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This is a pre-calculus introduction to limits
By the end of this lesson, the student will be able to compute a basic limit of a function using limit notation.
Pre-Class Assignment:
Students will create a poster titled "There is a Limit to….." or "There is No Limit to….". They will draw or cut out pictures from magazines. I always show them several posters created in classes before them or posters that I have created. For example: "There is No Limit to the Number of Pairs of Shoes One Can Have" or "There is a Limit to the Amount you can Spend on a Credit Card". (These posters will be hung up in the room or in the hall.)
Resources: Precalculus Textbook
Time Required: 90 minutes
Equipment:
Overhead projector or Interwrite Pad; graphics calculator (TI-84 recommended) and some method of posting a graph on a large screen (TI-Presenter/TI-84 calculator, Epson projector/document camera, etc.)
Outline:
Introduce the lesson by having students show their posters. Discuss how there is or is not a limit to the topic they picked.
Discuss how many mathematical problems involve the behavior of a function at a particular number. Present the students with the question: "What is the value of the function f(x) when x = c?" Make sure the students understand the concept by telling them that they are to find where "y" is when "x" is at a particular number. Then start explaining that the idea of
limit
is the behavior of a function (y-value) when the x-value is getting close to or near a particular number, rather than when x = a particular number.
Introduce the word
approach
. Start with an example such as:
Walk to a particular desk and stand close to it; then go back and walk to the same desk and sit down in it. Ask students what the difference is in those 2 different actions.
Another example would be to point out on the x-axis of getting close to a number and comparing that to actually getting to the number.
It is very important that students understand the difference between the terms approaching and actually getting there.
Give them the following examples and let them graph them on a graphics calculator:
A.
y=x 2
Ask them to find what x is getting close to when y=4.
B.
y=x 2 -x+2
Ask them to investigate the behavior for values of y as x nears 2 from the left. Point out that as 'x' gets closer to 2, 'y' gets closer to 4. Make a
Table of Values
like the following:
x
f(x)
1.0
2.000000
1.5
2.750000
1.8
3.440000
1.9
3.710000
1.95
3.852500
1.995
3.985025
1.999
3.997001
Then make a chart as 'x' approaches 2 from the right. Point out that as 'x' gets closer to 2 from, 'y' again gets closer to 4.
x
f(x)
3.0
8.000000
2.5
5.750000
2.2
4.640000
2.05
4.152000
2.01
4.030100
2.005
4.015025
2.001
4.003001
C.
y = sin x
Discuss that as x is getting closer to pi that y is approaching 0.
Then give them the formal definition of limit from the textbook.
Calculator Exploration:
Have students create a Table of Values for functions such as:
y = sin x/x y= (.1x 4 -.8x 3 -2x)/(x-4)
and discuss that is happening to the y-value when x is getting closer to 0.
Introduce the notation for limit as x is approaching a specific number. You can also re-introduce the
infinity symbol
and discuss what is happening to a specific function as 'x approaches infinity'. Demonstrate how to write the limit notation as x is approaching infinity.
Cooperative Learning Activity:
Number the students from 1-7, put them in their respective groups and give them several problems to work from the textbook using the limit notation. Then have each group to come up with their own examples to be presented tomorrow at the beginning of class. |
College Math
About This Course
"College Math covers a survey of college-level mathematics topics from basic algebra to consumer math. Emphasis is placed on the breadth of topics rather than depth. This introductory math class also prepares students for other math classes that may be required for graduation."
Course Details
Code / Name: ASMA 103 − College Mathematics (3 credits)
Description: Designed to equip the student with mathematical reasoning skills and to introduce the student to a diversity of mathematical areas. Topics will include problem solving, set theory, data interpretation, the real number system, introduction to algebra, functions, consumer mathematics and an introduction to statistics.
Note: This class is not open to students who have previously obtained a waiver of three mathematics credits of the general education requirements or have been awarded three credits by CLEP examination. |
*These are notes + solutions to herstein problems(second edition TOPICSINALGEBRA) on groups,subgroups and direct products.It is a cute pdf print of a MS word doc which explains er..:P Group theory ... (25,26 I got discouraged inspite of what herstein had to say :P (see exercises on finite
I rst taught an abstract algebra course in 1968, using Herstein's Topicsin ... hard to improve on his book; the subject may have become broader, with applications to computing and other areas, but Topics contains the core of ... high school algebra, to be able to work with coe cients that ...
The syllabus includes topics for PhD entrants too and so contains material which may often be found only in MSc courses and not BSc courses in the country. ... Herstein (iii) Basic Algebra, Jacobson (iv) Abstract Algebra, Dummit and Foote Complex Analysis.
I.N. Herstein, TopicsinAlgebra, John Wiley & Sons, 1975. Dummit and Foote, Abstract Algebra, John Wiley & Sons, 2004. ... Topics: The following is a tentative list of topics that will be covered in the course. The chapter numbers
... INTRODUCTION TO MODERN ALGEBRA II, SPRING 2014 Classroom: JWB 208 Time: MWF 12:55 { 1:45 PM ... Herstein's book. But if you do not have Artin's book, ... variety of topics and examples that fall under the general heading of rings and elds: ...
Text: TopicsinAlgebra, Second Edition, by I. N. Herstein, John Wiley & Sons. Prerequisite: MT816 or the consent of the instructor. ... three others are the easier algebra book of Herstein (used recently at BC for the undergraduate course, Math 301), ...
Definition 5.1: An Algebra over a field K, is a set A, which is a vector space over K along with a vector multiplication, ⊗, such that for a,b,c ∈ A and λ ∈ K, ... Herstein, I.N., TopicsinAlgebra, 2ed, John Wiley and Sons, 1975. 2.
... In Math 331 we will study the very basics of modern algebra (including groups, ... These topics are very well covered in the first part of the above textbook. The other books covering these topics are Abstract Algebra (Second Edition) by I.N. Herstein, A First Course in Abstract Algebra by M. |
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We're Sorry Sold Out text is for courses that are typically called (Introductory) Differential Equations, (Introductory) Partial Differential Equations, Applied Mathematics, Fourier Series and Boundary Value Problems. The text is appropriate for two semester courses: the first typically emphasizes ordinary differential equations and their applications while the second emphasizes special techniques (like Laplace transforms) and partial differential equations. The texts follows a "traditional" curriculum and takes the "traditional" (rather than "dynamical systems") approach.Differential Equations is a text that follows a traditional approach and is appropriate for a first course in ordinary differential equations (including Laplace transforms) and a second course in Fourier series and boundary value problems. Note that some schools might prefer to move the Laplace transform material to the second course, which is why we have placed the chapter on Laplace transforms in its location in the text. Ancillaries like Differential Equations with Mathematica and/or Differential Equations with Maple would be recommended and/or required ancillaries depending on the school, course, or instructor.Typical students in the text's audience are capable of reading the text. This assists them in learning how to read technical material in other texts.The exercise sets are far superior to the competition. They are more carefully graded than any competitor, including a wider range of easy and difficult exercises.New applications and extended projects have been added |
textbook is an introduction to the concept of factorization and its application to problems in algebra and number theory. With the minimum of prerequisites, the reader is introduced to the notion of rings, fields, prime elements and unique factorization. The author shows how concepts can be applied to a variety of examples such as factorizing polynomials, finding determinants of matrices and Fermat's 'two-squares theorem'. Based on an undergraduate course given at the University of Sheffield, Dr Sharpe has included numerous examples which demonstrate how frequently these ideas are useful in concrete, rather than abstract, settings. The book also contains many exercises of varying degrees of difficulty together with hints and solutions. Second and third year undergraduates will find this a readable and enjoyable account of a subject lying at the heart of much of mathematics. less |
Description of Life Of Fred: Pre-Algebra 2 With Economics - Grades 6-8 by Z Twist Books
Featuring humor, easy-to-understand explanations, and silly illustrations, Life of Fred is guaranteed to make your math studies come alive! Each text is written as a novel, including a hilarious story line based on the life of Fred Gauss. As Fred encounters the need for math during his daily exploits, he learns the methods necessary to solve his predicaments – plus loads of other interesting facts! Filled with plenty of solved examples, each book is self-teaching and reusable – perfect for families full of learners.
Introduce your students to Fred today and see how his fun, lighthearted approach to learning is revolutionizing mathematics!
Life of Fred: Pre-Algebra 2 With Economics covers the following concepts:
Domain and codomain of a function
Conversion factors
Steps in Solving Word Problems
How Not to Bore Your Horse If You Are a Jockey
One-to-one Functions
Unit Analysis
Key to a Successful Business
Five Qualities that Money Should Have
The Tulip Mania in Holland
Definitions of Capitalism, Socialism, and Communism
Payday Loans
The Tragedy of the Commons
Partnerships
Cardinality of a Set
Four Ways to Kill Competition
Freedom vs. Liberty
Why We Have a High Standard of Living
Tariffs
Demand Curves
Venn Diagrams
Ricardo's Law of Comparative Advantage
Product:
Life Of Fred: Pre-Algebra 2 With Economics - Grades 6-8
Vendor:
Z Twist Books
Minimum Grade:
6th Grade
Maximum Grade:
8th Grade
Weight:
1.65 pounds
Length:
10.25 inches
Width:
7 inches
Height:
0.9 inches
Subject:
Math
Learning Style:
Kinesthetic, Visual
Teaching Method:
Charlotte Mason, Unit Study, Unschooling
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General Question
Were you taught linear programming in high school?
I never studied linear programming and just picked up the basics of it on my own. I have been doing some online tutoring and I have worked with students doing linear programming in two dimensions. This seems like a fairly advanced topic for high school. I know just enough to help the students. Did you learn linear programming? Did they give an explanation for why it works? For the two dimensional case it is not difficult to give an intuitive explanation. I think that if the subject is taught then some sort of explanation should be given.
Yes, I took three years of it in high school from 1981 through 1983. We had one of the first, and by far the largest, high school computer labs in the area with sixteen TRS-80's and three Apple II+ machines. It was weird to switch to object-oriented programming years later.
Good lord, no. To be clear, I assume you are talking about optimization, as opposed to what @ragingloli and @ru2bz46 are talking about, which I believe is non-structured or procedural programming. Linear programming is indeed an advanced topic for high school.
@LostInParadise I've never even really done much with it, just had an overview and let Excel do the heavy lifting. If the kids are in an advance math/comp sci type of program, and are really ready for it, great for them. I can't imagine it's standard curriculum material. And honestly I can't imagine the program so advanced that there isn't some other more foundational topic they could cover. But hey, I did very poorly in math in high school. It wasn't until much later when I went to college a second time that I found out I wasn't actually just naturally bad at math.
I did not learn linear programming in high school, nor was it available to learn. However, the concepts are simple enough that it could be taught in high school, in my opinion. I mean as far as matrix methods and simple constraint optimization. I think the theory would be better saved for after calculus and I think real optimization problems (requiring Simplex or something) should be saved for more experience with differential equations.
But solving a three variable system with a matrix? Yep, they can do that.
@ragingloli and @ru2bz46 I think you mean sequential or functional programming as opposed to object-oriented programming. I've never heard it called linear programming (though that makes sense semantically) before. I could be wrong though.
My teacher for algebra 1 in 9th grade taught us really rudimentary 2-D linear programming. Although I don't think that this is typical of high school algebra courses, I think my teacher was just a little unusual. The only other optimization methods i learned in high school were from calculus. |
Thinking Mathematically - With 2 CDs - 4th edition
Summary: This general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life. Demonstrates how mathematics can be applied to readers' lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the ...show moresolution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question. A highly readable reference for anyone who needs to brush up their mathematics skillsCD Missing. Free State Books. Never settle for less.
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"Graphing Calculator Template - DOC"
Please download to view full document
7773747013822177
Technology Integration Unit
Template
Title: Investigating Slope with a Graphing Calculator
Overview of the Technology Integration Unit (Brief Description): This lesson is an
introductory activity on the effects of slope on the graphs of equations. It uses graphing
calculators to help students make the connection between changes in equations and
corresponding changes in graphs.
Grade Level: High School (General)
Time allotment: 1 Hour
Curriculum Area: Mathematics
Learning Objectives:
- Students will understand the effect of changing the x-coefficient of a linear equation on
its graph.
- Students will gain initial familiarity with the slope-intersect form of an equation.
- Students will become more skilled in calculator use.
Curriculum Standards
Delaware Mathematics Standard #1
Students will develop their ability to SOLVE PROBLEMS by engaging in
developmentally appropriate problem-solving opportunities in which there is a need to
use various approaches to investigate and understand mathematical concepts; to
formulate their own problems; to find solutions to problems from everyday situations; to
develop and apply strategies to solve a wide variety of problems; and to integrate
mathematical reasoning, communication and connections.
Delaware Mathematics Standard #3
Students will develop their ability to REASON MATHEMATICALLY by solving
problems in which there is a need to investigate significant mathematical ideas in all
content areas; to justify their thinking; to reinforce and extend their logical reasoning
abilities; to reflect on and clarify their own thinking; to ask questions to extend their
thinking; and to construct their own learning.
Delaware Mathematics Standard #7
Students will develop an understanding of ALGEBRA by solving problems in which
there is a need to progress from the concrete to the abstract using physical models,
equations and graphs; to generalize number patterns; and to describe, represent and
analyze relationships among variable quantities.
Specifically, this lesson corresponds with subgoals:
7.90 model relationships among quantities using symbols and expressions;
7.91 develop appropriate symbol sense to use algebraic technology;
7.92 use tables and graphs to interpret expressions, equations and inequalities;
7.93 describe relationships between variable quantities verbally, symbolically and
graphically (including slope as a rate of change);
7.94 translate and make connections from narrative to table, graph and function;
7.98 explore algebraic relationships using technology
Technology Standards
1 Basic operations and concepts
• Students demonstrate a sound understanding of the nature and operation of
technology systems.
• Students are proficient in the use of technology.
Performance indicators:
1. Demonstrate an understanding of concepts underlying hardware, software, and
connectivity, and of practical applications to learning and problem solving
2 Technology productivity tools
• Students use technology tools to enhance learning, increase productivity, and
promote creativity.
• Students use productivity tools to collaborate in constructing technology-
enhanced models, prepare publications, and produce other creative works.
Performance indicators:
1. Research and evaluate the accuracy, relevance, appropriateness,
comprehensiveness, and bias of electronic information sources concerning real-world
problems.
Resources
Hardware
One graphing calulator (any type, TI-82 through TI-86 or equivalent) for each student. If
less, have students work in pairs.
One overhead graphing calculator projector.
OR, 1 computer for each student (use web-based GC) and one computer with projector
diplay.
Software
None
Relevant Material
Handout for student responses
Bookmarked Sites
(if using computer instead of graphing
calculator)
Activities
To begin the lesson, use existing warm-up, review of previous lesson, or start
immediately with instruction.
To begin (this section can be removed if students are very familiar with using graphing
calculators), the teacher should graph the equation y=2x on the overhead calculator
display. Students should graph the same equation on their calculators along with her.
This will give a chance for students to go through the sequence of graphing.
Next, the teacher should ask the students to describe the graph, asking questions like
"Would you describe this line as steep?" "Where does it cross the x-axis? The y-axis?"
Now, the teacher should either use the table function on the calculator to display a table
of values for the equation, or construct one by hand. The teacher can then draw
comparisons between the equation, table, and graph (such as the fact that there is a 2 in
the equation and the y-values in the table go up by 2 each time, as well as x- and y-
intercepts). The students should draw the graph and table and write their observations in
their notes.
At this point, the students will engage in guided practice using the worksheet. The
teacher should circulate around the room looking for misconceptions, technology issues,
exemplary student responses to share, etc. The students can complete the worksheets
individually or in pairs.
For homework, the students could be asked to predict what the graph of each of the
following equations would look like. They should draw their prediction and then make a
table for each euqation to check their prediction. If the table and graph do not match,
they should sketch a new graph based on their table.
Equations:
y = 5x
y = (1/2)x
y = 0.7x
2y = x
Assessment
Students can be assessed by collecting their handouts at the beginning of the next class
period with their homework (they may need the handouts to complete the homework).
Responses could be graded based on depth of mathematical conversation, use of good
mathematics language, and the completeness of graphs and tables.
Interdisciplinary Connections
This lesson could be connected to a science lesson on graphs or fundamental equations.
For example, if students were studying physics, a connection could be made to graphs of
the equation F=MA, by asking what effect the changing the mass would have on the force
needed for acceleration.
Helpful Notes
This lesson could easily be modified to explore the effect of y-intercepts on graphs of
linear equation (by simply changing the prompts on the worksheet) or even to explore the
effects of different parameters on quadratic and cubic graphs. This lesson could also be
adapted for use in a "Families of Functions" unit in trig/precalc by varying the type of
equation as well as the parameters.
Theory to Practice Connection
This lesson reflects a constructivist learning theory because it asks students to interact
with an environment (the graphing calculator) to construct their own understanding about
the effects of these parameters. If students work on the handout in groups, the social
aspect of learning could be highlighted. In addition, this lesson could be considered to
reflect the situative perspective because it is authentic in two ways. First, it mirrors the
kinds of activities that mathematicians do to figure out problems, and second, it
encourages the use of thinking skills which are authentic to everyday |
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There's the girl who loves to draw. There's the drummer who can't sit still long enough to choke down the basics.
It's those students McHenry hopes will get help from an experimental new approach to teach algebra. Known as aesthetic computing, the approach encourages students to express equations as pictures or stories. It was pioneered not by teachers or education experts, but rather by a computer science professor with a background in simulating complex systems and a fondness for obtuse terms like "multimodeling."
McHenry, one of about a dozen teachers who attended a recent workshop at the University of Florida introducing aesthetic computing, was intrigued enough to say she would give it a try with her 11th- and 12th-graders at North Marion High School in Ocala.
"Hopefully," she said, "this is something my students can do."
Teachers have struggled to engage students' interest in math for as long as math has been taught. All the more so, perhaps, with algebra. Getting already distracted teenagers to focus on abstract, to-their-eyes useless, algebraic concepts can be Herculean, teachers say.
Many educators have tried, but few have come up with memorable solutions. As Tim Ballas, another teacher at the workshop, said, "I'm looking for anything right now that will give my students insights into concepts they will not grasp."
Paul Fishwick, a UF computer and information science engineering professor, has spent nearly a decade cogitating aesthetic computing, a term he coined. He teaches a popular UF undergraduate class on the concept, and his ideas have generated a buzz in academic circles. His course book, a 22-chapter compendium of his and others' thoughts on the concept, will be published by MIT Press later this year.
Fishwick's focus has been on university-level scholarship. But a National Science Foundation grant prompted him to rethink aesthetic computing for younger students. Based on his ideas, he and two secondary-school teachers acting as consultants, Katie Indarawis and Jodee Alice Rose, wrote an introductory curriculum for middle and high school.
The recent workshop was the first introduction of the curriculum to classroom teachers.
"To be honest," Fishwick told the crowd, which also included computer engineering students and UF education professors, "if we can't interest you in this, there's no way we can get it to the students."
The basic idea of aesthetic computing is to make abstract ideas or algebraic formulas "real" through drawings, sculptures or computer graphics — the way concepts in geometry, for example, can come to life in the plans for a house. Fishwick has also likened it to how the graphical user interface changed computer operating systems. When early versions first appeared on the Apple Macintosh, the system made long-opaque operations familiar only to computer experts accessible to lay computer users, popularizing the personal computer.
But just as house plans have dimensions that conform to geometry's rules, so the algebraic representations in aesthetic computing are meant to be rigorous and accurate.
Indarawis and Rose spent a good part of the workshop explaining the method, which involves unpacking a traditional equation into its parts and operations, and then repackaging it in very nontraditional form, one that resembles a diagram with circles and lines. The final step is the fun one: depicting the re-formed equation as a piece of art or a short story.
Rose took the teachers through the process with the equation for a straight line. The final result was "Dorm Life," a picture that included a stereo, lava lamp, power strip, several cords and plugs and an electricity bill. Each object represented a variable or operation in the equation, while the bill represented the solution, or result.
The teachers tried their hand next, seeking to dissect and then depict the Pythagorean Theorem. McHenry wound up with a garden. Ballas, who teaches at a technical high school with a culinary arts program, created a menu. They and the other teachers struggled with reforming the equations in the aesthetic computing mold, but they seemed intrigued.
Julie Edison, a teacher at Dunnellon High, said high school students do not get enough hands-on learning. Aesthetic computing may quench that need, no small achievement for an area that seems hopelessly abstract.
"This idea gives the students a structure to use with any concept, from the real fundamental all the way through trigonometry and higher math," she Millions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don't come easily, it gets worse: Most preschoolers ... full story
Sep. 16, 2013 — Researchers have found high school students in the United States achieve higher scores on a standardized mathematics test if they study from a curriculum known as integrated ... full story
Mar. 21, 2013 — Ninth-graders in Chicago Public Schools have significantly benefited from double-dose algebra (attending two consecutive periods of the subject) in ways that were not easily observable in the ... full story
ScienceDaily features breaking news and videos about the latest discoveries in health, technology, the environment, and more -- from major news services and leading universities, scientific journals, and research organizations. |
Elementary Statistics - With CD - 6th edition
Summary: Elementary Statistics is appropriate for a one-semester introductory statistics course, with an algebra prerequisite. ES has a reputation for being thorough and precise, and for using real data extensively. Students find the book readable and clear, and the math level is right for the diverse population that takes the introductory statistics course. The text thoroughly explains and illustrates concepts through an abundance of worked out examples.
Very good condition book with only light signs of previous use |
MATHEMATICS
Mathematics 8
Duration: two semesters per year
Eighth grade mathematics is the study of pre-algebra. One way to explain pre-algebra is to describe it
as a bridge between arithmetic and algebra. As you cross this bridge, you will move from working
problems that are mostly numerical to solving those that require more advanced reasoning skills. This
requires working with variables and unknowns. You will be introduced to a variety of reasoning skills
that are practical because you will use them in other disciplines and outside of school.
Algebra I
Duration: two semesters per year
Prerequisite: Current math teacher signature based on four quarter grades of B or higher in Pre-Algebra
along with a homework completion score equal to A- or higher (students who do not exhibit a strong
work ethic with consistent homework completion are not accepted into this course). NWEA RIT score
of 245 along with earning ISTEP plus (+) on their last State ISTEP exam. Open registration is not
available for this course.
Algebra I provides a formal development of algebraic skills and concepts used in a wide range of
problem-solving situations. The concept of functions is emphasized. Topics include: operations with
polynomials, quadratic equations, slope intercept, statistics and probability, exponents and exponential
functions, system of equations, and radical functions. Desire alone does not qualify a student for this
course.
This course is specifically for students who find math interesting and are self-motivated within the
context of their daily studies. The pace of the subject matter is accelerated and requires additional
homework on a weekly basis. Students must be self-disciplined and able to accept a challenging
schedule that moves at a faster rate of speed than the regular 8th grade math course. A demanding
course of this type requires a very strong skill set in Pre-Algebra mathematics.
IMPORTANT:
To advance to 9th grade Geometry, a student must have a full understanding of algebraic skills.
To take Geometry as a Freshman:
1) Students should receive 1st/2nd semester grade(s) of "B+" or higher or 2nd Semester exam
grade of "A-"or higher.
2) Students must score a minimum of 250 on the math section of their NWEA Test.
3) Student must pass the Indiana Exit Exam in May (Algebra 1 End of Course Assessment)
Middle School Algebra I credits will be awarded at the end of the Freshman year in High School
following successful completion Geometry
Students earning a "C+ or lower" in the first quarter will be placed on probationary status for the 2nd
quarter. All students with C+ or lower on the 1st Semester grade will be re-evaluated prior to starting
the second semester of Algebra I. Following a teacher recommendation and parent approval, they will
be moved to a Pre-Algebra course. Students who exhibit a consistent pattern of incomplete homework
assignments will be removed from the course by teacher recommendation.
Geometry Honors
Duration: two semesters per year
Prerequisite: teacher recommendation, NWEA RIT of 250, successful completion of Algebra I, and a
pass on the Alg I ECA(end of course assessment).
Geometry provides students with experiences that deepen the understanding of shapes and their
properties. Properties and relationships of geometric figures include the study of: angles, lines, planes,
congruent and similar triangles, trigonometric ratios, polygons, circles, and spatial drawings. An
understanding of proof logic is developed.
Recommendation: Due to the importance of fully understanding basic Algebra skills, students
taking Algebra I in 7th grade should receive 1st and 2nd semester grades of at least a "B+" or a
spring final exam grade of at least an "A-" to take Geometry Honors as an 8th grader; otherwise,
Algebra I should be retaken.
Middle School Algebra I and Geometry credits will be awarded following successful completion of
year long Algebra II at CHS. Additionally, Geometry Honors receives one additional grade point in
GPA calculations.
SCIENCE
Science 8
Duration: two semesters per year
Eighth grade Science is a two-semester course that incorporates the Indiana Academic Standards for
Science. The students will use science notebooks to help develop, practice, and refine their science
understanding, while also enhancing reading, writing, mathematics and communications. These
notebooks contain information about the students' classroom experiences and students are encouraged
to use them as scientists would, before, during, and after all investigations. They are a place where
students formulate and record their questions, make predictions, record data, procedures, and results,
compose reflections, and communicate findings. Some studies include: weather and the atmosphere,
chemistry, and genetics.
High Interest Science 8
Duration: two semesters per year
Prerequisite: teacher recommendation
Students will cover the same standards and requirements found in 8th grade Science Class. Students'
skills will be expanded, creativity challenged, and independence fostered. Possessing a high interest in
science curriculum is of primary importance. Additionally, students must have a strong performance
on assessments (ISTEP, NWEA, and classroom tests) and have teacher recommendation.
SOCIAL STUDIES
Social Studies 8
Duration: two semesters per year
Eighth grade Social Studies will consist of two semesters of United States History. This course will
follow the path of American History from exploration to the Civil War. Students will need to
demonstrate an understanding of the interaction between political, economic and social forces within a
country or region. Skills such as note taking, identifying main ideas and concepts, and summarizing
are essential to this course. The course deals more with why events and situations exist as opposed to
the collection of statistics. Students must be able to analyze data and draw conclusions. Much of the
material in this course is interactive and requires several steps in the analysis process. Following
directions is completely essential.
High Interest Social Studies 8
Duration: two semesters per year
Prerequisite: teacher recommendation
Students will cover the same standards and requirements found in 8th grade Social Studies Class.
Students' skills will be expanded, creativity challenged, and independence fostered. Possessing a high
interest in social studies curriculum is of primary importance. Additionally, students must have a
strong performance on assessments (ISTEP, NWEA, and classroom tests) and have teacher
recommendation.
ENGLISH
English 8
Duration: two semesters per year
Students will study all basic areas of communication: listening, reading, writing, and speaking.
Listening techniques will be emphasized in conjunction with note taking and outlining. The study of
literature will encompass works of diverse genres and authors. Students will study the use of common
literary devices. Writing skills will be developed through the study of grammar and the use of the 6+1
Traits of Writing model. Students will have the opportunity to enhance their writing skills through the
study and implementation of the four basic writing formats: narrative, persuasive, descriptive, and
expository. Research skills and techniques will be applied to at least one full-length research report. A
study of vocabulary will include the influence of historical events on our language and a study of
Greek and Latin word roots, prefixes, and suffixes. In all areas of the curriculum, expectations will be
set to better prepare the students for high school.
Accelerated English
Duration: two semesters per year
Prerequisite: teacher recommendation
Students will cover the same curriculum as the English 8 students; however, accelerated classes will
also include various assignments, discussions, and explorations, which will expand students' skills and
challenge their creativity and foster their independence.
ROTATION PERIOD
Keyboarding and Computer Applications 8
Duration: one nine week period
Keyboarding offers the learning skills of the touch system for alphabetic and numeric keys. Students
continue to master skills in word processing and spreadsheets. They will learn to incorporate
hyperlinks and other multimedia features in Power Point.
Art 8
Duration: one nine week period
These courses are designed to make the student aware of the basic art elements as well as to develop a
more knowledgeable appreciation of the aesthetic quality of the art objects he/she creates in various
media. The areas of study include: color and design, drawing, painting, construction, and art history.
The objectives of these lessons have been planned to build upon each other, using information learned
in an earlier project to aid them in creating a qualitative art object. The principles that a student
learned in the seventh grade will be reinforced and advanced during this course.
Personal Finance 8
Duration: one nine week period
In this 9-week class, eighth graders will study basic individual financing. Curriculum will enhance the
student's understanding of income, banking, loans, and other fundamental elements of personal
finance. Basic accounting concepts are applied to "real life" situations in order to improve financial
decision making skills of students.
Technology Education 8
Duration: one nine week period
This course is structured around the State Introductory Indicators for Technology Education at the
middle school level. An introductory activity-based course in which students are introduced to the
importance of technology and the principles used to design, produce, use, and assess it. The students
develop both individual and group abilities needed to participate in and contribute to society.
SPANISH
Spanish 1
Duration: two semesters per year
For a Carroll Middle 7th grader to sign up for Spanish 1 in 8th grade, he or she must meet at least 2 of
the 3 criteria:
1. Pass 7th grade ISTEP English
2. Obtain a 1st semester grade of a C- or better in English 7.
3. Obtain his/her current English teacher recommendation.
Students who successfully complete Spanish I at CMS will be awarded high school credits following
successful completion of year long Spanish II at CHS.
Note: CMS reserves the right to move students to Charger Time who are enrolled in Spanish but
failing Core classes during their 8th grade year.
Level 1 foreign language courses provide instruction enabling students to discuss the many reasons for
learning languages and to develop an understanding of the people who speak them. Students are able
to apply effective strategies for language learning and show a willingness to experience various aspects
of the culture. Within this context, the course provides students with opportunities to:
Respond to and give oral directions and commands and to make routine requests in the classroom and
in public places;
Understand and use appropriate forms of address in courtesy expressions and be able to tell about daily
routines and events;
Ask and answer simple questions and participate in brief guided conversations related to their needs
and interests;
Read isolated words and phrases in a situational context, such as menus, signs, and schedules;
Comprehend brief written directions and information;
Read short narrative texts on simple topics;
Write familiar words and phrases in appropriate contexts and respond in writing to various stimuli.
They will learn:
About nonverbal communication, such as gestures and body language
About awareness of current events in the cultures
The major holidays and geographical features of the countries being studied
Greeting and leave taking behaviors in a variety of social situations
The appropriate way to respond to introductions and use courtesy behaviors
Appropriate etiquette in a variety of social settings
CHARGER TIME
Charger Time 8
Duration: two semesters per year
Charger Time is a structured study hall. It is also used as a time for students to receive homework
help. Charger Time also serves as an extension of the classroom for projects that can't be finished
during regular classroom time. It can also provide extra time for test taking and assignments.
PHYSICAL EDUCATION
Physical Education 8
Duration: 18 weeks (two 9-week sessions)
Building on the seventh grade program, the physical education program is designed to further develop
gross and fine motor skills and to improve cardiovascular and muscular fitness. This is accomplished
by participation in team sports, individual sports and lifetime and recreational sports. Basic rules are
taught and tested so the students have a working knowledge of each activity. Team sports include:
Soccer, Speedball, Lacrosse, Flag Football, Basketball, Volleyball, Softball, Floor Hockey, Team
Handball, Ultimate Frisbee and Badminton. Individual and recreational sports include: Table Tennis,
Tennis and Pickleball. As an integrated portion of the physical educational curriculum, students will
be involved in the L.I.F.E. (Learning Involves Fitness and Education) Program. This program will
concentrate on individual fitness for all students, by using aerobic fitness equipment and strength
training equipment. Students' fitness and skills proficiency are developed by alternating between the
L.I.F.E Room and the team sport activities on a rotating basis.
HEALTH
Health 8
Duration: 18 weeks (two 9-week sessions)
This course builds on the seventh grade curriculum by expanding topics covered during that year.
Emphasis is placed on nutrition, fitness, safety skills, advanced first aid and the use and abuse of
alcohol and other drugs. Review of the changes of puberty, abstinence, consequences of sexual
behavior before marriage such as pregnancy, STDs and HIV/AIDS will be discussed. Positive ways to
deal with stress and grief will also be covered.
MUSIC
Band –8
Duration: two semesters per year
Prerequisite: seventh grade band instruction and/or Instructor approval
This band rehearses daily for one period, stressing total musicianship. Grades are based on practice
time, written tests, playing tests and attendance at performances. The band performs at concerts
throughout the year and at the ISSMA band contest.
Choir – 8
Duration: two semesters per year
Prerequisite: an enjoyment of singing and willingness to learn more musical concepts
Open to any eighth grade student who enjoys singing and would like to learn a variety of vocal styles.
Emphasis is placed on developing note reading, ear training skills, and students are taught to sing
three-part harmony plus develop good tone production skills. Grades are based on singing tests,
written tests, and attendance at performances and ISSMA contest. The choir performs at concerts
throughout the year.
ELECTIVES
General Music 8
Duration: one semester
This course, through the use of hands-on music technology and listening, develops basic music literacy
in the areas of music theory and appreciation. Students will study musical concepts with hands-on
musical experiences.
Foundations in Business
Duration: one semester
Students in Foundations in Business explore various career opportunities in the business world. This
course is recommended for students with an interest in business and related concepts. They will
explore the use of computer technology in business as well as other practical business skills. They will
also develop a basic understanding of the fundamentals of economics.
Peer Tutoring
Duration: one semester
Students in Peer Tutoring participate in assisting special needs students for one period each day. This
experience provides students with the ability to explore opportunities in the field of special education.
Classroom grades are based upon both participation and reflective writing assignments.
All courses will reflect the Indiana Standards |
Discrete Mathematics
9780198534273
ISBN:
0198534272
Publisher: Oxford University Press, Incorporated
Summary: This text is a carefully structured, coherent, and comprehensive course of discrete mathematics. The approach is traditional, deductive, and straightforward, with no unnecessary abstraction. It is self-contained including all the fundamental ideas in the field. It can be approached by anyone with basic competence in arithmetic and experience of simple algebraic manipulations. Students of computer science whose curric...ulum may not allow the study of many ancillary mathematics courses will find it particularly useful. Mathematics students seeking a first approach to courses such as graph theory, combinatorics, number theory, coding theory, combinatorial optimization, and abstract algebra will also enjoy a clear introduction to these more specialized fields. The main changes to this new edition are to present descriptions of numerous algorithms on a form close to that of a real programming language. The aim is to enable students to develop practical programs from the design of algorithms. Students of mathematics and computer science seeking an eloquent introduction to discrete mathematics will be pleased by this work.
Biggs, Norman L. is the author of Discrete Mathematics, published under ISBN 9780198534273 and 0198534272. Fifty three Discrete Mathematics textbooks are available for sale on ValoreBooks.com, twelve used from the cheapest price of $0.23, or buy new starting at $87 |
Inequalities
Mathematica uses a large number of original algorithms to provide automatic systemwide support for inequalities and inequality constraints. Whereas equations can often be solved in terms of numbers, even representing solution sets for inequalities is only made possible by Mathematica's symbolic capabilities. |
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MATH201: Intermediate College Algebra
Course Overview
This course introduces intermediate college algebra concepts and their applications through problem-solving and the use of equations. Course content covers key topics that include:
Algebraic expressions
Solving equations and inequalities
Graphs of linear equations
Polynomials
Rational expressions and equations
Systems of equations
Radical expressions and equations
Quadratic equations
Students will use MyMathLab (a Pearson Education product) that offers an online learning tool to help master mathematical concepts. Each week, students will select resources in MyMathLab that they will use to learn the related math concept, complete exercises with their new knowledge, and then take a short quiz on the concepts they have just learned. Two tests are presented during the course – the first covering the concepts learned in the first four modules of the course and the second covering the concepts learned in the remainder of the course.
Course Learning Objectives
Evaluate and simplify algebraic expressions by collecting like terms and using the rules for the order of operations.
Solve equations with no solutions and with an infinite number of solutions.
Solve for inequalities and graph inequalities on a number line.
Graph linear equations, finding the y-intercept; find the slope of a line using two points on a line; and, find a slope of a line from an equation.
Make use of polynomials, rational expressions, exponents, and radicals in the problem-solving process.
Solve for systems of equations using the substitution and elimination methods.
Solve quadratic equations by factoring, completing the square, and using the quadratic formula.
Special Requirements
A graphing calculator is required to complete homework assignments.
This course requires the use of MyMathLab (MML), which offers online exercises, quizzes, tests, and numerous resources for learning new math concepts. Note: At this time MyMathLab will not work with an Apple iPad. If your only option is to use an iPad, contact your SSA.
It is very important that you purchase the access code through the JIU bookstore, as the MML access code you will receive is customized for the work you will complete at JIU.
If you took MATH101 at JIU, the access code you used in MATH101 can be used in this course (MATH201) as well; your access code remains active for 12 months from the time you first used it.
There is no need to purchase a second MML access code for this MATH201 course, unless your code has expired. |
What math course is ita that would be just about any. I am pretty sure the starting level for that is integrations and approximations...that class was always a little challenging!
for like the first week or so we reviewed some trig stuff then went into the harder calc stuff.
Can't imagine what next quarter's class will be like...
Quote:
Originally Posted by Wooster
If I was the TA I would have given extra quizes.
heh. The class would probably be throwing pencils at you. Heheh always thought calc was easier than trig, as long as you understand trig, calc has a lot of shortcuts you can take (as long as you can understand the long way around). I just think it looks a whole lot more intimidating than it is.
I remember when I took numbers theory and combinatorics, the concepts required you to really let go of a lot of preconceived thoughts you had about numbers (which was the hard part) but at their core, they really weren't all that hard...people just look at you funny when you study because you talk to yourself a lot :-D
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hm. There are some parts that are easier. But still kinda hard nonetheless. Yea. It is really helpful to know those shortcuts.
Hahaha. I remember someone who said that too. Talking to yourself a lot |
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In processing plants, routine jobs such as recording temperatures, checking flow rates, and monitoring pressure require the use of mathematics. Most of this math is not complicated. It is, however, important to recall and understand these fundamentals to better understand the "how" and "why" of everyday tasks in the plant. This course is designed to review basic math and how it applies to the work of a plant operator. |
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Note: Students who wish to take a mathematics course for credit must take the course for six weeks. Other students may take the course for fewer weeks for enrichment.
JUNIOR HIGH MATH SKILLS
This is an individual math review class for Junior High students who need to brush up on basic math skills. Division and multiplication of whole numbers will be reviewed. Fractions, decimals, percents, and simple word problems will be introduced.
After a thorough review of fractions, decimals, and percents, the course will cover positive and negative numbers, the use of variables and other topics prerequisite to an Algebra I course. This course would be suitable for a 7th or 8th grade student who
will be starting Algebra I in the fall.
The Algebra I course will deal with the following topics: basic operations of addition, subtraction, multiplication and division of monomials and polynomials; linear equations and heir graphs; systems of equations and quadratic equations, and word problems.
This course covers standard Euclidian Geometry. Applications are integrated within Coordinate Geometry. The majority of time will be spent emphasizing deductive reasoning and mathematical proofs as applied to triangles, perpendicular and parallel lines, and polygons.
This course will begin with an introduction to relations and functions and their inverses. These basic ideas will then be applied to various types of functions and relations met within Algebra and Trigonometry. The concepts of Algebra II will be studied in the first four weeks, and those of Trigonometry in the last two weeks. |
Presented by the University of Illinois at Urbana-Champaign, this page allows visitors access to courses designed with Mathematica to "achieve a better conceptual understanding of the material while still gaining a good...
This survey, created by Milo Schield of Augsburg College, assesses statistical literacy. The survey focuses on the general use of informal statistics in everyday situations: reading and interpreting tables and graphs...
This website provides a syllabus for a course that uses "writing to reinforce the learning and understanding of mathematical concepts, while also using mathematics and statistics as a topic for the essays." Students...
Before entering the main portion of this interactive introduction to statistics, visitors will have to answer a few quick questions on polls. It's actually a bit fun, and it serves as a nice introduction to the site....
Teaching statistics can be tough for even experienced instructors, so it is nice to learn about the Statistical Understanding Made Simple (SUMS) website. Created by researchers at the University of Glasgow, the site... |
--BrooksBook Description:Brooks/Cole Pub Co292
Book Description:Cengage Learning, Inc, United States, 2011. Hardback. Book Condition: New. 3rd. 277 x 221 mm. Brand New Book. This best selling author team explains concepts simply and clearly, without glossing over difficult points. Problem solving and mathematical modeling are introduced early and reinforced throughout, providing students with a solid foundation in the principles of mathematical thinking. Comprehensive and evenly paced, the book provides complete coverage of the function concept, and integrates a significant amount of graphing calculator material to help students develop insight into mathematical ideas. The authors attention to detail and clarity, the same as found in James Stewart s market-leading Calculus book, is what makes this book the market leader. Bookseller Inventory # AAS9780840068132 |
1IntroductionGraph theory may be said to have its begin-ning in 1736 when E ULER considered the (gen-eral case of the) Königsberg bridge problem:Does there exist a walk crossing each of theseven bridges of Königsberg exactly once? (So-lutio Problematis ad geometriam situs perti-nentis, Commentarii Academiae Scientiarum Impe-rialis Petropolitanae 8 (1736), pp. 128-140.) It took 200 years before the first book on graph theory was written. This was "The-orie der endlichen und unendlichen Graphen" ( Teubner, Leipzig, 1936) by K ÖNIG in1936. Since then graph theory has developed into an extensive and popular branch ofmathematics, which has been applied to many problems in mathematics, computerscience, and other scientific and not-so-scientific areas. For the history of early graphtheory, seeN.L. B IGGS , R.J. L LOYD AND R.J. W ILSON, "Graph Theory 1736 – 1936", ClarendonPress, 1986. There are no standard notations for graph theoretical objects. This is natural, be-cause the names one uses for the objects reflect the applications. Thus, for instance, ifwe consider a communications network (say, for email) as a graph, then the comput-ers taking part in this network, are called nodes rather than vertices or points. On theother hand, other names are used for molecular structures in chemistry, flow chartsin programming, human relations in social sciences, and so on. These lectures study finite graphs and majority of the topics is included inJ.A. B ONDY, U.S.R. M URTY, "Graph Theory with Applications", Macmillan, 1978.R. D IESTEL, "Graph Theory", Springer-Verlag, 1997.F. H ARARY, "Graph Theory", Addison-Wesley, 1969.D.B. W EST, "Introduction to Graph Theory", Prentice Hall, 1996.R.J. W ILSON, "Introduction to Graph Theory", Longman, (3rd ed.) 1985. In these lectures we study combinatorial aspects of graphs. For more algebraic topicsand methods, seeN. B IGGS, "Algebraic Graph Theory", Cambridge University Press, (2nd ed.) 1993.C. G ODSIL , G.F. R OYLE, "Algebraic Graph Theory", Springer, 2001.and for computational aspects, seeS. E VEN, "Graph Algorithms", Computer Science Press, 1979.
3 In these lecture notes we mention several open problems that have gained respectamong the researchers. Indeed, graph theory has the advantage that it contains easilyformulated open problems that can be stated early in the theory. Finding a solutionto any one of these problems is another matter. Sections with a star (∗) in their heading are optional.Notations and notions• For a finite set X, |X | denotes its size (cardinality, the number of its elements).• Let [1, n] = {1, 2, . . . , n},and in general, [i, n] = {i, i + 1, . . . , n}for integers i ≤ n.• For a real number x, the floor and the ceiling of x are the integers ⌊ x⌋ = max{k ∈ Z | k ≤ x} and ⌈ x⌉ = min{k ∈ Z | x ≤ k}.• A family {X1 , X2 , . . . , Xk } of subsets Xi ⊆ X of a set X is a partition of X, if X= Xi and Xi ∩ X j = ∅ for all different i and j . i∈[1,k]• For two sets X and Y, X × Y = {( x, y) | x ∈ X, y ∈ Y }is their Cartesian product, and X △Y = ( X Y ) ∪ (Y X )is their symmetric difference. Here X Y = { x | x ∈ X, x ∈ Y }. /• Two integers n, k ∈ N (often n = |X | and k = |Y | for sets X and Y) have the sameparity, if both are even, or both are odd, that is, if n ≡ k (mod 2). Otherwise, theyhave opposite parity. Graph theory has abundant examples of NP-complete problems. Intuitively, aproblem is in P 1 if there is an efficient (practical) algorithm to find a solution to it. Onthe other hand, a problem is in NP 2 , if it is first efficient to guess a solution and thenefficient to check that this solution is correct. It is conjectured (and not known) thatP = NP. This is one of the great problems in modern mathematics and theoreticalcomputer science. If the guessing in NP-problems can be replaced by an efficientsystematic search for a solution, then P=NP. For any one NP-complete problem, if itis in P, then necessarily P=NP. 1 Solvable – by an algorithm – in polynomially many steps on the size of the problem instances. 2 Solvable nondeterministically in polynomially many steps on the size of the problem instances.
1.1 Graphs and their plane figures 41.1 Graphs and their plane figuresLet V be a finite set, and denote by E(V ) = {{u, v} | u, v ∈ V, u = v} .the 2-sets of V, i.e., subsets of two distinct elements.D EFINITION . A pair G = (V, E) with E ⊆ E(V ) is called a graph (on V). The elementsof V are the vertices of G, and those of E the edges of G. The vertex set of a graph Gis denoted by VG and its edge set by EG . Therefore G = (VG , EG ). In literature, graphs are also called simple graphs; vertices are called nodes or points;edges are called lines or links. The list of alternatives is long (but still finite). A pair {u, v} is usually written simply as uv. Notice that then uv = vu. In order tosimplify notations, we also write v ∈ G and e ∈ G instead of v ∈ VG and e ∈ EG .D EFINITION . For a graph G, we denote νG = |VG | and ε G = | EG | .The number νG of the vertices is called the order of G, and ε G is the size of G. For anedge e = uv ∈ G, the vertices u and v are its ends. Vertices u and v are adjacent orneighbours, if uv ∈ G. Two edges e1 = uv and e2 = uw having a common end, areadjacent with each other.A graph G can be represented as a plane figure bydrawing a line (or a curve) between the points u and v1 v3 v6v (representing vertices) if e = uv is an edge of G.The figure on the right is a geometric representationof the graph G with VG = {v1 , v2 , v3 , v4 , v5 , v6 } and v2 v4 v5E G = { v1 v2 , v1 v3 , v2 v3 , v2 v4 , v5 v6 }. Often we shall omit the identities (names v) of the vertices in our figures, in whichcase the vertices are drawn as anonymous circles. Graphs can be generalized by allowing loops vv and parallel (or multiple) edgesbetween vertices to obtain a multigraph G = (V, E, ψ), where E = {e1 , e2 , . . . , em } isa set (of symbols), and ψ : E → E(V ) ∪ {vv | v ∈ V } is a function that attaches anunordered pair of vertices to each e ∈ E: ψ(e) = uv.Note that we can have ψ(e1 ) = ψ(e2 ). This is drawn in bthe figure of G by placing two (parallel) edges that con-nect the common ends. On the right there is (a draw-ing of) a multigraph G with vertices V = {a, b, c}and edges ψ(e1 ) = aa, ψ(e2 ) = ab, ψ(e3 ) = bc, and a cψ(e4 ) = bc.
1.1 Graphs and their plane figures 5Later we concentrate on (simple) graphs.D EFINITION . We also study directed graphs or digraphsD = (V, E), where the edges have a direction, that is, theedges are ordered: E ⊆ V × V. In this case, uv = vu. The directed graphs have representations, where the edges are drawn as arrows.A digraph can contain edges uv and vu of opposite directions. Graphs and digraphs can also be coloured, labelled, and weighted:D EFINITION . A function α : VG → K is a vertex colouring of G by a set K of colours.A function α : EG → K is an edge colouring of G. Usually, K = [1, k] for some k ≥ 1. If K ⊆ R (often K ⊆ N), then α is a weight function or a distance function.Isomorphism of graphsD EFINITION . Two graphs G and H are isomorphic, denoted by G ∼ H, if there exists =a bijection α : VG → VH such that uv ∈ EG ⇐⇒ α(u)α(v) ∈ E Hfor all u, v ∈ G. Hence G and H are isomorphic if the vertices of H are renamings of those of G.Two isomorphic graphs enjoy the same graph theoretical properties, and they are oftenidentified. In particular, all isomorphic graphs have the same plane figures (exceptingthe identities of the vertices). This shows in the figures, where we tend to replace thevertices by small circles, and talk of 'the graph' although there are, in fact, infinitelymany such graphs.Example 1.1. The following graphs are v2 v3 2 4isomorphic. Indeed, the required iso- v5 1morphism is given by v1 → 1, v2 → 3,v3 → 4, v4 → 2, v5 → 5. v1 v4 3 5Isomorphism Problem. Does there exist an efficient algorithm to check whether any twogiven graphs are isomorphic or not? n The following table lists the number 2( 2 ) of all graphs on a given set of n vertices,and the number of all nonisomorphic graphs on n vertices. It tells that at least forcomputational purposes an efficient algorithm for checking whether two graphs areisomorphic or not would be greatly appreciated. n 1 2 3 4 5 6 7 8 9 graphs 1 2 8 64 1024 32 768 2 097 152 268 435 456 236 > 6 · 1010 nonisomorphic 1 2 4 11 34 156 1044 12 346 274 668
1.1 Graphs and their plane figures 6Other representationsPlane figures catch graphs for our eyes, but if a problem on graphs is to be pro-grammed, then these figures are, to say the least, unsuitable. Integer matrices are idealfor computers, since every respectable programming language has array structuresfor these, and computers are good in crunching numbers.Let VG = {v1 , . . . , vn } be ordered. The adjacency ma-trix of G is the n × n-matrix M with entries Mij = 1 or Mij = 0 according to whether vi v j ∈ G or vi v j ∈ G. / 0 1 1 0 1 1 0 0 1 1For instance, the graph in Example 1.1 has an adja- 1 0 0 1 0cency matrix on the right. Notice that the adjacency 0 1 1 0 0matrix is always symmetric (with respect to its diag-onal consisting of zeros). 1 1 0 0 0 A graph has usually many different adjacency matrices, one for each ordering ofits set VG of vertices. The following result is obvious from the definitions.Theorem 1.1. Two graphs G and H are isomorphic if and only if they have a common adja-cency matrix. Moreover, two isomorphic graphs have exactly the same set of adjacency matri-ces. Graphs can also be represented by sets. For this, let X = {X1 , X2 , . . . , Xn } be afamily of subsets of a set X, and define the intersection graph GX as the graph withvertices X1 , . . . , Xn , and edges Xi X j for all i and j (i = j) with Xi ∩ X j = ∅.Theorem 1.2. Every graph is an intersection graph of some family of subsets.Proof. Let G be a graph, and define, for all v ∈ G, a set Xv = {{v, u} | vu ∈ G }.Then Xu ∩ Xv = ∅ if and only if uv ∈ G. ⊔ ⊓ Let s( G ) be the smallest size of a base set X such that G can be represented as anintersection graph of a family of subsets of X, that is, s( G ) = min{| X | | G ∼ GX for some X ⊆ 2X } . =How small can s( G ) be compared to the order νG (or the size ε G ) of the graph? It wasshown by K OU , S TOCKMEYER AND W ONG (1976) that it is algorithmically difficult todetermine the number s( G ) – the problem is NP-complete.Example 1.2. As yet another example, let A ⊆ N be a finite set of natural numbers,and let G A = ( A, E) be the graph with rs ∈ E if and only if r and s (for r = s) have acommon divisor > 1. As an exercise, we state: All graphs can be represented in the formG A for some set A of natural numbers.
1.2 Subgraphs 71.2 SubgraphsIdeally, given a nice problem the local properties of a graph determine a solution.In these situations we deal with (small) parts of the graph (subgraphs), and a solu-tion can be found to the problem by combining the information determined by theparts. For instance, as we shall later see, the existence of an Euler tour is very local, itdepends only on the number of the neighbours of the vertices.Degrees of verticesD EFINITION . Let v ∈ G be a vertex a graph G. The neighbourhood of v is the set NG (v) = {u ∈ G | vu ∈ G } .The degree of v is the number of its neighbours: dG (v) = | NG (v)| .If dG (v) = 0, then v is said to be isolated in G, and if dG (v) = 1, then v is a leaf of thegraph. The minimum degree and the maximum degree of G are defined as δ( G ) = min{dG (v) | v ∈ G } and ∆( G ) = max{dG (v) | v ∈ G } . The following lemma, due to E ULER (1736), tells that if several people shakehands, then the number of hands shaken is even.Lemma 1.1 (Handshaking lemma). For each graph G, ∑ d G ( v) = 2 · ε G . v∈ GMoreover, the number of vertices of odd degree is even.Proof. Every edge e ∈ EG has two ends. The second claim follows immediately fromthe first one. ⊔ ⊓ Lemma 1.1 holds equally well for multigraphs, when dG (v) is defined as the num-ber of edges that have v as an end, and when each loop vv is counted twice. Note that the degrees of a graph G do not determine G. Indeed, there are graphsG = (V, EG ) and H = (V, E H ) on the same set of vertices that are not isomorphic, butfor which dG (v) = d H (v) for all v ∈ V.
1.2 Subgraphs 8SubgraphsD EFINITION . A graph H is a subgraph of a graph G, denoted by H ⊆ G, if VH ⊆ VGand E H ⊆ EG . A subgraph H ⊆ G spans G (and H is a spanning subgraph of G), ifevery vertex of G is in H, i.e., VH = VG . Also, a subgraph H ⊆ G is an induced subgraph, if E H = EG ∩ E(VH ). In thiscase, H is induced by its set VH of vertices. In an induced subgraph H ⊆ G, the set E H of edges consists of all e ∈ EG such thate ∈ E(VH ). To each nonempty subset A ⊆ VG , there corresponds a unique inducedsubgraph G [ A] = ( A, EG ∩ E( A)) .To each subset F ⊆ EG of edges there corresponds a unique spanning subgraph of G, G [ F ] = (VG , F ) . G subgraph spanning induced For a set F ⊆ EG of edges, let G − F = G [ EG F ]be the subgraph of G obtained by removing (only) the edges e ∈ F from G. In partic-ular, G −e is obtained from G by removing e ∈ G. Similarly, we write G + F, if each e ∈ F (for F ⊆ E(VG )) is added to G. For a subset A ⊆ VG of vertices, we let G − A ⊆ G be the subgraph induced byVG A, that is, G − A = G [VG A] ,and, e.g., G −v is obtained from G by removing the vertex v together with the edgesthat have v as their end.Reconstruction Problem. The famous open problem, Kelly-Ulam problem or the Re-construction Conjecture, states that a graph of order at least 3 is determined up to isomor-phism by its vertex deleted subgraphs G −v (v ∈ G): if there exists a bijection α : VG → VHsuch that G −v ∼ H −α(v) for all v, then G ∼ H. = =
1.3 Paths and cycles 11 A discrete graph is 0-regular, and a complete graph Kn is (n − 1)-regular. In par-ticular, ε Kn = n(n − 1)/2, and therefore ε G ≤ n(n − 1)/2 for all graphs G that haveorder n. Many problems concerning (induced) subgraphs are algorithmically difficult. Forinstance, to find a maximal complete subgraph (a subgraph Km of maximum order)of a graph is unlikely to be even in NP.Example 1.4. The graph on the right is the Petersengraph that we will meet several times (drawn differ-ently). It is a 3-regular graph of order 10.Example 1.5. Let k ≥ 1 be an integer, and consider the set B k of all binary stringsof length k. For instance, B3 = {000, 001, 010, 100, 011, 101, 110, 111}. Let Qk be thegraph, called the k-cube, with VQk = B k , where uv ∈ Qk if and only if the strings uand v differ in exactly one place. 110 111The order of Qk is νQk = 2k ,the number of binary 100 101strings of length k. Also, Qk is k-regular, and so, by the 010 011handshaking lemma, ε Qk = k · 2k−1 . On the right wehave the 3-cube, or simply the cube. 000 001Example 1.6. Let n ≥ 4 be any even number. We show by induction that there existsa 3-regular graph G with νG = n. Notice that all 3-regular graphs have even order bythe handshaking lemma. x yIf n = 4, then K4 is 3-regular. Let G be a 3-regulargraph of order 2m − 2, and suppose that uv, uw ∈ EG . w vLet VH = VG ∪ { x, y}, and E H = ( EG {uv, uw}) ∪ u{ux, xv, uy, yw, xy}. Then H is 3-regular of order 2m.1.3 Paths and cyclesThe most fundamental notions in graph theory are practically oriented. Indeed, manygraph theoretical questions ask for optimal solutions to problems such as: find ashortest path (in a complex network) from a given point to another. This kind ofproblems can be difficult, or at least nontrivial, because there are usually choices whatbranch to choose when leaving an intermediate point.WalksD EFINITION . Let ei = ui ui+1 ∈ G be edges of G for i ∈ [1, k]. The sequence W =e1 e2 . . . ek is a walk of length k from u1 to uk+1 . Here ei and ei+1 are compatible in thesense that ei is adjacent to ei+1 for all i ∈ [1, k − 1].
1.3 Paths and cycles 12 We write, more informally, k W : u1 − u2 − . . . − u k − u k+1 → → → → or W : u1 − u k+1 . →Write u − v to say that there is a walk of some length from u to v. Here we under- →⋆stand that W : u − v is always a specific walk, W = e1 e2 . . . ek , although we sometimes ⋆ →do not care to mention the edges ei on it. The length of a walk W is denoted by |W |.D EFINITION . Let W = e1 e2 . . . ek (ei = ui ui+1 ) be a walk. W is closed, if u1 = uk+1 . W is a path, if ui = u j for all i = j. W is a cycle, if it is closed, and ui = u j for i = j except that u1 = uk+1 . W is a trivial path, if its length is 0. A trivial path has no edges. For a walk W : u = u1 − . . . − uk+1 = v, also → → W −1 : v = u k+1 − . . . − u1 = u → →is a walk in G, called the inverse walk of W. A vertex u is an end of a path P, if P starts or ends in u. The join of two walks W1 : u − v and W2 : v − w is the walk W1 W2 : u − w. →⋆ ⋆ → ⋆ →(Here the end v must be common to the walks.) Paths P and Q are disjoint, if they have no vertices in common, and they areindependent, if they can share only their ends. Clearly, the inverse walk P−1 of a path P is a path (the inverse path of P). The joinof two paths need not be a path.A (sub)graph, which is a path (cycle) of lengthk − 1 (k, resp.) having k vertices is denoted byPk (Ck , resp.). If k is even (odd), we say that thepath or cycle is even (odd). Clearly, all paths oflength k are isomorphic. The same holds for cy- P5 C6cles of fixed length.Lemma 1.3. Each walk W : u − v with u = v contains a path P : u − v, that is, there is a ⋆ → ⋆ →path P : u − v that is obtained from W by removing edges and vertices. ⋆ →Proof. Let W : u = u1 − . . . − uk+1 = v. Let i < j be indices such that ui = u j . → →If no such i and j exist, then W, itself, is a path. Otherwise, in W = W1 W2 W3 : u − ⋆ →ui − u j − v the portion U1 = W1 W3 : u − ui = u j − v is a shorter walk. By ⋆ → →⋆ ⋆ → →⋆repeating this argument, we obtain a sequence U1 , U2 , . . . , Um of walks u − v with ⋆ →|W | > |U1 | > · · · > |Um |. When the procedure stops, we have a path as required.(Notice that in the above it may very well be that W1 or W3 is a trivial walk.) ⊔ ⊓
1.3 Paths and cycles 13D EFINITION . If there exists a walk (and hence a path) from u to v in G, let k dG (u, v) = min{k | u − v} →be the distance between u and v. If there are no walks u − v, let dG (u, v) = ∞ by ⋆ →convention. A graph G is connected, if dG (u, v) < ∞ for all u, v ∈ G; otherwise, itis disconnected. The maximal connected subgraphs of G are its connected compo-nents. Denote c( G ) = the number of connected components of G .If c( G ) = 1, then G is, of course, connected. The maximality condition means that a subgraph H ⊆ G is a connected compo-nent if and only if H is connected and there are no edges leaving H, i.e., for every ver-tex v ∈ H, the subgraph G [VH ∪ {v}] is disconnected. Apparently, every connected /component is an induced subgraph, and ∗ NG (v) = {u | dG (v, u) < ∞}is the connected component of G that contains v ∈ G. In particular, the connectedcomponents form a partition of G.Shortest pathsD EFINITION . Let G α be an edge weighted graph, that is, G α is a graph G togetherwith a weight function α : EG → R on its edges. For H ⊆ G, let α( H ) = ∑ α( e) e∈ Hbe the (total) weight of H. In particular, if P = e1 e2 . . . ek is a path, then its weight isα( P) = ∑k=1 α(ei ). The minimum weighted distance between two vertices is i dα (u, v) = min{α( P) | P : u − v} . G ⋆ → In extremal problems we seek for optimal subgraphs H ⊆ G satisfying specificconditions. In practice we encounter situations where G might represent• a distribution or transportation network (say, for mail), where the weights on edges are distances, travel expenses, or rates of flow in the network;• a system of channels in (tele)communication or computer architecture, where the weights present the rate of unreliability or frequency of action of the connections;• a model of chemical bonds, where the weights measure molecular attraction.
2Connectivity of Graphs2.1 Bipartite graphs and treesIn problems such as the shortest path problem we look for minimum solutions thatsatisfy the given requirements. The solutions in these cases are usually subgraphswithout cycles. Such connected graphs will be called trees, and they are used, e.g., insearch algorithms for databases. For concrete applications in this respect, seeT.H. C ORMEN , C.E. L EISERSON AND R.L. R IVEST, "Introduction to Algorithms",MIT Press, 1993.Certain structures with operations are representable +as trees. These trees are sometimes called constructiontrees, decomposition trees, factorization trees or grammatical · ytrees. Grammatical trees occur especially in linguistics,where syntactic structures of sentences are analyzed. x +On the right there is a tree of operations for the arith- y zmetic formula x · (y + z) + y.Bipartite graphsD EFINITION . A graph G is called bipartite, if VG has a partition to two subsets X andY such that each edge uv ∈ G connects a vertex of X and a vertex of Y. In this case,( X, Y ) is a bipartition of G, and G is ( X, Y )-bipartite.A bipartite graph G (as in the above) is complete (m, k)-bipartite, if | X | = m, |Y | = k, and uv ∈ G for all u ∈ Xand v ∈ Y.All complete (m, k)-bipartite graphs are isomorphic. LetKm,k denote such a graph.A subset X ⊆ VG is stable, if G [ X ] is a discrete graph. K2,3 The following result is clear from the definitions.Theorem 2.1. A graph G is bipartite if and only if VG has a partition to two stable subsets.Example 2.1. The k-cube Qk of Example 1.5 is bipartite for all k. Indeed, considerA = {u | u has an even number of 1′ s} and B = {u | u has an odd number of 1′ s}.Clearly, these sets partition B k , and they are stable in Qk .
2.1 Bipartite graphs and trees 18BridgesD EFINITION . An edge e ∈ G is a bridge of the graph G,if G −e has more connected components than G, that is,if c( G −e) > c( G ). In particular, and most importantly,an edge e in a connected G is a bridge if and only if G −eis disconnected.On the right (only) the two horizontal lines are bridges. We note that, for each edge e ∈ G, e = uv is a bridge ⇐⇒ u, v in different connected components of G −e .Theorem 2.3. An edge e ∈ G is a bridge if and only if e is not in any cycle of G.Proof. (⇒) If there is a cycle in G containing e, say C = PeQ, then QP : v − u is a ⋆ →path in G −e, and so e is not a bridge. (⇐) If e = uv is not a bridge, then u and v are in the same connected componentof G −e, and there is a path P : v − u in G −e. Now, eP : u − v − u is a cycle in G ⋆ → → → ⋆containing e. ⊔ ⊓Lemma 2.1. Let e be a bridge in a connected graph G. (i) Then c( G −e) = 2.(ii) Let H be a connected component of G −e. If f ∈ H is a bridge of H, then f is a bridge of G.Proof. For (i), let e = uv. Since e is a bridge, the ends u and v are not connected inG −e. Let w ∈ G. Since G is connected, there exists a path P : w − v in G. This is a ⋆ →path of G −e, unless P : w − u → v contains e = uv, in which case the part w − u is ⋆ → →⋆a path in G −e. For (ii), if f ∈ H belongs to a cycle C of G, then C does not contain e (since e is inno cycle), and therefore C is inside H, and f is not a bridge of H. ⊔ ⊓TreesD EFINITION . A graph is called acyclic, if it has no cycles. An acyclic graph is alsocalled a forest. A tree is a connected acyclic graph. By Theorem 2.3 and the definition of a tree, we haveCorollary 2.1. A connected graph is a tree if and only if all its edges are bridges.Example 2.3. The following enumeration result for trees has many different proofs,the first of which was given by C AYLEY in 1889: There are nn−2 trees on a vertex set V ofn elements. We omit the proof.
2.1 Bipartite graphs and trees 19 On the other hand, there are only a few trees up to isomorphism: n 1 2 3 4 5 6 7 8 trees 1 1 1 2 3 6 11 23 n 9 10 11 12 13 14 15 16 trees 47 106 235 551 1301 3159 7741 19 320 The nonisomorphic trees of order 6 are: We say that a path P : u − v is maximal in a graph G, if there are no edges e ∈ G →⋆for which Pe or eP is a path. Such paths exist, because νG is finite.Lemma 2.2. Let P : u − v be a maximal path in a graph G. Then NG (v) ⊆ P. Moreover, if →⋆G is acyclic, then dG (v) = 1.Proof. If e = vw ∈ EG with w ∈ P, then also Pe is a path, which contradicts the /maximality assumption for P. Hence NG (v) ⊆ P. For acyclic graphs, if wv ∈ G, thenw belongs to P, and wv is necessarily the last edge of P in order to avoid cycles. ⊔ ⊓Corollary 2.2. Each tree T with νT ≥ 2 has at least two leaves.Proof. Since T is acyclic, both ends of a maximal path have degree one. ⊔ ⊓Theorem 2.4. The following are equivalent for a graph T. (i) T is a tree. (ii) Any two vertices are connected in T by a unique path.(iii) T is acyclic and ε T = νT − 1.Proof. Let νT = n. If n = 1, then the claim is trivial. Suppose thus that n ≥ 2. (i)⇒(ii) Let T be a tree. Assume the claim does not hold, and let P, Q : u − v →⋆be two different paths between the same vertices u and v. Suppose that | P| ≥ | Q|.Since P = Q, there exists an edge e which belongs to P but not to Q. Each edge of Tis a bridge, and therefore u and v belong to different connected components of T −e.Hence e must also belong to Q; a contradiction. (ii)⇒(iii) We prove the claim by induction on n. Clearly, the claim holds for n = 2,and suppose it holds for graphs of order less than n. Let T be any graph of order nsatisfying (ii). In particular, T is connected, and it is clearly acyclic.
2.1 Bipartite graphs and trees 20 Let P : u − v be a maximal path in T. By Lemma 2.2, we have dT (v) = 1. In this ⋆ →case, P : u − w − v, where vw is the unique edge having an end v. The subgraph ⋆ → →T −v is connected, and it satisfies the condition (ii). By induction hypothesis, ε T −v =n − 2, and so ε T = ε T −v + 1 = n − 1, and the claim follows. (iii)⇒(i) Assume (iii) holds for T. We need to show that T is connected. Indeed,let the connected components of T be Ti = (Vi , Ei ), for i ∈ [1, k]. Since T is acyclic, soare the connected graphs Ti , and hence they are trees, for which we have proved that|Ei | = |Vi | − 1. Now, νT = ∑k=1 |Vi |, and ε T = ∑k=1 |Ei |. Therefore, i i k k n − 1 = εT = ∑ (|Vi | − 1) = ∑ |Vi | − k = n − k , i=1 i=1which gives that k = 1, that is, T is connected. ⊔ ⊓Example 2.4. Consider a cup tournament of n teams. If during a round there are kteams left in the tournament, then these are divided into ⌊k⌋ pairs, and from eachpair only the winner continues. If k is odd, then one of the teams goes to the nextround without having to play. How many plays are needed to determine the winner? So if there are 14 teams, after the first round 7 teams continue, and after the secondround 4 teams continue, then 2. So 13 plays are needed in this example. The answer to our problem is n − 1, since the cup tournament is a tree, where aplay corresponds to an edge of the tree.Spanning treesTheorem 2.5. Each connected graph has a spanning tree, that is, a spanning graph that isa tree.Proof. Let T ⊆ G be a maximum order subtree of G (i.e., subgraph that is a tree). IfVT = VG , there exists an edge uv ∈ EG such that u ∈ T and v ∈ T. But then T is not / /maximal; a contradiction. ⊔ ⊓Corollary 2.3. For each connected graph G, ε G ≥ νG − 1. Moreover, a connected graph G isa tree if and only if ε G = νG − 1.Proof. Let T be a spanning tree of G. Then ε G ≥ ε T = νT − 1 = νG − 1. The secondclaim is also clear. ⊔ ⊓Example 2.5. In Shannon's switching game a positive player P and a negative playerN play on a graph G with two special vertices: a source s and a sink r. P and N al-ternate turns so that P designates an edge by +, and N by −. Each edge can be des-ignated at most once. It is P's purpose to designate a path s − r (that is, to designate →⋆all edges in one such path), and N tries to block all paths s − r (that is, to designate ⋆ →at least one edge in each such path). We say that a game ( G, s, r) is
2.1 Bipartite graphs and trees 21• positive, if P has a winning strategy no matter who begins the game,• negative, if N has a winning strategy no matter who begins the game,• neutral, if the winner depends on who begins the game. rThe game on the right is neutral. s L EHMAN proved in 1964 that Shannon's switching game ( G, s, r) is positive if and onlyif there exists H ⊆ G such that H contains s and r and H has two spanning trees with noedges in common. In the other direction the claim can be proved along the following lines. Assumethat there exists a subgraph H containing s and r and that has two spanning treeswith no edges in common. Then P plays as follows. If N marks by − an edge fromone of the two trees, then P marks by + an edge in the other tree such that thisedge reconnects the broken tree. In this way, P always has two spanning trees for thesubgraph H with only edges marked by + in common. In converse the claim is considerably more difficult to prove. There remains the problem to characterize those Shannon's switching games( G, s, r) that are neutral (negative, respectively).The connector problemTo build a network connecting n nodes (towns, computers, chips in a computer) itis desirable to decrease the cost of construction of the links to the minimum. This isthe connector problem. In graph theoretical terms we wish to find an optimal span-ning subgraph of a weighted graph. Such an optimal subgraph is clearly a spanningtree, for, otherwise a deletion of any nonbridge will reduce the total weight of thesubgraph. Let then G α be a graph G together with a weight function α : EG → R + (posi-tive reals) on the edges. Kruskal's algorithm (also known as the greedy algorithm)provides a solution to the connector problem.Kruskal's algorithm: For a connected and weighted graph G α of order n:(i) Let e1 be an edge of smallest weight, and set E1 = {e1 }.(ii) For each i = 2, 3, . . . , n − 1 in this order, choose an edge ei ∈ Ei−1 of smallest / possible weight such that ei does not produce a cycle when added to G [ Ei−1 ], and let Ei = Ei−1 ∪ {ei }. The final outcome is T = (VG , En−1 ).
2.1 Bipartite graphs and trees 22 By the construction, T = (VG , En−1 ) is a spanning tree of G, because it contains nocycles, it is connected and has n − 1 edges. We now show that T has the minimumtotal weight among the spanning trees of G. Suppose T1 is any spanning tree of G. Let ek be the first edge produced by thealgorithm that is not in T1 . If we add ek to T1 , then a cycle C containing ek is created.Also, C must contain an edge e that is not in T. When we replace e by ek in T1 , westill have a spanning tree, say T2 . However, by the construction, α(ek ) ≤ α(e), andtherefore α( T2 ) ≤ α( T1 ). Note that T2 has more edges in common with T than T1 . Repeating the above procedure, we can transform T1 to T by replacing edges, oneby one, such that the total weight does not increase. We deduce that α( T ) ≤ α( T1 ). The outcome of Kruskal's algorithm need not be unique. Indeed, there may existseveral optimal spanning trees (with the same weight, of course) for a graph.Example 2.6. When applied to the weightedgraph on the right, the algorithm produces the se- 3 2 v1 v2 v3quence: e1 = v2 v4 , e2 = v4 v5 , e3 = v3 v6 , e4 = v2 v3 1and e5 = v1 v2 . The total weight of the spanning 1 2 2tree is thus 9. 4Also, the selection e1 = v2 v5 , e2 = v4 v5 , e3 = v5 v6 , 2 1 2e4 = v3 v6 , e5 = v1 v2 gives another optimal solu- v4 v5 v6tion (of weight 9). 3Problem. Consider trees T with weight functions α : ET → N. Each tree T of order nhas exactly (n) paths. (Why is this so?) Does there exist a weighted tree T α of order n such 2that the (total) weights of its paths are 1, 2, . . . , (n)? 2In such a weighted tree T α different paths have 1 4different weights, and each i ∈ [1, ( n)] is a weight 2 5of one path. Also, α must be injective. 2 8No solutions are known for any n ≥ 7.TAYLOR (1977) proved: if T of order n exists, then necessarily n = k2 or n = k2 + 2 forsome k ≥ 1.Example 2.7. A computer network can be presented as a graph G, where the verticesare the node computers, and the edges indicate the direct links. Each computer v hasan address a(v), a bit string (of zeros and ones). The length of an address is the numberof its bits. A message that is sent to v is preceded by the address a(v). The Hammingdistance h( a(v), a(u)) of two addresses of the same length is the number of places,where a(v) and a(u) differ; e.g., h(00010, 01100) = 3 and h(10000, 00000) = 1. It would be a good way to address the vertices so that the Hamming distanceof two vertices is the same as their distance in G. In particular, if two vertices wereadjacent, their addresses should differ by one symbol. This would make it easier fora node computer to forward a message.
2.2 Connectivity 23 010A graph G is said to be addressable, if it has an 000 110 111addressing a such that dG (u, v) = h( a(u), a(v)). 100 We prove that every tree T is addressable. Moreover, the addresses of the vertices of T canbe chosen to be of length νT − 1. The proof goes by induction. If νT ≤ 2, then the claim is obvious. In the caseνT = 2, the addresses of the vertices are simply 0 and 1. Let then VT = {v1 , . . . , vk+1 }, and assume that dT (v1 ) = 1 (a leaf) and v1 v2 ∈ T. Bythe induction hypothesis, we can address the tree T −v1 by addresses of length k − 1.We change this addressing: let ai be the address of vi in T −v1 , and change it to 0ai .Set the address of v1 to 1a2 . It is now easy to see that we have obtained an addressingfor T as required. The triangle K3 is not addressable. In order to gain more generality, we modifythe addressing for general graphs by introducing a special symbol ∗ in addition to0 and 1. A star address will be a sequence of these three symbols. The Hammingdistance remains as it was, that is, h(u, v) is the number of places, where u and vhave a different symbol 0 or 1. The special symbol ∗ does not affect h(u, v). So, h(10 ∗∗01, 0 ∗ ∗101) = 1 and h(1 ∗ ∗ ∗ ∗∗, ∗00 ∗ ∗∗) = 0. We still want to have h(u, v) =dG (u, v).We star address this graph as follows: v3 a(v1 ) = 0000 , a(v2 ) = 10 ∗ 0 , a(v3 ) = 1 ∗ 01 , a(v4 ) = ∗ ∗ 11 . v1 v2These addresses have length 4. Can you design a v4star addressing with addresses of length 3? W INKLER proved in 1983 a rather unexpected result: The minimum star addresslength of a graph G is at most νG − 1. For the proof of this, see VAN L INT AND W ILSON, "A Course in Combinatorics".2.2 ConnectivitySpanning trees are often optimal solutions to problems, where cost is the criterion.We may also wish to construct graphs that are as simple as possible, but where twovertices are always connected by at least two independent paths. These problems oc-cur especially in different aspects of fault tolerance and reliability of networks, whereone has to make sure that a break-down of one connection does not affect the func-tionality of the network. Similarly, in a reliable network we require that a break-downof a node (computer) should not result in the inactivity of the whole network.
2.2 Connectivity 26 Conversely, we use induction on m = νG + ε G to show that if S = {w1 , w2 , . . . , wk }is a (u, v)-separating set of the smallest size, then G has at least (and thus exactly) kindependent paths u − v. →⋆ The case for k = 1 is clear, and this takes care of the small values of m, requiredfor the induction. (1) Assume first that u and v have a common neighbour w ∈ NG (u) ∩ NG (v). Thennecessarily w ∈ S. In the smaller graph G −w the set S {w} is a minimum (u, v)-separating set, and the induction hypothesis yields that there are k − 1 independentpaths u − v in G −w. Together with the path u − w − v, there are k independent ⋆ → → →paths u − v in G as required. →⋆ ∗ (2) Assume then that NG (u) ∩ NG (v) = ∅, and denote by Hu = NG−S (u) and ∗Hv = NG−S (v) the connected components of G −S for u and v. (2.1) Suppose next that S NG (u) and S NG (v).Let v be a new vertex, and define Gu to be the graphon Hu ∪ S ∪ {v } having the edges of G [ Hu ∪ S] to-gether with vwi for all i ∈ [1, k]. The graph Gu is con- wknected and it is smaller than G. Indeed, in order for ... vS to be a minimum separating set, all wi ∈ S have w2to be adjacent to some vertex in Hv . This shows that u w1ε Gu ≤ ε G , and, moreover, the assumption (2.1) rulesout the case Hv = {v}. So | Hv | ≥ 2 and νGu < νG . If S′ is any (u, v)-separating set of Gu , then S′ will separate u from all wi ∈ S S′ inG. This means that S′ separates u and v in G. Since k is the size of a minimum (u, v)-separating set, we have |S′ | ≥ k. We noted that Gu is smaller than G, and thus by theinduction hypothesis, there are k independent paths u − v in Gu . This is possible ⋆ →only if there exist k paths u − wi , one for each i ∈ [1, k], that have only the end u in →⋆common. By the present assumption, also u is nonadjacent to some vertex of S. A symmetricargument applies to the graph Gv (with a new vertex u), which is defined similarlyto Gu . This yields that there are k paths wi − v that have only the end v in common. ⋆ →When we combine these with the above paths u − wi , we obtain k independent →⋆paths u − wi − v in G. →⋆ →⋆ (2.2) There remains the case, where for all (u, v)-separating sets S of k elements,either S ⊆ NG (u) or S ⊆ NG (v). (Note that then, by (2), S ∩ NG (v) = ∅ or S ∩NG (u) = ∅.) Let P = e f Q be a shortest path u − v in G, where e = ux, f = xy, and Q : y − v. →⋆ ⋆ →Notice that, by the assumption (2), | P| ≥ 3, and so y = v. In the smaller graph G − f ,let S′ be a minimum set that separates u and v. If |S′ | ≥ k, then, by the induction hypothesis, there are k independent paths u − v →⋆in G − f . But these are paths of G, and the claim is clear in this case.
2.2 Connectivity 27 If, on the other hand, |S′ | < k, then u and v are still connected in G −S′ . Every pathu − v in G −S′ necessarily travels along the edge f = xy, and so x, y ∈ S′ . →⋆ /Let Sx = S′ ∪ { x} and Sy = S ′ ∪ {y} .These sets separate u and v in G (by the above fact), and they have size k. By ourcurrent assumption, the vertices of Sy are adjacent to v, since the path P is shortestand so uy ∈ G (meaning that u is not adjacent to all of Sy ). The assumption (2) yields /that u is adjacent to all of Sx , since ux ∈ G. But now both u and v are adjacent to thevertices of S′ , which contradicts the assumption (2). ⊔ ⊓Theorem 2.8 (M ENGER (1927)). A graph G is k-connected if and only if every two verticesare connected by at least k independent paths.Proof. If any two vertices are connected by k independent paths, then it is clearthat κ ( G ) ≥ k. In converse, suppose that κ ( G ) = k, but that G has vertices u and v connected by atmost k − 1 independent paths. By Theorem 2.7, it must be that e = uv ∈ G. Considerthe graph G −e. Now u and v are connected by at most k − 2 independent paths inG −e, and by Theorem 2.7, u and v can be separated in G −e by a set S with |S| = k − 2.Since νG > k (because κ ( G ) = k), there exists a w ∈ G that is not in S ∪ {u, v}. Thevertex w is separated in G −e by S from u or from v; otherwise there would be a pathu − v in ( G −e)−S. Say, this vertex is u. The set S ∪ {v} has k − 1 elements, and it ⋆ →separates u from w in G, which contradicts the assumption that κ ( G ) = k. This provesthe claim. ⊔ ⊓ We state without a proof the corresponding separation property for edge connec-tivity.D EFINITION . Let G be a graph. A uv-disconnecting set is a set F ⊆ EG such thatevery path u − v contains an edge from F. ⋆ →Theorem 2.9. Let u, v ∈ G with u = v in a graph G. Then the maximum number of edge- ⋆disjoint paths u − v equals the minimum number k of edges in a uv-disconnecting set. →Corollary 2.4. A graph G is k-edge connected if and only if every two vertices are connectedby at least k edge disjoint paths.Example 2.9. Recall the definition of the cube Qk from Example 1.5. We show thatκ ( Qk ) = k. First of all, κ ( Qk ) ≤ δ( Qk ) = k. In converse, we show the claim by induction.Extract from Qk the disjoint subgraphs: G0 induced by {0u | u ∈ B k−1 } and G1induced by {1u | u ∈ B k−1 }. These are (isomorphic to) Qk−1 , and Qk is obtained fromthe union of G0 and G1 by adding the 2k−1 edges (0u, 1u) for all u ∈ B k−1 .
2.2 Connectivity 28 Let S be a separating set of Qk with |S| ≤ k. If both G0 −S and G1 −S were con-nected, also Qk −S would be connected, since one pair (0u, 1u) necessarily remains inQk −S. So we can assume that G0 −S is disconnected. (The case for G1 −S is symmet-ric.) By the induction hypothesis, κ ( G0 ) = k − 1, and hence S contains at least k − 1vertices of G0 (and so |S| ≥ k − 1). If there were no vertices from G1 in S, then, ofcourse, G1 −S is connected, and the edges (0u, 1u) of Qk would guarantee that Qk −Sis connected; a contradiction. Hence |S| ≥ k.Example 2.10. We have κ ′ ( Qk ) = k for the k-cube. Indeed, by Whitney's theorem,κ ( G ) ≤ κ ′ ( G ) ≤ δ( G ). Since κ ( Qk ) = k = δ( Qk ), also κ ′ ( Qk ) = k.Algorithmic Problem. The connectivity problems tend to be algorithmically difficult.In the disjoint paths problem we are given a set (ui , vi ) of pairs of vertices for i =1, 2, . . . , k, and it is asked whether there exist paths Pi : ui − vi that have no vertices in ⋆ →common. This problem was shown to be NP-complete by K NUTH in 1975. (However,for fixed k, the problem has a fast algorithm due to R OBERTSON and S EYMOUR (1986).)Dirac's fansD EFINITION . Let v ∈ G and S ⊆ VG such that v ∈ S /in a graph G. A set of paths from v to a vertex in S iscalled a (v, S)-fan, if they have only v in common. ∗ ... ∗ vTheorem 2.10 (D IRAC (1960)). A graph G is k-connected ∗ Sif and only if νG > k and for every v ∈ G and S ⊆ VG with /|S| ≥ k and v ∈ S, there exists a (v, S)-fan of k paths.Proof. Exercise. ⊔ ⊓Theorem 2.11 (D IRAC (1960)). Let G be a k-connected graph for k ≥ 2. Then for any kvertices, there exists a cycle of G containing them.Proof. First of all, since κ ( G ) ≥ 2, G has no cut vertices, and thus no bridges. Itfollows that every edge, and thus every vertex of G belongs to a cycle. Let S ⊆ VG be such that |S| = k, and let C be a cycle of G that contains themaximum number of vertices of S. Let the vertices of S ∩ VC be v1 , . . . , vr listed inorder around C so that each pair (vi , vi+1 ) (with indices modulo r) defines a pathalong C (except in the special case where r = 1). Such a path is referred to as a segmentof C. If C contains all vertices of S, then we are done; otherwise, suppose v ∈ S is noton C. It follows from Theorem 2.10 that there is a (v, VC )-fan of at least min{k, |VC |}paths. Therefore there are two paths P : v − u and Q : v − w in such a fan that end →⋆ ⋆ →in the same segment (vi , vi+1 ) of C. Then the path W : u − w (or w − u) along C →⋆ ⋆ →contains all vertices of S ∩ VC . But now PWQ−1 is a cycle of G that contains v and allvi for i ∈ [1, r]. This contradicts the choice of C, and proves the claim. ⊔ ⊓
3Tours and Matchings3.1 Eulerian graphsThe first proper problem in graph theory was the Königsberg bridge problem. In gen-eral, this problem concerns of travelling in a graph such that one tries to avoid usingany edge twice. In practice these eulerian problems occur, for instance, in optimiz-ing distribution networks – such as delivering mail, where in order to save time eachstreet should be travelled only once. The same problem occurs in mechanical graphplotting, where one avoids lifting the pen off the paper while drawing the lines.Euler toursD EFINITION . A walk W = e1 e2 . . . en is a trail, if ei = e j for all i = j. An Euler trail ofa graph G is a trail that visits every edge once. A connected graph G is eulerian, if ithas a closed trail containing every edge of G. Such a trail is called an Euler tour. Notice that if W = e1 e2 . . . en is an Euler tour (and so EG = {e1 , e2 , . . . , en }), alsoei ei+1 . . . en e1 . . . ei−1 is an Euler tour for all i ∈ [1, n]. A complete proof of the followingEuler's Theorem was first given by H IERHOLZER in 1873.Theorem 3.1 (E ULER (1736), H IERHOLZER (1873)). A connected graph G is eulerian ifand only if every vertex has an even degree.Proof. (⇒) Suppose W : u − u is an Euler tour. Let v (= u) be a vertex that occurs ⋆ →k times in W. Every time an edge arrives at v, another edge departs from v, andtherefore dG (v) = 2k. Also, dG (u) is even, since W starts and ends at u. (⇐) Assume G is a nontrivial connected graph such that dG (v) is even for all v ∈G. Let W = e1 e2 . . . en : v0 − vn with ei = vi−1 vi ⋆ →be a longest trail in G. It follows that all e = vn w ∈ G are among the edges of W, for,otherwise, W could be prolonged to We. In particular, v0 = vn , that is, W is a closedtrail. (Indeed, if it were vn = v0 and vn occurs k times in W, then dG (vn ) = 2(k − 1) + 1and that would be odd.) If W is not an Euler tour, then, since G is connected, there exists an edge f = vi u ∈G for some i, which is not in W. However, now e i + 1 . . . e n e1 . . . e i fis a trail in G, and it is longer than W. This contradiction to the choice of W provesthe claim. ⊔ ⊓
3.1 Eulerian graphs 30Example 3.1. The k-cube Qk is eulerian for even integers k, because Qk is k-regular.Theorem 3.2. A connected graph has an Euler trail if and only if it has at most two verticesof odd degree.Proof. If G has an Euler trail u − v, then, as in the proof of Theorem 3.1, each vertex ⋆ →w ∈ {u, v} has an even degree. / Assume then that G is connected and has at most two vertices of odd degree. If Ghas no vertices of odd degree then, by Theorem 3.1, G has an Euler trail. Otherwise,by the handshaking lemma, every graph has an even number of vertices with odddegree, and therefore G has exactly two such vertices, say u and v. Let H be a graphobtained from G by adding a vertex w, and the edges uw and vw. In H every vertexhas an even degree, and hence it has an Euler tour, say u − v − w − u. Here the ⋆ → → →beginning part u − v is an Euler trail of G. →⋆ ⊔ ⊓The Chinese postmanThe following problem is due to G UAN M EIGU (1962). Consider a village, where apostman wishes to plan his route to save the legs, but still every street has to bewalked through. This problem is akin to Euler's problem and to the shortest pathproblem. Let G be a graph with a weight function α : EG → R + . The Chinese postmanproblem is to find a minimum weighted tour in G (starting from a given vertex, thepost office). If G is eulerian, then any Euler tour will do as a solution, because such a tourtraverses each edge exactly once and this is the best one can do. In this case the weightof the optimal tour is the total weight of the graph G, and there is a good algorithmfor finding such a tour:Fleury's algorithm:• Let v0 ∈ G be a chosen vertex, and let W0 be the trivial path on v0 .• Repeat the following procedure for i = 1, 2, . . . as long as possible: suppose a trail Wi = e1 e2 . . . ei has been constructed, where e j = v j−1 v j . Choose an edge ei+1 (= e j for j ∈ [1, i ]) so that(i) ei+1 has an end vi , and(ii) ei+1 is not a bridge of Gi = G −{e1 , . . . , ei }, unless there is no alternative. Notice that, as is natural, the weights α(e) play no role in the eulerian case.Theorem 3.3. If G is eulerian, then any trail of G constructed by Fleury's algorithm is anEuler tour of G.Proof. Exercise. ⊔ ⊓
3.2 Hamiltonian graphs 31 If G is not eulerian, the poor postman has to walk at least one street twice. Thishappens, e.g., if one of the streets is a dead end, and in general if there is a street cornerof an odd number of streets. We can attack this case by reducing it to the eulerian caseas follows. An edge e = uv will be duplicated, if it is added to G parallel to an existingedge e′ = uv with the same weight, α(e′ ) = α(e). 4 3 4 3 4 3 3 3 3 3 2 1 2 2 2 1 2 2 2 1 2 Above we have duplicated two edges. The rightmost multigraph is eulerian. There is a good algorithm by E DMONDS AND J OHNSON (1973) for the constructionof an optimal eulerian supergraph by duplications. Unfortunately, this algorithm issomewhat complicated, and we shall skip it.3.2 Hamiltonian graphsIn the connector problem we reduced the cost of a spanning graph to its minimum.There are different problems, where the cost is measured by an active user of thegraph. For instance, in the travelling salesman problem a person is supposed to visiteach town in his district, and this he should do in such a way that saves time andmoney. Obviously, he should plan the travel so as to visit each town once, and sothat the overall flight time is as short as possible. In terms of graphs, he is lookingfor a minimum weighted Hamilton cycle of a graph, the vertices of which are thetowns and the weights on the edges are the flight times. Unlike for the shortest pathand the connector problems no efficient reliable algorithm is known for the travellingsalesman problem. Indeed, it is widely believed that no practical algorithm exists forthis problem.Hamilton cyclesD EFINITION . A path P of a graph G is a Hamilton path,if P visits every vertex of G once. Similarly, a cycle C isa Hamilton cycle, if it visits each vertex once. A graph ishamiltonian, if it has a Hamilton cycle. Note that if C : u1 → u2 → · · · → un is a Hamilton cycle, then so is ui → . . . un →u1 → . . . ui−1 for each i ∈ [1, n], and thus we can choose where to start the cycle.Example 3.2. It is obvious that each Kn is hamiltonian whenever n ≥ 3. Also, as iseasily seen, Kn,m is hamiltonian if and only if n = m ≥ 2. Indeed, let Kn,m have a
3.2 Hamiltonian graphs 32bipartition ( X, Y ), where | X | = n and |Y | = m. Now, each cycle in Kn,m has evenlength as the graph is bipartite, and thus the cycle visits the sets X, Y equally manytimes, since X and Y are stable subsets. But then necessarily | X | = |Y |. Unlike for eulerian graphs (Theorem 3.1) no good characterization is known forhamiltonian graphs. Indeed, the problem to determine if G is hamiltonian is NP-complete. There are, however, some interesting general conditions.Lemma 3.1. If G is hamiltonian, then for every nonempty subset S ⊆ VG , c( G − S ) ≤ |S | .Proof. Let ∅ = S ⊆ VG , u ∈ S, and let C : u − u be a Hamilton cycle of G. Assume →⋆G −S has k connected components, Gi , i ∈ [1, k]. The case k = 1 is trivial, and hencesuppose that k > 1. Let ui be the last vertex of C that belongs to Gi , and let vi be thevertex that follows ui in C. Now vi ∈ S for each i by the choice of ui , and v j = vt forall j = t, because C is a cycle and ui vi ∈ G for all i. Thus |S| ≥ k as required. ⊔ ⊓Example 3.3. Consider the graph on the right. In G,c( G −S) = 3 > 2 = |S| for the set S of black ver-tices. Therefore G does not satisfy the condition ofLemma 3.1, and hence it is not hamiltonian. Interest-ingly this graph is ( X, Y )-bipartite of even order with|X | = |Y |. It is also 3-regular.Example 3.4. Consider the Petersen graph on the right,which appears in many places in graph theory as acounter example for various conditions. This graphis not hamiltonian, but it does satisfy the conditionc( G −S) ≤ |S| for all S = ∅. Therefore the conclusionof Lemma 3.1 is not sufficient to ensure that a graph ishamiltonian. The following theorem, due to O RE, generalizes an earlier result by D IRAC (1952).Theorem 3.4 (O RE (1962)). Let G be a graph of order νG ≥ 3, and let u, v ∈ G be such that dG (u) + dG (v) ≥ νG .Then G is hamiltonian if and only if G + uv is hamiltonian.Proof. Denote n = νG . Let u, v ∈ G be such that dG (u) + dG (v) ≥ n. If uv ∈ G, thenthere is nothing to prove. Assume thus that uv ∈ G. / (⇒) This is trivial since if G has a Hamilton cycle C, then C is also a Hamiltoncycle of G + uv. (⇐) Denote e = uv and suppose that G + e has a Hamilton cycle C. If C does notuse the edge e, then it is a Hamilton cycle of G. Suppose thus that e is on C. We maythen assume that C : u − v − u. Now u = v1 − v2 − . . . − vn = v is a Hamilton ⋆ → → → → →
3.3 Matchings 353.3 MatchingsIn matching problems we are given an availability relation between the elements ofa set. The problem is then to find a pairing of the elements so that each element ispaired (matched) uniquely with an available companion. A special case of the matching problem is the marriage problem, which is statedas follows. Given a set X of boys and a set Y of girls, under what condition can eachboy marry a girl who cares to marry him? This problem has many variations. One ofthem is the job assignment problem, where we are given n applicants and m jobs,and we should assign each applicant to a job he is qualified. The problem is thatan applicant may be qualified for several jobs, and a job may be suited for severalapplicants.Maximum matchingsD EFINITION . For a graph G, a subset M ⊆ EG is a matching of G, if M contains noadjacent edges. The two ends of an edge e ∈ M are matched under M. A matchingM is a maximum matching, if for no matching M ′ , | M | < | M ′ |.The two vertical edges on the right constitute a match-ing M that is not a maximum matching, although youcannot add any edges to M to form a larger matching.This matching is not maximum because the graph hasa matching of three edges.D EFINITION . A matching M saturates v ∈ G, if v is anend of an edge in M. Also, M saturates A ⊆ VG , if itsaturates every v ∈ A. If M saturates VG , then M is aperfect matching.It is clear that every perfect matching is maximum.On the right the horizontal edges form a perfect match-ing.D EFINITION . Let M be a matching of G. An odd pathP = e1 e2 . . . e2k+1 is M-augmented, if• P alternates between EG M and M (that is, e2i+1 ∈ G − M and e2i ∈ M), and• the ends of P are not saturated.Lemma 3.3. If G is connected with ∆( G ) ≤ 2, then G is a path or a cycle.Proof. Exercise. ⊔ ⊓ We start with a result that gives a necessary and sufficient condition for a match-ing to be maximum. One can use the first part of the proof to construct a maximum
3.3 Matchings 36matching in an iterative manner starting from any matching M and from any M-augmented path.Theorem 3.8 (B ERGE (1957)). A matching M of G is a maximum matching if and only ifthere are no M-augmented paths in G.Proof. (⇒) Let a matching M have an M-augmented path P = e1 e2 . . . e2k+1 in G.Here e2 , e4 , . . . , e2k ∈ M, e1 , e3 , . . . , e2k+1 ∈ M. Define N ⊆ EG by / N = ( M {e2i | i ∈ [1, k]}) ∪ {e2i+1 | i ∈ [0, k]} .Now, N is a matching of G, and | N | = | M | + 1. Therefore M is not a maximummatching. (⇐) Assume N is a maximum matching, but M is not. Hence | N | > | M |. Considerthe subgraph H = G [ M △ N ] for the symmetric difference M △ N. We have d H (v) ≤2 for each v ∈ H, because v is an end of at most one edge in M and N. By Lemma 3.3,each connected component A of H is either a path or a cycle. Since no v ∈ A can be an end of two edges from N or from M, each connectedcomponent (path or a cycle) A alternates between N and M. Now, since | N | > | M |,there is a connected component A of H, which has more edges from N than fromM. This A cannot be a cycle, because an alternating cycle has even length, and itthus contains equally many edges from N and M. Hence A : u − v is a path (of →⋆odd length), which starts and ends with an edge from N. Because A is a connectedcomponent of H, the ends u and v are not saturated by M, and, consequently, A is anM-augmented path. This proves the theorem. ⊔ ⊓Example 3.5. Consider the k-cube Qk for k ≥ 1. Each maximum matching of Qk has2k−1 edges. Indeed, the matching M = {(0u, 1u) | u ∈ B k−1 }, has 2k−1 edges, and itis clearly perfect.Hall's theoremFor a subset S ⊆ VG of a graph G, denote NG (S) = {v | uv ∈ G for some u ∈ S} .If G is ( X, Y )-bipartite, and S ⊆ X, then NG (S) ⊆ Y. The following result, known as theTheorem 3.9 (H ALL (1935)). Let G be a ( X, Y )-bipartite graph. Then G contains a matchingM saturating X if and only if |S| ≤ | NG (S)| for all S ⊆ X. (3.2)
3.3 Matchings 38Applications of Hall's theoremD EFINITION . Let S = {S1 , S2 , . . . , Sm } be a family of finite nonempty subsets of a setS. (Si need not be distinct.) A transversal (or a system of distinct representatives) ofS is a subset T ⊆ S of m distinct elements one from each Si . As an example, let S = [1, 6], and let S1 = S2 = {1, 2}, S3 = {2, 3} and S4 ={1, 4, 5, 6}. For S = {S1 , S2 , S3 , S4 }, the set T = {1, 2, 3, 4} is a transversal. If we addthe set S5 = {2, 3} to S , then it is impossible to find a transversal for this new family. The connection of transversals to the Marriage Theorem is as follows. Let S = Yand X = [1, m]. Form an ( X, Y )-bipartite graph G such that there is an edge (i, s)if and only if s ∈ Si . The possible transversals T of S are then obtained from thematchings M saturating X in G by taking the ends in Y of the edges of M.Corollary 3.2. Let S be a family of finite nonempty sets. Then S has a transversal if and onlyif the union of any k of the subsets Si of S contains at least k elements.Example 3.6. An m × n latin rectangle is an m × n integer matrix M with entriesMij ∈ [1, n] such that the entries in the same row and in the same column are different.Moreover, if m = n, then M is a latin square. Note that in a m × n latin rectangle M,we always have that m ≤ n. We show the following: Let M be an m × n latin rectangle (with m < n). Then M canbe extended to a latin square by the addition of n − m new rows. The claim follows when we show that M can be extended to an (m + 1) × n latinrectangle. Let Ai ⊆ [1, n] be the set of those elements that do not occur in the i-thcolumn of M. Clearly, | Ai | = n − m for each i, and hence ∑i∈ I | Ai | = | I |(n − m) forall subsets I ⊆ [1, n]. Now |∪i∈ I Ai | ≥ | I |, since otherwise at least one element fromthe union would be in more than n − m of the sets Ai with i ∈ I. However, each rowhas all the n elements, and therefore each i is missing from exactly n − m columns. ByMarriage Theorem, the family { A1 , A2 , . . . , An } has a transversal, and this transversalcan be added as a new row to M. This proves the claim.Tutte's theoremThe next theorem is a classic characterization of perfect matchings.D EFINITION . A connected component of a graph G is said to be odd (even), if it hasan odd (even) number of vertices. Denote by codd ( G ) the number of odd connectedcomponents in G. Denote by m( G ) be the number of edges in a maximum matching of a graph G.
3.3 Matchings 40 For perfect matchings we have the following corollary, since for a perfect matchingwe have m( G ) = (1/2)νG .Theorem 3.11 (T UTTE (1947)). Let G be a nontrivial graph. The following are equivalent. (i) G has a perfect matching.(ii) For every proper subset S ⊂ VG , codd ( G −S) ≤ |S|. Tutte's theorem does not provide a good algorithm for constructing a perfectmatching, because the theorem requires 'too many cases'. Its applications are mainlyin the proofs of other results that are related to matchings. There is a good algorithmdue to E DMONDS (1965), which uses 'blossom shrinkings', but this algorithm is some-what involved.Example 3.7. The simplest connected graph that has no perfect matching is the pathP3 . Here removing the middle vertex creates two odd components.The next 3-regular graph (known as the Sylvestergraph) does not have a perfect matching, because re-moving the black vertex results in a graph with threeodd connected components. This graph is the smallestregular graph with an odd degree that has no perfectmatching. Using Theorem 3.11 we can give a short proof of P ETERSEN's result for 3-regulargraphs (1891).Theorem 3.12 (P ETERSEN (1891)). If G is a bridgeless 3-regular graph, then it has a perfectmatching.Proof. Let S be a proper subset of VG , and let Gi , i ∈ [1, t], be the odd connectedcomponents of G −S. Denote by mi the number of edges with one end in Gi and theother in S. Since G is 3-regular, ∑ dG (v) = 3 · νGi and ∑ d G ( v) = 3 · |S | . v ∈ Gi v∈SThe first of these implies that mi = ∑ d G ( v ) − 2 · ε Gi v ∈ Giis odd. Furthermore, mi = 1, because G has no bridges, and therefore mi ≥ 3. Hencethe number of odd connected components of G −S satisfies 1 t 1 t≤ ∑ m i ≤ 3 ∑ d G ( v) = |S | , 3 i=1 v∈Sand so, by Theorem 3.11, G has a perfect matching. ⊔ ⊓
3.3 Matchings 41Stable MarriagesD EFINITION . Consider a bipartite graph G with a bipartition ( X, Y ) of the vertex set.In addition, each vertex x ∈ G supplies an order of preferences of the vertices ofNG ( x). We write u < x v, if x prefers v to u. (Here u, v ∈ Y, if x ∈ X, and u, v ∈ X, ifx ∈ Y.) A matching M of G is said to be stable, if for each unmatched pair xy ∈ M /(with x ∈ X and y ∈ Y), it is not the case that x and y prefer each other better thantheir matched companions: xv ∈ M and y < x v, or uy ∈ M and x <y u. We omit the proof of the next theorem.Theorem 3.13. For bipartite graphs G, a stable matching exists for all lists of preferences.Example 3.8. That was the good news. There is a catch, of 4 8course. A stable matching need not saturate X and Y. For 3 7instance, the graph on the right does have a perfect match- 2 6ing (of 4 edges). 1 5 Suppose the preferences are the following: 1: 5 2: 6 < 8 < 7 3: 8 < 5 4: 7 < 5 5: 4 < 1 < 3 6: 2 7: 2 < 4 8: 3 < 2Then there is no stable matchings of four edges. A stable matching of G is the follow-ing: M = {28, 35, 47}, which leaves 1 and 6 unmatched. (You should check that thereis no stable matching containing the edges 15 and 26.)Theorem 3.14. Let G = Kn,n be a complete bipartite graph. Then G has a perfect and stablematching for all lists of preferences.Proof. Let the bipartition be ( X, Y ). The algorithm by G ALE AND S HAPLEY (1962)works as follows. Procedure. Set M0 = ∅, and P( x) = ∅ for all x ∈ X. Then iterate the following process until all vertices are saturated: Choose a vertex x ∈ X that is unsaturated in Mi−1 . Let y ∈ Y be the most preferred vertex for x such that y ∈ P( x). / (1) Add y to P( x). (2) If y is not saturated, then set Mi = Mi−1 ∪ { xy}. (3) If zy ∈ Mi−1 and z <y x, then set Mi = ( Mi−1 {zy}) ∪ { xy}. First of all, the procedure terminates, since a vertex x ∈ X takes part in the iter-ation at most n times (once for each y ∈ Y). The final outcome, say M = Mt , is aperfect matching, since the iteration continues until there are no unsaturated verticesx ∈ X.
3.3 Matchings 42 Also, the matching M = Mt is stable. Note first that, by (3), if xy ∈ Mi and zy ∈ M jfor some x = z and i < j, then x <y z. Assume the that xy ∈ M, but y < x z for somez ∈ Y. Then xy is added to the matching at some step, xy ∈ Mi , which means thatz ∈ P( x) at this step (otherwise x would have 'proposed' z). Hence x took part in theiteration at an earlier step Mk , k < i (where z was put to the list P( x), but xz was notadded). Thus, for some u ∈ X, uz ∈ Mk−1 and x <z u, and so in M the vertex z ismatched to some w with x <z w. Similarly, if x <y v for some v ∈ X, then y <v z for the vertex z ∈ Y such thatvz ∈ M. ⊔ ⊓
4Colourings4.1 Edge colouringsColourings of edges and vertices of a graph G are useful, when one is interested inclassifying relations between objects. There are two sides of colourings. In the general case, a graph G with a colouringα is given, and we study the properties of this pair G α = ( G, α). This is the situation,e.g., in transportation networks with bus and train links, where the colour (buss, train)of an edge tells the nature of a link. In the chromatic theory, G is first given and then we search for a colouring that thesatisfies required properties. One of the important properties of colourings is 'proper-ness'. In a proper colouring adjacent edges or vertices are coloured differently.Edge chromatic numberD EFINITION . A k-edge colouring α : EG → [1, k] of a graph G is an assignment of kcolours to its edges. We write G α to indicate that G has the edge colouring α. A vertex v ∈ G and a colour i ∈ [1, k] are incident with each other, if α(vu) = i forsome vu ∈ G. If v ∈ G is not incident with a colour i, then i is available for v. The colouring α is proper, if no two adjacent edges obtain the same colour: α(e1 ) =α(e2 ) for adjacent e1 and e2 . The edge chromatic number χ′ ( G ) of G is defined as χ′ ( G ) = min{k | there exists a proper k-edge colouring of G } . A k-edge colouring α can be thought of as a partition {E1 , E2 , . . . , Ek } of EG , whereEi = {e | α(e) = i }. Note that it is possible that Ei = ∅ for some i. We adopt asimplified notation G α [ i 1 , i 2 , . . . , i t ] = G [ Ei1 ∪ Ei2 ∪ · · · ∪ Ei t ]for the subgraph of G consisting of those edges that have a colour i1 , i2 , . . . , or it . Thatis, the edges having other colours are removed.Lemma 4.1. Each colour set Ei in a proper k-edge colouring is a matching. Moreover, for eachgraph G, ∆( G ) ≤ χ′ ( G ) ≤ ε G .Proof. This is clear. ⊔ ⊓
4.1 Edge colourings 44Example 4.1. The three numbers in Lemma 4.1 can be equal. This happens, for in-stance, when G = K1,n is a star. But often the inequalities are strict. A star, and a graph with χ′ ( G ) = 4.Optimal colouringsWe show that for bipartite graphs the lower bound is always optimal: χ′ ( G ) = ∆( G ).Lemma 4.2. Let G be a connected graph that is not an odd cycle. Then there exists a 2-edgecolouring (that need not be proper), in which both colours are incident with each vertex v withdG (v) ≥ 2.Proof. Assume that G is nontrivial; otherwise, the claim is trivial. (1) Suppose first that G is eulerian. If G is an even cycle, then a 2-edge colouringexists as required. Otherwise, since now dG (v) is even for all v, G has a vertex v1 withdG (v1 ) ≥ 4. Let e1 e2 . . . et be an Euler tour of G, where ei = vi vi+1 (and vt+1 = v1 ).Define 1, if i is odd , α( ei ) = 2, if i is even .Hence the ends of the edges ei for i ∈ [2, t − 1] are incident with both colours. Allvertices are among these ends. The condition dG (v1 ) ≥ 4 guarantees this for v1 . Hencethe claim holds in the eulerian case. (2) Suppose then that G is not eulerian. We define a new graph G0 by adding avertex v0 to G and connecting v0 to each v ∈ G of odd degree.In G0 every vertex has even degree including v0 (bythe handshaking lemma), and hence G0 is eulerian. Lete0 e1 . . . et be an eulerian tour of G0 , where ei = vi vi+1 . 2 1By the previous case, there is a required colouring α of 1 2 v0G0 as above. Now, α restricted to EG is a colouring of Gas required by the claim, since each vertex vi with odd 2 1degree dG (vi ) ≥ 3 is entered and departed at least oncein the tour by an edge of the original graph G: ei−1 ei . ⊔ ⊓D EFINITION . For a k-edge colouring α of G, let cα (v) = |{i | v is incident with i ∈ [1, k]}| .
4.1 Edge colourings 45A k-edge colouring β is an improvement of α, if ∑ c β ( v) > ∑ cα ( v) . v∈ G v∈ GAlso, α is optimal, if it cannot be improved. Notice that we always have cα (v) ≤ dG (v), and if α is proper, then cα (v) = dG (v),and in this case α is optimal. Thus an improvement of a colouring is a change towardsa proper colouring. Note also that a graph G always has an optimal k-edge colouring,but it need not have any proper k-edge colourings. The next lemma is obvious.Lemma 4.3. An edge colouring α of G is proper if and only if cα (v) = dG (v) for all verticesv ∈ G.Lemma 4.4. Let α be an optimal k-edge colouring of G, and let v ∈ G. Suppose that thecolour i is available for v, and the colour j is incident with v at least twice. Then the connectedcomponent H of G α [i, j] that contains v, is an odd cycle.Proof. Suppose the connected component H is not an odd cycle. By Lemma 4.2, Hhas a 2-edge colouring γ : E H → {i, j}, in which both i and j are incident with eachvertex x with d H ( x) ≥ 2. (We have renamed the colours 1 and 2 to i and j.) We obtaina recolouring β of G as follows: γ ( e ), if e ∈ H , β( e) = α ( e ), if e ∈ H . /Since d H (v) ≥ 2 (by the assumption on the colour j) and in β both colours i and jare now incident with v, c β (v) = cα (v) + 1. Furthermore, by the construction of β,we have c β (u) ≥ cα (u) for all u = v. Therefore ∑u∈ G c β (u) > ∑u∈ G cα (u), whichcontradicts the optimality of α. Hence H is an odd cycle. ⊔ ⊓Theorem 4.1 (K ÖNIG (1916)). If G is bipartite, then χ′ ( G ) = ∆( G ).Proof. Let α be an optimal ∆-edge colouring of a bipartite G, where ∆ = ∆( G ). Ifthere were a v ∈ G with cα (v) < dG (v), then by Lemma 4.4, G would contain an oddcycle. But a bipartite graph does not contain such cycles. Therefore, for all vertices v,cα (v) = dG (v). By Lemma 4.3, α is a proper colouring, and ∆ = χ′ ( G ) as required. ⊔⊓Vizing's theoremIn general we can have χ′ ( G ) > ∆( G ) as one of our examples did show. The followingimportant theorem, due to V IZING, shows that the edge chromatic number of a graphG misses ∆( G ) by at most one colour.Theorem 4.2 (V IZING (1964)). For any graph G, ∆( G ) ≤ χ′ ( G ) ≤ ∆( G ) + 1.Proof. Let ∆ = ∆( G ). We need only to show that χ′ ( G ) ≤ ∆ + 1. Suppose on thecontrary that χ′ ( G ) > ∆ + 1, and let α be an optimal (∆ + 1)-edge colouring of G.
4.1 Edge colourings 46 We have (trivially) dG (u) < ∆ + 1 < χ′ ( G ) for all u ∈ G, and soClaim 1. For each u ∈ G, there exists an available colour b(u) for u. Moreover, by the counter hypothesis, α is not a proper colouring, and hence thereexists a v ∈ G with cα (v) < dG (v), and hence a colour i1 that is incident with v at leasttwice, say α(vu1 ) = i1 = α(vx) . (4.1)Claim 2. There is a sequence of vertices u1 , u2 , . . . such that α(vu j ) = i j and i j+1 = b(u j ) .Indeed, let u1 be as in (4.1). Assume we have already found the vertices u1 , . . . , u j ,with j ≥ 1, such that the claim holds for these. Suppose, contrary to the claim, that vis not incident with b(u j ) = i j+1 .We can recolour the edges vuℓ by iℓ+1 for ℓ ∈ [1, j], and urobtain in this way an improvement of α. Here v gains a u r −1new colour i j+1 . Also, each uℓ gains a new colour iℓ+1 .. .(and may loose the colour iℓ ). Therefore, for each uℓ ei- . . i r = i t +1 . i r −1ther its number of colours remains the same or it in- it v utcreases by one. This contradicts the optimality of α, and u2 i2proves Claim 2. i1 i1 u1 xLet t be the smallest index such that for some r < t,it+1 = ir . Such an index t exists, because dG (v) is finite. ur u r −1Let β be a recolouring of G such that for 1 ≤ j ≤ r − 1, ..β(vu j ) = i j+1 , and for all other edges e, β(e) = α(e). . i r = i t +1 . . . irClaim 3. β is an optimal (∆ + 1)-edge colouring of G. it v utIndeed, c β (v) = cα (v) and c β (u) ≥ cα (u) for all u, since u2 i3each u j (1 ≤ j ≤ r − 1) gains a new colour ji+1 although i2 i1 u1it may loose one of its old colours. x urLet then the colouring γ be obtained from β by re- u r −1colouring the edges vu j by i j+1 for r ≤ j ≤ t. Now, .. .vut is recoloured by ir = it+1 . . . i r +1 . irClaim 4. γ is an optimal (∆ + 1)-edge colouring of G. v ir ut u2 i3Indeed, the fact ir = it+1 ensures that ir is a new colour i2 i1incident with ut , and thus that cγ (ut ) ≥ c β (ut ). For all u1other vertices, cγ (u) ≥ c β (u) follows as for β. x
4.2 Ramsey Theory 47 By Claim 1, there is a colour i0 = b(v) that is available for v. By Lemma 4.4, theconnected components H1 of G β [i0 , ir ] and H2 of G γ [i0 , ir ] containing the vertex v arecycles, that is, H1 is a cycle (vur −1 ) P1 (ur v) and H2 is a cycle (vur −1 ) P2 (ut v), whereboth P1 : ur −1 − ur and P2 : ur −1 − ut are paths. However, the edges of P1 and P2 ⋆ → →⋆have the same colours with respect to β and γ (either i0 or ir ). This is not possible,since P1 ends in ur while P2 ends in a different vertex ut . This contradiction provesthe theorem. ⊔ ⊓Example 4.2. We show that χ′ ( G ) = 4 for the Petersen graph. Indeed, by Vizing'theorem, χ′ ( G ) = 3 or 4. Suppose 3 colours suffice. Let C : v1 − . . . − v5 − v1 be → → →the outer cycle and C ′ : u1 − . . . − u5 − u1 the inner cycle of G such that vi ui ∈ EG → → →for all i. Observe that every vertex is adjacent to all colours 1, 2, 3. Now C uses one colour(say 1) once and the other two twice. This can be done uniquely (up to permutations): 1 2 3 2 3 v1 − v2 − v3 − v4 − v5 − v1 . → → → → → 2 3 1 1 1Hence v1 − u1 , v2 − u2 , v3 − u3 , v4 − u4 , v5 − u5 . However, this means that 1 → → → → →cannot be a colour of any edge in C ′ . Since C ′ needs three colours, the claim follows.Edge Colouring Problem. Vizing's theorem (nor its present proof) does not offer anycharacterization for the graphs, for which χ′ ( G ) = ∆( G ) + 1. In fact, it is one of thefamous open problems of graph theory to find such a characterization. The answeris known (only) for some special classes of graphs. By H OLYER (1981), the problemwhether χ′ ( G ) is ∆( G ) or ∆( G ) + 1 is NP-complete. The proof of Vizing's theorem can be used to obtain a proper colouring of G withat most ∆( G ) + 1 colours, when the word 'optimal' is forgotten: colour first the edgesas well as you can (if nothing better, then arbitrarily in two colours), and use the proofiteratively to improve the colouring until no improvement is possible – then the proofsays that the result is a proper colouring.4.2 Ramsey TheoryIn general, Ramsey theory studies unavoidable patterns in combinatorics. We con-sider an instance of this theory mainly for edge colourings (that need not be proper).A typical example of a Ramsey property is the following: given 6 persons each pair ofwhom are either friends or enemies, there are then 3 persons who are mutual friendsor mutual enemies. In graph theoretic terms this means that each colouring of theedges of K6 with 2 colours results in a monochromatic triangle.
4.2 Ramsey Theory 49Ramsey's theoremD EFINITION . Let α be an edge colouring of G. A subgraph H ⊆ G is said to be (i-)monochromatic, if all edges of H have the same colour i. The following theorem is one of the jewels of combinatorics.Theorem 4.4 (R AMSEY (1930)). Let p, q ≥ 2 be any integers. Then there exists a (smallest)integer R( p, q) such that for all n ≥ R( p, q), any 2-edge colouring of Kn → [1, 2] contains a1-monochromatic K p or a 2-monochromatic Kq . Before proving this, we give an equivalent statement. Recall that a subset X ⊆ VGis stable, if G [ X ] is a discrete graph.Theorem 4.5. Let p, q ≥ 2 be any integers. Then there exists a (smallest) integer R( p, q)such that for all n ≥ R( p, q), any graph G of order n contains a complete subgraph of orderp or a stable set of order q. Be patient, this will follow from Theorem 4.6. The number R( p, q) is known as theRamsey number for p and q. It is clear that R( p, 2) = p and R(2, q) = q. Theorems 4.4 and 4.5 follow from the next result which shows (inductively) thatan upper bound exists for the Ramsey numbers R( p, q).Theorem 4.6 (E RDÖS and S ZEKERES (1935)). The Ramsey number R( p, q) exists for allp, q ≥ 2, and R( p, q) ≤ R( p, q − 1) + R( p − 1, q) .Proof. We use induction on p + q. It is clear that R( p, q) exists for p = 2 or q = 2, andit is thus exists for p + q ≤ 5. It is now sufficient to show that if G is a graph of order R( p, q − 1) + R( p − 1, q),then it has a complete subgraph of order p or a stable subset of order q. Let v ∈ G, and denote by A = VG ( NG (v) ∪ {v}) the set of vertices that are notadjacent to v. Since G has R( p, q − 1) + R( p − 1, q) − 1 vertices different from v, either| NG (v)| ≥ R( p − 1, q) or | A| ≥ R( p, q − 1) (or both). Assume first that | NG (v)| ≥ R( p − 1, q). By the definition of Ramsey numbers,G [ NG (v)] contains a complete subgraph B of order p − 1 or a stable subset S of orderq. In the first case, B ∪ {v} induces a complete subgraph K p in G, and in the secondcase the same stable set of order q is good for G. If | A| ≥ R( p, q − 1), then G [ A] contains a complete subgraph of order p or a stablesubset S of order q − 1. In the first case, the same complete subgraph of order p isgood for G, and in the second case, S ∪ {v} is a stable subset of G of q vertices. Thisproves the claim. ⊔ ⊓
4.2 Ramsey Theory 52The values R2 (K2,n ) are known for n ≤ 16, and in general, R2 (K2,n ) ≤ 4n − 2. Thevalue R2 (K2,17 ) is either 65 or 66. Let Wn denote the wheel on n vertices. It is a cycle Cn−1 , where a vertex v withdegree n − 1 is attached. Note that W4 = K4 . Then R2 (W5 ) = 15 and R2 (W6 ) = 17. For three colours, much less is known. In fact, the only nontrivial result for com-plete graphs is: R3 (K3 ) = 17. Also, 128 ≤ R3 (K4 ) ≤ 235, and 385 ≤ R3 (K5 ), butno nontrivial upper bound is known for R3 (K5 ). For the square C4 , we know thatR3 (C4 ) = 11. Needless to say that no exact values are known for Rk (Kn ) for k ≥ 4 and n ≥ 3. It follows from Theorem 4.4 that for any complete Kn , there exists a graph G(well, any sufficiently large complete graph) such that any 2-edge colouring of Ghas a monochromatic (induced) subgraph Kn . Note, however, that in Corollary 4.2the monochromatic subgraph Hi is not required to be induced. The following impressive theorem improves the results we have mentioned in thischapter and it has a difficult proof.Theorem 4.9 (D EUBER, E RDÖS, H AJNAL, P ÓSA, and R ÖDL (around 1973)). Let H beany graph. Then there exists a graph G such that any 2-edge colouring of G has an monochro-matic induced subgraph H.Example 4.4. As an application of Ramsey's theorem, we shortly describe Schur'stheorem. For this, consider the partition {1, 4, 10, 13}, {2, 3, 11, 12}, {5, 6, 7, 8, 9} ofthe set N13 = [1, 13]. We observe that in no partition class there are three integerssuch that x + y = z. However, if you try to partition N14 into three classes, then youare bound to find a class, where x + y = z has a solution. S CHUR (1916) solved this problem in a general setting. The following gives a shortproof using Ramsey's theorem.For each n ≥ 1, there exists an integer S(n) such that any partition S1 , . . . , Sn of N S(n) hasa class Si containing two integers x, y such that x + y ∈ Si . Indeed, let S(n) = R(3, 3, . . . , 3), where 3 occurs n times, and let K be a completeon N S(n) . For a partition S1 , . . . , Sn of N S(n) , define an edge colouring α of K by α(ij) = k, if |i − j| ∈ Sk .By Theorem 4.8, K α has a monochromatic triangle, that is, there are three verticesi, j, t such that 1 ≤ i < j < t ≤ S(n) with t − j, j − i, t − i ∈ Sk for some k. But(t − j) + ( j − i) = t − i proves the claim. There are quite many interesting corollaries to Ramsey's theorem in various partsof mathematics including not only graph theory, but also, e.g., geometry and algebra,seeR.L. G RAHAM , B.L. R OTHSCHILD AND J.L. S PENCER, "Ramsey Theory", Wiley, (2nded.) 1990.
4.3 Vertex colourings 534.3 Vertex colouringsThe vertices of a graph G can also be classified using colourings. These colouringstell that certain vertices have a common property (or that they are similar in somerespect), if they share the same colour. In this chapter, we shall concentrate on propervertex colourings, where adjacent vertices get different colours.The chromatic numberD EFINITION . A k-colouring (or a k-vertex colouring) of a graph G is a mappingα : VG → [1, k]. The colouring α is proper, if adjacent vertices obtain a different colour:for all uv ∈ G, we have α(u) = α(v). A colour i ∈ [1, k] is said to be available for avertex v, if no neighbour of v is coloured by i. A graph G is k-colourable, if there is a proper k-colouring for G. The (vertex)chromatic number χ( G ) of G is defines as χ( G ) = min{k | there exists a proper k-colouring of G } .If χ( G ) = k, then G is k-chromatic. Each proper vertex colouring α : VG → [1, k] provides a partition {V1 , V2 , . . . , Vk }of the vertex set VG , where Vi = {v | α(v) = i }.Example 4.5. The graph on the right, which is often called awheel (of order 7), is 3-chromatic.By the definitions, a graph G is 2-colourable if and only if itis bipartite. Again, the 'names' of the colours are immaterial:Lemma 4.5. Let α be a proper k-colouring of G, and let π be any permutation of the colours.Then the colouring β = πα is a proper k-colouring of G.Proof. Indeed, if α : VG → [1, k] is proper, and if π : [1, k] → [1, k] is a bijection, thenuv ∈ G implies that α(u) = α(v), and hence also that πα(u) = πα(v). It follows thatπα is a proper colouring. ⊔ ⊓Example 4.6. A graph is triangle-free, if it has no subgraphs isomorphic to K3 . Weshow that there are triangle-free graphs with arbitrarily large chromatic numbers. The following construction is due to G RÖTZEL: Let G be any triangle-free graphwith VG = {v1 , v2 , . . . , vn }. Let G t be a new graph obtained by adding n + 1 newvertices v and u1 , u2 , . . . , un such that G t has all the edges of G plus the edges ui v andui x for all x ∈ N (vi ) and for all i ∈ [1, n].
4.3 Vertex colourings 54Claim. G t is triangle-free and it is k + 1-chromatic Indeed, let U = {u1 , . . . , un }. We show first that G t is triangle-free. Now, U isstable, and so a triangle contains at most (and thus exactly) one vertex ui ∈ U. If{ui , v j , vk } induces a triangle, so does {vi , v j , vk } by the definition of G t , but the lattertriangle is already in G; a contradiction. For the chromatic number we notice first that χ( G t ) ≤ (k + 1). If α is a properk-colouring of G, extend it by setting α(ui ) = α(vi ) and α(v) = k + 1. Secondly, χ( G t ) > k. Assume that α is a proper k-colouring of G t , say with α(v) =k. Then α(ui ) = k. Recolour each vi by α(ui ). This gives a proper (k − 1)-colouring toG; a contradiction. Therefore χ( G t ) = k + 1. Now using inductively the above construction starting from the triangle-freegraph K2 , we obtain larger triangle -free graphs with high chromatic numbers.Critical graphsD EFINITION . A k-chromatic graph G is said to be k-critical, if χ( H ) < k for all H ⊆ Gwith H = G. In a critical graph an elimination of any edge and of any vertex will reduce thechromatic number: χ( G −e) < χ( G ) and χ( G −v) < χ( G ) for e ∈ G and v ∈ G. EachKn is n-critical, since in Kn −(uv) the vertices u and v can gain the same colour.Example 4.7. The graph K2 = P2 is the only 2-critical graph. The 3-critical graphs areexactly the odd cycles C2n+1 for n ≥ 1, since a 3-chromatic G is not bipartite, and thusmust have a cycle of odd length.Theorem 4.10. If G is k-critical for k ≥ 2, then it is connected, and δ( G ) ≥ k − 1.Proof. Note that for any graph G with the connected components G1 , G2 , . . . , Gm ,χ( G ) = max{χ( Gi ) | i ∈ [1, m]} . Connectivity claim follows from this observation. Let then G be k-critical, but δ( G ) = dG (v) ≤ k − 2 for v ∈ G. Since G is critical,there is a proper (k − 1)-colouring of G −v. Now v is adjacent to only δ( G ) < k − 1vertices. But there are k colours, and hence there is an available colour i for v. If werecolour v by i, then a proper (k − 1)-colouring is obtained for G; a contradiction. ⊔ ⊓ The case (iii) of the next theorem is due to S ZEKERES AND W ILF (1968).Theorem 4.11. Let G be any graph with k = χ( G ). (i) G has a k-critical subgraph H. (ii) G has at least k vertices of degree ≥ k − 1.(iii) k ≤ 1 + max H ⊆ G δ( H ).
5Graphs on Surfaces5.1 Planar graphsThe plane representations of graphs are by no means unique. Indeed, a graph G canbe drawn in arbitrarily many different ways. Also, the properties of a graph are notnecessarily immediate from one representation, but may be apparent from another.There are, however, important families of graphs, the surface graphs, that rely onthe (topological or geometrical) properties of the drawings of graphs. We restrict our-selves in this chapter to the most natural of these, the planar graphs. The geometryof the plane will be treated intuitively. A planar graph will be a graph that can be drawn in the plane so that no twoedges intersect with each other. Such graphs are used, e.g., in the design of electrical(or similar) circuits, where one tries to (or has to) avoid crossing the wires or laserbeams. Planar graphs come into use also in some parts of mathematics, especially ingroup theory and topology. There are fast algorithms (linear time algorithms) for testing whether a graph isplanar or not. However, the algorithms are all rather difficult to implement. Most ofthem are based on an algorithm designed by A USLANDER AND PARTER (1961) seeSection 6.5 ofS. S KIENA, "Implementing Discrete Mathematics: Combinatorics and Graph Theorywith Mathematica", Addison-Wesley, 1990.DefinitionD EFINITION . A graph G is a planar graph, if it has aplane figure P( G ), called the plane embedding of G,where the lines (or continuous curves) correspond-ing to the edges do not intersect each other except attheir ends.The complete bipartite graph K2,4 is a planar graph.D EFINITION . An edge e = uv ∈ G is subdivided, when it is replaced by a pathu − x − v of length two by introducing a new vertex x. A subdivision H of a graph → →G is obtained from G by a sequence of subdivisions.
5.1 Planar graphs 62 The following result is clear.Lemma 5.1. A graph is planar if and only if its subdivisions are planar.Geometric propertiesIt is clear that the graph theoretical properties of G are inherited by all of its planeembeddings. For instance, the way we draw a graph G in the plane does not changeits maximum degree or its chromatic number. More importantly, there are – as weshall see – some nontrivial topological (or geometric) properties that are shared bythe plane embeddings. We recall first some elements of the plane geometry. Let F be an open set of theplane R × R, that is, every point x ∈ F has a disk centred at x and contained in F.Then F is a region, if any two points x, y ∈ F can be joined by a continuous curve thepoints of which are all in F. The boundary ∂( F ) of a region F consists of those pointsfor which every neighbourhood contains points from F and its complement. Let G be a planar graph, and P( G ) one of its plane embeddings. Regard now eachedge e = uv ∈ G as a line from u to v. The set (R × R ) EG is open, and it is dividedinto a finite number of disjoint regions, called the faces of P( G ).D EFINITION . A face of P( G ) is an interior face, if it isbounded. The (unique) face that is unbounded is called theexterior face of P( G ). The edges that surround a face F con- F3stitute the boundary ∂( F ) of F. The exterior boundary is the F2boundary of the exterior face. The vertices (edges, resp.) on F1the exterior boundary are called exterior vertices exterior F0edges, resp.). Vertices (edges, resp.) that are not on the exte-rior boundary are interior vertices interior edges, resp.). Embeddings P( G ) satisfy some properties that we accepts at face value.Lemma 5.2. Let P( G ) be a plane embedding of a planar graph G. (i) Two different faces F1 and F2 are disjoint, and their boundaries can intersect only on edges. (ii) P( G ) has a unique exterior face.(iii) Each edge e belongs to the boundary of at most two faces.(iv) Each cycle of G surrounds (that is, its interior contains) at least one internal face of P( G ). (v) A bridge of G belongs to the boundary of only one face.(vi) An edge that is not a bridge belongs to the boundary of exactly two faces.
5.1 Planar graphs 63 If P( G ) is a plane embedding of a graph G, then so is any drawing P′ ( G ) which isobtained from P( G ) by an injective mapping of the plane that preserves continuouscurves. This means, in particular, that every planar graph has a plane embedding insideany geometric circle of arbitrarily small radius, or inside any geometric triangle.Euler's formulaLemma 5.3. A plane embedding P( G ) of a planar graph G has no interior faces if and only ifG is acyclic, that is, if and only if the connected components of G are trees.Proof. This is clear from Lemma 5.2. ⊔ ⊓ The next general form of Euler's formula was proved by L EGENDRE (1794).Theorem 5.1 (Euler's formula). Let G be a connected planar graph, and let P( G ) be any ofits plane embeddings. Then νG − ε G + ϕ = 2 ,where ϕ is the number of faces of P( G ).Proof. We shall prove the claim by induction on the number of faces ϕ of a planeembedding P( G ). First, notice that ϕ ≥ 1, since each P( G ) has an exterior face. If ϕ = 1, then, by Lemma 5.3, there are no cycles in G, and since G is connected, itis a tree. In this case, by Theorem 2.4, we have ε G = νG − 1, and the claim holds. Suppose then that the claim is true for all plane embeddings with less than ϕ facesfor ϕ ≥ 2. Let P( G ) be a plane embedding of a connected planar graph such thatP( G ) has ϕ faces. Let e ∈ G be an edge that is not a bridge. The subgraph G −e is planar with a planeembedding P( G −e) = P( G )−e obtained by simply erasing the edge e. Now P( G −e)has ϕ − 1 faces, since the two faces of P( G ) that are separated by e are merged intoone face of P( G −e). By the induction hypothesis, νG− e − ε G−e + ( ϕ − 1) = 2, andhence νG − (ε G − 1) + ( ϕ − 1) = 2, and the claim follows. ⊔ ⊓ In particular, we have the following invariant property of planar graphs.Corollary 5.1. Let G be a planar graph. Then every plane embedding of G has the samenumber of faces: ϕ G = ε G − νG + 2Maximal planar graphsLemma 5.4. If G is a planar graph of order νG ≥ 3, then ε G ≤ 3νG − 6. Moreover, if G hasno triangles C3 , then ε G ≤ 2νG − 4.
5.1 Planar graphs 64Proof. If G is disconnected with connected components Gi , for i ∈ [1, k], and if theclaim holds for these smaller (necessarily planar) graphs Gi , then it holds for G, since νG νG εG = ∑ ε Gi ≤ 3 ∑ νGi − 6k = 3νG − 6k ≤ 3νG − 6 . i=1 i=1It is thus sufficient to prove the claim for connected planar graphs. Also, the case where ε G ≤ 2 is clear. Suppose thus that ε G ≥ 3. Each face F of an embedding P( G ) contains at least three edges on its boundary∂( F ). Hence 3ϕ ≤ 2ε G , since each edge lies on at most two faces. The first claimfollows from Euler's formula. The second claim is proved similarly except that, in this case, each face F of P( G )contains at least four edges on its boundary (when G is connected and ε G ≥ 4). ⊔ ⊓ An upper bound for δ( G ) for planar graphs was achieved by H EAWOOD.Theorem 5.2 (H EAWOOD (1890)). If G is a planar graph, then δ( G ) ≤ 5.Proof. If νG ≤ 2, then there is nothing to prove. Suppose νG ≥ 3. By the handshakinglemma and the previous lemma, δ( G ) · νG ≤ ∑ dG (v) = 2ε G ≤ 6νG − 12 . v∈ GIt follows that δ( G ) ≤ 5. ⊔ ⊓Theorem 5.3. K5 and K3,3 are not planar graphs.Proof. By Lemma 5.4, a planar graph of order 5 has at most 9 edges, but K5 has 5vertices and 10 edges. By the second claim of Lemma 5.4, a triangle-free planar graphof order 6 has at most 8 edges, but K3,3 has 6 vertices and 9 edges. ⊔ ⊓D EFINITION . A planar graph G is maximal, if G + e is nonplanar for every e ∈ G. /Example 5.1. Clearly, if we remove one edge from K5 , the result is a maximal planargraph. However, if an edge is removed from K3,3 , the result is not maximal!Lemma 5.5. Let F be a face of a plane embedding P( G ) that has at least four edges on itsboundary. Then there are two nonadjacent vertices on the boundary of F.Proof. Assume that the set of the boundary vertices of F induces a complete sub-graph K. The edges of K are either on the boundary or they are not inside F (since Fis a face.) Add a new vertex x inside F, and connect the vertices of K to x. The resultis a plane embedding of a graph H with VH = VG ∪ { x} (that has G as its inducedsubgraph). The induced subgraph H [K ∪ { x}] is complete, and since H is planar, wehave |K | < 4 as required. ⊔ ⊓
5.1 Planar graphs 65 By the previous lemma, if a face has a boundary of at least four edges, then anedge can be added to the graph (inside the face), and the graph remains to be planar.Hence we have provedCorollary 5.2. If G is a maximal planar graph with νG ≥ 3, then G is triangulated, that is,every face of a plane embedding P( G ) has a boundary of exactly three edges.Theorem 5.4. For a maximal planar graph G of order νG ≥ 3, ε G = 3νG − 6 .Proof. Each face F of an embedding P( G ) is a triangle having three edges on itsboundary. Hence 3ϕ = 2ε G , since there are now no bridges. The claim follows fromEuler's formula. ⊔ ⊓Kuratowski's theoremTheorem 5.5 will give a simple criterion for planarity of graphs. This theorem (due toK URATOWSKI in 1930) is one of the jewels of graph theory. In fact, the theorem wasproven earlier by P ONTRYAGIN (1927-1928), and also independently by F RINK ANDS MITH (1930). For history of the result, seeJ.W. K ENNEDY, L.V. Q UINTAS , AND M.M. S YSLO, The theorem on planar graphs.Historia Math. 12 (1985), 356 – 368. The graphs K5 and K3,3 are the smallest nonplanar graphs, and, by Lemma 5.1, ifG contains a subdivision of K5 or K3,3 as a subgraph, then G is not planar. We provethe converse of this result in what follows. ThereforeTheorem 5.5 (K URATOWSKI (1930)). A graph is planar if and only if it contains no subdi-vision of K5 or K3,3 as a subgraph. We prove this result along the lines of T HOMASSEN (1981) using 3-connectivity.Example 5.2. The cube Qk is planar only for k = 1, 2, 3. Indeed, the graph Q4 containsa subdivision of K3,3 , and thus by Theorem 5.5 it is not planar. On the other hand, eachQk with k ≥ 4 has Q4 as a subgraph, and therefore they are nonplanar. The subgraphof Q4 that is a subdivision of K3,3 is given below. 0000 1010 1001 0100 1110 1101 0001 0011 1000 1100 0010
5.1 Planar graphs 66D EFINITION . A graph G is called a Kuratowski graph, if it is a subdivision of K5 orK3,3 .Lemma 5.6. Let E ⊆ EG be the set of the boundary edges of a face F in a plane embedding ofG. Then there exists a plane embedding P( G ), where the edges of E are exterior edges.Proof. This is a geometric proof. Choose a circle that contains every point of the planeembedding (including all points of the edges) such that the centre of the circle isinside the given face. Then use geometric inversion with respect to this circle. Thiswill map the given face as the exterior face of the image plane embedding. ⊔ ⊓Lemma 5.7. Let G be a nonplanar graph without Kuratowski graphs of minimal total sizeε G + νG . Then G is 3-connected.Proof. By the minimality assumption, G is connected. We show then that G is 2-connected. On the contrary, assume that v is a cut vertex of G, and let A1 , . . . , Ak bethe connected components of G −v.Since G is minimal nonplanar with respect to ε G , thesubgraphs Gi = G [ Ai ∪ {v}] have plane embeddings A1 A2P( Gi ), where v is an exterior vertex. We can glue theseplane embeddings together at v to obtain a plane em-bedding of G, and this will contradict the choice of G. Assume then that G has a separating set S = {u, v}. Let G1 and G2 be any sub-graphs of G such that EG = EG1 ∪ EG2 , S = VG1 ∩ VG2 , and both G1 and G2 contain aconnected component of G −S. Since G is 2-connected (by the above), there are pathsu − v in G1 and G2 . Indeed, both u and v are adjacent to a vertex of each connected →⋆component of G −S. Let Hi = Gi + uv. (Maybe uv ∈ G.)If both H1 and H2 are planar, then, by Lemma 5.6, theyhave plane embeddings, where uv is an exterior edge.It is now easy to glue H1 and H2 together on the edge H1 H2uv to obtain a plane embedding of G + uv, and thusof G. We conclude that H1 or H2 is nonplanar, say H1 . Now ε H1 < ε G , and so, by theminimality of G, H1 contains a Kuratowski graph H. However, there is a path u − v →⋆in H2 , since G2 ⊆ H2 . This path can be regarded as a subdivision of uv, and thus Gcontains a Kuratowski graph. This contradiction shows that G is 3-connected. ⊔ ⊓Lemma 5.8. Let G be a 3-connected graph of order νG ≥ 5. Then there exists an edge e ∈ Gsuch that the contraction G ∗ e is 3-connected.Proof. On the contrary suppose that for any e ∈ G, the graph G ∗ e has a separatingset S with |S| = 2. Let e = uv, and let x = x(uv) be the contracted vertex. Necessarilyx ∈ S, say S = { x, z} (for, otherwise, S would separate G already). Therefore T =
5.1 Planar graphs 67{u, v, z} separates G. Assume that e and S are chosen such that G − T has a connectedcomponent A with the least possible number of vertices.There exists a vertex y ∈ A with zy ∈ G. (Otherwise{u, v} would separate G.) The graph G ∗ (zy) is not 3- Bconnected by assumption, and hence, as in the above, uthere exists a vertex w such that R = {z, y, w} separates A T vG. It can be that w ∈ {u, v}, but by symmetry we can z ysuppose that w = u. Since uv ∈ G, G − R has a connected component B such that u, v ∈ B. For each /y′ ∈ B, there exists a path P : u − y′ in G −{z, w}, since G is 3-connected, and hence ⋆ →this P goes through y. Therefore y′ is connected to y also in G − T, that is, y′ ∈ A, andso B ⊆ A. The inclusion is proper, since y ∈ B. Hence | B| < | A|, and this contradicts /the choice of A. ⊔ ⊓ By the next lemma, a Kuratowski graph cannot be created by contractions.Lemma 5.9. Let G be a graph. If for some e ∈ G the contraction G ∗ e has a Kuratowskisubgraph, then so does G.Proof. The proof consists of several cases depending on the Kuratowski graph, andhow the subdivision is made. We do not consider the details of these cases. Let H be a Kuratowski graph of G ∗ e, where x = x(uv) is the contracted vertex fore = uv. If d H ( x) = 2, then the claim is obviously true. Suppose then that d H ( x) = 3or 4. If there exists at most one edge xy ∈ H such that uy ∈ G (or vy ∈ G), then oneeasily sees that G contains a Kuratowski graph. There remains only one case, where H is a subdivision of K5 , and both u and vhave 3 neighbours in the subgraph of G corresponding to H. In this case, G containsa subdivision of K3,3 . ⊔ ⊓ v2 v4 v2 v4 x u v v1 v3 v1 v3Lemma 5.10. Every 3-connected graph G without Kuratowski subgraphs is planar.Proof. The proof is by induction on νG . The only 3-connected graph of order 4 is theplanar graph K4 . Therefore we can assume that νG ≥ 5. By Lemma 5.8, there exists an edge e = uv ∈ G such that G ∗ e (with a contractedvertex x) is 3-connected. By Lemma 5.9, G ∗ e has no Kuratowski subgraphs, andhence G ∗ e has a plane embedding P( G ∗ e) by the induction hypothesis. Consider
5.2 Colouring planar graphs 68the part P( G ∗ e)− x, and let C be the boundary of the face of P( G ∗ e)− x containingx (in P( G ∗ e)). Here C is a cycle of G (since G is 3-connected). Now since G −{u, v} = ( G ∗ e)− x, P( G ∗ e)− x is a plane embedding of G −{u, v},and NG (u) ⊆ VC ∪ {v} and NG (v) ⊆ VC ∪ {u}. Assume, by symmetry, that dG (v) ≤dG (u). Let NG (v) {u} = {v1 , v2 , . . . , vk } in order along the cycle C. Let Pi,j : vi − v j ⋆ →be the path along C from vi to v j . We obtain a plane embedding of G −u by drawing(straight) edges vvi for 1 ≤ i ≤ k.(1) If NG (u) {v} ∈ Pi,i+1 (i + 1 is taken modulo k) for some i, then, clearly, G has aplane embedding (obtained from P( G )−u by putting u inside the triangle (v, vi , vi+1 )and by drawing the edges with an end u inside this triangle). y(2) Assume there are y, z ∈ NG (u) {v} such that y ∈ Pij uand z ∈ Pij for some i and j, where y, z ∈ {vi , v j }. Now, / / z{u, vi , vi+1 } ∪ {v, z, y} form a subdivision of K3,3. vBy (1) and (2), we can assume that NG (u) {v} ⊆ NG (v).Therefore, NG (u) {v} = NG (v) {u} by the assumption udG (v) ≤ dG (u). Also, by (1), dG (v) = dG (u) > 3. But now vu, v, v1 , v2 , v3 give a subdivision of K5 . ⊔ ⊓Proof of Theorem 5.5. By Theorem 5.3 and Lemma 5.1, we need to show that eachnonplanar graph G contains a Kuratowski subgraph. On the contrary, suppose thatG is a nonplanar graph that has a minimal size ε G such that G does not contain aKuratowski subgraph. Then, by Lemma 5.7, G is 3-connected, and by Lemma 5.10, itis planar. This contradiction proves the claim. ⊔ ⊓Example 5.3. Any graph G can be drawn in the plane so that three of its edges neverintersect at the same point. The crossing number ×( G ) is the minimum number ofintersections of its edges in such plane drawings of G. Therefore G is planar if andonly if ×( G ) = 0, and, for instance, ×(K5 ) = 1. We show that ×(K6 ) = 3. For this we need to show that ×(K6 ) ≥ 3. For the equal-ity, one is invited to design a drawing with exactly 3 crossings. Let X (K6 ) be a drawing of K6 using c crossings so that two edges cross at mostonce. Add a new vertex at each crossing. This results in a planar graph G on c + 6vertices and 2c + 15 edges. Now c ≥ 3, since ε G = 2c + 15 ≤ 3(c + 6) − 6 = 3νG − 6.5.2 Colouring planar graphsThe most famous problem in the history of graph theory is that of the chromaticnumber of planar graphs. The problem was known as the 4-Colour Conjecture formore than 120 years, until it was solved by A PPEL AND H AKEN in 1976: if G is a
5.2 Colouring planar graphs 69planar graph, then χ( G ) ≤ 4. The 4-Colour Conjecture has had a deep influence onthe theory of graphs during the last 150 years. The solution of the 4-Colour Theoremis difficult, and it requires the assistance of a computer.The 5-colour theoremWe prove H EAWOOD's result (1890) that each planar graph is properly 5-colourable.Lemma 5.11. If G is a planar graph, then χ( G ) ≤ 6.Proof. The proof is by induction on νG . Clearly, the claim holds for νG ≤ 6. By Theo-rem 5.2, a planar graph G has a vertex v with dG (v) ≤ 5. By the induction hypothesis,χ( G −v) ≤ 6. Since dG (v) ≤ 5, there is a colour i available for v in the 6-colouring ofG −v, and so χ( G ) ≤ 6. ⊔ ⊓ The proof of the following theorem is partly geometric in nature.Theorem 5.6 (H EAWOOD (1890)). If G is a planar graph, then χ( G ) ≤ 5.Proof. Suppose the claim does not hold, and let G be a 6-critical planar graph. Re-call that for k-critical graphs H, δ( H ) ≥ k − 1, and thus there exists a vertex v withdG (v) = δ( G ) ≥ 5. By Theorem 5.2, dG (v) = 5.Let α be a proper 5-colouring of G −v. Such a colouring v5 v1exists, because G is 6-critical. By assumption, χ( G ) > 5,and therefore for each i ∈ [1, 5], there exists a neigh- v v2 P13bour vi ∈ NG (v) such that α(vi ) = i. Suppose theseneighbours vi of v occur in the plane in the geometric v4 v3order of the figure. Consider the subgraph G [i, j] ⊆ G made of colours i and j. The vertices vi andv j are in the same connected component of G [i, j] (for, otherwise we interchange thecolours i and j in the connected component containing v j to obtain a recolouring ofG, where vi and v j have the same colour i, and then recolour v with the remainingcolour j). Let Pij : vi − v j be a path in G [i, j], and let C = (vv1 ) P13 (v3 v). By the geometric ⋆ →assumption, exactly one of v2 , v4 lies inside the region enclosed by the cycle C. Now,the path P24 must meet C at some vertex of C, since G is planar. This is a contradiction,since the vertices of P24 are coloured by 2 and 4, but C contains no such colours. ⊔ ⊓ The final word on the chromatic number of planar graphs was proved by A PPELAND H AKEN in 1976.Theorem 5.7 (4-Colour Theorem). If G is a planar graph, then χ( G ) ≤ 4. By the following theorem, each planar graph can be decomposed into two bipar-tite graphs.
5.2 Colouring planar graphs 70Theorem 5.8. Let G = (V, E) be a 4-chromatic graph, χ( G ) ≤ 4. Then the edges of G canbe partitioned into two subsets E1 and E2 such that (V, E1 ) and (V, E2 ) are both bipartite.Proof. Let Vi = α−1 (i ) be the set of vertices coloured by i in a proper 4-colouring α ofG. The define E1 as the subset of the edges of G that are between the sets V1 and V2 ;V1 and V4 ; V3 and V4 . Let E2 be the rest of the edges, that is, they are between the setsV1 and V3 ; V2 and V3 ; V2 and V4 . It is clear that (V, E1 ) and (V, E2 ) are bipartite, sincethe sets Vi are stable. ⊔ ⊓Map colouring∗The 4-Colour Conjecture was originally stated for maps. In the map-colouring prob-lem we are given several countries with common borders, and we wish to coloureach country so that no neighbouring countries obtain the same colour. How manycolours are needed? A border between two countries is assumed to have a positive length – in par-ticular, countries that have only one point in common are not allowed in the mapcolouring. Formally, we define a map as a connected planar (embedding of a) graph with nobridges. The edges of this graph represent the boundaries between countries. Hencea country is a face of the map, and two neighbouring countries share a common edge(not just a single vertex). We deny bridges, because a bridge in such a map would bea boundary inside a country.The map-colouring problem is restated as follows:How many colours are needed for the faces of a planeembedding so that no adjacent faces obtain the samecolour.The illustrated map can be 4-coloured, and it can-not be coloured using only 3 colours, because ev-ery two faces have a common border. Let F1 , F2 , . . . , Fn be the countries of a map M, and define a graph G with VG ={v1 , v2 , . . . , vn } such that vi v j ∈ G if and only if the countries Fi and Fj are neighbours.It is easy to see that G is a planar graph. Using this notion of a dual graph, we canstate the map-colouring problem in new form: What is the chromatic number of a planargraph? By the 4-Colour Theorem it is at most four. Map-colouring can be used in rather generic topological setting, where the mapsare defined by curves in the plane. As an example, consider finitely many simpleclosed curves in the plane. These curves divide the plane into regions. The regions are2-colourable.
5.2 Colouring planar graphs 71That is, the graph where the vertices corre- 2spond to the regions, and the edges correspond 1 1to the neighbourhood relation, is bipartite. To 1 2see this, colour a region by 1, if the region is in- 2 1 2 1side an odd number of curves, and, otherwise, 2 2 1colour it by 2.History of the 4-Colour TheoremThat four colours suffice planar maps was conjectured around 1850 by F RANCISG UTHRIE, a student of D E M ORGAN at University College of London. During thefollowing 120 years many outstanding mathematicians tried to solve the problem,and some of them even thought that they had been successful. In 1879 C AYLEY pointed out some difficulties that lie in the conjecture. The sameyear A LFRED K EMPE published a paper, where he claimed a proof of the 4CC. Thebasic idea in K EMPE's argument (known later as Kempe chains) was the same as laterused by H EAWOOD to prove the 5-Colour Theorem, (Theorem 5.6). For more than 10 years K EMPE's proof was considered to be valid. For instance,TAIT published two papers on the 4CC in the 1880's that contained clever ideas, butalso some further errors. In 1890 H EAWOOD showed that K EMPE's proof had seri-ous gaps. As we shall see in the next chapter, H EAWOOD discovered the numberof colours needed for all maps on other surfaces than the plane. Also, he provedthat if the number of edges around each region is divisible by 3, then the map is4-colourable. One can triangulate any planar graph G (drawn in the plane), by adding edgesto divide the faces into triangles. B IRKHOFF introduced one of the basic notions (re-ducibility) needed in the proof of the 4CC. In a triangulation, a configuration is a partthat is contained inside a cycle. An unavoidable set is a set of configurations such thatany triangulation must contain one of the configurations in the set. A configurationis said to be reducible, if it is not contained in a triangulation of a minimal counterexample to the 4CC. The search for avoidable sets began in 1904 with work of W EINICKE, and in 1922F RANKLIN showed that the 4CC holds for maps with at most 25 regions. This numberwas increased to 27 by R EYNOLDS (1926), to 35 by W INN (1940), to 39 by O RE ANDS TEMPLE (1970), to 95 by M AYER (1976). The final notion for the solution was due to H EESCH, who in 1969 introduceddischarging. This consists of assigning to a vertex v the charge 6 − dG (v). From Euler'sformula we see that for the sum of the charges, we have ∑(6 − dG (v)) = 12. vNow, a given set S of configurations can be proved to be unavoidable, if for a triangu-lation, that does not contain a configuration from S, one can 'redistribute' the chargesso that no v comes up with a positive charge.
5.2 Colouring planar graphs 72 According to H EESCH one might be satisfied with a set of 8900 configurations toprove the 4CC. There were difficulties with his approach that were solved in 1976by A PPEL AND H AKEN. They based the proof on reducibility using Kempe chains,and ended up with an unavoidable set with over 1900 configurations and some 300discharging rules. The proof used 1200 hours of computer time. (K OCH assisted withthe computer calculations.) A simplified proof by R OBERTSON , S ANDERS , S EYMOURAND T HOMAS (1997) uses 633 configurations and 32 discharging rules. Because ofthese simplifications also the computer time is much less than in the original proof. The following book contains the ideas of the proof of the 4-Colour Theorem.T.L. S AATY AND P.C. K AINEN, "The Four-Color Problem", Dover, 1986.List colouringD EFINITION . Let G be a graph so that each of its vertices v is given a list (set) Λ(v) ofcolours. A proper colouring α : VG → [1, m] of G is a (Λ-)list colouring, if each vertexv gets a colour from its list, α(v) ∈ Λ(v). The list chromatic number χℓ ( G ) is the smallest integer k such that G has a Λ-listcolouring for all lists of size k, |Λ(v)| = k}. Also, G is k-choosable, if χℓ ( G ) ≤ k. {1, 2} {1, 3} {2, 3}Example 5.4. The bipartite graph K3,3 is not 2- y1 y2 y3choosable. Indeed, let the bipartition of K3,3 be( X, Y ), where X = { x1 , x2 , x3 } and Y = {y1 , y2 , y3 }.The lists for the vertices shown in the figure show x1 x2 x3that χℓ (K3,3 ) > 2. {1, 2} {1, 3} {2, 3} Obviously χ( G ) ≤ χℓ ( G ), since proper colourings are special cases of list colour-ings, but equality does not hold in general. However, it was proved by V IZING (1976)and E RDÖS , R UBIN AND TAYLOR (1979) that χ ℓ ( G ) ≤ ∆( G ) + 1 .For planar graphs we do not have a '4-list colour theorem'. Indeed, it was shown byV OIGT (1993) that there exists a planar graph with χℓ ( G ) = 5. At the moment, thesmallest such a graph was produced by M IRZAKHANI (1996), and it is of order 63.Theorem 5.9 (T HOMASSEN (1994)). Let G be a planar graph. Then χℓ ( G ) ≤ 5. In fact, T HOMASSEN proved a stronger statement:Theorem 5.10. Let G be a planar graph and let C be the cycle that is the boundary of theexterior face. Let Λ consist of lists such that |Λ(v)| = 3 for all v ∈ C, and |Λ(v)| = 5 for all /v ∈ C. Then G has a Λ-list colouring α.Proof. We can assume that the planar graph G is connected, and that it is given bya near-triangulation; an embedding, where the interior faces are triangles. (If the
5.2 Colouring planar graphs 73boundary of a face has more than 3 edges, then we can add an edge inside the face.)This is because adding edges to a graph can only make the list colouring more diffi-cult. Note that the exterior boundary is unchanged by a triangulation of the interiorfaces. The proof is by induction on νG under the additional constraint that one of thevertices of C has a fixed colour. (Thus we prove a stronger statement than claimed.)For νG ≤ 3, the claim is obvious. Suppose then that νG ≥ 4. Let x ∈ C be a vertex, for which we fix a colour c ∈ Λ( x). Let v ∈ C be a vertexadjacent to x, that is, C : v → x − v. →⋆Let NG (v) = { x, v1 , . . . , vk , y}, where y ∈ C, and vi are vordered such that the faces are triangles as in the figure. y xIt can be that NG (v) = { x, y}, in which case xy ∈ G. v1Consider the subgraph H = G −v. The exterior bound- vk v2 ...ary of H is the cycle x → v1 → · · · → vk → y − x. ⋆ →Since |Λ(v)| = 3, there are two colours r, s ∈ Λ(v) thatdiffer from the colour c of x. We define new lists for H as follows: Λ′ (vi ) ⊆ Λ(vi ) {r, s} such that |Λ′ (vi )| = 3for each i ∈ [1, k], and otherwise Λ′ (z) = Λ(z).Now νH = νG − 1, and by the induction hypothesis (with c still fixed for x), H has aΛ′ -list colouring α. For the vertex v, we choose α(v) = r or s such that α(v) = α(y).This gives a Λ′ -list colouring for G. Since Λ′ (z) ⊆ Λ(z) for all z, we have that α is aΛ-list colouring of G. ⊔ ⊓Straight lines and kissing circles∗We state an interesting result of WAGNER, the proof of which can be deduced fromthe above proof of Kuratowski's theorem. The result is known as Fáry's Theorem.Theorem 5.11 (WAGNER (1936)). A planar graph G has a plane embedding, where the edgesare straight lines. This raises a difficult problem:Integer Length Problem. Can all planar graphs be drawn in the plane such that theedges are straight lines of integer lengths?We say that two circles kiss in the plane, if they inter- ·sect in one point and their interiors do not intersect. Fora set of circles, we draw a graph by putting an edge be- · · ·tween two midpoints of kissing circles. · ·The following improvement of the above theorem isdue to K OEBE (1936), and it was rediscovered indepen- · ·dently by A NDREEV (1970) and T HURSTON (1985).Theorem 5.12 (K OEBE (1936)). A graph is planar if and only if it is a kissing graph of circles.
5.2 Colouring planar graphs 74 Graphs can be represented as plane figures in many different ways. For this, con- sider a set S of curves of the plane (that are continuous between their end points). The string graph of S is the graph G = (S, E), where uv ∈ E if and only if the curves u and v intersect. At first it might seem that every graph is a string graph, but this is not the case. It is known that all planar graphs are string graphs (this is a trivial result). Line Segment Problem. A graph is a line segment graph if it is a string graph for a set L of straight line segments in the plane. Is every planar graph a line segment graph for some set L of lines? Note that there are also nonplanar graphs that are line segment graphs. Indeed, all complete graphs are such graphs. The above question remains open even in the case when the slopes of the lines are +1, −1, 0 and ∞. +1 −1 A positive answer to this 4-slope problem for pla- 0 ∞ nar graphs would prove the 4-Colour Theorem. The Minor Theorem∗ D EFINITION . A graph H is a minor of G, denoted by H G, if H is isomorphic to a graph obtained from a subgraph of G by successively contracting edges. A recent result of R OBERTSON AND S EYMOUR (1983-2000) on graph minors is (one of) the deepest results of graph theory. The proof goes beyond these lectures. Indeed, the proof of Theorem 5.13 is around 500 pages long. e G a subgraph a contraction Note that every subgraph H ⊆ G is a minor, H G. The following properties of the minor relation are easily established: (i) G G, (ii) H G and G H imply G ∼ H, =(iii) H L and L G imply H G. The conditions (i) and (iii) ensure that the relation is a quasi-order, that is, it is reflexive and transitive. It turns out to be a well-quasi-order, that is, every infinite sequence G1 , G2 , . . . of graphs has two graphs Gi and Gj with i < j such that Gi Gj .
5.2 Colouring planar graphs 75Theorem 5.13 (Minor Theorem). The minor order is a well-quasi-order on graphs. Inparticular, in any infinite family F of graphs, one of the graphs is a (proper) minor of another. Each property P of graphs defines a family of graphs, namely, the family of thosegraphs that satisfy this property.D EFINITION . A family F of graphs is said to be minor closed, if every minor H of agraph G ∈ F is also in F . A property P of graphs is said to be inherited by minors,if all minors of a graph G satisfy P whenever G does. The following families of graphs are minor closed: the family of (1) all graphs, (2)planar graphs (and their generalizations to other surfaces), (3) acyclic graphs.The acyclic graphs include all trees. However, the family of trees is not closed undertaking subgraphs, and thus it is not minor closed. More importantly, the subgraphorder of trees (T1 ⊆ T2 ) is not a well-quasi-order. WAGNER proved a minor version of Kuratowski's theorem:Theorem 5.14 (WAGNER (1937)). A graph G is nonplanar if and only if K5 G or K3,3G.Proof. Exercise. ⊔ ⊓ R OBERTSON AND S EYMOUR (1998) proved the Wagner's conjecture:Theorem 5.15 (Minor Theorem 2). Let P be a property of graphs inherited by minors.Then there exists a finite set F of graphs such that G satisfies P if and only if G does nothave a minor from F . One of the impressive application of Theorem 5.15 concerns embeddings of graphson surfaces, see the next chapters. By Theorem 5.15, one can test (with a fast algo-rithm) whether a graph can be embedded onto a surface. Every graph can be drawn in the 3-dimensional space without crossing edges. Anold problem asks if there exists an algorithm that would determine whether a graphcan be drawn so that its cycles do not form (nontrivial) knots. This problem is solvedby the above results, since the property 'knotless' is inherited by minors: there existsa fast algorithm to do the job. However, this algorithm is not known!Hadwiger's Problem. H ADWIGER conjectured in 1943 that for every graph G, Kχ( G) G,that is, if χ( G ) ≥ r, then G has a complete graph Kr as its minor. The conjecture is trivialfor r = 2, and it is known to hold for all r ≤ 6. The cases for r = 5 and 6 follow fromthe 4-Colour Theorem.
5.3 Genus of a graph 765.3 Genus of a graphA graph is planar, if it can be drawn in the plane without crossing edges. A planeis an important special case of a surface. In this section we study shortly drawinggraphs in other surfaces. There are quite many interesting surfaces many of which are rather difficult todraw. We shall study the 'easy surfaces' – those that are compact and orientable.These are surfaces that have both an inside and an outside, and can be entirely char-acterized by the number of holes in them. This number is the genus of the surface.There are also non-orientable compact surfaces such as the Klein bottle and the pro-jective plane.Background on surfacesWe shall first have a quick look at the general surfaces and their classification withoutgoing into the details. Consider the space R3 , which has its (usual) distance functiond( x, y) ∈ R of its points. Two figures (i.e., sets of points) A and B are topologically equivalent (or homeo-morphic) if there exists a bijection f : A → B such that f and its inverse f −1 : B → Aare continuous. In particular, two figures are topologically equivalent if one can bedeformed to the other by bending, squeezing, stretching, and shrinking without tear-ing it apart or gluing any of its parts together. All these deformations should be suchthat they can be undone. A set of points X is a surface, if X is connected (there is a continuous line insideX between any two given points) and every point x ∈ X has a neighbourhood that istopologically equivalent to an open planar disk D ( a) = { x | dist ( a, x) < 1}. We deal with surfaces of the real space, and in this case a surface X is compact, ifX is closed and bounded. Note that the plane is not compact, since it it not bounded.A subset of a compact surface X is a triangle if it is topologically equivalent to atriangle in the plane. A finite set of triangles Ti , i = 1, 2, . . . , m, is a triangulation ofX if X = ∪m 1 Ti and any nonempty intersection Ti ∩ Tj with i = j is either a vertex or i=an edge. The following is due to R ADÓ (1925).Theorem 5.16. Every compact surface has a triangulation. Each triangle of a surface can be oriented by choosing an order for its vertices upto cyclic permutations. Such a permutation induces a direction for the edges of the tri-angle. A triangulation is said to be oriented if the triangles are assigned orientationssuch that common edges of two triangles are always oriented in reverse directions. Asurface is orientable if it admits an oriented triangulation. Equivalently, orientability can be described as follows.Theorem 5.17. A compact surface X is orientable if and only if it has no subsets that aretopologically equivalent to the Möbius band.
5.3 Genus of a graph 77In the Möbius band (which itself is not a surface accordingthe above definition) one can travel around and return to thestarting point with left and right reversed. A connected sum X#Y of two compact surfaces is obtained by cutting an opendisk off from both surfaces and then gluing the surfaces together along the boundaryof the disks. (Such a deformation is not allowed by topological equivalence.) The next result is known as the classification theorem of compact surfaces.Theorem 5.18 (D EHN AND H EEGAARD (1907)). Let X be a compact surface. Then (i) if X is orientable, then it is topologically equivalent to a sphere S = S0 or a connected sum of tori: Sn = S1 #S1 # . . . #S1 for some n ≥ 1, where S1 is a torus.(ii) if X is nonorientable, then X is topologically equivalent to a connected sum of projective planes: Pn = P#P# . . . #P for some n ≥ 1, where P is a projective plane. It is often difficult to imagine how a figure (say, a graph) can be drawn in a sur-face. There is a helpful, and difficult to prove, result due to R ADÓ (1920), stating thatevery compact surface (orientable or not) has a description by a plane model, whichconsists of a polygon in the plane such that• each edge of the polygon is labelled by a letter,• each letter is a label of exactly two edges of the polygon, and• each edge is given an orientation (clockwise or counter clockwise).Given a plane model M, a compact surface is obtained by gluing together the edgeshaving the same label in the direction that they have. a a a a a b b b b b a b b a a b Sphere Torus Klein bottle Projective plane From a plane model one can easily determine if the surface is oriented or not. Itis nonoriented if and only if, for some label a, the edges labelled by a have the samedirection when read clockwise. (This corresponds to the Möbius band.) A plane model, and thus a compact surface, can also be represented by a (circular)word by reading the model clockwise, and concatenating the labels with the conven-tion that a−1 is chosen if the direction of the edge is counter clockwise. Hence, thesphere is represented by the word abb−1 a−1 , the torus by aba−1 b−1 , the Klein bottleby aba−1 b and the projective plane by abb−1 a.
5.3 Genus of a graph 78These surfaces, as do the other surfaces, have manyother plane models and representing words as well.A word representing a connected sum of two surfaces,represented by words W1 and W2 , is obtained by con-catenating these words to W1 W2 . By studying the rela-tions of the representing words, Theorem 5.18 can beproved. Klein bottle Drawing a graph (or any figure) in a surface can be elaborated compared to draw-ing in a plane model, where a line that enters an edge of the polygon must continueby the corresponding point of the other edge with the same label (since these pointsare identified when we glue the edges together). a e4 e5Example 5.5. On the right we have drawn K6 inthe Klein bottle. The black dots indicate, where e3 e1the lines enter and leave the edges of the plane b e2 e2 bmodel. Recall that in the plane model for the e1 e3Klein bottle the vertical edges of the squarehave the same direction. e4 e5 aSphereD EFINITION . In general, if S is a surface, then a graph G has an S-embedding, if Gcan be drawn in S without crossing edges.Let S0 be (the surface of) a sphere. According to the nexttheorem a sphere has exactly the same embeddings as dothe plane. In the one direction the claim is obvious: if G isa planar graph, then it can be drawn in a bounded area ofthe plane (without crossing edges), and this bounded areacan be ironed on the surface of a large enough sphere. ¡ ¢£Clearly, if a graph can be embedded in one sphere, then it can be embedded in anysphere – the size of the sphere is of no importance. On the other hand, if G is em-beddable in a sphere S0 , then there is a small area of the sphere, where there are nopoints of the edges. We then puncture the sphere at this area, and stretch it open untilit looks like a region of the plane. In this process no crossings of edges can be created,and hence G is planar.Another way to see this is to use a pro-jection of the sphere to a plane:
5.3 Genus of a graph 79Theorem 5.19. A graph G has an S0 -embedding if and only if it is planar. Therefore instead of planar embeddings we can equally well study embeddings ofgraphs in a sphere. This is sometimes convenient, since the sphere is closed and it hasno boundaries. Most importantly, a planar graph drawn in a sphere has no exteriorface – all faces are bounded (by edges). If a sphere is deformed by pressing or stretching, its embeddability properties willremain the same. In topological terms the surface has been distorted by a continuoustransformation. ¡Torus ¢£Consider next a surface which is obtained from the sphereS0 by pressing a hole in it. This is a torus S1 (or an ori-entable surface of genus 1). The S1 -embeddable graphsare said to have genus equal to 1.Sometimes it is easier to consider handles than holes: a torus S1 can be deformed (by ¡ ¡ ¡a continuous transformation) into a sphere with a handle. ¢£ ¢£ ¢£If a graph G is S1 -embeddable, then it can be drawn in any one of the above surfaces ¡ ¡without crossing edges. ¢£ ¢£ 1 1 5Example 5.6. The smallest nonplanar graphs K5 and 2 6K3,3 have genus 1. Also, K7 has genus 1 as can be seen 3from the plane model (of the torus) on the right. 7 4 1 1
5.3 Genus of a graph 80GenusLet Sn (n ≥ 0) be a sphere with n holes in it. The drawing of an S4 can already bequite complicated, because we do not put any restrictions on the places of the holes(except that we must not tear the surface into disjoint parts). However, once again an ¡¢ ¡¢Sn can be transformed (topologically) into a sphere with n handles. £¤¥ £¤¥D EFINITION . We define the genus g( G ) of a graph G as the smallest integer n, forwhich G is Sn -embeddable. For planar graphs, we have g( G ) = 0, and, in particular, g(K4 ) = 0. For K5 , wehave g(K5 ) = 1, since K5 is nonplanar, but is embeddable in a torus. Also, g(K3,3 ) = 1. The next theorem states that any graph G can be embedded in some surface Snwith n ≥ 0.Theorem 5.20. Every graph has a genus. ¡¢This result has an easy intuitive verification. Indeed, £¤¥consider a graph G and any of its plane (or sphere)drawing (possibly with many crossing edges) such thatno three edges cross each other in the same point (sucha drawing can be obtained). At each of these crossingpoints create a handle so that one of the edges goes be- ¡¢low the handle and the other uses the handle to crossover the first one. £¤¥We should note that the above argument does not de-termine g( G ), only that G can be embedded in some Sn .However, clearly g( G ) ≤ n, and thus the genus g( G ) ofG exists.The same handle can be utilized by several edges.
5.3 Genus of a graph 81Euler's formula with genus∗The drawing of a planar graph G in a sphere has the advantage that the faces of theembedding are not divided into internal and external. The external face of G becomesan 'ordinary face' after G has been drawn in S0 . In general, a face of an embedding of G in Sn (with g( G ) = n) is a region of Snsurrounded by edges of G. Let again ϕ G denote the number of faces of an embeddingof G in Sn . We omit the proof of the next generalization of Euler's formula.Theorem 5.21. If G is a connected graph, then νG − ε G + ϕ G = 2 − 2g( G ) . If G is a planar graph, then g( G ) = 0, and the above formula is the Euler's formulafor planar graphs.D EFINITION . A face of an embedding P( G ) in a surface is a 2-cell, if every simpleclosed curve (that does not intersect with itself) can be continuously deformed to asingle point.The complete graph K4 can be embedded in a torus such that it has a face that is nota 2-cell. But this is because g(K4 ) = 0, and the genus of the torus is 1. We omit theproof of the general condition discovered by Y OUNGS:Theorem 5.22 (Y OUNGS (1963)). The faces of an embedding of a connected graph G in asurface of genus g( G ) are 2-cells.Lemma 5.12. For a connected G with νG ≥ 3 we have 3ϕ G ≤ 2ε G .Proof. If νG = 3, then the claim is trivial. Assume thus that νG ≥ 4. In this case weneed the knowledge that ϕ G is counted in a surface that determines the genus of G(and in no surface with a larger genus). Now every face has a border of at least threeedges, and, as before, every nonbridge is on the boundary of exactly two faces. ⊔ ⊓Theorem 5.23. For a connected G with νG ≥ 3, 1 1 g( G ) ≥ ε G − (νG − 2) . 6 2Proof. By the previous lemma, 3ϕ G ≤ 2ε G , and by the generalized Euler's formula,ϕ G = ε G − νG + 2 − 2g( G ). Combining these we obtain that 3ε G − 3νG + 6 − 6g( G ) ≤2ε G , and the claim follows. ⊔ ⊓ By this theorem, we can compute lower bounds for the genus g( G ) without draw-ing any embeddings. As an example, let G = K8 . In this case νG = 8, ε G = 28, and so 5g( G ) ≥ 3 . Since the genus is always an integer, g( G ) ≥ 2. We deduce that K8 cannotbe embedded in the surface S1 of the torus. If H ⊆ G, then clearly g( H ) ≤ g( G ), since H is obtained from G by omittingvertices and edges. In particular,
5.3 Genus of a graph 83Theorem 5.29. For each g ≥ 1, there exists a graph G with genus g( G ) = g so that 7+ 1 + 48g χ( G ) = . 2 If a nonplanar graph G can be embedded in a torus, then g( G ) = 1, and χ( G ) ≤⌊(7 + 1 + 48g)/2⌋ = 7. Moreover, for G = K7 we have that χ(K7 ) = 7 and g(K7 ) =1.Three dimensions∗Every graph can be drawn without crossing edges in the 3-dimensional space. Sucha drawing is called spatial embedding of the graph. Indeed, such an embedding canbe achieved by putting all vertices of G on a line, and then drawing the edges indifferent planes that contain the line. Alternatively, the vertices of G can be put in asphere, and drawing the edges as straight lines crossing the sphere inside. A spatial embedding of a graph G is said to have linked cycles, if two cycles ofG form a link (they cannot be separated in the space). By C ONWAY and G ORDON in1983 every spatial embedding of K6 contains linked cycles. It was shown by R OBERTSON , S EYMOUR AND T HOMAS (1993) that there is a setof 7 graphs such that a graph G has a spatial embedding without linked cycles if andonly if G does not have a minor belonging to this set. This family of forbidden graphs was originally found by S ACHS (without proof),and it contains K6 and the Petersen graph. Every graph in the set has 15 edges, whichis curious. For further results and proofs concerning graphs in surfaces, seeB. M OHAR AND C. T HOMASSEN, "Graphs on Surfaces", Johns Hopkins, 2001.
6Directed Graphs6.1 DigraphsIn some problems the relation between the objectsis not symmetric. For these cases we need directedgraphs, where the edges are oriented from one vertexto another.As an example consider a map of a small town. Canyou make the streets one-way, and still be able to drivefrom one house to another (or exit the town)?DefinitionsD EFINITION . A digraph (or a directed graph) D = (VD , ED ) consists of the verticesVD and (directed) edges ED ⊆ VD × VD (without loops vv). We still write uv for(u, v), but note that now uv = vu. For each pair e = uv define the inverse of e ase−1 = vu (= (v, u)). Note that e ∈ D does not imply e−1 ∈ D.D EFINITION . Let D be a digraph. Then A is its• subdigraph, if VA ⊆ VD and E A ⊆ ED ,• induced subdigraph, A = D [ X ], if VA = X and E A = ED ∩ ( X × X ).The underlying graph U ( D ) of a digraph D isthe graph on VD such that if e ∈ D, then theundirected edge with the same ends is in U ( D ). A digraph D is an orientation of a graph G, if G = U ( D ) and e ∈ D impliese −1∈ D. In this case, D is said to be an oriented graph. /D EFINITION . Let D be a digraph. A walk W = e1 e2 . . . ek : u − v of U ( D ) is a directed ⋆ →walk, if ei ∈ D for all i ∈ [1, k]. Similarly, we define directed paths and directed cyclesas directed walks and closed directed walks without repetitions of vertices. The digraph D is di-connected, if, for all u = v, there exist directed paths u − v ⋆ →and v − u. The maximal induced di-connected subdigraphs are the di-components →⋆of D.
6.1 Digraphs 86Proof. Let k = χ( G ) and let α be a proper k-colouring of G. As usual the set of coloursis [1, k]. We orient each edge uv ∈ G by setting uv ∈ D, if α(u) < α(v). Clearly, the soobtained orientation D has no directed paths of length ≥ k − 1. ⊔ ⊓D EFINITION . An orientation D of an undirected graph G is acyclic, if it has no di-rected cycles. Let a( G ) be the number of acyclic orientations of G. The next result is charming, since χ G (−1) measures the number of proper colour-ings of G using −1 colours!Theorem 6.3 (S TANLEY (1973)). Let G be a graph of order n. Then the number of the acyclicorientations of G is a( G ) = (−1)n χ G (−1) ,where χ G is the chromatic polynomial of G.Proof. The proof is by induction on ε G . First, if G is discrete, then χ G (k) = kn , anda( G ) = 1 = (−1)n (−1)n = (−1)n χ G (−1) as required. Now χ G (k) is a polynomial that satisfies the recurrence χ G (k) = χ G−e (k) −χ G∗e (k). To prove the claim, we show that a( G ) satisfies the same recurrence. Indeed, if a( G ) = a( G − e) + a( G ∗ e) (6.1)then, by the induction hypothesis, a( G ) = (−1)n χ G−e (−1) + (−1)n−1 χ G∗e (−1) = (−1)n χ G (−1) .For (6.1), we observe that every acyclic orientation of G gives an acyclic orientationof G −e. On the other hand, if D is an acyclic orientation of G −e for e = uv, it extendsto an acyclic orientation of G by putting e1 : u → v or e2 : v → u. Indeed, if D has nodirected path u − v, we choose e2 , and if D has no directed path v − u, we choose →⋆ ⋆ →e1 . Note that since D is acyclic, it cannot have both ways u − v and v − u. →⋆ ⋆ → We conclude that a( G ) = a( G − e) + b, where b is the number of acyclic orienta-tions D of G −e that extend in both ways e1 and e2 . The acyclic orientations D thatextend in both ways are exactly those that contain neither u − v nor v − u as a directed path. ⋆ → ⋆ → (6.2) Each acyclic orientation of G ∗ e corresponds in a natural way to an acyclic orienta-tion D of G −e that satisfies (6.2). Therefore b = a( G ∗ e), and the proof is completed. ⊔ ⊓
6.1 Digraphs 87One-way trafficEvery graph can be oriented, but the result may not be di-connected. In the one-way traffic problem the resulting orientation should be di-connected, for otherwisesomeone is not able to drive home. R OBBINS' theorem solves this problem.D EFINITION . A graph G is di-orientable, if there is a di-connected oriented graph Dsuch that G = U ( D ).Theorem 6.4 (R OBBINS (1939)). A connected graph G is di-orientable if and only if G hasno bridges.Proof. If G has a bridge e, then any orientation of G has at least two di-components(both sides of the bridge). Suppose then that G has no bridges. Hence G has a cycle C, and a cycle is alwaysdi-orientable. Let then H ⊆ G be maximal such that it has a di-orientation D H . IfH = G, then we are done.Otherwise, there exists an edge e = vu ∈ G such that u vu ∈ H but v ∈ H (because G is connected). The edge e / eis not a bridge and thus there exists a cycle P′ P Q C ′ = ePQ : v − u − w − v ⋆ → → → ⋆ win G, where w is the last vertex inside H. In the di-orientation D H of H there is a directed path P′ : u − w. Now, we orient ⋆ →e : v − u and the edges of Q in the direction Q : w − v to obtain a directed cycle → →⋆eP ′ Q : v − u − w − v. In conclusion, G [V ∪ V ] has a di-orientation, which → ⋆ → →⋆ H Ccontradicts the maximality assumption on H. This proves the claim. ⊔ ⊓Example 6.1. Let D be a digraph. A directed Euler tour of D is a directed closed walkthat uses each edge exactly once. A directed Euler trail of D is a directed walk thatuses each edge exactly once. The following two results are left as exercises.(1) Let D be a digraph such that U ( D ) is connected. Then D has a directed Euler tour if andonly if dD (v) = dO (v) for all vertices v. I D(2) Let D be a digraph such that U ( D ) is connected. Then D has a directed Euler trail if andonly if dD (v) = dO (v) for all vertices v with possibly excepting two vertices x, y for which I D I ( v) − dO ( v)| = 1.|d D D The above results hold equally well for multidigraphs, that is, for directed graphs,where we allow parallel directed edges between the vertices.Example 6.2. The following problem was first studied by H UTCHINSON AND W ILF(1975) with a motivation from DNA sequencing. Consider words over an alphabetA = {a1 , a2 , . . . , an } of n letters, that is, each word w is a sequence of letters. In the case
6.1 Digraphs 88of DNA, the letters are A, T, C, G. In a problem instance, we are given nonnegativeintegers si and rij for 1 ≤ i, j ≤ n, and the question is: does there exist a word w inwhich each letter ai occurs exactly si times, and ai is followed by a j exactly rij times. For instance, if n = 2, s1 = 3, and r11 = 1, r12 = 2, r21 = 1, r22 = 0, then the worda1 a2 a1 a1 a2 is a solution to the problem. Consider a multidigraph D with VD = A for which there are rij edges ai a j . Itis rather obvious that a directed Euler trail of D gives a solution to the sequencingproblem.TournamentsD EFINITION . A tournament T is an orientation of a complete graph.Example 6.3. There are four tournaments of four vertices that are not isomorphic witheach other. (Isomorphism of directed graphs is defined in the obvious way.)Theorem 6.5 (R ÉDEI (1934)). Every tournament has a directed Hamilton path.Proof. The chromatic number of Kn is χ(Kn ) = n, and hence by Theorem 6.1, a tour-nament T of order n has a directed path of length n − 1. This is then a directed Hamil-ton path visiting each vertex once. ⊔ ⊓ The vertices of a tournament can be easily reached from one vertex (sometimescalled the king).Theorem 6.6 (L AUDAU (1953)). Let v be a vertex of a tournament T of maximum outdegree. ⋆Then for all u, there is a directed path v − u of length at most two. →Proof. Let T be an orientation of Kn , and let dO (v) = d be the maximum outdegree in TT. Suppose that there exists an x, for which the directed distance from v to x is at leastthree. It follows that xv ∈ T and xu ∈ T for all u with vu ∈ T. But there are d verticesin A = {y | vy ∈ T }, and thus d + 1 vertices in {y | xy ∈ T } = A ∪ {v}. It followsthat the outdegree of x is d + 1, which contradicts the maximality assumption madefor v. ⊔ ⊓Problem. Ádám's conjecture states that in every digraph D with a directed cycle thereexists an edge uv the reversal of which decreases the number of directed cycles. Here the newdigraph has the edge vu instead of uv.
6.1 Digraphs 89Example 6.4. Consider a tournament of n teams that play once against each other, andsuppose that each game has a winner. The situation can be presented as a tournament,where the vertices correspond to the teams vi , and there is an edge vi v j , if vi won v jin their mutual game.D EFINITION . A team v is a winner (there may be more than one winner), if v comesout with the most victories in the tournament. Theorem 6.6 states that a winner v either defeated a team u or v defeated a teamthat defeated u. A ranking of a tournament is a linear ordering of the teams vi1 > vi2 > · · · >vin that should reflect the scoring of the teams. One way of ranking a tournamentcould be by a Hamilton path: the ordering can be obtained from a directed Hamiltonpath P : vi1 − vi2 − . . . − vin . However, a tournament may have several directed → → →Hamilton paths, and some of these may do unjust for the 'real' winner.Example 6.5. Consider a tournament of six teams1, 2, . . . , 6, and let T be the scoring digraph as in the 1 2figure. Here 1 − 2 − 4 − 5 − 6 − 3 is a di- → → → → →rected Hamilton path, but this extends to a directed 6 3Hamilton cycle (by adding 3 − 1)! So for every team →there is a Hamilton path, where it is a winner, and in 5 4another, it is a looser. Let s1 ( j) = dO ( j) be the winning number of the team j (the number of teams Tbeaten by j). In the above tournament, s1 (1) = 4, s1 (2) = 3, s1 (3) = 3, s1 (4) = 2, s1 (5) = 2, s1 (6) = 1 .So, is team 1 the winner? If so, is 2 or 3 next? Define the second-level scoring for eachteam by s2 ( j ) = ∑ s1 ( i ) . ji∈ TThis tells us how good teams j beat. In our example, we have s2 (1) = 8, s2 (2) = 5, s2 (3) = 9, s2 (4) = 3, s2 (5) = 4, s2 (6) = 3 .Now, it seems that 3 is the winner,but 4 and 6 have the same score. We continue bydefining inductively the mth-level scoring by sm ( j) = ∑ s m −1 (i ) . ji∈ TIt can be proved (using matrix methods) that for a di-connected tournament with atleast four teams, the level scorings will eventually stabilize in a ranking of the tournament:there exits an m for which the mth-level scoring gives the same ordering as do the
6.2 Network Flows 90(m + k)th-level scorings for all k ≥ 1. If T is not di-connected, then the level scoringshould be carried out with respect to the di-components. In our example the level scoring gives 1 − 3 − 2 − 5 − 4 − 6 as the ranking → → → → →of the tournament.6.2 Network FlowsVarious transportation networks or water pipelines are conveniently represented byweighted directed graphs. These networks usually possess also some additional re-quirements. Goods are transported from specific places (warehouses) to final loca-tions (marketing places) through a network of roads. In modelling a transportationnetwork by a digraph, we must make sure that the number of goods remains thesame at each crossing of the roads. The problem setting for such networks was pro-posed by T.E. Harris in the 1950s. The connection to Kirchhoff's Current Law (1847) isimmediate. According to this law, in every electrical network the amount of currentflowing in a vertex equals the amount flowing out that vertex.FlowsD EFINITION . A network N consists of 4• an underlying digraph D = (V, E), 5• two distinct vertices s and r, called the source and 5 2 the sink of N, and s 4 2 r 4• a capacity function α : V × V → R + (nonnegative 6 5 real numbers), for which α(e) = 0, if e ∈ E. / 4Denote VN = V and EN = E. Let A ⊆ VN be a set of vertices, and f : VN × VN → R any function such thatf (e) = 0, if e ∈ N. We adopt the following notations: / [ A, A] = {e ∈ D | e = uv, u ∈ A, v ∈ A} , / f + ( A) = ∑ f ( e) and f − ( A) = ∑ f ( e) . e ∈[ A,A] e ∈[ A,A ]In particular, f + (u) = ∑ f (uv) and f − (u) = ∑ f (vu) . v∈ N v∈ ND EFINITION . A flow in a network N is a function f : VN × VN → R + such that 0 ≤ f (e) ≤ α(e) for all e, and f − (v) = f + (v) for all v ∈ {s, r} . /Example 6.6. The value f (e) can be taught of as the rate at which transportation actu-ally happens along the channel e which has the maximum capacity α(e). The secondcondition states that there should be no loss.
6.2 Network Flows 95Proof. A network N with source s and sink r is obtained by setting the capacitiesequal to 1. The claim follows from Lemma 6.5 and Corollary 6.10. ⊔ ⊓Corollary 6.3. Let u and v be two vertices of a graph G. The maximum number of edge- ⋆disjoint paths u − v equals the minimum number of edges, whose removal destroys all the → ⋆connections u − v from G. →Proof. Consider the digraph D that is obtained from G by replacing each (undirected)edge uv ∈ G by two directed edges uv ∈ D and vu ∈ D. The claim follows then easilyfrom Corollary 6.2. ⊔ ⊓ The next corollary is Menger's Theorem for edge connectivity.Corollary 6.4. A graph G is k-edge connected if and only if any two distinct vertices of G areconnected by at least k independent paths.Proof. The claim follows immediately from Corollary 6.3. ⊔ ⊓Seymour's 6-flows∗D EFINITION . A k-flow ( H, α) of an undirected graph G is an orientation H of G to-gether with an edge colouring α : E H → [0, k − 1] such that for all vertices v ∈ V, ∑ α( e) = ∑ α( f ) , (6.3) e = vu ∈ H f = uv∈ Hthat is, the sum of the incoming values equals the sum of the outgoing values. Ak-flow is nowhere zero, if α(e) = 0 for all e ∈ H. In the k-flows we do not have any source or sink. For convenience, let α(e−1 ) =−α(e) for all e ∈ H in the orientation H of G so that the condition (6.3) becomes ∑ α( e) = 0 . (6.4) e = vu ∈ H 1 1 2Example 6.8. A graph with a nowhere zero 4-flow. 2 1 3 2 The condition (6.4) generalizes to the subsets A ⊆ VG in a natural way, ∑ α( e) = 0 , (6.5) e ∈[ A,A]since the values of the edges inside A cancel out each other. In particular,
6.2 Network Flows 96Lemma 6.6. If G has a nowhere zero k-flow for some k, then G has no bridges.Tutte's Problem. It was conjectured by T UTTE (1954) that every bridgeless graph hasa nowhere zero 5-flow. The Petersen graph has a nowhere zero 5-flow but does nothave any nowhere 4-flows, and so 5 is the best one can think of. Tutte's conjectureresembles the 4-Colour Theorem, and indeed, the conjecture is known to hold for theplanar graphs. The proof of this uses the 4-Colour Theorem. In order to fully appreciate Seymour's result, Theorem 6.11, we mention that itwas proved as late as 1976 (by J AEGER) that every bridgeless G has a nowhere zerok-flow for some integer k. S EYMOUR's remarkable result reads as follows:Theorem 6.11 (S EYMOUR's (1981)). Every bridgeless graph has a nowhere zero 6-flow.Proof. Omitted. ⊔ ⊓D EFINITION . The flow number f ( G ) of a bridgeless graph G is the least integer k forwhich G has a nowhere zero k-flow.Theorem 6.12. A connected graph G has a flow number f ( G ) = 2 if and only if it is eulerian.Proof. Suppose G is eulerian, and consider an Euler tour W of G. Let D be the orien-tation of G corresponding to the direction of W. If an edge uv ∈ D, let α(e) = 1. SinceW arrives and leaves each vertex equally many times, the function α is a nowherezero 2-flow. Conversely, let α be a nowhere zero 2-flow of an orientation D of G. Then neces-sarily the degrees of the vertices are even, and so G is eulerian. ⊔ ⊓Example 6.9. For each 3-regular bipartite graph G, we have f ( G ) ≤ 3. Indeed, letG be ( X, Y )-bipartite. By Corollary 3.1, a 3-regular graph has a perfect matching M.Orient the edges e ∈ M from X to Y, and set α(e) = 2. Orient the edges e ∈ M from /Y to X, and set α(e) = 1. Since each x ∈ X has exactly one neighbour y1 ∈ Y suchthat xy1 ∈ M, and two neighbours y2 , y3 ∈ Y such that xy2 , xy3 ∈ M, we have that /f ( G ) ≤ 3.Theorem 6.13. We have f (K4 ) = 4, and if n > 4, then 2 if n is odd , f (Kn ) = 3 if n is even .Proof. Exercise. ⊔ ⊓ |
MA 125 Intermediate Algebra Daniels, JamesTo provide the student with a more in depth understanding of algebra, the student will discuss, practice and collaborate on following algebraic procedures in practical real life applications. There will be emphasis on the properties and operations with the real number system and their use in evaluating algebraic expressions, solving systems of equation, determining maximum and minimum values, finding areas and dimensions of figures and solving rate problemsThe final grade will be determined using the weighted averages of three components:
Quizzes – 53%; Homework – 12%; Final Exam – 35%
Grading:
Your course grade will be determined after weighing your scores in homework, quizzes and the final exam. A letter grade is then assigned:
90-100A
80-89 B
70-79 C
60-69 D
0-59 F (or more than 3 Unexcused Absences)
Late Submission of Course Materials:
No late work accepted beyond one week after chapter completed.
Classroom Rules of Conduct:
Students are expected to attend all classes, be on time and prepared. Classes missed for legitimate reasons are excusable. The student will make up all work for classes missed.
Course Topic/Dates/Assignments:
Date
Chap/Sect
Topic
24 Oct
1
Review
26
2.1, 2
31
2.3-5
2 Nov
2.6, 7
Quiz 1
7
3
9
4
14
5.1-3
Quiz 2
16
5.4, 5
21
6.1, 2
23
6.3, 4
Quiz 3
28
6.5
30
7.1
Quiz 4
5 Dec
7.2, 3
7
7.4
14
Final Exam Review
16
Cumulative |
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Overview
This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the |
Books a la Carte are unbound, three-hole-punch versions of the textbook. This lower cost option is easy to transport and comes with same access code or media that would be packaged with the bound book.
Pirnot's Mathematics All Around offers the supportive and clear writing style that you need to develop your math skills. By helping to reduce your math anxiety, Pirnot helps you to understand the use of math in the world around you. You appreciate that the author's approach is like the help you would receive during your own instructors' office hours.
The Fifth Edition increases the text's emphasis on developing problem-solving skills with additional support in the text and new problem-solving questions in MyMathLab. Quantitative reasoning is brought to the forefront with new Between the Numbers features and related exercises. Since practice is the key to success in this course, exercise sets are updated and expanded. MyMathLab offers additional exercise coverage plus new question types for problem-solving, vocabulary, reading comprehension, and more. |
Comment:
I found this site really useful. I looked up coin tossing under probibility and learned how tossing a coin is like tossing a...
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Comment:
I found this site really useful. I looked up coin tossing under probibility and learned how tossing a coin is like tossing a two sided die,and the Bernolli Distribution, how the patterns of the toss can be somewhat predicted. For example, it is more likely that you would toss TTH (tails, tails, heads) before you toss THT. Under the "history and terminolgy" page, I learned about the following terms, end of proof(QED),a latin abbreviation that means "that which has been demonstrated״. Another term I had never heard of was pairwise, it means taking all unordered 2 subsets of a given set. Another is star that is a term used in geometry. Porism, which means proportion and called theorm in mathmatical terms.There were others I knew but needed to review, like unknown and complex variables. The examples of each term were pretty straightforward, and something an instructor could easily expand upon.Technical Remarks:By using the "history and terminology page" I could find the definition to many terms I didn't know or thought were interesting. Sometimes after reading about foreign term/concept, I recognized other terms with in that concept that I was familiar with and saw how it was connected. Then it was more applicable for me. It was also a great page for expanding my mathematical vocabulary. The more I learn about math the more confidence I will have, and the more I will want to learn. I think this page was a great tool for students and instuctors to use. For students, it allows them to explore on their own, learn about concepts, write down some questions they may have and bring those to class. For the instructor, it is a great way to see examples of how the concepts are being explained and the way the material is being presented. It could also be used as a great refresher course for instuctors. There was alot of terminology and it is a great resource for both students and instructors.
Comment:
i liked this site because iam really into the probabilites of things because of statistics. i play baseball so probabilities...
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Comment:
i liked this site because iam really into the probabilites of things because of statistics. i play baseball so probabilities are very interesting when it comes to statistics. unfortunatly this is a part of math that not too many people are interested in. i think one of the main reasons i like probabilites so much is because everything else is so confusing to me.
Author:
belardes
(Student)
Date Added:
Mar 10, 2003 |
The Iowa Algebra Aptitude Test, Fifth Edition, provides educators with
the most up-to-date assessment of algebra readiness available. IAAT Fifth
Edition is designed and developed according to NCTM standards published in 2000.
Tests
IAAT Fifth Edition is divided into four parts:
Part 1: Pre-Algebraic Number Skills and Concepts – How well
students understand the mathematical skills and concepts necessary for success in
an algebra course
Part 2: Interpreting Mathematical Information – How well
students can learn new materials presented in graphs or text
IAAT, Fifth Edition, was developed by The University of Iowa to meet the
highest standards for test quality. The Manual for Test Use, Interpretation, and
Technical Support is available for purchase to guide educators in the use
of IAAT results. Item norms are also available in this manual. |
GeoGebra
you're a mathematics educator, you'll find the GeoGebra site to be a perfect addition to your stable of online resources. On the top of the homepage, visitors can look through sections that include About, Download, Community, and Materials. Most visitors will want to download the GeoGebra application as it is the primary way to utilize the 4,400 learning activities offered on the site. After completing the download of this free, interactive geometry, algebra, statistics, and calculus software, click on over to the Featured Materials area. Here visitors can look over the newest materials, check out the Best Worksheets, and wander through the popular tags. Mathematics educators have contributed many of the items here, and a small sample includes "The 11 Patterns of the Cube" and "Introduction to Linear Equations." Finally, visitors can also use the Community area to ask questions and share resources.Thu, 3 Oct 2013 10:27:37Multivariable Calculus
series of videos, created by Michael Hutching of the University of California, Berkeley, introduces parametric equations and polar coordinates. Vectors in 2- and 3-dimensional Euclidean spaces. Partial derivatives. Multiple integrals. Vector calculus. They also discuss theorems such as Green, Gauss, and Stokes.Wed, 15 Dec 2010 03:00:02 authored by Denis Auroux of Massachusetts Institute of Technology, covers vector and multi-variable calculus. It is the second semester in the freshman calculus sequence. Topics include vectors and matrices, partial derivatives, double and triple integrals, and vector calculus in 2 and 3-space.Wed, 8 Dec 2010 03:00:02 -0600Raindrops
for the Connected Curriculum Project, the purpose of this module is to study differential equations that model falling objects subject to air resistance -- for example, raindrops -- and to develop Euler's Method, a tool for approximating solutions of initial value problems. This is one lesson within a much larger set of learning modules hosted by Duke University.Tue, 29 Jun 2010 03:00:01 -0500Radioactive Decay
by William Barker and David Smith for the Connected Curriculum Project, this module develops a mathematical model for decay of radioactive substances, and a technique for deciding whether quantitative data fits the model or not. This is one within a much larger set of learning modules hosted by Duke University.Mon, 28 Jun 2010 03:00:02 -0500Where is the Ozone Layer?
by Dale Harak and Anita Salem for the Connected Curriculum Project, the purpose of this project is to investigate the question "Where is the ozone layer?" by examining the change in photon flux (a measure of sunlight intensity) as a function of altitude. This is one within a much larger set of learning modules hosted by Duke University.Thu, 24 Jun 2010 03:00:01 -0500The Spread of AIDS
by Lang Moore and David Smith for the Connected Curriculum Project, this module enables the user to study the growth pattern of a given set of data and to derive and check a model formula for the observed growth pattern. This is one of a much larger set of learning modules hosted by Duke University.Thu, 17 Jun 2010 03:00:02 -0500Taylor Polynomials I
by Lang Moore and David Smith for the Connected Curriculum Project, this is a module using differentiation to find coefficients of polynomial approximations to functions that are not polynomials. This is one of a much larger set of learning modules hosted by Duke University.Tue, 15 Jun 2010 03:00:01 -0500Mathematica Tutor: Calculus
by Joshua Holden, Lang Moore, David Smith and Jim Tomberg, this is a Mathematica Tutorial for calculus. This is one of a much larger set of resources hosted by Duke University.Fri, 11 Jun 2010 03:00:01 -0500 |
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An Introduction to the History of Algebra: Solving Equations from Mesopotamian Times to the Renaissance (Mathematical World)
Book Description: This book does not aim to give an exhaustive survey of the history of algebra up to early modern times but merely to present some significant steps in solving equations and, wherever applicable, to link these developments to the extension of the number system. Various examples of problems, with their typical solution methods, are analyzed, and sometimes translated completely. Indeed, it is another aim of this book to ease the reader's access to modern editions of old mathematical texts, or even to the original texts; to this end, some of the problems discussed in the text have been reproduced in the appendices in their original language (Greek, Latin, Arabic, Hebrew, French, German, Provençal, and Italian) with explicative notes |
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). |
What math is required for physics?
I have been aiming to venture further into my understanding of quantum mechanics( Past classical mechanics).I see many thread
s
What math is required for physics?Physics: Algebra (although many of the formulas you use are either derived or proven using calculus)
High School AP Physics B: Algebra (plus a good understanding of Conceptual Physics)
High School AP Physics C, Mechanics and E&M: Algebra and some Calculus (derivatives and anti-derivatives and integrals of polynomials and simple trig functions. Chain rule and U-substitution and rarely integrals involving 1/x, differential equations [separation of variables only, i.e. F=ma becomes F=m (dv/dt)]). If you don't know ANY calculus, you can still get a 4/5 on the AP C exams provided you ACE algebra stuff.
College Physics if you're NOT a Physics or Engineering major: Pretty much the same as High School regular Physics.
College Physics if you're ARE a Physics or Engineering major: Not too much difference from AP C High School Physics! It's a little tougher, and you'll be competing against a higher caliber of students. You'll also see some thermo and fluids thrown in.
College Physics if you ARE going for a Masters: Know your math. Know ALL your math. But make sure you're VERY good with Conceptual Physics. Otherwise all the math in the world will do you no good at this level.
College Physics including Quantum: "differential equations" ARE part of Calculus. If you're a Physics major, or Engineering major, or an Astronomy major (assuming your university offers it as a separate major)
To get a "deeper understanding of the universe" at ANY level, you simply need to learn more math. To understand the universe as no one before you ever has understood it, you need to make up your own math like Newton did! (he co-invented Calculus to prove Earth can be simplified as a point mass).Dunno - I agree that pushing formulas won't get anyone far but that's the same with math in general.
I've not been terribly impressed with courses called "conceptual physics" as a foundation for more advanced study. But that may not be what you mean.
Physicists do seem to think of math differently to mathematicians though - they tend to be less formal or rigorous, treating it like a language. I think that's the central point: got used to using math concepts to describe things in nature. It's why we set lots of wordy problems and use lots of pictures.
By "pushing formulas" I am talking about the approach where you memorize the "correct equation" or formula to use for each situation ... you put the numbers in the equation and get the right answers out.
Think of math as a language and "pushing formulas" as the "tourist phrasebook" version of that language.
I've not been terribly impressed with courses called "conceptual physics" as a foundation for more advanced study. But that may not be what you mean...
That is what I mean. The course Conceptual Physics was popularized by Paul Hewitt. I took the course from him at City College of San Francisco in 1982. Now it seems that high schools are moving in the direction of offering Conceptual Physics to freshmen. I teach three levels of high school Physics: Conceptual Physics, "regular" Physics, and AP C, Mechanics. We try to use Conceptual Physics to get the freshmen, who come from many different grade schools, and even many different countries on the same page. Basic formula manipulation, how to use lab equipment such as balances (thanks for your calibration help in the other thread!), how to write lab reports, unit conversion, introduction to scientific notation, etc. Even if they have trouble with some of this material, at least they've been introduced to it, and will hopefully have an easier time with it when they see it again at a more level in regular Physics or Chemistry.
I taught Integrated Physical Science at a different high school. I actually liked it better than Conceptual Physics for the freshmen. It was a little bit of Physics, a little bit of Astronomy, and a little bit of Earth Science thrown together. But I was told that Integrated Physical Science doesn't transfer to the UC colleges, while Conceptual Physics does. It's too bad. Unless I deviate a little from the syllabus (which I always do!), students will go through our high school not knowing why summer is warmer than winter, and not knowing why the moon goes through phases.
Simon Bridge, are you trying to say: unless you want to conduct expirements and calculate how much one H atom heats up when hit by a gamma ray, formulas prove no use?
Nope.
I'm saying you won't get far just by memorizing algorithms. You need to develop an understanding of the principles that underlie the formulas and equations. I take it you are not saying that if you have the formulas you don't need to understand them?
For instance - how does it make sense to talk about a single H atom "heating up" as a result of the interaction with a gamma ray?
The answer may just be a matter of finding the right formula - but science education is supposed to train you to solve problems where there is no known correct formula or the correct one is in dispute.
I think you'll see it more clearly if you follow the introductory physics threads in PF.
Simon Bridge, are you trying to say: unless you want to conduct expirements and calculate how much one H atom heats up when hit by a gamma ray, formulas prove no use?
This shows that you really have not learned any substantial amount of physics yet. You are still in the mentality of thinking that all one needs is a "formula", and then one only needs to do a plug-and-chug.
To answer your question, the minimum amount of math required to do physics is whatever is in Mary Boas's "Mathematical Methods in the Physical Sciences" text. That is as clear as an unambiguous of an answer that I can give. Whether you believe it or not, that's your problem.
Group Theory, Differential Geometry, Hilbert Space, and Topology are also important to modern physics. No one ever seems to mention those. You pretty much have to be a mathematician to get into theoretical physics.
I have been aiming to venture further into my understanding of quantum mechanics( Past classical mechanics).I see many threads |
Welcome to the Department of Mathematics
Mathematics is the language of science and is an essential part of work done in the natural sciences, engineering, economics, and other areas. Since ancient times and in many civilizations, mathematics has been central in human thought and critical to many aspects of intellectual development and progress. The Department of Mathematics offers courses and degree programs reflecting the place and various roles of mathematics in modern civilization. |
This unique text/reference reviews algorithms for the exact or approximate solution of shortest-path problems, with a specific focus on a class of algorithms called rubberband algorithms. Discussing each concept and algorithm in depth, the book includes mathematical proofs for many of the given statements. Topics and features: provides theoretical... more...
This is the first study of the history of Diophantine analysis and the theory of numbers from Ab? K?mil to Fermat (9th-17th century). It thus offers an elaborate and detailed overview on a fundamental chapter on classical mathematical thought and its relation to algebra and Diophantus? Arithmetica. more...
This volume contains 21 research and survey papers on recent developments in the field of diophantine approximation. This includes contributions to Wolfgang Schmidt's subspace theorem and its applications to diophantine equations and to the study of linear recurring sequences. The articles are either in the spirit of more classical diophantine... more... |
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What topics are included in the Math Placement Assessment?
PART 1: Elementary Algebra and Logical Reasoning Prime factorization of an integer Rounding an integer Laws of exponents, particularly for integer exponents Radicals Simplify expressions Evaluate an expression Distance formula (Pythagorean theorem) Scientific notation Simple linear equations Simple story problems: age, area, cost, constant speed, average of a set of numbers, business (tax, profit, discount) Recognizing the value of a million, a billion, a trillion Understanding perimeter and area of simple figures Triangles: number of degrees in the sum of the angles; obtuse and acute angles; Pythagorean theorem; finding area of a right triangle Circles: finding area and circumference Percentages Straight lines: slope, y-intercept, x-intercept Linear inequalities Parallel and perpendicular lines
PART 2: Intermediate Algebra Factoring polynomials Division of polynomials Absolute value Simple inequalities Language of functions Quadratic formula Meaning of the discriminant: number of real roots, number of complex roots Finding the vertex of a parabola; finding maximum and minimum values of quadratics Systems of two equations in two unknowns Direct and inverse variation Story problems |
An Early Functions Approach
Bob Blitzer's unique background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus ...Show synopsisBob Blitzer's unique background in mathematics and behavioral sciences, along with his commitment to teaching, inspired him to develop a precalculus series that gets readers engaged and keeps them engaged. Presenting the full scope of the mathematics is just the first step. Blitzer draws in the reader with vivid applications that use math to solve real-life problems. These applications help answer the question "When will I ever use this?" Readers stay engaged because the book helps them remain focused as they study. The three-step learning system-See It, Hear It, Try It-makes examples easy to follow, while frequent annotations offer the support and guidance of an instructor's voice. Every page is interesting and relevant, ensuring that readers will actually use their textbook to achieve success! Prerequisites: Fundamental Concepts of Algebra; Functions and Graphs; Equations and Inequalities; Polynomial and Rational Functions; Exponential and Logarithmic Functions; Systems of Equations and Inequalities; Matrices and Determinants; Conic Sections; Sequences, Induction, and Probability For all readers interested in college algebra |
Publisher Comments:
This Super Review can be used as a supplement to your high school or college textbook, or as a handy guide for anyone who needs a fast review of the subject.
• Comprehensive, yet concise coverage - review covers the material that is typically taught in a beginning-level math and pre-algebra course. Each topic is presented in a clear and easy-to-understand format that makes learning easier.
• Packed with practice - each review lesson is packed with practice questions and answers for each topic. Practice what youve learned and build your basic math and pre-algebra skills, so youll be ready for any problem you encounter on your next quiz or |
books.google.com - Studying math is often a source of great anxiety for children and teenagers. It also proves troublesome for parents, as many are reminded of their own struggles with the subject and feel lost when trying to tackle it again years later in an effort to aid their offspring. Help Your Kids with Math is designed... Your Kids with Math: A Visual Problem Solver for Kids and Parents
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I am a college student, and I was having trouble with math. This book is very well written and the steps are easy to follow. I have never had algebra, statistics or number theory. After doing the problems in the book, I am now ready to take on my college math. I would recommend this book for anyone having trouble understanding math. I forgot to mention, I am 59 years young, and have been out of school for many years, I truly needed something to help me to understand the college math. I recommend this also for parents that need to help their children with their math. |
Exceptionally clear and accessible, Pat McKeague's best-selling texts offer all the review, drill, and practice students need to develop solid mathematical proficiency and confidence. McKeague's attention to detail, exceptional writing style, and organization of mathematical concepts make teaching enjoyable and learning accessible. The seventh edition of this popular series gives the book a greater emphasis on the connections between equations, tables, and graphs and by incorporating pedagogy that addresses collaborative learning and terminology. |
Cliffs Quick Review for Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented wri...show moreters who know how to cut to the chase -- and zero in on the essential information you need to succeed. ...show less
Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched Studies.
2001 Paperback Our goal with every sale is customer satisfaction, so pleas eSusies Books Garner, NC
2001 Paperback This book looks good. It is like any used book you would expect to find in a used book shop |
MATH-221 Real Nbr Sys/Meth for Elem Sch
This course provides prospective elementary school teachers with background needed for teaching elementary mathematics. Both content and methodology relevant to school mathematics are considered. Topics covered include the real number system and its sub-systems. Pedagogical issues addressed include the nature of mathematics and of mathematics learning and the role of problem solving and the impact of technology in the elementary school mathematics curriculum. Prerequisites: Education 102. This course meets mathematics core. |
David B. Spangler outlines powerful diagnostic and NCTM- and Common Core State Standards-aligned RTI strategies for analyzing student errors and provides specific interventions for each error pattern. more...
This volume supports the belief that a revised and advanced science education can emerge from the convergence and synthesis of several current scientific and technological activities including examples of research from cognitive science, social science, and other discipline-based educational studies. The anticipated result: the formation of science... more...
CliffsQuickReview course guides cover the essentials of your toughest subjects. Get a firm grip on core concepts and key material, and test your newfound knowledge with review questions. Whether you're new to limits, derivatives, and integrals or just brushing up on your knowledge of the subject, CliffsQuickReview Calculus can help. This guide covers... more...
This unique resource provides 190 high-interest, ready-to-use activities to help students master basic math skills— including whole numbers, decimals, fractions, percentages, money concepts, geometry and measurement, charts and graphs, and pre-algebra— for use with students of varying ability levels. All activities are classroom-tested... more...
This friendly self-help workbook covers mathematics essential to first-year undergraduate scientists and engineers. In the second edition of this highly successful textbook the author has completely revised the existing text and added a totally new chapter on vectors. 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. Math problems arranged
according to how long they should take to solve ? 5, 10, and 15 minutes.
Each section of the book explains how a strategy is used and provides examples
to show application to various types of problems. The strategies include:
guess and check, listing, find a pattern, working backwards, and... more...
Presents
a set of problem solving techniques for thinking about the puzzles and questions
of mathematics. Each of the eight techniques is accompanied by practice
questions to consolidate the skill. The book also contains 100 additional
problems for students to attempt as their problem solving skills increase. more... |
Norwood, PA Geometry can guide, but students must do the work.Introduction to the basics of symbolic representation and manipulation of variables. Liberal use of concrete examples. May require review of arithmetic concepts, including fractions and decimals. |
(Second 4 Weeks) This is a traditional lecture section that will cover Unit 6 only. Computer access will still be required for homework and quizzes. Class dates: February 11 - March 17 The student will learn to perform operations on exponential expressions and polynomials. Students will also learn techniques to factor polynomials and use these techniques to solve polynomial equations. Emphasis should be on learning all the different factoring methods, and solving application problems using polynomial equations. Credit is not applicable toward graduation. Prerequisite(s): MTE 5 or qualifying placement score. Lecture 1 hour per week. |
Written for teenagers, English in Mind creates an inspiring learning experience for secondary students. Everything, from the choice of imaginative topics, texts and exercises to the attractive design is perfectly matched to students' interests, age and ability. It provides a solid basis for effective language learning through a strong focus on grammar and vocabulary. Flexibility and support are offered in the form of photocopiable activities and tests in the Teacher's Resource Pack, 'EiMTV' DVDs, Workbook CD-ROMs and the extensive free worksheets, tests, wordlists and resources on the website. Each level of the course provides 80-90 hours of work with the possibility of extension. It can be used with mixed-ability classes. The Starter level is for complete beginners and Level 1 is for elementary students and contains a 16-page starter section to revise key language. Levels 2 to 5 take students from pre-intermediate to advanced level.
The aim of this book is to present some ideas, methods and topics in elementary combinatorial geometry. Even if most of the book can be understood without any mathematical background, so it is accessible for 12-14 years old children too, we recommend it to high school students and college students as an introduction to a few topics in combinatorial (or convex) geometry (counting problems, pigeonhole principle, Helly's theorem, Sperner lemma). Our approach is basically an elementary one, but it is useful to know some combinatorial techniques (such as the product rule, mathematical induction, recursive counting, double counting, etc.) and some basic notions (number of permutations, binomial coefficients). These notions appear in the subject index, so they can be identified easily. Some problems (Helly's theorem, Sperner lemma, etc.) are formulated both in R^2 and in R^3 to help the understanding of analogies and to show some directions of generalization. Almost all of these problems can be solved even in R^n, but we presented only their 2 or 3 dimensional version, for a better understanding of the main ideas, without the complications of the tools necessary in higher dimensions. Some problems we present here are classical ones, some are from regional contests and from different levels of the mathematical olympiads and some are extracted from articles (and formulated in low dimension). The main purpose was not to create an exhaustive col1ection, but to offer a quick and short overview, to present a few properties which have surprising applications also in higher mathematics (the Sperner lemma) and to show some interesting ways of generalizations. So this book wants to be a kind of bridge between elementary problems and university courses etc....
Transforming Teacher Education: What Went Wrong with Teacher Training, and How We Can Fix It Publisher: Stylus Publishing | English | April 2010 | ISBN: 1579224377 | PDF | 288 pages | 1.8Mb
We need the best research, theories, and practices to make the kind of changes needed in the field of teacher education. This volume helps us take a major step in this direction.... Teachers, prospective teachers, educational planners, and policy-makers will all benefit from engaging this book. Hopefully they will engage the chapters with open minds, and begin to challenge 'business as usual' approaches to teacher education
Review ... students and researchers in elementary particle physics should seriously consider having this book in their personal library, partly to be able to look at the many pictures and brief biographies of their favorite elementary particle physicists, and partly to learn Veltman's incisive thinking on obscure or unsettled areas in particle physics ... there are plenty of popular books on dark matter, dark energy, black holes, and cosmology, but relatively few straightforward yet fascinating popular books on elementary particle physics, so in this sense Veltman's book is exactly what is needed. -- American Journal of Physics
Summary: This excellent series of ELT resource books gives any teacher, a wealth of photocopiable graded games and activities to liven up any class. You can use the material as suggested by the author or make up your own memory games, quizzes etc. It certainly helps my students.
Test Your English Vocabulary in Use Elementary 2nd edition can be used on its own or with the companion volume English Vocabulary in Use Elementary 2nd edition. It is a handy book of tests covering the vocabulary practised in English Vocabulary in Use Elementary 2nd edition.
English Practice & Progress: Grammar (Elementary) contains over fifty lively grammar activities for Elementary lever students. Activities include quizzes, wordsearches, games and crosswords. This book is ideal for consolidating or recycling key Elementary grammar structures and can be used in class or at home.
Processes of Inquiry: Inservice Teacher Educators Research Their Practice
2011 | 212 | ISBN: 9460915302 , 9460915299 | PDF | 2 Mb
This book presents inservice teacher educators' accounts of systematic inquiry into their practice in a variety of contexts throughout New Zealand. The importance of purposeful networks of practice at all levels of a system in supporting education change and improvement is a theme across the chapters. The contributors describe the challenges and successes associated with working in professional learning and development in ways that aim to improve outcomes for teacher educators, teachers and students. Their accounts illuminate the importance of a research and development approach that enables the generation and application of new knowledge and, more importantly, enables all contributors to be learners. Each of the authors describes their role in investigating the effectiveness of inservice teacher educator practice, as part of the overall project that endeavoured to improve practice for the future. Included are processes created for Maori (indigenous) settings where cultural metaphors were used to frame investigations of practice. The book makes an important contribution to our knowledge base about effective inservice teacher educator practice and its influence on classroom practice. The book will appeal to teacher educators interested in examining the fit between their practices and their goals in helping teachers to build knowledge and practice, including those working in indigenous settings. It will also be of interest to policy makers and evaluators involved in system-level change. ...a well organised and carefully argued text that offers compelling evidence for an integrated approach to project management, practice, research and evaluation (J. John Loughran, Series Editor).... |
This applet simulates an experiment to study Newton's Second Law. A mass sliding on a horizontal surface is connected by a string to a hanging mass. The value of the two masses and the coefficient of friction can be set...
Go with Alice to a wonderland of math! This website utilizes Lewis Carroll?s bright universe and most-recognizable character in order to teach mathematical concepts. Many students may feel as though they have s...
Add song and dance to your mathematics lessons with this exercise, in which students discover the various covert mathematical relations hidden in the lyrics to the song ?The Twelve Days of Christmas.? In addition to ...
This website applies virtual reality to calculus in order to illustrate mathematical concepts more clearly to students. While many courses utilize computers via computer algebra systems and graphing tools to...
This handy program will solve any function equation entered into its screen, and provide tips for solving a similar problem on paper. After providing the answer, the program then provides further information on the key... |
Angwin Calculus Philosophy is that mathematics provides a critical foundation for other learning. Young people can grasp advanced concepts early on. Creative approaches to Mathematics education that work well for gifted students can help students of all levels and backgrounds succeed in Math. |
Math Word Problems for Dummies
Overview
Covers percentages, probability, proportions, and more
Get a grip on all types of word problems by applying them to real life
Are you mystified by math word problems? This easy-to-understand guide shows you how to conquer these tricky questions with a step-by-step plan for finding the right solution each and every time, no matter the kind or level of problem. From learning math lingo and performing operations to calculating formulas and writing equations, you'll get all the skills you need to succeed!
Discover how to:
* Translate word problems into plain English
*
Brush up on basic math skills
*
Plug in the right operation or formula
*
Tackle algebraic and geometric problems
*
Check your answers to see if they work
Author Information
Mary Jane Sterling is the author of Algebra I For Dummies, Algebra Workbook For Dummies, Algebra II For Dummies, Algebra II Workbook For Dummies, and several other For Dummies titles. She has been a Professor of Mathematics at Bradley University in Peoria, Illinois, for more than 30 years.
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9780470245583
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Using this PowerPoint template makes it easy to create your own professional-looking Jeopardy games. Five categories with five levels of difficulty are provided in the template. Complete instruction... More: lessons, discussions, ratings, reviews,...
Explore retirement issues and consumer credit with this activity, which involves online calculators, interviews, journaling, and group presentations. This activity was used as a project to accompany a... More: lessons, discussions, ratings, reviews,...
A free web-based function graphing tool. Graph up to three different functions on the same axes.
Functions can refer to up to three independent variables controlled by sliders. As you move the... More: lessons, discussions, ratings, reviews,...
Zoom Algebra is a Computer Algebra System App for TI-83 Plus and TI-84 Plus graphing calculators. Its patent-pending interface is visual and easy to use, with many little shortcuts. For example, ... More: lessons, discussions, ratings, reviews,...
Guided activities with the Graph Explorer applet, in which students explore how the graph of a linear function relates to its formula, and learn to graph and edit functions in the applet, including on... More: lessons, discussions, ratings, reviews,...
This workbook provides students with an interesting way to explore math and science concepts. It is a revision of the earlier Texas Instruments publication Real-World Math with the CBL 2 System: Activ... More: lessons, discussions, ratings, reviews,...
The SimCalc Project aims to democratize access to the Mathematics of Change for mainstream students by combining advanced simulation technology with innovative curriculum that begins in the early g... More: lessons, discussions, ratings, reviews,...
Lots of real-world data, with descriptions of environmental and mathematical implications; stored in a variety of formats for easy download. Catalogued by mathematical topic and by environmental topi... More: lessons, discussions, ratings, reviews,...
GeoGebra is a free and multi-platform dynamic mathematics software for schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do construc... More: lessons, discussions, ratings, reviews,...
Linear regression is a powerful technique for fitting data as students begin mathematical modeling. This is a quick introduction to the technique using an interactive spreadsheet that gives students ... More: lessons, discussions, ratings, reviews,...
Students are given an input like f(3) and asked to move a point to the location on a given graph. Feedback is given. Students are also given the output, and asked to place the point properly, for ex... More: lessons, discussions, ratings, reviews,...
The eNLVM enhances the applets available through the National Library of Virtual Manipulatives (NLVM). It allows teachers to embed applets in lessons, make up questions to which students can submit an... More: lessons, discussions, ratings, reviews,...
With this one-variable function grapher applet and function evaluator, users can rotate axis/axes, change scale, and translate by using mouse or by entering data. The web site also contains informatio... More: lessons, discussions, ratings, reviews,...
This tool lets you plot functions, polar plots, and 3D with just a suitable web browser (within the IE, FireFox, or Opera web browsers), and find the roots and intersections of graphs. In addition, yo... More: lessons, discussions, ratings, reviews,...
This workbook provides high school students with "hands-on" motion activities using the Texas Instruments Calculator-Based Ranger (CBR). The CBR is a stand-alone motion data collection device that sen... More: lessons, discussions, ratings, reviews,...
This Graphing Calculator website contains hundreds of examples of how to best use the graphing calculator with specific classroom examples. Unlike using a manual, teachers and students can see how |
Title
Author
Date of Award
Degree Type
Degree Name
Department
Teaching and Leadership
Advisor(s)
Joanna O. Masingila
Keywords
College students, Algebra students, Variable
Subject Categories
Education | Science and Mathematics Education
Abstract
The concept of variable is a central idea in mathematics at all levels. Understanding of this concept provides the basis for the transition from arithmetic to algebra, lays the foundation for the understanding of functions and all advanced mathematics, and is necessary in real-world problem solving. However, research indicates that students have difficulty understanding the concept of variable.
The purpose of this study was to develop a better understanding of college students' conception of variable. The research questions that guided this study were: (a) What are students' conceptions of variable? (b) What effect do the choices of literal symbol have on students' performance? (c) What roles of variables do students use? and (d) How do students actively change from one definition of role of variable to another within the same problem?
This study was based on three frameworks. Philipp (1992) presented a categorization of literal symbols: labels, constants, unknowns, generalized provided a framework to examine the different uses of variables in relation to the purpose for teaching algebra and the conception of algebra. This study also used Hiebert and Carpenters (1992) framework for thinking about understanding.
Thirteen student enrolled in a college course (algebra or precalculus) were asked to answer a set of problems using a think-aloud procedure. In-class observation and analysis of the textbook, assignments and exams were used to categorize the set of problems.
I found that students recognized and used, with little difficulty, variables as unknowns and as generalized numbers. This result is related to the roles of variable that students learned in the classroom. I also found that students' level of confidence in approaching standard algebra problems was higher for familiar type of problems than for non-familiar type of problems. Thus, students need to be exposed to more situations that involve the different aspect of the concept of variable.
As expected, the more advanced math course, the more sophistication on the solutions, and less difficulty on using variables. Also, students talked more about their reasoning and were more articulated in their explanations for their solutions. I also present a model of solving word problems and a framework for algebraic variables.
Access
Surface provides description only. Full text is available to ProQuest subscribers. Ask your Librarian for assistance. |
-
Statistics reveal that more than two-thirds of students fear mathematics as a subject. Needless to mention, most students loathe mathematics assignments. However, it is good to know that mathematics is one of the most interesting subjects, and if students take the suggestions and tips mentioned in this article, then they are sure to be able to present a decent assignment. |
Basic Topology
9780387908397
ISBN:
0387908390
Pub Date: 1983 Publisher: Springer Verlag
Summary: In this broad introduction to topology, the author searches for topological invariants of spaces, together with techniques for calculating them. Students with knowledge of real analysis, elementary group theory, and linear algebra will quickly become familiar with a wide variety of techniques and applications involving point-set, geometric, and algebraic topology. Over 139 illustrations and more than 350 problems of ...various difficulties will help students gain a rounded understanding of the subject.
Armstrong, M. A. is the author of Basic Topology, published 1983 under ISBN 9780387908397 and 0387908390. Eight hundred two Basic Topology textbooks are available for sale on ValoreBooks.com, one hundred seventy four used from the cheapest price of $32.92, or buy new starting at $55.52 272 pages, Printed in Black And White, Have same content as... [more]ALTERNATE EDITION: New Softcover International Edition, 272 |
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). |
The Pythagorean Relationship - MAT-956What does A-squared plus B-squared equals C-squared really mean? After teaching with hands-on activities, video demonstrations, animations, and comics, your students will be able to answer that question and apply the relationship in problem solving situations. This course is built around core propositions from the National Board for Professional Teaching Standards as well as national content standards.
Connect With
Testimonial
"I really enjoyed this course! It has helped broaden my approach to teaching. I have learned many new practical applications and I'm very excited about implementing them into my classroom! My professor was always very helpful and prompt to answer my questions." |
Full course description
Algebra+ is a 10-week online course designed for students who have successfully completed high school algebra but who placed into pre-college level mathematics at their local college or university. This course is for refreshing their math skills with a review of pre-college level algebra. After successfully completing this course, the goal would be to retake your college's placement exam, or take the placement exam for the first time, and place into a college level mathematics course like College Algebra or Precalculus.
The plus in Algebra+ indicates a unique aspect of this course, the inclusion of "the rest of the story" about the things you learned in algebra. What is often not clear when one takes the standard algebra course is that most of the problems in the text have been carefully constructed to make sure they can be solved using the limited techniques taught in the course. The problems encountered in the real world tend to be messy problems for which the textbook approaches are very tedious or simply won't work. During this course, in addition to the standard textbook techniques, we will show you how to leverage technology to solve the more general problems you may encounter in your application courses and how to use technology to turbocharge your learning experience.
Randy Anderson
Professor
Randy Anderson is currently the Chair of the Mathematics Department at Peninsula College in Port Angeles, Washington. He received his M.S. in Mathematics from Northern Arizona University (1997) and his B.S. in Applied Mathematics from The University of Texas at Dallas (1990). He has taught mathematics classes at the community college level since 1992. |
Combinatorial Problems and Exercises
9780821842621
ISBN:
0821842625
Pub Date: 2007 Publisher: American Mathematical Society
Summary: The main purpose of this book is to provide help in learning existing techniques in combinatorics. The most effective way of learning such techniques is to solve exercises and problems. This book presents all the material in the form of problems and series of problems.
Lovász, László is the author of Combinatorial Problems and Exercises, published 2007 under ISBN 9780821842621 and 0821842625. Two hundred fif...ty seven Combinatorial Problems and Exercises textbooks are available for sale on ValoreBooks.com, one hundred three used from the cheapest price of $73.31, or buy new starting at $85.61 |
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Test prep
College Arithmetic Brush Up
Use this to brush up if you're preparing for an Arithmetic placement exam for college. This set of practice exercises is a quick list of some of the toughest concepts in Arithmetic! Challenge yourself to practice all the exercises. If you find it challenging, that's good! It means you're learning about what you need to learn :)
If you find an area where you need to go back and review more, you can use your Learning Dashboard to help you build mastery and explore foundations you need to review. |
Geometry - Second Edition is a clear presentation of the essentials of geometry for the high school student. Topics include: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Transformations. Volume 2 includes the last 6 chapters: Similarity, Right Triangle Trigonometry, Circles, Perimeter and Area, Surface Area and Volume, and Rigid Holy Grail of mathematics revealed as a truly 17th-century numerical and geometrical proof as a letter by Fermat to a colleague. This will withstand all challenges Advanced Probability and Statistics-Second Edition is a clear presentation of the basic topics in statistics and probability, but finishes with the rigorous topics an advanced placement course requires. Volume 2 includes the last 7 chapters and covers the following topics: Sampling Distributions and Estimations, Hypothesis Testing, Regression and Correlation, Chi-Square, Analysis of Variance and F-Distribution, and Non-Parametric Statistics. It also includes a collection of resources in the final chapter Middle School Math Grade 6 covers the fundamentals of fractions, decimals, and geometry. Also explored are units of measurement, graphing concepts, and strategies for utilizing the book's content in practical situations. Volume 1 includes the first 6 chapters Basic Probability and Statistics – A Short Course is an introduction to theoretical probability and data organization. Students learn about events, conditions, random variables, and graphs and tables that allow them to manage data Calculus textbook is a college-level, introductory textbook that covers the fascinating subject of Calculus. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.This textbook covers:Building Blocks of Calculus -- Precalculus Review, Functions and Models, LimitsDerivatives and Integrals -- Derivatives, Applications of Differentiation, Integrals, Applications of IntegrationInverse Functions and Advanced Integration -- Inverse Functions: Exponential, Logarithmic, and Trigonometric Functions, Techniques of Integration, Further Applications of IntegrationDifferential Equations, Parametric Equations, and Sequences and Series -- Differential Equations, Parametric Equations and Polar Coordinates, Infinite Sequences and SeriesAdvanced Topics in Single-Variable Calculus and an Multivariable Calculus -- Vectors and the Geometry of Space, Vector Functions, Partial Derivatives, Multiple Integrals, Vector Calculus, Second-Order Linear Equations Statistics textbook is a college-level, introductory textbook that covers the fascinating subject of Statistics, a branch of mathematics often highly relevant to everyday situations. Boundless works with subject matter experts to select the best open educational resources available on the web, review the content for quality, and create introductory, college-level textbooks designed to meet the study needs of university students.This textbook covers:Statistics and Statistical Thinking – OverviewStatistics in Practice -- Observational Studies, Controlled ExperimentsVisualizing Data -- The Histogram, Graphing DataFrequency Distributions -- Frequency Distributions for Quantitative Data, Frequency Distributions for Qualitative DataDescribing, Exploring, and Comparing Data -- Central Tendancy, Measures of Relative Standing, The Law of Averages, Further Considerations for DataMeasures of Variation -- Describing VariabilitySampling -- Populations and Samples, Sample Surveys, Sampling Distributions, Errors in Sampling, Sampling ExamplesProbability -- What Are the Chances?, Probability Rules, More About ChanceProbability and Variability -- Discrete Random Variables, The Binomial Random Variable, Other Random VariablesContinuous Random Variables -- The Normal Curve, Normal Approximation for Data, Measurement Error, Expected Value and Standard Error, Normal Approximation for Probability HistogramsCorrelation and Regression -- Correlation, More About Correlation, Regression, The Regression Line, R.M.S. Error for Regression, Multiple RegressionEstimation and Hypothesis Testing -- Estimation, Confidence Intervals, Hypothesis Testing, Hypothesis Testing with Two Samples, Comparing More than Two Means, The F-Test, Analysis of Variance (Repeated Measures)Other Hypothesis Tests -- The T-Test, The Chi-Squared Test, Tests for Ranked Data, Nonparametric StatisticsA Closer Look at Tests of Significance -- Which Test?, A Closer Look at Tests of Significance designed Geometry FlexBook is a clear presentation of the essentials of geometry for the high school student. Topics include: Proof, Congruent Triangles, Quadrilaterals, Similarity, Perimeter & Area, Volume, and Algebra FlexBook is an introduction to algebraic concepts for the high school student. Topics include: Equations & Functions, Real Numbers, Equations of Lines, Solving Systems of Equations & Quadratic Equations.' |
MATH 818 - Basic Math. for Health Science
at Cañada College
for Fall 2012 (CRN : 93494)
This course is offered online and it is recommended you complete a short self-assessment to ensure readiness before taking an online course. Most Mathematics courses have prerequisites that are listed as part of the course description in the Schedule of Classes. Before registering for a Mathematics course, be sure you have completed the stated prerequisite.
Note to all Algebra students:
The Math Department uses a single textbook for the Elementary and Intermediate Algebra sequence. This allows students to complete the Algebra sequence in three different ways: a four semester sequence of MATH 111, 112, 122, and 123 each covering one fourth of the book; a two semester sequence of MATH 110 and MATH 120 each covering half of the book; or a combination of the above. Please see your counselor to be sure you take the correct course.
Online, are courses where the instructor and student are separated by distance for the entire course and can interact exclusively through the assistance of communication technology. The course is conducted through a class website, which may include multimedia material and links to other online resources. Students interact with the instructor and other students through posted class discussions, direct individual communication and assignments (which may include group work). Testing may be done online via proctoring arrangements or other means. Instructors require no mandatory on-campus meetings. .
Recommended orientation meeting for MATH 818 OLH: Orientation for MATH 818 OLH will be held online through CCCConfer.org. Instructor will email students (@my.smccd.edu account) one week before the orientation for details. |
Students
will use mathematical analyasis, scientific inquiry, and engineering
design, as appropriate, to pose questions, seek answers, and develop
solutions.
Students
will access, generate, process, and transfer information using appropriate
technologies.
Students
will apply the knowledge and thinking skills of mathematics, science,
and technology to address real-life problems and make informed decisions.
BENCHMARK
STANDARDS CONTENT STANDARDS
Students
will use mathematical modeling/multiple represaentation to provide
a means of presenting, interpreting, communicating and connecting
mathematical information and relationships
Students
will usae graphing utilities to model real-world problems with linear
equations and tables to analyze mathematical phenomena.
PERFORMANCE
STANDARDS
Students
will be able to take raw data and express it in the form of a table
and as a linear equation
Using
the models they created, students will make decisions and predictions
of future events
Students
will use a graphing calculator to help them to form mathematical models
and to make informed decisions.
CONTENT STANDARDS
The TI-82
calculator will be instrumental in helping the students graph a given
a set of coordinates Cartesian Plane, determine the slope of a line
numerical and graphical methods, and information obtained in the local
or newspapers students will develop linear models use these models
to evaluate situations and make predictions for the future.
PERFORMANCE
MEASURES
Students
will be marked on a scale ranging from 1 to 4. The grades will be assigned
in the following fashion:
1 Completing
task A
2 Completing
tasks A, 8 and C
3 Completing
tasks A, 6, C and D
4 completing
tasks A, B, C, D, E and F
A Students
will be able to graph a set of coordinates and connect the points
and create a picture
B Students
will be able to find the slope of a line numerically
C Given
a picture of a variety of lines, students will be able to determine
if the slopes are negative, zero or if the lines have an undefined
slope
D Given
equations of lines, the students will be to determine if the lines
are parallel or perpendicular
E Given
real-life data, students will represent them in a tabular format
F Given
real-life data, students will create a model to represent the data
and to make predictions.
ENABLING ACTIVITES
This two week lesson plan will cover material concerning
coordinate graphing and the creation of linear equations. The content
of this unit will be developed using the TI-82 calculator. Time
will be sent having the students and graph points in the Cartesian Plane
using the calculator. The students will be introduced to the concept of
the slope of a line. They will be asked to represented slopes of
different lines both numerically and graphically. Slopes of lines which
are parallel and perpendicular will be discussed, as well as lines that
have slopes which are positive, negative, zero and lines whose slopes
do not exist. The students will be introduced to the concept of writing
equations of lines. They will then be given situations from real-life
problems and they will be asked to represent this data in both a tabular
form and as a linear model. Using the model which they created they will
be asked make predictions concerning possible outcomes of future events.
WORKSHEET 1
1.
Simplify the expressions in both the x and y column and place the answers
in a second t-chart
2.
Using your TI-82, graph the coordinates which you found in question 1.
3.
Which coordinates are located in Quadrant I? II? III? IV?
4.
Change the type of graph from one where you just points to one where you
connect the points. What have you drawn?
WORKSHEET II
You
are in need of a new car since your old car has died, so on Monday you
stop at a dealership which offering a 42 month lease. The salesperson
tells you the terms of the lease will require you to pay $159 a month.
In addition you will be required to pay $1000 down, a $ bank fee and a
$200 security deposit.
Write
an equation which will represent the terms of this lease.
Using
the equation which you just formed, how money will you have paid at
the end of:
6
months
15 months
26 months
36 months
42 months
On
Tuesday you decide to try another dealership who is offering another deal
on the exact same car you yesterday. This salesperson offers you
a 36 month lease for $199 a month. The security deposit here is
$400 and bank fee is $798.
Write
an equation which will represent the terms of this lease.
Using
the equation which you just formed, how money will you have paid at
the end of
6
months
15 months
26 months
36 months.
Using
your graphing calculator, graph each equation and determine at what
point have you paid the same amount of money on both of the leases.
WORKSHEET III
According
to the National Association of Scholars, the average number of instructional
days at a college in the year 1914 was 204. During the academic
year 1993, the number of days had decreased to 156. If you represent
this information as order pairs in the form (year, days in class)
Determine
the slope of the line which can be drawn between these two points.
Using
the slope which you just found, write the equation of the line which
can be drawn between these two points(Round all decimals to ten-thousandths).
Use this
equation to answer the following
What is
the coordinate of the x-intercept?
What is
the coordinate of the y-intercept?
The slope
of this line is
Using
this equation and assuming that nothing differs, in what year will
a student not have to go to class?
While
the number of days that a student spends in formal classes have decreased,
the average tuition paid by have increased. In 1914 the average
daily cost was $7.lS
while in 1993 the average cost was $100. If you this information
as an order pair in the form (year, cost)
Determine
the slope of the line which can be drawn between these two points.
Using
this slope, write the equation of the line which can be drawn between
these two points(Round all decimals to the nearest ten-thousandths).
Using
this equation, determine
The x
- intercept.
The y
- intercept.
The slope
of this line is
If the
rate of inflation continues, in what will students be paying an average
of $200 a day for college?
WORKSHEET IV
You
are a little short of cash so you decide to turn your hobby of making
holiday ornaments into a money proposition by selling them at a flea market.
The first day you are there you sell 3S items and you make a profit $20.
The next time that you sell the items you sell lS2 the profit you make
is $254. The third time you sell ornaments you sell 98 of them and
the profit you make is $146.
Express
this information in a t-chart.
Use,the
TI-82 calculator and form a linear which can be used to express this
information.
If you
sold 163 ornaments, how much profit would you have made?
You need $600 to cover the cost of your car and insurance this month.
How many ornaments do you need to sell to cover these two expenses? |
Abstracts of studies on how much mathematics is used in various occupations: Mathematics in Automobile Production; Proportional Reasoning by Nurses; Modeling the Mathematics of Banking; Mathematical Models as Seen by...
"This year's theme is mathematics and biology. Two important examples of how mathematics is used in biology are: in developing computer models of the human heart, which are helping us to understand how the heart works...
Programs for students about to start first and second year courses in Mathematics. Students can test themselves to see whether they are ready for intermediate algebra, college algebra, business calculus, calculus I, II,...
Plexagon is a new modular geometry based on the radial pleating of polygonal plates, and their joining or co-bracing into 3D modules. This engineering approach can form a very practical basis for modular greenhouses... |
Basic Geometry for College Students: An Overview of the Fundamental Concepts of Geometry - 2nd edition
Summary: Intended to address the need for a concise overview of fundamental geometry topics. Sections 1-7 introduce such topics as angles, polygons, perimeter, area, and circles. In the second part of the text, Sections 8-11 cover congruent and similar triangles, special triangles, volume, and surface area Basic Geometry for College Students: An Overview of the Fundamental Concepts of Geometry:
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Library and Electronic Media
Mathematics Faculty and students have access to a monograph and journal collection which is housed in the James Earl Carter Library on the Georgia Southwestern campus. Other resources needed for research and study can be obtained through the Library's efficient interlibrary loan system. Students and faculty also can use the University System of Georgia's on-line, virtual library, GALILEO, using terminals in offices, the library, dormitories, and in numerous computer labs located around the campus. Galileo's numerous databases allow users to retrieve current journal articles on-screen, as printable files, or as portable document (pdf) files which may be viewed using Adobe Acrobat Reader software. The Library's monograph and journal collection, supplemented by Galileo and other Web resources give our faculty and students an active window on evolving mathematical topics. In addition, many faculty supplement their lectures with personal web sites on Southwestern's Faculty Web. Some of the sites were developed the old fashion way, using HTML, while others use the full power of such on-line courseware as WebCT. You can get the feel for designing an on-line course yourself by visiting one of the many WebCT tutorial sites on the net, for example, the OFFICIAL site.
Computer Labs and Software
Mathematics students can use the new Mathematics and Science PC computer lab, housed on the third floor of the Roney Building, and just next door to the Math and Science Building. Our students can also use lab facilities in the School of Computer and Information Science (PC and SunSparc workstations), and the Power Macintosh lab in the School of Education. Our small campus makes all these lab facilities easily accessible from dormitories, classrooms, and parking areas. All computer labs are on the campus network, and have internet access via Peachnet.
Mathematics students at Georgia Southwestern start using technology as a means of exploring mathematics immediately. Our College Algebra, Precalculus and Calculus I courses all require familiarity with a graphing calculator such as the TI-83. In more advanced courses, students use a variety of software packages including Maple, Minitab, Mathlab, Mathematica, Macsyma, MacMath, ODE Architect, Interactive Differential Equations, and IDL. Majors are required to take at least two computer programming courses: Introduction to Structured Programming, and Advanced Structured Programming, which are currently focused on C++. The Department also encourages students to gain additional computational experience in other disciplines, either by taking a minor in computer science, or by taking any of a variety of classes with a strong modeling and visualization element like the Department of Geology and Physics' courses in Remote Satellite Sensing and Geographic Information Systems.
Teaching Facilities
Mathematics courses are taught in classrooms in the Math and Science Building, in the Business/History and Political Science Building, in labs in the School of Computer and Information Science and in the School of Education, and in a state of the art Electronic Classroom in the Roney Building. Smaller classes are often taught in faculty offices, or in classrooms in which a small computer network has been set up to give students access to high speed computers and sophisticated computer algebra system (CAS) software. |
Fundamentals of Precalculus - 2nd edition
Summary: ''Fundamentals of Precalculus'' is designed to review the fundamental topics that are necessary for success in calculus. Containing only five chapters, this text contains the rigor essential for building a strong foundation of mathematical skills and concepts, and at the same time supports students' mathematical needs with a number of tools newly developed for this revision. A student who is well acquainted with the material in this text will have the necessary skills, understanding,...show more and insights required to succeed in calculus. ...show less
Chapter 3 Trigonometric Functions 3.1 Angles and Their Measurements 3.2 The Sine and Cosine Functions 3.3 The Graphs of the Sine and Cosine Functions 3.4 The Other Trigonometric Functions and Their Graphs 3.5 The Inverse Trigonometric Functions 3.6Right Triangle Trigonometry 3.7Identities 3.8Conditional Trigonometric Identities 3.9The Law of Sines and the Law of Cosines478980 |
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface Preface Many students find "Trigonometry" to be a difficult topic in a diffi- cult subject "Math". Yet most students have no difficulty with com- puter games, and enjoy playing them even though many of these games have lots of pieces to manipulate, and are subjected to com- plex rules. For example, a simple game like "Tetris" has seven differ- ent pieces; and the player has to orientate and manipulate each piece in turn, as it falls. The objective is to construct a solid wall with all the random pieces — and to do it, racing against the clock. "Trigonometry", or "Trig" for short, can be thought of as an intellec- tual equivalent of "Tetris". There are six main pieces to manipulate. Three of them, sine, cosine, and tangent, are most important — that is why this branch of Math is called Trigonometry; the "tri" refers to three functions, three angles and three sides of a triangle. And there is only one simple rule — Logic. Trig can be thought of as a game that involves the logical manip- ulation of various trig pieces to achieve different identities and equations, and to solve numerical problems. Trig can also be viewed as a non-numerical equivalent of the number game "Sudoku". The logic and the arrangement of the digits 1 to 9, is now applied to the six trig pieces — sine, cosine, tangent, cosecant, secant and cotangent. Tetris and Sudoku are both simple games that give lots of fun and pleasure. Trig is also a simple game, but with a vital difference — knowledge of it has invaluable applications in Math, surveying, building, ix
September 17, 2007 22:30 Book:- Trig or Treat (9in x 6in) 03˙preface x Trig or Treat navigation, astronomy, and other branches of science, engineering, and technology. Adults, children and students can play Sudoku and Tetris for hours on end. So they should have little difficulty playing "Trig", if they derive similar fun and pleasure from it. Albert Einstein said: "Everything should be made as simple as possible, But not simpler". This book seeks to make Trig as simple as possible, by treating it as a game — albeit, an intellectual game — as interesting and stimulating as Tetris and Sudoku. Mastering of Trig will not only give mental and intel- lectual satisfaction and pleasure, but it will also lead to beneficial results in one's future career and life. This book is the third math book∗ that I have written for my two grand- children, Kathryn and Rebecca, ages 5 and 7, respectively. The challenge that I set for myself here is to explain Trig so simply that my seven-year- old granddaughter, Rebecca, can understand it. Indeed she has been able to do some of the "Level-One-Games". My hope is that in the coming years both Kathryn and Rebecca would be able to play the "Level-Two-Games" and the "Level-Three-Games" in this book also. I thank Lim Sook Cheng and her excellent team at World Scientific Publishing for the production of this book; and Zee Jiak Gek for her metic- ulous reading and critique of all the details in the manuscript. ∗ The other two books are " The Pleasures of Pi,e and Other Interesting Numbers", and "Are You the King or Are You the Joker?".
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 04˙introduction Introduction The approach taken in this book is to treat Trig as a game. Beginning with only the definition of sine, the superstar of Trig, the book introduces the reader slowly to the basics of Trig. Then, by applying simple logic, the two co-stars, cosine and tangent, are introduced. Thereafter three supporting starlets, named the reciprocals — cosecant, secant and cotangent — are added. With these six pieces, the application of simple logic, arithmetic and algebra will give countless Trig equations called identities. Played like jigsaw puzzles, Tetris and Sudoku, moving the Trig pieces around to give different identities can be a lot of fun. As with other games and puzzles, practice can lead to greater skill and mental agility. About 300 games (proofs) are provided in this book to give fun (and confidence) to readers who want to try their hands (and work their brains) on these intellectual games. The numerous games are broadly grouped into three overlapping levels — Level-One-Games (Easy Proofs), Level-Two-Games, (Less-Easy Proofs) and Level-Three-Games (Not-so- Easy Proofs). For the first time ever, a "Concordance of Trigonometric Identities" has been created. Trigonometric identities are given a 6-digit code, which enables readers (and students) to have easy reference to the identity to be proved, and to locate rapidly the proof in the Encyclopedia of Trigonomet- ric Identity Proofs (TIPs) in the Appendix. Readers are welcome to look at the identities in the Concordance first, and try their hand at proving any of the identities, prior to looking at the xiii
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 04˙introduction xiv Trig or Treat detailed proofs in the Encyclopedia. (Some identities which may appear simple, may be difficult to prove; conversely, some complex-looking iden- tities may turn out to be relatively easy!) The games provide the challenge to readers to match their skills, and progress up the ladder of increasing intellectual agility. If you are really good in Trig, then the speed of proving the identities is the speed with which you write out the proofs, i.e. your brain works faster than your brawn (hand). Have fun with Trig!
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 3 Measuring Angles "The sun rises in the east, and sets in the west". Similarly, the measur- ing of angles begins "in the east" (0 ◦ ), goes counterclockwise, up into the overhead sky at noon (90◦ ) and sets in the west (180◦ ). People in many ancient civilisations (including the Babylonians, Mesopotamians and the Egyptians) used a numbering system based on 60 called the sexagesimal system. This resulted in the convention of 360 ◦ (60◦ × 6) for the angle round a point. This convention for measuring angles continues to the present day, despite the widespread use of the metric system based on decimals (10's). Another sexagesimal legacy from the past is the use of 60 seconds in a minute, and 60 minutes in an hour.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 5 Sine Over the centuries, many civilisations used calculations based on right- angled triangles and the relationships of their sides for various purposes, including the building of monuments such as palaces, temples, and pyra- mids and other tombs for their rulers. Some of these mathematical tech- niques were also applicable to the study of the stars (astronomy) which led to calender making. The origins of Trig are lost in the mist of antiquity. One of the earliest recorded reference to the concept of "the sine of an angle" — "jya" — was found in a sixth century Indian math book. This word was later translated into "jiba" or "jaib" in Arabic. A further translation into Latin converted the word into "sinus", meaning a bay or curve, the same meaning as "jaib". This was further simplified in the 17th century into English — "sine", and abbreviated as "sin" (but always pronounced as "SINE"and not "SIN".) Sine is simply the name of a specific ratio: length of the opposite side of angle (A) sine of an angle (A) = length of the hypotenuse This definition is often abbreviated to O sin A = H You cannot do Trig if you cannot remember the definition of sin! There are many simple ways of remembering. How about: 1. O/H lang SINE?∗ 2. O/H, it's so SINple? Can you create your own mnemonics? ∗ Sounds like "Auld Lang Syne", the universally popular song sung at the stroke of midnight on New Year's Day.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 6 Trig or Treat Cosine (90−A)° se nu pote Hy A 90° Adjacent side of angle A Adjacent cos A = Hypotenuse A = H Tangent Opposite side of angle A A 90° Adjacent side of angle A Opposite tan A = Adjacent = O A sin A tan A = cos A
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 7 Cosine and Tangent The complementary angle to the angle A in a right-angled triangle is the third angle, with the value of (90 − A) ◦ , because the three angles of a tri- angle sum to a total of 180◦ . The term "co-sine" was derived from the phrase "the sine of the complementary angle" co-sine A = sine of complementary angle of A = sine(90 − A)◦ length of the adjacent side of angle A ∴ cos A = length of the hypotenuse A = H Tangent∗ is defined as the ratio of sin A/ cos A sin A ∴ tan A = cos A O = H A H O = A ∗ This ratio (tangent) should be distinguished from the line which touches a circle, which is also called tangent in geometry.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 8 Trig or Treat Reciprocals 1 cosec A = sin A 1 sec A = cos A 1 cot A = tan A sin A tan A = cos A cos A cot A = sin A
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 9 Reciprocals The superstar "sin" and its two co-stars (cos and tan) make up the three key players in Trig. Their definitions and their relationships are essential for all problems in Trig. Hence it is important that they be committed to memory. Three more trig terms — the supporting cast — are also used. These are known as the reciprocals, and are best remembered as the reciprocals of sin, cos and tan. 1 = cosec A (cosecant) sin A 1 = sec A (secant) cos A 1 = cot A (cotangent) tan A These reciprocals are rarely used in applications in science, engineering and technology. But for intellectual gymnastics (and in examinations!), these reciprocals are often used in equations and identities.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 11 Pythagoras' Theorem∗ The most well-known theorem in Math, which practically every student has learnt, is the Pythagoras' Theorem, named after the Greek mathematician Pythagoras (∼580–500 BC). This theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the two other sides. This lengthy statement can be represented accurately in mathematical terms: a2 + b 2 = c 2 where a and b are lengths of the two sides, and c is the length of the hypotenuse, the side facing the right-angle. The most famous right-angled triangle is the "3-4-5 triangle": 32 + 4 2 = 5 2 (9 + 16 = 25) A less famous sister is the "5-12-13 triangle" (5 2 + 122 = 132 ; 25 + 144 = 169). Recent research has shown that many civilisations, including the Baby- lonian, the Egyptian, the Chinese and the Indian civilisations, indepen- dently knew about the relationship between the squares of the three sides of the right-angled triangle, in some cases, centuries before Pythagoras was born. (This illustrates a truism in Math, that often, your discoveries based on your own efforts, may have been preceded by others. However this does not diminish in any way, the pleasure, excitement and sense of achievement that you experienced — the so-called "eureka effect". Indeed, it proves that you have a mathematical mind, capable of the same deep thoughts as the ancient heroes of Math.) ∗ A theorem is simply a mathematical statement whose validity has been proven by meticulous mathematical reasoning.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 The Basics of Trigonometry 13 Trig Equivalent of Pythagoras' Theorem One of the most important of Trig identities ∗ is the trig equivalent of the Pythagoras' Theorem. The proof is simple: a b sin A = cos A = c c a2 b2 sin2 A + cos2 A = + c2 c2 a2 + b 2 = c2 c2 by Pythagoras' Theorem = c2 a2 + b 2 = c 2 =1 sin2 A + cos2 A ≡ 1 A simpler visual proof can be obtained by using a special right-angled triangle with a hypotenuse of unit length (1). Then the length of the opposite side is now equal to sin A, and the length of the adjacent side is equal to cos A (see figure opposite). Then by Pythagoras' Theorem: sin2 A + cos2 A ≡ 12 sin2 A + cos2 A ≡ 1 . This unity trig identity is the simplest and the most important of all trig identities. It is also extremely useful in helping to solve trig problems. Whenever you see sin 2 A or cos2 A, always consider the possibility of using this identity to simplify further. ∗ Anidentity is a mathematical equation that is true for all values of the angle A. It does not matter whether A = 30◦ , 60◦, 90◦ etc, whether it is acute or obtuse, etc. The symbol (≡) is used to show that the two sides of an equation are identical.
September 17, 2007 19:21 Book:- Trig or Treat (9in x 6in) 05˙level-1 14 Trig or Treat We can also derive two other identities: dividing sin2 A cos2 A 1 + 2 ≡ by sin2 A 2 sin A sin A sin2 A 1 + cot2 A ≡ cosec2 A dividing sin2 A cos2 A 1 + ≡ by cos2 A cos 2A cos 2A cos2 A tan2 A + 1 ≡ sec2 A. After this simple introduction, you are now ready to play Level-One- Games (Easy Proofs), some of which seven-year-old Rebecca could play. The general approach for playing the games (proving the identities) is to: 1. start with the more complex side of the identity (usually the left hand side (LHS)); 2. eyeball the key terms, and think in terms of sin and cos of the angle; 3. engage in some mental gymnastics — rearranging and simplifying; 4. whilst at all times, keeping the terms in the right hand side — the final objective — in mind. Like a guided missile, your logic and math manipulation of the LHS should lead you to zoom in to the RHS. Have Fun!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2 Compound Angles 17 Sine of the Sum and the Difference of Angles This is the beginning of Level-Two-Games. The sin of compound angles (i.e. angles that are the sum or the difference of two other angles) can be expressed in terms of trig functions of the single angles. For example, sin of (A + B) can be expressed as a combination of the sum and products of the sin and cos of A and B separately. Similarly, since we know that sin(−A) = − sin A, and cos(−A) = cos A, we can derive sin(A − B). By judicious substitution (using B = A), sin(A + B) can be changed to sin 2A and then into sin A (using A = 2(A/2)). In earlier days, before calculators and computers were available, knowl- edge of the trig functions of compound angles was invaluable in practical workplace calculations. Knowing the basics for 0 ◦ , 30◦ , 45◦ , 60◦ and 90◦ , one could work out the values of trig functions of 15 ◦ and 22.5◦ and other such angles by way of these functions. In those days, Trig was both "pure math" and "applied math", useful in many professions involving science, engineering and architecture. Today, where the pressing of a few buttons on a calculator or computer will give answers for all such calculations, Trig is largely "pure math" — a mental pursuit, an intellectual game. But what a beautiful game it is, especially when you are immersed in the proving of the vast number of identities!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2 Compound Angles 19 Cosine of the Sum and Difference of Angles Cos functions require special attention as occassionally they act in a con- trarian, counter-intuitive manner — this arises largely from the fact that cos(−A) = cos A. This is the first function where such counter-intuitive behaviour of cos shows itself. Although the LHS of the cos function is for the sum of two angles, the RHS shows a difference of the two products. Students are usually careless and are not sufficiently sensitive to such minor (???) intricacies in Math. Unfortunately such minor (???) inattention can be very costly in examina- tions because they lead to wrong answers and major (???) losses in marks!
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2 Compound Angles 21 Tangent of the Sum and the Difference of Two Angles The tan formulas for the sum and difference of two angles derive directly from the sin and cos formulas. The double angle formula, tan(2A), and the half angle formula, tan A, have proven to be extremely valuable in many mathematical proofs, and have resulted in sophisticated methods for the calculation of π to a large number of decimal places. (Would you believe that π has been calculated to 1.24 trillion decimal places — yes, 1,240,000,000,000 decimals?)
September 12, 2007 19:17 Book:- Trig or Treat (9in x 6in) 06˙level-2 Compound Angles 23 The sin, cos and tan of compound angles, and their "double angle" and "half angle" formulas provide the basis for many of the Level-Two-Games. Together with the most important identity: sin2 A + cos2 A ≡ 1 , the 12 identities on the opposite page, make up the total of the key Trig functions. Most other Trig identities can be derived from these "12 + 1" key identities. The typical student (or Trig player) is expected to know these "12 + 1" key functions (very lucky if you know them instinctively, and very un- lucky if you don't). With these "12 + 1" key functions, Level-Two-Games (Less-Easy Proofs) should prove to be easy also. Within Level-Two-Games, the proofs begin with the simpler ones and progress upwards in difficulty. Have Fun!
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Angles in a Triangle 27 Trig Identities Involving All Three Angles in a Triangle Some special trig identities apply only when all three angles in a triangle are involved. For such identities, the additional constraint of (A + B +C) = 180◦ is a critical one. For such identities the relationship between the three angles is always necessary for simplification, and sometimes result in beautiful identities as seen in some of the examples in Level-Three-Games.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Angles in a Triangle 29 The Sum and Difference of Sine Functions Often in Math, the addition or subtraction of similar equations, or the re- arrangement of sums and differences can lead to new insights and new equations which may be of special value. The equations in the earlier pages are the sin and cos formulas of com- pound angles. Here we are looking at the sum and difference of the trig functions of such compound angles, and after further simplification, we derive new relationships. Knowing these new relationships provide greater flexibility and agility in the manipulation of the trig building blocks, and enables new equations or identities to be proved.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Angles in a Triangle 31 The Sum and Difference of Cosine Functions Similar addition and subtraction of the cos functions for compound angles give similar equations for the sum and difference of cos functions. While these functions were extremely useful before calculators and computers were available, they have fallen into disuse in modern times except for purposes of "examinations", to test the student's versatility. You are now ready to play Level-Three-Games — the "Not-So-Easy" Proofs.∗ Have Fun! ∗ There are no simple sum and difference formulas for tan!
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Numerical Values of Special Angles 39 Value of Sine in the Four Quadrants The values of the sine of angles in the first and second quadrants (i.e. between 0◦ and 180◦ ) are always positive as seen in the graph on the opposite page, rising from 0 for sin 0 ◦ to a maximum value of 1 for sin 90◦ . Similarly the values of the sine of angles in the third and fourth quad- rants (i.e. between 180◦ and 360◦ ) are all negative, going to a minimum of −1 for sin 270◦ , and returning to 0 for sine 360◦ . The brain remembers pictures better than equations or words; so com- mit the graph on the opposite page to memory; and this would prove to be extremely valuable in solving trig problems. This graph is well-known in Math as "the sinusoidal curve" or the "sine curve", for short. A quick sketch of the sinusoidal curve (done in 10 seconds) will provide a good guide for ensuring that the correct values of sine in the different quadrants are obtained. It is also important to remember that the sine of a negative angle is the negative of the sine of the angle. sin(−A) = − sin A.
September 17, 2007 22:20 Book:- Trig or Treat (9in x 6in) 07˙level-3 Numerical Values of Special Angles 41 Cosine and Tangent By similar consideration, the values of the cos and tan of angles in the four quadrants can be obtained. Again it is easier to remember the pictures (graphs on the opposite page). Always remember that cos 0◦ = 1, so the graph for cos always begins at 1, and go to −1 for cos 180◦ . (The cos graph for 0◦ to 360◦ looks like a hole in the ground). Again, remember: cos(−A) = cos A. The cos function has this unusual feature and hence cos functions need special attention (i.e. be extra careful with cos functions). For tan, the value goes from tan 0◦ = 0 all the way to the indeterminate value for tan 90◦ . Interestingly, the third quadrant is an exact replica of the first quadrant, and the fourth an exact replica of the second. Again, remember: tan(−A) = − tan A
September 14, 2007 1:0 Book:- Trig or Treat (9in x 6in) 08˙appendix The Concordance of Trigonometric Identities The six-digit code for the Concordance is based on the number of trig functions on the LHS of the identity e.g. 123 000 means that the LHS has 1 sin, 2 cos, 3 tan and no cosec, sec and cot functions. On the rare occasion when you cannot find the identity in the Concordance, use the functions on the RHS to determine the code.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips 74 Trig or Treat An Example for Proving a Trig Identity 1 + sin A ≡ sin A 1 + cosec A Eyeballing and Mental Gymnastics 1. Start with the more complex side, normally the LHS; this is more amenable to simplification. 2. Consider simplifying tan, cot, and the reciprocal functions to sin and cos. 3. Consider the use of common denominators. 4. Rearrange and simplify through cancellation of common terms, if available. Let us explore this first game (proof) together. Eyeballing the identity, we see that the more complex side is indeed the LHS (occassionally the RHS is the more complex; then it may be preferable to begin with the RHS. On rare occassions both the LHS and the RHS are complex; then one can explore simplifying both side to a common set of terms). We note the cosec term in the denominator and remember that cosec = 1/ sin. Generally speaking it is easier to work with sin and cos than with their reciprocals as it makes rearrangement, simplification and cancellation easier. With a reciprocal term in the denominator, we expect to use common denominators prior to rearrangement, simplification and cancellation. If all goes well, and no careless mistakes are made, we should end up with "sin A" which is the target objective on the RHS. It is useful to begin a proof by writing down accurately the LHS. If a mistake is made here, no amount of effort will give the required identity.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 09˙tips 76 Trig or Treat sin A + cos A ≡ 1 + tan A cos A Eyeballing and Mental Gymnastics∗ 1. t = s/c 2. rearrange and simplify. sin A + cos A LHS = cos A = tan A + 1 ≡ RHS. ∗ Some teachers prefer that students always write out the trig functions to include the angle i.e. sin A instead of sin. In the "Eyeballing and Mental Gymnastics" sections, and in the short explanatory notes, we will use the abbreviations: s = sin, c = cos, and t = tan. Such abbreviations reflect the mental process in action, and conveys a sense of speed with eyeballing and mental gymnastics taking place.
September 12, 2007 19:18 Book:- Trig or Treat (9in x 6in) 10˙less-easy Level-Two-Games 319 This sin 2A equation is one of the amazing equations in Trigonometry where a slight difference (from minus to plus sign in the denominator) changes the tangent identity for the double angle to the sine identity for the same double angle.
September 12, 2007 19:19 Book:- Trig or Treat (9in x 6in) 13˙about-author About the Author Dr Y E O Adrian graduated from the University of Singapore with first class Honours in Chemistry in 1966, and followed up with a Master of Science degree in 1968. He received his Master of Arts and his Doctor of Philosophy degrees from Cambridge University in 1970, and did post-doctoral research at Stanford University, California. For his research, he was elected Fellow of Christ's College, Cambridge and appointed Research Associate at Stanford University in 1970. His career spans fundamental and applied research and development, academia, and top appointments in politics and industry. His public ser- vice includes philanthropy and sports administration. Among his numer- ous awards are the Charles Darwin Memorial Prize, the Republic of Singapore's Distinguished Service Order, the International Olympic Committee Centenary Medal, and the Honorary Fellowship of Christ's College, Cambridge University. |
TCV.8.EE.2 (2) Evaluate square roots of small
perfect squares and cube roots of small perfect cubes.
TCV.8.EE.2 (3) Know that the square root of any
number that is not a perfect square is irrational (√2
is irrational)
TCV.8.EE.3 (1) Express large and small numbers in
scientific notation
TCV.8.EE.3 (2) Use scientific notation to compare
two quantities
TCV.8.EE.4 (1) Perform operations with numbers
expressed in scientific notation, including problems where both
decimal and scientific notation are used
TCV.8.EE.4 (2) Use scientific notation and choose
units of appropriate size for measurements of very large or very small
quantities
MCC8.EE.7a. Give examples of linear equations in
one variable with one solution, infinitely many solutions, or no
solutions. Show which of these possibilities is the case by
successively transforming the given equation into simpler forms, until
an equivalent equation of the form x=a, a=a, or a=b results (where a
and b are different numbers)
MCC8.EE.7b Solve linear equations with rational
number coefficients, including equations whose solutions require
expanding expressions using the distributive property and collecting
like terms
TCV.8.NS.1 Distinguish between rational and
irrational numbers and the difference in their decimal expansions
TCV.8.NS.2 Use rational approximations of
irrational numbers to compare the size of irrational numbers, locate
them approximately on a number line diagram, and estimate the value of
expressions
Prerequisites: As identified by the
CCGPS Frameworks
computation with whole numbers and decimals,
including application of order of operations |
This complete course in Calculus for beginners is one of MIT's OpenCourseWare offerings. It includes nearly a dozen Java...
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This complete course in Calculus for beginners is one of MIT's OpenCourseWare offerings. It includes nearly a dozen Java applets to illustrate some of the concepts covered; there is a corresponding set of Flash applets with accompanying audioAn engaging basic calculus website. Two noteworthy resources are Flash tutorials and an archived Problem of the Week...
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An engaging basic calculus website. Two noteworthy resources are Flash tutorials and an archived Problem of the Week section. The tutorials explain elementary calculus concepts and techniques. Each Problem of the Week is rated by difficulty with an accompanying detailed solution. Other features of note include the yearly Superbowl of High School Calculus, Calculus & Music, and calculus "Cheat Sheets."
This site contains the learning materials for eleven online courses that are currently taught at Carnegie Mellon...
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This site contains the learning materials for eleven online courses that are currently taught at Carnegie Mellon University. Students anywhere may use these materials as learning resouces. Instructors at other learning institutions may use and even base their own courses on these materials free of charge. An instructor can create an account on OLI, select and sequence course materials, and take advantage of OLI's tracking of student progress. While all courses come in the "open and free" version, at least one course (Logic and Proofs) also comes in a version that requires payment of a fee to OLI. The available couses at this time are: Biology, Calculus, Causal Reasoning, Chemistry, Economics, Emperical Research Methods, French, Logic and Proofs, Physics, Statistics, and Statics and SeriesThe University of California, Irvine Extension, supported by generous grants from the |
This is an informative app consisting of interesting facts, tricks and tips in mathematics. Must have for any school/college going student. It'll help you solve calculations quickly. You can learn tricks in mathematics. Impress your friends with this new knowledge about fun facts in maths.
We have searched for this information from various sources and putting it in a single app, so that you can get it all in one place only. This saves your time to find this info yourself. We will be adding few more stuff as we find it. If you have any suggestions, or something to add in the app, mail us at manas.gajare@dexterltd.com
We have developed many tools for students having mathematics as one of the subjects. e.g. Differentiate It, Numerical Calculus, Interpolation Calculator, Graph Plotter, Equation solver, Matrix Calculator..do check them out if you are interested tools |
books.google.ca - This text intends to provide the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. The geometry of surfaces is an ideal starting point... of Surfaces |
CK-12 Basic Geometry Concepts
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Description
CK-12's Basic Geometry FlexBook is designed to present students with geometric principles in a simpler, more graphics-oriented course. Students will explore geometry at a slower pace with an emphasis placed on visual aids and approachability. |
Mathematics is all around us. We rarely find a place where mathematics is not applied. It not helps us in interpreting and solving many problems but also trains mind to handle abstractness. At the level of class IX and X the syllabi focus on these two comparing quantities, statistics, probability, A.P. geometrical and algebraic concepts have been introduced to make the learner capable of applying these concepts in problems of day to day life. In addition to these topics many other topics like extended number systems, trigonometry etc. have also been included in order to prepare learners to take up undergraduate mathematics after passing class X. Keeping these goals in mind the chapters in Xam Idea have been designed. Class X has another significant importance, that is, the students has to face first public examination to have a certificate. In most of the schools, the terminal examinations upto class IX are generally conducted by schools only. But at the level of class X, students has to attempt a general question paper for which the evaluation is done by external examiners. The students require proper training and skill to handle such situations. Keeping in view all the other requirements. The textual material, examples and exercises have been presented in Xam Idea which not only help in developing conceptual understanding of various topics, but also imparts skills to solve a question as per the requirement of the board examinations many imported problems of your textbooks have been solved and similar problems have been presented in exercise. At this stage many students are at the verge of deciding the career which they want to up for their future. Many of you have already started preparing for competition examinations to achieve your goals.
The Xam Idea provides you execellent support, to prepare yourself for forthcoming challenges in life. For this purpose graded examples and exercises have been presented for such brilliant students in the form of "Purging the Puzzles" and Higher Order Thinking Skills (HOTS).
This category of problems contain slightly different problems which really force learners for brain learning.
If you struck up some where while solving these problems or may be from your books, online support of national and international mathematics experts is provided by Xam Idea.
In nutsher we can say that this concept is suitable for both types of students (i) preparing for examinations and (ii) preparing for competition examinations.
You can now start your journey to explore many new ideas for class X mathematics through on line Xam Idea. |
״TouchCalc is a comfortable calculator program and offers several different modes.- The scientific mode offers all the usual...
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״TouchCalc is a comfortable calculator program and offers several different modes.- The scientific mode offers all the usual functions and operations like the basic arithmetical operations, power, logarithm, roots, trigonometry etc.- The bit/integer mode offers logical operations (AND, OR, XOR, >>, etc.) on bit level. All values can be displayed as binary, octal, decimal or hexadecimal numbers. Calculations can be done in 8, 16 32 or 64 bits.- In the statistics mode you can create a sample by adding several values and then calculate mean, median, quantil values, variance, standard deviation, range etc.Colorized keys will help to distinguish between the different categories of the functions and operations.״This is a free app
'Taking calculus? Then you need the Wolfram Calculus Course Assistant. This definitive app for calculus--from the world...
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'This app covers the following topics applicable to Calculus, AP Calculus AB, AP Calculus BC, Calculus I, and Calculus II:- Evaluate any numeric expression or substitute a value for a variable.- Plot basic, parametric, or polar plots of the function(s) of your choice.- Determine the limit of a function as it approaches a specific value.- Differentiate any function or implicit function.- Find the critical points and inflection points of a function.- Identify the local and absolute extrema of a function.- Integrate a function, with or without limits.- Sum a function given a lower and upper bound.- Find the closed form of a sequence or generate terms for a specific sequence.'This app costs $3.99
'Taking multivariable calculus? Then you need the Wolfram Multivariable Calculus Course Assistant. This definitive app for...
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'Taking multivariable calculus? Then you need the Wolfram Multivariable Calculus Course Assistant. This definitive app for multivariable calculus--from the world leader in math software--will help you work through your homework problems, ace your tests, and learn calculus concepts. Forget canned examples! The Wolfram Multivariable Calculus Course Assistant solves your specific multivariable problems on the fly, providing step-by-step guidance for limits, derivatives, integrals, and much more.This app covers the following topics applicable to Multivariable Calculus, Advanced Calculus, and Vector Calculus:- Evaluate any numeric expression, or substitute a value for a variable- Plot 2D or 3D functions of your choice- Determine the limit of a function as it approaches a specific value or values- Differentiate any single or multivariable function- Find the critical points and saddle points of a function- Calculate the gradient of a function- Identify the local extrema of a function- Find the single, double, or triple integral of a function- Determine the dot or cross product of two vectors- Calculate the divergence or curl of a vector fieldStay up to date with the latest version, and see the additions of directional derivatives, line integrals, surface integrals, arc length, and curvature!'This app costs $4.99 |
Heart of Mathematics, Manipulative Kit (New) - 3rd edition
Summary: An excellent resource for the general reader, the third edition succeeds at uncovering the mathematics inherent in the world around them in an accessible way. It introduces them to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking. The emphasis is placed on mathematical methods of investigation as well. Mindscape exercises are also included for the development of application, problem-solving, and argumentation ski...show morells. Visualizations techniques are also integrated throughout the book to make key concepts easier for the average reader to understand. ...show less
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You are bidding on a text book / study guide:Collins - GCSE Maths 2 tier-higher for AQA B it is as new condition - no rips, tears, highlighting or turned corners (in fact it's hardly been read as it w...
AQA GCSE Mathematics Higher revision book - 112 pages of all you need maths revision textbook. This has been used, there's some light wear on the cover including curled corners and a few light marks -... |
A unit about economics and the family structure, designed to give students knowledge about family finances (opening a checking or savings account) and to improve their math skills for use in everyday life.
The purpose of this study was to examine the factors that effect actual construction time in the context of food sector projects in Bangladesh. One of the factors of particular interest is political unrest. The data for...
Back in 1968, Professor Joe Konhauser at Macalester College started a tradition by creating a math problem for his students every week. Since that time, this long-standing tradition has migrated to the web, and along...
Created by the Cornell University Center for Hospitality Research and managed by Sage Publications, this e-journal provides ?applied research theories about important industry trends and timely topics in lodging, r... |
Featured Research
from universities, journals, and other organizations
Math goes viral in the classroom
Date:
December 14, 2009
Source:
University of Alberta
Summary:
At least a dozen Alberta high-school calculus classrooms were exposed to the West Nile virus recently. Luckily, it wasn't literally the illness. Educators used the virus as a theoretical tool when they designed materials for use in an advanced high-school math course.
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At least a dozen Alberta high-school calculus classrooms were exposed to the West Nile virus recently.
Luckily, however, it wasn't literally the illness. University of Alberta education professor Stephen Norris and mathematics professor Gerda de Vries used the virus as a theoretical tool when they designed materials for use in an advanced high-school math course. The materials allow students to use mathematical concepts learned in their curriculum to determine the disease's reproductive number, which determines the likelihood of a disease spreading.
The approach is a marriage of science and math, subjects the researchers say seem to exist in separate worlds at a secondary-school level, but that when brought together can effectively bring real-world scenarios into the classroom to enhance learning and understanding.
Not to mention answering that ages old high-school student question: "why do I need to know this?"
"This piece was designed to satisfy an optional unit in Math 31 (Calculus), for which there are no materials, so we said, 'let's fill the gap,'" said Norris. "These materials show a real application of mathematics in the biology curriculum for high-school students."
Norris and de Vries chose a published academic math paper on the transmission of the West Nile virus and modified it -keeping the science intact, but making it readable and practical for high-school calculus students.
The information and equations in the original paper dealing with disease transmission were then used as the basis for calculus math problems to be solved by the students. Students were presented with a variety of materials that covered topics and concepts such as rate of change, exponential growth-decay models, and models for the carriers of the virus, including mosquitoes and infectious and susceptible birds. The students' mathematical skills were then put to use in determining the spread of the disease using various parameters, which included variables such as biting rate and the probability of infection.
Norris underlines that the project challenged the students to see and understand science in a different fashion from what they learn inside the science curricula. He points out that high-school classroom scientific experiments are "proven" science and have been around for at least 300 years, in many cases. For the students to discover that real scientists often work with some assumptions that they know to be false in order to reach their conclusions was certainly an eye-opening realization for them, he says.
"There's no way out of the fact that the knowledge you gain from science is imperfect; it's tentative and subject to change," said Norris. "I think that's what struck the students between the eyes."
Both researchers agree that this form of collaborative, interdisciplinary learning can take place across all subject areas. De Vries and Norris are currently working on another project that focuses on population genetics that will fit into Grade 12 biology and math courses.
"It's mathematics in the real world. Kids are always asking, 'why am I learning this,'" she said. "All of a sudden the mathematics that kids have learned comes together in a project like thisOct. 11, 2013 — Writing instruction in US classrooms is "abysmal" and the Common Core State Standards don't go far enough to address glaring gaps for students and teachers, an education scholar journals, and research organizations. |
Math Analysis Honors
Course Description Math Analysis Honors is an accelerated course meant to serve as an extension of Algebra 2. Topics include functions, series, sequences, matrices, complex numbers, conic sections, polar and parametric equations, vectors, applications of trigonometry, and an introduction to Calculus. Since this course is designed to prepare students for Calculus, the focus will be on problem solving using mathematical models to represent real world situations. Scientific and graphing calculators are used as tools to assist in the development of concepts. |
Unlike most engineering maths texts, this book does not assume a firm grasp of GCSE maths, and unlike low-level general maths texts, the content is tailored specifically for the needs of engineers. The result is a unique book written for engineering students, which takes a starting point below GCSE level. Basic Engineering Mathematics is therefore ideal for students of a wide range of abilities, and especially for those who find the theoretical side of mathematics difficult.All students taking vocational engineering courses who require fundamental knowledge of mathematics for engineering and do not have prior knowledge beyond basic school mathematics, will find this book essential reading. The content has been designed primarily to meet the needs of students studying Level 2 courses, including GCSE Engineering and Intermediate GNVQ, and is matched to BTEC First specifications. However Level 3 students will also find this text to be a useful resource for getting to grips with the essential mathematics concepts needed for their study, as the compulsory topics required in BTEC National and AVCE / A Level courses are also addressed.The fourth edition incorporates new material on adding waveforms, graphs with logarithmic scales, and inequalities – key topics needed for GCSE and Level 2 study.JohnBird's approach is based on numerous worked examples, supported by 600 worked problems, followed by 1050 further problems within exercises included throughout the text. In addition, 15 Assignments are included at regular intervals. Ideal for use as tests or homework, full solutions to the Assignments are supplied in the accompanying Instructor's Manual, available as a free download for lecturers from
In this book JohnBird and Carl Ross introduce mechanical principles and technology through examples and applications - enabling students to develop a sound understanding of the principles needed by professional engineers and technicians. No previous background in engineering is assumed and theoretical concepts are supported by over 600 problems and worked examples.This completely new text is designed to match a wide range of pre-degree courses, and provide an accessible introduction for undergraduates with no previous background in engineering studies. The authors have ensured syllabus-match for the leading UK courses at this level: AVCE optional units Mechanical Engineering Principles and Further Mechanical Engineering Principles, and the new BTEC National unit: Mechanical Principles.
02.07.2012 [Telegraph.co.uk] - It got national recognition and, on the back of that, I won a scholarship to Cambridge. In my first week I got a knock on the door from JohnBird, who was one of the big directors at Cambridge. He put me in a play in which I was miscast badly. I was so
20.06.2012 [The Nation. (blog)] - Distinguished Mark Twain scholars Tom Quirk and JohnBird wrote to bemoan the damage that the closing of the press's Mark Twain and His Circle series will do to Twain scholarship. All ten of the editors of the Collected Works of Langston Hughes issued
14.06.2012 [LoughboroughEcho.net] - ... and every single colleague helping with the team challenge days and the Collect 4 Christmas appeal. JohnBird, managing director of ThankQ, said: "Every single person in the company has been chivvied, cajoled and press-ganged into helping!"
20.06.2012 [Weekly Times Now] - He replaces JohnBird, who left the company in March after 14 years as chief executive. Send your horticulture news to Sandra Godwin, The Weekly Times, PO Box 1728, Swan Hill 3585, email godwins@theweeklytimes.com.au or fax to (03) 5032 9532.
JohnBird is a director/cinematographer whose work includes independent films, music videos, fashion and documentaries.
He's traveled worldwide and won numerous awards. Click on the thumbnails to view a gallery of his work. |
Graphing
Author:
Unknown
ISBN-13:
9780768202342
ISBN:
0768202345
Publisher: Schaffer Publications, Frank
Summary: Help students succeed in math! Math Minders provide students with the self-confidence they need to succeed in math. Students learn one step at a time, reviewing skills learned in earlier grades, then moving to skills appropriate for their grade level. They progress gradually, giving them the constant feeling of success! Vocabulary is kept at a level appropriate for each grade level to help ensure success. Fun and sim...ple formats help maintain a high level of student interest. Perfect for home or school, or to reinforce any existing math program copy that has been read, but remains in excellent condition. Pages are intact and are not marred by notes or highlighting. The spine remains undamaged. We make every effort [more]
A copy that has been read, but remains in excellent condition. Pages are intact and are not marred by notes or highlighting. The spine remains undamaged. We make every effort to identify defects, but as we're only human, we do occasionally miss one. If you get a problem item be sure to contact us.[less] |
Complete Book of Algebra and Geometry Grades 5-6
9780769643304
ISBN:
0769643302
Publisher: Carson-Dellosa Publishing, LLC
Summary: The Complete Book of Algebra and Geometry offers children in grades 5-6 easy-to-understand lessons in higher math concepts, skills, and strategies. This best-selling, 352 page workbook teaches children how to understand algebraic and geometric languages and operations. Children complete a variety of activities that help them develop skills and then complete lessons that apply these skills and concepts to everyday sit...uations. Including a complete answer key this workbook features a user friendly format perfect for browsing, research, and review. Basic Skills Include: -Order of Operations -Numbers -Variables -Expressions -Integers -Powers -Exponents -Points -Lines -Rays -Angles -Area Over 4 million in print! The best-selling "Complete Book series" offers a full complement of instruction, activities, and information about a single topic or subject area. Containing over 30 titles and encompassing preschool to grade 8 this series helps children succeed in every subject area!
Carson-Dellosa Publishing Staff is the author of Complete Book of Algebra and Geometry Grades 5-6, published under ISBN 9780769643304 and 0769643302. Seventy Complete Book of Algebra and Geometry Grades 5-6 textbooks are available for sale on ValoreBooks.com, twenty eight used from the cheapest price of $10.83, or buy new starting at $46.86.[read more]
Ships From:Charlotte, NCShipping:Standard, ExpeditedComments:Book is completely readable and ready to use. Extremely FAST shipping with FREE tracking. To help... [more] [.[less] |
calculus
The branch of mathematics that deals with (1) the rate of change of quantities
(which can be interpreted as the slopes of curves), known as differential
calculus, and (2) the length, area, and volume of objects, known
as integral calculus. It can be seen as an extension of
analytical geometry.
Calculus was one of the most important developments in mathematics and also
in physics, much of which involves studying how quickly one quantity changes
with respect to another. It is no coincidence that one of the founders of
calculus was the brilliant English physicist Isaac Newton;
another was Gottfried Leibniz. William Fogg
Osgood (1864–1943) said: "The calculus is the greatest aid we have
to the application of physical truth in the broadest sense of the word."
Although students nowadays learn the differential calculus first, the integral
calculus has older roots.
Differential calculus
Consider the functiony = f(x).
This may be plotted as a curve on a set of
Cartesian coordinates. Assuming
that the curve is not a straight line, tangents
to it at different points will have different gradients.
Two points close together on the curve, (x, y) and (x+δx,
y + δy), where δx and δy
mean very small distances in the x- and y-directions,
will usually have tangents of similar, though not identical, gradients.
The gradient of the line passing through these two points is given by
[(y + δy) - y]/[(x
+ δx) - x],
and is the same as that of a tangent to the curve somewhere between the
two points. The smaller δx is, the closer the two gradients
will be; and if δx is infinitely small, they will be identical.
This limit as x
0 is the derivative
dy/dx
or f '(x)
of the function and is given by (putting f(x) in place
of y):
f'(x) = lim (δx0)
[f(x + δx) - f(x)]
/ δx.
This formula will give the gradient of the curve for f(x)
at any value of x. For example, in the parabolay = x 2 the derivative at x = a
is given by
Hence we can say that f '(x) = 2x for f(x)
= x 2. This process is termed differentiation.
In general if f(x) = x 2 then f '(x) = z.xz - 1, and the
second derivative
f ''(x) or d 2x/dx
2,
the result of differentiating again, is (z - 1).z.x
z - 2, and so on.
Integral calculus
The derivative of f (x) gives the instantaneous rate of
change of f(x) for a particular value of x. Now,
consider the plot of y = f '(x), and assume that
f '(x) = f(x) = 0 when x = 0. A very
thin strip with one vertical side the f '(x)-axis and
the other the line joining the points (δx, 0) and (δx,
f '(δx)), will have an area of approximately δx,
f '(δx). A second thin strip drawn next to it will
have an area of roughly δx, f '(2δx),
and so on. The area between the curve and the x-axis from x
= 0 to x = a will therefore be roughly δx.f
'(δx) + δx.f '(2δx)
+ ... + δx.f '(a + δx) +
δx.f '(a). If one plots on a different
graph x against (area of strip 1), (area of strips 1 + 2), area
of strips 1 + 2 + 3), etc., one finds that one is plotting a close approximation
to y = f(x). This reverse of differentiation
is called integration, and f (x) is the integral of f
'(x). We find, too, that the integral of x z
is
x z + 1 / (z + 1)
which is what one would expect. This is the indefinite integral of x
z since we have not specified how much of the curve we wish to
consider, and we must add a constant, c, since the derivative of
any constant is 0. As integration is the sum of the areas of the strips
described, we symbolize it by an extended S. For example, the integral from
0 to 1 of x 2 is written as |
Books
Trigonometry
-- More than 44,000 copies sold of second edition-- More than 230,000 students are enrolled in trigonometry courses-- Required study for all mathematics majors-- Hundreds of practical problems solved step by step-- Complements most popular textbooks
Student's Solution Manual Complete, worked-out solutions are given for odd-numbered exercises and chapter review exercises and all chapter test exercises in a volume available for purchase by students. In addition, a practice chapter test and cumulative review exercises are provided for each chapter. Videotapes A new videotape series has been developed to accompany Trigonometry, Sixth Edition. In a separate lesson for each section of the book, the series covers all objectives, topics, and problem-solving techniques within the text. Interactive Mathematics Tutorial Software with Management System is an innovative package that is objective-based, self-paced, and algorithm driven to provide unlimited opportunity for review and practice.
A modern approach to college algebra and right-triangle trigonometry is supported by optional graphics calculator material. The accessible writing style, and realistic applications from a variety of disciplines reinforce the student-oriented approach of this text. "Graphics Calculator Alert" boxes and "Graphics Calculator Power User's Corner" boxes, along with graphing calculator exercises show students how to get the most out of technology. A split-screen format for algebraic procedures, rules, and properties offers simultaneous examples of the mathematical concepts to facilitate students understanding. "Progress Checks" follow most examples and enable students to assess their comprehension. Exceptional explanation in Chapter 2 (Sections 2.2 and 2.4) on how to solve word problems help students master this important skill. "Warnings" reinforce good mathematical habits by pointing out incorrect practices most commonly found in homework and on exams.
This text aims to teach students to view questions from various perspectives, analyze problems carefully, reformulate problems in more familiar terms, and recognize that most mathematical problems require significantly more thinking than writing. |
Subject: Agriculture, Food, and Natural Resources (9 - 12), or Mathematics (9 - 12) Title: Determining Percent of Seed Germination-Enhancing mathematics in the career/technical classroom and providing relevance in the mathematics classroom Description: In this project, students will conduct an experiment to determine the percentage of seed that will germinate in a given time frame. Students will produce charts and graphs to represent the results algebraically. Mathematics and career/technical students can participate in this lab experiment together, or the activity can be conducted separately in either classroom. The mathematics teacher will want to adapt the student handouts to provide additional emphasis on mathematics and to remove emphasis on the career/technical.
This integrated lesson is the result of collaboration between Chip Blanton, an agriculture teacher, and Greg Pendergrass, a math teacher (Fort Payne HS). For information about using the eight step model for developing integrated projects developed by the Southern Regional Education Board, contact Leslie Carson at leslie.carson@sreb.org. For information about the project implementation, contact Chip Blanton (cblanton@ftpayk12.org) or Greg Pendergrass (gpendergrass@ftpayk12.org).
Subject: Mathematics (9 - 12) Title: Systems of Linear Inequalities Project Description: The systems of linear inequalities project was designed to be used in an Algebra IB class after a preliminary lesson on systems of linear inequalities. The project is to be graded per group based on the work completed and presentation to the class. Each group is required to use a graphing calculator in its presentation.This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation. Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: Math is Functional Description: This lesson is a technology-based activity in which students extend graphing of linear functions to the use of spreadsheet software. After students have become proficient in constructing a table of values, students are able to efficiently graph equations with more extensive computational requirements. Furthermore, inquiry and discovery about slope and y-intercept will help students conceptualize material normally presented in Algebra I textbooks.
Subject: Mathematics (9 - 12), or Technology Education (9 - 12) Title: You Mean ANYTHING To The Zero Power Is One? Description: This lesson is a technology-based project to reinforce concepts related to the Exponential Function. It can be used in conjunction with any textbook practice set. Construction of computer models of several Exponential Functions will promote meaningful learning rather than memorization.
Subject: Mathematics (6 - 12) Title: Swimming Pool Math Description: Students will use a swimming pool example to practice finding perimeter and area of different rectangles.
Thinkfinity Lesson Plans
Subject: Mathematics Title: Automobile Mileage: Years Since 1990 vs. MileageAdd Bookmark Description: This lesson, one of a multi-part unit from Illuminations, is another representation of the automobile mileage data used in previous lessons of this unit. This lesson provides another opportunity for students to analyze how changing the independent variable in a set of data can result in a different least squares regression line. Students can then use the new equation to make some of the same predictions they made in Lessons Five and Six. This lesson incorporates an interactive regression line applet. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Counting Embedded FiguresAdd Bookmark Description: In this Illuminations lesson, students look for patterns in an embedded-square problem. After looking at the patterns, students form generalizations for the pattern. This activity sharpens students algebraic thinking and visualization skills. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Subject: Health,Mathematics Title: Make a Conjecture Add Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students explore rates of change and accumulation in context. They are asked to think about the mathematics involved in determining the amount of blood being pumped by a heart. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Investigating Pick's TheoremAdd Bookmark Description: In this unit of three lessons, from Illuminations, students rediscover Pick's Theorem, which they were likely introduced to in middle school, and use algebra to determine the coefficients of the equation. They explore the concept of change as a mechanism for finding the coefficients of Pick's Theorem. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Graph Chart Add Bookmark Description: Using this reproducible chart, from an Illuminations lesson, students record the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. This resource can also be used as an overhead transparency so students can share their results with their classmates. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics,Science Title: Inclined PlaneAdd Bookmark Description: In this multiple-day activity, from Illuminations, students time balls rolling down inclines of varying lengths and heights. They then try to make inferences about the relationships among the variables involved. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Subject: Mathematics Title: Flowing Through MathematicsAdd Bookmark Description: This student interactive, from Illuminations, simulates water flowing from a tube through a hole in the bottom. The diameter of the hole can be adjusted and data can be gathered for the height or volume of water in the tube at any time. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Automobile Mileage: Comparing and ContrastingAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students compare and contrast their findings from previous lessons of the unit. This lesson allows students the time they need to think about and discuss what they have done in the previous lessons. This lesson provides the teacher with another opportunity to listen to student discourse and assess student understanding. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Traveling DistancesAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students interpret the meaning of the slope and y-intercept of a graph of real-life data. By examining the graphical representation of the data, students relate the slope and y-intercept of the least squares regression line to the real-life data. They also interpret the correlation coefficient of the resulting least squares regression line. This lesson incorporates an interactive regression line applet. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Exploring Linear DataAdd Bookmark Description: In this lesson, from Illuminations, students model linear data in a variety of settings. Students can work alone or in small groups to construct scatterplots, interpret data points and trends, and investigate the notion of line of best fit. Thinkfinity Partner: Illuminations Grade Span: 6,7,8,9,10,11,12
Subject: Mathematics Title: The Effects of OutliersAdd Bookmark effect of outliers on a regression line and easily see their significance. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Bathtub Water LevelsAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students examine real-life data that illustrates a negative slope. Students interpret the meaning of the negative slope and y-intercept of the graph of the real-life data. By examining the graphical representation of the data, students relate the slope and y-intercept of the least squares regression line to the real-life data. They also interpret the correlation coefficient of the least squares regression line. This lesson incorporates an interactive regression line applet. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Apple Pie Recording ChartAdd Bookmark Description: This reproducible activity sheet, from an Illuminations lesson, prompts students to use strings and rulers to measure and record the distance around several round objects, as well as the distance across the middle of those objects. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Subject: Mathematics Title: Graph ChartAdd Bookmark Description: This reproducible transparency, from an Illuminations lesson, contains the answers to the similarly named student activity in which students identify the independent and dependent variables, the function, symbolic function rule and rationale for a set of graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Think of a Graph Add Bookmark Description: This reproducible transparency, from an Illuminations lesson, asks students to sketch a graph in which the side length of a square is graphed on the horizontal axis and the perimeter of the square is graphed on the vertical axis. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Least Squares RegressionAdd Bookmark Description: In this nine-lesson unit, from Illuminations, students interpret the slope and y-intercept of least squares regression lines in the context of real-life data. Students use an interactive applet to plot the data and calculate the correlation coefficient and equation of the least squares regression line. These lessons develop skills in connecting, communicating, reasoning, and problem solving as well as representing fundamental ideas about data. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Escape from the Tomb Activity Add Bookmark Description: This reproducible activity sheet, from an Illuminations lesson, includes instructions and questions for a mathematical adventure game. In the game, students are given a problem in which two bowls are suspended from the ceiling by springs, and one bowl is lower than the other. Students must work out how many items should be placed in each bowl so that the heights of the bowls are the same. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics Title: Graphing What Add BookmarkSubject: Mathematics Title: Automobile Mileage: Age vs. MileageAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students plot data about automobile mileage and interpret the meaning of the slope and y-intercept of the least squares regression line. By examining the graphical representation of the data, students analyze the meaning of the slope and y-intercept of the line and put those meanings in the context of the real-life application. This lesson incorporates an interactive regression line applet. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Subject: Mathematics,Science Title: Finding Our Top SpeedAdd Bookmark Description: This Illuminations lesson sets the stage for a discussion of travel in the solar system. By considering a real-world, hands-on activity, students develop their understanding of time and distance. The mathematics necessary for the lesson relate to measuring time and distance as well as graphing to portray the data collected. Thinkfinity Partner: Illuminations Grade Span: 6,7,8
Subject: Mathematics Title: Gallery WalkAdd Bookmark Description: In this lesson, one of a multi-part unit from Illuminations, students view the work of other students in the class and explain their own work. Students move from graph to graph during this time and, without any talking, view the work of their classmates. While on their Gallery Walk, the students think about the data their classmates plotted and whether or not the information accompanying each graph seems appropriate. Each pair of students then stands next to their own work and explains one of the graphs. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12 |
for Welders
Math for Welders is a combination text and workbook designed to help welding students learn and apply basic math skills. The basic concept behind each math operation is explained at the opening of the unit. Next, students are given clear instruction for performing the operation. Each unit includes a variety of weldingrelated practice problems to reinforce what the students have learned. The practice problems are identical to the types of problems the students will be required to solve in a welding shop. In addition to teaching basic math concepts, the problems give students a preview of the types of challenges they will face in a work environment. This helps the students develop solid troubleshooting skills that will serve them throughout their careers as weld |
and Trigonometry: A Unit Circle Approach With an emphasis on problem solving and critical thinking, Mark Dugopolski's College Algebra and Trigonometry: A Unit Circle Approach, Sixth Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find carefully placed learning aids and review tools to help them do the math. 0321916492 / 9780321916495 College Algebra and Trigonometry: A Unit Approach Plus NEW MyMathLab with Pearson eText -- Access Card Package Package consists of: 0321431308 / 9780321431301 MyMathLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321916522 / 9780321916525 College Algebra and Trigonometry: A Unit Circle Approach |
Discrete Mathematics
9780130890085
ISBN:
0130890081
Edition: 5 Pub Date: 2000 Publisher: Prentice Hall PTR
Summary: For one or two term introductory courses in discrete mathematics. This best-selling book provides an accessible introduction to discrete mathematics through an algorithmic approach that focuses on problem- solving techniques. This edition has woven techniques of proofs into the text as a running theme. Each chapter has a problem-solving corner that shows students how to attack and solve problems.
Johnsonbaug...h, Richard is the author of Discrete Mathematics, published 2000 under ISBN 9780130890085 and 0130890081. Thirty Discrete Mathematics textbooks are available for sale on ValoreBooks.com, twenty seven used from the cheapest price of $2.60, or buy new starting at $40.43 |
This concise article of twenty pages takes you on a short tour on how to solve algebraic equations using MATLAB. The presentation covers both numerical and symbolic (analytical) solution of equations. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach."
This concise article of forty pages takes you on a short tour on how to handle matrices using MATLAB. Topics covered include how to generate matrices in MATLAB, different operations on matrices, and how to handle matrix, vector, and scalar quantities. Various MATLAB functions associated with matrices are also explored.
This concise article takes you on a short tour on how to plot graphs using MATLAB. The presentation covers both two-dimensional and three-dimensional graphs. This article is taken form the bestselling book "MATLAB for Beginners: A Gentle Approach." |
Algebra for College Students - 9th edition
Summary: Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; practice the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundamental problem solving skills necessary for future mathematics c...show moreourses. Algebraic ideas are developed in a logical sequence, and in an easy-to-read manner, without excessive vocabulary and formalism. The open and uncluttered design helps keep students focused on the concepts while minimizing distractions. Problems and examples reference a broad range of topics, as well as career areas such as electronics, mechanics, and health, showing students that mathematics is part of everyday life94 +$3.99 s/h
Good
Bernie Hornbuckle AL Fort Myers, FL
2010-01-01 Hardcover Good Super Buy! --Clean and Neat, but may contain some markings/highlighting--Tight binding, but the cover is worn--Choose priority (exped. ) shipping for quickest delivery--Bu...show morey with confidence, our feedback speaks for itself! ...show less
$12.664622.80 |
Derivatives and integrals of trigonometric, exponential, and logarithmic functions are studied, with an introduction to differential equations. Special integration techniques, including substitution, integration by parts and partial fractions are studied. After reviewing the basic concepts of Geometry, including lines, angles, and triangles, the focus of the course is proof. |
CollegeClearly written and comprehensive, the ninth edition of Gustafson and Frisk's popular book provides in-depth and precise coverage, incorporated into a framework of tested teaching strategy. The authors combine carefully selected pedagogical features and patient explanation to give students a book that preserves the integrity of mathematics, yet does not discourage them with material that is confusing or too rigorous. Long respected for its ability to help students quickly master difficult problems, this book also helps them develop the skills they'll need in future courses and in everyday life. |
More About
This Textbook
Overview
The power that analysis, topology and algebra bring to geometry has revolutionised the way geometers and physicists look at conceptual problems. Some of the key ingredients in this interplay are sheaves, cohomology, Lie groups, connections and differential operators. In Global Calculus, the appropriate formalism for these topics is laid out with numerous examples and applications by one of the experts in differential and algebraic geometry. Ramanan has chosen an uncommon but natural path through the subject. In this almost completely self-contained account, these topics are developed from scratch. The basics of Fourier transforms, Sobolev theory and interior regularity are proved at the same time as symbol calculus, culminating in beautiful results in global analysis, real and complex. Many new perspectives on traditional and modern questions of differential analysis and geometry are the hallmarks of the book. The book is suitable for a first year graduate course on Global |
Piscataway PrealgebraUnlike in earlier basic math classes, the problems usually have more than one step. Beginning students often run into trouble when they try to skip steps, or forget which order to do the steps in. Mastering algebra requires doing lots of problems, often with someone prompting you as to what step to do in which order |
one of the worst teachers you could possibly take. You can't understand a word he says and he shows up late to every class. You have to teach yourself everything in the class. He should retire as soon as he can.
Where do I begin with him. For one im in class right now typing this review because I have nothing else to do. He doesnt follow the book at all and just goes off on whatever he feels like. YOU MUST TEACH YOURSELF so be prepared. He seems like a good guy but he should definitely retire immediately, make that he should have retired years ago.
He can be very unclear and often teaches things in the most round-about difficult way possible. He would often explain Algebra using reasoning from a higher level course. His mannerisms are both cute and annoying. Try to teach yourself the material before coming to class. I would never really grasp the concept until I went home and taught myself.
The class is hard and the material doesn't come naturally. But he does do a good job explaining the concepts. And he's willing to help anyone, who's struggling, when they attend his office hours. He might come off as craggy but if you put in the work, you'll do fine.
cutest man alive. "never do my fingers leave my hands". If you pay attention in class, or even just GO TO CLASS and pay the slightest bit of attention, you will pass the tests. Hes really old and has a lot of experience.
You need to read to book before going to class so that you understand what he is talking about because he is not always clear. Can go off on tangents about certain things that you do not need to know. Helpful if you have questions...he's pretty funny too |
Chapter 1. Math Review
Chapter 1. Math Review
Basic math skills are needed in order to calculate everything from personal checkbook balances to entries on business financial statements. This chapter is a review of the basic math skills that are utilized throughout this book. If needed, you can work through all of Chapter 1 to prepare for the math you will encounter. You may find that you only need to brush up on a few of the skills in this chapter, or you may prefer to come back to this chapter as you work through other chapters and encounter skills on which you need a refresher. |
Basic Operations
Basic operations are the building blocks and rules of math. They are like learning the rules of the road in Driver's Ed. You will be expected to know these basics down cold. We'll make sure that you do.
Pre-Algebra
Pre-algebra is the math that will help guide you through the rest of your life, when shopping, watching sports, managing a budget, and just chillin'. Honestly, this math will probably be more essential to your everyday life than calculus, so drink it up and let it soak in. |
This module deals with the mathematical elements essential to understanding physics courses, namely the study of real functions, derivation and integration of a function with one and several real variables, the development of a function, some elements of numerical calculations and, finally, solving a system of linear equations. Learning activities of different difficulty levels are developed with formal assess- ments. Moreover, online word and useful links enable students to study certain topics in detail. Finally, the students will also be able to use software such as "Microsoft Excel 2000" and "Maxima". |
Kirk Trigsted revolutionized the way this course is taught when he created College Algebra, a completely clickable eText 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 Algebra & Trigonometry, Trigsted continues to innovate with a revised design that improves navigation and usability, expanded videos, and increased animation coverage. All multimedia elements, exercises, feedback, and content are written by the author himself so that the students have a consistent voice throughout.
Description:
Dwyer and Gruenwald's COLLEGE ALGEBRA AND TRIGONOMETRY, SECOND EDITION, is
a graphing calculator dependent text with a contemporary flair. Building upon a traditional table of contents, the authors integrate the best of modern math pedagogy including the rule of ...
Description:
Dugopolski''s College Algebra and Trigonometry: A Unit Circle Approach, Fifth
Edition gives students the essential strategies to help them develop the comprehension and confidence they need to be successful in this course. Students will find enoughcarefully placed learning aids ... |
This algebra lesson from Illuminations has students collect data for rolling objects of differing sizes in order to further understand periodic phenomena. They will then create two sinusoidal graphs of the data....
This lesson plan involves comparing different methods to determine the shortest route when traveling from Cleveland to Boston. Students will be given the opportunity to interpret data presented in table and graph format...
This lesson uses the example of successive discounts at a retail store to demonstrate numeric, algebraic, and graphical representations of compositions of mathematical functions. Students will get the opportunity to |
This learning object from Wisc-Online covers solving systems of linear equations using the addition or subtraction method. The unit looks at the common solution to simultaneous linear equations (also referred to as...
This learning object from Wisc-Online covers the properties of equality as related to algebraic equations. The unit's activities include defining the terminology and properties of equality associated with algebraic...
This learning object from Wisc-Online covers solving systems of linear equations using the substitution method. The unit looks at the common solution to two or more linear equations in two variables. Practice questions...
This learning object from Wisc-Online covers trade discount word problems. The lesson teaches a method of solving these problems which requires students to memorize only one equation. Example problems are included.
This math unit from Illuminations includes 4 lessons which use iteration, recursion and algebra to model and analyze a population of fish. Graphs, equations, tables and technological tools are used in each lesson. Each... |
Algebra I
This course provides a thorough introduction to the language of algebra, including its symbols and the axioms and laws which govern its structure. Emphasis is given to the understanding and manipulation of all manner of algebraic expressions, from performing standard operations to factoring polynomials and simplifying radical expressions. Among the primary goals are competence in solving linear equations and inequalities in one variable, systems of linear equations in two variables, and simple quadratic equations. Experience is provided in graphing in the Cartesian plane and in applying algebraic methods to the solution of practical problems. |
Formulas in mathematics, physics and chemistry for high school. The formulas are categorized by subject, course and range. For mathematics are all the formulas for the national sample in a custom view, broken down by course.
Favourite Mark formulas, courses and fields and see them in a separate tab. The last displayed formulas can be accessed from another tab. Through flexible search function, you can easily access all the formulas, courses and fields. Search can be accessed in any view by scrolling down a search bar.
Each formula can be viewed in portrait or landscape mode, choose the one that best suits theFormula Friend is the most complete database of formulas and data for students, scientists, and engineers. It is designed to help the user quickly find formulas and data to assist in calculations. It covers everything from integration tables to valence shell electron pair repulsion theory to relativistic quantum mechanics. Features include: -2,000+ equations -30,000+ unit conversions in the built-in unit converter -300+ common scientific constants -an interactive periodic table of elements with over 90 properties per element |
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