithmetic sequences may be introduced through geometric models. For example, Unit 1 Boardworks High Identifying Double Tree
and geometric sequences ... with mathematics. the number of beams required to make a steel truss in the following pattern is an Patterns School Algebra CC Similarities and
recursively and [arithmetic arithmetic sequence. PowerPoints Differences Sequence of Bounces
sequences] with an explicit HS.MP.5. Use Note Taking Activity - Modeling Motion
appropriate tools Geometric Fractals
formula, use them to model Summarizing
strategically. Spreading Doom
situations, and translate Finding the Next Number Cooperative
between the two forms.* HS.MP.8. Look for in a Sequence: Recursive Learning Using Cabri Geometry to
Arithmetic sequences are also found in patterns for integers which can be used
and express Sequences Nonlinguistic Create Fractals
to justify and reinforce rules for operations. For example in completing this
regularity in Representations
pattern
repeated reasoning. 5 * 4 = _______ Arithmetic Sequences Vocabulary
CT.9-12.1.C.1.a.(1) Development
Identify, describe, create
PowerPoint
and generalize numeric, 5 * 3 = _______
Recursive and Explicit Forms
geometric, and statistical of Arithmetic Sequences
patterns with tables, 5 * 2 = _______
graphs, words and Geometric Sequences
symbolic rules. 5 * 1 = _______
Lesson on Explicit
5 * 0 = _______ Geometric Sequence
Formula
5 * -1 = _______
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
95 * -2 = _______
5 * -3 = _______
students will find an arithmetic sequence with a common difference of -5. This
pattern illustrates that the product of a positive integer and a negative integer is
negative.
Compound interest is a good example of a geometric sequence. For example,
"You just won first prize in a poetry writing contest.
If you take the $500 you won and invest it in a mutual fund
earning 8% interest per year, about how long will it take for
your money to double? "
Fractals are another example of geometric sequences. For example in creating the
Sierpinski triangle the number of unshaded triangles in each stage form a geometry
sequence. In this case the explicit rule, T= 3n, is readily determined. In general,
however in this unit we focus on recursive rules for geometric sequences since they
will be revisited in Unit 7 in conjunction with exponential functions, where the
explicit rule will be given greater emphasis.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
10 1. Use units as a way HS.MP.4. Model In all problem situations the answer should be reported with appropriate units. In Unit 2 Boardworks High Identifying
to understand problems and with mathematics. situations involving money, answers should be rounded to the nearest cent. When Linear School Algebra CC Similarities and
to guide the solution of data sets involve large numbers (e.g. tables in which quantities are reported in the Equations PowerPoints Differences
multi-step problems; HS.MP.5. Use millions) the degree of precision in any calculation is limited by the degree of and Note Taking
appropriate tools precision in the data. These ideas are introduced in the solution to contextual Inequalities Mining Solar Data
choose and interpret units Summarizing
strategically. problems in Unit 2 and reinforced throughout the remainder of the course.
consistently in formulas; Modeling Lesson: Having Cooperative
choose and interpret the HS.MP.6. Attend Kittens Learning
scale and the origin in to precision. Nonlinguistic
graphs and data displays. Lesson; Density Representations
Vocabulary
CT.9-12.2.C.2.a.(1) Select Lesson: Estimations and Development
and use appropriate methods . Approximations: The
for computing to solve Money Munchers
problems in a variety of
contexts. Mathline Activity III --
Pairwise Comparisons
CT.9-12.3.C.3.a.(1) Select
appropriate units, scales,
degree of precision, and
strategies to determine length,
angle measure, perimeter,
circumference and area of
plane geometric figures.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
11 2. Define appropriate HS.MP.4. Model Unit 2 Boardworks High Identifying
quantities for the purpose of with mathematics. Linear School Algebra CC Similarities and
descriptive modeling. Equations PowerPoints Differences
HS.MP.6. Attend and Note Taking
CT.9-12.2.C.1.a.(2) Select to precision. Inequalities Lesson: Cosmic Summarizing
and use an appropriate form of Measurements
Cooperative
number (integer, fraction, Learning
decimal, ratio, percent, Lesson: Location Location
Location It's Not Just Real Nonlinguistic
exponential, scientific
Estate Representations
notation, irrational) to solve
Vocabulary
practical problems involving
Development
order, magnitude, measures,
labels, locations and scales.
N-Q 3. Choose a level of HS.MP.5. Use Unit 2 Boardworks High Identifying
accuracy appropriate to appropriate tools Linear School Algebra CC Similarities and
limitations on measurement strategically. Equations PowerPoints Differences
and Note Taking
when reporting quantities.
HS.MP.6. Attend Inequalities
Summarizing
CT.9-12.2.C.2.a.(3) Develop to precision.
Cooperative
and use a variety of strategies Learning
to estimate values of formulas, Nonlinguistic
functions and roots; to Representations
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
12recognize the limitations of Vocabulary
estimation; and to judge the Development
implications of the results.
A-SSE 1. Interpret HS.MP.1. Make Understanding the order of operations is essential to unpacking the meaning of a Unit 2 Boardworks High Identifying First Degree Monomials and
expressions that represent a sense of problems complex algebraic expression and to develop a strategy for solving an equation. Linear School Algebra CC Similarities and Binomials
quantity in terms of its and persevere in Equations PowerPoints Differences
solving them. and Note Taking
context.* Using Stations to Explore
Inequalities Summarizing
a. Interpret parts of an HS.MP.2. Reason Algebra Expressions
abstractly and Cooperative
expression, such as Grades 9-12, Math,
quantitatively. Learning
Differentiation
terms, factors, and Nonlinguistic
coefficients. HS.MP.4. Model Representations
A-SSE Animal Populations
b. Interpret complicated with mathematics. Vocabulary
Lesson: Rags to Riches or
expressions by viewing Development
Using the commutative, associative and distributive properties enables students to find Riches to Rags?
one or more of their HS.MP.7. Look
equivalent expressions, which are helpful in solving equations.
parts as a single for and make use Lesson: Interpreting
entity... of structure. Algebraic Expressions
CT.9-12.1.C.2.a.(3) Comparing Investments
Recognize and explain the
meaning of the slope and x- Representing Polynomials
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
13and y-intercepts as they
relate to a context, graph,
table or equation.
CT.9-12.1.C.3.a.(1) Model
and solve problems with
linear, quadratic and
absolute value equations and
linear inequalities.
A-CED 1. (part) Create HS.MP.2. Reason Here are some examples where students can create equations and inequalities. Unit 2 Boardworks High Identifying Border Patrol
equations and inequalities in abstractly and Linear School Algebra CC Similarities and
quantitatively. a. (Two step equation) The bank charges a monthly fee of $2.25 for your Equations PowerPoints Differences Dinner Party
one variable and use them Dad's checking account and an additional $1.25 for each transaction with his
to solve problems. Include debit card, whether used at an ATM machine or by using the card to make a and Note Taking
HS.MP.4. Model Eileen's Work Week:
equations arising from linear purchase. He noticed a transaction charge of $13.50 on this month's Inequalities Summarizing
with mathematics. To Babysit or Not To Solving Systems of
... functions statement. He is trying to remember how many times he used the debit card. Cooperative Inequalities
Babysit?
HS.MP.5. Use Can you use the information on the statement help him figure out how many Learning
CT.9-12.1.C.3.a.(1) Model appropriate tools transactions he made? Nonlinguistic Exponential Functions
and solve problems with strategically. b. (Equations with variables on both sides) Willie and Malia have been Representations
linear, quadratic and hired by two different neighbors to pick-up mail and newspapers while they Vocabulary
absolute value equations and are on vacation. Willie will be paid $7 plus $3 per day. Malia will be paid Development
linear inequalities. $10 plus $3 per day. By which day will they have earned the same amount of
money?
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
14 (Equations which require using the distributive property) Jessica
wanted to buy 7 small pizzas but she only had four $2 off coupons. So, she
bought four with the discount and paid full price for the other three, and the
ill came to $44.50. How much was each small pizza?
d. (Inequality) The student council has set aside $6,000 to purchase the
shirts. (They plan to sell them later at double the price.) How many shirts
can they buy at the price they found online if the shipping costs are $14?
A-CED 4. Rearrange HS.MP.2. Reason Begin with a familiar formula such as one for the perimeter of a rectangle: Unit 2 Boardworks High Identifying Inverse of Two Temps
formulas to highlight a abstractly and p = 2l + 2w. Consider this progression of problems: Linear School Algebra CC Similarities and
quantity of interest, using quantitatively. Equations PowerPoints Differences
the same reasoning as in (a) Values for the variables l and w are given. We can find p by substituting and Note Taking
solving equations. For HS.MP.4. Model for l and w and evaluating the expression on the right side. Inequalities A-CED Equations and Summarizing
example, rearrange Ohm's law with mathematics. Formulas
V = IR to highlight resistance (b) Values for the variables l and p are given. We can find w by substituting Cooperative
R. HS.MP.5. Use for l and p and solving the equation for w. Learning
Lesson: Solving Literal
appropriate tools Equations
CT.9-12.1.C.3.a.(2) strategically. (c) We can find a formula for w in terms of l and p, by following the same Nonlinguistic
Determine equivalent steps as in (b) above to solve for w. This gives us a general method for Representations
Lesson: Optimization
representations of an HS.MP.7. Look finding w when the other variables are known. (Check this new formula by Vocabulary
Problems: Boomerangs
algebraic equation or for and make use showing that it gives the correct value for w when the values of l and p from Development
inequality to simplify and of structure. (b) are substituted. )
solve problems
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
15 1. Explain each step HS.MP.2. Reason For two-step equations, flow charts may be used to help students "undo" the order of Unit 2 Boardworks High Identifying Linear Equations: Using
in solving a simple equation abstractly and operations to find the value of a variable. For example, this flow chart may be used to Linear School Algebra CC Similarities and Algebra
as following from the quantitatively. solve the equation 4x – 2 = 30. Equations PowerPoints Differences
and Note Taking One Step at a Time
equality of numbers
HS.MP.3. Inequalities The Yo-Yo Problem
asserted at the previous Summarizing
Construct viable Solving Linear Equation
step, starting from the arguments and Cooperative
assumption that the original critique the Learning
equation has a solution. reasoning of Nonlinguistic
Construct a viable argument others. Representations
to justify a solution method. Vocabulary
HS.MP.7. Look Development
CT.9-12.1.C.3.a.(2) for and make use Then students learn to solve equations by performing the same operation (except for
Determine equivalent of structure. division by zero) on both sides of the equal sign.
representations of an algebraic
equation or inequality to
simplify and solve problems.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
16 3. Solve linear HS.MP.2. Reason 4x – 2 = 30 Unit 2 Boardworks High Identifying Linear Inequalities
equations and inequalities in abstractly and +2 +2 Linear School Algebra CC Similarities and
one variable, including quantitatively. Equations PowerPoints Differences Linear Inequalities: Using
4x = 32 and Note Taking Algebra
equations with coefficients
HS.MP.7. Look 4 4 Inequalities Elementary Algebra:
represented by letters. Summarizing
for and make use Solving Linear Equations in One Step Equations and
of structure. x =8 One Variable Cooperative Inequalities
CT.9-12.1.C.3.a.(1) Model Learning
and solve problems with Nonlinguistic
HS.MP.8. Look In solving multiple-step equations students should realize that there may be several Working with Numbers
linear, quadratic and Representations
for and express valid solution paths. and Letters in Algebra
absolute value equations regularity in Vocabulary
and linear inequalities. repeated For example here are two approaches to the equation 3x + 10 = 7x – 6. Development
reasoning.
3x + 10 = 7x – 6 3x + 10 = 7x – 6
–3x –3x –7x –7x
10 = 4x – 6 –4x + 10 =–6
+6 +6 –10 –10
16 = 4x –4x = –16
4 4 –4 –4
4 = x x = 4
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
17 2. Create equations HS.MP.2. Students may collect data from water that is cooling using two thermometers, one Unit 3 Boardworks High Identifying A Linear Picture
in two or more variables to Reason abstractly measuring Celsius, the other Fahrenheit. From this they can create of the relationship Functions School Algebra CC Similarities and
represent relationships and and show that it can be modeled with a linear function. PowerPoints Differences Distance vs. Time to School
quantitatively. Note Taking
between quantities; graph
Home for the Holidays
equations on coordinate axes Summarizing
HS.MP.4. Model Lesson Plan 1: The Phone
with labels and scales with Bill Problem - Linear Cooperative
mathematics. Learning
Functions
CT.9-12.1.C.2.a.(1) Nonlinguistic
Represent functions and Representations
HS.MP.5. Use
relations on the
coordinate plane. appropriate tools Vocabulary
strategically. Development
CT.9-12.1.C.3.a.(1) Model
and solve problems with
linear, quadratic and
absolute value equations
and linear inequalities.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
18 10. Understand that HS.MP.2. The graph below shows the height of a hot air balloon as a function Unit 3 Boardworks High Identifying
the graph of an equation in Reason abstractly of time. Explain what the point (50, 300) on this graph represents. Functions School Algebra CC Similarities and
two variables is the set of all and PowerPoints Differences
its solutions plotted in the quantitatively. Note Taking
coordinate plane, often Summarizing
HS.MP.4. Model
forming a curve (which with Cooperative
could be a line). mathematics. Learning
Nonlinguistic
CT.9-12.1.C.2.a.(4) Evaluate Representations
and interpret the graphs of Vocabulary
linear, exponential and Development
polynomial functions.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
19 1. Understand that a HS.MP.2. The domain of a function given by an algebraic expression, unless otherwise specified, Unit 3 Boardworks High Identifying Back in Time
function from one set (called Reason abstractly is the largest possible domain. Functions School Algebra CC Similarities and
the domain) to another set and PowerPoints Differences Function or Relation?
(called the range) assigns to quantitatively. Mapping diagrams may be used to introduce the concepts of domain and range. The Note Taking
vertical line test may be used to determine whether a graph represents a function. Using a Lottery to Illustrate Functions
each element of the domain Summarizing
Functions
exactly one element of the Grades 9-12, Math, Cooperative Relations and Functions
range. If f is a function and x Functions Learning
is an element of its domain, Nonlinguistic
then f(x) denotes the output Function Notation: Intro / Representations
of f corresponding to the Evaluation Vocabulary
Development
input x. The graph of f is the
Interpreting Functions
graph of the equation y =
f(x).
Vertical Line Test
CT.9-12.1.C.1.a.(3) Identify
the characteristics of
functions and relations,
including domain and
range.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
20 2. Use function HS.MP.2. Examples: Unit 3 Boardworks High Identifying Back in Time
notation, evaluate functions Reason abstractly Functions School Algebra CC Similarities and
for inputs in their domains, and If f(x) = x2 + 4x – 12, find f(2) PowerPoints Differences Evaluate Me?
and interpret statements that quantitatively. Note Taking
Let f(x) = 10x – 5; find f(1/2), f(-6), f(a) Domain Representations Function Notation
use function notation in Summarizing
terms of a context. If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = Function Notation: Cooperative
487,000 and P(10)-P(9) = 5,900. Evaluation Learning
CT.9-12.1.C.1.a.(3) Identify Nonlinguistic
the characteristics of Evaluating Functions Representations
functions and relations, PowerPoint Vocabulary
including domain and Development
range.
F-IF 4. For a function that HS.MP.2. Students may be given graphs to interpret or produce graphs given an expression or Unit 3 Boardworks High Identifying Advanced Algebra
models a relationship Reason abstractly table for the function, by hand or using technology. Functions School Algebra CC Similarities and Nomograph
between two quantities, and Examples: PowerPoints Differences
quantitatively. Note Taking Composing Music
interpret key features of
A rocket is launched from 180 feet above the ground at time t = 0. The function that Math Tasks: Mathematically
graphs and tables in terms of Summarizing
HS.MP.4. Model models this situation is given by Planning for Prom
the quantities and sketch with Cooperative
h = – 16t2 + 96t + 180, where t is measured in seconds and h is height above Distance - Time Graphs
graphs showing key features mathematics. What is Your Weather? Learning
the ground measured in feet.
given a verbal description of
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
21the relationship. Key features HS.MP.5. Use 1. What is a reasonable domain restriction for t in this context? Lesson: Nonlinguistic Maximum, minimum,
include: intercepts; intervals appropriate tools Real-World Quadratics Representations increasing, decreasing
where the function is increasing, strategically. 2. Determine the height of the rocket 2 seconds after it was launched. Graphing Vocabulary
decreasing, positive, or Development
HS.MP.6. 3. Determine the maximum height obtained by the rocket.
negative....* Attend to
precision. 4. Determine the time when the rocket is 100 feet above the ground.
CT.9-12.1.C.2.a.(2) Identify
an appropriate symbolic
representation for a 5. Determine the time at which the rocket hits the ground.
function or relation
displayed graphically or 6. How would you refine your answer to the first question based on your
verbally. response to the fifth question?
CT.9-12.1.C.2.a.(3) It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the
Recognize and explain the storm was over, with a total rainfall of 3 inches. It didn't rain for the rest of the day.
meaning of the slope and Sketch a possible graph for the number of inches of rain as a function of time, from
x- and y-intercepts as they midnight to midday
relate to a context, graph,
table or equation
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
22 5. Relate the domain of HS.MP.2. Students may explain orally, or in written format, the existing relationships. Unit 3 Boardworks High Identifying Domain & Range
a function to its graph and, Reason abstractly Functions School Algebra CC Similarities and
where applicable, to the and PowerPoints Differences Domain and Range of
quantitative relationship it quantitatively. Note Taking Graphs
describes. For example, if Lesson:
Summarizing
the function h(n) gives the HS.MP.4. Model Domain
number of person-hours it with Cooperative
takes to assemble n engines mathematics. Learning
in factory, then the positive Nonlinguistic
integers would be an HS.MP.6. Representations
appropriate domain for the Attend to Vocabulary
function.* precision Development
F-IF 7b. Graph square root, HS.MP.5. Use Emphasis in this unit is distinguishing the graphs of linear and non-linear functions; Unit 3 Boardworks High Identifying Absolute Value Function
cube root, and piecewise- appropriate tools linear functions are emphasized in Unit 4; non-linear functions are studied in more Functions School Algebra CC Similarities and Match
defined functions, including strategically. detail in later units. PowerPoints Differences
Note Taking Absolute Value and
step functions and absolute
HS.MP.6. Math Task: Piecewise Functions
value functions. Summarizing
Attend to Cell Phone Costs
precision. Cooperative Behaviors-Square Root
CT.9-12.1.C.2.a.(1) Learning
Represent functions and Lesson: Functions
Piecewise Functions Nonlinguistic
relations on the
coordinate plane. Representations
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
23Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
24 9. Compare properties HS.MP.6. Students may be asked to match graphs with tables or equations with which they may Unit 3 Boardworks High Identifying Getting Started with the TI
of two functions each Attend to represent, and to explain their reasoning. Functions School Algebra CC Similarities and Navigator: What's My Rule
represented in a different precision. PowerPoints Differences
way (algebraically, Note Taking
HS.MP.7. Look CT.9-12.1.C.1.a.(1) Identify, describe, create and generalize numeric, geometric Sorting Functions
graphically, numerically in and statistical patterns with tables, Summarizing
for and make use
tables, or by verbal of structure. Comparing Functions Cooperative
descriptions). ,,, Learning
Nonlinguistic
CT.9-12.1.C.2.a.(1) Representations
Represent functions and Vocabulary
relations on the Development
coordinate plane.
CT.9-12.1.C.2.a.(2) Identify
an appropriate symbolic
representation for a
function or relation
displayed graphically or
verbally.
graphs, words and
symbolic rules.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
25 6. Calculate and HS.MP.2. Reason For linear functions, the rate of change is also the slope of the line. Unit 4 Boardworks High Identifying Constant Rate of Change
interpret the average rate of abstractly and Linear School Algebra CC Similarities and
change of a function quantitatively. Example: A cell phone company uses the function y = 0.03x + 14.99 to determine Functions PowerPoints Differences Dog Days or Dog Years
the monthly charge, y, in dollars, for a customer using the phone for x minutes. Note Taking
(presented symbolically or
HS.MP.4. Model Interpret the slope of this function in the context of this problem and indicate the Lesson: Exploring Linear
as a table) over a specified Summarizing
with mathematics. appropriate units. Rate of Change Relationships -- Stacking
interval. Estimate the rate of Cooperative Cups and Walking Rates
change from a graph.* HS.MP.5. Use Mall Resource Learning
appropriate tools Nonlinguistic
CT.9-12.1.C.2.a.(3) strategically. Marathon Scenario Representations
Recognize and explain the Vocabulary
meaning of the slope and Development
Linear vs. Exponential
x- and y-intercepts as they
relate to a context, graph,
Venn Diagrams
table or equation.
CT.9-12.1.E.2.a.(3)
Recognize that the slope
of the tangent line to a
curve represents the rate
of change.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
26. Graph functions HS.MP.5. Use Example: Find the x-intercept and the y-intercept for each of these lines. Unit 4 Boardworks High Identifying Graphic Line Designs
expressed symbolically and appropriate tools Linear School Algebra CC Similarities and
show key features of the strategically. y = –3x + 18 Functions PowerPoints Differences How Many Drivers?
graph, by hand in simple y – 7 = 2/3(x + 4) Note Taking Investigate the Slope-
cases and using technology HS.MP.6. Attend 6x + 4y = 96 Lesson Intercept Form of a Line
Summarizing
for more complicated cases. to precision Understanding Graphs of
a. Graph linear functions Linear and Exponential Cooperative
and show intercepts. Functions Learning
Nonlinguistic
Warm-Ups Representations
CT.9-12.1.C.2.a.(3) Vocabulary
Recognize and explain the Option 1 Development
meaning of the slope and Option 2
x- and y-intercepts as they Option 3
relate to a context, graph,
table or equation. Equation and Graph
Matching
Day 2 Extension 2
Exit Slip
UDL Checklist
Linear Functions
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
27. Write a function HS.MP.2. Reason For linear functions, the slope intercept form easily reveals the slope and the y- Unit 4 Boardworks High Identifying
defined by an expression in abstractly and intercept. From the standard form one can easily determine both intercepts. The Linear School Algebra CC Similarities and
different but equivalent quantitatively. point-slope form focuses attention on a particular point on the graph and is directly Functions PowerPoints Differences
related to the definition of slope. Note Taking
forms to reveal and explain
HS.MP.7. Look Forming Quadratics
different properties of the Summarizing
for and make use
function. of structure. NCTM Illuminations Cooperative
(F-IF.8.b) Learning
CT.9-12.1.C.3.a.(2) Nonlinguistic
Determine equivalent Representations
Analyzing the Data
representations of an Vocabulary
Exploring
algebraic equation or Development
Measurement,
inequality to simplify and
Sequences, and Curves
solve problems.
with Stringed
Instruments: Pre-
Activity Questions
Graphical Analysis
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
28 1. Distinguish between HS.MP.3. Given several tables of values, determine which represent linear functions, and Unit 4 Boardworks High Identifying Accelerates Returns
situations that can be Construct viable explain why. Linear School Algebra CC Similarities and
modeled with linear arguments and Functions PowerPoints Differences Add Them Up
critique the CT.9-12.1.C.3.a.(1) Model and solve problems with linear, quadratic and Note Taking
functions [and with absolute value equations
reasoning of Lesson: Behaviors-Exponential
exponential functions]. CT.9-12.2.C.2.b.(2) Solve problems using direct and inverse variation. Summarizing
others. Comparing Linear and Function
Exponential Functions Cooperative
a. Prove that linear Learning
HS.MP.4. Model Buying Your First New
functions grow by equal Nonlinguistic
with mathematics. Input Output Lesson Plan 1: Overrun Car!
differences over equal Input Output x y Representations
by Skeeters -
intervals... over equal 1 8 1 3 0 1 Vocabulary
HS.MP.5. Use Exponential Growth Linear Functions: Slope as
intervals. 2 6 1 2 Development
appropriate tools 2 4 Rate of Change
b. Recognize situations in 3 4 4 3
strategically. 3 6 Lesson Plan 2: Skeeter
which one quantity changes 4 2
HS.MP.7. Look 4 9 7 4 Populations and
at a constant rate per unit
for and make use 5 0 5 13 8 5 Exponential Growth
interval relative to
of structure.
another....
Proving that linear
functions grow by equal HS.MP.8. Look
differences over equal for and express
intervals and that regularity in
exponential functions repeated
grow by equal factors over reasoning.
equal intervals are not
assessed on CAPT.
and linear inequalities.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
29 2. Construct linear ... HS.MP.4. Model Example: Unit 4 Boardworks High Identifying Generating Recursive
functions, including with mathematics. Sara's starting salary is $32,500. Each year she receives a $700 raise. Write a Linear School Algebra CC Similarities and Sequences to Explore
arithmetic ... sequences, sequence in explicit form to describe the situation. Draw a graph representing this Functions PowerPoints Differences Exponential Patterns
HS.MP.8. Look situation. Is this function discrete or continuous? Note Taking
given a graph, a description
for and express Math Tasks: Generating Recursive
of a relationship, or two Summarizing
regularity in Setting the Table Sequences to Explore
input-output pairs (include repeated Cooperative Linearity
reading these from a table). reasoning. Learning
Nonlinguistic
CT.9-12.1.C.2.a.(2) Identify Representations
an appropriate symbolic Vocabulary
representation for a Development
function or relation
displayed graphically or
verbally.
F-LE 5. Interpret the HS.MP.2. Reason Use technology to explore the effects of the parameters m and b in the linear Unit 4 Boardworks High Identifying Applications of
parameters in a linear ... abstractly and functions y = mx + b by holding first one parameter and then the other constant Linear School Algebra CC Similarities and Exponential Functions
function in terms of a quantitatively. while allowing the other one to vary. Functions PowerPoints Differences
Note Taking Behaviors-Linear Function
context.
HS.MP.4. Model Math.Montana.edu
Summarizing
with mathematics. Linear Models Exploring Linear Equations
Cooperative with Activity Center
Learning
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
301.a.(3) Identify Example: Mathematics Assessment Nonlinguistic
the characteristics of Project (MAP) Representations
functions and relations, A student club is raising funds to fight cancer. They raffle off a gift basket and sell
including domain and N08: Linear and Vocabulary
raffle tickets. The function that gives their profit is f(x) = 2x – 80, where x is the Development
range. Exponential Models
number of tickets sold.
NCTM Illuminations
a. How much did they spend on the gift basket? Egg Launch Contest
b. How much do they charge for each ticket? Graph Chart
c. If they raise the price of a ticket, which parameter will change, the slope or Graphing What
the y-intercept? Explain. Movie Lines
Pedal Power
Smokey Bear Takes
Algebra
Online Practice from IXL
Solve equations: Solve
linear equations: word
problems (Algebra - J.8)
Shodor
Multi-Function Data
Flyer
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
31b. Graph ... piecewise- HS.MP.5. Use Piecewise linear functions may model sets of data. For example the world Unit 5 Boardworks High Identifying
defined functions, including appropriate tools women's swimming records for the 100 meter free style exhibit two distinct Scatter School Algebra CC Similarities and
step functions and absolute strategically. linear trends; one for the years 1910-1936, the other for 1956-2010. Plots and PowerPoints Differences
Trend Note Taking
value functions.
HS.MP.6. Attend to Lines Math Task:
Summarizing
CT.9-12.1.C.2.a.(1) precision. Cell Phone Costs
Cooperative
Represent functions and
Lesson: Learning
relations on the coordinate
plane. Piecewise Functions Nonlinguistic
RepresentationsS-ID 1 Represent data with HS.MP.4. Model Students may compare and contrast the advantage of each of these representations. Unit 5 Boardworks High Identifying Data Collection for Bar
plots on the real number line with mathematics. Scatter School Algebra CC Similarities and Graphs
(dot plots, histograms, and Plots and PowerPoints Differences
HS.MP.5. Use Trend Note Taking Graph My Center
box plots).
appropriate tools Lines Sorting Functions
Summarizing
strategically. Intro to the Central Limit
Comparing Functions Cooperative Theorem
Learning
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
324.C.1.a.(1) Collect Nonlinguistic
real data and create Representations
meaningful graphical
representations of the data. Vocabulary
Development
S-ID 2. Use statistics HS.MP.2. Reason Students may use spreadsheets, graphing calculators and statistical software for Unit 5 Boardworks High Identifying Data and Statistical
appropriate to the shape of abstractly and calculations, summaries, and comparisons of data sets. Scatter School Algebra CC Similarities and Methods 1
the data distribution to quantitatively. Plots and PowerPoints Differences
compare center (median, Example: Trend Note Taking Measures of Central
HS.MP.3. Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, Lines Math Tasks: Tendency
mean) and spread Construct viable Summarizing
median and standard deviation. Explain how the values vary about the mean and Weighing In on the
(interquartile range, standard arguments and median. What information does this give the teacher? Baltimore Ravens Cooperative
deviation) of two or more critique the Learning
different data sets. reasoning of others. Lesson: Nonlinguistic
Introducing Standard Representations
HS.MP.4. Model
CT.9-12.4.C.2.a.(3) Deviation Vocabulary
with mathematics.
Determine and use Development
Standard Deviation Practice
measures of spread and HS.MP.5. Use
central tendency to appropriate tools Standard Deviation Practice
describe and compare strategically. Key
sets of data. HS.MP.7. Look for Pencast for Teachers:
and make use of Standard Deviation and
structure. Variance
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
33 3. Interpret differences HS.MP.2. Reason Example: Unit 5 Boardworks High Identifying Data Analysis: Home Runs
in shape, center, and spread abstractly and After the 2009-2010 NBA season LeBron James switched teams from the Scatter School Algebra CC Similarities and in Major League Baseball
in the context of the data quantitatively. Cleveland Cavaliers to the Miami Heat, and he remained the top scorer (in points Plots and PowerPoints Differences
sets, accounting for possible Trend Note Taking
HS.MP.3. per game) in his first year in Miami. Compare team statistics for Cleveland (2009- Is it Rare?
effects of extreme data points Lesson:
Construct viable 2010) and Miami (2010-2011) for all players who averaged at least 10 minutes per Lines Summarizing
(outliers). Comparing Data
arguments and game. Distributions Cooperative
CT.9-12.4.C.2.a.(3) critique the Comparing Distribution Learning
Determine and use reasoning of others. Using the 1.5 X IQR rule, determine for which team and year James's Resource Nonlinguistic
measures of spread and performance may be considered an outlier. Representations
HS.MP.4. Model
central tendency to Vocabulary
with mathematics. NCTM Lesson:
describe and compare sets Development
of data. Eruptions: Old Faithful
HS.MP.5. Use
Geyser
appropriate tools
strategically.
HS.MP.7. Look for
and make use of
structure.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
34 6. Represent data on HS.MP.2. Reason Students may use spreadsheets, graphing calculators, and statistical software to Unit 5 Boardworks High Identifying 'Snail Mail'' Postal Rate
two quantitative variables on abstractly and represent data, describe how the variables are related, fit functions to data, Scatter School Algebra CC Similarities and History
a scatter plot, and describe quantitatively. perform regressions, and calculate residuals. Plots and PowerPoints Differences
Trend Note Taking Balloons
how the variables are related. HS.MP.3. Example: Lines Math Tasks:
Construct viable Summarizing
a. Fit a function to the data; A Statistical Analysis of Chirp, Jump Scatter
arguments and Make a scatter plot of data showing the rise in sea level over the past century. Fit Facebook Cooperative
use functions fitted to data to critique the Learning
solve problems in the context a trend line and use it to predict the sea level in the year 2020.
reasoning of others. A Statistical Analysis of Nonlinguistic Forensic Math: Bones to
of the data. HS.MP.4. Model Representations
Facebook PowerPoint Height
with mathematics. Vocabulary
c. Fit a linear function for a
scatter plot that suggests a Rhinos and M&M's® Development
HS.MP.5. Use
linear association appropriate tools (Exponential Models)
strategically.
CT.9-12.4.C.1.a. Create the Weather: Rainfall
appropriate visual or HS.MP.7. Look for (Grades 8-11)
graphical representation of and make use of
real data. structure.
CT.9-12.4.C.1.a.(1) Collect HS.MP.8. Look for
real data and create and express
meaningful graphical regularity in
representations of the data. repeated reasoning.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
35 7. Interpret the slope HS.MP.1. Make Students may use spreadsheets or graphing calculators to create representations of Unit 5 Boardworks High Identifying 400 Meter World Records
(rate of change) and the sense of problems data sets and create linear models. Scatter School Algebra CC Similarities and
intercept (constant term) of a and persevere in Plots and PowerPoints Differences Beebopper Shoe Store
solving them. Example: Trend Note Taking adapted from CPM
linear model in the context of
Alabama Lesson Plans Mathematics 1-Algebra 1
the data. Lines Summarizing
HS.MP.2. Reason Lisa lights a candle and records its height in inches every hour. The results Determining Percent
abstractly and recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), Cooperative Chirp, Jump Scatter
CT.9-12.1.C.2.a.(3) of Seed Germination- Learning
quantitatively. (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the candle's height (h) as a Enhancing
Recognize and explain the Nonlinguistic
function of time (t) and state the meaning of the slope and the intercept in terms mathematics in the
meaning of the slope and x- Representations
HS.MP.4. Model of the burning candle. career/technical
and y-intercepts as they with mathematics. Vocabulary
relate to a context, graph, classroom and
providing relevance in Development
table or equation. HS.MP.5. Use the mathematics
appropriate tools classroom
strategically.
Graphing is Great!
Exploring Slope
HS.MP.6. Attend to
precision. Show Me The Money -
Saving and Investing
My Peanut Butter is
Better Than Yours!
Investigating School
Safety and Slope
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
36 What is the slope of
the stairs in front of
the school?
AnalyzeMath.com
Linear Functions
CK12
Writing Linear
Equations
Learner.org
Linear Functions and
Slope
S-ID 8. Compute (using HS.MP.4. Model Example: Unit 5 Boardworks High Identifying Describing Bivariate Data
technology) and interpret the with mathematics. Scatter School Algebra CC Similarities and (Two Quantitative
correlation coefficient of a Collect height, shoe-size, and wrist circumference data for each student. Plots and PowerPoints Differences Variables)
HS.MP.5. Use Determine the best way to display the data. Answer the following questions: Is Trend Note Taking
linear fit.
appropriate tools there a correlation between any two of the three indicators? Is there a correlation Lines Lesson: Does a Correlation Exist?
Summarizing
No CAPT Correlation strategically. between all three indicators? What patterns and trends are apparent in the data? Correlation Coefficient
What inferences can be made from the data? Cooperative
HS.MP.8. Look for Correlation Frayer Model Learning
and express Nonlinguistic
regularity in Correlation Data Stations Representations
repeated reasoning. Vocabulary
Teacher Resource Development
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
37 9. Distinguish betweenHS.MP.3. Some data leads observers to believe that there is a cause and effect relationship Unit 5 Boardworks High Identifying
correlation and causation. Construct viable when a strong relationship is observed. Students should be careful not to assume Scatter School Algebra CC Similarities and
arguments and that correlation implies causation. The determination that one thing causes Plots and PowerPoints Differences
Causation is not specified critique the another requires a controlled randomized experiment. Trend Note Taking
in the CT standard. reasoning of others.
Lines Summarizing
Example: Alabama Lesson Plan
HS.MP.4. Model Cooperative
Lines of Best Fit
with mathematics. Diane did a study for a health class about the effects of a student's end-of-year Learning
math test scores on height. Based on a graph of her data, she found that there was a Nonlinguistic
Mathematics Assessment
HS.MP.6. Attend to direct relationship between students' math scores and height. She concluded that Representations
Project (MAP)
precision. "doing well on your end-of-course math tests makes you tall." Is this conclusion Vocabulary
justified? Explain any flaws in Diane's reasoning. L12: Interpreting Development
Statistics: A Case of
Muddying the Waters
N11: Interpreting
Categorical and
Quantitative Data
Khan Academy
Correlation and Causality:
Understanding why
correlation does not imply
causality
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
38 3. Represent HS.MP.2. Reason Example: Given two sets of data that can be modeled with linear functions, find the Unit 6 Boardworks High Identifying "You're Hired!"
constraints by equations or abstractly and intersection of the two trend lines, if it exists, and interpret the solution. For Systems of School Algebra CC Similarities and
inequalities, and by systems quantitatively. instance, if these trends continue, when will the women catch the men and Linear PowerPoints Differences Border Patrol
of equations and/or Equations Note Taking
what percentage of women will be earning $50,000 - $74,999? Eileen's Work Week:
HS.MP.4. Model
inequalities, and interpret Summarizing
with mathematics. Starting a Skateboard Business Solving Systems of
solutions as viable or Number % of men % of women Cooperative
of years Inequalities
nonviable options in a HS.MP.5. Use Lesson Plan 2: Hot Dog Learning
since earning earning Nonlinguistic
modeling context... appropriate tools Sales - Solving Linear
strategically. 2000 Equations and Inequalities Representations
$50,000 - $50,000 - $74,999
CT.9-12.1.C.3.a Manipulate Vocabulary
equations, inequalities and $74,999
LESSON: ROVERS ON Development
functions to solve
problems. 3 20.2 13.3 MARS
CT.9-12.1.C.3.a.(1) Model 4 20.5 14.2
and solve problems with
linear, quadratic and 5 20.7 15.1
absolute value equations
and linear inequalities.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
39 5. Prove that, given a HS.MP.2. Reason Example: Unit 6 Boardworks High Identifying FCAT Real World Solving
system of two equations in abstractly and Given that the sum of two numbers is 10 and their difference is 4, what are the Systems of School Algebra CC Similarities and Linear Equations/Systems
two variables, replacing one quantitatively. numbers? Explain how your answer can be deduced from the fact that they two Linear PowerPoints Differences
equation by the sum of that numbers, x and y, satisfy the equations x + y = 10 and x – y = 4. Equations Note Taking Linear Systems: Using
HS.MP.3. Algebra
equation and a multiple of the Summarizing
Construct viable Alabama Lesson Plans
other produces a system with arguments and Cooperative Solving Linear Systems by
Systems of Equations:
the same solutions. critique the Learning Graphing, Substitution, or
What Method Do You
reasoning of Prefer? Nonlinguistic Elimination
Proving that replacing one others. Representations
equation in a system by the Vocabulary
sum of that equation and a Mathematics Assessment
Development
multiple of the other Project (MAP)
produces a system with the L01: Optimization
same solutions is not Problems: Boomerangs
assessed on CAPT.
L03: Solving Linear
Equations in Two
Variables
N05: Reasoning with
Equations and
Inequalities
Escape from the Tomb
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
40 6. Solve systems of HS.MP.2. Reason The system solution methods can include but are not limited to graphical, Unit 6 Boardworks High Identifying Boats in Motion
linear equations exactly and abstractly and elimination/linear combination, substitution, and modeling. Systems can be written Systems of School Algebra CC Similarities and
approximately (e.g., with quantitatively. algebraically or can be Examples: Solve the system of equations: x+ y = 11 and Linear PowerPoints Differences Breakeven Analysis
graphs), focusing on pairs of 3x – y = 5. Use a second method to check your answer. Equations Note Taking
HS.MP.4. Model Lesson Plan 1: Left Hand, Graphing Systems at the
linear equations in two Summarizing
with mathematics. Your class is planning to raise money for a class trip to Washington, DC, by selling Right Hand - Solving Races
variables. your own version of Connecticut Trail Mix. You find you can purchase a mixture of Systems of Equations Cooperative
HS.MP.5. Use dried fruit for $3.25 per pound and a nut mixture for $5.50 per pound. The class Learning Graphing Systems of linear
CT.9-12.1.C.3.a.(3) Solve appropriate tools plans to combine the dried fruit and nuts to make a mixture that costs $4.00 per Nonlinguistic Equations
systems of two linear strategically. pound, which will be sold at a higher price to make a profit. You anticipate you will Representations
equations using algebraic need 180 pounds of trail mix. How many pounds of dried fruit and how many Vocabulary
or graphical methods. HS.MP.6. Attend pounds of mixed nuts do you need? Development
to precision.
represented in context. Students may use graphing calculators, programs, or applets
HS.MP.7. Look to model and find approximate solutions for systems of equations.
for and make use
of structure.
HS.MP.8. Look
for and express
regularity in
repeated
reasoning.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
41 11. Explain why the x- HS.MP.2. Reason Students need to understand that numerical solution methods (data in a table used to Unit 6 Boardworks High Identifying How Many Solutions?
coordinates of the points abstractly and approximate an algebraic function) and graphical solution methods may produce Systems of School Algebra CC Similarities and
where the graphs of the quantitatively. approximate solutions, and algebraic solution methods produce precise solutions Linear PowerPoints Differences Intersecting the Solution
equations y = f(x) and y = that can be represented graphically or numerically. Students may use graphing Equations Note Taking
HS.MP.4. Model Lesson:
g(x) intersect are the solutions calculators or programs to generate tables of values, graph, or solve a variety of Summarizing Nonlinear Systems of
with mathematics. Analyzing Exponential
of the equation f(x) = g(x); functions. Cooperative Equations
Functions
find the solutions HS.MP.5. Use Learning
approximately, e.g., using appropriate tools Nonlinguistic Which Garage Is Better?
UDL Checklist
technology to graph the strategically. Representations
functions, make tables of Vocabulary
HS.MP.6. Attend Development
values, or find successive
to precision.
approximations. Include cases
where f(x) and/or g(x) are HS.MP.7. Look
linear ...functions.* for and make use
of structure.
CT.9-12.1.C.2.a.(4) Evaluate
and interpret the graphs of
HS.MP.8. Look
linear, exponential and
for and express
polynomial functions.
regularity in
repeated
reasoning.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1(HS)
*Adapted from the Arizona Academic Content Standards.
42 1. Explain how the HS.MP.2. Reason Students may explain orally or in written format. Unit 7 Boardworks High Identifying
definition of the meaning of abstractly and Introduction School Algebra CC Similarities and
rational exponents follows quantitatively. to PowerPoints Differences
Exponential Note Taking
from extending the
HS.MP.3. Functions Alabama Lesson Plans
properties of integer Summarizing
Construct viable Just the facts!
exponents to those values, arguments and Exploring Order of Cooperative
allowing for a notation for critique the Operations and Learning
radicals in terms of rational reasoning of Properties of Real Nonlinguistic
exponents. For example, we others. Numbers Representations
define 51/3 to be the cube root Vocabulary
Hot Math Development
of 5 because we want (51/3)3 =
Rational Exponent
5(1/3)3 to hold, so (51/3)3 must
Problems
equal 5. (HotMath.com)43 Tree Core Models
Not to Fret
PDF Lesson
Rational Exponent
Lesson
Regents Prep
Discovering the Laws
of Rational Exponents
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
44
Waterbury Public Schools Mathematics Standards Articulated by Discipline
Algebra 1
Introductions 2. Rewrite HS.MP.7. Look Example Unit 7 Boardworks High Identifying Definitions and Laws of
expressions involving for and make use 1 1 1 Introduction School Algebra CC Similarities and Exponents
radicals and rational of structure. ab = (ab) = a b = a × b
2 2 2
to PowerPoints Differences
Exponential Note Taking Radicals
exponents using the
Functions
properties of exponents. Summarizing
Alabama Lesson Plans Investigating Laws of
Exponents and Cooperative Exponents
Division Learning
Just the facts! Nonlinguistic
Exploring Order of Representations
Operations and Vocabulary
Properties of Real Development
Numbers45 Exact Ratio
Not to Fret
PDF Lesson
Rational Exponent
Lesson
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
46SSE 1b. Interpret HS.MP.1. Make Students should understand the vocabulary for the parts that make up the whole Unit 7 Boardworks High Identifying
complicated expressions by sense of problems expression and be able to identify those parts and interpret there meaning in terms Introduction School Algebra CC Similarities and
viewing one or more of and persevere in of a context. to PowerPoints Differences
solving them. Exponential Note Taking
their parts as a single entity.
For example in the expression P(1+r)n, r may be the interest rate and 1 + r may be Functions
For example, interpret P(1+r)n Summarizing
HS.MP.2. Reason described as the "growth factor." Alabama Lesson Plans
as the product of P and a factor abstractly and Cooperative
Exponential Growth
not depending on P. quantitatively. Learning
and Decay
Nonlinguistic
Rags to Riches or
HS.MP.4. Model Representations
Riches to Rags?
with mathematics. Vocabulary
Mathematics Assessment Development
HS.MP.7. Look Project (MAP)
for and make use
L04: Increasing and
of structure.
Decreasing Quantities
by a Percent
L05: Modeling
Situations With
Linear Equations
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
47 L13: Sorting
Equations and
Identities
L16: Interpreting
Algebraic Expressions
A04: Circle Pattern
N02: Seeing
Structure in
Expressions
A-SSE 3c. Use the HS.MP.1. Make In this example the annual growth factor is 1.15 and the monthly growth factor is Unit 7 Boardworks High Identifying
properties of exponents to sense of problems 1.012. Introduction School Algebra CC Similarities and
transform expressions for and persevere in to PowerPoints Differences
exponential functions. For solving them. Exponential Note Taking
example the expression Functions Alabama Lesson Plans
Summarizing
1.15t can be rewritten as Discover the Roots of
[1.15 (1/12)](12t) ≈1.012(12t) to HS.MP.2. Reason a Polynomial Cooperative
reveal the approximate abstractly and Function Learning
equivalent monthly interest quantitatively. "Factoring by Mack" Nonlinguistic
rate if the annual rate is Exponential Growth
Representations
15%. and Decay
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
48 Factoring Fanatic Vocabulary
Development
Mathematics Assessment
Project (MAP)
L13: Sorting
Equations and
Identities
L20: Forming
Quadratics
A04: Circle Pattern
N02: Seeing
Structure in
Expressions
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
49 expressions(A-SSE.3.c)
Analyzing the Data
Exploring
Measurement,
Sequences, & Curves
with Stringed
Instruments: Pre-
Activity Questions
Graphical Analysis
Movement with
Functions
Numerical Analysis
Science Net Links (A-
SSE.3.c)
Frosty the Snowman
Meets His Demise:
An Analogy to
Carbon Dating
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
50 7. Graph functions HS.MP.5. Use Exponential functions are similar to linear functions in that the y-intercept often Unit 7 Boardworks High Identifying Analysis of a
expressed symbolically and appropriate tools represents a starting point. Introduction School Algebra CC Similarities and Bouncing Ball
show key features of the strategically. to PowerPoints Differences
In exponential growth models, as the independent variable increases, the dependent Exponential Note Taking Back and Forth---
graph, by hand in simple
HS.MP.6. Attend variable increases at continually increasing rates. Functions Analysis of Spring
cases and using technology Summarizing
to precision. Alabama Lesson Plans Motion
for more complicated In exponential decay models, as the independent variable increases, the dependent Cooperative
Predict the Future?
cases.* variable approaches zero asymptotically. Learning
Human slope
Nonlinguistic
Exponential Growth
e. Graph exponential ... Representations
and Decay
functions, showing Vocabulary
Math is Functional
intercepts and end Development
Graphing at all levels:
behavior... It's a beautiful thing!
Who am I? Find A
Polynomial From Its
Roots
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
51 L20: Forming
Quadratics
A16: Sorting
Functions
N06: Interpreting
Functions
NCTM Illuminations
(F-IF.7.e)
Exploring
Measurement,
Sequences, and
Curves with Stringed
Instruments: Pre-
Activity Questions
Graphical Analysis
Hearing Music,
Seeing Waves
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
52 8b. Use the properties HS.MP.2. Reason All exponential functions may be written in the form f(x) = abx. In a later course Unit 7 Boardworks High Identifying Behaviors-Exponential
of exponents to interpret abstractly and students may learn that they may also be written in the form Introduction School Algebra CC Similarities and Function
expressions for exponential quantitatively. f(x) = aebx. to PowerPoints Differences
Exponential Note Taking
functions. For example,
HS.MP.7. Look Functions NCTM Illuminations
identify percent rate of Summarizing
for and make use (F-IF.8.b)
change in functions such as y of structure. Cooperative
= (1.02) t, y = (0.97)t, y = Learning
Analyzing the Data
(1.01) 12t, y = (1.2) (t/10), Nonlinguistic
Exploring
and classify them as Representations
Measurement,
representing exponential Sequences, and Vocabulary
Curves with Stringed Development
functions.
Instruments: Pre-
CT.9-12.1.C.1.a.(4) Activity Questions
Describe and compare Smokey Bear Takes
properties and classes of Algebra
linear, quadratic and
exponential functions.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
53E.1.a.(4) Solve
problems involving
financial applications
including compound
interest, amortization of
loans, and investments.
CT.9-12.1.E.1.a.(1)
Describe and compare
properties and classes of
functions, including
exponential, polynomial
quadratic.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
54BF 2. Write ... geometric HS.MP.4. Model A recursive formula is An= r An-1. Unit 7 Boardworks High Identifying Double Tree
sequences both recursively with mathematics. Introduction School Algebra CC Similarities and
and with an explicit An explicit formula is An= A1 rn-1 to PowerPoints Differences Sequence of Bounces
HS.MP.5. Use Exponential Note Taking Activity - Modeling Motion
formula, use them to model
appropriate tools Functions Lesson: 7 Billion
situations, and translate Summarizing
strategically. Spreading Doom
between the two forms.* Cooperative
Lesson: Marathon Math
HS.MP.8. Look Learning Using Cabri Geometry to
CT.9-12.1.C.1.a.(1) for and express NCTM Illuminations Nonlinguistic Create Fractals
Identify, describe, create regularity in Representations
Lessons:
and generalize numeric, repeated Vocabulary
Counting the Trains
geometric, and statistical reasoning. Development
More Trains
patterns with tables,
graphs, words and Recursive and Exponential
symbolic rules. Rules
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
55 1. Distinguish HS.MP.3. Example: Common differences for world population growth show an increasing Unit 7 Boardworks High Identifying Accelerates Returns
between situations that can Construct viable trend, suggesting that an exponential model may be more appropriate than a linear Introduction School Algebra CC Similarities and
be modeled with linear arguments and one. to PowerPoints Differences Add Them Up
critique the Years since 1980 Population (billions) Common differences Exponential Note Taking
functions and with
reasoning of 0 4.453 --- Functions Lesson: Behaviors-Exponential
exponential functions. Summarizing
others. Comparing Linear and Function
1 4.529 .076 Exponential Functions Cooperative
a. Prove ... that exponential HS.MP.4. Model Learning Buying Your First New Car!
functions grow by equal with mathematics. 2 4.609 .080 Lesson Plan 1: Overrun Nonlinguistic
factors over equal by Skeeters - Exponential Representations Linear Functions: Slope as
HS.MP.5. Use Growth
intervals.... 3 4.690 .081 Vocabulary Rate of Change
appropriate tools
Development
Proving that linear
strategically. Lesson Plan 2: Skeeter
4 4.771 .081
functions grow by equal Populations and
HS.MP.7. Look
differences over equal 5 4.852 .081 Exponential Growth
intervals and that
for and make use
exponential functions of structure.
6 4.936 .084 Alabama Lesson Plans
grow by equal factors
over equal intervals are Predict the Future?
not assessed on CAPT. 7 5.022 .086 Show Me The Money
- Saving and Investing
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
56 Recognize situations in HS.MP.8. Look Example: Technetium-99m is a drug taken by a patient and then used to study Unit 7 Boardworks High
which a quantity grows or for and express tumors in the brain, lungs and other parts of the body. A patient takes a 1000- Introduction School Algebra CC
decays by a constant percent regularity in mg pill. The data below shows how much active ingredient remains in the to PowerPoints
rate per unit interval repeated body over 6-hour time intervals. Exponential
relative to another. reasoning. Functions
Shodor (F-LE.1.b,c)
Technetium-99m Decay
Reading Graphs
# of 6-hour Amount of Drug
Time Intervals Remaining (mg) Shodor (F-LE.1.a,b,c)
Function Flyer
0 1000 Graphit
Multi-Function Data
1 500 Flyer
Data Flyer
2 250
3 125
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
57 2. Construct linear HS.MP.4. Model Example: Determine an exponential function of the form f(x) = abx using data Unit 7 Boardworks High Identifying
and exponential functions, with mathematics. points from the table. Graph the function and identify the key characteristics of the Introduction School Algebra CC Similarities and
including arithmetic and graph. to PowerPoints Differences
HS.MP.8. Look Exponential Note Taking
geometric sequences, given
for and express x f(x) Functions Math Tasks:
a graph, a description of a Summarizing
regularity in 0 2 Setting the Table
relationship, or two input- repeated Cooperative
1 6
output pairs (include reasoning. Alabama Lesson Plans Learning
3 54
reading these from a table). Predict the Future? Nonlinguistic
Writing equations for Representations
CT.9-12.C.2.a.2.(1) Select parallel lines Vocabulary
and use appropriate Exponential Growth Development
methods for computing in and Decay
a variety of contexts. Density
What is the slope of
the stairs in front of
the school?
Marathon Math
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
58hematics Assessment
Project (MAP)
E11: Table Tiling
N08: Linear and
Exponential Models
NCTM Illuminations
How Did I Move?
How Should I Move?
Movement with
Functions
Shrinking Candles,
Running Water,
Folding Boxes
Shodor
Function Flyer
Graphit
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
59 Functions and
Vertical Line Test
Graphs and Functions
Recognizing Patterns
F-LE 3. Observe using HS.MP.2. Reason Example: Contrast the growth of f(x)=3x + 20 and g(x)=3x, Unit 7 Boardworks High Identifying
graphs and tables that a abstractly and for x = 0, 1, 2, 3, 4, 5. Introduction School Algebra CC Similarities and
quantity increasing quantitatively. to PowerPoints Differences
exponentially eventually Exponential Note Taking
exceeds a quantity Functions Summarizing
Alabama Lesson Plan
increasing linearly ... Exponential Growth Cooperative
and Decay Learning
CT.9-12.1.C.1.a.(4) Nonlinguistic
Describe and compare Representations
Mathematics Assessment
properties and classes of Vocabulary
Project (MAP)
linear, quadratic and Development
E11: Table Tiling
exponential functions.
N08: Linear and
Exponential Models
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
60 Shrinking Candles,
Running Water,
Folding Boxes
Shodor
Function Flyer
Graphit
Multi-Function Data
Flyer
Data Flyer
Slope Slider
Graphs and Functions
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
61 5. Interpret the HS.MP.2. Reason Example: A function of the form f(n) = P(1 + r)n is used to model the amount of Unit 7 Boardworks High Identifying
parameters in a ... abstractly and money in a savings account that earns 5% interest, compounded annually, where n Introduction School Algebra CC Similarities and
exponential function in quantitatively. is the number of years since the initial deposit. What is the value of r? What is the to PowerPoints Differences
meaning of the constant P in terms of the savings account? Exponential Note Taking
terms of a context.
HS.MP.4. Model Mathematics Assessment
Functions Summarizing
CT.9-12.1.C.1.a.(3) with mathematics. Project (MAP) Cooperative
Identify the N08: Linear and Learning
characteristics of Exponential Nonlinguistic
functions and relations, Models Representations
including domain and Vocabulary
NCTM Illuminations Development
range.
Egg Launch
Contest
Graph Chart
Graphing What
Movie Lines
Pedal Power
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
62N-RN 3. Explain why the HS.MP.2. Reason Since every difference is a sum and every quotient is a product, this includes Unit 8 Boardworks High Identifying
sum or product of two abstractly and differences and quotients as well. Explaining why the four operations on rational Quadratic School Algebra CC Similarities and
rational numbers is quantitatively. numbers produce rational numbers can be a review of students understanding of Functions PowerPoints Differences
fractions and negative numbers. Explaining why the sum of a rational and an and Note Taking
rational; that the sum of a
HS.MP.3. irrational number is irrational, or why the product is irrational, includes reasoning Equations
rational number and an Summarizing
Construct viable about the inverse relationship between addition and subtraction (as well as Google PDF
irrational number is arguments and multiplication and division) Why is the sum of a Cooperative
irrational; and that the critique the rational and irrational Learning
product of a nonzero reasoning of Example: if a, b, and c are all rational numbers then the solutions to the equation number irrational? Nonlinguistic
rational number and an others. (PROOF) Representations
b b 2 - 4ac
irrational number is x=- ± Vocabulary
ax 2 + bx + c = 0 are given by 2a 2a , MathForum.org Development
irrational.
Why is the product of
CT.9-12.2.C.1.a Extend the Recognizing that the first term is always rational, we may conclude that whenever a nonzero rational
understanding of number the second term is irrational then the solutions must also be irrational. number and an
to include integers, irrational number
rational numbers and real irrational?
numbers.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
63 functions standards include Mathematics Assessment
other properties in the Project (MAP)
same standard. CCSS
focuses primarily on N01: The Real
property of closure Number System
within number systems.
Wiki.Answers.com
Why is the sum of two
rational numbers
rational?
A-SSE 3. HS.MP.1. Make Three forms of the quadratic function reveal different features of its graph. Unit 8 Boardworks High Identifying The Box Method
sense of problems Quadratic School Algebra CC
Standard form: f (x) = ax + bx + c reveals the y intercept, (0, c).
2 Similarities and
a. Factor a quadratic and persevere in Functions PowerPoints Differences The Factor Connection
expression to reveal the solving them. and Note Taking
Vertex form: f (x) = a(x - h) + k reveals the vertex (h, k) and thus the
2
zeros of the function it Equations Math Tasks:
Summarizing
defines. HS.MP.2. Reason maximum or minimum value of the function. Building a Playground
abstractly and Cooperative
quantitatively. Virtual Algebra Tiles Learning
Factored form: f (x) = a(x - x1 )(x - x2 ) reveals the x-intercepts (x1,0) and
b. Complete the square in a
Nonlinguistic
quadratic expression to (x2,0) . Representations
GeoGebra Applet –
reveal the maximum or Completing the Square Vocabulary
minimum value of the Development
function it defines.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
64CCSS is more specific than
the CT standards.
Factoring quadratic
expressions to reveal the
zeros of the function it
defines is not assessed on
CAPT.
CCSS is more specific than
the CT standard.
Completing the square in
a quadratic expression to
reveal the maximum or
minimum value of the
function it defines is not
assessed on CAPT.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
65APR 1. Understand that HS.MP.4. Model The product of two binomials of degree one is generally a trinomial of degree two; Unit 8 Boardworks High Identifying Area - "FOILED" Again
polynomials form a system with mathematics. the process of expanding this product may be reversed to produce the factored form Quadratic School Algebra CC Similarities and
analogous to the integers, of the polynomial. Functions PowerPoints Differences Binomial Multiplication
namely, they are closed and Note Taking
Alabama Lesson Plans
under the operations of Equations Summarizing
What are You - Prime
addition, subtraction, and or Composite? Cooperative
multiplication; add, Learning
Polynomials Divided
subtract, and multiply by Monomials Nonlinguistic
polynomials. What's The Real Cost
Representations
of That Car? Vocabulary
Show Me The Money - Development
Saving and Investing
Addition, subtraction and "Like Terms", I Add
multiplication of
polynomials is not
Them
assessed on CAPT. Fortune Properties
Multiplying
Polynomials
Color this Polynomial
Simplified
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
66Mathematics Assessment
Project (MAP)
N03: Arithmetic with
Polynomials and
Rational Expressions
NCTM Illuminations
A Geometric
Investigation of (a +
b)2
A-CED 1. Create equations HS.MP.7. Look Example: Lava coming from the eruption of a volcano follows a parabolic path. The Unit 8 Boardworks High Identifying
and inequalities in one for and make use height h in feet of a piece of lava t seconds after it is ejected from the volcano is Quadratic School Algebra CC Similarities and
of structure. Functions PowerPoints
given by h(t) = -16t + 64t + 936 .
variable and use them to 2 Differences
and Note Taking
solve problems. Include After how many seconds does the lava reach its maximum height of 1000 feet? Equations To Babysit or Not To
equations arising from Summarizing
Babysit
...quadratic functions ... Cooperative
Learning
Nonlinguistic
Representations
Vocabulary
Development
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
67
absolute value equations
and linear inequalities.
A-CED 2. Create equations HS.MP.2. Reason Unit 8 Boardworks High Identifying
in two or more variables to abstractly and Quadratic School Algebra CC Similarities and
represent relationships quantitatively. Functions PowerPoints Differences
between quantities; graph and Note Taking
HS.MP.4. Model Equations Lesson Plan 1: The
equations on coordinate Summarizing
with mathematics. Phone Bill Problem -
axes with labels and scales. Linear Functions Cooperative
HS.MP.5. Use Learning
CT.9-12.1.C.2.a.(1) appropriate tools Nonlinguistic
Represent functions and Representations
strategically.
relations on the
coordinate plane. Vocabulary
Development
CT.9-12.1.C.3.a.(1) Model
and solve problems with
linear, quadratic and
absolute value equations
and linear inequalities.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
68REI 4. a. Use the HS.MP.2. Reason Students may solve by factoring, completing the square, and using the quadratic Unit 8 Boardworks High Identifying Area Of The Missing
method of completing the abstractly and formula. The zero product property is used to explain why the factors are set equal Quadratic School Algebra CC Similarities and Square
square to transform any quantitatively. to zero. Students should relate the value of the discriminant to the type of root to Functions PowerPoints Differences
quadratic equation in x into expect. A natural extension would be to relate the type of solutions to ax2 + bx + c and Note Taking Bridge On The River
HS.MP.7. Look = 0 to the behavior of the graph of y = ax2 + bx + c . Equations Lesson: Quad
an equation of the form (x – Summarizing
for and make use Solving by Completing the
p)2 = q that has the same of structure. Value of Nature of Roots Nature of Graph Square Cooperative
solutions. Derive the Discriminant Learning
quadratic formula from this HS.MP.8. Look b2–4ac = 0 One real root One x-intercept UDL Checklist Nonlinguistic
form. for and express b2–4ac > 0 Two real roots Two x-intercepts Representations
regularity in b2–4ac < 0 No real root Does not intersect x-axis Pencast for Teachers: Vocabulary
b. Solve quadratic repeated Deriving the Quadratic Development
equations by inspection reasoning. Formula
(e.g., for x2 = 49),
Interactive Quadratic Quiz
taking square roots,
completing the square,
the quadratic formula
and factoring, as
appropriate to the
initial form of the
equation.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
69 absolute
value equations and linear
inequalities.
Completing the square
and deriving the
quadratic formula are not
assessed on CAPT.
Solving quadratic
equations as appropriate
to the initial form of the
equation and recognizing
when the quadratic
formula gives complex
solutions is not assessed
on CAPT.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
70 4. For a function that HS.MP.2. Reason Example : Unit 8 Boardworks High Identifying
models a relationship abstractly and A rocket is launched from 180 feet above the ground at time t = 0. The function Quadratic School Algebra CC Similarities and
between two quantities, quantitatively. that models this situation is given by h = – 16t2 + 96t + 180, where t is measured in Functions PowerPoints Differences
interpret key features of seconds and h is height above the ground measured in feet. and Note Taking
HS.MP.4. Model Equations Math Tasks:
graphs and tables in terms Summarizing
with mathematics. 1. What is a reasonable domain restriction for t in this context? The Lacrosse Tournament
of the quantities, and sketch Cooperative
graphs showing key features HS.MP.5. Use 2. Determine the height of the rocket two seconds after it was launched. Grapher Learning
given a verbal description appropriate tools Nonlinguistic
of the relationship. Key strategically. 3. Determine the maximum height obtained by the rocket. Representations
features include: intercepts; Vocabulary
HS.MP.6. Attend 4. Determine the time when the rocket is 100 feet above the ground. Development
intervals where the function
to precision.
is increasing, decreasing, 5. Determine the time at which the rocket hits the ground.
positive, or negative;
relative maximums and How would you refine your answer to question 1 based on your response to
minimums; symmetries... * question 5?
CT.9-12.1.C.2.a.(2)
Identify an appropriate
symbolic representation
for a function or relation
displayed graphically or
verbally.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
712.a.(3)
Recognize and explain
the meaning of the slope
and x- and y-intercepts as
they relate to a context,
graph, table or equation
F-IF 7a. Graph ... quadratic HS.MP.5. Use Unit 8 Boardworks High Identifying
functions and show appropriate tools Quadratic School Algebra CC Similarities and
intercepts, maxima, and strategically. Functions PowerPoints Differences
and Note Taking
minima.
HS.MP.6. Attend Equations
Summarizing
CT.9-12.1.C.2.a.(3) to precision.
Cooperative
Recognize and explain Learning
the meaning of the slope Nonlinguistic
and x- and y-intercepts as Representations
they relate to a context,
Vocabulary
graph, table or equation.
Development
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
72 Minimuma. Use the process of HS.MP.2. Reason Unit 8 Boardworks High Identifying Quadratic Equations
factoring and completing abstractly and Quadratic School Algebra CC Similarities and
the square in a quadratic quantitatively. Functions PowerPoints Differences
function to show zeros, and Note Taking
HS.MP.7. Look Equations Lesson:
extreme values, and Summarizing
for and make use Factoring Trinomials Using
symmetry of the graph, and of structure. Algebra Tiles
interpret these in terms of a Algebra Tile Teacher Cooperative
context. Reference Learning
Nonlinguistic
Factoring and completing Factoring Algebra Tiles Representations
the square in a quadratic Yellow Vocabulary
equation is not assessed Factoring Algebra Tiles Development
on CAPT. Blue
Lesson Plan 1: The X
Factor - Trinomials and
Algebra Tiles
Interactive Algebra Tiles
NCTM Illuminations
activity: Completing the
Square with Algebra Tiles
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards.
73BF 3. Identify the effect HS.MP.4. Model Students should recognize these transformations of the parent graph f(x)= x2: Unit 8 Boardworks High Identifying Around the Vertex in 80
on the graph of replacing with mathematics. Quadratic School Algebra CC Similarities and Days
f(x) by f(x) + k, kf(x), f(kx), f(x)= x2 + k is a vertical translation Functions PowerPoints Differences
and f(x + k) for specific HS.MP.5. Use f(x) = (x + k)2 is a horizontal translation and Note Taking Changing Functions
appropriate tools f(x)= kx2 stretches the graph for k > 0 Equations Lesson Plan 1: Up,
values of k (both positive Summarizing
strategically. f(x)= –x2 reflects the graph over the x-axis. Down, Right, Left - Exploring
and negative); find the Function Families Cooperative Transformations of the
value of k given the graphs. HS.MP.7. Look Learning Quadratic Function
Experiment with cases and for and make use Nonlinguistic
illustrate an explanation of of structure. Representations
the effects on the graph Vocabulary
Development
using technology...
Identifying the effect on
the graph of
transformations of given
functions is not specified
in the CT standard.
Adapted from The Leadership and Learning Center "Rigorous Curriculum Design" Algebra 1*Adapted
from the Arizona Academic Content Standards |
Essentials of Trigonometry - With CD - 4th edition
Summary: Intended for the freshman market, this book is known for its student-friendly approach. It starts with the right angle definition, and applications involving the solution of right triangles, to help students investigate and understand the trigonometric functions, their graphs, their relationships to one another, and ways in which they can be used in a variety of real-world applications. The book is not dependent upon a graphing calculator.
Used - Good Hardcover. texbook only 4th Edition May contain highlighting/underlining/notes/etc. May have used stickers on cover. Ships same or next day. Expedited shipping takes 2-3 business days; sta...show morendard shipping takes 4-14 business days. ...show less
No guarantee for ancillary materials(Such as CDs, Online access code).Ships today or the next business day. Cover and binding show moderate wear. Text may contain moderate writing/highlightings but...show more is easily readable. 10-12. ...show less
$22.90 +$3.99 s/h
Good
Thrifty_Text Emeryville, CA
No guarantee for ancillary materials(Such as CDs, Online access code).Ships today or the next business day. Cover and binding show wear. Text may contain some writing/highlightings but nothing majo...show morer. 10-12. ...show less
$24.19 +$3.99 s/h
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Big Planet Books Burbank, CA
2005-09-14 Hardcover Good Expedited shipping is available for this item!
$29.51 +$3.99 s/h
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Very good.
$30.00 +$3.99 s/h
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Tin Can Mailman CA Arcata, CA
2006 Hardcover Good NO CD. Includes iLrn Tutorial and Studen Solutions Manual.
$30.00 +$3.99 s/h
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Textbook Barn Woodland Hills, CA
Hardcover Good 0534494234 Shipped from CA. This book may have a school stamp or sticker. Book may contain some highlighting and other markings throughout book and on binding. Book shows backpack we...show morear |
Mathematics helps us make sense of the world and reveals methods for using known information to find unknown information. We regularly study the most efficient methods for reaching solutions, but also realize that examining different solution methods helps develop more flexible problem solving skills.
Our department is focused on instilling students with enduring understandings in mathematics. We seek to help students become efficient users of algorithms who can articulate their thinking and understand how to apply mathematics in different contexts. For example, if you are taking a prescription medication and miss a dose, why do the instructions warn against simply taking a double dose? Or how does a country with over 300 million people in 50 states fairly apportion a mere 435 seats in the House of Representatives?
Mathematics is the language of the universe. We strive to help students become fluent users of this language.
Quote
"You understand how to think mathematically when you are resourceful, flexible, and efficient in your ability to deal with new problems in mathematics." |
Course Description
This course could be called Generalizations of Fundamental
Theorem of Calculus. We will define integration over surfaces
and curves and we will relate those integrals in a fashion similar to
the Fundamental Theorem of Calculus.
The whole material has been developed for direct
applications in physics and engineering.
Prerequisites
Definition of Riemann integral.
Some techniques of integration and differentiation.
Polar coordinates.
Curves, tangent (velocity) vectors to curves.
Partial derivatives.
Linear maps and their matrices.
Using determinants for computing area/volume of
parallelopipeds.
Using determinants of matrices of linear maps for computing the
area of the image of a region
under that linear map.
Text:
Calculus Early
Transcendentals, by C. Henry Edwards and David E. Penney
The text is required and is available at the bookstore. We will cover
material from chapters 13, 14 and some review from chapter 11. |
Mathematical Reasoning for Elementary School Teachers (6th Edition)
9780321693129
ISBN:
0321693124
Edition: 6 Pub Date: 2011 Publisher: Addison Wesley
Summary: Calvin T. Long is the author of Mathematical Reasoning for Elementary School Teachers (6th Edition), published 2011 under ISBN 9780321693129 and 0321693124. Seven hundred fifty two Mathematical Reasoning for Elementary School Teachers (6th Edition) textbooks are available for sale on ValoreBooks.com, two hundred sixty six used from the cheapest price of $29.89, or buy new starting at $5010/10)[ |
Synopses & Reviews
Publisher Comments:
Kirk Trigsted revolutionized the way this course is taught when he created College Algebra, a completely clickable ebook that was written from the ground up within MyMathLab. Recognizing that today's students start with the homework instead of reading the text, Trigsted created an online learning environment that is a seamless mix of exposition, videos, interactive animations, tutorials, and assessment. This approach leverages the power of MyMathLab and leads students to interact with course materials in a way that is proving to be more effective. With the Second Edition, Trigsted continues to innovate with a revised design that improves navigation and usability, expanded videos, and increased animation coverage.
Synopsis:
About the Author
Kirk Trigsted teaches mathematics at the University of Idaho and has been Director of the Polya Mathematics Center since its inception in 2001. Kirk has taught with MyMathLab for many years, and has contributed to the videos for several Pearson books. Kirk is also actively involved with the National Center for Academic Transformation (NCAT |
The Algebra Experience
1998 We want kinds in eighth and ninth grade to get help on Algebra. This will help them in school and in future years. This page is also great for mathematically advanced students. They can learn some Algebra and impress their teacher! Another objective is to have people discuss questions and get involved. Our main purpose is for students to learn! |
ics for Elementary Teachers: A Contemporary Approach
Students who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement ...Show synopsisStudents who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world. the text helps students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format. . |
books.google.com - The second edition of 'Discrete Mathematics for Computer Scientists' provides an accessible introduction to the subject for beginners and an unparalleled reference source for the more experienced programmer. It includes a thorough coverage of all areas of discrete mathematics, including logic, natural... Mathematics for Computer Scientist
Discrete Mathematics for Computer Scientist
The second edition of 'Discrete Mathematics for Computer Scientists' provides an accessible introduction to the subject for beginners and an unparalleled reference source for the more experienced programmer. It includes a thorough coverage of all areas of discrete mathematics, including logic, natural numbers, coding theory, combinatorics, sets, algebraic functions, partially ordered structures, graphs, formal machines and complexity theory, as well as a special emphasis on the central role of propositional and predicate logic. This edition incorporates an improved treatment of induction and coverage of more 'basic' algebra, while the style has been made more accessible. Chapter 6 has been replaced by two chapters on propositional logic and further logic.
About the author (1999)
John Truss has taught at Oxford University, Paisley College of Technology and currently at the University of Leeds. He has been a committee member of the British Logic Colloquium since 1990, and has recently been appointed an editor of the Journal of the London Mathematical Society. He wrote Foundations of Mathematical Analysis in 1997 and has authored 40 research papers. |
{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":93.98,"ASIN":"0321652746","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":92.89,"ASIN":"032119991X","isPreorder":0}],"shippingId":"0321652746::RLuZ1Y6QwB%2FYb5JjShcoagZlc1LDyGKm5mAh6ft7jlkdAJA%2FHmV69gr49YV6PR8VexcrkdbfjIKox%2BpTrx7yKIxjKKhd8Y%2FOlf1AoXeMY6I4U6G4C4de5A%3D%3D,032119991X::texp%2FPv8cJI4TBLFW2t%2BPE%2Fsd%2B0sFV2WMt9j4M3DrXOObaEG4ndcMhlWrYPF9R0gV1MOACcgzW%2BvciPvcjBDkuc4%2Bv76JdV%2B%2FPfWUCQ1VgAMw3xOTibuyEl Martin-Gay developed an acclaimed series of lecture videos to support developmental mathematics students. These highly successful videos originally served as the foundation materials for her texts. Today, the videos are specific to each book in her series. She has also created Chapter Test Prep Videos to help students during their most "teachable moment"—as they prepare for a test—along with Instructor-to-Instructor videos that provide teaching tips, hints, and suggestions for every developmental mathematics course, including basic mathematics, prealgebra, beginning algebra, and intermediate algebra.
Elayn is the author of 12 published textbooks and numerous multimedia interactive products, all specializing in developmental mathematics courses. She has participated as an author across a broad range of educational materials: textbooks, videos, tutorial software, and courseware. This offers an opportunity for multiple combinations for an integrated teaching and learning package, offering great consistency for the student.
More About the Author-- Her textbooks and acclaimed video program support Elayn's passion of helping every student to succeed.
This book was a required text for my Algebra course, it covers 3 different math courses, so it's worth the buy. I am taking my classes online and everything in this book is broken down and simple to follow. I got C's and D's in Algebra in high school, I aced my first test and find the lessons are step by step so you learn what you are doing and why! This is the best math textbook you could get! |
The reader is introduced to higher mathematics in an experimental way. He works with numerous interactive Java- simulations treating mathematical topics from number theory to infinitesimal calculus and partial differential equations. On the way he playfully learns the EJS simulation technique. Beyond the mathematics simulations the data pool contains... more...
Despite what we may sometimes imagine, popular mathematics writing didn't begin with Martin Gardner. In fact, it has a rich tradition stretching back hundreds of years. This entertaining and enlightening anthology--the first of its kind--gathers nearly one hundred fascinating selections from the past 500 years of popular math writing, bringing to... more...
Perfect for revision, this Study Guide concisely covers all the syllabus topics in a digestible format. With lots of opportunity to practise, examiner hints and past exam questions, it will fully prepare students for exams. more...
The easy way to brush up on the math skills you need in real life Not everyone retains the math they learned in school. Like any skill, your ability to speak "math" can deteriorate if left unused. From adding and subtracting money in a bank account to figuring out the number of shingles to put on a roof, math in all of its forms factors into daily... more...
The fundamental aim of this book is to communicate the knowledge necessary for a competent mathematical assessment of modern developments in the age of the Internet. Most crucially, this includes an understanding of very large graphs, calculating with large numbers, and calculating using prime number bases. more...
This photocopy master book, which has proven extremely popular over the years, provides a range of 30+ problem solving activities using strategies such as: Developing logical thinking; Using number concepts to develop logical thinking; Logical reasoning; Developing visual imagery; and Pattern perception using number. more...
Photocopy Master book. Includes problem solving strategies such as Guess and Check, Act It Out, Make A Model, Look for a Pattern, Construct a Table and so on. These strategies are applied to a range of interesting problem situations. Children will enjoy the variety of characters that provide an amusing element to the serious business of solving mathematical... more... |
Algebra 2
This course is designed to build on algebraic and geometric concepts. It develops advanced algebra skills such as systems of equations, advanced polynomials, imaginary and complex numbers, quadratics, and concepts and includes the study of trigonometric functions. It also introduces matrices and their properties. The content of this course are important for students' success on both the ACT and college mathematics entrance exams.
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Course Description: This course is the foundation for high
school mathematics courses. It is the bridge from the concrete to
the abstract study of mathematics. Topics include simplifying
expressions, evaluating and solving equations and inequalities,
and graphing linear and quadratic functions and relations. Real
world applications are presented within the course content and a
function's approach is emphasized.
NEISD No. 7018
Course: PREP
Credit: 1.0 (Local Credit)
Grade Placement: 9
Prerequisites: Students who were unsuccessful on eighth
grade math TAAS or who scored below 80% in eighth grade
math will be required to take this course.
Course Description: This course is designed to assist students
to successfully complete Algebra I and the math portion of the
TAAS. Students will prepare for the math portion of TAAS given in
the tenth grade and will be assisted in successfully completing
Algebra I. Objectives for the math TAAS will be taught using a
variety of methods. Students who were unsuccessful on the
eighth grade math section of TAAS or who have not passed eighth
grade math with an 80 or better average will be required to take
this course in place of Study Skills which is required of all
non-athlete freshmen.
NEISD No. 2323
Course: ALGEBRA II
Credit: 1.0
Grade Placement: 10-11
Prerequisites: Geometry or Algebra I and Teacher
Recommendation
Course Description: This course extends the topics first seen
in Algebra I and provides advanced skills in algebraic
operations. Additionally, linear and quadratic functions and
relations, conic sections, exponential and logarithmic functions,
graphing, and sequences and series will be explored.
NEISD No. 2324
Course: ALGEBRA II - HONORS
Credit: 1.0
Typical Grade Placement: 10-11
Prerequisites: Geometry Honors and Teacher Recommendation
Course Description: This course extends and deepens the topics
of the regular course.
NEISD No. 2325
Course: ALGEBRA II GIFTED/TALENTED
Credit: 1.0
Typical Grade Placement: 9
Prerequisites: Geometry GT and admission to district's GT
program
Course Description: This course extends and deepens the topics
of the honors course. Problem solving and higher-order thinking
skills are required as well as qualification on the district
Gifted Math matrix. |
Upon entering high school students should have completed all the necessary math courses available at the junior high level. Students should know basic math such as addition, subtraction, multiplications and division. They should also have an understanding of algebra, geometry, percentages, decimals, data analysis and probability, along with other junior high level mathematics. 9th grade is usually the grade where high school begins. If students on not versed in the math courses offered in junior high, they will not be prepared for the courses offered at the high school level...
There are certain expectations of students upon completion of the 9th grade. In the math area, is assumed that the students have mastered the previous grade math. It is impossible to advance in math without knowing the previous course study. Students are expected to reason, order, compare, and solve multi-step problems with rational and irrational numbers. They should also understand the concept of the complex number system, and be able to solve a number of problems, and use the coordinate system, Students are also expected to analyze argue, and support, conclusions based on statistical information, draw and explain graphs, design statistical experiments based on real world situations. In the geometry area. Students should be able to apply trigonometry to a problem about triangles. Problem solving for questions involving angles, derive the equation of a circle. And apply transformations and vectors to solve problems. Students in the 9th grad also need advance algebra which they should have been prepared for earlier. They should be able to investigate problems involving linear, quadratic, exponential, logarithmic, and rational functions. Students need to know how to pose and prove a variety of theorems use matrices for representing data, and master problems using a variety of polynomials.
These are only some of the subjects expected of students after completion of the 9th grade. It may seem a bit on the hard side to comprehend of this at such an early age, but it is essential to now this math in order to advance. No matter if you attend a public or private school, or you are home schooled, or studies online, this is expected of a 9th grade student before he moves ahead. These math studies will vary depending on the choice of study, but all of the institutions will expect the students to know this information, and some will expect even more. |
Mathematical Equation Evaluator is a mobile based student guiding application which is specially designed for local ordinary level students and advanced level students who are working in the different contexts related to complex mathematical formulas. The present process of solving complex mathematical problems is, writing down the mathematical formulae and simplifying them for solving the problem easily with a lesser number of variables. Assigning the given values to the final formula and calculating the final answer or the value by using a scientific calculator in step by step are the last two steps involved. The developed mobile application Mathematical Equation Evaluator system will eliminate the complexity and delays in the manual solving of the complex mathematical equations. The system users can enter the mathematical equation to be solved (or to be simplified) via the camera of the mobile phone in to the system. First the system will identify the equation string and detect the variables and the constants. Then the users can enter the known values for the variables in the equation. Finally, the system will generate the answer for the equation, with the step-by-step explanation displaying how the system has calculated the answer. The application is to be used by students after they have worked out the problem for answers. The application will show the answer and the steps of reaching the answer for the student to verify the result of the calculation attempt. In addition to that, the teachers can use this application to buildup answer scripts for mathematics papers. The most important feature of this project is the capability of inputting the equations via the mobile phone camera. Then, the students can scan the equation via the camera attached to the phone and evaluate the equation easily and quickly. Optical Character Recognition technology is used in implementing the above. Complex mathematical equations are solved by using predefined mathematical equations (axioms) and by simplifying the given equations in to smaller terms or simplifying the given equation to have lesser number of unknown variables. In this stage, it is required to follow a large set of steps to find the correct answer. Previously, a paper based system and calculators were used in solving the equation. But it's not efficient and can lead to errors, because, the equation to solve and values are on a paper and the calculations are processing in the calculator. There is more possibility to occur mistakes in data entering to the calculator from the paper and vice-versa. In addition to that, it will consume more time in doing the calculations. Another aspect that leads to difficulties in mathematical equation solving is that, for some mathematical equations, it's not provided a correct answer with it. Therefore the students and lectures may get confused with different potential answers. Then, some of the complex mathematical equations need using recursive methods in solving the final equation. So the person who is doing the calculation will be bored with the calculation task, and such long and recursive calculations are having a higher risk of getting incorrect evaluations during the intermediate steps. The developed solution, Mathematical equation evaluator is a mobile application, targeted for mobile devices and tablets (with Android operating system). This application aims to provide solutions to above mentioned problems that are faced by students in learning mathematics. The application summery is as follows. There are two types of information gathering techniques using by the application for receiving input data in to the system. First technique is capturing the equation via mobile device's camera and system automatically detecting the equation by using the OCR technology and the second technique is manually entering the equation details in to the system.
1. Capturing the mathematical equation via mobile device's camera and automatically detecting the equation by using the OCR technology 2. Support for manually entering the equation details in to the system via the keypad. 3. Final solution is displayed with steps of solving. 4. Variable to find value should be the only literal in left the hand side of the equal mark and it should be marked as ? (question mark) when asking for known values. |
The Mathematics Curriculum: Counting and Configurations what they could look like. There is then a detailed look at probability, graph theory as a network and also some topics for classroom |
The Akst/Bragg series' success is built around clear and concise writing, a side-by-side "teach by example" approach, and integrated applications throughout that help students achieve conceptual understanding. The user-friendly design offers a distinctive side-by-side format that pairs examples and their solutions with corresponding practice exercises. Students understand from the very beginning that doing math is an essential part of learning it. Motivational, real-world applications demonstrate how integral mathematical understanding is to a variety of disciplines, careers, and everyday situations. |
More About
This Textbook
Overview
Assuming minimal background on the part of students, this text gradually develops the principles of basic real analysis and presents the background necessary to understand applications used in such disciplines as statistics, operations research, and engineering. The text presents the first elementary exposition of the gauge integral and offers a clear and thorough introduction to real numbers, developing topics in n-dimensions, and functions of several variables. Detailed treatments of Lagrange multipliers and the Kuhn-Tucker Theorem are also presented. The text concludes with coverage of important topics in abstract analysis, including the Stone-Weierstrass Theorem and the Banach Contraction Principle |
Diagrams, charts and graphs
Diagrams, charts and graphs are used by all sorts of people to express information in a...
Di
Diagrams, charts and graphs
Introduction
This unit has two aims: firstly, to help you read and interpret information in the form of diagrams, charts and graphs, and secondly, to give you practice in producing such diagrams yourself.
To start you will deal with interpreting and drawing diagrams to a particular scale. You will then learn to extract information from tables and charts. Finally you will learn to draw graphs using coordinate axes, which is a very important mathematical technique.
This unit is from our archive and is an adapted extract from Open mathematics (MU120i'm not so good in math since birth, but now with this free university i've realized that i have a good potential to interpret graphs, diagrams... thank you so much for the very interesting and challenging post of test here... and thank you for the best opportunity to keep on improving my mathematical ability... now you are making me in love with diagrams, charts, graphs...
Copyright & revisions
Publication details
Originally published: Friday, 10 |
Description: 1001 Algebra Problems offers those with math anxiety and others who need tutoring the hands-on practice they need. This useful manual providers users the tools they need to master algebra. This title helps users to prepare for exams, develop m/c strategies, apply algebra rules to application problems and build problem solving skills. Includes the most common algebra concepts from expressions to linear equalities to functions. |
Westport, Connecticut, U.S.A. 1996 Hard Cover Good The inside back cover page has been removed leaving a straight edge. Presentable library withdrawal in the plastic protective ...cover. Some shelf wear to the covers. Stamped by the library at the page edge. Reliable and still sturdy. The text flows clearly from beginning to end.Read moreShow Less
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Editorial Reviews
School Library Journal
Gr 6 Up-In addition to acting as a young person's introduction to genealogy, this text also serves as a good general source for the study of Mexican-American history. In fact, the first third of the book is devoted to developing readers' pride in their heritage. (The authors assume that all readers will be Mexican Americans.) One strength is the authors' discussion of the genealogy of nontraditional and adoptive families. The lengthy lists for further reading that appear at the end of each chapter are also beneficial; however, nearly all of the titles are adult publications. Lila Perl's The Great Ancestor Hunt (Clarion, 1990) is a better overall introduction to the subject, but it does not specifically address Mexican-American genealogy.-Denise E. Agosto, Midland County Public Library, TX
Booknews
A text for a one-semester post-calculus course, based largely on the recommendations of the Linear Algebra Curriculum Study Group, with the main differences being early treatment of Euclidean vector geometry, extended treatment of determinants prior to eigenvalues, and ongoing material about linear transformations. Material on LU factorization is part of a chapter on numerical methods, and the QR factorization is not discussed. Key concepts are presented as definitions and proved theorems. Includes worked examples, exercises, and MATLAB |
Welcome to your online math class! The following steps should help you get started. This is intended to be a very short overview of what is expected of you the first week of class. All details are found on documents posted on Blackboard.
Log on to Blackboard. Download each document under "Course Information" and print them out for future reference.. Doing this will help you decide if you are in the correct course. It will also provide you with your homework assignment, due dates, test schedule, minimum computer requirements, etc.
Complete the online orientation found on Blackboard. This is required!
Purchase your materials.
Log in to MyMathLab. To do this you will need your ACC email address set up. You will also need my courseID provided on the handout you printed from my Blackboard documents. Lastly, you will need an access code that will come with the MyMathLab materials you purchased.
Begin your lessons in MyMathLab. This is where the majority of your learning will take place.
Log in to Blackboard weekly to read any announcements from me. This is also where you will receive your homework and test grades. You should log on to Blackboard at least once a week. |
In this section of matrix algebra, we deal with more complex topics, beginning with an introduction to matrices. NOTE: This is a clip from a much larger (seven hours) tutorial. Instructor uses whiteboard.(41:42)
In this video, an instructor explains the procedure for multiplying two small matrices. He also provides an example to show when two matrices cannot be multiplied. The problems are shown on a white board. ( 3:56)
This two minute video provides instructions on how to use a calculator to identity matrix; a matrix that when multiplied by another matrix, it
generates the same exact answer. It has a diagonal of ones, and the rest are all zeros. This video shows ...a calculator close-up and may have to be repeated as it moves fairly rapidly |
What
is Infinity?
Infinity is a program intended for solving differential and algebraic
equations. Infinity surely does not possess such an
universality as, for example, Maple. But with its help you can solve
problems which would be a hard nut to crack for any existing
mathematical programs.
What are those problems?
Ordinary nonlinear nonautonomous differential equation systems.
The basic advantages of the
method realized in I compared to the methods realized in other mathematic
programs:
The result of the method application
is not only an approximate solution, but also the area containing
the unknown precise value of the desired solution;
The choice of a calculation step
is an assumption diagram internal procedure (the step length anticipates
the desired solution tendency);
In some cases, having carried out
the transformation within the framework of the analytical part
of the method, the solution is possible to be found in a closed
analytical species ( Example: Duck-solutions); |
College Algebra in Context, Fourth Edition, is ideal for students majoring in business, social sciences, and life sciences. The authors use modeling, applications, and real-data problems to develop skills, giving you the practice you need to become an adept problem solver in your future courses and career.
Algebra II is the fork in the road. Those who succeed in this second part of the algebra sequence are well on their way to precalculus, calculus, and higher mathematics, which open the door to careers in science, engineering, medicine, economics, information technology, and many other fields. And since algebraic thinking is found in almost every sphere of modern life, a thorough grounding in this abstract discipline is essential for many nontechnical careers as well, from law to business to graphic arts.
Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the 'bible of computer algebra', analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.
Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical, real-world explanations, this book will help you learn everything from natural numbers and exponents to solving systems of equations and graphing polynomials. |
Math 8/7 is the prealgebra program for students who have completed Math 7/6. It emphasizes the concepts and skills necessary for your child's success in upper-level mathematics courses, including scientific notation; statistics and probability; ratios and proportions; simplifying and balancing equations; factoring algebraic expressions; slope-intercept form; graphing linear inequalities; arcs and sectors; and the Pythagorean theorem. |
"This handbook offers comprehensive coverage of structural equation modeling (SEM), beginning with background issues, continuing through statistical underpinnings and steps in implementation, then moving into basic and advanced applications of SEM. In a single volume, it [...]
Consists of a series of peer-reviewed papers in stochastic processes, analysis, filtering and control, with particular emphasis on mathematical finance, actuarial science and engineering. This book is of interest to researchers and practitioners.
College Algebra and Trigonometry, Fifth Edition, by Lial, Hornsby, Schneider, and Daniels, engages and supports students in the learning process by developing both the conceptual understanding and the analytical skills necessary for success in mathematics. With the Fifth [...] [...]
Drawing on their decades of teaching experience, William Briggs and Lyle Cochran have created a calculus text that carries the teacher's voice beyond the classroom. That voice--evident in the narrative, the figures, and the questions interspersed in the narrative--is a [...]
Michael Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Serieshas evolved to meet today's course needs by [...]
The Graphs and Models series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students "see the math" through its focus on visualization and technology. These texts continue to maintain the features that have helped students succeed for years: focus on [...]
This is a book designed for an introductory course in mathematical modeling in engineering. The content of the book will be mainly built on the three main principles of Mechanics: Conservation of Mass, Conservation of Linear and Angular Momentum, Conservation of Energy. [...]
Cinderella.2, the new version of the well-known interactive geometry software, has become an even more versatile tool than its predecessor. The geometry component extends the functionality to such spectacular objects as dynamic fractals, and the software includes two major [...]
Using an original mode of presentation, and emphasizing the computational nature of the subject, this book explores a number of the unsolved problems that still exist in coding theory. A well-established, yet still highly relevant branch of mathematics, the theory of [...]
Mathematical Olympiad Treasures aims at building a bridge between ordinary high school exercises and more sophisticated, intricate and abstract concepts in undergraduate mathematics. The book contains a stimulating collection of problems in the subjects of algebra, [...]
" Statistical Pattern Recognition provides an introduction to statistical pattern theory and techniques, with material drawn from a wide range of fields, including the areas of engineering, statistics, computer science and the social sciences. The book describes techniques [...]
English;
Hardback
Released: November 2011
/
Delivery time: 1-3 weeks (not available at short notice)
An Introduction to Tensors and Group Theory for Physicists provides both an intuitive and rigorous approach to tensors and groups and their role in theoretical physics and applied mathematics. A particular aim is to demystify tensors and provide a unified framework for [...] |
0030103479
9780030103476
Elementary Linear Algebra with Applications:This edition strives to develop students' geometric intuition as a foundation for learning the concepts of span and linear independence. Applications are integrated throughout to illustrate the mathematics and to motivate the student. Numerical ideas and concepts using the computer are interspersed throughout the text; instructors can use these at their discretion. This textbook allows the instructor considerable flexibility to choose the applications and numerical topics to be covered according to his or her tastes and the students' needs.
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Rent Elementary Linear Algebra with Applications 3rd edition today, or search our site for Julia textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by CENGAGE Learning. |
Editorial Reviews
Review
From the reviews of the third edition:
The book is full of interesting things. The first half focuses on visualization and the math is relatively elementary (nothing beyond multivariable calculus). The second half works on much harder problems. —MAA Reviews
"Beginners will benefit from the book as will more experienced Mathematica programmers as the explorations build from simple introductory concepts … . The best feature of the book is that it teaches via interesting examples. … The writing is remarkably good. … The book contains a good number of color illustrations and is rich in Mathematica output. … interesting explorations with clear and engaging writing are hard to come by. I'm delighted to have this book in my collection." (Playing with Mathematica, June, 2011)
From the Publisher
"Mathematica in Action, 2nd Edition," is designed both as a guide to the extraordinary capabilities of Mathematica as well as a detailed tour of modern mathematics by one of its leading expositors, Stan Wagon. Ideal for teachers, researchers, mathematica enthusiasts, "Mathematica in Action" leads the reader on a gued tour of elementary calculus, high-precision number theory, fractals, turtle geometry, and numerical analysis. Includes diskette with "Mathematica" programs for DOS/Windows, Macintosh and Unix computer platforms.
--This text refers to the
Hardcover
edition.
Stan Wagon's long awaited 3'rd edition of Mathematica in Action fulfilled all my hopes. Every page is filled with deep insight into both topic covered and the related Mathematica implementation. Here is just a small sample of some of the things you will read about and learn:
1) How to effectively plot unstable functions. 2) How to speed up animations 3) Techniques for getting the most out of Mathematica's 3D plotting functions 4) Practical advice and examples using Mathematica's dynamic interactivity features (Manipulate, Dynamic, etc.) 5) Beautiful ways to visualize the behavior of Differential Equations. 6) An entire chapter dedicated to optimization and another to computational geometry (I have only skimmed these at this time)
Naturally the book revisits a lot material from the 2nd edition but there is enough new material to reward owners of older editions, especially with respect to the latest features of Mathematica 7.
The book has a slight leaning towards pure mathematics (whole chapters on such topics as Prime Numbers, Penrose Tiles, Fractals, Pi, Banach-Tarski Paradox and Riemann Zeta Function). Readers whose needs are more firmly rooted in the practical (statistics, finance, image processing, audio processing, parallel processing) might look at some of the other recent Mathematica publications (yes, that is shameful plug!) to fill out their Mathematica knowledge. However, even within the theoretical coverage there is priceless gems of Mathematica know-how that only an author like Stan (who has spent a significant portion of his life using and teaching Mathematica) could provide.
Two added bonuses:
- Many full color images and plots - The notebooks containing the code (but not text) for every chapter.
This is a very good book, but only in paper form. The 3e kindle edition has significant errors that do not appear in my 2e paper copy. Judging by the specific differences, this appears to be an artifact of the kindle conversion process.
While such errors are annoying in general writing, they render this particular book unusable. Terse Mathematica techniques do not work if just a single character is incorrect. |
A convenient single source for vital mathematical concepts, written by
engineers and for engineers Almost every discipline in electrical and computer engineering relies heavily on advanced mathematics. Modern Advanced Mathematics for Engineers builds a strong foundation in modern applied ...
Description:
Structures are defined by laws of composition, rules of generation,
and relations. The objects on which these laws operate may be numbers, geometric objects like points and lines, or abstract symbols. Algebra is the study of mathematical laws, with ... |
Mathematics
General Mathematics NC 5111/5112 - (9,10,11,12)
A two-semester course designed to review the basic operations of whole numbers, fractions, decimals, and directed numbers. Problem solving, rates, percents, statistics, and basic geometry are also discussed. If a student takes this course a subsequent year, mathematics life skills will be emphasized.
Pre-Algebra NC 5121 - (9,10,11,12)
Pre-Algebra is a two-semester course for students who need additional time to prepare for Algebra I. It reviews fundamentals of mathematics, reviews algebraic concepts previously introduced in other math courses, and introduces new math concepts. Topics covered include the four fundamental math operations, order of operation, fraction, decimals, percents, ratio, proportion, exponents, scientific notation, perimeter area, volume, solving equations and inequalities, graphing and translating real life problems into mathematical sentences.
Algebra I 5123 - (9)
A two-semester course in Algebra designed to give the student a complete foundation in algebraic structure and method. The course covers work in the areas of real numbers, simple and complex equation and problem solving, polynomials, fractions, factoring, graphing, rational and irrational expressions, and quadratic equations and the inequalities.
Geometry* 5133 - (9,10,11,12)
This two-semester course is designed to help students understand the basic structure of geometry, develop powers of special visualization while building their knowledge of the relationships among geometric elements, and grow in the understanding of the deductive method and in an appreciation of the need for precision of language. Students will also strengthen their algebraic skills, gain some knowledge of the methods of coordinate geometry and the way in which algebra and geometry complement each other, and experience the stimulation and satisfaction that comes from clear and creative thinking. (Prerequisite: Algebra I)
Algebra II* 5126 - (10,11,12)
This elective course is designed to give the student an in-depth study of algebraic concepts. The course covers a review of Algebra I, rational and irrational numbers, radicals, quadratic equations, complex numbers, graphing, exponents, logarithms, permutations, and combinations. (Prerequisite: Algebra I and Geometry)
Algebra II and Trigonometry* H 5127 - (10,11,12)
An elective course designed to give the student and in-depth study of algebraic concepts and an introduction to trigonometric functions. This course covers rational and irrational numbers, radicals, quadratic equations, complex numbers, graphing, exponents, logarithms, permutations, combinations, trigonometric functions and identities, polar coordinates, and vectors. (Prerequisite: Algebra I and Geometry)
Advanced Mathematics* H LTC 5157 - (11,12)
A two-semester course designed to prepare the student for modern courses in calculus, abstract algebra, and probability. Topics covered are logic, properties of the complete ordered field, mathematical induction sequences and series, algebra of vectors, plane analytical geometry of points and lines, linear and polynomial functions, exponential and logarithmic functions, and circular and trigonometric functions and their properties. (Prerequisite: Algebra II)
Statistics & Data Analysis* 5195 - (11,12)
This course is designed to provide the student with a comprehensive treatment of introductory statistics and probability in such areas as business, sociology, ecology, economics, education, medicine, psychology, and mathematics as well as in our every day life as consumers. Students in these courses must frequently demonstrate a knowledge of the language and methods of statistics. Methodology and applications have been integrated throughout the course. (Prerequisite: Algebra I and Geometry) Limit 25
Consumer Mathematics NC 5117 (12)
This course begins with an extensive look at federal and state income tax forms and filing procedures. Other items covered are auto ownership, purchasing goods, personal income, various areas of banking and banking services, investments, insurance, and housing cost.
Mathematics Connections NC 5118 (12)
This is a one-semester course, which is designed for the student who has completed Algebra 1 and or Geometry, but it is not a requirement. This math course will connect experiences with comprehensive mathematics. The course will use algebra, geometry, trigonometry, linear programming, and optimization techniques to solve problems. This course will show students mathematics is a vital, relevant, flexible tool for interacting with their world in many ways and at many levels, from the commonplace setting of everyday life and work to the frontiers of science and technology. |
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Axiomatic Set Theory by Patrick Suppes Geared toward upper-level undergraduates and graduate students, this treatment examines the basic paradoxes and history of set theory and advanced topics such as relations and functions, equipollence, more. 1960Sets, Sequences and Mappings: The Basic Concepts of Analysis by Kenneth Anderson, Dick Wick Hall This text bridges the gap between beginning and advanced calculus. It offers a systematic development of the real number system and careful treatment of mappings, sequences, limits, continuity, and metric spaces. 1963Product Description:
their classmates, and lecture on their results. This method offers a deeper understanding of the material, as well as a clearer view of what it means to do mathematics. An Outline of Set Theory is organized into three parts: the first presents definitions and statements of problems, the second offers suggestions for their solutions, and the third contains complete solutions. Topics include standard undergraduate set theory, as well as considerations of nonstandard analysis, large cardinals, and Goodstein's theorem. Drawn from the author's practical experience as Professor of Mathematics at Smith College, this text offers a novel and effective approach to teaching and learning the fundamentals of set theory |
9780321654274
ISBN:
0321654277
Edition: 3 Pub Date: 2010 Publisher: Addison Wesley
Summary: Beckmann, Sybilla is the author of Mathematics for Elementary Teachers with Activity Manual (3rd Edition), published 2010 under ISBN 9780321654274 and 0321654277. Two hundred seventy five Mathematics for Elementary Teachers with Activity Manual (3rd Edition) textbooks are available for sale on ValoreBooks.com, seventy eight used from the cheapest price of $9.36, or buy new starting at $169.62 |
... the real-life situa- tion into a geometric model, further analyzing this ... This
activity provides two real-world applications for bisectors that ... PROBLEM.
Students are confronted with a problem in which.
contrapositive) and be able to use them to describe real life situations. In
combination, these concepts will lead to .... postulates discussed earlier to solve
geometric problems. Construction exercises.
previously performed poorly in abstract math and science courses could achieve
high levels in those ... maintained academic rigor but introduced real-world
examples, applications, and problems, and ...
various real world applications. In robotics ... Various computer vision problems,
including pose estimation can be addressed .... To develop the practical
importance of geometric optimization algorithms. |
More About
This Textbook
Overview
James Stewart's well-received SINGLE VARIABLE CALCULUS: CONCEPTS AND CONTEXTS, Second Edition follows in the path of the other best-selling books by this remarkable author. The First Edition of this book was highly successful because it reconciled two schools of thought: it skillfully merged the best of traditional calculus with the best of the reform movement. This new edition continues to offer the balanced approach along with Stewart's hallmark features: meticulous accuracy, patient explanations, and carefully graded problems. The content has been refined and the examples and exercises have been updated. In addition, CALCULUS: CONCEPTS AND CONTEXTS, Second Edition now includes a free CD-ROM for students that contains animations, activities, and homework hints. The book integrates the use of the CD throughout by using icons that show students when to use the CD to deepen their understanding of a difficult concept.
In CALCULUS: CONCEPTS AND CONTEXTS, this well respected author emphasizes conceptual understanding - motivating students with real world applications and stressing the Rule of Four in numerical, visual, algebraic, and verbal interpretations. All concepts are presented in the classic Stewart style: with simplicity, character, and attention to detail. In addition to his clear exposition, Stewart also creates well thought-out problems and exercises. The definitions are precise and the problems create an ideal balance between conceptual understanding and algebraic skills.
Editorial Reviews
Booknews
In the new edition of this introductory text, Stewart (McMaster U.) presents all topics geometrically, numerically, algebraically, and verbally, for better conceptual understanding by students. Topics include functions and models; derivatives; differential equations; and infinite sequences and series. Annotation c. by Book News, Inc., Portland, Or 2001
Realistic review
I am currently a student using this text for my Calc I class. I am now in chapter 4 and have become truly disheartened in the fashion with which Mr. Stewart approaches calculus. Many of the definitions are not written for students but to impress professors. This book is written for the college level professor. Much of this text is difficult to read, and sorely lacks sufficient examples. The process in which the examples are provided is not linear, the first couple of examples for each section are easy, then they jump from simplistic to the insane. I am sorry but I feel this is a terrible text. I had to use my sisters text to suppliment the gaps that were in this text. What ever happened to good math text that give a reasonable explanation of how to go about understanding the concept and then solving it? Instead Mr. Stewart you are treating calculus as if it was for the mental elite. Math is a fun subject, reading text like this makes me understand why people hate math! student working on his second degree (BS in CmpEng)
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. |
Cartesian Free Download
A math software for calculation of coordinates. Cartesian coordinates, spherical coordinates und cylindrical coordinates can be transformed into each other. (5 languages, 2 and 3 dimensions) You can select between radiant and degrees, between 2 and 3 dimensions and use two ways to express the angle.
Software that is a tool to generate Cartesian X/Y-plots from scientific data. You can enter and calculate tabular data. View the changing graphs, including linear and non linear regression, interpolation, differentiation and integration, during entering. Education shareware Rt-Plot is listed in education tools a freeware software filed under mathematics, middle school, k-12 utilities.++).
Software that Both cartesian and polar coordinates are supported as well as parametric equations and inequalities. Fast rotation and translation of graphs with the mouse and with animations make graph visualization easy Education freeware Graphing Functions is listed in mathematics, middle school, k-12 This shareware is archived under mathematics, science, cad, graphics category This development tools shareware SignalLab .NET is categorized under development c and c++ and c# Education shareware 3D Grapher is listed in math, graph, education, business tools.
Calc3D is a calculator for vectors, matrices, complex numbers, quaternions and coordinates.
For objects like point, line, plane and sphere distances, intersections, volume, area of squeres, area of a triangle can be calculated.
(5 languages). Education shareware Calc 3D is listed in math tools Delphi 6. This shareware is archived under development components, libraries category.
This software is an easy-to-use software for 2D, 2.5D and 3D function graphing, animation and table data visualization. This education shareware Function Grapher is categorized under education mathematics.
The SignalLab is a set of Visual C++ components for fast Digital Signal Processing (DSP) and data visualization. They allow fast complex signal manipulations with zero lines of program code. Free for non-commercial use. SignalLab VC++ is a shareware software filed under development c and c++ and c# utilities.
CMATH makes fast complex-number math functions (cartesian and polar) available for Visual C++. This comprehensive library was written in Assembler for superior speed and accuracy. All functions may be called from C or C++.
Software that can be used for world wide professional coordinate transformations between international and national coordinate systems. It is clearly arranged and easy to use. Education demo AllTrans is listed in science tools.
Education shareware Calc 3D Pro is listed in mathematics tools EconoModeler is licensed as shareware and placed in education teaching, training tools section.
A programmable graphics calculator which lets you visualize expressions and formulas as graphs in a chart and can also create video clips from graphs. GraphiCal has over 50 built-in mathematical functions, can integrate and find roots and extrema. It is programmable, offers user defined variables and functions, programming interfaces for VB and VC and a very user-friendly graphics system. GraphiCal is licensed as shareware and placed in calculators, educational section This shareware is archived under business calculators, converters category. |
Arithmetic the Easy Way - 4th edition
Summary: Everybody uses arithmetic on virtually a daily basis, and this book serves as a handy brush-up for general readers while it also helps students master basic skills that they need before moving up to high-school-level math and beyond. It reviews addition, subtraction, multiplication, and division, then moves on to calculating with fractions, decimals, and percentages. A concluding chapter reviews units of measurement and word problems. Chapters are filled with short p...show moreractice exercises, all of which are answered at the back of the book. The book features many tables, charts, and line illustrations. Barron's Easy Way books focus on both practical and academic topics, presenting fundamental subject matter in clear, understandable language. Equally popular as self-teaching manuals and supplementary texts for classroom use, they are written to help students improve their grades and review subject matter before tests. They are also useful for introducing general readers to a new career-related skill. Easy Way titles cover virtually all subjects that are taught on advanced high school and college-101 levels. New subjects are periodically added, and existing titles are frequently updated to keep them timely and relevant to students' needs. Subject heads and key phrases are set in a second colorVeryGood
Yankee Clipper Books Windsor, CT
Very good condition book with only light signs of previous use. Sail the Seas of Value
$4.497575 |
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Contents
1. Rationale 1
2. Aims 1
3. Assessment Objectives 2
4. Syllabus Content: 3
Form 1 4
Form 2 10
Form 3 19
Form 4 28
Form 5 38
5. Information and Communication Technology (ICT) in Mathematics 48
Form 1 50
Form 2 54
Form 3 58
Form 4 62
Form 5 66
6. Implementing the Mathematics Syllabus 70
7. Annual Examinations 73
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
THE SECONDARY SCHOOLS MATHEMATICS CURRICULUM
1. RATIONALE 2. AIMS
Mathematics furnishes the prime means by which information can be Teachers should aim to enable candidates to
organised, communicated and manipulated. It is also an ever-expanding
understand and appreciate the place and purpose of Mathematics in
body of facts, skills, concepts and strategies used in the solution of a wide
range of problems. As a consequence, when implementing this syllabus, society and apply mathematical concepts to situations arising in
teachers of Mathematics should emphasize that: their own lives;
apply mathematical knowledge and understanding to solve
(i) Mathematics is useful. It equips children with the necessary problems;
knowledge to help them understand and interact with the world
around them. Moreover, it forms the basis of science, technology, think and communicate mathematically - precisely, logically and
architecture, engineering, commerce, industry and banking. It is creatively;
also increasingly being used in the medical sciences, biological
sciences, economics and geography. This pervasiveness makes develop a positive attitude to Mathematics, including confidence
Mathematics one of the most important subjects in the school and perseverance;
curriculum.
(Utilitarian Aspect of Mathematics Teaching and Learning) develop an ability to work independently and co-operatively when
doing Mathematics;
(ii) Mathematics is an evolving body of knowledge that is characterised
appreciate the interdependence of the different branches of
by its order, precision, conciseness and logic. It should offer the
Mathematics;
children intellectual challenge, excitement, satisfaction and wonder.
(Aesthetic Aspect of Mathematics Teaching and Learning) acquire a secure foundation for the further study of Mathematics;
use Mathematics across the curriculum;
make efficient, creative and effective use of appropriate technology
in Mathematics.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
3. ASSESSMENT OBJECTIVES
Assessment is based on develop a feel for numbers;
the candidate"s ability to recall, understand and apply mathematical develop and use a range of methods of computation, namely,
knowledge in a wide context; mental, pencil-and-paper, calculator and computer methods, and
apply these to a range of problems;
the candidate"s ability to understand and analyse a problem, select
an appropriate strategy, apply suitable knowledge and techniques develop and use a range of methods for approximation of numbers
to solve it, verify and interpret the results; and apply these to a range of problems;
the candidate"s ability to understand, interpret and evaluate develop and use a range of methods for estimation of measures and
mathematical ideas that are presented in oral, written and visual apply these to a range of problems;
forms.
explore a variety of situations which lead to the expression of
In particular, the candidate will be required to demonstrate the ability to relationships;
communicate, conjecture, reason and prove mathematically;
consider how relationships between number operations underpin the
understand the nature of numbers and make use of them; techniques for manipulating algebraic expressions;
understand the nature of algebraic relationships and make use of consider how algebra can be used to model real-life situations and to
them; solve problems;
understand the nature and properties of shape, space and measures explore shape and space through drawing and practical work;
and make use of them;
use computers to generate and transform graphic images and to
understand the nature of statistics and process, represent and solve problems;
interpret data;
formulate questions that can be solved using statistical methods;
understand the nature of probability and calculate the probabilities
of events. undertake purposeful inquiries based on data analysis;
During the course candidates should be given opportunities to engage in practical and experimental work in order to appreciate
principles which govern random events;
use calculators and computer software including spreadsheets,
LOGO, a dynamic geometry package and a computer algebra use investigative approaches;
system;
look critically at some of the ways in which representations of data
use computers as a source of large samples, as a tool for exploring can be misleading and conclusions can be uncertain.
graphical representations, and as a means for simulating events;
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
4. THE SYLLABUS CONTENT
The Secondary Schools syllabus is divided into two parts; core and
extension topics. Form 1 syllabus has been kept at core level.
This facilitates differentiated teaching and learning and the possibility of
implementing setting.
"differentiated instruction provides students with multiple
options for taking in information, understanding ideas, and
expressing what they learn. A differentiated classroom
provides multiple approaches and is proactive, qualitative,
student centred, organic, and a mix of whole-class, group, and
individual instruction."
(Tomlinson, C. A., 1995).
Core Topics is the content to be covered by all secondary school
students. This level allows the student to acquire the
basic mathematical skills at a slower pace.
Extension Topics is extension content to be covered by all those
students who intend to sit for the Mathematics
Matsec Paper IIB by the end of Form 5.
Revised Syllabus Reversed-out sections in the September 2003
syllabus are the revised or the additional topics to
the September 2000 syllabus.
Core topics printed in italics are topics which
have already been covered in the Extension
part in one of the previous years.
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Secondary Schools Mathematics Syllabus
Foundation and Intermediate Levels
FORM 1
Core
Unit 1: Number
1.1 Natural Numbers
1.1.1 Properties of whole numbers
Distinguish between even and odd numbers. It is important that students gain fluency with whole numbers. This
means that students should develop efficient mental and/or pencil and
Find the factors and multiples of numbers. paper methods for manipulating whole numbers based on a
1.1.2 Place value well-understood knowledge of number properties and relationships.
Express numbers less than 100 000 in words and figures and vice-versa.
Arrange a set of numbers in ascending/descending order. Fluency with whole numbers can also be obtained if students are given
1.1.3 Four rules opportunities
Add and subtract natural numbers. to use ICT to explore properties of numbers and the relationships
Know multiplication facts up to 10 10. between them,
to carry out investigative work related to number.
Multiply natural numbers by a single digit.
Divide numbers by a single digit with remainder.
1.1.4 Rounding
Round numbers to the nearest 10, 100, or 1000.
Carry out rough estimates to check the accuracy of calculations.
1.2 Decimal Numbers
1.2.1 Notion of decimal numbers The notion of decimal number is introduced through the pupils"
Read decimal numbers from scales. concrete experience with measurement, money and weight. For
1.2.2 Place value (up to two decimal places) example, the length of a pencil can be expressed in ten centimetres and
4 tenths of a centimetre, i.e. 10.4 cm.
Arrange a set of numbers in ascending/descending order.
1.2.3 Four rules
Add and subtract decimal numbers.
Multiply and divide decimal numbers by single digit natural numbers.
1.2.4 Rounding
Round numbers up to two decimal places.
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1.3 Fractions
1.3.1 Notion of fractions
Understand that a fraction is a part of a whole.
1.3.2 Fraction of a quantity
Find a fraction of a whole number. Restricted to proper fractions with denominator 2, 3, 4, … , 10, 100.
1.3.3 Ordering
Identify two equivalent fractions.
Compare the magnitude of two fractions.
1.3.4 Simplification
Reduce fractions to their lowest terms.
Change mixed to improper factions and vice-versa.
Express one quantity as a fraction of another.
1.3.5 Four rules
Add and subtract up to two proper fractions with the same denominator.
1.3.6 Relationship between fractions and decimals
Convert decimals to fractions. Decimals are restricted to two decimal places.
Convert fractions to decimals. Restricted to proper fractions with denominators being factors of 100.
Unit 2: Applications
2.1 Number
2.1.1 Simple problems
Solve simple problems involving whole numbers, fractions, decimals and
percentages.
2.2 Metric Measures
2.2.1 Units
Convert large units of length and weight to smaller units and vice-versa.
2.2.2 Four rules
Add, subtract, multiply and divide quantities of length and weight.
Solve simple problems involving four rules.
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2.3 Personal and household finance
2.3.1 The Maltese currency system
Convert Maltese Liri to cents and vice-versa.
Add, subtract, multiply and divide money quantities.
Solve simple problems involving money.
2.4 Time
2.4.1 Units
Distinguish between different units of time: sec., min., hour, day, week,
month, leap year.
Read and use a calendar.
2.4.2 The clock
Write times using 12-hour and 24-hour clock.
Convert 12-hour to 24-hour clock and vice-versa.
2.5 Percentages
2.5.1 Notion of percentage
Understand that a percentage is a fraction having 100 as denominator. Restricted to proper fractions with denominators being factors of 100.
Change fractions into percentages.
2.5.2 Percentage of a quantity
Express a quantity as a percentage of another.
Find the percentage of a quantity. Restricted to simple percentages: 25%, 50% and 75%.
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Unit 3: Shapes, Space and Measurement
3.1 Measurement
3.1.1 Perimeter of simple shapes Students should appreciate that the concept of measurement goes
Find the perimeter of simple shapes by adding the side lengths, which are beyond simple notions such as length and width. It is important that
all given. students learn to select the appropriate unit for the attribute being
3.1.2 Area of simple shapes. measured,
the notions of measurement are linked to other areas of the
Find the area of simple shapes drawn on squared paper.
mathematics curriculum such as number, shape and data handling.
Use units for area: mm2 and cm2.
3.1.3 Area of rectangles and squares Experience with squared paper, LOGO and Dynamic Geometry
Find the area of rectangles and squares using the formula: Software can enhance students" appreciation of concepts of
Area = length breadth measurement, angles and the properties of basic shapes.
Find the length/breadth in simple cases.
3.1.4 Volume of cube and cuboid Children should also be given opportunities to do extensive
investigative work involving both pencil and paper methods and ICT.
Understand notions of faces, edges and vertices.
Draw and distinguish the nets of cubes and cuboids.
Find the volume by counting cubes.
Find the volume of cubes and cuboids using the formula:
Volume = length breadth height.
Use units of volume: mm3 and cm3.
3.2 Angles
3.2.1 Angles as a measure of turn. LOGO provides an ideal environment for children to experience angle
Understand that a complete revolution is divided into 360. as a measure of turn, in both clockwise and anti-clockwise direction.
Define a right angle as a quarter turn. Besides turtle geometry gives children an opportunity to manipulate
angles of different sizes.
Identify right, acute, obtuse and reflex angles.
Estimate the size of an angle. A Dynamic Geometry Software can help pupils discover the properties
Main compass directions (NSEW) linked to a fraction of a revolution of vertically opposite angle, angles at a point and angles on a straight
(¼, ½, ¾). line.
3.2.2 The protractor
Use a protractor to measure and draw angles up to 180.
3.2.3 Calculations
Make calculations involving angles at a point, angles on a straight line and
vertically opposite angles.
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3.3 Lines
3.3.1 Straight lines
Construct a straight line using ruler and compasses.
Estimate the length of a straight line.
3.3.2 Perpendicular lines
Draw perpendicular lines using set squares and protractor.
3.3.3 Parallel lines
Identify parallel lines in geometric figures.
3.3.4 Line symmetry
Draw lines of symmetry.
Complete a figure to make it symmetrical about a given line.
Identify equal lines and angles through folding.
3.4 Shape
3.4.1 Tessellations By drawing simple shapes such as squares, rectangles and triangles
Cover a given area with tessellating shapes. using simple LOGO commands such as FD, BK, RT, LT and
Identify tessellating shapes. REPEAT, pupils get the opportunity to reflect upon the properties of
these shapes.
3.4.2 Squares and rectangles
Identify equal lines and angles through line symmetry. A Dynamic Geometry Software also provides a perfect environment by
Construct squares and rectangles using ruler and compasses for lengths and which pupils can discover the properties of triangles, squares and
protractor for angles. rectangles. By drawing these shapes and measuring their sides and
Identify and draw diagonals. angles, children can discover for themselves the properties of these
3.4.3 Triangles shapes.
Distinguish between scalene, isosceles and equilateral triangles.
Construct triangles given the length of the sides using ruler and compasses
only.
3.4.4 The circle
Identify parts of a circle: centre, radius, diameter, chord and circumference.
Form patterns made up from a number of circles.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 4: Algebra
4.1 Patterns and sequences
4.1.1 Geometric and number patterns Children should be given opportunities to use a spreadsheet to generate
Recognize geometric and number patterns. sequences of numbers, which they can describe both verbally and
Extend patterns and sequences of numbers. symbolically.
Describe simple patterns verbally.
4.2 Functions
4.2.1 Function machines At this stage pupils should realise that in algebra letters are no more
Describe functions verbally. than a device to represent numbers. They should be encouraged to
Represent simple functions symbolically. describe functions, both verbally and symbolically.
Construct simple function machines.
4.3 Graphs
4.3.1 Cartesian coordinates
Mark points in the first quadrant using an ordered pair of numbers known
as the x-coordinate and the y-coordinate.
Draw lines and shapes given the coordinates of their endpoints or vertices.
Unit 5: Data Handling
5.1 Statistics
Pupils should be given opportunities
5.1.1 Collection of data
to formulate questions about relevant issues, and answer these
Collect data using observations, surveys and experiments. questions by collecting data and presenting it in meaningful ways,
5.1.2 Frequency tables to use spreadsheets to display and analyse the collected data.
Compile a frequency table for ungrouped data.
Interpret data in simple frequency tables.
5.1.3 Charts and diagrams
Draw and interpret bar charts.
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FORM 2
Core Extension
Unit 1: Number
1.1 Integers
1.1.1 Properties of whole numbers 1.1.1 Properties of whole numbers Students should develop efficient mental
Find the common factors and common Find the least common multiple of up to three and/or pencil and paper methods for
manipulating whole numbers based on a
multiples of up to three numbers. numbers.
well-understood knowledge of number
Identify a prime number as a number having 2 properties and relationships.
distinct factors only (i.e. 1 and the number
itself) Fluency with whole numbers can also be
Write numbers as a product of their prime obtained if students are given
factors. opportunities
1.1.2 Notion of integers to use ICT to explore properties of
Understand negative integers through practical numbers and the relationships
examples. between them,
to carry out investigative work
Represent integers on the number line. related to number.
Arrange a set of integers in ascending/
descending order.
1.1.3 Four rules 1.1.3 Four rules Division by a two-digit number using
Add and subtract integers: Multiply numbers by a two-digit number. the repeated subtraction method.
e.g. +2 + 6, 2 + 6, +2 6, 2 6 Divide numbers by a two-digit number using e.g. 786 29
Understand the precedence of mathematical repeated subtraction.
786
operations.
( by 10) 290 10 29
Multiply positive integers by multiples of 10. 496
1.1.4 Rounding 290 10 29
Round positive integers to the nearest 10, 100 206
or 1000. (5 is half of ten) 145 5 29
Carry out rough estimates to check the 61
accuracy of calculations. ( by 1) 29 1 29
32
29 1 29
3
2
Ans: 27 rem 3
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1.2 Decimal Numbers
1.2.1 Four rules 1.2.1 Four rules
Multiply decimal numbers by 10 and 100. Multiply decimal numbers by two digit Multiplication/division by 100 can be
Divide decimal numbers by 10 and 100. numbers, e.g. 3.47 45, 5.28 7.1 seen a repeated multiplication/division
1.2.2 Rounding by 10. A calculator can be used to help
children establish a relationship between
Round numbers to a given number of decimal
multiplying/dividing by 10 and the
places. manner in which the number is
changing.
1.3 Fractions
1.3.1 Four rules 1.3.1 Four rules
Add and subtract up to two proper fractions Add and subtract two mixed numbers.
with different denominators. e.g. 6 2 2 1 , 2 1 3 .
3 4 5 4
1.3.2 Quantities
Find a fraction of a quantity,
e.g. 3 of Lm15.64.
4
1.3.3 Fractions and decimals Pupils should be aware of the decimal
Use the calculator to change fractions to equivalence of fractions with
decimals. denominators of 2, 4, 5 and 10.
1.4 Indices
1.4.1 Notion of indices 1.4.1 Notion of indices
Evaluate ax where a and x are positive Estimate the square roots of positive integers
integers. less than 100, e.g. 50 7 .
e.g. 53 = 125.
Find the cube roots of perfect cubes less than
Use appropriate language: "to the power of", 100 without a calculator.
"square", "cube".
Find the positive square root of perfect
squares less than or equal to 100.
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1.5 The calculator
1.5.1 Basic functions 1.5.1 Basic functions
Make efficient use of the +, , , , = buttons. Use the x2 and button on the calculator. The calculator should not be used only
Encourage students to use approximations to as a numerical computation tool, but as
check the reasonableness of calculator results. a tool which can help pupils to gain
insight into mathematical concepts. It is
Choose a reasonable approximation, e.g. also important that children learn to use
Expressing Lm6.6666667 as Lm6.67. the calculator appropriately and
efficiently while at the same time
employing suitable procedures to check
the result displayed on their calculator.
Unit 2: Applications
2.1 Number
2.1.1 Problems
Apply notions of integers, fractions and
decimals to practical situations.
2.2 Metric measures
2.2.1 Units 2.2.3 Problems
Understand the notions of ml and l. Solve problems involving length, weight and
Convert litres to millilitres and vice-versa. capacity.
Use the relationship 1000 cm3 = 1l.
2.2.2 Four rules
Add, subtract, multiply and divide quantities
of capacity.
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2.3 Time
2.3.1 The clock
Find time intervals in hours and minutes.
Understand and use timetables. Encourage the use of the time-line to
work out intervals of time.
2.4 Percentages
2.4.1 Percentages, fractions and decimals
Express a percentage as a fraction and as a
decimal.
2.4.2 Quantities
Find the percentage of a quantity,
e.g. 15% of 1.36.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 3: Shape, Space and Measurement
3.1 Measurement
3.1.1 Perimeter and area of a rectangle 3.1.1 Perimeter and area of a rectangle Pupils should appreciate that the concept
of measurement goes beyond simple
Use of the formula A l b to find A. Find the area of
notions such as length and width. It is
Find length and/or breadth in simple cases, a border, important that
given either the area or the perimeter. the net of a cube and cuboid pupils learn to select the appropriate
3.1.2 Area of triangle. Find the perimeter of compound shapes. unit for the attribute being
Understand that the area of a triangle is half measured;
the area of a rectangle. notions of measurement are linked
bh to other areas of the mathematics
Use the formula A . curriculum such as number, shape
2
and data handling.
3.1.3 The circle
children should be given
Understand that is the ratio between the opportunities to do extensive
circumference and diameter of a circle. investigative work involving both
Use the formulae C d and C 2 r to find pencil and paper methods, LOGO
the circumference of a circle. and Dynamic Gometry Software.
Use the button on the calculator to find the
circumference.
3.1.4 Volume of a cube and a cuboid 3.1.4 Volume of a cube and a cuboid
Use the formula V l b h to find the Use the formula V l b h to find the length,
volume of a cube and a cuboid. breadth or height.
3.2 Angles
3.2.1 Angles of a triangle 3.2.2 Angles of a Quadrilateral
Understand that the sum of the angles of a Use the angle sum property of a triangle to A Dynamic Geometry Software is an
triangle is 180. find the sum of the angles of a quadrilateral. ideal tool by which pupils can discover
the angle properties of parallel lines,
Find unknown angles in scalene, isosceles and Find unknown angles in quadrilaterals. triangles and quadrilaterals.
equilateral triangles.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
3.3 Lines
3.3.1 Parallel lines
Understand the relationship between
corresponding, alternate and interior angles.
Find unknown angles involving these
properties.
3.3.2 Symmetry 3.3.3 Scale drawing
Revision of line symmetry. Understand and interpret scale drawings.
Identify shapes having rotational symmetry. Draw simple scale drawings of familiar
Determine the order of rotational symmetry. objects.
3.4 Shape
3.4.1 Triangles
Construct a triangle given the length of one
side and two angles.
3.4.2 Parallelograms
Identify the symmetrical properties of the
parallelogram, the rhombus and the kite.
Draw these shapes using squared paper.
3.4.3 Cubes and cuboids 3.4.4 The circle
Construct cubes and cuboids from their nets. Draw the inscribed regular hexagon and
Identify properties relating to faces, vertices equilateral triangle in a given circle using
and edges. ruler and compasses only.
3.4.5 Prism and pyramid
Identify properties relating to faces, vertices
and edges.
Construct prisms and pyramids from their
nets.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 4: Algebra
4.1 Patterns and sequences
4.1.1 Geometric and number patterns.
Revise geometric and number patterns. It is important that students not only
recognize, extend and verbally describe
Extend/complete patterns and sequences of
geometric and numerical patterns, but
numbers. they should represent these patterns
Describe patterns verbally and represent symbolically and graphically.
them with tables and symbols.
Children should also be given
opportunities to use a spreadsheet to
generate sequences of numbers, which
they can describe both verbally and
represent symbolically.
4.2 Functions
4.2.1 Function machines
Write expressions for given functions.
e.g. writing 2x 1 for "double a number and
add 1 to it".
4.3 Equations
4.3.1 Linear equations 4.3.3 Linear equations
Form and solve simple linear equations Solve simple linear equations involving two Pupils should be given opportunities to
involving one operation. operations, e.g. 2x 3 7 . (Exclude examples make use of a Computer Algebra
x involving collecting literal terms, e.g. Software (CAS) and a spreadsheet to
e.g. x 3 7 , x 8 1 , 4x 24 , 5 introduce and reinforce the related
3 2x 3 7 3x .)
algebraic concepts. Pupils should
4.3.2 Formulae become familiar with authoring
Evaluate simple formulae with 2 positive expressions and equations and work
inputs. with them in a CAS and a spreadsheet.
e.g. If p 3q r find the value of p when
q = 4 and r = 5.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
4.4 Graphs
4.4.1 Cartesian coordinates
Mark points using an ordered pair of Students should be given opportunities
numbers in any quadrant. to use a spreadsheet and/or a CAS to
explore algebraic relationships both
Draw lines and shapes given the coordinates symbolically and graphically. For
of their endpoints or vertices. example by representing relationships of
4.4.2 Straight line graphs the form y mx graphically using a
Noting the relationship between the CAS, pupils can appreciate that by
coordinates, e.g. y 2 x , y x 3 . changing values of m the gradient of the
Draw a linear graph from a simple relation line will change accordingly.
involving, for example:
currency conversions,
variations of cost and weight.
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Unit 5: Data Handling
5.1 Statistics
5.1.1 Tables Pupils should be given opportunities
5.1.1 Tables
to formulate questions about
Compile a frequency table for discrete Construct and interpret information tables. relevant issues, and answer these
grouped and ungrouped data. questions by collecting data and
5.1.2 Charts and diagrams presenting it in meaningful ways,
Draw and interpret bar charts for both grouped to use spreadsheets to display and
and ungrouped data. analyse the collected data.
5.1.3 The mean
Understand that the mean is a value which
represents a set of data.
Compute the mean from a set of raw data.
5.1.4 The range It is important that pupils not only learn
Understand that the range describes the how to compute the mean and range but
variation of a set of data. understand its scope and its drawbacks.
Compute the range of a set of raw data
Understand that the probability of a successful
event is given by:
number of successful events
total number of possible outcomes
Find the probability from statistical data.
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FORM 3
Core Extension
Unit 1: Number
1.1 Real numbers
1.1.1 Properties of whole numbers
Students should develop efficient mental
Find the least common multiple of up to three
and/or pencil and paper methods for
numbers. manipulating whole numbers based on a
1.1.2 Four rules well-understood knowledge of number
Revise the four rules applied to whole numbers properties and relationships.
and decimals.
Add and subtract integers, including the Fluency with whole numbers can also be
subtraction of negative integers, obtained if students are given
e.g. 7(4). opportunities
to use ICT to explore properties of
Multiply numbers by a two-digit number.
numbers and the relationships
Divide numbers by a two-digit number using between them,
repeated subtraction. to carry out investigative work
Understand the precedence of mathematical related to number.
operations.
1.1.3 Accuracy of results
Round numbers to a given number of decimal
places.
Correct numbers to a given number of
significant figures.
1.1.4 Fractions
Add and subtract two mixed numbers.
e.g. 6 2 2 1 , 2 1 3 .
3 4 5 4
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1.2 Indices
1.2.1 Notion of indices
Estimate the square roots of positive integers
less than 100.
e.g. 50 7 .
Find the square root using a calculator.
Find the cube roots of perfect cubes less than
100, without using a calculator.
Find cube roots using a calculator.
1.2.2 Integral indices
Evaluate positive integral indices.
e.g. 2.53 15 .625 .
1.2.3 Standard form 1.2.4 Rules of indices
Express numbers in standard form and vice- Evaluate integral indices with and without
versa. calculator, e.g. 4 3 64 , 5 2 1 / 25 , 6 0 1
Understand and use the rules for addition,
subtraction and opening brackets.
e.g. 3 2 35 37 , 4 5 4 2 4 3 , (5 3 ) 2 5 6 .
1.3 The calculator
1.3.1 Basic commands 1.3.1 Basic commands
Use the x2 and button on the calculator. Use the xy, and x buttons.
y The calculator should not be used only
as a numerical computation tool, but as
Use the bracket buttons. a tool which can help pupils to gain
insight into mathematical concepts. It is
also important that children learn to use
the calculator appropriately and
efficiently while at the same time
employing suitable procedures to check
the result displayed on their calculator.
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Unit 2: Applications
2.1 Number
2.1.1 Problems
Apply notions of integers, fractions and
decimals to practical situations.
2.2 Metric measures
2.2.1 Problems
Solve problems involving length, weight and
capacity.
2.3 Percentages
2.3.1 Revision
2.3.2 Expressing a decimal as a percentage
2.3.3 Percentage increase and decrease
Apply percentages to problems of profit and
loss, discount, appreciation/ depreciation (year
by year up to 2 years), commission, v.a.t., etc.
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2.4 Ratio and proportion
2.4.1 Ratio 2.4.2 Map ratios
Use ratio notation to compare two or more Understand and use simple map ratios.
quantities. 2.4.3 Direct proportion
Write ratios in their simplest form. Solve problems using the unitary method.
Divide a quantity in a given ratio. Link cost with other measures.
Apply proportion to common rates: km/litre,
kg/m2, etc.
2.5 Distance and speed
2.5.1 Speed
Understand that speed is a comparison of
distance with time.
Express speed in km/h and m/s.
Calculate time/distance given the speed and
distance/time.
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Unit 3: Shape, Space and Measurement
3.1 Mensuration
3.1.1 Perimeter and area of a rectangle 3.1.2 Area of triangle
Find the area of bh Pupils should appreciate that the concept
Use the formula A to find the base or
a border, 2 of measurement goes beyond simple
the net of a cube and cuboid height. notions such as length and width. It is
Find the perimeter of compound shapes. important that
pupils should learn to select the
appropriate unit for the attribute
3.1.3 Area of parallelogram 3.1.3 Area of parallelogram being measured;
Find the area of a parallelogram by dividing it Use the formula A bh to find the base or notions of measurement are linked
into two equal triangles. height. to other areas of the mathematics
Use the formula A bh . curriculum such as number, shape
3.1.4 Area of trapezium and data handling.
Find the area of a trapezium by dividing it into children should be given
two triangles. opportunities to do extensive
3.1.5 Circumference and area of a circle 3.1.5 Circumference and area of a circle investigative work involving both
pencil and paper methods, LOGO
Revise the use of the formulae C d and Use C 2 r and A r 2 to find the radius of and Dynamic Geometry Software.
C 2 r to find the circumference of a circle. a circle.
Use A r to find the area of a circle.
2 Use C d to find the diameter.
3.1.6 Composite shapes
Find the area of composite shapes by dividing
them into simple shapes.
3.1.7 Volume of a cube and a cuboid 3.1.7 Volume of a cube and a cuboid
Use the formula V l b h to find the length, Use V Ah to find V, A or h.
breadth or height. Find the side of a cube given the volume.
Solve problems using the relations:
1 litre = 1000 cm3, 1 m3 = 1000 litres
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3.2 Angles
3.2.1 Revision 3.2.5 Bearings
3.2.2 Parallel lines Use the eight main compass directions.
Use the properties of corresponding, alternate Understand and use the notions of clockwise
and interior angles to show that lines are and anticlockwise.
parallel. Use three-figure bearings measured clockwise
3.2.3 Angles of a triangle from the north to describe the position of one
Notion of an exterior angle of a triangle. point from another.
Understand that the exterior angle of a triangle Find the distance and bearing from a scale
is equal to the two interior opposite angles. drawing or a map.
Solve simple problems involving the exterior
angle theorem.
3.2.4 Angles of a quadrilateral
Use the angle sum property of a triangle to
find the sum of the angles of a quadrilateral.
Find unknown angles in quadrilaterals.
3.3 Shapes
3.3.1 Scale drawing 3.3.5 Polygons
Understand and interpret scale drawings. Recognize by name polygons up to 8 sides Students should be given the
Draw simple scale drawings of familiar excluding the heptagon. opportunity to use LOGO and a
objects. Understand the symmetry properties of a Dynamic Geometry Software to
regular polygon. investigate the properties of
3.3.2 The circle
Construct a regular polygon. quadrilaterals and regular polygons.
Draw the inscribed regular hexagon and
equilateral triangle in a given circle using Find the sum of the interior/exterior angles of Pupils are expected to use a formula
ruler and compasses only. a polygon. such as (2n 4) right angles to find the
3.3.3 Quadrilaterals 3.3.6 Pythagoras theorem sum of the interior angles of a polygon.
Identify the properties of the square, Understand the theorem of Pythagoras
rectangle, parallelogram, rhombus, kite and through drawing and measurement.
trapezium through line and rotational Use this theorem to find the side/s of a right-
symmetry. angled triangle.
3.3.4 Triangle
Construct a triangle given two sides and the
included angle.
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3.4 Transformations
3.4.1 Enlargements
Use squared paper to enlarge simple shapes The lengths and areas of enlarged
by a positive integral scale factor without shapes can be investigated by means of
referring to the centre of enlargement. LOGO, a spreadsheet or a Dynamic
Find the scale factor given two similar Geometry Software.
shapes.
Find the lengths of an enlarged shape.
Unit 4: Algebra
4.1 Patterns and sequences
4.1.1 Generate a sequence from a given rule.
4.1.2 Establish a rule given a sequence (Students will
not be expected to write the nth term.).
4.1.3 Complete geometric patterns.
4.2 Algebraic expressions
4.2.1 Simplify algebraic expressions
Pupils should be given opportunities to
Collect like terms, e.g. 3x 5 2x 8 , make use of a Computer Algebra
4c 2d c 7d Software (CAS) and a spreadsheet to
Multiply terms, e.g. 6 9x , 3z 2z introduce and reinforce the related
4.2.2 Multiplication 4.2.3 Factorization algebraic concepts. Pupils should
Multiply a single term over a bracket, Taking out a single term common factor, become familiar with authoring
expressions and equations and work
e.g. 2 p(3 p 7) e.g. 6p214p with them in a CAS and a spreadsheet
4.2.4 Evaluation of expressions environment.
Evaluate simple expressions by substitution.
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4.3 Equations and Formulae
4.3.1 Linear equations 4.3.1 Linear equations
Form and solve simple linear equations Construct and solve simple linear equations
involving two operations. from given situations.
e.g. 2x 3 7 , 2x 3 7 3x , 2( x 5) 21 . e.g. A soft drink costs x cents and an 4x
cents. The total cost is 80 cents. Write down
an equation and find the value of x.
4.3.2 Formulae 4.3.2 Formulae
Revise evaluation of formulae by substitution. Construct simple formulae.
4.4 Graphs
4.4.1 Linear graphs
Draw a straight line graph by Scales for both axes are given and equal.
compiling a table of values from its
Students should be given opportunities
equation,
to use a spreadsheet and/or a CAS to
plotting the points from the table, explore algebraic relationships both
joining the points. symbolically and graphically.
Use the graph to find the value of one
coordinate given the other.
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Unit 5: Data Handling
5.1 Statistics
5.1.1 Information tables
Construct and interpret information tables. Pupils should be given opportunities
5.1.2 Mean, Range and Mode to formulate questions about
relevant issues, and answer these
Revise Mean and Range.
questions by collecting data and
Understand and find the Mode from raw data. presenting it in meaningful ways,
5.1.3 Frequency tables 5.1.3 Frequency tables to use spreadsheets to display and
Collect, classify and tabulate statistical data. Compile frequency tables for continuous analyse the collected data.
Revise compilation of a frequency table for grouped and ungrouped data with equal class
discrete grouped and ungrouped data. intervals.
5.1.4 Charts and diagrams
Draw and interpret pie charts 5.2.2 Probability of an event
Understand that the probability of a successful Find the probability that an event does not
event is given by: occur.
number of successful events
total number of possible outcomes
Find the probability from statistical data.
5.2.3 Possibility spaces
Find the probability of two independent
events by constructing a possibility space.
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FORM 4
Core Extension
Unit 1: Number
1.1 Real numbers
1.1.1 Four rules
Revise the four rules applied to whole numbers Students should develop efficient mental
and decimals. and/or pencil and paper methods for
1.1.2 Accuracy of results manipulating whole numbers based on a
Round numbers to a given number of decimal well-understood knowledge of number
places. properties and relationships.
Correct numbers to a given number of
Fluency with whole numbers can also be
significant figures. obtained if students are given
Round quantities to a given unit of measure, opportunities
e.g. to the nearest cent. to use ICT to explore properties of
1.1.3 Estimation numbers and the relationships
Estimate a result by rounding numbers to one between them,
significant figure. to carry out investigative work
1.1.4 Fractions 1.1.4 Fractions related to number.
Revise addition and subtractions of two Multiply and divide two fractions.
fractions. e.g. 3
4
11 , 4 2 2 5 , 21 7 , 8 5 3 10
8
3
2 8 4 4 3
1.2 Indices
1.2.1 Rules of indices
Evaluate integral indices,
e.g. 4 3 64 , 5 2 1 / 25 , 6 0 1
Understand and use the rules for
multiplication, division and brackets raised to
a power.
e.g. 3 2 35 37 , 4 5 4 2 4 3 , (5 3 ) 2 5 6 .
1.2.2 Standard form
Revise expressing numbers in standard form
4
and vice-versa.
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1.3 The calculator
1.3.1 Basic commands
Use the xy, and x buttons.
y The calculator should not be used as a
numerical computation tool, but as a
Use the bracket buttons. tool which can help students to acquire
skills and insight in solving problems.
Pupils should be encouraged to check
the accuracy of the calculator result by:
making rough estimates;
repeating calculations;
reversing the order of an operation.
Unit 2: Applications
2.1 Ratio and proportion
2.1.1 Notion of ratio
Revision.
2.1.2 Map ratios
Understand and use simple map ratios.
Find the map distance given the scale and the
actual distance.
Find the actual distance given the scale and
the map distance.
Determine the map scale from map and actual
distances.
2.1.3 Direct proportion
Solve problems using the unitary method.
Link cost with other measures.
Apply proportion to common rates:
km/litre, kg/m2, etc.
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2.2 Percentages
2.2.1 Revision
2.2.2 Percentage change 2.2.3 Interest
Determine percentage change as a one-step Calculate simple interest.
process. PTR
Use the formula I to find I.
Revise percentages to problems of profit and 100
loss, discount, appreciation/ depreciation (year
by year up to 2 years), commission, v.a.t., etc.
2.3 Distance, time and speed
2.3.1 Speed 2.3.1 Speed
Understand that speed is a comparison of Change units of speed.
distance with time. e.g. km/h to m/s and vice-versa.
Express speed in km/h and m/s. Calculate the average speed.
Calculate time/distance given the speed and Understand and interpret travel graphs.
distance/time.
2.4 Household and Personal Finance
2.4.1 Exchange rates 2.4.2 Personal finance
Convert one currency into the other. Solve problems relating to salary,
Solve simple problems involving exchange commission, overtime, income tax, etc.
rates. 2.4.3 Household finance
Solve problems relating to budgeting,
holidays, best buys, invoices, bills, etc.
2.5 Metric measures
2.5.1 Length, mass, capacity
Understand, convert and use metric units of
mass, length, capacity, area and volume to
solve practical problems.
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Unit 3: Shape, Space and Measurement
3.1 Measurement
3.1.1 Area of triangle.
bh
Use the formula A to find the base or Pupils should appreciate that the concept
2 of measurement goes beyond simple
height. notions such as length and width. It is
3.1.2 Area of parallelogram important that
Use the formula A bh to find the base or pupils should learn to select the
height. appropriate unit for the attribute
3.1.3 Circumference and area of a circle being measured;
notions of measurement are linked
Use C 2 r and A r 2 to find the radius of to other areas of the mathematics
a circle. curriculum such as number, shape
Use C d to find the diameter. and data handling.
3.1.4 Composite shapes children should be given
opportunities to do extensive
Find the area of composite shapes by dividing
investigative work involving both
them into simple shapes. pencil and paper methods, LOGO
3.1.5 Volume of a cube and a cuboid 3.1.6 Area of trapezium and Dynamic Geometry Software.
Use V Ah to find V, A or h. Use the formula A 1 (a b)h to find A or h.
2
Find the side of a cube given the volume.
Solve problems using the relations:
1 litre = 1000 cm3, 1 m3 = 1000 litres 3.1.7 Solids
Find the volume of a prism using
Volume = area of cross-section length
Find the surface area of a cylinder.
Find the volume of a cylinder using
Volume = r 2 h
3.1.8 Composite solids
Find the surface area and volume of
composite solids.
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3.2 Angles
3.2.1 Bearings 3.2.2 Angles of elevation and depression
Use the eight main compass directions. Use scale drawings to solve problems
Understand and use the notions of clockwise involving angles of elevation/depression in
and anticlockwise. simple practical situations.
Use three-figure bearings measured clockwise
from the north to describe the position of one
point from another.
Find the distance and bearing from a scale
drawing or a map.
3.3 Shapes
3.3.1 Polygons
Recognize by name polygons up to 8 sides LOGO and Dynamic Geometry
excluding the heptagon. Software give pupils ample
Understand the symmetry properties of a opportunities to explore the properties of
regular polygon. polygons.
Pupils are expected to use a formula
Construct a regular polygon.
such as (2n 4) right angles to find the
Find the sum of the interior/exterior angles of sum of the interior angles of a polygon.
a polygon.
3.3.2 Scale drawing
Determine lengths from a scale drawing.
Make a scale drawing from a sketch.
3.3.3 Pythagoras theorem 3.3.3 Pythagoras theorem
Understand the theorem of Pythagoras Use the converse of Pythagoras theorem to
through drawing and measurement. show that a triangle is right-angled.
Use this theorem to find the side/s of a right-
angled triangle.
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3.4 Transformations
3.4.1 Enlargements 3.4.1 Enlargements
Use squared paper to enlarge simple shapes by Draw enlargements on squared paper given the Pupils should be given opportunities to
a positive integral scale factor without centre of enlargement and a positive use LOGO, a Dynamic Geometry
referring to the centre of enlargement. (integral/fractional) scale factor. Software and a spreadsheet to explore
Find the scale factor given two similar shapes. Find the centre of enlargement and/or scale and investigate transformations.
Find the lengths of an enlarged shape. factor.
Understand that similar figures occur when one
is an enlargement of the other.
3.4.2 Reflections
Recognise, describe and construct reflections
in y x , y c , x c
3.4.3 Rotations
Recognise, describe and construct rotations
restricted to multiples of 90.
3.4.4 Translations
Recognise, describe and construct translations
described by a simple column vector.
3.5 Trigonometry
3.5.1 Trigonometric ratios 3.5.1 Trigonometric ratios
Understand the trigonometric relationships Use trigonometric ratios to solve problems
in a right-angled triangle. involving angles of elevation/depression.
Use the sine, cosine and tangent ratios to
solve a right-angled triangle.
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3.6 The Circle
3.6.1 Symmetry properties of the circle
Understand that Pupils should be given an opportunity to
Equal chords are equidistant from the use a Dynamic Geometry Software to
centre and its converse. explore the symmetry properties of the
The perpendicular bisector of a chord circle.
passes through the centre and its converse.
A tangent is perpendicular to the radius at
the point of contact.
Two tangents from an external point are
equal.
Solve problems involving the symmetry
properties of the circle and their converse.
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Unit 4: Algebra
4.1 Algebraic Expressions
4.1.1 Expressions
Revise algebraic expressions.
4.1.2 Factorization
Taking out a single term common factor,
e.g. 6p214p
4.2 Equations and Formulae
4.2.1 Linear equations
Construct and solve simple linear equations Pupils should be given opportunities to
from given situations. make use of a Computer Algebra
e.g. A soft drink costs x cents and an ice- Software (CAS) and a spreadsheet to
cream costs 4x cents. The total cost is 80 introduce and reinforce the related
cents. Write down an equation and find algebraic concepts. Pupils should
the value of x. become familiar with authoring
expressions and equations and work
4.2.2 Formulae
with them in a CAS and a spreadsheet
Construct simple formulae. environment.
Substitute numerical values to find one of the
variables.
Rearrange the formula with the subject
appearing only once.Secondary Schools Mathematics Syllabus - September 2003
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4.4 Graphs
4.4.1 Gradient
Understand that the gradient is a measure of
slope.
Calculate the gradient of a line from the
coordinates of two points on it.
Understand that parallel lines have equal
gradients.
4.4.2 The equation of a straight line
Revision
Recognize that equations of the form
y mx c represent straight lines.
Find the gradient and the y-intercept of a
straight line given its equation.
Find the equation of a straight line from its
graph.
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Unit 5: Data Handling
5.1 Statistics
5.1.1 Frequency tables
Revision Pupils should be given opportunities
5.1.2 Charts and diagrams to formulate questions about
relevant issues, and answer these
Revise bar charts and pie charts.
questions by collecting data and
Draw and interpret histograms with equal presenting it in meaningful ways,
intervals. to use spreadsheets to display and
5.1.3 Measures of central tendency 5.1.3 Measures of central tendency analyse the collected data.
Compute the median for a set of raw data. Compute the mean, median and mode from an
ungrouped frequency table. It is important that pupils not only learn
Understand the advantages/disadvantages of how to compute the mean, mode, and
these statistics. median but understand the aim of these
statistics.
5.2 Probability
5.2.1 Probability of an event
Find the probability that an event does not Pupils should be given opportunities to
occur. use a spreadsheet to simulate randomly
5.2.2 Possibility spaces occurring events such as the throw of a
Find the probability of two independent events die and the toss of a coin.
by constructing a possibility space.
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FORM 5
Core Extension
Unit 1: Number
1.1 Real numbers
1.1.1 Four rules Students should develop efficient mental
Revise the four rules applied to integers and and/or pencil and paper methods for
decimals. manipulating whole numbers based on a
well-understood knowledge of number
1.1.2 Fractions
properties and relationships.
Revision
Multiply and divide two fractions. Fluency with whole numbers can also be
e.g. 3 11 , 4 2 2 5 , 21 7 , 8 5 3 10 .
4
8
3
2 8 4 4 3 obtained if students are given
opportunities
1.1.3 Accuracy of results
to use ICT to explore properties of
Revise rounding numbers to a given number of numbers and the relationships
decimal places. between them,
Revise correcting numbers to a given number to carry out investigative work
of significant figures. related to number.
Revise rounding quantities to a given unit of
measure, e.g. to the nearest cent.
1.1.4 Estimation
Revise estimating a result by rounding
numbers to one significant figure.
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1.2 The calculator
1.2.1 Basic functions
Use a calculator efficiently. The calculator should not be used only
Use the memory facility for mixed operations as a numerical computation tool, but as
with numbers. a tool which can help pupils to gain
insight into mathematical concepts. It is
also important that children learn to use
the calculator appropriately and
efficiently while at the same time check
the accuracy of the calculator result by:
making rough estimates;
repeating calculations;
reversing the order of an operation.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 2: Applications
2.1 Ratio and proportion
2.1.1 Direct and inverse proportion
Revise direct proportion.
Understand and solve problems involving
inverse proportion.
2.2 Percentages
2.2.1 Revision
2.2.2 Percentage change
Carry out calculations involving reverse
percentages.
e.g. to find the cost price given the selling price
and percentage profit.
2.2.3 Interest 2.2.3 Interest
Calculate the simple interest. PTR
Use the formula I to find P, T or R.
PTR 100
Use the formula I to find I.
100
2.3 Household and personal finance
2.3.1 Personal finance
Solve problems relating to salary, commission,
overtime, income tax, etc.
2.3.2 Household finance
Solve problems relating to budgeting,
holidays, best buys, invoices, bills, etc.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 3: Shape, Space and Measurement
3.1 Mensuration
3.1.1 Revision
3.1.2 Area of trapezium. 3.1.3 Circumference and area of a circle It is important that
Use the formula A 1 (a b)h to find A or h.
2 Find the length of arc and area of sector as pupils should learn to select the
appropriate unit for the attribute
fractions of the circumference and area of a
being measured;
circle. notions of measurement are linked
3.1.4 Solids 3.1.4 Solids to other areas of the mathematics
Find the volume of a prism using Find the surface area of a pyramid curriculum such as number, shape
Volume = area of cross-section length and data handling.
Find the surface area of a cylinder. children should be given
Find the volume of a cylinder using opportunities to do extensive
investigative work involving both
Volume = r 2 h . pencil and paper methods, LOGO
3.1.5 Composite solids and Dynamic Geometry Software.
Find the surface area and volume of
composite solids.
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3.2 Angles, lines and shape
3.2.1 Angles and lines Pupils should be given opportunities to
Revise the basic facts about lines and angles. investigate the properties of shapes by
3.2.2 Constructions means of LOGO and a Dynamic
Carry out constructions based on Geometry Software.
measurement.
Estimate, measure and draw lines and
angles.
Construct parallel lines.
Construct simple geometrical figures from
given data.
Use straight edges and compasses to
construct
angles of 60 and 90,
the perpendicular bisector of a line
segment,
the perpendicular from a point to a line,
the bisector of an angle.
3.2.3 Shape
Revise the properties of the triangle,
quadrilateral, polygons and the circle.
3.2.4 Solution of right-angled triangles
Revise Pythagoras theorem.
Use the converse of Pythagoras theorem to
show that a triangle is right-angled.
Use scale drawings to solve problems
involving angles of elevation/depression in
simple practical situations.
3.2.5 Similar figures 3.2.5 Similar figures*
Understand that similar figures occur when Understand and use the AAA and common
one is an enlargement of the other. ratio property of sides to prove similarity of
Identify similar figures. triangles.
Find the scale factor and lengths from similar
figures using models and plans drawn to
scale.
Secondary Schools Mathematics Syllabus - September 2003
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
3.2.6 Congruent figures
Understand and identify congruent shapes.
Appreciate the uniqueness of triangles
satisfying SSS, SAS, ASA and RHS.
Use SSS, SAS, ASA and RHS to prove the
congruence of triangles.
3.3 Transformations
3.3.1 Enlargements 3.3.1 Enlargements Pupils should be given opportunities to
Draw enlargements on squared paper given Understand that enlargements preserve use LOGO, a Dynamic Geometry
the centre of enlargement and a positive Software and a spreadsheet to explore
angle and not necessarily length.
and investigate transformations.
(integral/fractional) scale factor.
Find the centre of enlargement and/or scale
factor.
3.3.2 Reflections
Recognise, describe and construct
reflections in y x , y c , x c
3.3.3 Rotations
Recognise, describe and construct rotations
restricted to multiples of 90
3.3.4 Translations 3.3.5 Congruency
Recognise, describe and construct
Understand the congruency of shapes under
translations described by a simple column
reflection, rotation and translation
vector.
3.4 Trigonometry
3.4.1 Trigonometric ratios
Revision
Use trigonometric ratios to solve problems
involving:
angles of elevation/depression;
bearings.
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3.5 The Circle
3.5.1 Symmetry properties of the circle 3.5.2 Angle properties of the circle
Pupils should be given an opportunity to
Understand that Understand that
use a Dynamic Geometry Software to
Equal chords are equidistant from the The angle subtended at the centre is explore the symmetry properties of the
centre and its converse. twice the angle subtended at the circle.
The perpendicular bisector of a chord circumference.
passes through the centre and its The angle in a semicircle is a right
converse. angle.
A tangent is perpendicular to the radius Angles in the same segment are equal.
at the point of contact. Opposite angles of a cyclic quadrilateral
Two tangents from an external point are are supplementary.
equal. The exterior angle of a cyclic
Solve problems involving the symmetry quadrilateral is equal to the interior
properties of the circle and their converse. opposite angle.
Solve problems involving the angle
properties of the circle (Reasons justifying
the use of these angle facts in simple words
are expected).
3.6 Loci
3.6.1 Loci in two dimensions
Determine the locus of points which
are at a fixed distance from a given
point.
are equidistant from two given points.
Apply these loci to practical situations.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 4: Algebra
4.1 Patterns and Sequences
4.1.1 Revision.
4.1.2 Use expressions to describe the nth term of a
simple sequence.
4.2 Functions, Equations and Formulae
4.2.1 Functions Pupils should be given opportunities to
Use function notation. make use of a Computer Algebra
e.g. f(x) = 7x 2 Software (CAS) and a spreadsheet to
introduce and reinforce the related
4.2.2 Linear equations algebraic concepts. Pupils should
Revision become familiar with authoring
4.2.3 Simultaneous equations expressions and equations and work
Solve two simultaneous linear equations in with them in a CAS and a spreadsheet
two unknowns algebraically by elimination environment.
and by substitution.
Solve two simultaneous linear equations in
two unknowns graphically.
4.2.4 Formulae
Revision.
Change the subject of a formula.
Use and construct formulae on a
spreadsheet.
Secondary Schools Mathematics Syllabus - September 2003
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta4.4 Graphs
4.4.1 Linear graphs 4.4.3 Applications Students should be given opportunities
Revision Interpret information presented in a variety to use a spreadsheet and/or a CAS to
of linear and non-linear graphs. explore algebraic relationships both
symbolically and graphically. For
e.g. distance-time graphs, velocity-time
example by representing relationships of
graphs, conversion graphs, graphs of length the form y mx graphically using a
4.4.2 Quadratic graphs against age.
CAS, pupils can appreciate that by
Construct table of values for quadratic changing values of m the gradient of the
functions. line will change accordingly.
Plot and draw graphs of such functions by
making use of
pencil and paper methods,
a spreadsheet,
a graphing package.
Read off values from graphs.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
Unit 5: Data Handling
5.1 Statistics
5.1.1 Information tables
Understand and interpret conversion tables, Pupils should be given opportunities
time-tables, currency tables, etc. to formulate questions about
relevant issues, and answer these
Construct and use tables.
questions by collecting data and
5.1.2 Frequency tables presenting it in meaningful ways,
Revision to use spreadsheets to display and
Compile frequency tables from continuous analyse the collected data.
grouped and ungrouped data with equal class
intervals. It is important that pupils not only learn
5.1.3 Charts and diagrams how to compute the mean, mode, and
Revision. median but understand the scope of
these statistics
Pictograms and misleading bar charts.
5.1.4 Measures of central tendency 5.1.4 Measures of central tendency
Compute the mean, median and mode from Revise computation of the range of a set of
an ungrouped frequency table. raw data.
Compute the range from an ungrouped
frequency table.
5.1.5 Measures of variation
Revise the range of a set of raw data.
5.2 Probability
5.2.1 Revision Pupils should be given opportunities to
use a spreadsheet to simulate randomly
occurring events such as the throw of a
die and the toss of a coin.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
5. INFORMATION AND COMMUNICATION TECHNOLOGY (ICT) IN MATHEMATICS
"Students learn by asking questions and by establishing connections . . . Students learn from everyday experiences through observation,
listening, investigation, experimentation and the comparison of what has been discovered with what is already known . . . Learning is an
organic process of invention and mental structuring and not a mechanical process of gathering information . . . Students need to change
their modes of knowing in an active manner. Teachers or learning systems must facilitate this process. A healthy education therefore
encourages . . . a pedagogy based on questioning . . . (and) on learning by doing . . . Students are not empty receptacles to be filled in . . ."
(National Minimum Curriculum – Principle 3: Stimulation of Analytical, Critical and Creative Thinking Skills – Ministry of Education -
December 1999)
". . . The computer is still being given lip service. Apart from a change in mentality, one must seriously consider which physical and
organizational changes are required of a school which seriously regards the computer as an indispensable learning vehicle." (National
Minimum Curriculum – Principle 13: The Importance of Learning Environments – Ministry of Education - December 1999)
"Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students'
learning." (Principles and Standards for School Mathematics– NCTM April 2000)
ICT has the potential to make a significant contribution to pupils" learning in Work with realistic sets of data (e.g. carry out experiments using
mathematics by helping them to: random samples generated through simulation)
Practise and consolidate number skills (e.g. use a spreadsheet to Explore, describe and explain patterns and relationships in
revise number skills) sequences and tables of numbers (e.g. use a formula on a
Experiment with, make hypotheses from, and discuss or explain spreadsheet to generate the given values)
relationships in shape and space (e.g. use dynamic geometry Develop skills of mathematical modelling through the exploration,
software to investigate angle properties in a circle) interpretation and explanation of data (e.g. choose an appropriate
Develop logical thinking and modify assumptions and strategies graphical representation to display information from a set of data)
through immediate feedback (e.g. plan a series of instructions to
draw a required shape or carry out a set of manipulations in a
When using ICT teachers need to keep in mind that:
spreadsheet)
Make connections within and across the different areas of
mathematics (e.g. relate a function and its graph)
Secondary Schools Mathematics Syllabus - September 2003 48
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
ICT is employed because it is the most effective way to achieve It cannot be emphasized strongly enough that the use of ICT in mathematics
lessons can only be effective if it is used within the contexts of good
teaching and learning objectives, not simply for reward or mathematics teaching. Teachers therefore will have to set clear objectives
motivation. for pupils" learning in mathematics and understand how the ICT used will
help to support their teaching and their pupils" learning.
They should avoid the use of ICT for simple or routine tasks which
would be better accomplished by other means. The following Internet sites are suitable for both students and teachers.
These sites contain sources of downloadable software and teaching and
Pupils are expected to use ICT to answer valid questions appropriate learning resources:
to the subject matter being taught.
MSW Logo
Teachers should therefore plan Dr Geo
WinGeom
the ways in which ICT will be used to meet the teaching and Graphmatica
learning objectives
the key questions to ask and opportunities for teacher intervention in
order to stimulate and direct pupils" learning
the organization and conduct of the lesson and how the lesson is to
be managed.
(the Curriculum Unit's website – click
on Links for Maths Website)
(the Education Division Schools'
website – for the mathematics' section
click on Secondary…Maths…Maths
Resources for the Maths Section)
The following is a scheme for the employment of ICT in each Form.
The scheme relates directly to the mathematical content for that particular form.
Secondary Schools Mathematics Syllabus - September 2003 49
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
FORM 1 1: 100 Square.xls Bqija.xls
Decimal + & .xls Fixed Perimeter.xls
Function Machine.xls Ladder.xls
Multiply Tables.xls Number Pyramid.xls
Place Value.xls Grocer.xls
Number Bonds.xls Rounding.xls
Writing Numbers in Words.xls
Gradually students should be given opportunities to learn, through Excel, how to
Construct simple bar charts.
(Jeans Program, which is sponsored by Sedqa, provides a similar exercise for Form 1 students.)
Secondary Schools Mathematics Syllabus - September 2003 50
1 become familiar with the following Logo primitives.
PenUp PU ShowTurtle ST Right RT Home HOME
PenDown PD Forward FD Left LT Repeat REPEAT
HideTurtle HT Back BK Clear Graphics CG
OPTIONAL: They may also learn how to use SetColor SETC, SetBackGround SETBG, SetShape SETSH, Heading (to change direction in which the turtle
points), FILL to colour a closed shape, CLEAN.
Typical activities using LOGO include:
Drawing simple shapes or pictures made up of a number of simple primitives;
e.g. letters of the alphabet, squares, rectangles, equilateral triangles, sheds, houses, . . .;
Angle rotation at a point;
Estimating perimeters of simple shapes, in turtle steps;
Drawing simple shapes or pictures requiring repeated patterns, using the REPEAT command;
Predicting outcomes on the screen given a list of primitives;
Debugging a list of primitives to produce a required shape or picture.
Secondary Schools Mathematics Syllabus - September 2003 51
1The students should know how to
Use the Author command;
Use F3 and F4 to copy highlighted expressions when using the Author command;
Edit expressions to correct mistakes or modify an expression;
Use the Algebra and the Plot windows.
A typical form 1 activity using Derive include:
Use of Cartesian coordinates to plot points.
e.g. Points can be authored: Author…Expression…[[3,2],[3,6],[7,6],[7,2],[3,2]]
Secondary Schools Mathematics Syllabus - September 2003 52
1 learn know how to use the basic tools:
POINTER (Pointer)
POINTS (Point, Point on Object, Intersection Point)
LINES (Line, Segment, Ray, Triangle, Regular Polygon)
CURVES (Circle)
CONSTRUCT (Perpendicular Line, Parallel Line, Midpoint, Compass)
TRANSFORM (Reflection) – this tool may be used for line (reflective) symmetry
MEASURE (Distance and Length, Area, Angle)
DISPLAY (Label, Mark Angle)
Optional: Any of the tools from the DRAW toolbox.
The students are expected to become familiar with the use of these tools in a natural way, through investigative tasks set by the teacher. Typical tasks include
discovering:
The angles" sum properties at a point.
The angles" sum properties on a straight line.
The construction of a triangle given the three sides.
The angles" sum property for any triangle.
Symmetry properties, if any, of triangles, squares and rectangles.
The students should be encouraged to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of the
appropriate mathematical language.
Secondary Schools Mathematics Syllabus - September 2003 53
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
FORM 2 2: 24h time.xls Coins.xls
Function Machine.xls Ladder.xls
Percentages.xls Place Value.xls
Rounding.xls
Gradually they should be given opportunities to learn how to
Carry out simple calculations involving the four rules;
Generate "function machines" (e.g. double a number and subtract 3 . . . =A1*2 – 3);
Enter numerical data into the spreadsheet and analyze this data using the built in functions AVERAGE, MAX, MIN and construct formula to calculate the
RANGE;
Secondary Schools Mathematics Syllabus - September 2003 54
2 Use the basic Logo primitives encountered in Form 1
Write simple procedures incorporating the Logo primitives.
e.g. TO SQUARE
REPEAT 4 [FD 60 RT 90]
END
Edit, save and load procedures.
Document a procedure.
Students should be given opportunities to become familiar with the following Logo primitives.
Wait WAIT Set colour SETC Set background SETBG Set shape SETSH
Fill FILL
Typical activities using LOGO include:
Write a procedure that will draw a particular given shape (e.g. square, rectangle, equilateral triangle);
Debug a procedure until it draws the required shape;
Predict what a given procedure will draw;
Secondary Schools Mathematics Syllabus - September 2003 55
2 Use the Author, Simplify Basic, Approximate and Substitute For commands;
Use F3 and F4 to copy highlighted expressions when using the Author command;
Edit expressions to correct mistakes or modify an expression;
Solving simple linear equations involving two operations;
Construct a simple formula or equation from a word problem and use it to solve the problem;
Author a set of ordered pairs of points and plot these points in the PLOT WINDOW;
Know how to view the ALGEBRA WINDOW and the PLOT WINDOW simultaneously or individually;
Know how to adjust scales to obtain the desired graphical picture.
OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options related to the Plot Window.
Secondary Schools Mathematics Syllabus - September 2003 56
2 use other new tools, namely:
LINES (Polygon)
CONSTRUCT (Measurement transfer)
MEASURE (Calculate)
DISPLAY (Animation)
DRAW (Show/Hide Axes, New Axes, Define Grid)
The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
Construction of lines, angles, parallels and perpendiculars;
Construction of the perpendicular to a line;
Constructing an equilateral triangle inscribed in a regular hexagon;
Compound constructions involving basic ones;
Discovering the relationship between the circumference and diameter of a circle to establish ;
Plotting points satisfying particular rules;
The angle sum property of a triangle.
The students should be encouraged to discuss, communicate and, whenever possible, explain their findings and record these in writing, by making use of the
appropriate mathematical language.
Secondary Schools Mathematics Syllabus - September 2003 57
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
FORM 3Pre-written spreadsheet lessons which can be used in Form 3: Decimal expansion.xls Ladder.xls
Percentages.xls Pie Chart.xls
Quadrilaterals.xls Number Pyramid
Ratio.xls Round.xls
Variables and F(x) Machines.xls
Gradually they should be given opportunities to learn how to
Consolidate previously acquired knowledge and skills associated with the spreadsheet;
Generate more complicated "function machines" (e.g. square a number, subtract 3 and divide the result by 5 . . . =(A1^2 – 3)/5);
Carry out calculations involving percentages on a spreadsheet, for example, to model "what happens if . . ." situations;
Generate a set of ordered pairs linearly related and construct a scatter and line graph to display this linear relationship.
Construct a simple pie chart.
Secondary Schools Mathematics Syllabus - September 2003 58
3 and Form 2.
Write simple procedures incorporating the Logo primitives.
e.g. TO TRIANGLE
REPEAT 3 [FD 100 RT 120]
END
Write procedures that call other procedures.
Discover the "total trip" theorem (the sum of the exterior angles of a polygon is 360).
Edit, save and load procedures.
Document a procedure.
Typical activities using LOGO include:
Write a procedure that will draw a particular given shape (e.g. square, rectangle, equilateral triangle, regular polygon, circle)
Write a procedure that will draw a parallelogram (or a rhombus) given the sides and the included angle.
Write procedures that draw composite shapes;
Debug a procedure until it draws the required shape;
Predict what a given procedure will draw.
Secondary Schools Mathematics Syllabus - September 2003 59
3 Familiarize themselves with those facilities provided by the software they used in the previous two years.
Use the EXPAND and SUBSTITUTE FOR VARIABLES features.
Use the FACTORIZE feature to a) appreciate that factorization is the reverse process of expansion and
b) to develop paper and pencil strategies that will enable them to factorize simple binomials.
Plot linear graphs, observe their characteristic properties and deduce information from them.
Use the software as a problem solving tool.
OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options available for the Plot Window.
Typical form 3 activities using Derive include:
Collecting like terms to check pencil and paper results;
Multiplying terms to check pencil and paper results;
Solving simple linear equations involving expanding brackets;
Substitution to find a variable in a formula;
Use of Cartesian coordinates to plot and join points.
Finding the value of one coordinate given the other.
Secondary Schools Mathematics Syllabus - September 2003 60
3 2 and use other new features,
namely:
POINTER (Dilate)
CHECK PROPERTY (Collinear, Parallel, Perpendicular)
DISPLAY (Comments, Numerical Edit)
TRANSFORM (Dilation)
DRAW (Thick, Dotted, Modify Appearance)
The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
Exterior angle of a triangle property;
Establishing the sum of the interior and exterior angles of a polygon;
Enlargements with a positive scale factor;
Discovering Pythagoras" Theorem 61
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
FORM 4.
Use the spreadsheet functions RAND(), INT, and IF to model and simulate real life situations.
Carry out purposeful enquiries to answer "what happens if . . ." type questions.
Carry out calculations involving percentages in real life contexts (e.g. related to tax, insurance, commission, simple interest, exchange rates, etc.).
Use the GOAL SEEK command to investigate feasibility of a real life situations.
Use INSERT NAME CREATE to create a variable, as a substitute to the usual cell referencing.
Analyse data, draw histograms and interpret the results (e.g. examination results).
Use the spreadsheet as a problem solving tool.
Pre-written spreadsheet lesson which can be used in Form 4: Dynamic Gradient.xls Quadrilaterals.xls
Rounding.xls Straight line.xls
Translations
Secondary Schools Mathematics Syllabus - September 2003 62
4 Form 2 and Form 3.
Write procedures that make use of variable inputs
Typical activities involving LOGO include:
Write a procedure that will accept one or more variables to draw, for example, regular polygons having a different number of sides and varying side
lengths.
Write procedures that perform simple transformations – reflection, enlargement, rotation and translation.
Projects involving variables.
Secondary Schools Mathematics Syllabus - September 2003 63
4 three years.
Use the correct procedures to change the subject of formulae.
Use the FACTORISE feature to a) appreciate that factorization is the reverse process of expansion and b) to develop paper and pencil strategies that will enable
them to factorise binomials and trinomials.
Use the SOLVE feature to solve quadratic equations and appreciate that the algebraic solutions and the graphical solutions of a quadratic equation are identical.
Plot linear and quadratic graphs, observe their characteristic properties and deduce information from them.
Solve problems leading to quadratic equations.
Use the software as a problem solving tool.
OPTIONAL: Know how to change the Plot Background, the colour of the Plots; know how to use other options available for the Plot Window.
Secondary Schools Mathematics Syllabus - September 2003 64
4 and 3 and use other new features,
namely:
POINTER (Rotate, Dilate, Rotate and Dilate)
TRANSFORM (Reflection, Rotation, Translation)
MEASURE (Calculate, Tabulate)
The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use. Typical tasks include:
Appreciating the properties of congruent triangles;
Exploring properties of similar triangles;
Discovering the symmetry properties of the circle;
Discovering and establishing the angle properties of the circle;
Using scale drawing to solve simple problems in 2-D;
Transformation activities 65
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
FORM 5;
Model and simulate real life situations;
Carry out purposeful enquiries to answer "what happens if . . ." type questions (through use of logic gates IF, AND and OR);
Construct travel graphs and deduce information from them;
Analyse data and draw histograms, cumulative frequency curves and box-plots;
Use the spreadsheet for modeling real-life situations and to solve problems.
By form 5 students should be able to carry out investigational work by writing down observations, recording and tabulating results, making predictions and carry
out tests on them (conjectures) and finally arrive at a generalised conclusion.
Secondary Schools Mathematics Syllabus - September 2003 66
5 2, 3 and 4;
Write procedures that make use of variable inputs.
Typical activities involving LOGO include projects involving variables.
Secondary Schools Mathematics Syllabus - September 2003 67
5 four years.
Carry out appropriate manipulations on expressions to change the subject of the formula.
Plot linear and quadratic functions.
Solve simultaneously linear equations graphically, using the Plot Window, and interpret the solutions.
Use the software as a general problem solving tool.
Secondary Schools Mathematics Syllabus - September 2003 68
5 , 3 and 4 and use other new features,
namely:
CONSTRUCT (Measurement Transfer, Locus)
CHECK PROPERTY (Collinear, Parallel, Perpendicular, Equidistant, Member)
Optional - DISPLAY (Comments, Numerical Edit, Fix/Free, Trace On/Off, Animation, Multiple Animation)
The students may become acquainted with these tools in a natural way when teachers set tasks requiring their use.
Typical tasks include:
Constructing loci in two dimensions;
Discovering symmetry properties;
Transformation activities;
Using the software as a general 69
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
6. IMPLEMENTING THE MATHEMATICS SYLLABUS
The teaching and learning process must reflect that the following b) COMMUNICATION
considerations are being taken into account:
Communication is a necessary component for learning, doing and
understanding mathematics. Communication in mathematics means that
a) TEACHING AND LEARNING MATHEMATICS one is able not only to use its vocabulary, notation and structure to express
ideas and relationships but also to think and reason mathematically. In
The quality of mathematics learning depends on the quality of the teaching. fact, communication is considered the means by which teachers and
What students learn is a reflection of the learning experiences that teachers students can share the processes of learning, doing and understanding
provide. On the other hand, teaching mathematics well is a complex task mathematics. Students should express their thinking and problem solving
and no simple recipes for helping all students to learn are available. processes in both written and oral formats, allowing code-switching when
Nevertheless effective mathematics teaching requires a serious commitment necessary. The clarity and completeness of students" communication can
to the development of students" understanding of mathematics. Effective indicate how well they understand the related mathematical concepts.
teachers know how to ask exploratory questions to reveal what the students Because teaching is communicating, teachers play a central role in
already know and plan and design appropriate tasks that build on that fostering students" mathematical communication and understanding.
knowledge. They reflect on their practice and engage themselves in Teachers should therefore provide students with opportunities to discuss
continuous self-improvement. mathematics, particularly during activities which involve exploration,
conjecturing, analysis and application of mathematical ideas. This
Teachers must also ensure that students learn mathematics with approach gives way to the most common pedagogy prevalent in most
understanding in order to be able to apply procedures, concepts and classes, namely that of "teaching by telling" and reinforces the alternative
processes effectively. Learning with understanding makes subsequent approach, that of "constructivist teaching". In constructivist teaching
learning easier. Mathematics makes more sense and is easier to remember students construct new knowledge by connecting it with experience.
and to apply when students connect new knowledge to existing knowledge Students should not be looked upon as empty vessels waiting to receive
in meaningful ways. Besides, learning with understanding creates information but rather have a certain amount of stored knowledge which
autonomous learners who can take control of their learning, become may be accessed and updated. Research has shown that students learn and
confident in their ability to tackle difficult problems, seek alternative retain more information when they are able to articulate what they know to
solution methods and learn to persevere. Such learners develop a feeling of others. Of more benefit is social constructivism which recognises that
accomplishment and eventually a willingness to pursue further engagement students are part of the learning communities in which language and
with mathematics. meaning can be shared among peers and teachers. Social constructivism
therefore provides a framework in which teachers can facilitate student
communication and in which learning is both corporate and individual.
Secondary Schools Mathematics Syllabus - September 2003 70
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
The ability to communicate mathematical ideas can be fostered if: c) PROBLEM SOLVING
Understanding of mathematics is developed through reflection and by
Learning to solve problems is one of the principal reasons for studying
organization and communication of ideas.
Mathematics. Problem solving therefore is the heart of the mathematics
curriculum. Consequently students should be capable of applying previously
Understandings and relationships between and among mathematical acquired knowledge to new and unfamiliar situations, to solve non-routine
concepts, procedures and symbols are communicated in writing and problems, to pose questions, to analyse situations, translate results and apply
speaking. This must be done at each stage of conceptual development, trial-and-error methods. Students should see alternative solutions to
whether concrete, pictorial or abstract, and in every area of mathematics. problems and be aware that a particular problem may have more than one
solution. Many different activities, such as gathering data, exploring
Mathematical situations are represented or described in a variety of ways patterns, making and testing conjectures, and justifying conclusions through
(e.g. verbal, concrete, pictorial, graphical, algebraic). logical arguments, are necessary to develop the students" mathematical
reasoning and ability to communicate about mathematics.
Positions on mathematical processes and solutions are defended through
sound argument. Problem Solving also develops critical thinking skills, particularly when:
The need for mathematical symbolism is demonstrated. Checks for reasonableness and completeness of results form an integral
part of the problem solving process. Incorrect solutions are analysed to
identify common errors in the problem-solving process.
The ability to read mathematics is emphasised.
Multi-step solutions and non-routine problems are posed on a regular
The ability to write mathematical problems from real-world situations is
basis.
emphasised.
Activities that require collecting, organising, manipulating data and
Proper and precise mathematical vocabulary and notation is stressed.
drawing inferences from that data are provided.
Communication skills are developed in small groups working together,
Group problem solving is encouraged so that the students will be able to
through listening, exploring, questioning, discussing and summarizing.
share responsibility for the product of the activity after having an
opportunity to discuss the results.
Activities are structured so that several strategies or techniques are
available for use in the solution process.
Inter-disciplinary projects are encouraged.
Strategies such as top-down analysis and stepwise refinement are used to
analyse and solve complex problems.
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Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
d) INFORMATION AND COMMUNICATION TECHNOLOGY (ICT) They should also be able to understand the ways in which information is
gathered by counting and measuring and presented in the form of graphs,
Calculators and computers are essential tools for teaching, learning and diagrams, charts and tables.
doing Mathematics. The potential that ICT has in making a significant
contribution to pupils" learning in mathematics has already been referred to Students should therefore be provided with opportunities that help them
in Section 5 of this document and need not be amplified any further. acquire a feel for the size of a number and know where it fits into the
However, as with any teaching tool, it can be used well or poorly. Teachers number system
therefore have to select or create mathematical tasks that take advantage of recall number facts (number bonds, multiplication tables, doubles and
what this new technology can do efficiently and well. halves)
figure out answers mentally
use a range of calculation strategies to calculate accurately and
e) DIFFERENTIATED TEACHING efficiently, both mentally and with pencil and paper
appreciate when it is appropriate to use a calculator
All students are entitled to learn Mathematics. To achieve this goal
check whether their answers are reasonable by using different strategies
expectations for students" learning must be raised. Teachers are therefore
required to develop effective methods to support the learning of mathematics make sensible estimates of measurements
by all students. This does not mean that every student should receive interpret graphs, diagrams, charts and tables and make predictions from
identical instruction; it means that reasonable and appropriate adjustments the information these display.
be made to promote access and attainment for all students.
Some students may need further support. For example, some students may g) ASSESSMENT
benefit from oral rather than written assessments. Others need more time to
complete certain tasks. Others may need additional resources and individual The purpose of assessment is not simply to certify the students" attainment at
attention. Those students with special interests or exceptional talent in the the end of a series of lessons. It also serves to inform teachers about their
subject may need enrichment material to challenge and engage them. The effectiveness. Consequently assessment can help teachers make important
school must therefore take care to accommodate the special needs of some decisions regarding their teaching and eventually adjust their teaching to
students without keeping back the learning of others by providing the enhance the students" learning. Very often assessment is based on tests or
necessary human and material resources. homeworks. Although these do contribute towards assessment, they are not
the only methods that can provide feedback. For example, through the use
of good tasks involving investigative work the students" level of proficiency
f) NUMERACY can be determined. Classroom discussions in which students present and
evaluate different approaches to the solution of a complex problem may also
One of the outcomes of this curriculum is to produce citizens who have be utilized to good effect. Other informal means, such as open-ended
confidence and competence with numbers and measures. They should be questions, performance tasks, observations and conversations, journals and
able to understand the number system, possess a repertoire of computational portfolios, can also give the teacher information about the students"
skills and be capable of solving number problems in a variety of contexts. progress. All the feedback that the teacher acquires from these different
forms of assessment can help the teacher to decide, for example, how and
Secondary Schools Mathematics Syllabus - September 2003 72
Mathematics Section – Curriculum Unit – The Mall – Floriana – Malta
when to revisit a particular topic or how to adapt tasks for students who are 7. ANNUAL EXAMINATIONS
either struggling or need enrichment or perhaps a challenge.
The Annual Examination will consist of two papers:
Formal assessments provide only one viewpoint on what students can do in a
very particular situation. Excessive reliance on such assessments may give i) Forms 1 to 3: A written non-calculator paper consisting of 10
an incomplete and perhaps a distorted picture of the students" performance. questions to be answered in 10 minutes, carrying a total of 10 marks.
Teachers need to be aware that different students show what they know and
can do in different ways, so when various forms of assessment are used each Forms 4 & 5: A written non-calculator paper consisting of 20
student will be allowed to show his or her best strengths. questions to be answered in 20 minutes, carrying a total of 20 marks.
Rulers, protractors and any other mathematical instrument will not be
allowed. Questions will typically involve number calculations,
approximations, estimations, data and graphical interpretations, application
of formulae, recall and application of properties of shapes and recall and
application of mathematical facts. To answer these questions, particularly
those involving numerical calculations, students are advised to choose and
use the more efficient techniques (mental and pencil and paper). They are
expected to have a range of strategies to aid mental calculations of unknown
facts from facts that can be rapidly recalled.
ii) Forms 1 to 3: A written paper consisting of 15 questions to be
answered in 1 hour 50 minutes carrying a total of 90 marks. Five
questions will carry 4 marks each, another five questions will carry 6
marks each and the remaining five questions 8 marks each. The use of
calculators is only allowed in forms 2 and 3.
Forms 4 & 5: A written calculator paper consisting of not more
than 13 questions to be answered in 1 hour 40 minutes, carrying a total
of 80 marks.
Besides testing knowledge, skills and understanding, questions testing the
use of ICT may also be set (e.g. to write down or complete a set of LOGO
instructions to draw a rectangle.)
The difficulty levels of the questions will be roughly set as follows:
Low 25 30%; Medium 40 – 45%; High 25 – 35%.
The syllabus content in forms 2 to 5 papers will be roughly 70% from the
core and 30% from the extension part.
Secondary Schools Mathematics Syllabus - September 2003 |
Transition to Advanced Mathematics--This text refers to an out of print or unavailable edition of this title.I've honestly never understood the negativity thrown towards this book.
This book introduces basic proof-writing techniques, naive set theory, cardinality, functions, and also includes a very brief introduction to abstract algebra and real analysis. I thought that the section on proofs was quite well done even though there are better textbooks which cover this topic more extensively with more examples. The examples are generally simple and are pulled almost exclusively from number theory, but they provide understanding. The set theory chapter is not very rigorous, but it meets the goals of the type of student who would take such a class. A more rigorous (i.e. axiomatic) introduction to set theory would require an entire semester, and the goal of this book is simply to provide students with just enough tools to prepare themselves for classes like Abstract/Modern Algebra, Real Analysis, and Set Theory, but not much more. Also, I consider the proofs to be generally well-written although there are occasional mistakes.
The chapters introducing abstract algebra and real analysis don't provide much information, but only serve to give students a taste of the fields. I felt that they should have either been omitted (since you won't learn much) or they should have been introduced with more rigor and explanation.
Some people have complained that the exercises were too hard. I felt that the opposite was true. All of the problems are either trivial or of medium difficulty. I think that the problem was that the proofs presented in the text were even easier. The proofs are generally well-written, but it would have been nice to see more challenging proofs to give the student a better idea of how to work the more challenging homework problems.
It's not the best or the worst text that you can find on this subject. At minimum, it's at least underrated (though overpriced).
20 of 25 people found the following review helpful
4.0 out of 5 starsRead Carefully and Think!26 Jan 2006
By Erik Wallace - Published on Amazon.com
Format:Hardcover
Firstly, I must clarify: other reviewers have said that the book claims 1 + 2 does not equal 3. They are not reading carefully enough. I have the 5th edition, and this is exactly what it says (on page 18 for the 5th edition).
"The sentence 'x1 is equal to x2 + x3' is an open sentence with three variables. If we denote this sentence by P(x1,x2,x3), then P(7,3,4) is true since 7 = 3 + 4, while P(1,2,3) is false."
Now to translate for them, in the case of P(1,2,3)
x1 = 1, x2 = 2 and x3 = 3. Thus the sentence says 1 = 2 + 3 which is false, not 1 + 2 = 3 as other reviewers have claimed. What is confusing is the order and the abstract expression. In the order given P(3,1,2) would say 3 = 1 + 2.
Now as for my rating, I do not claim that the book is ideal. But, it does fill a gap in math education, and there are not many other books like it. A good replacement would be an introduction to logic, or to set theory, but this book contains both. I hope we will see more books like this in the future, and if we do then we may eventually find one worth bragging about.
Currently, though, there is no such book that I know of, and believe me I have seen a lot of books. In the meantime this book can serve as a valuable reference. So, as per my title: read carefully and think! If you do that, then the contents of this book should be crystal clear to you.
7 of 8 people found the following review helpful
4.0 out of 5 starsNot a Bad Textbook9 Nov 2006
By Alethio Grapher - Published on Amazon.com
Format:Hardcover|Amazon Verified Purchase
We've been using this in our Introduction to Mathematical Proofs class (an upper-level undergraduate math class). I was a little nervous after seeing so many bad reviews here, but it's been a fine textbook. It's easy to learn the material from the book, and the question/problem sets are useful too. I don't have any other books of this type to compare it to, of course, but the layout of the text and material is clear, with definitions and properties given as needed. |
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Bill Harmon has loved science for as long as he can remember. After completing his B.S. in Chemistry, he returned to Florida where he gained experience teaching a variety of subjects: science, math, Latin, and computer courses. Now he works as a chemist in the Safety Services Office at BJU, teaches Distance Learning Physics and Algebra, and teaches Chemistry at Bob Jones Academy. He is currently pursuing an M.Ed. in Secondary Education. He and his wife Mary Ann have two children, Brian and Janette. His favorite Bible verse is II Timothy 3:14.
Author - BJU Press Distance Learning
Copyright Year - 2010
Format - Distance Learning by DVD
Grade Level - 7,8
Publisher - BJU Press
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Fundamentals of Math DVD Only
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Overview
MATHEMATICAL EXCURSIONS, Third Edition, teaches students that mathematics is a system of knowing and understanding our surroundings. For example, sending information across the Internet is better understood when one understands prime numbers; the perils of radioactive waste take on new meaning when one understands exponential functions; and the efficiency of the flow of traffic through an intersection is more interesting after seeing the system of traffic lights represented in a mathematical form. Students will learn those facets of mathematics that strengthen their quantitative understanding and expand the way they know, perceive, and comprehend their world. We hope you enjoy the journey.
Meet the Author
Richard Aufmann is the lead author of two bestselling developmental math series and a bestselling college algebra and trigonometry series, as well as several derivative math texts. He received a BA in mathematics from the University of California, Irvine, and an MA in mathematics from California State University, Long Beach. Mr. Aufmann taught math, computer science, and physics at Palomar College in California, where he was on the faculty for 28 years. His textbooks are highly recognized and respected among college mathematics professors. Today, Mr. Aufmann's professional interests include quantitative literacy, the developmental math curriculum, and the impact of technology on curriculum development.
Joanne Lockwood received a BA in English Literature from St. Lawrence University and both an MBA and a BA in mathematics from Plymouth State University. Ms. Lockwood taught at Plymouth State University and Nashua Community College in New Hampshire, and has over 20 years' experience teaching mathematics at the high school and college level. Ms. Lockwood has co-authored two bestselling developmental math series, as well as numerous derivative math texts and ancillaries. Ms. Lockwood's primary interest today is helping developmental math students overcome their challenges in learning math.
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APSC 171
WORKBOOK
Interactive Class Notes
prepared by
Leo Jonker
with the assistance of
Cesar Aguilar
Queen's University
Fall 2008
Foreword
These are so-called "incomplete class notes", or "interactive class notes",
prepared for your use in APSC 171. The notes reflect the way in which the
instructors will present the course material in the lectures. Gaps have been
left in the notes, some as blanks in formulas or text, others as problems or
unanswered questions. These blanks should be filled in by you as you study
the material in this course.
What's our reason for using interactive notes? They work especially
well if the instructor uses transparencies instead of the blackboard to present
the material. Transparencies are a good presentation medium. The organi-
zation of material is often clearer on transparencies than on blackboard; it is
easier to bring back examples, definitions and other refresher material; visual
cues (colour, size, point-form presentation) can help to high-light what is im-
portant; diagrams better because they have been prepared beforehand; the
instructor can face the class and stay aware of student reaction. The main
purposes of the interactive notes, however, is to transfer some of these same
advantages to you, the student. In fact, even in a class taught on the black-
board, the use of interactive notes should enhance the learning experience.
Interactive notes eliminate the need for tedious copying from the board or
overhead when you should be thinking. They also eliminate the frustration
of trying to, in a few minutes, copy a diagram that took the teacher a long
time or a lot of practice to make. The second purpose of interactive notes
is to encourage student involvement beyond passive listening and reading.
By reading the notes before class and trying to fill in the gaps and doing
the problems you will become committed to the material in a way that will
enhance the lectures.
i
ii
How should you use these notes? They are intended to complement
the textbook and to highlight what is important. By their organization into
weeks they also provide a guideline indicating approximately what we can
expect to cover each day. Read ahead in the text - even browsing will
be useful. Then look at the notes, and try to answer some of the
questions before you come to class. Bring the notes to class - you will
need them. Treat the notes as a workbook. At the end of the course, with
the blanks filled in and perhaps with the addition of some material of your
own, these interactive notes will constitute your class notes for the course.
Use the extra blank pages or the margins to write down your own questions.
Ask these questions in class. You can be sure that if a question seems
important to you it will seem so to at least ten others in the class. Additional
"Notes" pages are included at the end of each week for you to write things
that come up in class but are not part of the notes (such as additional
examples used by your instructor). You may also wish to bring to class some
coloured pens or pencils to highlight important ideas. Finally, keep a list
of your comments about the interactive notes themselves and let us know
what they are, so that suggestions for improvement can be incorporated for
the benefit of future classes.
The textbook: The textbook for the course this year is Calculus Early
Transcendentals, Edition 6e, by James Stewart. This is a new edition. Last
year we used Edition 5e. All references in these notes will be to the new
edition, followed by the corresponding reference to the old edition given in
square brackets.
Contents
Material to review on your own 1
Week 1 3
Course Unit: How do we describe a moving object? . . . . . . . . 4
1.1 Four Ways to Represent a Function . . . . . . . . . . . . . . . 5
10.1 & 13.1 Other Types of Functions . . . . . . . . . . . . . . . . 9
13.2 Velocity and Parametric Curves . . . . . . . . . . . . . . . . . 14
Week 2 27
3.3 [3.4] Derivatives of Trigonometric Functions . . . . . . . . . . . 28
13.2 & 13.4 Velocity and Acceleration . . . . . . . . . . . . . . . . 37
Week 3 49
Course Unit: More About Derivatives . . . . . . . . . . . . . . . . . 50
1.6 Inverse Trigonometric Functions . . . . . . . . . . . . . . . . . 54
3.5 [3.6] Implicit Differentiation . . . . . . . . . . . . . . . . . . . 62
3.6 [3.8] Logarithmic Differentiation . . . . . . . . . . . . . . . . . 67
Week 4 69
iii
iv CONTENTS
3.7 [3.3] Rates of Change in Science . . . . . . . . . . . . . . . . . 70
3.10 [3.11] Differentials and Linear Approximations . . . . . . . . 76
4.4 Indeterminate Forms and l'Hopital's Rule . . . . . . . . . . . . 85
Week 5 93
Course Unit: Integration . . . . . . . . . . . . . . . . . . . . . . . 94
5.1 & 5.2 What is an Integral? . . . . . . . . . . . . . . . . . . . . 96
Riemann Sums . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Area Under a Graph . . . . . . . . . . . . . . . . . . . . . . . 101
Week 6 117
5.3 The Fund. Theorem of Calculus, Part 2 . . . . . . . . . . . . . 118
5.2 Properties of Integrals . . . . . . . . . . . . . . . . . . . . . . . 126
5.4 The Indefinite Integral . . . . . . . . . . . . . . . . . . . . . . 131
Week 7 141
5.5 Substitution Rule . . . . . . . . . . . . . . . . . . . . . . . . . 142
6.1 Areas between Curves . . . . . . . . . . . . . . . . . . . . . . . 150
Week 8 159
6.2 Volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.3 Volumes by Cylindrical Shells . . . . . . . . . . . . . . . . . . 167
6.4 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Week 9 185
7.1 Integration by Parts . . . . . . . . . . . . . . . . . . . . . . . . 186
CONTENTS v
7.4 Partial Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . 193
Week 10 209
7.7 Approximate Integration . . . . . . . . . . . . . . . . . . . . . 210
7.8 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 216
Week 11 229
Course Unit: Differential Equations . . . . . . . . . . . . . . . . . . 230
9.1 Modeling with Differential Equations . . . . . . . . . . . . . . 231
Exponential Growth (see also 3.8 [9.4]) . . . . . . . . . . . . . 231
Harmonic Motion . . . . . . . . . . . . . . . . . . . . . . . . . 236
Week 12 249
9.3 Separable Differential Equations . . . . . . . . . . . . . . . . . 250
9.4 [9.5] The Logistic Differential Equation . . . . . . . . . . . . . 259
How to study in this course
Some of you, perhaps most of you, come in with good study habits. All
of you are excellent mathematics students - you must be, to get the marks
needed for admission to Queen's. However, being good at mathematics in
high school can be a real trap, for it can make you complacent about study
habits, and give you the false impression that you can rely on your facility for
calculations. You will find that in your university mathematics courses, the
volume and the pace is such that you cannot afford to do things in a hurry.
The only way to maintain the pace is to really understand the fundamental
ideas very well. Here are some do's and dont's that will help you achieve
this:
• Do not study the material by doing the homework questions, and then
checking the text only when you get stuck. Read the week's section of
the text first. Then read the interactive notes, and try the classroom
questions before you get to class. Go to the tutorials. Only after that
should you attempt the homework questions, as a final check on your
understanding.
• Read definitions and statements of theorems carefully. The powerful
ideas in mathematics are often subtle and depend on the precise use of
terms.
• Do not imagine that the course is about memorizing a bunch of for-
mulas. If you do not understand the concepts behind the formulas you
have gained very little.
• Do not do lots and lots of problems on a topic. It is much better to do
two or three and then to reflect on them.
Material to review on your own
Study Sections: 1.2, 1.3, 1.5, 1.6, 2.1-2.3 (to page 103
[108]), 2.7-2.9, 3.1, 3.2, 3.4 [3.5], and 3.6 [3.8] in the textbook
• Basic Functions and their Graphs (Polynomial, Power,
Rational, Trigonometric, Exponential, Logarithmic)
• Inverse Functions
• Derivatives and Rates of Change
• Derivatives of Polynomials, Exponential and Logarithmic
Functions
• Differentiation Rules (Product, Quotient, and Chain Rules)
The topics listed above represent what we expect you to know from the
start of this course. All of the topics are covered in the Ontario high
school curriculum, so in principle you should not need to review any of
this material. If you have a weak background in differential calculus, then
you should have reviewed the Quick and Easy Differential Calculus
Notes this summer, available on the APSC 171 website. These notes
introduce differential calculus in a highly intuitive manner, which fits well
with the tenor of the rest of the course. They also review some other
important topics, such as inverse functions and logarithmic functions, of
which we judged you were most likely to be unsure.
1
2 MATERIAL TO REVIEW ON YOUR OWN
Week 1
Study Sections: 1.1, 10.1 and 13.1 in the textbook
• Course Unit: Describing Moving Objects
• Different Ways of Representing Functions
• Vector-valued Functions, or Parametric Curves
3
4 WEEK 1
Course Unit: How do we describe a moving
object?
The first unit of the course will concern the description and study of moving
objects. These objects will move, not only along a line, but also in two-
dimensional and three-dimensional space. We will try to relate formulas to
geometric descriptions, and to problems of velocity, force and acceleration.
While this is happening, you will find that the skills you are learning in the
calculus course complement the mechanics you are studying in the first few
weeks of your physics course. By having one subject reinforce the other, we
hope that both will benefit.
To start this unit off, we begin with a problem, which we will come back to
at the close of the unit. Here is that problem.
Problem to be solved at the end of week 2, at the end of the first
unit of the course1 FOUR WAYS TO REPRESENT A FUNCTION 5
1.1 Four Ways to Represent a Function
A function is a rule or process that assigns to each
input a corresponding output.
A function can come in a variety of forms, as the following example will
illustrate.
Example 1. Suppose we have a wheel of radius 1, with a red dot at the top
of the wheel's rim and a green dot at the extreme right. If we turn the wheel
by an angle θ, the heights f (θ) and g(θ) of the two dots above (or below) a
horizontal line through the centre of the wheel vary with the angle. Discuss
the functions f (θ) and g(θ).
R
1 G
6 WEEK 1
1.1 FOUR WAYS TO REPRESENT A FUNCTION 7
8 WEEK 1
This example shows that there are (at least) four ways to represent a
function:
• Verbal description
• Table
• Graph
• Formula
A function is not necessarily given by a formula.
In every one of the descriptions provided, the function involves two varying
quantities, with one of them, the input, giving rise to, or "causing", the
other one, which is then called the output.
For this reason, a function may be thought of as a machine with input and
output.
x f (x)
input
f output
The input variable is also referred to as the "independent variable" and the
output as the "dependent variable".
In the above example the term "machine" can be taken quite literally: You
can imagine turning a crank that makes the wheel go around and stop at
the angle (the input) you choose, and then reading off the y-coordinate of
the green (or red) dot. This y-coordinate is the output. In other cases the
machine picture may have to be understood in a more figurative sense.
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 9
10.1 & 13.1 Other Types of Functions
The functions familiar to us from our high school courses are all functions
for which both the input and the output are single numbers. We do not have
to restrict ourselves to functions of that type, however.
Functions of several variables
For one thing, there is no reason why a function might not require more
than one input. For example, if we want to think of the temperature in
Ontario at noon today, the answer cannot be specified unless we are given a
precise location. That is, we need both a longitude (an x-coordinate) and a
latitude (a y-coordinate) before we can say what the temperature is at the
point (x, y). In this setting, the temperature is the output of a function of
two variables:
T = f (x, y) .
We can still think of this function as a machine, but this machine works only
if the input you give it is a pair of numbers:
(x, y) f (x, y) = T
input
f output
2 in R
in R
We will be studying such functions extensively in the second term.
The domain of a function is the set of inputs the function will accept (the
coordinates corresponding to locations in Ontario in this case). The range
is the set of the outputs it can produce (that is, the range of temperatures
in Ontario at noon today).
10 WEEK 1
Vector-valued functions
We can also vary the type of function under discussion by taking a single
number as an input, but admitting a pair of numbers as the output:
t r(t) = (x(t), y(t))
input
r output
in R in R2
This is precisely what we get if we think of the moving green dot in the
problem we have been discussing:
Example 2. How do the coordinates of the green dot vary as θ varies?
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 11
Suppose the dot is rotating at an even rate, say at one revolution per 2π
seconds. Then at time t the location of the dot is given by coordinates
(cos t, sin t), for at one revolution per 2π seconds, the angle θ (measured in
radians) is always precisely equal to the elapsed time. We can think of the
moving location G of the green dot as a vector-valued function
r(t) = (cos t, sin t) .
The input is time (t) and the output is a location (x, y) = (cos t, sin t), or
equivalently, the position vector cos t, sin t of that location. Another term
for such a function is (parametric) plane curve. This is a natural way to
describe it if we think of the green dot as a moving particle whose position
depends on the parameter time. Notice that we can also, if we want, describe
this vector-valued function by means of the two (ordinary) functions giving
the coordinates separately:
x(t) = cos t
y(t) = sin t
12 WEEK 1
Concept Question 1. If we wanted to write down the displacement of the
green dot between times t = 0.5 and t = 1.6, which of the following would be
the right expression?
A. cos(1.6 − 0.5), sin(1.6 − 0.5) ;
B. sin(0.5), cos(0.5) ;
C. cos(1.6) − cos(0.5), sin(1.6) − sin(0.5) ;
D. cos(1.6), sin(1.6) − cos(0.5), sin(0.5) .
10.1 & 13.1 OTHER TYPES OF FUNCTIONS 13
Example 3. Sketch and identify the curve defined by the parametric equa-
tions, x(t) = t2 − 2t and y(t) = t + 1. What is the displacement of a particle
moving along this curve from t = 0 to t = 3? Sketch this vector on your
diagram.
14 WEEK 1
13.2 Velocity and Parametric Curves
Just as we learned how to calculate the derivative of a function whose output
is a single real number, we also want to discuss the derivative of a function
whose output is a point in the plane (or, equivalently, the position vector for
that point). To do this we consider what we mean by velocity. We define
velocity as the rate of change of position. That is, it is obtained by dividing
a displacement by the length of time needed to produce that displacement.
For example, using again the function
r(t) = cos t, sin t ,
if we divide the displacement between times 0.5 and 1.6 by the length of that
time period, we get
1
cos(1.6) − cos(0.5), sin(1.6) − sin(0.5) .
1.6 − 0.5
This can be thought of as the average velocity of the dot between times
0.5 and 1.6 because if we traveled with this constant velocity for a time
period of 1.1 (i.e. from t = 0.5 to t = 1.6), we would produce exactly the
desired displacement. Notice that this is a vector in the same direction as
the displacement vector, but multiplied by the scalar 1/1.1. In other words,
this vector is a bit shorter than the displacement vector. Of course the actual
velocity of the dot changes continuously as it moves along the circle. We will
discuss this motion next.
13.2 VELOCITY AND PARAMETRIC CURVES 15
Concept Question 2. Suppose we replaced 1.6 by 0.5 + h and calculated
the average velocity. It would give us
1
cos(0.5 + h) − cos(0.5), sin(0.5 + h) − sin(0.5) .
h
If now we let h get smaller and smaller (h → 0) then this vector indicating
average velocity will
A. Shrink to a point, because the displacement vector gets smaller and smaller;
B. Get longer and longer because we are dividing the displacement by a
smaller and smaller number h;
C. Stay more or less the same length.
16 WEEK 1
We would like to study this process more closely. It is key to understanding
the concept of derivative for a vector-valued function (that is, for a plane
curve). Writing a for 0.5, for convenience, the average velocity becomes
1
[r(a + h) − r(a)]
h
which can also be written as
r(a + h) − r(a)
.
h
Our question about what this becomes as h → 0 is a question about
r(a + h) − r(a)
lim .
h→0 h
This is the definition of the derivative at a of the vector-valued function
r(t).
Notice that it looks the same as the definition of the derivative of an "ordi-
nary" function. The difference is that the numerator is a vector and not a
scalar quantity. If you imagine successive instances of average velocity as h
gets smaller and smaller, you might get something like this diagram:
In the diagram you see two vectors representing average velocities between
times 0.5 and 1.6 (h = 1.1) and between 0.5 and 0.8 (h = 0.3), as well as the
limit (the dashed arrow) to which these average velocity vectors converge as
13.2 VELOCITY AND PARAMETRIC CURVES 17
h → 0. This limit vector is called the derivative of the vector-valued func-
tion r(t) at t = a. This derivative represents the instantaneous velocity
of r(t) at t = a, and we write it as r′ (a):
1
r′ (a) = lim [r(a + h) − r(a)]
h→0 h
Example 4. Suppose a vector-valued function is given by the formula r(t) =
t, t2 . What is its derivative at time t = a?
18 WEEK 1
The length of the velocity vector is the speed.
The example on page 17 demonstrates a general principle:
When a vector-valued function is given by the formula
r(t) = x(t), y(t)
then its velocity at a can be calculated by taking derivatives of the
components:
r′ (a) = x′ (a), y ′ (a)
When a vector-valued function arises in a problem as the path of a moving
particle, it is sometimes helpful to draw that path. To do so is, in effect,
to draw the range of the function. One way to do that is to plot the points
(x(t), y(t)) for a set of values of t and then to connect these with a smooth
curve. Another method that is often useful is the method of eliminating
the parameter. To do this for the vector-valued function r(t) = t, t2 ,
begin with the two formulas
x=t
and
y = t2 ,
and use algebra to eliminate the parameter t. You may have used the same
idea already when you solved Example 3.
13.2 VELOCITY AND PARAMETRIC CURVES 19
Example 5. Use elimination of the parameter to describe the path r(t) =
t, t2 .
You can generate a picture of this curve (without the velocity vectors) very
easily, using the Maple command
> plot([t,tˆ2, t=−3.5..3.5]);
20 WEEK 1
Concept Question 3. A particle moves so that its location at time t is
1
given by r(t) = ( t3 − t, t), where the first coordinate measures distance
3
along a horizontal axis and the second coordinate measures vertical distance.
At what value(s) of t is the particle moving in a perfectly vertical direction?
A. At t = 1
B. At t = 0
√
C. At t = 0 and t = ± 3
D. At t = ±1
Parametric curves (or space curves) in three dimensions
A vector-valued function r(t) can also have a triple of numbers as its output,
as in r(t) = (x(t), y(t), z(t)). We can think of this triple as a set of coor-
dinates in three-dimensional space, so (x(t), y(t), z(t)) can be thought of as
describing the path of a moving point in three-space. For this reason, the
curve consisting of all outputs (x(t), y(t), z(t)) is also called a space curve.
Of course we can also think of r(t) as the position vector of that point. In
that case we would probably write r(t) = x(t), y(t), z(t) .
Paths in three-space are handled in the same way as paths in two-dimensional
space, especially when it comes to calculating the velocity.
13.2 VELOCITY AND PARAMETRIC CURVES 21
Example 6. A particle moves in such a way that its position at time t is
1 t − 1 2t
given by r(t) = 2 , , e . What is its velocity at time t = 1? Also,
t +1 t−5
what is the domain of this vector-valued function?
22 WEEK 1
Example 7. What does the particle do when t gets close to 5?
13.2 VELOCITY AND PARAMETRIC CURVES 23
For the preceding two questions it was important to understand interval
notation; let's review it here. (See also Appendix A in the textbook.)
(2, 7) means: { x | 2 < x < 7 };
[−5, −3) means: { x | − 5 ≤ x < −3 };
(−7, ∞) means: { x | − 7 < x < ∞ }.
Never write (−7, ∞], for ∞ is not a number, so you cannot include it at
the end of an interval.
Basic set notation is also discussed in Appendix A in the textbook: Consider
the sets S1 and S2 , let S3 be the set consisting of the elements of S1 together
with those of S2 , or S3 = S1 ∪ S2 ; S1 ∪ S2 is called the union of S1 and S2 .
S1 S2
We can also have a set with its elements consisting of those that belong to
both S1 and S2 . This is called the intersection of S1 and S2 , and can be
written as S4 = S1 ∩ S2 .
S1 S2
x ∈ S means "x is a member of the set S ".
24 WEEK 1
Example 8. When I sketch the curve
r(t) = cos(t) sin(t3 ), cos(t) cos(t3 ), sin(t)
using the Maple commands,
> with(plots):
> spacecurve([cos(t)∗cos(tˆ3),cos(t)∗sin(tˆ3), sin(t)], t=-4..4, num-
points=1000);
I get the following picture, which makes it look as if that curve stays on a
sphere centered at the origin. Prove that this is the case.
13.2 VELOCITY AND PARAMETRIC CURVES 25
Notes
26 WEEK 1
Week 2
Study Sections: 3.3 [3.4], 13.2 (first three pages) and
13.4 (first 4 pages) in the textbook
• Derivatives of Trigonometric Functions
• Acceleration and Parametric Curves
• Application to Motion Problems
27
28 WEEK 2
3.3 [3.4] Derivatives of Trigonometric Func-
tions
We would like to calculate the velocity of the moving green dot
r(t) = (cos t, sin t)
in Example 2. In order to do that we have to know how to calculate the
derivative of the functions sin and cos.
To begin our discussion of derivatives of these trigonometric functions, we
want to do three things first:
1. Define the "secondary trigonometric ratios" secant, cosecant and cotan-
gent,
2. Introduce the formulas for the sine and cosine of the sum of two angles,
YOU MUST MEMORIZE THESE
sin t
3. Investigate what happens to the expression as t gets closer and
t
closer to 0 (that is, as t → 0).
Introducing the secondary trigonometric ratios is best done using a pic-
ture. We will use x instead of t to denote the input variable.
c b
x
a
cos x = a/c sec x = c/a = 1/ cos x
sin x = b/c csc x = c/b = 1/ sin x
tan x = b/a = sin x/ cos x cot x = a/b = cos x/ sin x = 1/ tan x
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 29
We also need a formula that expresses the sine of the sum of two angles in
terms of the sines and cosines of those angle. Here is that formula:
Theorem W2.1 (see text pg. A29)
sin(x + y) = sin x cos y + cos x sin y
To see why this is so, consider a circle of radius 1, with angles x, y, w and z
drawn in as shown:
P
z
O x 1 R
y
w
Q
Note that P Q is perpendicular to OR.
Example 9. Finish the following calculations to prove the identity:
It is easy to see that
PR sin x
PR = = tan x =
1 cos x
QR
QR = = =
1
PQ =
OQ =
sin z = cos ( )
30 WEEK 2
Now use the Sine Law to calculate
sin(x + y)
=
tan x + tan y
Next multiply both sides by tan x + tan y and get
sin x sin y
sin(x + y) = cos x cos y + =
cos x cos y
Theorem W2.2 (see text pg. A29)
cos(x + y) = cos x cos y − sin x sin y
This identity follows from the one we just proved if we keep in mind that
if z = π − x as in the preceding discussion, then cos z = sin x. That is,
2
cos(π/2−x) = sin x and cos x = sin(π/2−x) The rest is just some calculation:
cos(x + y) = sin(π/2 − (x + y)) =
These theorems have the following simple corollary:
Corollary W1.1 (see text pg. A29)
sin(2x) = 2 sin(x) cos(x)
cos(2x) = cos2 (x) − sin2 (x)
MEMORIZE THESE FORMULAS
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 31
Theorem W2.3 (see text pg. 190 [212])
sin θ
lim =1
θ→0 θ
A rigorous proof of this limit is given in the textbook on pages 190-191 [212].
Here we will find this limit experimentally:
sin x
Consider the graph of the function f (x) = . Apart from the little circle
x
on the y-axis, this is what we get if we plot it in Maple, using the command:
> plot(sin(x)/x, x=-1..1);
1
−1.0 −0.5 0 0.5 1.0
Notice that the function f (x) = sin x is undefined at x = 0, because with
x
x = 0 we cannot do the calculation. This is indicated by the little circle,
which was drawn in by hand - Maple does not draw it for you. Whether
or not 0 is in the domain of f (i.e. whether or not "f is defined at 0") has
nothing to do with the limit. The graph certainly suggests that
sin x
lim = 1.
x→0 x
32 WEEK 2
The following analogy may help explain what we mean by the limit: If the
graph in the diagram represents a road, and the vertical line through 0 a
canal, and if the little circle indicates that there is no bridge at that point,
then the limit is the y-value where you would fall into the canal if you came
along the road.
Another (also not totally reliable) way to show that this limit is equal to 1
is to use a calculator to test the function at values close to zero. (See text,
page 91 [93].)
x sin x/x
± 0.5 0.95885108
± 0.1 0.99833417
± 0.001 0.99999983
Maple will even check the limit directly for you: Simply enter
> Limit(sin(x)/x, x=0)=limit(sin(x)/x, x=0);
and press "return". (Here limit(...) does the calculation, while the "inert"
form of the command, Limit(...), gets Maple to write out the expression
sin x
lim ) Of course, none of these three methods is completely conclusive.
x→0 x
For a conclusive argument you have to go to the proof given in the text on
pages 190-191 [212].
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 33
Concept Question 4. What do you think the limit is likely to be in the
cos θ − 1
Corollary on the next page? That is, what do you think lim is equal
θ→0 θ
to, and why?
A. 0 because the numerator goes to zero as θ → 0
B. 1 because the limit in the theorem we just did was 1
C. ∞ because the denominator goes to zero as θ → 0
D. We cannot tell without doing some work.
A degree of confirmation of the answer can be obtained using Maple, either
by graphing the expression, using the command
> plot((cos(x)-1)/x, x= −2..2);
or by asking Maple to take the limit, using the command
> Limit((cos(x)-1)/x, x=0)=limit((cos(x)-1)/x, x=0);
Try it yourselves!
34 WEEK 2
Corollary W4.1 (see text pg. 192 [213])
cos θ − 1
lim =
θ→0 θ
Example 10. Prove the corollary using the theorem
3.3 [3.4] DERIVATIVES OF TRIGONOMETRIC FUNCTIONS 35
d
Example 11. Prove the following theorem: dx sin x = cos x. Note this
is where you will need the trigonometric identity sin(x + y) = sin x cos y +
cos x sin y.
36 WEEK 2
Table of Derivatives of Trigonometric Functions
Once you have the derivatives of the sine and cosine functions you can use
differentiation rules to deduce the others. Try a few to make sure you can
do it:
d
1. dx
sin x = cos x ;
d
2. dx
cos x = − sin x ;
d
3. dx
tan x = sec2 x ;
d
4. dx
csc x = − csc x cot x ;
d
5. dx
sec x = sec x tan x ;
d
6. dx
cot x = − csc2 x.
MEMORIZE THESE!
13.2 & 13.4 VELOCITY AND ACCELERATION 37
13.2 & 13.4 Velocity and Acceleration
We will now return to the green dot of Example 2. Its path was given, as a
function of time, by the formula
r(t) = (cos t, sin t) .
We are now in a position to calculate its velocity:
r′ (t) = − sin t, cos t .
Here are some of those velocity vectors:
1
0.5
0
-1 -0.5 0 0.5 1
-0.5
-1
Now what would it mean to take a second derivative of a vector-valued func-
tion? In physics we associate the second derivative of position with acceler-
ation. In calculus, in high school, you will have learned to relate the second
derivative to the concavity of the graph of a function. We will see that both
points of view continue to make sense when we deal with vector-valued func-
tions: the second derivative of the path of a moving particle measures how
38 WEEK 2
fast and in which direction a particle is accelerating; and, when the acceler-
ation vector points in a direction different from that of the velocity, it also
indicates the "amount" by which the path curves. Think of the green dot as
a vehicle with you as a passenger. As you zip around the circle, you will find
yourself pushed against the side of the vehicle. This indicates that you are
accelerating in the direction in which the vehicle pushes against your body.
A greater speed or a tighter circle will produce a greater acceleration
Consider the picture, drawn above, of the moving green dot and its velocities
at t = 0, t = π/4, and t = 3. The second derivative is always the derivative
of the first derivative, so we should think about the rate at which the veloc-
ity changes. To do this we could begin with an average rate of change
of velocity over a short time period. So consider the velocities at t = 0
and t = π/4. We have to calculate the difference between these two vectors
(subtract one from the other) and then divide that by the time increase. To
think of this geometrically is a little difficult, for to do it we first have to
place the two velocity vectors at the same point, as in the following diagram:
1
0.5
0
-1 -0.5 0 0.5 1
-0.5
-1
13.2 & 13.4 VELOCITY AND ACCELERATION 39
The vector on the right of the second figure is the velocity at t = 0:√ ′ (0) √
r =
0, 1 ; the vector on the left is the velocity at t = π/4: r′ (π/4) = −1/ 2, 1/ 2 .
The vector at the top of the second figure is their difference, the vector
r′ (π/4) − r′ (0). Its coordinates are easily calculated:
√ √
r′ (π/4) − r′ (0) = −1/ 2, 1/ 2 − 1 = −0.707, −0.293 .
This vector represents the change from the velocity at t = 0 to the velocity
at t = π/4. The (average) rate of change between these times is found by
dividing this change in velocity (a vector) by the change in time (a scalar):
1
−0.707, −0.293 = −0.9, −.373 .
π/4
By this estimate, the acceleration of the green dot at time 0 is given ap-
proximately by the vector −0.9, −.373 . To get instantaneous, rather than
average, acceleration we should turn this into a limit:
1 ′
r′′ (0) = lim (r (h) − r′ (0))
h→0 h
On page 18 we saw that this instantaneous rate is calculated by taking deriva-
tives of the components of the vector-valued function r′ (t) = − sin t, cos t .
Example 12. Calculate the acceleration at t = 0.
40 WEEK 2
If
r = (x(t), y(t))
then
r′ = x′ (t), y ′(t)
and
r′′ = x′′ (t), y ′′ (t)
Notice that in the case of the green dot describing the circle at a constant
rate of rotation, we have r′′ (0) = −r(0). The acceleration is towards the
center! It is easy to see that (for uniform circular motion) this is true at any
value of t, not just t = 0:
d2 d2
r′′ (t) = cos t, 2 sin t = − cos t, − sin t = −r(t) .
dt2 dt
Example 13. Suppose you are a passenger in the back seat of a taxi that is
speeding along so that its location at time t (seconds) is r(t) = (100t2 , 10t),
measured in meters. What does the path of the taxi look like; what is your
acceleration at time t = 0; and given that your mass is 70 kilograms, and
assuming that you forgot to put on your seat belt, with what force is the car
door pressing against you at that instant?
13.2 & 13.4 VELOCITY AND ACCELERATION 41
Concept Question 5. As t increases from 0 to 10, which of the following
alternatives best describes what happens?
A. The force of the door against your body stays the same
B. The force of the door against your body decreases, but the force of the
back of your seat against your body increases
C. The force of the door against your side remains the same, but you find
yourself sliding forward towards the back of the seat in front of you
The calculation of acceleration in three-dimensional space is exactly the same
as in two dimensions, for exactly the same reasons. To find the acceleration
all you have to do is differentiate each of the (three) components twice:
If the position of a moving particle is given by
r(t) = (x(t), y(t), z(t)) ,
then its acceleration is given by
r′′ (t) = x′′ (t), y ′′(t), z ′′ (t) .
42 WEEK 2
Example 14. The acceleration due to gravity is 9.8 meters per second per
second. Suppose that between times t = −10 and t = 10, measured in seconds,
a stunt plane follows the path
r(t) = (200t, 5t3, 800 − 5t2 )
without going upside down or banking significantly. Will the pilot ever be
lifted off his seat? Notice that in this problem, and in all cases un-
less indicated otherwise, the third component (the z-component)
is assumed to measure vertical distance
13.2 & 13.4 VELOCITY AND ACCELERATION 43
Example 15. Describe the parametric curve
r(t) = (2 cos t, sin t) .
44 WEEK 2
Concept Question 6. Given the answer to the preceding question, describe
the following parametric curve:
r(t) = (2 cos t, sin t, t) .
A. It is an ellipse tilted in the direction of the x-axis
B. It is an ellipse tilted in the direction of the y-axis
C. It is a helix
D. It is a curve in the plane that spirals outward
13.2 & 13.4 VELOCITY AND ACCELERATION 45
Solution of the problem stated at the start of this unit
We now complete the first unit in the course by revisiting the problem posed
at its beginning, on page 4 What does that tell us about the way this object moves?
If we ignore the last coordinate, and focus on the first two, we are effectively
looking at the projection of the path of the particle on the (x, y)-plane. In
other, words, if we imagine the sun shining from the "end" of the z-axis
(i.e. infinitely far away in that direction), then the first two coordinates
describe the movement of the object's shadow on the (x, y)-plane. But we
saw in earlier examples that (cos t, sin t) rotates around a circle of radius 1
centered at the origin at a constant rate of one revolution per unit time. The
z-coordinate ln(t + 1) simply tells us how far the object is above (or below)
the (x, y)-plane. Sketch the graph of ln t below:
3
2
1
0
-3 -2 -1 0 1 2 3
-1
-2
-3
46 WEEK 2
What is the effect of changing ln t to ln(t + 1)? Sketch the graph of ln(t + 1).
3
2
1
0
-3 -2 -1 0 1 2 3
-1
-2
-3
NOTE: You can plot both of these graphs in Maple, with the commands
> plot(ln(t), t=0..5, scaling=constrained);
> plot(ln(t+1), t=0..5,scaling=constrained);
A question that springs to mind is "if this function indicates the height of
the object at time t, what is the greatest height reached by it? " This is really
a question about the range of the function ln t. Does it have a horizontal
asymptote as t → ∞ or is every height eventually exceeded by the object?
From the origins of ln t as the inverse of the exponential function ex we know
that the range of ln t is the same as the domain of ex which, we know, is
the entire real line. Thus this object keeps rising as it circles, and eventually
reaches any height whatsoever, even though it rises more an more slowly, and
thus takes longer and longer to achieve a given increase in height.
2. What is its speed at, say, time t = 0?
13.2 & 13.4 VELOCITY AND ACCELERATION 47
3. When will you experience the greatest acceleration?
4. If you were to fall off at time t = 1, what would happen to you, especially
in the case of zero gravity?
48 WEEK 2
Notes
Week 3
Study Sections: 1.6, 3.5 [3.6], 3.6 [3.8] in the textbook
• Course Unit: More Derivative Techniques and Applications
• Inverse Trigonometric Functions
• Implicit Differentiation
• Derivatives of Inverse Trigonometric Functions
• Logarithmic Differentiation
49
50 WEEK 3
Course Unit: More About Derivatives
In the next two weeks, we return to functions of a single variable and ex-
pand our knowledge of differential calculus. This is a continuation of the
material that you should already know from high school, and which is cov-
ered in the Quick and Easy Differential Calculus Notes. Here we will learn
more derivative techniques and see how derivatives are used to model many
physical processes. We will also learn how derivatives can be used to approx-
imate functions; such approximations can be used to estimate experimental
measurement errors.
Let's begin with a question to motivate a discussion of inverse trigonometric
functions, the first topic in this section:
Example 16. A projectile is fired from the origin, with angle of elevation α
(radians) and speed 60 meters per second.
α
Assuming that air resistance is negligible and that the only force on the pro-
jectile is gravity (producing a downward acceleration of g = 9.8 meters per
second per second), answer the following questions:
1. Find a formula for the position of the projectile at time t.
2. How far down range will the projectile land?
3. At what angle α should the projectile be launched so that it will land
300 meters down range?
COURSE UNIT: MORE ABOUT DERIVATIVES 51
52 WEEK 3
COURSE UNIT: MORE ABOUT DERIVATIVES 53
54 WEEK 3
1.6 Inverse Trigonometric Functions
We will now apply our understanding of inverse functions to discuss inverse
trigonometric functions more carefully. We would like to be able to talk
about the inverse of the sine function, for example, as we did when we solved
Example 16. As you will have learned in high school, and as reviewed in
the Quick and Easy Calculus notes available through the course website, a
function must be one-to-one in order for it to have an inverse function. In
the case of the function f (x) = sin x, one can clearly observe by graphing
the function that it does not possess this property (does not pass horizontal
line test) and that it therefore does not have an inverse. How then can we
find an inverse for f (x) = sin x? What did the calculator think it was doing
when we used the sin−1 button? We get around this problem by analyzing
f (x) = sin x only from − π to π . The graph of the sin function from − π to
2 2 2
π
2
is shown in the following graph:
f (x) = Sin x
1
−π
2
π
2
−1
For convenience we could call this new function Sin (x), where
Sin (x) = sin(x)
provided − π ≤ x ≤ π .
2 2
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 55
Sin (x) satisfies the horizontal line test and therefore has an inverse function
which we call the inverse sine function and denote it as
sin−1 (x).
Note that
1
sin−1 (x) = .
sin(x)
In fact the practice of writing the inverse of the sine function as sin−1 is a
bad habit that we have to get used to. It is too well-established to change; in
particular, that is how the function is indicated on calculators. As mentioned
earlier, the inverse of sin is also often written as
arcsin(x) ,
and its graph is the following one:
π
2
f (x) = arcsin x
−1 1
−π
2
The domain of arcsin is: [−1, 1] .
The range of arcsin is: − π , π .
2 2
56 WEEK 3
Note that, as always, the graph can be produced using Maple. Simply enter
the command > plot(arcsin(x), x=-2..2, scaling=constrained);
Since arcsin undoes what sin does, and vice-versa, the following equations
are true, but only for the specified values of x:
sin−1 (sin x) = x, for − π ≤ x ≤ π
2 2
sin(sin−1 x) = x, for −1 ≤ x ≤ 1.
Example 17. What is the value of arcsin(0.5)?
Example 18. sin(−7π/5) = 0.951, so what is the value of sin−1 (0.951)?
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 57
The inverse of cosine is obtained by a calculation similar to the way the
inverse of sine was determined. We analyze cosine from 0 to π; this is shown
in the following graph:
1 f (x) = Cos x
0
π
−1
For convenience, we could call this new function Cos (x) where
Cos (x) = cos(x)
provided 0 ≤ x ≤ π.
Cos (x) satisfies the horizontal line test and therefore has an inverse function
which we call the inverse cosine function and denote it as
cos−1 (x),
noting that
1
cos−1 x = .
cos x
The inverse cosine function is also often written as
arccos x,
and it is graphed on the following page.
58 WEEK 3
f (x) = arccos x
π
−1 1
The domain of arccos is: [−1, 1].
The range of arccos is: [0, π].
Concept Question 7. When you enter cos−1 2 on your calculator, it ob-
jects. Why is that?
A. The numbers involved are too large for the calculator to handle
B. The calculator does not understand this business of taking the inverse
using only part of the cosine function
C. The cosine function does not really have an inverse
D. The number 2 is outside the domain of the function arccos
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 59
Concept Question 8. What is the value of arccos(cos(7π/6))?
A. 7π/6
B. 210
C. 5π/6
60 WEEK 3
The inverse of tangent is determined in the same way, only analyzing it from
− π to π , this is shown in the following graph:
2 2
f (x) = T an x
−π
2
π
2
As done before, we name this portion of the tan function T an (x), where
T an(x) = tan x
provided − π < x < π .
2 2
T an (x) satisfies the horizontal line test and therefore has an inverse, which
we call the inverse tangent function and denote it as
tan−1 x,
once again noting that
1
tan−1 x = .
tan x
1.6 INVERSE TRIGONOMETRIC FUNCTIONS 61
The inverse tangent function is also often written as
arctan x,
and its graph is the following one:
π
f (x) = tan−1 x
2
−π
2
This graph can be produced in Maple, using the command
> plot(arctan(x), x=-5..5, scaling=constrained);
If you want the horizontal asymptotes as well, use the command
> plot([arctan(x), Pi/2, -Pi/2], x=-5..5, scaling=constrained);
The domain of arctan is: (−∞, ∞).
The range of arctan is: (− π , π ).
2 2
Example 19. What is the value of arctan(1)?
62 WEEK 3
3.5 [3.6] Implicit Differentiation
The relation between an "input" x and "output" y may be given explicitly.
Here is an example: √
y = 1 − x2 .
This same relation can be given implicitly:
x2 + y 2 = 1.
When the relation is given implicitly, there are often several y's for a given x,
unless you add some information, e.g. y ≥ 0. Sometimes an implicit relation
can be turned into an explicit relation by algebra, as in the above example,
although usually this cannot be done! For example, consider the relation:
cos(x − y) = y sin x.
dy
You might think that to calculate dx you must first write y explicitly in
terms of x. In this section we learn that this is not necessary. Hence the
term "Implicit Differentiation".
Consider the following example:
√
x 1 + y + y x − 1 = x + 5.
In this relation, think of y as somehow dependent on x. That is y = y(x); in
other words y is a function of x. To help you think this way, you can write
the expression in this way:
√
x 1 + y(x) + y(x) x − 1 = x + 5.
Now differentiate both sides:
dy
1 dx dy √ 1 1
1 + y(x) + x · · + · x − 1 + y(x) · · √ = 1.
2 1 + y(x) dx 2 x−1
dy
Next, collect terms involving dx
and go back to writing y instead of y(x):
x √ dy y
√ + x−1 =1− 1+y− √ .
2 1+y dx 2 x−1
3.5 [3.6] IMPLICIT DIFFERENTIATION 63
dy
Solving for dx
gives:
√
dy 1 − 1 + y − 2√y
x−1
= √ .
dx √x + x−1
2 1+y
As in the example just demonstrated, it is typical of implicit differentiation
that the answer involves both x and y.
You can think of √
x 1+y +y x −1 = x +5;
or, to express it more completely in set notation,
√
{(x, y) ∈ R2 | x 1 + y + y x − 1 = x + 5} ,
as some sort of curve in the (x, y)-plane.
(2,3)
Of course, this curve is not presented to us as a parametric curve (as a
vector valued function). It may be possible to find a way to parametrize this
particular set of points, but it is probably difficult. Fortunately we do not
need to. Note that (x, y) = (2, 3) satisfies the equation, therefore (2, 3) lies
on the curve. We can calculate the slope at the point (2, 3) by substituting
dy
for x = 2 and y = 3 into the formula obtained earlier for dx . This works out
dy
to dx = − 5 . This tells us that the slope of the curve (that is, the tangent
3
line to the curve) at (2,3) is equal to − 5 . Note that we are not speaking of
3
a velocity here, for to have a velocity you first need a parametrization. If
we did have a parametrization for this curve - that is, if we had a particle
moving along this curve - then its velocity as it passes through the point
(2, 3) would lie along the tangent line whose slope we just calculated.
64 WEEK 3
Example 20. Find the points where the ellipse x2 − xy + y 2 = 3, crosses the
x-axis and show that the tangent lines at these points are parallel.
3.5 [3.6] IMPLICIT DIFFERENTIATION 65
Example 21. Let y = sin−1 x, and differentiate the equation sin(sin−1 x) =
d
x, or sin(y) = x, to get a formula for sin−1 (x).
dx
Notice that Maple calculates derivatives for you. In this case you can do it
in two ways: You can work with the function arcsin directly and enter
> D(arcsin);
or you can start with the expression arcsin(x) (that is; put the variable
in), and enter
> diff(arcsin(x), x);
66 WEEK 3
Here is a list of the derivatives of the inverse trigonometric functions. Mem-
orize the first three of these. They will be used a lot in the course.
d 1
sin−1 x = √ ;
dx 1 − x2
d 1
tan−1 x = ;
dx 1 + x2
d 1
cos−1 x = −√ ;
dx 1 − x2
d 1
sec−1 x = √ .
dx x x2 − 1
Example 22. Differentiate y = tan−1 (sin x).
3.6 [3.8] LOGARITHMIC DIFFERENTIATION 67
3.6 [3.8] Logarithmic Differentiation
dy
Example 23. Find when y = (x2 + 1)x .
dx
Maple is not fazed by logarithmic derivatives. Try
> diff((x∧2+1)∧x, x);
68 WEEK 3
Notes
Week 4
Study Sections: 3.7 [3.3], 3.10 [3.11], 4.4 in the text-
book
• Rates of Change in Science
• Differentials and Linear Approximations
• l'Hopital's Rule
69
70 WEEK 4
3.7 [3.3] Rates of Change in Science
A large part of the difficulty in applying mathematical methods to applied
problems is the matter of translating the English-language formulation of a
problem or of a scientific law into a mathematical sentence. It is especially
important to recognize a derivative when it appears in a discussion and to
know how to express it mathematically.
Example 24. "When water drains out of a tank, the rate of flow is propor-
tional to the square root of the volume of water left". Translate this into a
mathematical sentence.
3.7 [3.3] RATES OF CHANGE IN SCIENCE 71
Concept Question 9. Newton's Law of Cooling states that the rate at
which an object cools off (or heats up) is proportional to the difference between
the temperature of the object and the temperature of its surroundings. We
want to translate this into a mathematical sentence for a cup of coffee placed
in a room at 20 o C. What variable quantities must be given names so that
we can do this translation?
A. The temperature T of the object, and the time t.
B. The temperature T , the time t, and the rate R at which the object cools
off.
C. The temperature T , the time t, and the difference s between the tem-
perature of the surroundings and the temperature of the object
D. The temperature T , the time t, the difference s between the temperature
of the surroundings and the temperature of the object, and the rate R
at which the object cools.
72 WEEK 4
Example 25. Isothermal compressibility measures the compressibility of a
gas kept at constant temperature. It is given by the formula:
1 dV
β=− ,
V dP
(see page 200 in the text book), where V is volume and P is pressure.
• Why is there a derivative?
• Why a minus sign?
• Why the factor 1/V ?
3.7 [3.3] RATES OF CHANGE IN SCIENCE 73
Example 26. In a certain chemical reaction a substance A is transformed
into a substance B. Because the reaction is autocatalytic, the product B is
produced at a rate that is proportional to the product of the concentrations of
A and B. Assuming that the concentrations of A and B always add up to
K, answer the following questions:
• Translate this into a mathematical sentence.
• Sketch a rough graph of the concentration of B over time.
74 WEEK 4
Many laws of nature, when modeled by mathematics, take the form of a
differential equation. A differential equation is an equation that in-
volves an unknown function and some of its derivatives. To solve a
differential equation is to find a formula for the unknown function. There
will be a unit on differential equations at the end of this term.
Example 27. Under certain circumstances (abundant resources) the rate of
growth of a population is proportional to the size of the population. Translate
this into a differential equation.
3.7 [3.3] RATES OF CHANGE IN SCIENCE 75
Example 28. (Hooke's Law) Suppose a wooden block is attached to a hor-
izontal spring on a frictionless table. One end of the spring is fixed to the
table. Then the acceleration of the block is proportional to the distance by
which the spring is extended from its equilibrium position. Translate this into
a differential equation.
0
76 WEEK 4
3.10 [3.11] Differentials and Linear Approxi-
mations
When you look at a very small part of a (differentiable) function, it looks
linear.
• Zoom into a graph and it will look like a line.
• Zoom into a table (i.e. check nearby values) and the values go up by
(roughly) equal amounts.
Consider the function y = f (x). What do we mean by saying that f ′ (x0 )
is the slope of f at x0 ? It means that if we change x from x0 to x0 + ∆x,
resulting in a change of y from y0 = f (x0 ) to y0 + ∆y, then the ratio between
the changes is (approximately) f ′ (x0 ). In other words,
∆y ≈ f ′ (x0 )∆x .
The idea behind derivatives is that this approximate relationship becomes
precise in the limit as the changes in the variables go to zero. In other words,
as ∆x and ∆y get smaller and smaller, the ratio between them becomes more
and more precisely equal to f ′ (x0 ). Leibniz (see page 170 in the textbook),
who along with Newton invented calculus, liked to think of the quantities
as "infinitesimals" - infinitely small changes whose sizes can nevertheless be
compared to each other. To indicate their status as infinitely small limiting
quantities, we use the symbols dx and dy in place of ∆x and ∆y. They
are then referred to as differentials. Because they describe the limiting
situation, the relationship between them becomes exact:
dy = f ′ (x0 )dx .
You can also think of the relationship between ∆x, ∆y, dx and dy in terms
of the relationship between the graph of f and its tangent line at x0 : The
tangent line is what the graph becomes as you zoom in. Therefore, if we let
∆x = dx then we can think of ∆y as the corresponding change in the value
given by f , and dy as the corresponding increase given by the tangent line.
When we zoom in (that is when ∆x is very small) then
∆y ≈ dy.
The following diagram illustrates this relationship between the small changes:
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 77
y = f (x)
slope = f ′ (x0 )
dy
∆y
x0 x0 + ∆x
dx = ∆x
In summary, we have the equation relating the differentials
dy = f ′ (x0)dx.
and the equation relating small changes in the independent and dependent
variables:
∆y ≈ f ′ (x0)∆x.
when ∆x is small. Notice that in this formula we are merely repeating
something we have known from the time we first learned to differentiate:
∆y
∆x
≈ f ′ (x0).
78 WEEK 4
√
Example 29. Use differentials to get an approximate value for 4.03.
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 79
√
The method of using differentials to approximate 4.03 can be summarized
as follows:
√
• We have a function f (x) (in this case x) and a point a (in this case
4) where we know the value f (a).
• We have a nearby point x (in this case 4.03).
• We want an approximate value for f (x).
To solve this problem we let
∆x = x − a,
∆y = f (x) − f (a).
We know that
∆y ≈ f ′ (a) · ∆x.
Thus
f (x) − f (a) ≈ f ′ (a)(x − a).
That is
f (x) ≈ f (a) + f ′ (a)(x − a).
This is the linear approximation or the tangent line approximation of
f at a.
The right hand side,
L(x) = f (a) + f ′ (a)(x − a),
80 WEEK 4
is called the linearization of f at a. Think of this formula as a recipe for
a linear function L(x) which is very nearly equal to f (x) as long as x is close
to a. The ingredients that you have to supply are written in boldface letters.
Example 30. The function y = L(x) is a linear equation, so it represents a
line. What line is it and why?
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 81
Concept Question 10. If e.5 is approximated by using the tangent line
to the graph of f (x) = ex at (0, 1), and we know that f ′ (0) = 1, then the
approximation is1
A. 0.5
B. 1 + e.5
C. 1 + .5
1
We thank the Good Question project at for
this problem.
82 WEEK 4
Example 31. The edge of a cube was found to be 30 cm with a possible
error in measurement of 0.1 cm. Use differentials to estimate the maximum
possible error in computing the surface area of the cube.
3.10 [3.11] DIFFERENTIALS AND LINEAR APPROXIMATIONS 83
Concept Question 11. A function f and its derivative have the following
values at 0 and 10:
f (0) = 2 f (10) = 5
f ′ (0) = −1 f ′ (10) = 1
What is a good estimate for f (0.2)?
A. (0.02) × (5 − 2) + 2 = 2.06
B. 2
C. (0.2) × (−1) = −0.2
D. (−1) × (0.2) + 2 = 1.8
84 WEEK 4
Example 32. Find the linearization of f (x) = 1/x at a = 4.
Example 33. Find the equation of the tangent line to the hyperbola y = 1/x
at x = 4.
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 85
4.4 Indeterminate Forms and l'Hopital's Rule
l'Hopital's Rule is a convenient method for calculating limits. It is also a nice
application of linear approximations. Here is an example to demonstrate how
it works: Suppose we want to find the following limit:
sin x
lim .
x→0 x
Both sin x and x have limit 0. Therefore we would end up with the expression
0
.
0
This expression is called an indeterminate form. When an indetermi-
nate form of this type arises in calculating limits, l'Hopital's Rule says
that you should differentiate the numerator and denominator separately.
(Caution: Do not confuse this with the quotient rule!)
If we were to apply l'Hopital's Rule to our example, that is if we were to
differentiate the numerator and denominator, we would end up with
cos x
.
1
If we then take the limit of this new expression we get
cos x
lim = 1.
x→0 1
l'Hopital's Rule says that this limit is equal to the limit we originally began
with, namely
sin x
lim = 1.
x→0 x
Theorem W4.1 L'Hopital's Rule (see text pg. 299 [308])
If lim f (x) = 0 and lim g(x) = 0. Then
x→a x→a
f (x) f ′ (x)
lim = lim ′
x→a g(x) x→a g (x)
provided the latter exists.
86 WEEK 4
ex − 1
Example 34. Find lim .
x→0 sin x
Maple can handle limits like this quite well: Enter
> limit((exp(x)-1)/sin(x), x=0);
2
ex
Example 35. Find lim 2 .
x→0 x − 2
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 87
Why does this method work?
Consider the following diagrams, which show generic functions f and g whose
limits as x → a are both 0 (in fact, f (a) = g(a) = 0):
f (x)
g lim =?
x→a g(x)
a
f
Now if we zoom in near a, the graphs of the functions will begin to resemble
their tangent lines at a:
g
x
a
f
That is, for x close to a, we have these linearizations, represented by the two
straight lines in the picture:
f (x) ≈ f (a) + f ′ (a)(x − a) = f ′ (a)(x − a) ,
g(x) ≈ g(a) + g ′(a)(x − a) = g ′(a)(x − a) .
88 WEEK 4
Notice that we used the fact that f (a) = g(a) = 0. Therefore we have
f (x) f ′ (a)(x − a) f ′ (a)
≈ ′ = ′ .
g(x) g (a)(x − a) g (a)
Since all of these approximations get better and better as x → a, we conclude
that
f (a) f ′ (a)
lim = ′ .
x→a g(a) g (a)
If this were a more rigorous mathematics course, we would find that there
are several points at which this proof is not totally clear, but it captures the
essence of why l'Hopital's Rule works the way it does. For a complete treat-
ment covering the subtleties that have been glossed over in our discussion,
check Appendix F in the textbook.
l'Hospital's Rule can also be used when we get an indeterminate of the form
±∞
,
±∞
0
instead of , and when a limit is taken for x → ∞.
0
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 89
ex
Example 36. Calculate lim 3 .
x→∞ x
Maple does this limit for you if you enter
> limit(exp(x)/x∧3, x=infinity);
90 WEEK 4
Notes
4.4 INDETERMINATE FORMS AND L'HOPITAL'S RULE 91
92 WEEK 4
Week 5
Study Sections: 5.1 and 5.2 in the textbook
• What is an Integral?
• The integral as a way to calculate an accumulated total
• Riemann Sums
• Connection to Areas
93
94 WEEK 5
Course Unit: Integration
If we had to summarize the first four weeks of the course, we would have
to say that the focus was on differentiation. We learned derivatives of some
new functions (at least for most of us) and we learned to differentiate and
interpret vector-valued functions.
All differentiation problems ask the same basic question: At what rate does
a process change, and how does that rate of change relate to other charac-
teristics of the process? This entire discussion of rates of change depended on
one key idea: No matter how complicated a time-dependant process is, when
you examine change over a sufficiently small interval, the behaviour seems
linear; that is, over a very small time interval, the rate seems constant. The
analysis was always directly or indirectly related to this simple ideaIn the remainder of the course we will study integration. Again, the analysis
is made possible by the observation that on a very small scale all processes
look linear. This time, though, we will use this fact to see how regarding
a process as an accumulation of infinitely many small linear steps allows us
to calculate the accumulated total even when the rate of accumulation is
far from linear. Integration is always in some way about finding the
total at the end of a process of accumulation.
COURSE UNIT: INTEGRATION 95
Suppose we want to find the volume of the hat pictured above. The (solid)
shape pictured is the shape you get if you take the region between the graph
of the function 1/(x2 + 3x + 2) and the x-axis, bounded by the y-axis and the
line x = 1, and rotate it around the vertical axis. Initially it may seem that
this question has nothing do to with a process of accumulation. However,
there are several ways to turn this into an accumulation.
One option is to think of this figure as a stack of infinitely many infinitely
thin disks of decreasing radius piled on top of each other. This would allow us
to think of its volume as the accumulated total of the volumes of these disks.
If each disk is thin enough (and infinitely thin is very thin), and because they
have a circular cross-section, we can think of these disks as (infinitely thin)
cylinders. Since we know how to calculate the volume of a circular cylinder,
we may be able to use this analysis to arrive at a calculation for the total
volume. This method is described more fully at the end of week 7.
A second option is to think of the solid figure as a nested set of cylindri-
cal shells. Imagine the solid separated into infinitely many infinitely thin
cylindrical shells. Think of growth rings in the trunk of a tree; or imagine
a very fine set of concentric circular cookie cutters of various radii. Each
concentric cylindrical shell has a height determined by its distance from the
centre, and radius equal to that distance. This will allow us to determine
the volume (area multiplied by infinitely small thickness) of a typical shell.
We can think of the volume of the solid figure as the accumulated total of
these small volumes. This method of cylindrical shells is the one that turns
out to work best for this solid. The method of cylindrical shells is discussed
in Week 8. For a complete solution of this problem you should go to Week
9.
On the way, we will examine a lot of other problems that relate more obvi-
ously to a process of accumulation, and use integration to solve them.
96 WEEK 5
5.1 & 5.2 What is an Integral?
Concept Question 12. Suppose you are travelling by car on a long stretch
of isolated road. Your car is old. The speedometer works, but the odometer
is broken. You make the observations indicated on the following table:
time t (hours) 0 0.25 0.5 1 1.25 1.5
speed v (km/h) 55 60 65 65 60 50
You want to use the information in the table to estimate how far you travelled
in 1.5 hours. Which of the following calculations makes the most sense to
that end?
1
A. (55 + 60 + 65 + 65 + 60 + 55) × 1.5
6
B. 0.25 × 55 + 0.25 × 60 + 0.5 × 65 + 0.25 × 65 + 0.25 × 60
C. 0.25 × (55 + 60 + 65 + 65 + 60 + 50)
D. 55 ÷ 0.25 + 60 ÷ 0.25 + 65 ÷ 0.5 + 65 ÷ 0.25 + 60 ÷ 0.25
5.1 & 5.2 WHAT IS AN INTEGRAL? 97
98 WEEK 5
Here is a principle that we will see again and again as we learn about inte-
gration:
To say that we are studying the accumulation of a quantity s that accu-
mulates at a varying rate v(t) is to say
• first of all that
ds
= v(t) ;
dt
and secondly that
• an estimate of the total accumulated as t varies from a to b can
be obtained by sampling v in each of a number of subintervals
of the total interval [a, b] and adding the products of rate times
subinterval-length (v × ∆t) for each of these intervals.
We would be able to estimate total distance more accurately if we sampled
the speed of the car more frequently.
Example 37. For example, suppose we had sampled the speed every 6 min-
utes, and that we measured the speeds
v(0), v(0.1), v(0.2), · · · v(1.4), v(1.5)
How could we combine these numbers to obtain a better estimate of the dis-
tance travelled from time 0 to time 1.5?
5.1 & 5.2 WHAT IS AN INTEGRAL? 99
100 WEEK 5
Riemann Sums
Both the left sum and the right sum are examples of Riemann sums. A
Riemann sum is any sum obtained in the process of trying to calculate the
total of an accumulation that takes place at rate v(t) by partitioning an
interval a ≤ t ≤ b into n subintervals:
a = t0 < t1 < t2 < · · · < tn−1 < tn = b ,
and then choosing a representative point t∗ in (or on the edge of) the ith
i
interval for each i. Each term of the Riemann sum has the form
rate × subinterval-length
where the rate (for the ith subinterval) is taken as v(t∗ ) and the subinterval
i
length is the length of time represented by that subinterval, namely ∆ti =
ti − ti−1 . In effect, you are pretending that the rate of accumulation is v(t∗ )
i
not just at that one point of time, but throughout the ith subinterval. Thus
the Riemann sum representing the accumulated total is
v(t∗ ) × ∆t1 + v(t∗ ) × ∆t2 + v(t∗ ) × ∆t3 + · · · + v(t∗ ) × ∆tn ,
1 2 3 n
or n
v(t∗ ) × ∆ti
i
i=1
5.1 & 5.2 WHAT IS AN INTEGRAL? 101
Area Under a Graph
Another problem that gives rise to a Riemann Sum occurs when calculating
the area under a graph. Suppose we want to calculate the area under the
1
graph of f (x) = 1+x2 between 0 and 1.
1
f (x) =
1 1 + x2
−1 1
If the function were constant, the answer would be obvious:
Area = (function value) × (length of interval).
In general we get an approximate value by dividing the interval into (n) sub-
intervals and sampling the function once in each sub-interval: f (x∗ ), where
i
x∗ is a sample point in the the interval [xi−1 , xi ]:
i
0 = x0 < x1 < x2 < . . . < xn = 1.
If each interval [ xi−1 , xi ] is very small, then the function f (x) is "practically"
constant on [ xi−1 , xi ]. Therefore the area under the graph of f (x) between
xi−1 and xi is approximately
f (x∗ ) · (xi − xi−1 ) .
i
height ∆xi =width
We could then add these approximate values of the area under f for each
sub-interval, producing another "Riemann Sum":
n
f (x∗ ) · ∆xi .
i
i=1
102 WEEK 5
We can represent this process in terms of a picture:
1 �
∗
f (x1 )�
x0 = 0 x1 x2 x3 1
x∗
1 x∗
2 x∗
3 x∗
4
1
e.g. if x1 = 4 , x2 = 1 ,
2
and x3 = 3 ,
4
we could choose x∗ = 1 ,
1 8
x∗ = 3 ,
2 8
x∗ =
3
5
8
and x∗ = 7 .
4 8
Therefore,
1 64
f (x∗ ) = f
1 = ,
8 65
3 64
f (x∗ ) = f
2 = ,
8 73
5 64
f (x∗ ) = f
3 = ,
8 89
7 64
f (x∗ ) = f
4 = .
8 113
So the approximate area under f (x) between 0 and 1 is
64
Area ≈ 65
·1+
4
64
73
· 1
4
+ 64
89
·1+
4
64
113
· 1
4
Area ≈ 0.7867.
5.1 & 5.2 WHAT IS AN INTEGRAL? 103
We have calculated a Riemann Sum for a partition of [ 0, 1 ] into 4 equal
sub-intervals, choosing each "sample point" in the center of the sub-interval.
This is known as the "midpoint sum", or "midpoint rule".
We could have chosen the left end of each sub-interval instead (x∗ = xi−1 )
1
and we would get the "left sum". Similarly we could choose the right end of
each interval and calculate the "right sum". In the following picture the left
sum is indicated by solid lines and the right sum by dashed lines.
1
1 f (x) =
1 + x2
1
Because this function f (x) is decreasing on [ 0, 1 ], it follows that the
Right sum ≤ Area ≤ Left sum.
Summary
In certain problems that involve the accumulation of some quantity over an
x-interval [ a, b ] (e.g. gas used, area), an approximate answer may be found
by partitioning the interval and pretending that on each subinterval the
rate f (x) at which the accumulation takes place is constant. The obvious
calculation then produces a Riemann Sum:
n
f (x∗ )∆xi ,
i
i=1
where f (x∗ )
i represents the "rate" of accumulation on the interval, and ∆xi
the interval length.
104 WEEK 5
The true value of the accumulated total can be found in principle by doing
the calculation over and over again with finer partitions (i.e. ∆xi → 0 or
b−a
equivalently, as n → ∞ since n = ). In other words the Riemann Sums
∆xi
n
f (x∗ )∆xi ,
i
i=1
converge to the true value of the accumulated quantity. The true value is
known as the "Integral", and is denoted by the symbol
b
f (x) dx.
a
This is read as the "Integral of f (x) from a to b". It is the limit of the
approximating Riemann Sums. f (x) is called the integrand and a and b are
known as the limits of integration.
Notice how the symbol for the integral resembles the expression for the Rie-
mann sums that approximate it: The integral sign resembles an elon-
gated S, for "sum"; The index i and its limits 1 and n are replaced by a and
b; f (x∗ ) becomes f (x); and ∆xi becomes dx. In terms of developing your
i
intuition, it is not unreasonable to think of the integral as an infinite
(Riemann) sum of infinitely many small increments.
Thus, in the examples we have discussed so far,
1.5
The total distance traveled in Example 37 is v(t) dt
0
1
1
The total area discussed on page 101 is dx
0 1 + x2
In each case, the Riemann sums we computed are approximations to these
integrals. Notice that in the first case, the integrand is the rate of accu-
mulation of the underlying process. We will see in a moment that the
second case can also be interpreted that way.
5.1 & 5.2 WHAT IS AN INTEGRAL? 105
Concept Question 13. Water is pouring into a reservoir at a varying rate
of f (t) cubic meters per hour. The total volume (measured in cubic meters)
of water in the reservoir at time t (hours) is g(t). Then which of the following
statements are necessarily true?
5
1. The amount of water in the reservoir at time t = 5 is equal to f (t) dt
0
2. The increase in the amount of water in the reservoir between times 0
5
and 5 is f (t) dt
0
3. At time t = 3, water is flowing into the reservoir at a rate equal to
g ′ (3)
4. The increase in the amount of water in the reservoir between times 0
and 5 is g(5) − g(0)
A. 1 and 4 only
B. 2 and 4 only
C. 3 and 4 only
D. 2, 3 and 4 only
E. all
106 WEEK 5
Here is a more formal definition of the integral concept: Let P be a partition
of [ a, b ],
P : a = x0 ≤ x1 ≤ . . . xn−1 ≤ xn = b.
Choose x∗ in [xi−1 , xi ] and let ||P || = max {∆xi }. Then the integral of f
i
from a to b is defined to be
b n
f (x) dx = lim f (x∗ ) · ∆xi
i
a ||P ||→0
i=1
if the limit exists.
The function f is called integrable on [ a, b ] if this limit exists; that is, if
successive approximations get closer and closer to a fixed amount (which we
then decree to be the integral), and don't fluctuate erratically or go off to
infinity. You might think that this would always be the case. You would be
nearly correct in supposing this, but there are cases when a function does not
have an integral. For example suppose the function has a vertical asymptote
inside the interval [a, b]. In that case you would not expect to be able to
talk about the area under its graph. Or what about a really wild function
that is highly discontinuous (lots of sudden increases and decreases in value -
an extreme example is the function that takes the value 1 at rational inputs
and 0 at irrational inputs)? Fortunately, many (essentially all 'reasonable')
functions are integrable:
Theorem W5.1 (see text pg. 368 [380])
If f is continuous on [a, b], then f is integrable on [a, b].
The role of Riemann sums
1. They are needed to say what we mean by an integral.
2. They enable us to decide which integral is appropriate in a word
problem.
3. They can also be used to give an approximate value of the integral,
as seen in the next example.
5.1 & 5.2 WHAT IS AN INTEGRAL? 107
2
Example 38. Use the midpoint rule with n = 4 to approximate x2 dx.
0
108 WEEK 5
You can get Maple to draw the diagram associated with the midpoint rule
for this integral by entering
> with(student): (this enables the extra procedures you need)
> middlebox(x∧2, x=0..2, 4);
You can then produce this Riemann sum by typing
> middlesum(x∧2, x=0..2, 4);
and find the value of this sum by typing
> evalf(middlesum(x∧2, x=0..2, 4));
Concept Question 14. The answer we found in the preceding question is
A. An underestimate because the first rectangle is very small
B. An overestimate because this is an increasing function.
C. An underestimate because the graph of f is concave upward
D. An overestimate because we used the midpoints
5.1 & 5.2 WHAT IS AN INTEGRAL? 109
The Relationship Between the Integral and the Area
Under a Graph
We started our discussion of integration by stressing the fact that an integral
problem is always at heart a problem in which something accumulates. We
were able to interpret the distance and speed problem that way, and were
were able to present the gas consumption problem that way. In each case
the analysis resulted in a limit of Riemman sums, in which each term was
the product of the rate of accumulation and a time interval. So far we have
not discussed the problem of the area between the input axis and the graph
of a function that way. We want to show next that an area problem can also
be regarded as an accumulation problem in the same way; and conversely,
that any accumulation problem can be thought of as a problem of finding
the area between the input axis and the graph of the function that gives the
rate of accumulation.
It helps if we think of the function f as being positive for the moment. To
b
examine the process of accumulation implied in an area calculation a f (x) dx
we should identify an intermediate stage in the process that gives rise to
the total area. We do this by considering the area from the left end a of
the interval to an intermediate point u, which will be considered variable.
Suppose the area under the graph and above the interval [a, u] is thought of
as a total, called G(u) that accumulates as we let u increase gradually from
a to b. That is,
u
G(u) = f (x) dx,
a
where a ≤ u ≤ b. If this is the total that accumulates, what represents the
rate of accumulation? In other words, what is G′ (u)?
110 WEEK 5
Example 39. From first principles determine what G′ (u) is.
5.1 & 5.2 WHAT IS AN INTEGRAL? 111
In effect we have proved one of the most important theorems in the theory
of integration. A more rigorous calculus course would pay closer attention
to subtleties we have glossed over in our discussion, but our presentation
contains the central idea in the proof.
Theorem W5.2 Fundamental Theorem of Calculus, Part1 (see text
pg. 381 [396])
u
If f (x) is continuous on [ a, b ], then G(u) = a f (x) dx
is continuous as well as differentiable, and G′ (u) = f (u).
Another way of saying that f is the derivative of G is that G is an anti-
derivative of f . This theorem confirms what we saw in several examples: f
is the rate at which G accumulates. In the case of the distance problem and
the problem of gas consumption, the integrand was the rate of accumulation
already by virtue of the physics of the situation. In the case of an area
problem we just proved it assuming f is a positive function. The proof
works equally well if the function is not positive everywhere.
However, the relationship between the integral and area is a bit more com-
plicated when the function f is negative at some places on the interval [a, b].
The Riemann sum for a partition a = x0 < x1 < x2 < · · · < xn−1 < xn = b
has the form n
f (x∗ ) ∆xi .
i
i=1
Here is a picture representing this sum:
112 WEEK 5
f (x∗ ) ∆xi > 0
i
x
a b
f (x∗ ) ∆xi < 0
i
Each term in the Riemann sum represents the area of a corresponding rect-
angle as f (x∗ ) ∆xi . That makes sense, except that it means that the integral
i
does not exactly calculate the area between the graph and the x-axis, for
whenever the graph dips below the x-axis, the corresponding terms in the
Riemann sum are negative. That is, areas of little rectangles get subtracted.
b
The conclusion is that the integral f (x) dx is equal to A − B + C in the
a
example in the diagram.
A C
x
a b
B
Integral = A − B + C
When a function dips below the axis, the integral and the area between
the graph an the axis are no longer the same thing.
5.1 & 5.2 WHAT IS AN INTEGRAL? 113
There is a general principle that when you do an integral to calculate a total
quantity that accumulates at a varying rate, then the integral (between x = a
and x = b say) can be thought of as the area under the graph (or better:
between the horizontal axis and the graph) of the function expressing the
rate of accumulation. This principle is repeated more explicitly in the "Net
Change Theorem" discussed on page 131.
You may have already encountered some quantities in high school physics
that were represented as areas under graphs, without knowing exactly why
these quantities could be thought of as areas under graphs and thus, as
integrals. Such examples may include:
• displacement as the area under a velocity-time graph
Here displacement, s = v(t) dt, where velocity v(t) represents the
rate at which displacement is accumulated, i.e. v(t) = s′ (t).
• (change in) velocity as the area under an acceleration-time graph
Here velocity, v = a(t) dt, where acceleration a(t) represents the rate
at which velocity is accumulated, i.e. a(t) = v ′ (t).
• work as the area under a power-time graph
Here work, W = P (t) dt, where power P (t) represents the rate at
which work is accumulated, i.e. P (t) = W ′ (t).
In each of these cases, you were likely asked to compute a certain quantity
(such as displacement or work) by finding the area under a simple curve.
Justification for doing so, beyond the fact that the units worked out, was
likely limited or missing. Only with an understanding of accumulation (and
thus, integration) does it make sense to connect these quantities with areas.
In the next few weeks, we will see further examples where basic physical defi-
nitions, such as displacement = velocity × time, or work = force × distance,
applicable to simple situations of constant velocity or force, when general-
ized to situations of varying velocity or force, take the form of integrals. You
should see those discussions as explanations of these statements that connect
displacement, velocity, and work to areas under graphs.
114 WEEK 5
Notes
5.1 & 5.2 WHAT IS AN INTEGRAL? 115
116 WEEK 5
Week 6
Study Sections: 5.2-5.4 in the textbook
• The Second Part of the Fundamental Theorem of Calculus
• Integral Calculations using Antiderivatives
• Properties of Integrals
• The Indefinite Integral
117
118 WEEK 6
5.3 The Fund. Theorem of Calculus, Part 2
When we are looking for the value of an integral, calculating Riemann sums
is a lot of work; and when you have calculated one, it is not necessarily
obvious how much your answer differs from the true value (that is, from the
integral).
Fortunately, the Fundamental Theorem of Calculus provides a very simple
and clever way to do many integral calculations directly. The Second part
of the Fundamental Theorem of Calculus (see the next page) shows how it
does that.
Consider the example of the area under the graph of the function f (x) = x2
between x = 0 and x = 2. We calculated a Riemann sum for that when
we did Example 38. The Fundamental Theorem of Calculus, Part 1, tells us
that if we let u
G(u) = x2 dx ,
0
then
G′ (u) = u2 ;
That is, G(u) is an anti-derivative of u2 .
Example 40. Does this information help us get a formula for G(u)?
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 119
Once we have a formula for G, we can calculate the integral immediately, for
2
G(2) = x2 dx .
0
Example 41. Calculate G(2)
Theorem W6.1 Fundamental Theorem of Calculus, Part 2 (see
text pg. 384 [398])
If f is continuous and F is an anti-derivative of f , then
b
f (x) dx = F (b) − F (a).
a
The idea is that if somehow we can come up with the formula F for a function
whose derivative is the integrand f , then to get the integral of f over the
interval [a, b], all we have to do is plug in and subtract F (b) − F (a). This
idea is important enough that we should do the proof.
120 WEEK 6
The proof hinges on the fact that if you have a formula for one anti-derivative
F (x) of a function f (x) then you can immediately write down what the other
anti-derivatives look like, for they will all have the form F (x) + C where C
is an unspecified constant.
Example 42. Prove that if F (x) and G(x) both have the same derivative f ,
then they differ by a constant.
Example 43. Prove the Fundamental Theorem of Calculus, Part 2
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 121
Using the Fundamental Theorem of Calculus, Part 2
To use Part 2 of the Fundamental Theorem of Calculus, we need to know
formulas for the anti-derivatives of functions. We already know quite a few.
Example 44. Complete the following table of basic anti-derivatives by asking
yourself the question, "f (x) is the derivative of what function, F (x)?".
Function f (x) Anti-derivative F (x)
xn (n = 1)
1
x
ex
cos x
sin x
sec2 x
1
√
1 − x2
1
1 + x2
122 WEEK 6
Example 45. Find the derivative of ln(|x|) by considering x > 0 and x < 0
separately.
x3 − 3x + 5
Example 46. Find the most general anti-derivative of f (x) = .
x
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 123
Concept Question 15. Suppose we want to calculate the area under one
section of the graph of sin x, the part from 0 to π.
1
0.8
0.6
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3
Then we should calculate
A. cos(π)
B. cos(0) − cos(π)
C. cos(π) − cos(0)
D. sin(π) − sin(0)
124 WEEK 6
4 √
Example 47. Calculate x dx.
0
Notation
The expression F (b) −F (a) comes up so often that there is a special notation
for it. It is written as b
F (x) or [F (x)]b
a
a
Applying this notation to the previous example gives
4
4√
2 3/2 2 2 16
x dx = x = (4)3/2 − (0)3/2 = .
0 3 3 3 3
0
5.3 THE FUND. THEOREM OF CALCULUS, PART 2 125
π/3
Example 48. Calculate 3 sec2 θ dθ.
π/4
Maple does integrals very easily. You can refer to the Maple notes for Week
6 to see some of the ways it can do integration. For this integral, all you
have to do is to enter
> int(3*(sec(x))∧2, x=Pi/4..Pi/3);
Note that Pi has to be capitalized!
126 WEEK 6
5.2 Properties of Integrals
Before we go on to refine our skill at calculating integrals, we should first
reflect on some basic properties of integrals that derive from their origins as
limits of more and more accurate Riemann sums.
b a
1. If a > b then f (x) dx = − f (x) dx (by convention).
a b
b
2. If a = b then f (x) dx = 0.
a
b
3. c dx = c (b − a).
a
b b b
4. f (x) ± g(x) dx = f (x) dx ± g(x) dx.
a a a
b b
5. c f (x) dx = c f (x) dx.
a a
b c b
6. f (x) dx = f (x) dx + f (x) dx.
a a c
Properties 3-6 need proofs; they depend on the analogous properties for
Riemann Sums. See pages 373-375 [387-390] in the textbook for careful
proofs of all of these properties, as well as for properties 7-10 appearing
later.
5.2 PROPERTIES OF INTEGRALS 127
Example 49. Prove Property 4.
128 WEEK 6
2 4
Example 50. Given that f (x) dx = 3 , f (x) dx = 2 and
0 1
4 2
f (x) dx = 4, what is the value of f (x) dx ?
0 1
5.2 PROPERTIES OF INTEGRALS 129
Other Properties of Integrals
b
7. If f (x) ≥ 0 for a ≤ x ≤ b then f (x) dx ≥ 0.
a
b b
8. If f (x) ≥ g(x) for a ≤ x ≤ b then f (x) dx ≥ g(x) dx.
a a
b
9. If m ≤ f (x) ≤ M for a ≤ b then m (b − a) ≤ f (x) dx ≤ M (b − a).
a
b b
10. f (x) dx ≤ |f (x)| dx.
a a
The following diagrams illustrate the ideas of Properties 8 and 9:
f
g
a b
M
f
m
a b
In the second diagram, it can be observed that the area under f (x) in [ a, b ]
is greater than the area under m in [ a, b ] but less than the area under M.
130 WEEK 6
Example 51. Let f (x) be as shown
A C E
a b
B D
Sketch |f (x)| and so verify Property 10.
5.4 THE INDEFINITE INTEGRAL 131
5.4 The Indefinite Integral
The second part of the Fundamental Theorem of Calculus,
b
f (x) dx = F (b) − F (a),
a
reduces the problem of doing an integral to that of finding the anti-derivative
F . For that reason the anti-derivative is often called the indefinite integral
and written as
f (x) dx = F (x) .
The actual integral is often referred to as the definite integral to distinguish
it from the indefinite integral. Notice that these terms are confusing. The
definite integral is a number (the outcome of an integration) while
the indefinite integral is a function (the antiderivative needed to
calculate a definite integral). Note also that f (x) = F ′ (x), so f gives
the rate of change of F . Notice that this observation was made much earlier,
when we started our discussion of integration. For we saw then that when
an integral is associated with a process of accumulation then the rate of
accumulation is always precisely the integrand. We can express it this way:
b
F ′ (x) dx = F (b) − F (a).
a
"The integral of a rate of change is the total change". The textbook calls
this the Net Change Theorem. (See page 394 [408].)
132 WEEK 6
u
Concept Question 16. If G(u) = F ′ (x) dx, then
a
A. F is the derivative of G
B. G is the derivative of F
C. F and G are equal
D. F and G differ by a constant
Example 52. Suppose water is flowing into a tank at a rate given by r(t) =
200 − 4t L/min. How much water is added to the tank during the first 10
minutes of filling?
5.4 THE INDEFINITE INTEGRAL 133
√ 3
Example 53. Find t− √
t
dt.
Note: When you are asked to find an indefinite integral, as in the previous
question, it is important to add the "+C" to indicate that there is more than
one anti-derivative, and that all of them differ from each other by a constant.
Finding antiderivatives is surprisingly difficult. You would think that if we
know how to find derivatives then we should know how to find antiderivatives.
In fact, however, the latter is much more difficult, as illustrated next.
1
Example 54. Calculate dx
x3 +1
134 WEEK 6
Guess and Check
Next week we will study the "Substitution Rule", or "Method of Substitu-
tion", the first of a list of techniques for finding anti-derivatives. To prepare
us for the Method of Substitution, let me explain an informal method I call
"guess and check".
Example 55. Find cos(5x) dx.
5.4 THE INDEFINITE INTEGRAL 135
Example 56. Find cos(x2 ) dx.
What if we were given the problem x cos(x2 ) dx?
136 WEEK 6
So why does "guess and check" work in one case but not the other ?
In the case of cos(5x) dx, we were successful essentially because we were
able to rewrite it as
1
5 5 cos (5x) dx.
derivative of "outer function"
"inner function" 5x
In other words, we were able to interpret the integrand as the result of a
chain-rule differentiation:
1 du
5 dx
cos ( u ) dx.
5 5x
The same does not work for
cos(x2 ) dx.
You cannot turn this integral into
2x cos( x2 ) dx.
du u
dx
5.4 THE INDEFINITE INTEGRAL 137
Concept Question 17. Which of the following anti-differentiations can be
done by the guess-and-check method, do you think?
3 2
1. x2 ex dx 2. x2 ex dx
2 2
3. xex dx 4. ex dx
5. x2 ex dx
A. 1
B. 5
C. 2, 3, and 4
D. 1 and 3
E. 4
138 WEEK 6
Example 57. Calculate cos(x) esin(x) dx.
Next week we will turn these ideas into a formal procedure called
"Substitution".
5.4 THE INDEFINITE INTEGRAL 139
Notes
140 WEEK 6
Week 7
Study Sections: 5.5, and 6.1 in the textbook
• The Method of Substitution
• Areas
141
142 WEEK 7
5.5 Substitution Rule
We want to formalize the "guess and check" method we introduced at the
end of last week. That will give us the Method of Substitution. Consider
the previous example:
esin(x) cos(x) dx.
Here is the procedure:
1. Let u = sin x. (a guess)
2. Differentiate u: du = cos x dx.
3. Rewrite the integral entirely in terms of u and du:
eu du.
If this step cannot be carried out, it means that you should go
back and try a different choice of u!
4. Solve this anti-derivative:
eu du = eu + C.
Again, if this step cannot be carried out, it means that you
should go back and try a different choice of u!
5. Rewrite u in terms of x:
eu = esin x .
Therefore
esin(x) cos(x) = esin x + C .
5.5 SUBSTITUTION RULE 143
3x
Example 58. Calculate dx .
1 + x2
144 WEEK 7
5
Example 59. Calculate dx.
1 + 9x2
The method of substitution is really all about the chain rule. We will learn
many methods of integration, but the method of substitution is the first
one you should think of first when you encounter an integral question. You
should always determine what the "inner function" is and then try to see if
the derivative of this "inner function" is in the integrand. If it is, chances are
the method of substitution will easily solve the problem. Once you do many
examples using this method, you will become so familiar with the outlined
procedure that you will not need to follow it step by step. It will become so
automatic that your answers will be one-liners.
5.5 SUBSTITUTION RULE 145
Concept Question 18. What substitution(s) could you choose for the in-
definite integral
x cos(1 + x2 ) dx ?
(1.) u = cos(1 + x2 ) (2.) u = x2 (3.) u = sin(1 + x2 )
(4.) u = 1 + x2 (5.) u = x
A. Option 1
B. Options 1 and 3
C. Options 2 and 3
D. Options 2, 3, and 4
E. All options
146 WEEK 7
4 1 1
Example 60. Calculate 1 x2
1+ x
dx.
5.5 SUBSTITUTION RULE 147
Example 61. Calculate tan x dx.
148 WEEK 7
π
Example 62. Find 0
4
tan x dx.
Special procedure for using substitution on a definite integral:
u
In the last example, we evaluated F (cos x) at x = π and x = 0 and sub-
4
tracted. But when x = π then u = cos( π ) = √2 and when x = 0 then
4 4
1
u = cos(0) = 1. These are the u-values that correspond to x = 0 and
1
x = π . So the solution can also be expressed as F √2 − F (1), instead of
4
F cos π − F (cos(0) . In other words
4
x= π u= √1
4 2 1
tan x dx = − du,
x=0 u=1 u
√
= − ln |1/ 2| + ln |1|,
√
= − ln(1/ 2).
That is, when we use the method of substitution on a definite integral, we
can omit Step 5 provided we change the limits of integration when we change
variables.
5.5 SUBSTITUTION RULE 149
π 2 /4
√
cos( t)
Example 63. √ dt
π 2 /9 t
150 WEEK 7
6.1 Areas between Curves
We are now ready to use integration in some simple applications.
Example 64. Find the area between y = x and y = x2 .
6.1 AREAS BETWEEN CURVES 151
Here we do this problem a second time, using a more intuitive and informal
method called Short-cut Riemann sums:
152 WEEK 7
Example 65. Find the area between x = 1 − y 2 and x = y 2 − 1.
6.1 AREAS BETWEEN CURVES 153
Example 66. Find the area between y = x3 − 2x and y = −x2 .
154 WEEK 7
6.1 AREAS BETWEEN CURVES 155
Concept Question 19. Suppose we want to find the area between the curves
x = y − y 3 and x = y 2 − y. Using the diagram shown below, determine which
of the integrals or integral combinations will give the correct answer.
1.5
1 (0,1)
y
0.5
x
0 1 2 3 4 5 6
0
-0.5
-1
-1.5
(6,-2)
-2
1
A. ((y 2 − y) − (y − y 3)) dy
−2
1
B. ((y − y 3 ) − (y 2 − y)) dy
−2
0 1
C. (y 2 − y) dy + (y − y 3) dy
−2 0
0 1
D. ((y − y 3 ) − (y 2 − y)) dy + ((y 2 − y) − (y − y 3)) dy
−2 0
0 1
E. ((y 2 − y) − (y − y 3)) dy + ((y − y 3) − (y 2 − y)) dy
−2 0
156 WEEK 7
6.1 AREAS BETWEEN CURVES 157
Notes
158 WEEK 7
Week 8
Study Sections: 6.2, 6.3 and 6.4 in the textbook
• Volumes by slicing
• Volumes by Cylindrical Shells
• Work
159
160 WEEK 8
6.2 Volumes
In the integration problems considered in this section the accumulated total
is a volume. In each of the examples, this total is obtained by regarding
the solid figure as a stack of infinitely thin slices. For this reason it can be
referred to as the "method of slices". If the slices have a simple shape (say
if they are circular so that they can be thought of as infinitely thin cylinders)
so that we can obtain a simple formula for the volume of a typical slice, then
we can "add" (that is, integrate) the volumes of these slices to get a total
volume for the figure.
Example 67. Find the volume of the solid obtained by rotating the triangle
bounded by x = 0, y = 0 and x + y = 1 about the x-axis.
6.2 VOLUMES 161
162 WEEK 8
Example 68. Find the volume of the solid obtained by rotating the region
bounded by x = 0 and x = y − y 2 about the y-axis.
6.2 VOLUMES 163
Concept Question 20. Suppose a vase is such that when you fill it with
water up to depth h (measured in cm) the surface area of the water in the
vase is A(h) square centimeters. Then if you fill the vase up to 30 cm, the
integral that is equal to the volume of water in the vase is
30
A. hA(h) dh
0
30
B. 2πA(h) dh
0
30
C. A′ (h) dh
0
30
D. A(h) dh
0
164 WEEK 8
Example 69. Find the volume of the solid obtained by rotating the region
bounded by y = x3 and x = y 2 about the x-axis.
6.2 VOLUMES 165
Example 70. Find the volume of a ball of radius r.
166 WEEK 8
6.3 VOLUMES BY CYLINDRICAL SHELLS 167
6.3 Volumes by Cylindrical Shells
In this kind of application of integration, the accumulated total is a volume
consisting of infinitely many infinitely thin concentric cylindrical shells. The
goal is to find an expression for the volume of a typical shell, and then to
add (that is, integrate) these volumes to get the total. (Note that Example
69 done using washers can also be done using cylindrical shells.)
Example 71. Find the volume of the solid produced by rotating the region
bounded by y = x and y = 4x(1 − x) about the y-axis.
168 WEEK 8
6.3 VOLUMES BY CYLINDRICAL SHELLS 169
Concept Question 21. Suppose we have a function f (x) that is positive
on the interval [1, 2] and suppose that D is the region between the graph of
f , the lines x = 1, x = 2, and the x-axis.
D
1 2
If we rotate the region D about the y-axis to form a solid S, which of the
following integrals represents the volume of S?
2 2
A. f (x) dx B. 2πf (x) dx
1 1
2 2
C. 2πxf (x) dx D. πx2 f (x) dx
1 1
1
E. πx2 f (x) dx
0
170 WEEK 8
Example 72. Find the volume generated by rotating about the line x = −1
the region that lies in the first quadrant and is bounded by y = x2 , y = 4
and x = 0 .
6.3 VOLUMES BY CYLINDRICAL SHELLS 171
172 WEEK 8
Concept Question 22. We learned earlier, on page 165 that the volume of
4
a solid ball of radius r is π r 3 . Suppose we have a formula for the area of
3
the surface of a sphere of radius r. Imagine it is A(r) square units. Then
we can think of a ball of radius R as an accumulation of concentric spherical
shells each of thickness dr, starting at the center of the ball and going out
towards its outer shell. Which of the following are then true?
r R
4 4
1. π r3 = A(u) du 2. π R3 = A(r) dr
3 0 3 0
r
d 4 d 4
3. A(r) = π r 3 dr 4. A(r) = π R3
dr 0 3 dr 3
d 4
5. A(r) = π r3
dr 3
A. All of these
B. None of them
C. 2
D. 1, 2, and 5
E. 5
6.4 WORK 173
6.4 Work
The basic formula for work is the product of force times distance. If force
is measured in Newtons and distance in meters, the answer is in Joules. In
the case of a problem that requires an integral for its solution, this means
that we have to find a way to divide the process into infinitely small steps so
that on each step the amount of work may be computed as a simple product
of force and distance.
Example 73. When a particle is x meters from the origin, a force measuring
cos πx N acts on it. How much work is done by moving the particle from
3
x = 1 to x = 2 ?
174 WEEK 8
Example 74. Hooke's Law tells us that when a spring is extended, the force
(measured in Newtons) with which the spring pulls back is equal to product
of the distance by which the spring has been extended x (beyond its relaxed
length) and a constant k characterizing the stiffness of the spring, where k is
called the "spring constant". Calculate the work done in extending a spring
from x = 1 cm to x = 3 cm. We will assume that the units of k are N/cm so
that the final answer has units of N cm. What is the general formula for the
work done in extending a spring x units from its relaxed position?
6.4 WORK 175
Example 75. An aquarium 2 m long, 1 m wide and 1 m deep is full of water.
Find the minimum amount of work needed to pump half of the water out of
the aquarium.
176 WEEK 8
6.4 WORK 177
Comments on the aquarium problem:
1. Strictly speaking, the assumptions behind the problem are highly ide-
alized. In any real situation the water would come out of the hose with
some amount of kinetic energy, and this extra energy adds to the work
done. In addition to that a certain amount of work is needed to over-
come internal friction in the water, friction in the pump, and so on.
To calculate the minimum amount of work required is to ignore these
effects. Even if in real life the amount of work required is always some-
what more than what this calculation tells, it is nevertheless helpful to
know that it gives the absolute minimum that could be reached.
2. The method used really hinges on the conservation of energy: energy
gained = work done. We calculated this work by calculating the in-
crease in (potential) energy in the horizontal slabs of water.
3. To minimize the work needed, we imagine the pumping done "slowly"
so no kinetic energy is created.
4. In principle, if we knew what happened to each particle of water, we
could do a more detailed and "realistic" analysis. It would require
knowing where the hose is placed (on the bottom of the aquarium or
higher) and a calculation of the work done on or by each individual
water particle as it is pushed down the tube and then up again, or
(in other cases) as it sinks closer to the bottom of the aquarium. In
practice this picture becomes far too complicated to use. The power
of the principle of energy conservation is in its ability to simplify the
problem.
But then we would have to take into account the downward pressure
of the water, which "helps" the pump. The final answer would be the
same.
178 WEEK 8
Concept Question 23. A large cylindrical tank is filled with water. There
is a drain in the center of the bottom of the tank, two meters above the surface
of a lake. A hose is attached to the drain, and the tank is allowed to empty
through the hose onto the surface of the lake. We want to calculate the loss
of potential energy of the water as it runs from the tank to the surface of the
lake. We realize that the problem requires an integral - we have to think of
the water as a parametrized family of "pieces" so that for each such piece the
energy loss can be calculated. How should we choose those "pieces" of water
and why?
A. Horizontal slabs because that's what we did in the preceding question
B. Horizontal slabs because all the points in a horizontal slab are the same
distance above the surface of the lake
C. Horizontal slabs because when it is at rest water surface is always hor-
izontal
D. Cylindrical shells because the tank is cylindrical
E. Cylindrical shells because each such shell is at a constant radius from
the center, where the drain is located
6.4 WORK 179
Example 76. The parabola y = x2 is rotated about the y-axis, and filled
with water to the level y = 3. How much work is required to pump the water
out through a hole located at y = 4? (Assume all scales are in meters.)
180 WEEK 8
Concept Question 24. The angular momentum of a point mass rotating
about an axis at ω radians per second is defined as the product of the mass,
the square of the distance from the centre, and the angular velocity ω (that
is, mr 2 ω). Suppose we have a large metal cylinder of uniform density rotat-
ing about its axis. If we want to use an integral to calculate the total angular
momentum of the cylinder we have to think of the cylinder in terms of a para-
metric family of "pieces" chosen in such a way that the angular momentum
of each piece is easily calculated. How should we do that, and why?
A. Horizontal slices of vertical thickness dy, because the volumes of these
slices are given by π r 2 dy
B. Horizontal slices of vertical thickness dy, because the cylinder is rota-
tionally symmetric
C. Cylindrical shells because the object is a cylinder
D. Cylindrical shells because on each such shell the radius to the axis is the
same at every point, and thus the angular momentum of such a shell is
easily calculated.
6.4 WORK 181
182 WEEK 8
Notes
6.4 WORK 183
184 WEEK 8
Week 9
Study Sections: 7.1 and 7.4 in the textbook
• Integration by Parts
• Partial Fractions
185
186 WEEK 9
7.1 Integration by Parts
In Week 6 we learned that the key to integration is the ability to find
anti-derivatives. In Week 7 we learned the first technique for finding anti-
derivatives: The Method of Substitution. To allow us to apply integration
to a variety of problems, we need additional techniques for finding anti-
derivatives. Integration by Parts is the first of these.
The main idea of Integration by Parts is to use the Product Rule in reverse.
d
f (x) g(x) = f ′ (x) g(x) + f (x) g ′(x).
dx
If we integrate the expression on the right:
f ′ (x) g(x) dx + f (x) g ′(x) dx = f (x) g(x),
which is what we started with. If we let u = f (x) and v = g(x) then
du = f ′ (x) dx,
dv = g ′ (x) dx.
Therefore the equation can be written in the form
v du + u dv = uv,
u dv = uv − v du.
given simpler
The main goal of Integration by Parts is to end up with a simpler integral,
v du, than the given integral, u dv.
7.1 INTEGRATION BY PARTS 187
For example, consider the integral xex dx. Let
u=x dv = ex dx (this is our guess)
∴ du = dx ∴ v = ex
dif f erentiated integrated
Thus
xex dx = xex − ex dx = xex − ex .
original simpler
After the first step, if the new integral is much more complicated than the
original one, go back and set u and dv equal to something else so that you end
up with an easier integral to solve. You can avoid this by mentally checking
what v du will be before actually choosing what u and dv are.
Example 77. Calculate x sin(4x) dx.
188 WEEK 9
Analysis of our solution to Example 77:
Could we have foreseen that our choice of u and dv would work? Let's review
what we did:
x sin(4x) dx
differentiate integrate
1 cos(4x)
leaving out sign
and constants
So the new integral would be something like
cos(4x) dx.
What would of happened if we would have chosen u and dv to equal something
else?
dv
x sin(4x) dx
u
integrate differentiate
x2 cos(4x)
ignore constants for
this "rough work"
The new integral would be something like
x2 cos(4x) dx.
It is worse! This is obviously a lot more complicated to solve. Integration
by Parts is very easy once you learn to make smart choices for u and dv.
Practice a lot.
7.1 INTEGRATION BY PARTS 189
2
Example 78. Calculate x3 ex dx.
190 WEEK 9
Example 79. Calculate ln x dx. Use your answer to find the area under
the graph of ln x between x = 1 and x = 2.
7.1 INTEGRATION BY PARTS 191
1
Example 80. Calculate 2
0
sin−1 (x) dx.
192 WEEK 9
7
x5
Concept Question 25. If you want to do the integral √ dx
0 1 + x3
using integration by parts then you should choose
1
A. u = x5 and dv = √ dx
1 + x3
x
B. u = x4 and dv = √ dx
1 + x3
x2
C. u = x3 and dv = √ dx
1 + x3
x3
D. u = x2 and dv = √ dx
1 + x3
x2
E. u = √ and dv = x3 dx
1+x 3
7.4 PARTIAL FRACTIONS 193
7.4 Partial Fractions
This is the last of the techniques of integration (that is, techniques for anti-
differentiation) covered in this course. If you look in the textbook you will see
that there are many other methods we do not include. With the availability
of computer packages such as Maple, much of the hard work in finding anti-
derivatives has been automated. The techniques we have covered are included
because a certain amount of facility with a few basic methods is needed to
know how to get problems to the point where computer packages can take
over.
The method of Partial Fractions is purely algebraic. It consists of a series
of algorithmic steps that simplify the integrand so that an anti-derivative can
be easily found. For example, consider the integral
x+5
dx.
x2 + x − 2
At first glance this integral seems difficult to solve, but by using the method
of Partial Fractions, we can break the integrand into
2 1 x+5
− = 2 .
x−1 x+2 x +x−2
Check for yourself that this equation is true. We can then write the integral
as
2 1
dx − dx.
x−1 x+2
This integral is now very easy to solve.
Our goal then is to find this kind of Partial Fraction Decomposition.
This method is used only for expressions of the form
P (x)
(rational function),
Q(x)
where P and Q are polynomials.
A proper rational function is one for which the degree of P is strictly less
than the degree of Q. To be able to use the method of Partial Fractions, we
194 WEEK 9
must first make sure that the integrand is a proper rational function; this
is our first step:
Step 1.
Turn P (x) into an expression involving a proper rational function and a
Q(x)
polynomial:
P (x) R(x)
= S(x) + .
Q(x) Q(x)
The method we use to turn P (x) into a proper rational function is Long
Q(x)
Division. For example, to solve
x2 − 4x + 3
dx,
x−5
we do the following:
x+1 S(x)
x−5 x2 − 4x + 3 P (x)
Q(x) x2 − 5x
x+3
x−5
8 R(x)
so,
P (x) = S(x) · Q(x) + R(x)
P (x) R(x)
∴ = S(x) + .
Q(x) Q(x)
Thus with our example we get
x2 − 4x + 3 8
dx = (x + 1) + dx
x−5 x−5
x2
= + x + 8 ln |x − 5| + C.
2
7.4 PARTIAL FRACTIONS 195
x2 + 1
Example 81. Find dx.
x2 − 1
196 WEEK 9
The next next step (after we have turned the integral into one involving a
proper rational) consists of four cases, distinguished by what happens when
you factor the denominator.
Step 2. CASE I:
The denominator is the product of distinct linear factors. Say
Q(x) = (a1 x + b1 )(a2 x + b2 ) . . . (ak x + bk ).
The goal is to look for numbers A1 , . . . , Ak so that
R(x) A1 A2 Ak
= + + ...+ .
Q(x) a1 x + b1 a2 x + b2 ak x + bk
"Partial Fractions"
x
Example 82. Find dx.
x2 + 3x + 2
7.4 PARTIAL FRACTIONS 197
198 WEEK 9
3x2 − 2
Example 83. Find dx.
(x − 1)(x − 2)(x + 1)
7.4 PARTIAL FRACTIONS 199
We are now at the point where we can complete the problem we presented
on page 96 to introduce integration. Before going on to the second case of
the method of partial fractions, we will solve that problem.
Example 84. That is, find the volume of the "hat-shaped" solid you get
when you take the region between the graph of the function 1/(x2 + 3x + 2)
and the x-axis, bounded by the y-axis and the line x = 1, and rotate it around
the vertical axis200 WEEK 9
7.4 PARTIAL FRACTIONS 201
Step 2. CASE II:
The polynomial Q(x) is a product of linear factors, some of which are re-
peated. Rule: If (ax + b) occurs to the power r, then instead of one term,
put down the following r terms:
A1 A2 Ar
+ 2
+ ...+ .
ax + b (ax + b) (ax + b)r
dx
Example 85. Find .
x2 (x− 1)2
202 WEEK 9
7.4 PARTIAL FRACTIONS 203
Why do we have to learn these rules for setting these problems up?
It is natural at this point to wonder why we have to set up the question
precisely as we did. Why could we not reduce the number of variables by
letting
1 B C D
= 2+ + ?
x2 (x − 1)2 x x − 1 (x − 1)2
If we did this we would get the equations
C =0
B−C +D =0
2B =0
B =1
You can see immediately that this system of equations does not have a so-
lution. That is, there do not exist numbers B, C, and D that satisfy all four
equations at once.
1
Concept Question 26. Suppose we want to integrate dx.
(x − 1)(x + 2)2
How should we set up the partial fractions?
A B A B C
(A.) + (B.) + +
x−1 x+2 x−1 x+2 x+2
A B A B C
(C.) + (D.) + +
x − 1 (x + 2)2 x − 1 x + 2 (x + 2)2
A B C D
(E.) + 2
+ +
x − 1 (x − 1) x + 2 (x + 2)2
204 WEEK 9
1
Example 86. Find dx.
(x − 1)(x + 2)2
7.4 PARTIAL FRACTIONS 205
Step 2. CASES III AND IV:
The polynomial Q(x) contains quadratic factors. We will omit these
cases.
206 WEEK 9
Notes
7.4 PARTIAL FRACTIONS 207
208 WEEK 9
Week 10
Study Sections: 7.7 and 7.8 in the textbook
• Approximate Integration
• Improper Integrals
209
210 WEEK 10
7.7 Approximate Integration
Approximate Integration is very important when we cannot find a formula
for the anti-derivative of a given function. For example, the functions in the
integrals
2 √
e−x dx and 1 + x3 dx
do not have a formula for their anti-derivatives.
We have already learned and used a method for calculating an approximate
value for an integral: Riemann Sum.
1 x
Suppose we wanted to know how close we could get to calculating 0 x2 +1
dx
by using a simple Riemann Sum. By integrating, we know that
1
x 1
dx = ln 2 = 0.34657 . . . .
0 x2 +1 2
x
Here are some values for f (x) = between 0 and 1:
x2 +1
x 0 0.25 0.5 0.75 1
f (x) 0 0.23529 . . . 0.4 0.48 0.5
In the next few examples we will see that, even using the same set of numbers,
some ways of combining them gives more accurate results than others.
7.7 APPROXIMATE INTEGRATION 211
Example 87. Calculate the R.S. using two intervals and their midpoints.
(This is the "Midpoint Rule".)
Example 88. Calculate the R.S. using four intervals and their left-end
points. (This is the "left end-point approximation".)
212 WEEK 10
Example 89. Calculate the R.S. using four intervals and their right-end
points. (This is the "right end-point approximation".)
Example 90. Calculate the average of the left- and right-end point approx-
imations.
7.7 APPROXIMATE INTEGRATION 213
The general formula for the left end-point approximation is
∆x(f (x0 ) + f (x1 ) + . . . + f (xn−1 )),
and the general formula for the right end-point approximation is
∆x(f (x1 ) + f (x2 ) + . . . + f (xn )).
This average of the left end-point and right end-point approximations is
known as the "Trapezoidal Rule". The Trapezoidal Rule is best under-
stood with a picture:
f (x)
xi−1 xi
From the picture, we can observe that the area of the trapezoid in the ith
subinterval is
f (xi−1 ) + f (xi )
Area = ( xi − xi−1 ) · .
2
This area is equal to the average of the areas of the rectangles with base
( xi − xi−1 ) and heights f (xi−1 ) and f (xi ). We can write the Trapezoidal
Rule in terms of the formula:
∆x
f (x0 ) + 2f (x1 ) + 2f (x2 ) + · · · + 2f (xn−1 ) + f (xn ) .
2
You should memorize this formula.
214 WEEK 10
b
In practice, when you have to integrate a f (x) dx, you would like to know
beforehand how many intervals to divide [ a, b ] into to get the desired ac-
curacy. In other words, we would like to know the "error" made in approx-
b
imating a f (x) dx using the Trapezoidal Rule. The error is the difference
b
between the value of a f (x) dx and the approximated value obtained using
the Trapezoidal Rule. There are theorems for this, but we will not use them
in this course. The accuracy of the Trapezoidal Rule depends not only on the
number of intervals but also on the second derivative. The following theorem
gives a formula for calculating the error made when using the Trapezoidal
Rule.
Theorem W9.1 Error Bounds (see text pg. 499 [515])
If |f ′′(x)| ≤ K on [ a, b ], then the error ET made
b
in approximating a f (x) dx using the Trapezoidal
Rule with n partitions is
K(b − a)3
|ET | ≤
12n2
Why does this Error Bound theorem involve the second derivative?
You can 'sort of' see why the second derivative is important when estimating
the error. If you go back to the figure on page 213, you will see that the
error, the difference between the sum of the areas of the trapezoids and the
true area under the curve is just the sum of the areas of curved segments
such as the one above the trapezoid in the figure. If the second derivative is
zero, the graph is a straight line, and the graph will coincide with the tops
of the trapezoids, allowing no error at all. When the second derivative is
not equal to zero, it measures the amount by which the graph is curved (the
rate at which the slope changes). Therefore, if the second derivative is very
large, it is curved a lot, and each segment will have a rather large area, thus
producing a rather large error.
7.7 APPROXIMATE INTEGRATION 215
Simpson's Rule is an even better method. It is based on approximation by
pieces of parabola:
y
a b x
The formula for Simpson's Rule is: (n must be even)
∆x
f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + 2f (x4 ) + · · ·
3
· · · + 2f (xn−2 ) + 4f (xn−1 ) + f (xn ) .
You should MEMORIZE this formula.
Example 91. Apply Simpson's Rule to the previous example.
216 WEEK 10
7.8 Improper Integrals
So far in our study of integration, we have dealt with functions that were
always continuous on the interval that we were integrating on. When we
integrated f (x) over [ a, b ], the function f (x) did not have a discontinuity in
[ a, b ], and a and b were always finite. In this section we will introduce the
concepts of
1. integrating f (x) over an infinite interval, and
2. integrating f (x) over an interval [ a, b ] where the f (x) contains a dis-
continuity in [ a, b ].
Hence the term "Improper Integrals".
Improper Integrals of Type 1: Infinite Intervals
These integrals contain ∞ and/or −∞ in their limits of integration:
∞ b
f (x) dx or f (x) dx.
a −∞
For example, consider the integral
∞
1
dx.
1 x2
By calculating this integral, we will be finding the area under f (x) = x12
between x = 1 to x = ∞ and we will be able to show that its value is finite.
1
f (x) =
x2
...
1
7.8 IMPROPER INTEGRALS 217
∞
1
Example 92. Using the rule stated below, calculate dx, and so justify
1 x2
the statement made on the previous page.
∞ t
Rule : f (x) dx = lim f (x) dx .
a t→∞ a
Example 93. Is the area under the graph of f (x) = e−3x , extending from
x = 0 to infinity, finite? If so, what is its value?
218 WEEK 10
A nice way to explore improper integrals of Type I is to ask how the family
of functions 1/xp , for various positive values of p, relate to each other, and
what happens to them when we integrate them from 1 to ∞. Here are some
examples of such functions, generated by the Maple command
> plot([1/sqrt(x), xˆ(-1), xˆ(-2)], x= 0.. 6, y= 0.. 4);
4
3
y 2
1
0
0 1 2 3 4 5 6
x
Notice what goes wrong when you leave out the y= 0.. 4 part from the Maple
command. As you can see, all these functions cross at the point (1,1).
Concept Question 27. Which of the following expectations seems reason-
able to you?
∞
A. 1/xp dx is more likely to be finite when p is small than when p is
1
large, for then the anti-derivative will be smaller.
∞
B. 1/xp dx is more likely to be finite when p is small than when p is
1
large, for then the graph of 1/xp is closer to the horizontal axis.
∞
C. 1/xp dx is more likely to be finite when p is large than when p is
1
small, for then the graph of 1/xp is closer to the horizontal axis.
∞
D. 1/xp dx is more likely to be finite when p is large than when p is
1
small, for then the graph of 1/xp is farther from the vertical axis.
7.8 IMPROPER INTEGRALS 219
∞
Example 94. Calculate 1/xp dx
1
220 WEEK 10
∞
3
Example 95. Find x2 e−x dx . (# 14 in the text)
−∞
7.8 IMPROPER INTEGRALS 221
Improper Integrals of Type 2: Discontinuous Integrand
b
These are integrals, a f (x) dx, where f has a vertical asymptote somewhere
in [ a, b ]. For example, consider the integral
1
1
dx.
0 x2
1
Even though the function f (x) = x2 has a discontinuity in the specified
interval (at x = 0), we will learn techniques that will show that this integral
diverges to infinity. That is, the area under f (x) = x12 between x = 0 and
x = 1 is infinite.
.
.
.
1
f (x) =
x2
1
222 WEEK 10
Like Type 1 problems, the solution technique for Type 2 problems involves
taking a limit. In this case, since the function is undefined at the x location
of its vertical asymptote, say at x = a, we replace a with t and then take the
limit as t approaches a.
1
1
Example 96. Calculate dx and so verify that the area under the graph
0 x2
1
of 2 from x = 0 to x = 1 is infinite.
x
7.8 IMPROPER INTEGRALS 223
4
1
Example 97. Calculate dx.
0 x2 +x−6
You can easily miss that this is an improper integral!.
224 WEEK 10
7.8 IMPROPER INTEGRALS 225
Example 98. Newton's Law of Gravitation states that two bodies with masses
m1 and m2 attract each other with a force of magnitude
m1 m2
F =G
r2
where r is the distance between the two bodies and G = 6.67×10−11N ·m2 /kg2
is the gravitational constant. Assuming that the Earth's mass is 5.98 × 1024
kg and that its radius is 6.37 × 106 meters, calculate the work required to
propel a 1000-kg satellite out of the Earth's gravitational field. In problems
like this it is correct to act as if the mass of the Earth were concentrated at
its center.
226 WEEK 10
Notes
7.8 IMPROPER INTEGRALS 227
228 WEEK 10
Week 11
Study Sections: 9.1, 3.8 [9.4] in the textbook
• Course Unit: Differential Equations
• Modeling with Differential Equations
• Exponential Growth
• Harmonic Motion
229
230 WEEK 11
Course Unit: Differential Equations
As pointed out on several other occasions in the course, most laws of nature
and science, once they are translated into mathematics, take the form of a
differential equation.
A differential equation is quite unlike the other equations we have studied.
For one thing, it involves derivatives, often even second or higher derivatives.
But this is not the most important difference between a differential equation
and the kinds of equations we have been used to and continue to study
in our courses. In those more familiar problems involving equations, the
goal is to find the number x or perhaps the pair of numbers, x and y, that
constitutes the solution of the equation or set of equations. By contrast, in
a differential equation the unknown is the function f itself, and
not the variable! A related characteristic of a differential equation is that
when we write down a differential equation involving a variable x (and there
is always at least one variable), the equation is understood to hold for
all values of the variable. Thus we are looking for a particular f that
satisfies the equation for all values of x.
Concept Question 28. Which of the following functions is a solution to
the differential equation x′′ (t) = −36x(t)?
A. x(t) = −6t3
B. x(t) = cos(6t + 2)
C. x(t) = e−6t
D. x(t) = −e−6t
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 231
9.1 Modeling with Differential Equations
This week we will focus on two particular differential equations and find
their solutions. The first of these is the differential equation that gives rise
to exponential growth.
Exponential Growth (see also 3.8 [9.4])
Suppose we are in a situation where, for a certain species, the population of
size y grows in such a way that its rate of growth is always 3 times as big as
the size of the population at that moment, we would translate that as
dy
= 3y
dt
Here y is (the output of) a function of the variable t. We are looking for
a formula for y in terms of t, which satisfies the differential equation for all
values of t.
Example 99. Find a function y = f (t) that satisfies this differential equa-
tion. HINT: If you cannot think of it immediately, try to do it first for the
dy
differential equation = y.
dt
232 WEEK 11
What we have discovered in this problem is the origin of exponential growth!
When population size is given by the formula y = ket we have a population
that grows exponentially, as most of you will have learned in high school.
Exponential growth is not something that falls out of the sky as a peculiar
preference exhibited by populations! There is a clear and simple explanation
for exponential growth, and it centers on this differential equation: The rate
at which young are born into a population is in direct proportion to the size
of the population. The ratio k between the number of offspring per unit
time and the size of the population is called "fecundity", and depends on the
organism. It is proverbially higher for rabbits than for humans. In any case
when we translate "the rate at which young are born into a population is in
direct proportion to the size of the population" into mathematics, we get
dy
= ky .
dt
The fact that the solutions for this differential equation all have the form y =
Aekt is the reason why, in the presence of abundant resources, populations
grow exponentially.
Are there other functions that satisfy this differential equation?
We promised to show that functions of the form y = ekt are the only solutions
to the differential equation
dy
= ky .
dt
The next exercise will do that for us:
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 233
Example 100. Suppose y(t) is a solution of the differential equation
dy
= ky.
dt
d y(t)
(a) Evaluate .
dt ekt
(b) What does your calculation in part (a) tell you about the most general
form a solution of the differential equation can take ?
234 WEEK 11
Theorem: If a function y(t) satisfies the differential equation
dy
= ky(t)
dt
then y(t) necessarily has the form
y(t) = Aekt
for some constant A. The constant A is the value taken by y at t = 0.
Sometimes, a differential equation can be turned into the equation for expo-
nential growth by means of an appropriate substitution. Newton's Law of
Cooling is an example;
Example 101. Newton's Law of Cooling claims that if a hot cup of coffee
is left on the table, then the rate of cooling is always proportional to the
difference between the temperature of the coffee and ambient temperature in
the room. Write this down as a differential equation, assuming the room
temperature at 20◦ .
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 235
Example 102. By concentrating on the temperature difference rather than
the temperature, turn this into an exponential growth question, and then solve
this problem: If it takes 5 minutes for the temperature of the coffee to go from
100◦ to 80◦ , how hot will the coffee be after 10 minutes?
236 WEEK 11
Harmonic Motion
A very different differential equation arises when we study the motion of an
object at the end of a spring, as in the next diagram:
0 x
In this diagram, the grey object on the left is attached to the table, while
the block on the right is allowed to slide without friction. 0 indicates where
the right end of the spring is when the system is at rest. Thus x indicates
the distance by which the spring is extended. Hooke's law tells us that the
force exerted by the spring on the block has magnitude proportional to the
extension x, and is of course in the opposite direction:
F = −kx .
Here k is the "spring constant", a positive constant that describes the stiffness
of the spring. If the spring is very stiff, k is large, for then a small extension
is enough to produce a large force. Using Newton's second law, this leads to
the equation
d2 x
m 2 = −kx ,
dt
where m is the mass of the block and t is time. Notice that this, too, is a
differential equation. It involves a quantity, x, that is a function of time,
t, and it gives a relation between the function and one of its derivatives.
This differential equation is inherently more complicated than the one for
exponential growth, because it involves the second derivative of the function
x = x(t). Remember that, when we study a differential equation, the goal is
to find a formula for the unknown function. In this case we want an expression
for x in terms of t. Of course you expect k to occur in this formula as well
and, as in the case of the solution to the differential equation for exponential
growth, we should not expect the formula to be uniquely determined by the
differential equation, but to depend on one (or more) arbitrary constants.
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 237
To find a formula for x(t), we will first consider a special case of the differ-
ential equation by assuming that k = 1 and m = 1. Then we have
d2 x
= −x .
dt2
Example 103. This differential equation invites us to find a function x(t)
whose second derivative is its own negative. Try to find such a function.
d2 x
We have found a set of solutions for the differential equation = −x. Does
dt2
d2 x
this help us discover solutions for the differential equation m 2 = −kx? To
dt
help us find solutions to this, we could begin by putting the two positive
constants m and k together, and then giving the name ω to the square root
of their ratio:
d2 x k
2
= − x = −ω 2 x .
dt m
Example 104. Find a solution to this differential equation.
238 WEEK 11
Are there other solutions? You may have noticed that if A and φ are
constants, then x(t) = A cos(ωt+ φ) is a solution and yet has a different form
from the one we discovered. You can check easily that this function satisfies
the differential equation. Does this mean that this is a new type of solution?
In fact it does not, for we will now show that these two only look like different
functions, and that, using appropriate trigonometric identities, we can show
that each B cos(ωt) + C sin(ωt) can be transformed into A cos(ωt + φ) for
suitable A and φ, and conversely.
First of all, you should note that allowing A to be negative in the expression
x(t) = A cos(ωt + φ) is redundant, for if we add π to φ we change the sign of
the expression:
A cos(ωt + φ + π) = −A cos(ωt + φ)
The reason for this is that cos(x + π) = − cos(x) for any x. Thus, we may
as well assume that A is chosen to be positive.
Now recall the trigonometric identity proved in Week 2 on page 30:
cos(a + b) = cos a cos b − sin a sin b .
If we apply this with a = ωt and with b = φ, we obtain
A cos(ωt + φ) = A [cos(ωt) cos(φ) − sin(ωt) sin(φ)]
= [A cos(φ)] cos(ωt) + [−A sin(φ)] sin(ωt) .
Notice that t is the only variable in this expression; the other letters represent
constants. The question before us is: Can B and C be chosen so that this
is exactly the same as B cos(ωt) + C sin(ωt) for all t? Clearly we can, by
simply letting
B = A cos(φ) = A cos(−φ)
C = −A sin(φ) = A sin(−φ)
Conversely, we could ask whether we can always solve for A and φ in terms
of B and C. We are not interested in knowing an actual formula so much
as knowing that it is possible, so that we can say with confidence that the
two families of solutions we found are really one and the same - that for
each expression in one family there is a corresponding one (defining the same
function) in the other. To see that this can always be done, suppose a number
φ and a (positive) number A are given. The we can imagine a line segment
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 239
of length A emanating from the origin and making the angle −φ with the
horizontal axis:
−φ
A
Example 105. What are the coordinates of the endpoint of the line segment?
This shows that for any A and any φ the associated numbers B and C are just
the coordinates of the endpoint of the line segment we drew; and, conversely,
if B and C are given, then we let A be the length of the segment from the
origin to the point (B, C) and φ the angle it makes with the horizontal axis.
Thus we have shown geometrically that we can always solve for B and C if
we know A and φ, and conversely.
240 WEEK 11
The upshot of this discussion is that
The family of functions
B sin(ωt) + C cos(ωt) ,
where B and C are arbitrary constants, and the family of functions
A cos(ωt + φ) ,
where φ is an arbitrary constant and A an arbitrary positive constant,
are really the same family. Each member of this family of functions is a
solution to the differential equation
x′′ (t) = −ω 2 x(t)
arising from Hooke's Law.
By another, rather long, argument it can be shown that there are no solutions
arising from Hooke's Law, other than the ones in this family of functions.
We will simply assume this.
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 241
What do these solutions look like? We should now describe the func-
tions in this family of solution, for they tell us how the block will move under
the influence of the spring.
To do this it is easiest to work with the expression
A cos(ωt + φ) .
We begin by observing that these functions are all periodic:
Concept Question 29. x(t) = A cos(ωt + φ) is periodic with period
A. 2π
B. ω
C. φ/ω
D. 2π/ω
242 WEEK 11
We can study it in terms of transformations applied to the function cos(t).
We know what the graph of the cosine function looks like:
cos(t)
1
−2π 0 2π
As a first step in transforming this to the function A cos(ωt + φ), we now add
φ to the variable t. We learned in high school that this will have the effect
of moving the graph to the left by the amount φ:
cos(t + φ)
1
−2π −φ 0 2π
As a second step, we now multiply the variable t by the constant ω. You
will remember that the effect of this is to compress the graph towards the
vertical axis (or to expand it away from the vertical axis if ω < 1):
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 243
cos(ωt + φ)
1
−2π φ
−ω 2π
As a final step, we multiply the function by the constant A. This will have
the effect of stretching the graph vertically by that factor:
A cos(ωt + φ
A
1
−2π φ
−ω 2π
The upshot of this discussion is that A cos(ωt + φ) is a"sinusoidal" function
(that is, it is shaped like a sine or cosine function) with amplitude A.
The function is periodic with period 2π/ω, for if we increase t by that
much, the input into the cosine function is increased by exactly 2π. In
terms of the motion of the block, it means that it will vibrate back and
forth, between extreme positions −A and A, and that takes 2π/ω time units
between successive instants at position A. The motion described by the
function A cos(ωt + φ) is known as simple harmonic motion.
244 WEEK 11
Concept Question 30. Recall that in an application to spring motion ω =
k
, where k is the stiffness of the spring and m is the mass of the sliding
m
block. Suppose the motion of the block is started by releasing the block (at
zero velocity) from a non-equilibrium position. Which of the following will
change the time it takes for the block to go through one cycle of its periodic
motion?
A. The block is given a small push in the direction away from the equilib-
rium position
B. The block is released closer to the equilibrium position
C. A weight is attached to the top of the block
Example 106. Suppose that for a block attached to a spring (with spring
constant k) on a frictionless table the potential energy is 0 when the block is
in the equilibrium position, what is the potential energy when the extension
of the spring is x?
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 245
Example 107. If an object undergoes harmonic motion, so that its position
function is a solution of the differential equation
k
x′′ (t) = − x(t)
m
, prove from the differential equation that
m(x′ (t))2 + k(x(t))2 = a constant
What does this equation tell you physically?
246 WEEK 11
Notes
9.1 MODELING WITH DIFFERENTIAL EQUATIONS 247
248 WEEK 11
Week 12
Study Sections: 9.3, 9.4 [9.5] in the textbook
• Separable Differential Equations
• The Logistic Equation
249
250 WEEK 12
9.3 Separable Differential Equations
Last week we looked at two very particular differential equations, and found
the solutions for both of them. Our methods were ad hoc: There is no
indication how the techniques we used to find these solutions might be used
to solve other differential equations. Eventually we want to learn techniques
that can be applied to a wide variety of differential equations. Such a study
of differential equations is not taken up seriously until second year. This
week we will conclude the course by demonstrating a method for finding
solutions for one relatively simple class of differential equations that includes
exponential growth and Newton's Law of Cooling as special cases.
A differential equation is said to be of first order when it involves first
derivatives only. The simplest kind of first order equation is one in which the
first derivative can be isolated on one side of the equation. This will produce
an equation of the form
dy
= f (x, y)
dx
where f (x, y) indicates an expression involving x and y but no derivatives.
This type of equation has a very nice visualization. Think of the unknown
function y as measured along the dependent axis, and x as independent
variable in the usual way. We are looking for a solution function y(x). The
differential equation tells you that when x has a given value and y has a
particular value, then the slope of the function you are trying to find is equal
to the value of f (x, y). For example, suppose the differential equation is
dy
= xy
dx
and that a solution y(x) of this equation has the value 3 when x = 2, then
the slope of the graph of y(x) at x = 2 is 6. This observation allows us to
think of finding a solution to this kind of differential equation in terms of a
picture.
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 251
Example 108. On the following grid, draw the slopes for this differential
equation at a number of points. The result is know as the slope field of the
differential equation.
4
3
2
1
0
-1
-2
-3
-4
-4 -3 -2 -1 0 1 2 3 4
Maple is particularly well suited to draw a slope field. The Maple Notes
discuss this more fully in the section for Week 11. Here are the commands
for producing a slope field for the differential equation y ′ = x2 /100:
with(DEtools):
and then
DEplot(diff(y(x),x)=x∧2/100,y, x=-10..10, y=-10..10,
arrows=LINE);
252 WEEK 12
10
y(x)
5
K
10 K 5 0 5 10
x
K 5
K
10
You can use this slope field to sketch the graphs of some solutions of the
differential equation.
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 253
A separable differential equation is a differential equation in which the
expression for the derivative can be isolated and written as a product of
two functions, one involving only the independent variable and the other
involving only the dependent variable. That is, it is a separable differential
equation if it can be written in the form
dy
= g(x)f (y) .
dx
Concept Question 31. Which of the following are separable?
du
1. dt
= u cos(t)
2. x′ (t) = kt
dy
3. 4 dx = xy
4. T ′ (t) = T (t) − 20
A. All of them
B. Number 3. only
C. Numbers 1. and 3.
254 WEEK 12
Example 109. Suppose the function f (y) has a root at y0 ; that is, suppose
f (y0) = 0. Use this information to find a solution of the differential equation
dy
= g(x)f (y) that takes the value y0 when x = 0.
dx
This example allows us to focus on the values of y for which f (y) = 0. For
such values of y it is convenient to define h(y) = 1/f (y). The differential
equation then becomes
dy g(x)
= .
dx h(y)
The method for solving separable equations hinges on our ability to separate
the variables as follows:
h(y)dy = g(x)dx .
Writing it this way, invites integrating both sides:
h(y)dy = g(x)dx .
Before we go on, we should justify this formal pushing around of symbols. Do
we have any justification for believing that doing this will lead to a solution
for the differential equation? An indefinite integral is an anti-derivative (and
always involves an arbitrary constant), so the last equation can be thought
of as follows: Suppose we have found anti-derivatives H(y) = h(y)dy and
G(x) = g(x) dx (each hiding an arbitrary constant of course). Now put
these equal to each other, H(y) = G(x), and solve this for y in terms of x.
Do we know that y = y(x) will then be a solution to the differential equation?
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 255
Example 110. Use implicit differentiation on H(y) = G(x) to show that y
is a solution to the differential equation
dy g(x)
= .
dx h(y)
Now that we have provided this justification, the method for substitution of
variables can be summarized as follows:
Separable differential equations: Separate the variables to produce
an equality of differential expressions
h(y)dy = g(x)dx ,
Find anti-derivatives of both sides (thus introducing an arbitrary con-
stant) giving
H(y) = G(x)
and solve for y in terms of x.
256 WEEK 12
Example 111. Solve the differential equation
(x2 + 1)y ′ = xy .
9.3 SEPARABLE DIFFERENTIAL EQUATIONS 257
Example 112. In your second term physics course you will learn how to
analyze electric circuits. In particular, you will learn that the electric current
I in the circuit shown below
R
E L
switch
will satisfy the differential equation
dI
L + RI = E(t)
dt
Find an expression for the current (measured in Amperes) when the resistance
R = 12Ω (Ohms), the inductance L = 4 henries (H), and the power source E
gives a constant voltage of 60 volts (V), and the switch is turned on at time
t = 0.
258 WEEK 12
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 259
9.4 [9.5] The Logistic Differential Equation
Last week we discussed the exponential growth model, which is valid when
there is no limit on resources. Under those circumstances, a population is
likely to grow at a rate proportional to its size. The logistic differential
equation provides a more versatile model for population growth, one that
includes the effect of limited resources. To model this situation, we imagine
that the environment has a certain carrying capacity K. If the population
should ever reach the level K, it will not be able to grow any bigger. The
Logistic differential equation is simplest model that has this feature:
dP P
= kP 1− .
dt K
Notice that this differential equation resembles the differential equation for
exponential growth. The constant k can still be thought of as "fecundity",
and if we let the carrying capacity K become infinite, we get exponential
growth.
Concept Question 32. Assuming that K and k are positive constants,
when will the function P (t) decrease?
A. When P is greater than K.
B. When P is greater than K/2
C. When t is negative
D. When t < P/K
260 WEEK 12
Example 113. At what value of P does P increase most rapidly?
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 261
Example 114. (Taken from Stewart) The Pacific halibut fishery has been
modeled by the differential equation
dy y
= ky 1 −
dt K
where y(t) is the biomass (the total mass of the members of the population)
in kilograms at time t (measured in years), the carrying capacity is estimated
to be K = 8 × 107 kg, and k = 0.71 per year.
(a) If y(0) = 2 × 107 kg, find the biomass a year later.
(b) How long will it take for the biomass to reach 4 × 107 kg?
262 WEEK 12
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 263
264 WEEK 12
Notes
9.4 [9.5] THE LOGISTIC DIFFERENTIAL EQUATION 265 |
10th Grade Workbook
After a full year of algebra studies, the student enters tenth grade ready to take a big step forward: learning how to write deductive proofs. There is a great variety of topics to be explored in this workbook. While it is possible to cover most of these topics in the tenth grade year, a class may instead spend most of the year on the geometry topics, and then perhaps return in eleventh grade to a couple of the topics found in the second half of this workbook. Either way, this workbook promises to increase the students' sense of wonder at the power and beauty of mathematics. |
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Math
Calculus
Topics covered in the first two or three semesters of college calculus. Everything from limits to derivatives to integrals to vector calculus. Should understand the topics in the pre-calculus playlist first (the limit videos are in both playlists)
Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).
The Calculus BC AP exam is a super set of the AB exam. It covers everything in AB as well as some of the more advanced topics in integration, sequences and function approximation. This tutorial is great practice for anyone looking to test their calculus mettle! |
ALGEBRA offers a practical approach to the study of beginning algebra concepts, consistent with the needs of today's student. The authors place special emphasis on the worked examples in each section, treating them as the primary means of instruction, since students rely so heavily on examples to complete assignments. Many of the applications (both within the examples and exercises) are also uniquely designed so that students have an experience that is more true to life--students must read information as it appears in headline news sources and extract only the relevant information needed to solve a stated problem. This promotes the text's focus on developing better study habits, problem solving and critical thinking skills along with orienting students to think and reason mathematically. Through Elementary Algebra, students will not only be better prepared for future math courses, they will be better prepared to solve problems and answer questions they encounter in their own lives. |
Math.NET aims to provide a self contained clean framework for symbolic mathematical (Computer Algebra System) and numerical/scientific computations, including a parser and support for linear algebra, complex differential analysis, system solving and more |
This is a Stand Alone Instructinal Resource (StAIR) designed to teach and test students on their understanding of different...
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This is a Stand Alone Instructinal Resource (StAIR) designed to teach and test students on their understanding of different triangle classifications. It could be used as initial instruction, or as an effective review tool at the end of a unit/lesson on triangle classification.
CTAP Region IV's Middle School Math Project is a series of web pages providing links to technology resources for classroom...
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CTAP Region IV's Middle School Math Project is a series of web pages providing links to technology resources for classroom teachers and coaches to improve teaching and learning of middle school mathematics, including Algebra One
Housed with the Digital Mathematics Archive ( this site contains a complete...
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Housed with the Digital Mathematics Archive ( this site contains a complete photographic reproduction of the 1847 edition of Oliver Byrne's "The Elements of Euclid." Included are extensivie correlating commentaries from Thomas L. Heath and David Joyce, borrowed from their own online editions of the Elements. The Byrne edition is noted from its use of color illustrations and a minimum of text. |
Analyze, Visualize, Simulate: Mathematica for University Research
Michael Morrison
In this Wolfram Mathematica Virtual Conference 2011 course, learn why Mathematica is used for academic research with a look at its programming language, support for parallel computing, and multiple publishing and deployment options.
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This course explores Mathematica's built-in tools for creating visualizations from functions or data. You'll learn how to customize plots with styles, labels, and other features that are common across the visualization functions.
Report Generation allows you to create documents quickly and easily using Wolfram Finance Platform documents. Get an overview of the features in this talk from the Wolfram Finance Platform Virtual Seminar.
Report Generation allows you to create documents quickly and easily using Wolfram Finance Platform documents. Get an overview of the features in this talk from the Wolfram Finance Platform Virtual Seminar.
This Wolfram Mathematica Virtual Conference 2011 course provides an introduction to the new GPU functionality in Mathematica 8 and shows examples of how to deploy your solution and scale across GPUs and machines.
Mathematica's powerful tools for building graphics are tightly integrated into its high-level programming language. This introductory course from the Wolfram Mathematica Virtual Conference 2012 covers topics related to using Mathematica for creating 2D and 3D computer graphics.
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Discover functional programming with Mathematica. In this course from the Wolfram Mathematica Virtual Conference 2011, you'll learn the core constructs, functions, and evaluation rules of a dynamic, functional language. See how to transition your thinking from procedural to functional programming.
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Learn how to harness the power of Wolfram|Alpha in the classroom. This Wolfram Technology for STEM Education: Virtual Conference for Education talk gives an introduction to Wolfram|Alpha mobile apps and how to use them effectively for a variety of subjects.
See how Mathematica connects to external programs using MathLink and other technologies built on top of MathLink, including J/Link, .NET/Link, DatabaseLink, and LibraryLink, in this talk from the Wolfram Virtual Conference Spring 2013.
With products like Wolfram|Alpha, Wolfram Demonstrations Project, and CDF, Wolfram Research brings algebra into the twenty-first century. This Wolfram Technology for STEM Education: Virtual Conference for Education talk showcases these great products and demonstrates their use in a classroom setting. |
College Algebra and Trigonometry - 3rd edition
Summary: Focusing on helping students to develop both the conceptual understanding and the analytical skills necessary to experience success in mathematics, we present each mathematical topic in this text using a carefully developed learning system to actively engage students in the learning process. We have tried to address the diverse needs of today's students through a more open design, updated figures and graphs, helpful features, careful explanations of topics, and a com...show moreprehensive package of supplements and study aids. Students will benefit from the text's student-oriented approach. We believe instructors will particularly welcome the new Annotated Instructor's Edition, which provides answers in the margin to almost all exercises, plus helpful Teaching Tips.
Features
Real-Life Applications We are always on the lookout for interesting data to use in real-life applications. As a result, we have updated or incorporated many new or examples and exercises from fields such as business, pop culture, sports, life sciences, and environmental studies that show the relevance of algebra to daily life. Applications that feature mathematical modeling are labeled with a Modeling head. All applications are titled, and a comprehensive Index of Applications is included at the back of the text.
Use of Technology As in the previous edition, we have integrated the use of graphing calculators where appropriate, although graphing technology is not a central feature of this text. We continue to stress that graphing calculators are an aid to understanding and that students must master the underlying mathematical concepts first. We have included graphing calculator solutions for selected examples and continue to mark all graphing calculator notes and exercises that use graphing calculators with an icon for easy identification and added flexibility. This graphing calculator material is optional and can be omitted without loss of continuity.
Cautions and Notes We often give students warnings of common errors and emphasize important ideas in Caution and Note comments that appear throughout the exposition.
Looking Ahead to Calculus These margin notes offer glimpses of how the algebraic topics currently being studied are used in calculus.
Connections This boxed feature provides connections to the real world or to other mathematical concepts, historical background, and thought-provoking questions for writing, class discussion, or group work.
Relating Concepts Exercises Appearing in selected exercise sets, these sets of problems help students tie together topics and develop problem-solving skills as they compare and contrast ideas, identify and describe patterns, and extend concepts to new situations. These exercises make great collaborative activities for pairs or small groups of students.
New To This Edition
Examples We have added even more examples in this edition. Step-by-step solutions to examples are now easily identified using a Solution head. We have carefully polished all solutions and incorporated more side comments and explanations, including helpful section references to previously-covered material. Selected examples continue to provide graphing calculator solutions alongside traditional algebraic solutions. The graphing calculator solutions can be easily omitted if desired.
Now try Exercises To actively engage students in the learning process, each example now concludes with a reference to one or more parallel, odd-numbered exercises from the corresponding exercise set. In this way, students are able to immediately apply and reinforce the concepts and skills presented in the examples.
Exercise Sets We have taken special care to respond to the suggestions of users and reviewers and have added many new exercises to this edition based on their feedback. As a result, the text includes more problems than ever to provide students with ample opportunities to practice, apply, connect, and extend concepts and skills. We have included writing exercises and optional graphing calculator problems as well as multiple-choice, matching, true/false, and completion problems. Concept Check problems, which focus on mathematical thinking and conceptual understanding, were well received in the previous edition and have been expanded in this edition.
Solutions to Selected Exercises Exercise numbers enclosed in a blue circle indicate that a complete solution for the problem is included at the back of the text. These new solutions are given for selected exercises that extend the skills and concepts presented in the section examples. There are approximately 3 to 5 per section.
Summary Exercises These new sets of in-chapter exercises in most chapters provide students with the all-important mixed review problems they need to synthesize concepts and select appropriate solution methods.
Function Boxes Beginning in Chapter 2, functions are a unifying theme throughout the remainder of the text. To that end, special function boxes offer a comprehensive, visual introduction to each class of function and also serve as an excellent resource for student reference and review throughout the course. Each function box includes a table of values alongside traditional and calculator graphs, as well as the domain, range, and other specific information about the function.
Chapter Reviews Each Chapter ends with an expanded Summary, featuring a section-by-section list of Key Terms, New Symbols, and a Quick Review of important Concepts, presented alongside corresponding all-new Examples. A comprehensive set of Review Excercises and a Chapter Test are also provided.
Quantitative Reasoning Now appearing at the end of each chapter, these problems enable students to apply algebraic concepts to real-life situations, such as financial planning for retirement or determining the value of a college education. A photo highlights each problem.
Glossary As an additional student study aid, a comprehensive glossary of key terms from throughout the text is provided at the back of the book.
New Annotated Instructor's Edition --This special edition of the text provides answers to almost all text exercises in color on the page that the exercise appears. This provides the instructor with immediate access to answers without searching the back of the book.
The Law of Sines. The Law of Cosines. Vectors, Operations, and the Dot Product. Applications of Vectors. Trigonometric (Polar) Form of Complex Numbers; Products and Quotients. DeMoivre's Theorem; Powers and Roots of Complex Numbers. Polar Equations and Graphs. Parametric Equations, Graphs, and Applications.Penntext Downingtown, PA
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Analyze the relationships among proportions, constant rates, and linear functions.
Interpret the meaning of the slope and the intercepts of a linear equation that models a real-world situation.
Select the linear equation that best models a real-world situation.
0027 Apply mathematical knowledge and reasoning to communicate multiple solutions in detail to a problem involving two or more of the following subareas: Numbers and Operations, Functions and Algebra, Geometry and Measurement, and Statistics and Probability.
(Refer to objectives 0016 through 0026 and associated descriptive statements.)
Measurement, Geometry, Stats, and Probability
0023 Understand and apply concepts of measurement. For example:
Estimate and calculate measurements using customary, metric, and nonstandard units of measurement.
Use unit conversions and dimensional analysis to solve measurement problems.
Derive and use formulas for calculating the lengths, perimeters, areas, volumes, and surface areas of geometric shapes and figures.
Determine how the characteristics (e.g., area, volume) of geometric figures and shapes are affected by changes in their dimensions.
Solve a variety of measurement problems (e.g., time, temperature, rates, average rates of change) in real-world situations.
0024 Understand and apply concepts of geometry. For example:
Classify and analyze polygons using attributes of sides and angles, including real-world applications.
Classify and analyze three-dimensional figures using attributes of faces, edges, and vertices.
Analyze and apply geometric transformations (e.g., translations, rotations, reflections, dilations); relate them to concepts of symmetry, similarity, and congruence; and use these concepts to solve problems. |
Elements of Number Theory (Dover Phoenix Editions)
Synopses & Reviews
Publisher Comments:
"A very welcome addition to books on number theory."—Bulletin, American Mathematical Society
Clear and detailed in its exposition, this text can be understood by readers with no background in advanced mathematics; only a small part requires a working knowledge of calculus. One of the most valuable characteristics of this book is its stress on learning number theory by means of demonstrations and problems. More than 200 problems and full solutions appear in the text, plus 100 numerical exercises. Some of these exercises deal with estimation of trigonometric sums and are especially valuable as introductions to more advanced studies. Translation of 1949 Russian edition.
Synopsis:
Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition.
Synopsis:Synopsis:
Related Subjects
Clear, detailed exposition that can be understood by readers with no background in advanced mathematics. More than 200 problems and full solutions, plus 100 numerical exercises. 1949 edition.
"Synopsis"
by Hold All,"Synopsis"
by Ingram, |
Elementary Numerical Analysis
9780471433378
ISBN:
0471433373
Edition: 3 Pub Date: 2003 Publisher: Wiley
Summary: Offering a clear, precise, and accessible presentation, complete with MATLAB programs, this new Third Edition of Elementary Numerical Analysis gives students the support they need to master basic numerical analysis and scientific computing. Now updated and revised, this significant revision features reorganized and rewritten content, as well as some new additional examples and problems. The text introduces core areas... of numerical analysis and scientific computing along with basic themes of numerical analysis such as the approximation of problems by simpler methods, the construction of algorithms, iteration methods, error analysis, stability, asymptotic error formulas, and the effects of machine arithmetic.
Kendall Atkinson is the author of Elementary Numerical Analysis, published 2003 under ISBN 9780471433378 and 0471433373. Six hundred sixty eight Elementary Numerical Analysis textbooks are available for sale on ValoreBooks.com, one hundred ten used from the cheapest price of $58.68, or buy new starting at $68 |
...
Show More integrates applications and examples throughout and includes historical notes in many chapters. The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry's usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers. * Realistic applications integrated throughout the text, including (but not limited to): - Symmetries of artistic patterns- Physics- Robotics- Computer vision- Computer graphics- Stability of architectural structures- Molecular biology- Medicine- Pattern recognition* Historical notes included in many chapters* Instructor's Manual with solutions available for all adopters of the text |
Eighth Grade Mathematics Curriculum
This course is aligned with the Common Core Standards for 8th grade mathematics and focuses on three critical areas.
First, the students will learn to formulate expressions and equations, show the association of data with a linear equation, and to solve linear equations. The students will become comfortable using the linear equation, y = mx + b. They will understand that m represents the rate of change of the two variables. In addition they will understand the graphs are lines through the origin. The students will become comfortable using a linear equation to describe the relationship between two values in bivariate data. They will also be able to solve problems with one linear equation and systems with two linear equations.
Second, the students will learn to understand functions and to use a function to describe quantitative relationships. They will understand that functions help to describe situations where one quantity is dependent on another. Third, the students will learn to assess two- and three-dimensional shapes using distance, angle, and similarity using ideas about distance and angles and how they behave. The students will understand the Pythagorean Theorem and be able to explain why it is true. They will learn to use the theorem to find distances between points on the coordinate plane, to find lengths, and to analyze triangles. The students will complete their study of volume by learning to solve for the volume of cones, cylinders, and spheres. |
two halves, the first covering an undergraduate semester and the second a masters semester. It is thorough and pitched at the right level for those who need to master the basics and then move on to more advanced applications of the theory, but does not really make an effort to aid the reader's understanding by giving reasons for following developmental paths, or by providing any intuitive rationales that would have helped to see the machinery as it works rather just how it is put together.
This is the kind of book I've looked for as an undergraduate while taking my complex variables course. This book covers the fundamentals of complex analysis - analytic functions, integration, differentiation, infinite series and products, and residue calculus - using the oh-so-familiar but oh-so-very-effective method of theorem-proof-examples with short discussions inbetween. Since this text is so well organized this alone makes it a valuable asset. But this only comprises half of the book - the other half introduces uses of complex variables, including fourier series and transforms, asymptotic expansions, ordinary differential equations, Green's functions in potential theory and many other things - a real goldmine of information, for a very low price, too! Granted, you won't find graduate-level discussions here, but this isn't what this book is meant for - undergraduates. This isn't to say this book isn't rigorous - terms such as "normal convergence" and "compact set" do occur in the book. All-in-all, this has everything you'll need unless you intend to pursue higher-level courses in complex variables.
10 of 10 people found the following review helpful
5.0 out of 5 starsEssential Text16 April 2003
By A Customer - Published on Amazon.com
Format:Paperback
A true treasure, this text outlines in a simple, concise and yet approachable manner for the undergraduate student, the topic of complex variables and functions. It also not only outlines the theory (in the first half) but also presents a wealth of applications from potential flow theory to asymptotic expansions. It must be noted that the development of the applications is rather rudimentary and I would not recommend this book as a reference in any of these subjects. One essential book for the undergraduate or graduate student in engineering, engineering-physics, etc, with strong inclincations for applied mathematics.
6 of 7 people found the following review helpful
4.0 out of 5 starsnice book, good for self-learning1 Dec 2009
By Liguo Kong - Published on Amazon.com
Format:Paperback|Amazon Verified Purchase
I got this book and learned complex variable function from it. I like this book very much. It is well-organized. I read the first part of the book (theory part) carefully, and find it quite clear and approachable. For the application part, I read the Fourier and Laplace chapters and again enjoyed them, but I was confused when reading the last chapter, "asymptotic expansions".
A little background in mathematical analysis might be helpful, but not necessary. As a person in applied science, I think the rigorousness is enough (maybe not for pure-math people). Overall, it is a good and practical book for self-learning. |
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Starting at $268Elementary and Intermediate Algebra for College Students
Mymathlab Mystatlab Student Access Card
Mymathlab Spine Sticker
Student's Solutions Manual for Elementary and Intermediate Algebra for College Students
Worksheets for Classroom or Lab Practice for Elementary and Intermediate Algebra for College Students
Summary
This package consists of the textbook plus an access kit for MyMathLab/MyStatLab. TheAngel author teammeets the needs of todayrs"s students to identify important points at a glance. The updated examples use color to highlight the variables and important notation to clearly illustrate the solution process. MyMathLabprovides a wide range of homework, tutorial, and assessment tools that make it easy to manage your course online.
Table of Contents
Preface
To the Student
1. Real Numbers
1.1 Study Skills for Success in Mathematics
1.2 Problem Solving
1.3 Fractions
1.4 The Real Number System
1.5 Inequalities
Mid-Chapter Test: Sections 1.1—1.5
1.6 Addition of Real Numbers
1.7 Subtraction of Real Numbers
1.8 Multiplication and Division of Real Numbers
1.9 Exponents, Parentheses, and the Order of Operations
1.10 Properties of the Real Number System
Chapter 1 Summary
Chapter 1 Review Exercises
Chapter 1 Practice Test
2. Solving Linear Equations
2.1 Combining Like Terms
2.2 The Addition Property of Equality
2.3 The Multiplication Property of Equality
2.4 Solving Linear Equations with a Variable on Only One Side of the Equation
Mid-Chapter Test: Sections 2.1—2.4
2.5 Solving Linear Equations with the Variable on Both Sides of the Equation
2.6 Formulas
2.7 Ratios and Proportions
Chapter 2 Summary
Chapter 2 Review Exercises
Chapter 2 Practice Test
Cumulative Review Test
3. Applications of Algebra
3.1 Changing Application Problems into Equations
3.2 Solving Application Problems
Mid-Chapter Test: Sections 3.1—3.2
3.3 Geometric Problems
3.4 Motion, Money, and Mixture Problems
Chapter 3 Summary
Chapter 3 Review Exercises
Chapter 3 Practice Test
Cumulative Review Test
4. Graphing Linear Equations
4.1 The Cartesian Coordinate System and Linear Equations in Two Variables
4.2 Graphing Linear Equations
4.3 Slope of a Line
Mid-Chapter Test: Sections 4.1—4.3
4.4 Slope-Intercept and Point-Slope Forms of a Linear Equation
Chapter 4 Summary
Chapter 4 Review Exercises
Chapter 4 Practice Test
Cumulative Review Test
5. Exponents and Polynomials
5.1 Exponents
5.2 Negative Exponents
5.3 Scientific Notation
Mid-Chapter Test: Sections 5.1—5.3
5.4 Addition and Subtraction of Polynomials
5.5 Multiplication of Polynomials
5.6 Division of Polynomials
Chapter 5 Summary
Chapter 5 Review Exercises
Chapter 5 Practice Test
Cumulative Review Test
6. Factoring
6.1 Factoring a Monomial from a Polynomial
6.2 Factoring by Grouping
6.3 Factoring Trinomials of the Form ax2 + bx + c, a = 1
6.4 Factoring Trinomials of the Form ax2 + bx + c, a ≠ 1
Mid-Chapter Test: Sections 6.1—6.4
6.5 Special Factoring Formulas and a General Review of Factoring
6.6 Solving Quadratic Equations Using Factoring
6.7 Applications of Quadratic Equations
Chapter 6 Summary
Chapter 6 Review Exercises
Chapter 6 Practice Test
Cumulative Review Test
7. Rational Expressions and Equations
7.1 Simplifying Rational Expressions
7.2 Multiplication and Division of Rational Expressions
7.3 Addition and Subtraction of Rational Expressions with a Common Denominator and Finding the Least Common Denominator
7.4 Addition and Subtraction of Rational Expressions
Mid-Chapter Test: Sections 7.1—7.4
7.5 Complex Fractions
7.6 Solving Rational Equations
7.7 Rational Equations: Applications and Problem Solving
7.8 Variation
Chapter 7 Summary
Chapter 7 Review Exercises
Chapter 7 Practice Test
Cumulative Review Test
8. Functions and Their Graphs
8.1 More on Graphs
8.2 Functions
8.3 Linear Functions
Mid-Chapter Test: Sections 8.1—8.3
8.4 Slope, Modeling, and Linear Relationships
8.5 The Algebra of Functions
Chapter 8 Summary
Chapter 8 Review Exercises
Chapter 8 Practice Test
Cumulative Review Test
9. Systems of Linear Equations
9.1 Solving Systems of Equations Graphically
9.2 Solving Systems of Equations by Substitution
9.3 Solving Systems of Equations by the Addition Method
9.4 Solving Systems of Linear Equations in Three Variables
Mid-Chapter Test: Sections 9.1—9.4
9.5 Systems of Linear Equations: Applications and Problem Solving
9.6 Solving Systems of Equations Using Matrices
9.7 Solving Systems of Equations Using Determinants and Cramer's Rule
Chapter 9 Summary
Chapter 9 Review Exercises
Chapter 9 Practice Test
Cumulative Review Test
10. Inequalities in One and Two Variables
10.1 Solving Linear Inequalities in One Variable
10.2 Solving Equations and Inequalities Containing Absolute Values
Mid-Chapter Test: Sections 10.1—10.2
10.3 Graphing Linear Inequalities in Two Variables and Systems of Linear Inequalities
Chapter 10 Summary
Chapter 10 Review Exercises
Chapter 10 Practice Test
Cumulative Review Test
11. Roots, Radicals, and Complex Numbers
11.1 Roots and Radicals
11.2 Rational Exponents
11.3 Simplifying Radicals
11.4 Adding, Subtracting, and Multiplying Radicals
Mid-Chapter Test: Sections 11.1—11.4
11.5 Dividing Radicals
11.6 Solving Radical Equations
11.7 Complex Numbers
Chapter 11 Summary
Chapter 11 Review Exercises
Chapter 11 Practice Test
Cumulative Review Test
12. Quadratic Functions
12.1 Solving Quadratic Equations by Completing the Square
12.2 Solving Quadratic Equations by the Quadratic Formula
12.3 Quadratic Equations: Applications and Problem Solving
Mid-Chapter Test: Sections 12.1—12.3
12.4 Factoring Expressions and Solving Equations That Are Quadratic in Form |
Esta investigação visa analisar o modo como a resolução de tarefas de natureza exploratória e investigativa, envolvendo o uso da calculadora da gráfica, contribui para a compreensão e aprendizagem das funções quadráticas dos alunos. A metodologia insere-se no paradigma interpretativo e segue uma abordagem qualitativa, baseada em estudos de caso. Foram seleccionados, de uma escola secundária, uma turma do 10.º ano de escolaridade (científico-humanístico) e dois alunos desta turma. A recolha de dados recorreu a duas entrevistas clínicas realizadas individualmente a dois alunos, uma antes outra depois da unidade de ensino "Funções quadráticas", complementada por observação de aulas, registos áudio, resoluções de tarefas de investigação e relatórios escritos produzidos pelos alunos. Os resultados obtidos mostram que os alunos revelam diversas dificuldades na compreensão do conceito de função em diferentes representações e essas dificuldades não foram superadas após a realização da unidade de ensino. Também permitem concluir que os alunos sabem identificar as propriedades da função nas representações gráfica e algébrica revelando, portanto, que reificaram algumas propriedades da função afim e da função quadrática. A realização de tarefas de investigação, por parte dos alunos, possibilitou a utilização de vários processos característicos da actividade matemática. No entanto, na resolução de problemas, alguns utilizaram principalmente processos algébricos e usaram processos gráficos apenas quando a natureza da tarefa proporciona. Outros usaram também processos gráficos com a ajuda da calculadora. Os processos matemáticos utilizados durante o trabalho investigativo foram influenciados pela natureza da tarefa, conhecimento adquirido, experiência prévia e competência em usar a calculadora gráfica. This study aims at analyzing how the resolution of exploratory and research tasks using the graphic calculator contributes to the understanding and learning of quadratic functions by students. The methodology is based on the interpretative paradigm and follows a qualitative approach on case studies. A grade-ten class (science and humanities course) from a secondary school and two students from this class were selected. The collecting of data involved two clinical interviews to two students separately: one before the teaching unit "Quadratic functions", the other afterwards. Class observation, audio recordings, resolution of research tasks and written reports by the students also contributed to the data collection. The results show that the students revealed several difficulties in what regards understanding the concept of function in different representations. Those difficulties were not overcome after the study unit was taught. They also indicate that students were able to identify the properties of the function in the graphic and algebraic representations and thus understood some of the properties of the linear and quadratic functions. When students carried out research tasks, they used several distinctive mathematical processes. However, when solving problems, some used mainly algebraic processes and only when the nature of the task allowed it, did they use graphic processes. Others also used graphic processes with the help of the graphic calculator. The mathematical processes used during research were influenced by the nature of the task, the previous practice and ability to use the graphic calculator. |
EXAMINER TIPS for IGCSE Mathematics 0580 Cambridge Students. EXAMINER TIPS for IGCSE Mathematics 0580. How to use these tips. These tips highlight some common mistakes made by students. They are collected under
General Mathematics: Revision and Practice - David Rayner - Google. General Mathematics: Revision and Practice is a comprehensive resource for self-study or teacher-led courses to take GCSE and Standard Grade candidates right up to A*
HOW TO PREPARE YOUR EXTENDED ABSTRACT FOR THE INTERNATIONAL. Proccedings of the International Symposium on Rehabilitation of Tropical Rainforest Ecosystems 2011 HOW TO PREPARE YOUR EXTENDED ABSTRACT FOR THE INTERNATIONAL
Instructions on Preparing an Extended Abstract for the. plate will help us prepare the Volume of Abstracts with ease, which [2] O. U¨ gur, S. A. Author, Instructions on Preparing an Extended Abstract fo? r the International
Teachers Guide to Teaching Mathematics for English Language Learners. It is recognized that some Spanish-speaking groups prefer to be called benefit English Language Learners. Teachers guide created by the SHSU MELL Group, November 2005
DAIRY SHEEP BASICS FOR BEGINNERS David L. Thomas. crucial problem because many rendering companies which collect dead stock at the farm will no longer pick-up sheep carcasses. Other disposal options to investigate |
Algebra I This course allows students to develop a solid foundation in basic algebra skills and concepts. Topics include algebraic vocabulary,properties and their operations, linear sentences, lines and distance,slopes and lines, exponents and powers, polynomials, and systems of equations. Further, Algebra I allows students to develop mathematical power through problem solving strategies, reasoning activities and cooperative learning projects using appropriate tools.
Geometry This course reviews basic algebraic concepts and then introduces the elements of inductive and deductive reasoning as they relate to the study of geometry. Geometry topics include perpendicular lines,parallel lines and planes, congruent triangles, similar polygons,circles, arcs, triangles, geometric constructions and loci coordinate geometry, and areas and volumes of various types of figures. Through the use of geometry, students become problem solvers who are able to meet the demands of tomorrow's world. Prerequisite: Algebra I. Algebra II This course consists of a review of Algebra I topics and further develops the concepts of polynomials, factoring, relations, functions,solutions of linear equations, rational, irrational and complex numbers. The course then introduces the study of quadratic equations,logarithms, and elementary trigonometry. Algebra II allows students to enhance their creative thinking by interpreting the application of algebraic principles to related technology and scientific use.
Algebra II Honors This is an accelerated course of study with an in-depth development of the topics listed in Algebra II. The course is also technology rich with project-based learning including a summer technology component and a collaboration of student, CTE instructor and academic instructor in the creation of a video showing the relationship of mathematical concepts to a student's specific program of study.
Pre-Calculus *
This course consists of a review of the concepts taught in Algebra II and geometry as they relate to the principles of trigonometry.Development of the relationship between functions and their graphs is explored with extensive use of the graphing calculator incorporated throughout the course. Systems of linear equations and inequalities,including matrices are covered with application to technology where possible. After completing Pre-Calculus, students have a strong foundation for work in calculus and problem-solving applications necessary in a technical field. Prerequisite: Algebra II and Geometry.
Pre-Calculus Honors *
This is an accelerated course of study with an in-depth development of the topics listed in Pre-Calculus.
Calculus
This course introduces elementary topics of Calculus including limits, continuity and curve sketching. It also applies differentiation to minimum and maximum and related rate problems and integration to surface areas and volumes. It is recommended for students who will need to take calculus in college. Prerequisite: Pre-Calculus.
Probability and Statistics *
Probability emphasizes simulations of real world problems that involve students in experimenting, collecting, organizing and using data. Statistics emphasizes "making sense of data" by exploring and organizing relevant data in a variety of ways. Students learn about modeling trends and predicting the behavior of systems over time.Extensive use of the statistical calculator and applications that are relevant to the student's occupational program are incorporated.Prerequisite: Algebra II and Geometry.
Algebra I Math Lab 9
This math class provides an individual approach to Algebra I along with reinforcement of basic math skills. IEP goals are targeted with progress being closely monitored so that instruction can be adjusted as needed. It is a progressively challenging class. Algebra I topics taught are: simplifying expressions, plotting coordinate plan, solving slope, order of operations, linear equations and data analysis.
Geometry Math Lab 10
This math class provides an individual approach to Geometry along with reinforcement of basic math skills. IEP goals are targeted with progress being closely monitored so that instruction can be adjusted as needed. It is a progressively challenging class. Geometry topics taught are: basic geometry concepts (points, lines, distance, midpoints)angles, perpendicular and parallel lines and triangles.
Algebra II Math Lab 11
This math class provides an individual approach to Algebra II along with reinforcement of basic math skills. IEP goals are targeted with progress being closely monitored so that instruction can be adjusted as needed. It is a progressively challenging class. Algebra II topics taught are: exponents, scientific notation, probability, quadratic functions and equations, polynomials and factoring, and rational expressions and equations.
Consumer Math Lab 12
This math class provides an individual approach to consumer math along with reinforcement of basic math skills. IEP goals are targeted with progress being closely monitored so that instruction can be adjusted as needed. It is a progressively challenging class. Consumer math topics taught are: expenses (wants and needs), making a budget,salary (gross, net, deductions) retirement, stock market, using a checking account, finding suitable housing and transportation.
Integrated Math
This course provides a structured approach to a variety of topics such as ratios, percents, equations, inequalities, geometry, graphing and probability and statistics. A solid foundation in these topics with real-world applications to the more abstract algebraic concepts can be found throughout the text. Various activity labs in each chapter ensures students receive the visual and special instruction necessary to conceptualize these abstract concepts, better preparing them for life in the work force. |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). |
Program of Study: Mathematics
Math Placement and Prerequisite Requirements: What are the prerequisite requirements and how can I meet them?
The important thing to remember about placement and prerequisites is that the prerequisite for each course has been developed with the sole purpose of ensuring that students have the skills they need to be successful in the course they select. A student's placement into a math course may be determined using one or more methods and is related to how a student demonstrates that they meet a course's prerequisite requirements.
CLC has two types of requirements that affect enrollment in math courses.
1. Basic Algebra Readiness: Incoming students will need to demonstrate Basic Algebra Readiness before enrolling in certain courses at CLC. These courses may be in math or other science or technology-related fields. In the past, Basic Algebra Readiness was called Math Proficiency. Both terms mean that a student possesses a certain level of competency in arithmetic, which includes problem solving involving integers, fractions, ratios, decimals, and percents. See all of the different ways that a student may demonstrate Basic Algebra Readiness.
2. Prerequisites: Students must also demonstrate that they meet the mathematics prerequisite for the specific course they wish to take.
Different ways that all or part of a prerequisite can be met
For many math courses, the prerequisite may be met all or in part by submitting a transcript to the EMPS Division Office from an accredited college with successful grades in certain courses or by obtaining successful scores on certain nationally accepted exams.
Previous college coursework may fulfill prerequisites for math courses.
Submit the prerequisite check form with your transcripts (official or unofficial) together via:
Or in person: Room B101 at Grayslake Campus, College of Lake County, 19351 W. Washington St., Grayslake, IL 60030
Form and transcript MUST be submitted together via email, mail, fax, or in person. If not submitted together the request will not be processed.
It is possible to earn college credits in mathematics through taking certain nationally accepted exams such as CLEP, DANTES, and Advanced Placement exams. Credits earned through this method can be used to meet all or part of a prerequisite for a CLC mathematics course. Download the credit by exam document for information regarding these exams and earning credits at CLC. |
Channels: Primary and Secondary Education
In this video, get a quick introduction to the Wolfram Education Portal, which features teaching and learning tools created with Mathematica and Wolfram|Alpha, including a dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more Chinese Portuguese Russian Spanish audio.
Mathematica gives students the power to manipulate interactive graphics and develop complex data models. High-school teacher Abby Brown shares the success she experiences by using Mathematica in her classroom.
This screencast provides an overview of functionality in Mathematica 7 that makes it easy for educators to integrate into precollege, community college, and higher-education classrooms. It includes many examples of Mathematica's use for education that can be implemented immediately.
In this Wolfram Mathematica Virtual Conference 2011 course, learn different ways to use Mathematica to enhance your calculus class, such as using interactive models and connecting calculus to the real world with built-in datasets.
computerbasedmath.org is a project to build a new math curriculum with computer-based computation at its heart. In this talk from the Wolfram Technology Conference 2011, Conrad Wolfram discusses the concept, progress, and plans.
Wolfram|Alpha generates answers to questions in real time by doing computations on its own vast internal knowledge base—making it a valuable resource for education. In this video, educators of all levels share advice on getting started with Wolfram|Alpha.
Mathematica can be used to enhance course management systems by helping teachers easily communicate ideas, give students immediate feedback, and link real-world datasets to textbook examples. Learn more in this screencast.
See how Wolfram technologies like Mathematica and Wolfram|Alpha enhance math education. The video features visual examples of course materials, apps, and other resources to help teachers and students cover math from algebra to calculus to statistics and beyond.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Japanese audio.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Portuguese audio.
This tutorial screencast encourages users to work along in Mathematica 7 as they learn the basics to create their first notebook, calculations, visualizations, and interactive examples. Includes Spanish audio. |
Science Books
Practical Algebra: A Self-Teaching Guide, 2nd Edition
Practical Algebra If you studied algebra years ago and now need a refresher course in order to use algebraic principles on the job, or if you're a student who needs an introduction to the subject, here's the perfect book for you.
Practical Algebra is an easy and fun-to-use workout program that quickly puts you in command of all the basic concepts and tools of algebra.
With the aid of practical, real-life examples and applications, you'll learn:The basic approach and application of algebra to problem solvingThe number system (in a much broader way than you have known it from arithmetic)Monomials and polynomials; factoring algebraic expressions; how to handle algebraic fractions; exponents, roots, and radicals; linear and fractional equationsFunctions and graphs; quadratic equations; inequalities; ratio, proportion, and variation; how to solve word problems, and moreAuthors Peter Selby and Steve Slavin emphasize practical algebra throughout by providing you with techniques for solving problems in a wide range of disciplines—from engineering, biology, chemistry, and the physical sciences, to psychology and even sociology and business administration.
Step by step, Practical Algebra shows you how to solve algebraic problems in each of these areas, then allows you to tackle similar problems on your own, at your own pace.
Self-tests are provided at the end of each chapter so you can measure your mastery..
For more information about the title Practical Algebra: A Self-Teaching Guide, 2nd Edition, read the full description at Amazon.com, or see the following related books:
Algebra for Dummies — One of the most commonly asked questions in a mathematics classroom is, "Will I ever use this stuff in real life?" Some teachers can give a good, ... > read more
Physics — Improving the Game When it comes to teaching and learning physics, most pedagogical innovations were pioneered in Cutnell and Johnson |
Ontario
Ontario grade 10 math curriculum
Alignments coming soon
IXL's grade 10 skills will
be aligned to the 2005 Ontario Curriculum soon!
Until then, you can view a complete list of
grade 10
objectives below.
Be sure to check out the
unlimited
math practice problems in IXL's
282
grade 10 skills.
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MA.10. Principles of Mathematics (Academic)
MA.10.1 Mathematical process expectations.2.1 Overall Expectations
MA.10.2.1.1 determine the basic properties of quadratic relations;
MA.10.2.1.3 solve quadratic equations and interpret the solutions with respect to the corresponding relations;
MA.10.2.1.4 solve problems involving quadratic relations.
MA.10.2.2 Investigating the Basic Properties of Quadratic Relations
MA.10.2.22.2.2 determine, through investigation with and without the use of technology, that a quadratic relation of the form y = ax² + bx + c (a "not equal to" 0) can be graphically represented as a parabola, and that the table of values yields a constant second difference (Sample problem: Graph the relation y = x² - 4x by developing a table of values and plotting points. Observe the shape of the graph. Calculate first and second differences. Repeat for different quadratic relations. Describe your observations and make conclusions, using the appropriate terminology.);
MA.10.2.2 and use the appropriate terminology to describe them;
MA.10.2.2.4 compare, through investigation using technology, the features of the graph of y = x² and the graph of y = 2 to the x power, and determine the meaning of a negative exponent and of zero as an exponent (e.g., by examining patterns in a table of values for y = 2 to the x power; by applying the exponent rules for multiplication and division).
MA.10.2.3 Relating the Graph of y = x² and Its Transformations
MA.10.2.3.1 identify, through investigation using technology, the effect on the graph of y = x² of transformations (i.e., translations, reflections in the x-axis, vertical stretches or compressions) by considering separately each parameter a, h, and k [i.e., investigate the effect on the graph of y = x² of a, h, and k in y = x² + k, y = (x - h)², and y = ax²];
MA.10.2.3.2 explain the roles of a, h, and k in y = a(x - h)² + k, using the appropriate terminology to describe the transformations, and identify the vertex and the equation of the axis of symmetry;
MA.10.2.4.3 determine, through investigation, and describe the connection between the factors of a quadratic expression and the x-intercepts (i.e., the zeros) of the graph of the corresponding quadratic relation, expressed in the form y = a(x - r)(x - s);
MA.10.2.4.4 interpret real and non-real roots of quadratic equations, through investigation using graphing technology, and relate the roots to the x-intercepts of the corresponding relations;
MA.10.2.4.7 explore the algebraic development of the quadratic formula (e.g., given the algebraic development, connect the steps to a numerical example; follow a demonstration of the algebraic development [student reproduction of the development of the general case is not required]);
MA.10.2.5 Solving Problems Involving Quadratic Relations
MA.10.2.5.1 determine the zeros and the maximum or minimum value of a quadratic relation from its graph (i.e., using graphing calculators or graphing software) or from its defining equation (i.e., by applying algebraic techniques);
MA.10.2.5.2 solve problems arising from a realistic situation represented by a graph or an equation of a quadratic relation, with and without the use of technology (e.g., given the graph or the equation of a quadratic relation representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?).
MA.10.3 Analytic Geometry
MA.10.3.1 Overall Expectations
MA.10.3.1.1 model and solve problems involving the intersection of two straight lines;
MA.10.3.1.2 solve problems using analytic geometry involving properties of lines and line segments;
MA.10.3.2.2 solve problems that arise from realistic situations described in words or represented by linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: The Robotics Club raised $5000 to build a robot for a future competition. The club invested part of the money in an account that paid 4% annual interest, and the rest in a government bond that paid 3.5% simple interest per year. After one year, the club earned a total of $190 in interest. How much was invested at each rate? Verify your result.).
MA.10.3.3 Solving Problems Involving Properties of Line Segments
MA.10.3.3.1 develop the formula for the midpoint of a line segment, and use this formula to solve problems (e.g., determine the coordinates of the midpoints of the sides of a triangle, given the coordinates of the vertices, and verify concretely or by using dynamic geometry software);
MA.10.3.3.2 develop the formula for the length of a line segment, and use this formula to solve problems (e.g., determine the lengths of the line segments joining the midpoints of the sides of a triangle, given the coordinates of the vertices of the triangle, and verify using dynamic geometry software);
MA.10.3.3.3 develop the equation for a circle with centre (0, 0) and radius r, by applying the formula for the length of a line segment;
MA.10.3.3.4 Determine the radius of a circle with centre (0, 0), given its equation; write the equation of a circle with centre (0, 0), given the radius; and sketch the circle, given the equation in the form x² + y² = r²;
MA.10.3.3.5 solve problems involving the slope, length, and midpoint of a line segment (e.g., determine the equation of the right bisector of a line segment, given the coordinates of the endpoints; determine the distance from a given point to a line whose equation is given, and verify using dynamic geometry software).
MA.10.3.4 Using Analytic Geometry to Verify Geometric Properties
MA.10.3.4.1 determine, through investigation (e.g., using dynamic geometry software, by paper folding), some characteristics and properties of geometric figures (e.g., medians in a triangle, similar figures constructed on the sides of a right triangle);
MA.10.3.4.2 verify, using algebraic techniques and analytic geometry, some characteristics of geometric figures (e.g., verify that two lines are perpendicular, given the coordinates of two points on each line; verify, by determining side length, that a triangle is equilateral, given the coordinates of the vertices);
MA.10.3.4.3 plan and implement a multi-step strategy that uses analytic geometry and algebraic techniques to verify a geometric property (e.g., given the coordinates of the vertices of a triangle, verify that the line segment joining the midpoints of two sides of the triangle is parallel to the third side and half its length, and check using dynamic geometry software; given the coordinates of the vertices of a rectangle, verify that the diagonals of the rectangle bisect each other).
MA.10.4 Trigonometry
MA.10.4.1 Overall Expectations
MA.10.4.1.1 use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;
MA.10.4.1.2 solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
MA.10.4.1.3 solve problems involving acute triangles, using the sine law and the cosine law.
MA.10.4.2.1 verify, through investigation (e.g., using dynamic geometry software, concrete materials), the properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);
MA.10.4.2.2 describe and compare the concepts of similarity and congruence;
MA.10.44.3 Solving Problems Involving the Trigonometry of Right Triangles
MA.10.44.3.2 determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
MA.10.4.3.3 solve problems involving the measures of sides and angles in right triangles in real-life applications (e.g., in surveying, in navigating, in determining the height of an inaccessible object around the school), using the primary trigonometric ratios and the Pythagorean theorem.
MA.10.4.4.1 explore the development of the sine law within acute triangles (e.g., use dynamic geometry software to determine that the ratio of the side lengths equals the ratio of the sines of the opposite angles; follow the algebraic development of the sine law and identify the application of solving systems of equations [student reproduction of the development of the formula is not required]);
MA.10.4.4.2 explore the development of the cosine law within acute triangles (e.g., use dynamic geometry software to verify the cosine law; follow the algebraic development of the cosine law and identify its relationship to the Pythagorean theorem and the cosine ratio [student reproduction of the development of the formula is not required]);
MA.10.4.4.3 determine the measures of sides and angles in acute triangles, using the sine law and the cosine law (Sample problem: In triangle ABC, angle A = 35°, angle B = 65°, and AC = 18 cm. Determine BC. Check your result using dynamic geometry software.);
MA.10.4.4.4 solve problems involving the measures of sides and angles in acute triangles.
MA.10. Foundations of Mathematics (Applied)
MA.10.1 Mathematical process expectations1.7 Communicating
MA.10.1.7.1 communicate mathematical thinking orally, visually, and in writing, using mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
MA.10.2 Measurement and Trigonometry
MA.10.2.1 Overall Expectations
MA.10.2.1.1 use their knowledge of ratio and proportion to investigate similar triangles and solve problems related to similarity;
MA.10.2.1.2 solve problems involving right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
MA.10.2.1.3 solve problems involving the surface areas and volumes of three-dimensional figures, and use the imperial and metric systems of measurement.
MA.10.2.2 Solving Problems Involving Similar Triangles
MA.10.2.2.1 verify, through investigation (e.g., using dynamic geometry software, concrete materials), properties of similar triangles (e.g., given similar triangles, verify the equality of corresponding angles and the proportionality of corresponding sides);
MA.10.2.2.2 determine the lengths of sides of similar triangles, using proportional reasoning;
MA.10.22.3 Solving Problems Involving the Trigonometry of Right Triangles
MA.10.22.3.2 determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios and the Pythagorean theorem;
MA.10.2.3.3 the height attained.);
MA.10.2.3.4 the height attained.);
MA.10.2.3.5 describe, through participation in an activity, the application of trigonometry in an occupation (e.g., research and report on how trigonometry is applied in astronomy; attend a career fair that includes a surveyor, and describe how a surveyor applies trigonometry to calculate distances; job shadow a carpenter for a few hours, and describe how a carpenter uses trigonometry).
MA.10.2.4 Solving Problems Involving Surface Area and Volume, Using Imperial and Metric Systems of Measurement
MA.10.2.4.1 use the imperial system when solving measurement problems (e.g., problems involving dimensions of lumber, areas of carpets, and volumes of soil or concrete);
MA.10.2.4.2 perform everyday conversions between the imperial system and the metric system (e.g., millilitres to cups, centimetres to inches) and within these systems (e.g., cubic metres to cubic centimetres, square feet to square yards), as necessary to solve problems involving measurement (Sample problem: A vertical post is to be supported by a wooden pole, secured on the ground at an angle of elevation of 60°, and reaching 3 m up the post from its base. If wood is sold by the foot, how many feet of wood are needed to make the pole?);
MA.10.2.4.3 determine, through investigation, the relationship for calculating the surface area of a pyramid (e.g., use the net of a square-based pyramid to determine that the surface area is the area of the square base plus the areas of the four congruent triangles);
MA.10.2.4.4 solve problems involving the surface areas of prisms, pyramids, and cylinders, and the volumes of prisms, pyramids, cylinders, cones, and spheres, including problems involving combinations of these figures, using the metric system or the imperial system, as appropriate (Sample problem: How many cubic yards of concrete are required to pour a concrete pad measuring 10 feet by 10 feet by 1 foot? If poured concrete costs $110 per cubic yard, how much does it cost to pour a concrete driveway requiring 6 pads?).
MA.10.3.2.2 determine the value of a variable in the first degree, using a formula (i.e., by isolating the variable and then substituting known values; by substituting known values and then solving for the variable) (e.g., in analytic geometry, in measurement) (Sample problem: A cone has a volume of 100 cm³. The radius of the base is 3 cm. What is the height of the cone?);
MA.10.3.2.3 express the equation of a line in the form y = mx + b, given the form Ax + By + C = 0.
MA.10.3.3 Graphing and Writing Equations of Lines
MA.10.3.3.1 connect the rate of change of a linear relation to the slope of the line, and define the slope as the ratio m = rise/run;
MA.10.3.3.2 identify, through investigation, y = mx + b as a common form for the equation of a straight line, and identify the special cases x = a, y = b;
MA.10.3.3.3 identify, through investigation with technology, the geometric significance of m and b in the equation y = mx + b;
MA.10.3.3.4 identify, through investigation, properties of the slopes of lines and line segments (e.g., direction, positive or negative rate of change, steepness, parallelism), using graphing technology to facilitate investigations, where appropriate;
MA.10.3.3.5 graph lines by hand, using a variety of techniques (e.g., graph y = 2/3x - 4 using the y-intercept and slope; graph 2x + 3y = 6 using the x- and y-intercepts);
MA.10.3.3.6 determine the equation of a line, given its graph, the slope and y-intercept, the slope and a point on the line, or two points on the line.
MA.10.3.4 Solving and Interpreting Systems of Linear Equations
MA.10.3.4.1 determine graphically the point of intersection of two linear relations (e.g., using graph paper, using technology) (Sample problem: Determine the point of intersection of y + 2x = -5 and y = 2/3x + 3 using an appropriate graphing technique, and verify.);
MA.10.3.4.3 solve problems that arise from realistic situations described in words or represented by given linear systems of two equations involving two variables, by choosing an appropriate algebraic or graphical method (Sample problem: Maria has been hired by Company A with an annual salary, S dollars, given by S = 32 500 + 500a, where a represents the number of years she has been employed by this company. Ruth has been hired by Company B with an annual salary, S dollars, given by S = 28 000 + 1000a, where a represents the number of years she has been employed by that company. Describe what the solution of this system would represent in terms of Maria's salary and Ruth's salary. After how many years will their salaries be the same? What will their salaries be at that time?).
MA.10.4.3 Identifying Characteristics of Quadratic Relations
MA.10.4.34.3.2 determine, through investigation using technology, that a quadratic relation of the form y = ax² + bx + c (a "is not equal to" 0) can be graphically represented as a parabola, and determine that the table of values yields a constant second difference (Sample problem: Graph the quadratic relation y = x² - 4, using technology. Observe the shape of the graph. Consider the corresponding table of values, and calculate the first and second differences. Repeat for a different quadratic relation. Describe your observations and make conclusions.);
MA.10.4.3 using a given graph or a graph generated with technology from its equation, and use the appropriate terminology to describe the features;
MA.10.4.3.4 compare, through investigation using technology, the graphical representations of a quadratic relation in the form y = x² + bx + c and the same relation in the factored form y = (x - r)(x - s) (i.e., the graphs are the same), and describe the connections between each algebraic representation and the graph [e.g., the y-intercept is c in the form y = x² + bx + c; the x-intercepts are r and s in the form y = (x - r)(x - s)] (Sample problem: Use a graphing calculator to compare the graphs of y = x² + 2x - 8 and y = (x + 4)(x - 2). In what way(s) are the equations related? What information about the graph can you identify by looking at each equation? Make some conclusions from your observations, and check your conclusions with a different quadratic equation.).
MA.10.4.4.1 solve problems involving a quadratic relation by interpreting a given graph or a graph generated with technology from its equation (e.g., given an equation representing the height of a ball over elapsed time, use a graphing calculator or graphing software to graph the relation, and answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball hit the ground? Over what time interval is the height of the ball greater than 3 m?);
MA.10.4.4.2 solve problems by interpreting the significance of the key features of graphs obtained by collecting experimental data involving quadratic relations (Sample problem: Roll a can up a ramp. Using a motion detector and a graphing calculator, record the motion of the can until it returns to its starting position, graph the distance from the starting position versus time, and draw the curve of best fit. Interpret the meanings of the vertex and the intercepts in terms of the experiment. Predict how the graph would change if you gave the can a harder push. Test your prediction.). |
Core-Plus Mathematics
is a five-year project funded by the National Science Foundation to
develop student and teacher materials for a complete three-year high
school mathematics curriculum for all students, plus a fourth-year
course continuing the preparation of students for college mathematics.
The Contemporary Mathematics in Context curriculum builds upon the
theme of mathematics as sense-making. Throughout it acknowledges,
values, and extends the informal knowledge of data, shape, change,
and chance that students bring to situations and problems.
Each year Contemporary Mathematics in Context features strands
of algebra and functions, geometry and trigonometry, statistics
and probability, and discrete mathematics. These strands are connected
within and across units by fundamental ideas such as symmetry, function,
matrices, data analysis and curve-fitting. The strands also are
connected across units by mathematical habits of mind such as visual
thinking, recursive thinking, searching for and describing patterns,
making and checking conjectures, reasoning with multiple representations,
inventing mathematics, and providing convincing arguments. The strands
are linked further by the fundamental themes of data, representation,
shape, and change. Important mathematical ideas are continually
revisited through these connections so that students can develop
a robust understanding of mathematics. Numerical, graphics, and
programming/link capabilities of graphics calculators are being
capitalized on to enable students to develop versatile ways of dealing
with realistic situations. The curriculum is designed to make more
mathematics accessible to more students, while at the same time
challenging the most able students.
Differences in student performance and interest can be accommodated
by the depth and level of abstraction to which topics are pursued,
by the nature and degree of difficulty of applications, and by providing
opportunities for student choice of homework tasks and projects.
Instructional practices promote mathematical thinking through the
use of rich problem situations that involve students, both in collaborative
groups and individually, in investigating, conjecturing, verifying,
applying, evaluating, and communicating mathematical ideas. Comprehensive
assessment of student understanding and progress through both curriculum-embedded
assessment opportunities and supplementary assessment tasks enables
monitoring and evaluation of each student's performance in terms
of mathematical processes, content, and dispositions.
For more information about the program
and its implementation, contact: |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for). |
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