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You should now have a clear understanding of the twelve basic function families explored in this investigation. Think of all the things that you just learned about the practical applications of mathematics. What would our world be like without various functions, each filling a unique roll in our society?
This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory,... see more This site contains several hundred articles concerned with mathematics and physics. General topics include Number Theory, Combinatorics, Geometry, Algebra, Calculus & Differential Equations, Probability & Statistics, Set Theory & Foundations, Reflections on Relativity, History, and Physics. The articles under each general heading are highly varied, many are quite advanced, and there is no apparent organizational scheme. For example, under Calculus & Differential Equations there is a proof that pi is irrational, a examination of the Limit Paradox, a discussion of Ptolemy's Orbit, and an historical review of the cycloid among many other articles. Visitors can browse by topics or search by keyword. (Anyone with information on the identity of the site author please contact the MERLOT submitter.) Quoted from the site: EqWorld presents extensive information on solutions to various classes of ordinary differential,... see more Quoted from the site: EqWorld presents extensive information on solutions to various classes of ordinary differential, partial differential, integral, functional, and other mathematical equations. It also outlines some methods for solving equations, includes interesting articles, gives links to mathematical websites, lists useful handbooks and monographs, and refers to scientific publishers, journals, etc. This site will be kept up to date to include new equations with solutions and other useful information. ״Quadratic Formula Solver solves for the real (or complex) roots and the discriminant of quadratic equations.Quadratic... see more ״Quadratic Formula Solver solves for the real (or complex) roots and the discriminant of quadratic equations.Quadratic equations are frequently used in many high school and college level courses; including Algebra, Calculus, Chemistry, and Physics.״This is a free app This is a free resource hub dedicated to the learning student. I have dished out free question sets (with full detailed... see more This is a free resource hub dedicated to the learning student. I have dished out free question sets (with full detailed solutions) here, together with personally written summary notes for various topics including differentiation, integration, AP/GP, vectors, complex numbers etc. Hope it helps. Peace A Singaporean Maths site catering to the cambridge A level H2 maths syllabus; it alsocontains two large question/solution... see more A Singaporean Maths site catering to the cambridge A level H2 maths syllabus; it alsocontains two large question/solution portals -״The Question Locker" and "Beyond H2 maths״which are relevant to the general high school and early college maths student.
When considering a mathematical theorem one ought not only to know how to prove it but also why and whether any given conditions are necessary. All too often little attention is paid to to this side of the theory and in writing this account of the theory of real functions the authors hope to rectify matters. They have put the classical theory of real functions in a modern setting and in so doing have made the mathematical reasoning rigorous and explored the theory in much greater depth than is customary. The subject matter is essentially the same as that of ordinary calculus course and the techniques used are elementary (no topology, measure theory or functional analysis). Thus anyone who is acquainted with elementary calculus and wishes to deepen their knowledge should read this.
Schaum's Outline of Probability 40 million students have trusted Schaumís Outlines for their expert knowledge and helpful solved problems. Written by renowned experts in their respective fields, Schaumís Outlines cover everything from math to science, nursing to language. The main feature for all these books is the solved problems. Step-by-step, authors walk readers through coming up with solutions to exercises in their topic of choice. Outline format supplies a concise guide to the s... MOREtandard college course in probability
ALAN SELBY's mathematics appetizers range over arithmetic review problems, notions of what variables are, skills leading to algebra, painless theorem proving, complex numbers with some trig, the importance of slope (some calculus), a decimal perspective of error control and continuity (more calculus), and renaming the greater than sign (back to algebra). Advice on how to read, how to learn, why go to school, etc. is included. The tone is sometimes funny, and the writing is dense, rich, and intriguing. There are reflections on teaching, so these materials can be used in the classroom and as a place for teachers to learn. Alan's explanations of mathematical concepts using words and stories are particularly strong. Don't miss his Appetizers and Lessons for Logic, also on this site. -\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|-\-/-\|- - NEW AT THE FORUM - DYNAMIC ALGEBRA WITH NUCALC: SYMBOLIC ALGEBRA AND GRAPHING SOFTWARE FOR THE MACINTOSH Key Curriculum Press has provided us with sample pages from _Introducing Dynamic Algebra with NuCalc_, a book by Tim Erickson that draws from a wide range of math topics and explains how to use NuCalc in solving traditional and not-so-traditional high school math problems. You can access NuCalc files from the Web by downloading the NuCalc Web Helper application and configuring it as a helper application in your browser (instructions are included). These sample pages offer exercises to work through on transformations and the Fourier series. The book and software are available from Key Curriculum Press. If you have a Power Macintosh, you have the software already: it comes free on those machines, where it's called "Graphing Calculator." To find your particular interest, use the Search nctm-l link at the top right of the Archives page. Experiment by searching for the word "proof" (just the word, without the quotes) to find discussions of all kinds about the merits of proofs in Geometry - or look through the first thread that's returned by this search, 'Useful Problems', to get a feel for the depth and richness of the NCTM-L archive. The Forum's page of Public Discussions offers many more searchable math archives:
The Manga Guide to Linear Algebra By Shin Takahashi Illustrations by Iroha Inoue No Starch Press \| June 2012 \| ISBN-10: 1-59327-413-9 \| PDF \| 264 pages \| 56.2 mb Reiji wants two things in life: a black belt in karate and Misa, the girl of his dreams. Luckily, Misa's big brother is the captain of the university karate club and is ready to strike a deal: Reiji can join the club if he tutors Misa in linear algebra. Follow along in The Manga Guide to Linear Algebra as Reiji takes Misa from the absolute basics of this tricky subject through mind-bending operations like performing linear transformations, calculating determinants, and finding eigenvectors and eigenvalues. With memorable examples like miniature golf games and karate tournaments, Reiji transforms abstract concepts into something concrete, understandable, and even fun. As you follow Misa through her linear algebra crash course, you'll learn about: - Basic vector and matrix operations such as addition, subtraction, and multiplication - Linear dependence, independence, and bases - Using Gaussian elimination to calculate inverse matrices - Subspaces, dimension, and linear span - Practical applications of linear algebra in fields like computer graphics, cryptography, and engineering But Misa's brother may get more than he bargained for as sparks start to fly between student and tutor. Will Reiji end up with the girlΓÇöor just a pummeling from her oversized brother? Real math, real romance, and real action come together like never before in The Manga Guide to Linear Algebra. CONTENTS PREFACE..........................................xi PROLOGUE - LET THE TRAINING BEGIN!................1 1 - WHAT IS LINEAR ALGEBRA?.......................9 Z - THE FUNDAMENTALS.............................21 3 - INTRO TO MATRICES............................63 4 - MORE MATRICES................................85 5 - INTRODUCTION TO VECTORS.....................113 6 - MORE VECTORS................................131 7 - LINEAR TRANSFORMATIONS......................163 8 - EIGENVALUES ANP EIGENVECTOR.................205 EPILOGUE........................................231 ONLINE RESOURCES................................243 The Appendixes..................................243 Updates.........................................243 INDEX...........................................245 I have found that that Linear Algebra forms the foundation of just about every more advance form of mathematics. It is The Gateway - especially for the mathematics that underlies physics. This book is VERY user friendly. It is designed to construct UNDERSTANDING in the mind of its readers. That is the key - it is also what most standard LA texts fail to accomplish. I lived in Gilbert Strang (MIT). If I'd had this book as well, Strang would've been a cinch. Thank you very much to the original uploader and all seeders. Knowledge empowers one to traverse efficient paths toward understanding. It sets the mind free, even when one's life is imprisoned by the other forms of common human bondage.
The purpose of this study was to determine if integrating a unit on functions would benefit students. Previous studies have shown that integrating science and mathematics increases students' understanding of certain topics in science. Typically,... prediction of college mathematics grades has been widely researched. Results from other studies have shown that the prediction of college mathematics grades can vary depending on the set of predictor variables considered. For instance, some... In recent years, policymakers and academic experts have expressed alarm about the variance between Eastern and Western mathematics achievement. Research reveals that classrooms in the United States are lagging behind their Eastern counterparts.... The purpose of this study was to examine the effects of using Multiple Representations Charts in the classroom. This thesis was focused on the comparison of students' abilities to solve word problems using Multiple Representations Charts as anCan low achieving mathematics students succeed in the study of linear inequalities and linear programming through real world problem based instruction? This study sought to answer this question by comparing two groups of low achieving mathematics... Information about student performance was obtained by analyzing the performance of twenty-three fall sections of developmental mathematics taught at a state university. The independent variables included student performance on three predictorToday's special education teachers who work with students with difficulties in mathematics will need not only a deeper understanding of mathematics, but a thorough understanding in what the learning disabled student "sees" when approaching... This study examined whether different types of homework assignments had an impact on student achievement in Mathematics as well as on student completion of homework. The study was conducted over the course of 10 weeks and took place in a public is to determine if a personality test combined with a mathematics placement test can be used to identify students that may need additional interventions as early in the school year as possible. The school in which this examines the relationship between cognitive development and scores received on the Mathematics section of the Connecticut Academic Performance Test (CAPT). Individual interviews were conducted with 32 high school juniors who took theThere is a gap in the educational research that investigates the connection between the mathematics skills students learn in school and how they use these skills at the workplace. Students should be able to apply the mathematics skills they...
Crossing the River With Dogs Problem Solving for College Students 9781931914147 ISBN: 1931914141 Pub Date: 2003 Publisher: Springer Verlag Summary: Students who often complain when faced with challenging word problems will be engaged as they acquire essential problem solving skills that are applicable beyond the math classroom. The authors of Crossing the River with Dogs: Problem Solving for College Students:- Use the popular approach of explaining strategies through dialogs from fictitious students- Present all the classic and numerous non-traditional problem s...olving strategies (from drawing diagrams to matrix logic, and finite differences) - Provide a text suitable for students in quantitative reasoning, developmental mathematics, mathematics education, and all courses in between - Challenge students with interesting, yet concise problem sets that include classic problems at the end of each chapter With Crossing the River with Dogs, students will enjoy reading their text and will take with them skills they will use for a lifetime. Johnson, Ken is the author of Crossing the River With Dogs Problem Solving for College Students, published 2003 under ISBN 9781931914147 and 1931914141. One hundred twenty eight Crossing the River With Dogs Problem Solving for College Students textbooks are available for sale on ValoreBooks.com, twenty four used from the cheapest price of $1.64, or buy new starting at $81
Assessing Algebra in the Senior Phase: A Practical Guide by Saide OER Africa, Working at South Africa Institute for Distance Education on Apr 30, 2012 195 views This booklet was first developed in 2004, shortly after the introduction of the Revised National Curriculum Statement in South Africa. The intention of the booklet was to help teachers of senior phase ... This booklet was first developed in 2004, shortly after the introduction of the Revised National Curriculum Statement in South Africa. The intention of the booklet was to help teachers of senior phase (junior secondary) to integrate assessment into their teaching and learning. About Assessing Algebra in the Senior Phase: A Practical Guide T his booklet was first developed in 2004, shortly after the introduction of the Revised National Curriculum Statement in South Africa. The intention of the booklet was to help teachers of senior phase (junior secondary) to integrate assessment into their teaching and learning. Although the curriculum in South Africa has now changed, the purpose of assessment has not, and we think that teachers will find very useful the practical approach to planning, implementing and reflecting on assessment results. As the booklet is an Open Educational Resource, we invite teachers or teacher educators to take, use and adapt it for whatever curriculum they are currently teaching. All we request is that you acknowledge the document as originally developed by Saide. We would also like to hear what you think of this booklet, and especially how you use it. Please contact us at info@saide.org.za.This work is licensed under the Creative Commons Attribution 3.0 Unported licence.To1. IntroductionI n this booklet, we look at assessment in the Mathematics classroom. We focus particularly on algebra in the Senior Phase. Our main aim is to place assessment where it belongs – as an integral part of the whole teaching and learning cycle. Weuse some of the work of a teacher, Mrs Mothae, to illustrate some ideas about assessmentand to help you to reflect on your own assessment practice.We aim to provide you with resources to enable you to: Set tasks to assess whether a learner has achieved intended outcomes Assess learners' work, and record and interpret their results Use the results of assessment tasks to inform the ongoing teaching and learning processLook out for these icons: Task: Complete an activity Comment: An idea is explained or described. Reflect: Discuss and record your thoughts and ideas 32. The Mathematics Focus of this BookletIn this booklet we focus on an outcome that is common to many Maths curricula: The learner is able to recognise, describe and represent patterns and relationships, and solve problems using algebraic language and skills.This was Learning Outcome 2 in the South African National Curriculum Statement, and we think it is ameaningful way to introduce algebra in the Senior Phase. But what have patterns got to do with algebra? I thought algebra was solving equations with letters like x and y Well, patterns provide a meaningful reason to use letter symbols. When learners complete a pattern, and then describe it, this enables them to find a general rule for the pattern. Then they can use letter symbols to put that general rule into mathematical language. In the South African curriculum, learners are introduced to the teaching of algebra through patterns. Although there are other ways of introducing learners to algebra, this way has been found to be very successful. Have a look at Table 1 on the next page. The outcome is unpacked into so-called assessment standards for each of the grades in the Senior Phase. Do you notice how similar the Assessment Standards are across the grades? Do you notice how the Assessment Standards in each grade build onto the previous grade's work? In general terms, what would you need to teach your learners in order for them to achieve these Assessment Standards? What do the Assessment Standards expect learners to be able to do as they progress toward the outcome?4 1: Assessment standards for an introduction to algebraNote: This table gives learning outcome 2 with the relevant assessment standards from the South African National CurriculumStatement. It can, however, be adapted for other national curricula. Learning Outcome 2: The learner is able to recognise, describe and represent patterns and relationships, and solve problems using algebraic language and skills Grade 7 Grade 8 Grade 9 We know this when the learner: We know this when the learner: We know this when the learner: Investigates and extends numeric Investigates and extends numeric Investigates in different and geometric patterns looking and geometric patterns looking ways, a variety of geometric for a relationship or rules, for a relationship or rules, and numeric patterns and including patterns: including patterns: relationships by representing Represented in physical or Represented in physical or and generalising them and diagrammatic form diagrammatic form by explaining and justifying the rules that generate them Not limited to sequences Not limited to sequences (including patterns found in involving constant difference involving constant difference natural and cultural forms or ratio or ratio and patterns of the learner's Found in natural and Found in natural and own creation) cultural contexts cultural contexts Of the learner's own creation Of the learner's own creation Represents and uses Represented in tables Represented in tables relationships between Represented algebraically variables in order to determine input and/or Describes, explains and justifies Describes, explains and justifies output values in a variety of observed relationships or rules observed relationships or rules ways using: in own words in own words or algebraically verbal descriptions flow diagrams Represents and uses relationships Represents and uses relationships tables between variables in order to between variables in order to formulae, equations and determine input and/or output determine input and/or output expressions values in a variety of ways using: values in a variety of ways using: verbal descriptions verbal descriptions flow diagrams flow diagrams Constructs mathematical tables tables models that represent, formulae, equations and describe and provide Constructs mathematical models expressions solutions to problem that represent, describe and situations, showing provide solutions to problem Constructs mathematical models responsibility toward the situations, showing responsibility that represent, describe and environment and the health toward the environment and the provide solutions to problem of others (including health of others (including situations, showing responsibility problems within human problems within human rights, toward the environment and the rights, social, economic, social, economic, cultural and health of others (including cultural and environmental environmental contexts). problems within human rights, contexts). social, economic, cultural and environmental contexts). 5 When we looked at the Assessment Standards in Table 1, we identified several teaching and learning steps that are needed if learners are to achieve the standards: 1. Learners need to be able to complete patterns, by finding relationships between numbers. They also need to be able to complete patterns that are represented in tables. 2. Then they need to be able to describe these patterns verbally – this leads to 'generalising' or looking for the 'general rule'. 3. Then they need to be able to use letter symbols to write down their general rules in a mathematical way. They also sometimes need to use flow diagrams and graphs to do this. 4. Lastly, they need to use these steps to solve a problem. The example below illustrates a problem that takes learners from patterns, to making rules, to using variables (expressed as letter symbols) to solve a problem: Sharing out the bread Mrs Moroko bakes a loaf of bread. Drawing provided If she has to divide it between only herself and Mr Moroko, she would make 1 cut in the bread: However, if her daughter comes to visit her that day, she would need to make 2 cuts in the bread in order to share it between 3 of them: Her daughter brings her granddaughter with her! Now she needs to make 3 cuts in the bread to share it between 4 of them.6Complete this table: Number of people 2 3 4 7 10 ? Number of cuts 1 2 3 ? ? 14 Write down a sentence to explain what the pattern or general rule is for working out how many cuts Mrs Moroko must make. If Mrs Moroko wants to share the bread between people, how many cuts must she make? If Mrs Moroko makes cuts in the bread, to how many people can she offer a piece of bread? In this problem, there are two things that change – the number of cuts and the number of people. These are called the variables. Learners should be able to discover the general rule that the number of cuts is always one less than the number of people. They should be able to write this rule using letter symbols for the variables. This would be = -1 7 The Assessment Focus of this Booklet3.1 The Purpose of Assessment Even though the purposes for assessment of learners are many and varied, the main purpose is to give learners opportunities for growth and development. This includes assessing learners to facilitate their learning. What this means is that assessment is only effective and purposeful if it is an integral part of the entire teaching and learning cycle. This is what we focus on in this booklet.3.2 The teaching and learning cycle How do we make assessment an integral part of the teaching and learning process? The steps described below suggest how we can work with an ongoing cycle of planning, facilitating learning and assessing. 1. Plan In planning our work, we start by asking: What do the learners need to know? In other words, what outcomes do we want them to achieve? We ask ourselves: What learning opportunities should we provide for learners so that they can achieve the learning, and produce evidence to show that they have achieved the outcome? Planning involves three steps: Consult your term or year plan to see which learning outcomes/objectives to use for teaching and assessing. You need to plan your teaching to cover concrete, clearly defined assessment standards or sub-topics within the broad topic or outcome/objective. Plan a series of lessons, with teacher input and learner activities, that will give learners the opportunity to develop the knowledge and skills described in the assessment standards or sub-topics. Design an assessment task that gives learners a chance to prove (give evidence) that they have learnt and achieved the outcomes or mastered the topics you selected. 2. Facilitate Learning In this part of the cycle, learners have an opportunity to work toward the outcomes. The learners do the learning activities that you have chosen. This is an ongoing process that includes teaching (giving them input), mediating their learning, observing their progress and assisting them to achieve the outcomes. This step will equip the learners with the knowledge, skills and values needed in order to achieve the outcomes.8 AssessWhen the learners have had enough opportunity to acquire the skills and knowledge thatyou have planned, they are ready to complete an assessment task. The task should enableyou to assess how far the learners are in achieving the outcomes.In this part of the cycle you also: Record the results of the assessment. Analyse the group's results to see how they can be helped further towards achieving the outcomes and to inform your planning for the next step of learning needed. It is important to note trends in the class as a whole, their strengths and weaknesses. You provide feedback to learners to help them identify their own problem areas. If a learner can identify what her problem is, she can begin to work on improving her knowledge and skills. Feedback must be given in a spirit of encouragement and support, emphasising what the learner is able to do and building on this. Negative criticism can just entrench negative attitudes in learners. At times, report this assessment to parents, staff or others who need to know.4. Reflect and DecideIn this part of the cycle you use learners' responses to reflect on your own teaching, andto plan the details of the next step of the Learning Programme. You may need to think ofalternative ways to make the learning easier, provide more learning tasks or redo tasks toassist those learners who have not yet achieved the outcomes. Do you think that the cycle described above reflects your understanding of assessment? Are here any other steps you think should be included? In what ways do you think this cycle reflects what you currently do in your classroom? 9 Let's get Practical The Teaching and Lear ning Cycle 1. Plan Choose learning outcomes/topics and the more detailed assessment standards or sub-topics. What must learners know and be able to do? What are the outcomes? What learning opportunities can you provide? How can you assess what learners have achieved? 2. Facilitate Learning Teaching and learning takes place. Learners engage in learning activities. You assess learners informally while you teach, observe and mediate. 3. Assess The learners' achievement is assessed and results recorded and interpreted. Record your assessment. Have the learners achieved the intended outcomes? Identify strengths and weaknesses. Provide constructive feedback. 4. Reflect and Decide If the group of learners has satisfactory results, you can move to the next step in your plan. Individuals or the whole group may need concepts reinforced and another opportunity to be assessed in some way.10 4.1 The teaching and learning cycle in Mrs Mothae's class Meet Ms Sibongile Mothae. She teaches Maths at Rainbow Park Secondary School. This is how she has used the teaching and learning cycle."First, I used my year plan to plan in more detail for the term. We had planned to focus onLearning Outcome 2: The learner is able to recognise, describe and represent patterns andrelationships, and solve problems using algebraic language and skills for this term. Iagreed with the four key steps described on page 10 and used them in my planning.I looked at how this outcome was unpacked in my curriculum (see Table 1) across Grades 7, 8,and 9. I thought about what I wanted my Grade 8 learners to know and identifiedAssessment Standards that they needed to achieve. After designing learning activities, I setfour tasks that would assess whether learners had achieved these Assessment Standards atcertain key points in the programme." COMPLETE DESCRIBE AND USE LETTER SOLVE PATTERNS GENERALISE SYMBOLS PROBLEMS Task 1: Task 2: Task 3: Task 4:"For two weeks we worked with number and shape patterns in class. Thelearners completed patterns in a range of contexts and designed their ownpatterns, too. At first, I needed to help many of them. I also found that theyneeded to be reminded about how to use decimal fractions. I made notes abouttheir progress as I went along.""When I was confident that learners were ready to consolidate their work, I gavethem Assessment Task 1 to complete. Then I marked their work, recorded theirresults and gave feedback. I considered the strengths and weaknesses ofindividuals, and suggested what they needed to do to improve their work.""I then looked at the ways in which the class as a whole had managed thequestions in the task. I thought about why many had problems with certainquestions, and what I could do to help. I changed my original plan toaccommodate these ideas." 112 Assessment in Mrs Mothae's Maths class In the previous section we saw how Mrs Mothae used the teaching and learning cycle in her work. In the rest of this booklet we will consider in particular on Steps 3 and 4 of this cycle. In this section we pay greatest attention to step 3 – assessment. Our discussion focuses on Mrs Mothae's use of Assessment Task 1. ANALYSING ASSESSMENT TASK 1 All assessment tasks need to be carefully planned to: assess what was stated as the outcomes of the teaching and learning at the level that is expected from your grade. include enough questions that allow the "average" learner to succeed, with some challenging questions to extend the learners. provide opportunities for you to diagnose problem areas and consolidate previous Maths knowledge. Mrs Mothae tried to set tasks that would put all of these principles into practice! We have included Task 1 and the other three Assessment Tasks in appendices in a size suitable for you to copy for your own class if you wish. In this booklet, however, we will only consider how Mrs Mothae uses Assessment Task 1 to assess her learners, to analyse their strengths and weaknesses and to adjust her Learning Programme based on this information. For easy reference, we have shown Assessment Task 1 on the next page. Do Assessment Task 1 1. A good way for you to get to grips with Assessment Task 1 is for you to do it yourself. So, as a first step, you should read through the task and answer the questions yourself. 2. As you do the task, think about some of the problems that might face your learners when they do the task.12 13 Think About Assessment Task 1 1. Which questions In Assessment Task 1 did you find surprising, interesting, challenging or easy? 2. What can Assessment Task 1 tell you about the learner's ability to meet the Assessment Standards from Learning Outcome 2 (listed on page 5)? 3. What other Maths knowledge does the learner need in order to complete Assessment Task 1? 4. Suggest different ways in which you could use this assessment task in your classroom. 5. Match each of these statements about Task 1 correctly with the questions from Task 1. Note: There may be more than one answer for each statement and the same question from the task may apply to more than one of the statements. We have matched the first statement with the questions as an example. 6. Review your answers to questions 2 - 4 in the light of these statements. Learners need to understand simple fractions to be able to do these questions. Questions 1(c), (d) and (e) They also need to multiply and divide accurately. Learners need to understand decimals to be able to do these questions. Most learners in the Intermediate Phase should be able to complete these simple addition and subtraction patterns with little difficulty. To complete this pattern to the left, learners need to divide by 4. To complete the pattern to the left, learners need to multiply by 3; to complete the pattern to the right, they need to divide by 3 at each step. Learners must be able to add simple fractions. Even learners who are able to add quarters might not show the consistency in the pattern and fill in the zero (some may put one eighth as their answer)*. Learners must be able to add three digit numbers. There is a constant difference of 111 between consecutive numbers in the pattern. A good learner would show an understanding of place value and fill in 900 as the first number in the pattern. Some learners may have looked for the patterns in the digits instead of the whole numbers. The first digits are 7, 6, and 5; the second digits are 8,7 and 6 respectively. However, continuing the pattern in this way to the left would lead to the incorrect answer. This question needs an understanding of place value in large numbers. This question tests the learner's ability to be consistent in the pattern, but this time by subtracting zero from one to find the answer to the left of one. The result is that there are two consecutive '1s' in the pattern. In this question, learners have to show they can continue patterns in two dimensions or directions simultaneously. In these questions, the learner needs to explain the patterns, instead of just using the pattern. This is a necessary skill before being able to use variables in algebra. Learners have to extend shape patterns and match them to number patterns14 Here are some of our ideas about the questions asked on Assessment Task 1. Compare them with yours. Match each of these statements about Task 1 correctly with the questions from Task 1. We have given our ideas about question 5 first. We found that finding the links showed us some of the aspects of the Assessment Standards of Outcome 2 that Task 1 focuses on. They also give a sense of the other mathematical knowledge embedded in the questions. We found that by looking in this sort of detail at the questions in Task 1, we were better able to answer the more general questions that were asked about the task as a whole. We thought that:The statements and questions are linked like this:Learners need to understand simple fractions to be able to do these questions Questions 1(c), (d) and (e)They also need to multiply and divide accurately.Learners need to understand decimals to be able to do these questions Question 1(i)Learners continue simple addition and subtraction patterns. Most learners in Question 1(a) and (b)Intermediate Level should be able to complete these patterns with little difficulty.To complete this pattern to the left, learners need to divide by 4 Question 1(c)To complete the pattern to the left, learners need to multiply by 3; to complete Question 1(d)the pattern to the right, they need to divide by 3 at each stepLearners must be able to add simple fractions. Even learners who are able to add Question 1(e)quarters might not show the consistency in the pattern and fill in the zero(some may put one eighth as their answer).Learners must be able to add three digit numbers. There is a constant difference Question 1(f)of 111 between consecutive numbers in the pattern. A good learner would showan understanding of place value and fill in 900 as the first number in the pattern.Some learners may have looked for the patterns in the digits instead of the whole Question 1(f)numbers. The first digits are 7, 6, and 5; the second digits are 8,7 and 6 respectively.However, continuing the pattern in this way to the left would lead to the incorrectanswer.This question needs an understanding of place value in large numbers. Question 1(g)This question tests the learner's ability to be consistent in the pattern, but this timeby subtracting zero from one to find the answer to the left of one. The result isthat there are two consecutive '1s' in the pattern. Question 1(h)In this question, learners have to show they can continue patterns in two Question 2(a)dimensions or directions simultaneously.In these questions, the learner needs to explain the patterns, instead of just using Question 2(b) and 3the pattern. This is a necessary skill before being able to use variables in algebra.Learners have to extend shape patterns and match them to number patterns Questions 3 and 4 15 What can Assessment Task 1 tell you about the learner's ability to meet the Assessment Standards from Learning Outcome 2? We thought that: The task gives learners several opportunities to practise completing patterns, consolidate this ability and achieve what the first Assessment Standard for Grades 7 and 8 requires ('the learner investigates and extends numeric and geometric patterns looking for a relationship or rules'). The patterns in the task include some that are in 'diagrammatic form', some that do not involve a constant difference or ratio (question 1(i), (j), (k)) and some that are 'represented in tables'. These are some, but not all of the kinds of patterns noted under this first Assessment Standard. Questions 2(b) and 3 require the learner to describe an observed relationship in words. These questions link to the second Assessment Standard. If a learner can explain patterns using words, it gives an insight into their reasoning. Reflecting on how the class as a whole manages Questions 2(b) and 3 will help you when you teach this aspect to them. (Assessment Task 2 deals more thoroughly with this) If a learner performs well on this task, this may show that he or she is able to extend simple patterns. However, in this task alone, we have not gathered enough reliable and valid evidence yet to say that the learner has achieved Learning Outcome 2 as a whole. Can you see that several of the Assessment Standards have not been included? In your planning for the year, you will need to be sure that for each grade you teach, all the relevant Assessment Standards have been covered – but they do not all have to be covered in one assessment task. Remember how Mrs Mothae thought about this in her planning. What other Maths knowledge does the learner need in order to complete Assessment Task 1? We thought that: The number skills of earlier grades (covered in Learning Outcome 1) are needed to investigate and complete the patterns set in Assessment Task 1. We thought this was a good idea as it meant that the task can help identify learners with problems with these number skills. Some of the questions required learners to show that they understood that zero is a number in its own right. We think it is important that learners understand this, and that zero does not merely mean the absence of an amount. So, we thought it was a good idea to include several questions for which the answer is zero in patterns as there is a surprising reluctance to write zero as an answer.16 Some number concepts such as negative numbers and complex fractions, which are required in Grade 8, are not included in the task. The fraction knowledge expected here is simple. We thought this was a good idea as the main purpose of the exercise is to test pattern extension – and we would not have wanted to handicap learners who lacked more advanced fraction knowledge. When you are marking the learners' work, you need to keep the different skills and knowledge involved in mind. Although the primary focus in Task 1 is on learners' ability to recognise and complete patterns, at the same time it assesses learners' understanding of other mathematical knowledge and skills. You need to interpret learners' answers – to see exactly what it is they can and cannot do. Are they understanding patterning, but struggling with say, fractions? Are they able to add time accurately, but unable to see the patterns in the time question? Your analysis of this will help you know how to help the learners progress. Suggest different ways in which you could use this assessment task in your classroom.Mrs Mothae used this task to assess her Grade 8 learners after two weeks of learning and teaching. It providedher with a formal assessment of the learners' achievement. But it could be used in other ways, too. We thought that: It could be used as a worksheet rather than as a formal assessment task. Using the task in this way would give more opportunities to assist the learners when they have difficulties. We thought this would be particularly appropriate in a Grade 7 class. The task could also be used as a baseline assessment for Grade 9 learners who should be familiar with all the work covered. The task could provide a useful starting point for beginning algebra in Grade 9. It would give valuable information about the Grade 9 learners and their areas of difficulty. It is important to adapt learning materials and the way you use them according to the needs of your classes - and this is true, too, of Task 1 and your use of it. If you haven't yet covered a certain concept that is in Task 1 at a time when you want to use it for formal assessment, then you need to change the task accordingly. Leave out those items – replace them with something you have covered. Or you may decide to spend class time on that concept before giving learners the task. While you do need to be sure that all of the Assessment Standards for a grade are covered in the year, it is pointless rushing on to cover an Assessment Standard if you have not given learners enough opportunity to understand and achieve those standards which learners need to build the new standard on. Thus, if learners struggle with Assessment Standards from Outcome 1, such as those dealing with fractions, for example, they will probably not be able to recognise patterns involving these and you will need to consolidate this learning first. 17 MARKING ASSESSMENT TASK 1 I would like you to meet three of my learners, Ahmed, Thembeka and Pedro. They have completed Task 1 As we have shown, the way learners answer the questions in Task 1 will give valuable information about how well they have achieved the learning that preceded their doing the task. Before such an analysis is possible, though, the work has to be marked. In order to make sense of Mrs Mothae's work, we would like you to complete the task below Mark three learners' work Pedro, Thembeka and Ahmed's completed assessment tasks are shown on the next pages. Of course, there will be a greater variety of learner responses than those shown here in any class. However, these three represent many of the most likely learner responses. 1. Mark these learners' work as if they were in your class. Refer to the marking memorandum on page 36. To help each learner, write comments onto their scripts that will help them to see what they need to work on. Calculate their total mark out of 50.18 This work is licensed under the Creative Commons Attribution 3.0 Un20 21 Compare Mrs Mothae's marking with yours Thembeka – 33 Ahmed – 31 Pedro – 26 Compare Mrs Mothae's marking of the three learners' papers with yours. Although your marks may differ slightly from hers, they should be similar. We have shown the comments she made on Pedro's script. In what ways do you think they are helpful? How could they be made more helpful?22 2324 25 RECORDING AND INTERPRETING THE RESULTS Focusing on individual learners A formal assessment needs to be recorded as part of your measure of the learner's progress through the year. However, recording is not enough on its own. You need to use the results to understand each learner's areas of difficulty and to plan for the next cycle of teaching and learning. In marking the three learners' work, you have allocated them an overall mark for the task as a whole. How helpful is this? The questions below will help you think about this further. Thinking about using marks to record assessment 1. Does the learner's total mark indicate to what extent she is progressing towards achieving the outcome? In other words, does the mark show to what extent she is successfully completing patterns? 2. Does the learner's total mark indicate where she has managed well and where she has struggled? 3. Can the marks help you to identify misunderstandings or gaps in the learner's knowledge that might have led to incorrect answers? 4. Is there a way of marking this test that would show how the learner has done in each separate type of question? What could you do to record information about learner achievement that would be more helpful than a mark alone?26 EMrs Mothae found that the marks gave her a general idea of whether each learnercould complete patterns, but they did not show her a learner's areas of strength andweakness, or where the gaps in their knowledge might be.Mrs Mothae decided to identify and record how learners had done in the differentskills that were being tested in Task 1. She used these as criteria to help her assessto what extent her learners had achieved the outcomes she had set for the task. Wewill focus on choosing criteria and using them for recording assessment in moredetail in another issue of this series of booklets, but you may find it useful to thinkabout her system here.She noted 6 criteria that focus on patterning skills, and the questions that gave herevidence of learners' achievement of these. These are:1. Extend simple patterns to the right 1(a) – (j)2. Extend simple patterns to the left 1(a) – (j)3. Extend patterns horizontally and vertically 2(a)4. Describe simple patterns verbally 2(b) and 3(d)5. Complete a pattern in table form 3(a)6. Extend shape patterns 4Mrs Mothae also chose 4 criteria to help her assess whether any learners need helpwith an understanding of fractions, decimals, large numbers or time.7. Use fractions to complete patterns 1(c), (d), (e)8. Use larger numbers to complete patterns 1(f) and (g)9. Use place value and decimal numbers correctly 1(i)10. Extend patterns of time 1(j)Although this way of recording learner achievement looks complicated at first, it canbe efficient and is not as complicated as it seems! Mrs Mothae kept her class list nextto her and ticked off what her learners could do as she marked. Using this method,she decided not to use the marks at all! On the next page, you will see how MrsMothae completed her records for Pedro and some other learners.Using a more qualitative system of recording1. Use Mrs Mothae's system and your marking to complete the records for Ahmed and Thembeka in Table 2 on the next page.2. Then use the records and the learners' scripts to help you to identify each learner's strengths and weaknesses, and any gaps in knowledge or misunderstandings.3. Suggest what information this system gives about the learners that only recording marks does not. 27 2: Mrs Mothae's record sheet Thembeka Ahmed Natalie Pedro Kobus Portia The learner can 1. Extend simple patterns to the right 1(a) – (j) 2. Extend simple patterns to the left 1(a) – (j) 3. Extend patterns horizontally and vertically 2(a) 4. Describe simple patterns verbally 2(b) and 3(d) 5. Complete a pattern in table form 3(a) 6. Extend shape patterns Question 4 7. Use fractions to complete patterns 1(c), (d), (e) 8. Use large numbers to complete patterns 1(f) and (g) 9. Use place value and decimal numbers correctly 1(i) 10. Extend patterns of time 1(j) Each column gives you immediate information about how a particular learner has done on the set of criteria. This shows each learner's strengths and weaknesses across the range of skills and knowledge being assessed. It also gives you a picture of whether the learner has achieved what the whole task was assessing – can the learner extend and describe patterns? Each row indicates to you whether your class has grasped a particular skill or if they still need more work on this. Analysing assessment information Read what Mrs Mothae noticed about three of her learners. 1. Think about how her analysis compares with yours. 2. What would you say about Portia, Natalie and Kobus?28 Ahmed Understands patterning; struggles with fractions; needs to learn time notation; didn't notice that the pattern to the left is different from the pattern to the right in 2(b); struggled with patterning in table form and shape patterns. Thembeka Misunderstood that the pattern must be completed to the left as well in question 1 (c), (d), (e); struggled with large numbers; needs to be shown the pattern in (h); good work on shape patterns and tables (although needs help on 4(b)). Pedro Missed the pattern because he didn't look at all the numbers in 1(a) and (b); recognised patterns in (c) to (f) but wrote down rules instead of completing the pattern; needs to be reminded about conventions for decimal numbers and for time; stronger on finding the rule, shape patterns and completing the table.Mrs Mothae used the information about each learner she has gathered thus far to decide on a generalassessment of how well each learner is capable of completing and describing patterns (Learning Outcome 2)at this stage in her learning programme. She used ratings of 1 to 4 to record this general assessment of howwell each learner has achieved this outcome. She also used the information she gained from the assessmentof their work to write a comment about each learner's particular strengths and weaknesses. The table belowshows how she did this. Table 3: Mrs Mothae's rating sheet Thembeka Ahmed Natalie Pedro Kobus Portia Completing and describing number patterns 3 3 2 2 4 3 Completing and describing shape patterns 2 3 2 2 3 2 numbers & completing Can recognise number Comments Struggles with large and shape patterns patterns, but can't patterns to left complete them Needs practice using tables The South African National Curriculum Statement provides the following codes for recording learners' levels of achievement of outcomes. 4 = Learners performance has exceeded the requirements. 3 = Learners performance has satisfied the requirements. 2 = Learners performance has partially satisfied the requirements. 1 = Learners performance has not satisfied the requirements. 29 EFocusing on the group as a whole Mrs Mothae looked at her learners' results to see if she could identify trends within the class as a whole. These would help her to decide how to move forward. Finding trends 1. Which questions did the three learners, Pedro, Ahmed and Thembeka, all do correctly? What does this indicate about their knowledge and skills? 2. Which questions did these three learners do poorly? What does this indicate about knowledge and skills that they are lacking? It is difficult to analyse general trends from only three learners' work. When you do this exercise with a larger group, the trends will be clearer. However, even with only these learners' work to consider, some general trends are clear. All three learners identified most of the number patterns in question 1, although only Ahmed saw the pattern in 1(h). They all saw the patterns in the grid in question 2, although they confused multiplying and dividing when asked to describe the patterns. Two of the learners completed the table correctly. All three learners were unable to do 1(c) and (d) where they needed to move from whole numbers to fractions; Two learners struggled with large numbers in 1(g); Two learners struggled with writing time correctly in 1(j); They all struggled to complete the gaps on the left of the patterns correctly, although they were able to continue them to the right. All the learners struggled with the shape patterns. Thinking about what help learners need 1. In what areas of the work do you think the learners in Mrs Mothae's class need help? Have a look again at the grid showing which standards each of the 6 learners has achieved to see the trends more clearly. The lea rner ca eka n Ahmed 1. Exte Themb nd si the rig mple pattern Pedro Portia ht 1(a) Natalie – (j) s to 2. Exte Kobus There are three areas in which learners nd si the left mple pattern 1(a) – (j s to ) need help 3. Exte nd p and ve atterns hori rtically zontally 2(a) continuing patterns to the left; 4. Descr ibe sim ple pat verbal terns ly 2(b) continuing shape patterns 5. Com plete a and 3( d) form 3( pattern in table using fractions. 6. Exte a) nd shap Questi e patte on 4 rns 7. Use fractio pattern ns to co s 1(c), mplete (d), (e) 8. Use large comple numbers to te patte rns 1(f) 9. Use and (g place ) number value and d s correc ecimal tly 1(i) 10. Exte nd pat 1(j) terns o f time30USING THE ASSESSMENT RESULTS TO PLANAn essential part of the teaching and learning cycle is deciding how to use the knowledge you havegained from assessing the learners (Step 4 on pages 9 and 10) You need to do two things. Firstly, think about what the assessment suggests about how you might better teach this section of work another time. What strengths and weakness can you now anticipate, and how can you use this knowledge to plan your teaching better? Secondly, think about what you can do to help the learners whose work you have just assessed. How should you adjust your teaching plan to support their learning better? You cannot spend too long on going over work that has not been understood, but it can be damaging to the learners' progress to just ignore their difficulties. They are likely to struggle with the next section of the work. How can you help both individuals and the group as a whole? How can you extend those learners who are clearly able to do the work well? Think about what to do to support learning 1. Make suggestions about how you would address the problems that have been identified in the learners' work. 2. Compare your ideas with what Mrs Mothae did. This is what Mrs Mothae did to address these areas of concern: She showed the class how the patterns that they completed in the task continue in both directions. She helped the class see that the rule they find if they work to the right is the inverse of the rule they will use if they work to the left. She gave them some additional examples to work on in pairs to check that they had understood. She realised that she could not just give learners different versions of the same shape patterns, so she spent a whole lesson letting learners explore shape patterns by building the patterns with cardboard shapes, investigating their answers and explaining them in groups. This gave them a concrete experience of shape patterns. She had to remind them of previous work done with fractions and gave them a homework exercise on adding and subtracting simple fractions. She offered to make an extra time outside of classroom time to work with two learners who still struggled. She built up their confidence by working with simpler patterns first. She set some tasks to stretch the capabilities of the strongest learners, and arranged for them to work together in a group on some occasions when other groups of learners worked on simpler tasks. 31 THE CYCLE IS ON-GOING After some work such as that described above, Mrs Mothae was aware that she needed to move on to the next step in her plan – verbal and algebraic expression of patterns (ie generalisations). She continued with the planned classwork, but included more examples of shape patterns and patterns involving fractions. The learners worked in pairs and had to keep explaining what they were doing while they worked. They also began using letter symbols for variables to generalise the patterns. Mrs Mothae assessed them informally, mediating, assisting and observing their progress. By the time she used Assessment Task 2, most of her learners were able to explain and use number and shape patterns, although some still struggled with using variables. Those who still struggled with fractions attended extra lessons after school. In Assessment Task 2, Mrs Mothae set questions that asked learners to explain the relationship between different numbers in patterns. If you look at this task in Appendix 2, you will see that some of the same patterns of Assessment Task 1 are asked in a similar way in order to consolidate their knowledge. From question 2 on, the learners may start by using words to explain, but by the end should be able to show that they can use mathematical operations and variables (this addresses steps 2 and 3 identified on page 6). As before, if the learners' performance on Assessment Task 2 showed that they could do this work, then Mrs Mothae felt that she could move on to the work planned next, leading up to Assessment Task 3 and finally Assessment Task 4. Learners need to have worked with flow diagrams and graphs before they can complete Assessment Task 3, and Mrs Mothae made sure she gave them plenty of opportunity to do this, mediating their learning. Assessment Task 4 assesses learners' ability to apply their understanding to a problem, and again, learning opportunities were provided for this. While learners worked, Mrs Mothae checked on how they were doing, made informal notes about their progress and intervened to help and extend them where appropriate. As we described for Task 1, after each of the following assessment tasks Mrs Mothae checked more formally on what learners' work told about their achievement of the Assessment Standards their work was focusing on, and their strengths and weaknesses, and spent some time dealing with aspects of concern before moving on. She also built additional support of work already covered into her planned new work to help consolidate learning where she now realised this was needed. You may want to use these assessment tasks in your teaching and learning cycle, adapting them to suit your needs Conclusion We hope that you found this booklet useful. It showed you how one teacher tried to use assessment as part of the teaching and learning cycle. Mrs Mothae did not find it easy, but she enjoyed trying new ideas - and then changing them if she finds something that seems to work better. We hope that you will be encouraged to do the same.32 Appendices 1 - 4: The Assessment Tasks and Answers Task 1 34 Focuses on learners' ability to recognise and complete a variety of types of patterns in number and shape Task 2 38 Relates to learners' ability to express patterns in words and to complete a variety of patterns that are represented in tables Task 3 42 Assesses learners' ability to express relationships and generalisation using flow diagrams and graphs Task 4 46 Assesses learners' ability to apply their knowledge of patterning, generalisation and algebra to solve an integrated, 'real life' problem. 33 Legs3b) Explain how you would work out what the number of legs is for any number of pots. (2 marks) ........................................................................................................4 4 9 16 Pattern B 8 12 16 Pattern C 8 10 12 35 3 6 9 12 24 39 150 600 Legs3b) Explain how you would work out what the number of legs is for any number of pots. (2 marks) Multiply the number of pots by 34 1 4 9 16 25 Pattern B 4 8 12 16 20 Pattern C 6 8 10 12 14 37 1. Complete the patterns below a) ____; ____; 1; 4; 16; _______. b) _____; ______; _______; 3; 9; 27. c) 1; ____; 2; 4; 7; _____; _____; 22. d) 26; 18; 12; ______; ______; 6. 1 2 3 e) ____; 1 ; 2 ; 3 ; _____; _____. 5 5 5 f) ____; _____; 10h45; 11h30 12h15; ______; _____. g) _____; 4,055; 4,155; 4,255; ______. h) 2 250 m; 2 km; _____; _____; 1250 m; 1 km. i) _____; ______; 1; 3; 5; 7; _____. 2a) Explain how you would find: a) the number of sides on a given number of squares.......................................................................... b) the area of a square of any given side length................................................................................... 3) Three glasses can be poured from one litre of milk. How many glasses of milk can be poured from a given number of litres? ............................................................................ 4) One car needs five tyres including the spare. How many cars can be fitted with a given number of tyres? ............................................................................ 5) Four people can sit around a square table like this one.: a) How many people can sit around two tables set apart b) How many people can sit around 5 tables set apart c) Describe in words how you would find the number of people that could sit at any number of tables set apart. 6) If two of the same tables are pushed together, then 6 people can sit around them38 People b) Describe how you would find the number of people that could sit at any number of tables pushed together. ........................................................................................................7. Look at the pattern in the shapes and in the numbers of degrees on each shape. 120 90 60 51 Shape 1 Shape 2 Shape 3 Shape 4 Shape 5 120 90 60 51 a) Draw Shape 3 in the space above b) Find the number of degrees shown by the arrow in Shape 3 c) How many degrees would the arrow show if there were a Shape 6 d) How could you find the angle shown by the arrow in any shape in the same pattern? ..........................................................................................8. In this pattern, three matches form 1 triangle, five matches form 2 triangles. a) Draw and complete a table to show the number of matches needed to form 1, 2, 3, 4, 20 and 50 triangles b) Describe how you would find the number of matches needed to form any number of triangles in this way. c) Draw a line graph to show the 10 relationship between the number 9 Number of Matches 8 of triangles (horizontal axis) and 7 the number of matches (vertical axis). 6 You can use the grid given here. 5 4 3 2 1 0 1 2 3 4 5 Number of Triangles 39with answers and some of the working in bold 1. Complete the patterns below 1 1 a) 16 ; 4 ; 1; 4; 16; 64. 1 1 b) 9 ; 3 ; 1; 3; 9; 27. c) 1; 1; 2; 4; 7; 11; 16; 22. d) 26; 18; 12; 8; 6; 6. 1 2 3 4 e) 0; 1 5 ; 2 5 ; 3 5 ; 4 5 . f) 09h15; 10h00; 10h45; 11h30; 12h15; 13h00; 13h45. g) 3,955; 4,055; 4,155; 4,255; 4,355. h) 2 250 m; 2 km; 1750m; 1500m (or 1,5km); 1250 m; 1 km. i) -3; -1; 1; 3; 5; 7; 9. 2a) Explain how you would find: a) the number of sides on a given number of squares 4 x number of squares 2 b) the area of a square of any given side length Length x length (or ) 3) Three glasses can be poured from one litre of milk. How many glasses of milk can be poured from a given number of litres? 3 x numbers of litres 4) One car needs five tyres including the spare. How many cars can be fitted with a given number of tyres? Number of tyres divided by 5 5) Four people can sit around a square table like this one.: a) How many people can sit around two tables set apart? 2 x 4 = 8 people b) How many people can sit around 5 tables set apart? 5 x 4 = 20 people c) Describe in words how you would find the number of people that could sit at any number of tables set apart. Number of people is 4 times the number of tables 6) If two of the same tables are pushed together, then 6 people can sit around them40 4 6 8 10 18 32 People b) Describe how you would find the number of people that could sit at any number of tables pushed together. Learners answers will vary. One correct answer is: twice the number of tables and add 27. Look at the pattern in the shapes and in the numbers of degrees on each shape. 120 90 60 51 72 Shape 1 Shape 2 Shape 3 Shape 4 Shape 5 120 90 60 51 a) Draw Shape 3 in the space above b) Find the number of degrees shown by the arrow in Shape 3 360 = 45 c) How many degrees would the arrow show if there were a Shape 6? 8 d) How could you find the angle shown by the arrow in any shape in the same pattern? Angle will be 360 + 2 or 360 shape number number of sides8. In this pattern, three matches form 1 triangle, five matches form 2 triangles. a) Draw and complete a table to show the number of matches needed to form 1, 2, 3, 4, 20 and 50 triangles Number of triangles 1 2 3 4 20 50 Number of matches 3 5 7 9 41 101 b) Describe how you would find the number of matches needed to form any number of triangles in this way. (Number of triangles x 2)+1 or 1+(2 x number of triangles) c) Draw a line graph to show the 10 relationship between the number 9 of triangles (horizontal axis) and Number of Matches 8 the number of matches (vertical axis). 7 You can use the grid given here. 6 5 4 3 2 1 0 1 2 3 4 5 Number of Triangles 41 3 1. Work out: a) How many days in weeks? b) How many hours do you sleep in m days (If you sleep 8 hours each day) c) How many internal angles are there in t squares? d) What is the surface area of a cube with edge length ? e) How many legs on: i) p chickens? ii) q cows? iii) r snakes? f) A taxi holds people. How many people can fit into taxis? g) How many cuts do you make to divide a sausage into pieces? h) How many quarters in k? 2. Look at the two flow diagrams below. I. ÷3 +5 II. +5 ÷3 a) Put a circle around all the equations below that show Flow Diagram I + 5 ( + 5) = + 5 = = = ( )+ 5 ÷ 3 3 3 3 = + 5 ÷ 3 = ÷ 3 + 5 = ÷ (3 + 5) = 5 ÷ 3 b) In Flow Diagram I: i) if = 3, what is ? ii) If = -3, what is ? c) In Flow Diagram II: i) if = 3, what is ? ii) If = -3, what is ? 3. a) Draw a flow diagram to show = (2 + 3) 2 b) If = 9, what is ? c) If = 9, what is ?42 b) Complete the table below to show the number of little cubes needed to build a large cube. Edge length 1 2 3 4 5 Number of small cubes needed 1 c) How would you find the number of small cubes needed to create a larger cube of any edge length? d) On squared paper, draw a line graph to show the relationship between the edge length of a large cube (horizontal axis) and the number of little cubes you need to form a the large cube (vertical axis). You can use the grid given here. NUMBER OF SMALL CUBES (needed to make up a large cube) LENGTH OF EDGE OF LARGE CUBE (counted in small cubes) 43 27 small cubes b) Complete the table below to show the number of little cubes needed to build a large cube. Edge length 1 2 3 4 5 Number of small cubes needed 1 8 27 64 125 c) How would you find the number of small cubes needed to create a large cube 3 3 of any edge length? (Edge length) = d) On squared paper, draw a line graph to show the relationshop between the edge length of a large cube (horizontal axis) and the number of little cubes you need to form a the large cube (vertical axis). NUMBER OF SMALL CUBES (needed to make up a large cube) LENGTH OF EDGE OF LARGE CUBE (counted in small cubes) 45 ii) The total height of some cups stacked in this way is 17cm. How many cups are there in the stack? iii) Write an algebraic formula to calculate the height of cups stacked in this way. b) The same cups are stacked in the pattern shown here: i) Write an algebraic formula to find the height of cups stacked in this way ii) Find the difference in height between cups stacked as shown in (a) and cups stacked as shown in (b) around the 15cm edge of 17cm a shape 1cm 3cm46 17 15 Area 17 453. This famous number pattern below is called the Fibonacci sequence. To find the next number you add the previous two. 1 1 2 3 5 8 13 a) Find the next two numbers in the series. b) Here are more Fibonacci numbers. Find the missing two. 89 144 233 377 c) Use the same rule as the one for the Fibonacci series to find the missing numbers in the following serries. i) 2,17; 4,05; 6,22; 1 ii) 155 000; 2 2 million; iii) ; + ; 2 + ; 2 2 iv) 4 ; 6 ; v) 5; -2; 3; 47with answers and some of the working in bold 1 1 1 1 1 10 2 + (7 x 2 ) 10 2 + 3 2 = 14cm or 11 + (6 x 2 ) 11 + 3 =14cm ii) The total height of some cups stacked in this way is 17cm. How many cups are there in the stack? 13 cups iii) Write an algebraic formula to calculate the height of cups stacked in this way. 1 1 -1 = 10 2 + 2 or 11 + ( 2 )cm b) The same cups are stacked in the pattern shown here: i) Write an algebraic formula to find the height of cups stacked in this way 11 ii) Find the difference in height between cups stacked as shown in (a) and cups stacked as shown in (b) 1 1 11 - (10 2 + 2 ). around the 15cm edge of 17cm a shape 1cm 3cm There are 17 possible rectangles altogether. We have not drawn them here, but their dimensions are shown in the table. Learners will have chosen 10 from this set, but we have shown them all so as to include whichever any learner chooses. They do not have to show both sides – but we have done so to provide the range of answers possible in 2c. All learners should include the rectangle of sides 9 x 9 cm, as this will give the square asked for in the question.48 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Area 17 32 45 56 65 72 77 80 81 80 77 72 65 56 45 32 17 As increases from 1 to 9, area increases. As increases for > 9, the area decreases.3. This famous number pattern below is called the Fibonacci sequence. To find the next number you add the previous two. 1 1 2 3 5 8 13 a) Find the next two numbers in the series. 21 34 b) Here are more Fibonacci numbers. Find the missing two. 55 89 144 233 377 610 c) Use the same rule as the one for the Fibonacci series to find the missing numbers in the following serries. i) 1,88; 2,17; 4,05; 6,22; 10,27 1 ii) 2 345 000; 155 000; 2 2 million; 2 655 000 iii) ; ; + ; 2 + ; 3 +2 2 2 2 2 iv) 2 ; 4 ; 6 ; 10 v) -7; 5; -2; 3; 1 4950 This work is licensed under the Creative Commons Attribution 3.0 Unported licence. To view a copy of this licence visit
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Consumer Mathematics Description It captures the attention of teenagers immediately with Unit One, Buying a Car, and proceeds to units on budgeting; banking; investing; keeping tax records; purchasing food, clothing, and a home; and more! The text presents a Scriptural view of working, tithing, saving, paying taxes, and budgeting time and money and gives a positive introduction to the American free-enterprise system. Each information-packed unit contains sample problems for students to follow as well as an abundance of practice problems. An analytical skills section in each chapter challenges students to analyze practical problems or opportunities they could soon be facing. The colorful, attractively designed text is a joy to use. An abundance of charts, graphs, and illustrations spark student interest. The Skills and Review Exercises workbook is designed to accompany the text. It gives students the practice they need to master arithmetic skills
Help With Math Problems Help With Math Problems Friends math is a very vast subject and it plays an important role in our daily life. It is the study of space, relation, structure, change, and many other topics of pattern. It becomes an essential part in all... [More] Help With Math Problems Friends math is a very vast subject and it plays an important role in our daily life. It is the study of space, relation, structure, change, and many other topics of pattern. It becomes an essential part in all the areas which includes engineering, social science, medicine, natural science etc. Calculus is one of the most important part of mathematics. So today we are going to study about the calculus and how to take help with math solvers to solve our math problems. Calculus is the most important, interesting and most complicated topic in the world of mathematics, as various online calculus tutors are available to remove its complex nature and solve our calculus problems. Calculus plays an important role not only in the area of maths but also in other areas like physics, economics etc. Calculus is basically a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Basically it is the study of Rates of Change .
In implementing the Algebra 2 and Trigonometry process and content performance indicators, it is expected that students will identify and justify mathematical relationships, formally and informally. The intent of both the process and content performance indicators is to provide a variety of ways for students to acquire and demonstrate mathematical reasoning ability when solving problems. Local curriculum and local/state assessments must support and allow students to use any mathematically correct method when solving a problem. Throughout this document the performance indicators use the words investigate, explore, discover, conjecture, reasoning, argument, justify, explain, proof, and apply. Each of these terms is an important component in developing a student's mathematical reasoning ability. It is therefore important that a clear and common definition of these terms be understood. The order of these terms reflects different stages of the reasoning process. Investigate/Explore - Students will be given situations in which they will be asked to look for patterns or relationships between elements within the setting. Discover - Students will make note of possible patterns and generalizations that result from investigation/exploration. Conjecture - Students will make an overall statement, thought to be true, about the new discovery. Reasoning - Students will engage in a process that leads to knowing something to be true or false. Argument - Students will communicate, in verbal or written form, the reasoning process that leads to a conclusion. A valid argument is the end result of the conjecture/reasoning process. Justify/Explain - Students will provide an argument for a mathematical conjecture. It may be an intuitive argument or a set of examples that support the conjecture. The argument may include, but is not limited to, a written paragraph, measurement using appropriate tools, the use of dynamic software, or a written proof. Proof - Students will present a valid argument, expressed in written form, justified by axioms, definitions, and theorems. Apply - Students will use a theorem or concept to solve an algebraic or numerical problem.
This introduction to algorithms solution book is a facsimile reprint and may contain imperfections such as marks introduction to algorithms solution, notations introduction to algorithms solution, marginalia and flawed pages introduction to algorithms solution.
Math Study Books were created by Stella Germain, a mathematics teacher in New York State who has been teaching math for over 13 years. These study books were created to serve as either a supplement or alternative to traditional textbooks, bringing math to students at a level they can easily understand. Currently, 3 study books are available in each of the areas of Integrated Algebra and Algebra 2 with Trigonometry. To learn more about the study books, visit the products page or order today!
From basic math to precalculus, Microsoft Mathematics 4.0 can help you visualize and see mathematical concepts as you've never seen them before. This free downloadable tool includes step-by-step instructions and explains fundamental concepts. The wide range of tools to help students with complex mathematics includes a full-featured graphing calculator that's designed to work just like a hand-held calculator and ink handwriting support to recognize hand-written problems. The Step-by-Step Equation SolverStudents can use this to learn how to solve difficult math problems. Graphing calculatorIts full features and large two-dimensional and enhanced three-dimensional color graphs can better illustrate problems and concepts. Formulas and Equations LibraryStudents will find more than 100 commonly used equations and formulae to help identify and apply equations.
Precalculus Functions And Graphs 9780495108375 ISBN: 0495108375 Edition: 11 Pub Date: 2007 Publisher: Thomson Learning Summary: Clear p...rovides calculator examples, including specific keystrokes that show you how to use various graphing calculators to solve problems more quickly. Perhaps most important-this book effectively prepares you for further courses in mathematics. Swokowski, Earl W. is the author of Precalculus Functions And Graphs, published 2007 under ISBN 9780495108375 and 0495108375. One hundred twenty seven Precalculus Functions And Graphs textbooks are available for sale on ValoreBooks.com, seven used from the cheapest price of $11.64, or buy new starting at $283.31
Book Description: This unusually well-written, skillfully organized introductory text provides an exhaustive survey of ordinary differential equations — equations which express the relationship between variables and their derivatives. In a disarmingly simple, step-by-step style that never sacrifices mathematical rigor, the authors — Morris Tenenbaum of Cornell University, and Harry Pollard of Purdue University — introduce and explain complex, critically-important concepts to undergraduate students of mathematics, engineering and the sciences.The book begins with a section that examines the origin of differential equations, defines basic terms and outlines the general solution of a differential equation-the solution that actually contains every solutionThe theory of differential equations and their application. An abundance of solved problems and practice exercises enhances the value of Ordinary Differential Equations as
For more than two thousand years a familiarity with mathematics has been regarded as an indispensable part of the intellectual equipment of every cultured person. Today, unfortunately, the traditional place of mathematics in education is in grave danger. The teaching and learning of mathematics has degenerated into the realm of rote memorization, theThis best-selling guide--which has sold more than 340,000 copies since its first publication--has been thoroughly updated throughout to correspons to current advanced calculus courses. A complete and comprehensive review of the subject, this updated edition features important new chapters on topology and Laplace transforms and essential new theorems,... more... Large IT organizations increasingly face the challenge of integrating various web services, applications, and other technologies into a single network. The solution to finding a meaningful large-scale architecture that is capable of spanning a global enterprise appears to have been met in ESB, or Enterprise Service Bus. Rather than conform to the... more... Features an introduction to advanced calculus and highlights its inherent concepts from linear algebra Advanced Calculus reflects the unifying role of linear algebra in an effort to smooth readers' transition to advanced mathematics. The book fosters the development of complete theorem-proving skills through abundant exercises while also promoting
Concept Explanations, Sample Problems, Must-Know Shortcuts. Topics Introduction to Matrices Some functions are not easily written as a formula. On a graph, a step function looks like a flight of stairs. The graphs of step functions have lines with an open circle on one end and a closed circle on the other to indicate inclusion, like number line inequality graphs. A rounding step function tells us to round a decimal number to the next whole integer or the previous whole integer. Study Your Way Easy Help. Fun Teachers. Expert teachers who know their stuff bring personality & fun to every video.
My class just finished module 1 in our textbooks. Since it was the beginning of the year it was basically easy stuff and reviews but we still learned. Now that we are done with module 1 we can start to learn more advanced math. In module 1 there were six sections. Section 1 was about data displays, section 2 was about sequences and exponents, and in section 3 we learned about probability. Section 4 was about problem solving and section 5 and 6 were about problem solving and order of operations. In section 1 we learned about tallying data, and bar and line graphs. When you tally data you are supposed to gather the data you want and tally how many times it comes up. A bar graph is a graph that is used to display data that falls into distinct categories. A line graph is graph that shows how data changes over time. That is what we learned in section 1 in module 1. In section 2 we learned about modeling sequences and exponents, squares, and cubes. A sequence is an ordered list of numbers or objects called terms. We learned about term numbers and symbols also. Exponents show a pattern of repeated multiplication. Exponents make it simpler to write out as well. Instead of writing out 3.3.3.3 you could just write 34 or three to the fourth power. In section 3 we were taught about probability and theoretical probability. Probability is experiments that predict the outcome of an event. Probability is also a number from 0 to 1 that tells you how likely something will happen. Section 4 was about problem solving. The textbook gave us four steps to problem solving. Step 1: understand the problem. Step 2: Make a Plan. Step 3: Carry out the plan. Step 4: look back, or check out work. Those were all very helpful to me, especially when I got stuck on a difficult math equation. This was definitely an important chapter to me. Section 5 in the textbook taught the class about evaluating solutions and making connections. Section 6 was about the order of operations. The order of operations helps when you do long mathematical equations. It lets you break down long equations into sections to make it smaller and easier to figure out since it is better laid out. Since we started school that is what we have been learning for the first trimester.
Earthquake Shaking and Damage Eric Baer, Highline Community College This student homework and problem set has students quantitatively earthquake hazard, shaking and damage. Three-Point Problem by Simultaneous Linear Equations William Frangos, James Madison University Students are introduced to the use of linear algebra in an intuitive and accessible way, through classroom activity and homework set. The familiar three-point problem is cast in terms of three dimensional analytic geometry, fostering understanding of mathematical models for simple geometric forms.
You are here Mathematics of Choice: Or, How to Count Without Counting Publisher: Mathematical Association of America Number of Pages: 202 Price: 21.95 ISBN: 0883856158 This text is an engaging, even addictive, introduction to basic combinatorics. Written in a fun and inviting manner, reader interest is amplified by the author's infectious enthusiasm. This is an excellent introduce to combinations and permutations. First published in 1975, before computers and calculators were assumed to be at hand, the exercises in this book can all be done by hand on paper. Students finishing high school or in their first year of college will find this work an excellent adjunct to textbooks and lectures. The work is arranged in a logical progression beginning with the definitions and motivations for factorials, combinations, and permutations. From there the reader moves to binomial coefficients, power sets, and Fibonacci numbers. The effect of repetitions on combinations makes a natural prelude in Chapter Four to the Inclusion-Exclusion Principle and the groundwork for basic probability. From partitions of integers the author moves into a brief and basic, yet cogent and enlightening, explanation of generating functions and some applications for them. The book also includes the Pigeonhole Principle, induction, recursion, and allied topics. Tom Schulte teaches mathematics at Oakland Community College in Michigan.
BetterCalculator7 by Jim Anderson Abstract BetterCalculator 7 Freeware updated 5/21/2012 This Calculator was first developed in 2005 out of neccessity. Many, many hours of coding have gone into it's development. At that time, it was the best looking and most practical Calculator available on any computer. As new operating systems have developed, I've tried to keep pace. As of this update, it works fine on my Windows7 system. This is my contribution to mankind. Please feel free to use it and make it available for others to enjoy. No, I'm not a computer geek... -Jim Anderson- Description CALCULATOR OPERATING INSTRUCTIONS: Help screens are built into your Calculator. Place your cursor over any key or function button and Right Click your mouse button. You'll see a help screen pop up that explains what the key or function does and how to use it. The left hand side of the Calculator is about the same as an ordinary pocket Calculator. You can use your mouse or keyboard to enter numbers and most operands. Left Click your mouse to press a key, Right Click for help with that key or function. Remember, you can copy, save and print out important calculations using the COPY function. You can also add your own text into the RECORD screen simply by placing your cursor anywhere in the RECORD screen and typing your text where you want it.
Mathematics for Plumbers and Pipefitters updated for optimal learning, Mathematics for Plumbers and Pipefitters, 7E remains a trusted resource for essential math concepts in the plumbing and pipefitting trades. With an emphasis on real-world examples that will prepare readers to successfully transfer their knowledge to on-the-job situations, this book utilizes the most currently used fitting materials to demonstrate key concepts. Simplified, clear explanations and a straightforward approach, combined with new units on changes of state, pressure and heat, and mechanical advantage... MORE, make this an ideal tool for anyone entering the field.
How To Teach with DWFK Precalculus Chapter 5: Analytic Trigonometry This is a very important chapter for those who wish to pursue college mathematics, but not just because of the trigonometric material. The importance of equivalent expressions, the emphasis on proof, and the positive reinforcement gained from the ability to switch representations — all so important in analytic trigonometry — come as close to anything in the high school curriculum to approximating what matters to the real practitioners of mathematics. For this reason, we place deliberate emphasis on trigonometric identities as mathematical proofs and motivate them as such in the text. Also, because their proofs are as important as their applications, we have placed the Law of Sines and the Law of Cosines in this section, along with the triangle area formulas. Section 5.1 Fundamental Identities Objectives You will be able to justify the fundamental identities and use them to simplify trigonometric expressions. You will be able to use them to solve certain trigonometric equations. Key Ideas Cofunction identities Odd-even identities Domain of validity Pythagorean identities Identity Trigonometric equation Study Tips The identities that follow directly from the triangle ratios or from the unit circle are called the Fundamental Identities. (Note that we define the concept of "identity" at the beginning of the chapter.) Eventually, you will need to memorize some identities whether they understand them or not; however, they should all be able to understand the Fundamental Identities. Technology Tips Graphing both sides of an identity to see if the graphs match can be a little tedious, but it is a nice way for you to verify their answers. Several other technology tips are shown in the section. Realistically, this section could be taught quite comfortably with no technology at all. Top Section 5.2 Proving Trigonometric Identities Objectives You will be able to decide whether an equation is an identity and will be able to prove identities analytically. Key Ideas Proof of an identity Word ladder Study Tips Some people have lamented the decreased emphasis on proofs in the curriculum since the advent of certain education reforms, but it is only certain kinds of proofs that have been de-emphasized. Axiomatic algebra proofs (characteristic of the "New Math" of the 60's) have been de-emphasized because they did not contribute to student understanding, and two-column proofs in geometry were de-emphasized because they gave a misleading idea of what mathematical proofs look like. Really, trigonometric identities are the ideal introduction to proofs for beginners, as they actually read like proofs in higher mathematics courses. They have a structure that beginners can understand, and you can actually produce them on your own. This textbook embraces identity proofs as pedagogically important and teaches them as such. For example, this kind of identity verification is never suggested: Identity: "Proof": Sometimes this sort of a thing is followed by a check sign, but it does not render any more educationally acceptable a mode of proof that begins with assuming what was to be proved and ends with a tautology. (We demonstrate a preferable alternative to this approach in Example 5.) Our emphasis is on constructing a logical path from what is known to what must be shown (as in the word ladders at the start of the section), as this is what mathematical proofs really look like. Technology Tips The main use of technology in this section is to provide graphical support for what is an identity and graphical refutation of what is not. You will not need to do this for all identities we ask them to prove. In particular, exercises 7–47 are true identities, and all that we require are the proofs. Section 5.3 Sum and Difference Identities Objectives You will understand the derivations of, and be able to apply, the formulas for the cosine, sine, and tangent of a difference or sum. Key Ideas Angle sum formula Reduction formula Study Tips There is some debate about whether modern precalculus students need to memorize these (and other) formulas or not. The debate is not about whether students should memorize things, but rather about what they should be required to memorize and why. (These formulas can be stored on their calculators.) Regardless of where one stands on this debate, precalculus students ought to see how these formulas are derived. Moreover, calculus students will use these formulas later, and it saves time if one does not have to scroll down a calculator screen to find them. Technology Tips Most teachers realize that you can store these (and other) formulas on your graphing calculators, either as text screens or embedded in programs. At first glance you might think that this capability gives you an advantage on tests, but you need to consider that teachers are now less inclined to award ten points on an exam for simply stating a formula. A side effect of technology is that students and teachers alike are faced with finding more creative incentives for memorization. Top Section 5.4 Multiple-Angle Identities Objectives You will understand the derivations of, and be able to apply, the double-angle, half-angle, and power reducing identities. Study Tips The comments in the previous section about memorization apply equally well here. You will definitely use these formulas in calculus, especially the power-reducing formulas, which are the keys to finding certain antiderivatives. Technology Tips The trigonometric equations found in this section are intended as applications of the identities in the section, so solving them with calculators defeats their purpose. The instructions in the exercises should be carefully followed. Top Section 5.5 The Law of Sines Objectives You will be able to understand the proof of the Law of Sines and will be able to use the formula to solve a variety of problems. Key Ideas Law of Sines Solving Triangles Study Tips The Law of Sines could equally well have appeared in the previous chapter, but we felt that it deserved to be among the identities. As with the other identities in this section, the derivation of the formula is a very good thing for precalculus students to see. Technology Tips Application problems of the kind featured in this section have always been part of a trigonometry course, but today's students can arrive at actual answers far more quickly (and more accurately!) than students of previous generations, thanks to technology. This is a good thing, as it allows you to concentrate on the formula rather than on the grubbiness of the computations. The main use of the calculator in this section is actually as a machine that calculates. As they say, go figure. Top Section 5.6 The Law of Cosines Objectives You will be able to understand the proof of the Law of Cosines and will be able to use the formula to solve a variety of problems. You will know Heron's formula and be able to use it to find areas of triangles and to solve appropriate application problems. Key Ideas Dihedral angle Heron's formula Law of Cosines Study Tips You ought to be able to follow the proof of the Law of Cosines, although the proof of Herons' formula might be a little heavy for some. (It's not that complicated, just a little intimidating with those five variables being pushed around.) Technology Tips If you are a good trigonometry student who likes to write calculator programs, you might enjoy the challenge of writing a calculator program that will solve triangles, using the Law of Sines and the Law of Cosines. The user would input three parts of a triangle (SSS or ASA or AAS) and the program would announce all six parts, having computed the missing ones by applying one of the laws. Let alone the difficulty of the programming, there are some subtle difficulties in the mathematics that make it hard to come up with a program that is correct in all cases. Good programmers should try to make the program "idiot-proof," i.e., designed to recognize and reject input values that will not determine a triangle (angles that sum to more than 180°, side lengths that do not satisfy the triangle inequality, etc.).
Precal Trigonometric Functions; Analytic Trigonometry; Analytic Geometry; Counting and Probability; A Preview of Calculus; and more. For individ... MOREuals with an interest in learning Precalculus as it applies to their everyday lives. For undergraduate courses in Precalculus. A proven motivator for students of diverse mathematical backgrounds, these texts explore mathematics within the context of real life using understandable, realistic applications consistent with the abilities of any student. Graphing techniques are emphasized, including a thorough discussion of polynomial, rational, exponential, and logarithmic functions and conics. The use of a graphing calculator is optional. (All chapters end with a Chapter Review and a Project at Motorola). Preface. List of Applications. 1. Graphs. The Viewing Rectangle. Using a Graphing Utility to Graph Equations. Using a Graphing Utility to Locate Intercepts and Check for Symmetry. Using a Graphing Utility to Solve Equations. Square Screens. Using a Graphing Utility to Graph Inequalities. Using a Graphing Utility to Solve Systems of Linear Equations. Using a Graphing Utility to Graph a Polar Equation. Using a Graphing Utility to Graph Parametric Equations.
Product Description Math is often viewed as a completely neutral subject; 1+1=2 is the same around the world, isn't it? Yet this handy book reveals how the rules of a covenant-keeping God allows math to function, as it points out the evidence's of God's hand in creation, the functionality of everyday, practical math, how to view and teach math biblically, resources and curriculum to use. 97 pages, softcover. Related Products Product Reviews Unfortunately the subject of the philosophy of math is beyond the reach of this Christian author. She has no clue as to the nature of numbers and ascribes all kinds of man-made ideas to God without any Biblical support. My college professor told me that 90% of peer reviewed articles are trash. After 30 years, I see that 99.99% of books are trash. This book belongs in that vast trash bin. February 28, 2008 Katherine's excellent book is easy to read. It will not only help you understand God's purpose for math and how math testifies of God, but it will also give you some practical suggestions for implementing what you learn into your home school program. In addition, she rates current math curriculum as to how well they present math in the context of God's Word and includes a sample idea notebook for using math in your daily lives. Congratulations, Katherine, on a job well done! November 11, 2007
Solving Absolute Value Inequalities This video shows how to solve, graph the solutions set, and write solution in interval notation. The instructor demonstrates the steps necessary to correctly solve this type of problem. Several examples are shown. The instructor's conversational tone is easy to follow and very thorough. Author Linear Inequalities in Two Variables, Part IV Mr. Khan shows how to graph Linear Inequalities in Two Variables using the problem Y is less than 3X + 5. Mr. Khan uses the Paint Program (with different colors) to illustrate his points. Sal Khan was the recipient of the 2009 Microsoft Tech Award in Education. (03:03) Author(s): No creator set License information Related content No related items provided in this feed Graphing Systems of Inequalities, Part 3 The instructor in this video, Sal Khan, continues to discuss how to graph inequalities. In this segment, he places a calculator on the screenGraphing Systems of Inequalities, Part 2 The instructor in this video, Sal Khan, in an easy, conversational tone, continues to discuss how to graph inequalitiesTerms for Intro to Judaism class session 3 This is the beginning of a series of videos on defining terms for introduction to Judaism. Models of Judaism, denominations, and spiritual terms are discussed. Terms can be interesting to help understand Judaism. Author(s): No creator set License information Related content No related items provided in this feed Intro to Judaism #4 Terms Jewish life/synagoue This is the beginning of a series of videos on defining terms for introduction to Judaism. Terms for the Jewish life and ritual objects. Terms can be interesting to help understand Judaism. Author(s): No creator set License information Related content No related items provided in this feed Introduction to Judaism This video shows an introduction to what Judaism is by Dr Catanzaro. There is some historical background and a definition of "who are the Jews?" There is reference to modern Judaism, too. Author(s): No creator set License information Related content No related items provided in this feed Probability (Part 8) - Introduction to Bayes' Theorem This is a Featured Video on You Tube. The instructor in this video, Sal Khan, continues his discussion of probability and offers an introduction toIntroduction: Paper Heart Garland: Supplies Learn how to make a paper heart garland. This introduction to the multi-step project explains what the project requires. It require three colors of craft paper, glue, glitter, safety pins, ribbon and scissors. Author(s): No creator set License information Related content No related items provided in this feed Use the Bar Graph to Determine Population (PSSA~ Math 7th-Grade) PSSA~ Math 7th-Grade -- Bar Graph Question. The instructor uses a computer software program to demonstrate how to solve a word problem over bar graphs. The PSSA (Pennsylvania System of School Assessment) is given annually in PA in some grades in K-12. However, the example contained in this video appears on many standardized state-mandated tests. AuthorLine Integral - Proof of Green's Theorem, Part 1 The instructor talks a little too fast in this videoLine Integrals - Proof of Green's Theorem (Part 2) In this second installment, the instructor continues to talks a little too fastAmerica the Story of Us: Life in Jamestown Find out what life was like in the Jamestown settlement. As for all YouTube videos block the comments before showing. A three minute recreation with insights into why the colony was established. Very good introduction. Author(s): No creator set License information Related content No related items provided in this feed Visit to a Museum in England- Learn about Ancient worlds In this programme Year 7 pupils are treated to a visit to the World Museum in Liverpool as an introduction to Key Stage 3 history. Teacher Gareth Rogers from Weatherhead High School on the Wirral chose a day of Egyptian-themed sessions to make best use of the collections his local museum has to offer. On the trip they find out about mummification through role-playing, handle Egyptian treasures and practice writing hieroglyphics. (Professional video) Author(s): No creator set License information Related content No related items provided in this feed Marshmallows Second-graders create and discuss a bar graph based on the number of marshmallows they estimate each person in their class would eat on a camping trip. After discussing their results, students determine how many bags of marshmallows to take. NCTM standards: concepts of whole number operations, statistics and probability, reasoning, problem solving. Author(s): No creator set License information Related content No related items provided in this feed Graphing Linear Inequalities in Two Variables, Part III In this video, Sal Khan demonstrates how to graph a linear inequality. Mr. Khan uses the Paint Program (with different colors) to illustrate his points. Sal Khan is the recipient of the 2009 Microsoft Tech Award in Education. (03:20Against All Odds-What Is Statistics? What Is Statistics? Using historical anecdotes and contemporary applications, this introduction to the series explores the vital links between statistics and our everyday world. The program also covers the evolution of the discipline.' Due to licensing agreements, online viewing of the videos for this resource is restricted to network connections in the United States and Canada. < Author(s): No creator set License information Related content No related items provided in this feed Against All Odds - 6. Time Series 'Statistics can reveal patterns over time. Using the concept of seasonal variation, this program shows ways to present smooth data and recognize whether a particular pattern is meaningful. Stock market trends and sleep cycles are used to explore the topics of deriving a time series and using the 68-95-99.7 rule to determine the control limits.' Author(s): No creator set
Precalculus 9780471756842 ISBN: 0471756849 Pub Date: 2010 Publisher: Wiley Summary: This title offers a clear writing style that helps reduce any maths anxiety readers may have while developing their problem-solving skills. It incorporates parallel word and math boxes that provide detailed annotations which follow a multi-modal approach. Young, Cynthia Y. is the author of Precalculus, published 2010 under ISBN 9780471756842 and 0471756849. Four hundred sixty three Precalculus textbooks are ...available for sale on ValoreBooks.com, one hundred fourteen used from the cheapest price of $59.05, or buy new starting at $113 Marks in pencil, covers scuffed/scratched, minimal shelf wear, Item is intact, but may show shel... [more] MarksMarks in pencil, covers scuffed/scratched, minimal shelf wear, Item is intact, but may show shelf wear. Pages may include notes and highlighting. May or may not include suppl [more] Marks marked/wavy due to contact with moisture/coffee
College Algebra plus MyMathLab/MyStatLabMike Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts... MORE. In the Ninth Edition, College Algebra has evolved to meet today's course needs, building on these hallmarks by integrating projects and other interactive learning tools for use in the classroom or online. 6.9 Building Exponential, Logarithmic, and Logistic Functions from Data 7. Analytic Geometry 7.1 Conics 7.2 The Parabola 7.3 The Ellipse 7.4 The Hyperbola 8. Systems of Equations and Inequalities 8.1 Systems of Linear Equations: Substitution and Elimination 8.2 Systems of Linear Equations: Matrices 8.3 Systems of Linear Equations: Determinants 8.4 Matrix Algebra 8.5 Partial Fraction Decomposition 8.6 Systems of Nonlinear Equations 8.7 Systems of Inequalities 8.8 Linear Programming 9. Sequences; Induction; the Binomial Theorem 9.1 Sequences 9.2 Arithmetic Sequences 9.3 Geometric Sequences; Geometric Series 9.4 Mathematical Induction 9.5 The Binomial Theorem 10. Counting and Probability 10.1 Sets and Counting 10.2 Permutations and Combinations 10.3 Probability Appendix: Graphing Utilities 1 The Viewing Rectangle 2 Using a Graphing Utility to Graph Equations 3 Using a Graphing Utility to Graph Equations Locating Intercepts and Checking for Symmetry 4 Using a Graphing Utility to Solve Equations 5 Square Screens 6 Using a Graphing Utility to Graph Inequalities 7 Using a Graphing Utility to Solve Systems of Linear Equations Michael Sullivan, Emeritus Professor of Mathematics at Chicago State University, received a Ph.D. in mathematics from the Illinois Institute of Technology. Mike taught at Chicago State for 35 years before recently retiring. He is a native of Chicago's South Side and divides his time between a home in Oak Lawn IL and a condo in Naples FL. Mike is a member of the American Mathematical Society and the Mathematical Association of America. He is a past president of the Text and Academic Authors Association and is currently Treasurer of its Foundation. He is a member of the TAA Council of Fellows and was awarded the TAA Mike Keedy award in 1997 and the Lifetime Achievement Award in 2007. In addition, he represents TAA on the Authors Coalition of America. Mike has been writing textbooks for more than 35 years and currently has 15 books in print, twelve with Pearson Education. When not writing, he enjoys tennis, golf, gardening, and travel. Mike has four children: Kathleen, who teaches college mathematics; Michael III, who also teaches college mathematics, and who is his coauthor on two precalculus series; Dan, who is a sales director for Pearson Education; and Colleen, who teaches middle-school and secondary school mathematics. Twelve grandchildren round out the family.
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: 049582948X43.76 +$3.99 s/h New Textbookcenter.com Columbia, MO Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book $52.25 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 New Book. Shipped from US. Standard Shipping is 4 to 14 business days. Expedited Shipping is 3 to 6 business days. Established seller since 2000. $55.77 +$3.99 s/h New PaperbackshopUS Secaucus, NJ New Book. Shipped from US within 4 to 14 business days. Established seller since 2000 $5657.95 +$3.99 s/h New Textbook Barn Woodland Hills, CA PAPERBACK New 049582948X Premium Books are Brand New books direct from the publisher sometimes at a discount. These books are NOT available for expedited shipping and may take up to 14 business day...show mores to receive. ...show less $62.60
Linear Algebra For introductory sophomore-level courses in Linear Algebra or Matrix Theory. This text presents the basic ideas of linear algebra in a manner that ...Show synopsisFor introductory sophomore-level courses in Linear Algebra or Matrix Theory. This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof.Hide synopsis Description:New. When it comes to learning linear algebra, engineers trust...New. When it comes to learning linear algebra, engineers trust Anton. The tenth edition presents the key concepts and topics along with engaging and contemporary applications. The chapters have been reorganized to bring up some of the more abstract topics... Elementary Linear Algebra I bought this book because the author and picture of cover is same at the book store. However, it turned out to be an international student version which is not even authorized to sale in U.S. Most basic is same but the problems are different from U.S. version so I am stuck with this book now
FIRST COURSE IN COMPUTATIONAL PHYSICS by DEVRIES No options of this product are available. Rent Our Price: $44.41 Term: Description Computers and computation are extremely important components of physics and should be integral parts of a physicist s education. Furthermore, computation physics is reshaping the way calculations are made in all areas of physics. Intended for the physics and engineering students who have completed the introductory physics course, A First Course in Computational Physics, Second Edition covers the different types of computational problems using MATLAB with exercises developed around problems of physical interest. opics such as root finding, Newton-Cotes integration, and ordinary differential equations are included and presented in the context of physics problems. A few topics rarely seen at this level such as computerized tomography, are also included. Within each chapter, the student is led from relatively elementary problems and simple numerical approaches through derivations of more complex and sophisticated methods, often culminating in the solution to problems of significant difficulty. The goal is to demonstrate how numerical methods are used to solve the problems that physicists face.
Discrete Math Track Track Content Welcome to the Discrete Math refresher course. This refresher is designed to refresh (or create afresh) your intuition about the basic tools of discrete math and graph theory. We expect to cover the bread and butter of discrete mathematics, while also reserving time to discuss a few active research questions in our brief tour through the territory. In the end, you should find yourself with a deepened understanding of combinatorics, sorting, graph theory fundamentals, breadth- and depth-first search, minimum spanning trees, flow algorithms, complexity theory and notation, approximation algorithms, and current research areas in discrete math and graph theory. We take the view that algorithms and analysis are easiest to internalize when coupled with applications, and approach each new technique with an example firmly in mind. We will not shy away from proofs, but will do our best to nurture the novice while exerting the expert.
In the mathematics faculty at Tauhara we aim to ensure that every student has access to the level of mathematics tuition that they need to make sense of the world around them. Our programmes are tailored to meet the learning needs of our students, with every care taken to ensure that students are taught at their curriculum level and pace. In a broad sense Mathematics is the study of patterns - be they natural for measurement, or data for investigation, numerical for generalisation or even physical for manipulation. Joy of Mathematics is a fascination with these patterns, the way they work and the possibilities they can lead to. Mathematics is the language of the universe: our aim is that all students attain a level of fluency in this language before moving out to find their own places in our world. A first goal for us and every student is the attainment of at least eight (8) numeracy/mathematics credits so that NCEA Level One can be achieved. Other goals include: - 14 credits in Mathematics at Level One or higher for University Entrance - monitored improvement for every junior student - improvement in Merit and Excellence pass rates Mathematics is compulsory up to Year 11, but encouraged beyond that if achievement is appropriate at earlier levels. Subjects and Levels Taught Mathematics - Junior Years 9-10 Mathematics - NCEA Level 1 - Achievement Standards Course Year 11 Mathematics - NCEA Level 1 - Unit Standards Course Year 11 Mathematics - NCEA Level 2 - Achievement Standards Course Year 12 Mathematics - NCEA Level 2 - Unit Standards Course Year 12 Calculus Year 13 Statistics and Modelling Year 13 NCEA Level 1 Mathematics 11 Maths 1: Achievement Standard Course, aimed at students definitely intending to continue their mathematics study through the senior school and onto university. 11 Maths 2: Unit Standard course, tailored for those students not intending to pursue Maths in tertiary education, or beyond Year 11 except in special cases. NCEA Level 2 Mathematics 12 Maths 1: Achievement Standard Course, for those intending to study mathematics at NCEA Level 3 in Year 13. 12 Maths 2: Unit Standard course, tailored for those students intending to finish their mathematics study at Year 12. NCEA Level 3 Mathematics: The Big Stuff Level 3 Mathematics with Calculus: Achievement Standard Course vital for those intending to pursue careers in Mathematics, Engineering, Architecture or the Sciences Level 3 Statistics and Modelling: Achievement Standard course vital for those intending to pursue a wide variety of academic pathways, including careers in Medicine, Finance or the Sciences Faculty Staff Mrs Janice Lyke Head of Faculty Mathematics Mr Ross Shaw Senior Programme Overview Mr Varghese Painuthara BYOD/Technology/Competition Ms Juliette Jones TIC Year 10/12 Mr Jason Butterfield Mathematics Ms Amita Ganda Mathematics Mr Frans Paulussen Mathematics Special Activities Learning in the Mathematics faculty is supplemented by additional activities such as after school tuition, competitions, project work, homework and web-based resources.
Get the Popular EducationMath Cheat Sheets FREE 09 Dec 2011 15:07:17 +0000The best Collection of Math cheat sheets and quick reference cards<br><br>Many Cheat Sheets in your mobile to have the formulas wherever and whenever you want.<br><br>With this application you can use your travel time to study, or just have it as a quick reference when needed.<br><br>NOTE: App can be moved to the SD Card!!<br><br>Contents:<br>=========<br>Algebra. Elementary techniques for factoring binomials and trinomials<br>Algebra. Exponent laws and factoring tips<br>Algebra. Solving quadratic equations by completing the square<br>Algebra. College Algebra quick reference<br>Algebra. Solution of the 3rd degree polynomial equation<br>Algebra. Solution of the 4th degreee polynomial equation<br>Trig. Basic trig identities<br>Trig. Law of sines cosines etc and other triangle formulas<br>Trig. Graphs of the trig functions<br>Trig. Inverse trig functions<br>Trig. Power reducing formulas for powers of sines and cosines<br>Trig. Graph paper for plotting in polar coordinates<br>Trig. Two unit circles with trig funcion values<br>Trig. Single unit circle with trig function values<br>Calculus. Basic differentiation formulas and some useful trig identities<br>Calculus. Basic differentiation and integration formulas<br>Calculus. Definitions and theorems pertaining to Riemann sums and definite integrals<br>Calculus. A quick reference sheet on Taylor polynomials and series<br>Calculus. A summary of convergence tests<br>Calculus. Guidelines for evaluating integrals involving powers of sines and cosines<br>Calculus. Guidelines for evaluating integrals involving powers of secants and tangents<br>Calculus. Standard forms for conic sections<br>Calculus. Common infinite series<br>Calculus. Trigonometric substitution<br>Calculus. Cylindrical coordinates<br>Calculus. Spherical coordinates<br>Calculus. Hyperbolic functions<br>Calculus. Applications of integrals<br>Calculus. Applications of integrals<br>Calculus. Common ordinary differential equations<br>Calculus. Common ordinary differential equations<br>Calculus. Common ordinary differential equations<br>Calculus. Undetermined coefficients and variation of parameters<br>Calculus. Vector formulas<br>Calculus. Simple summary of cylindrical and spherical coordinates<br>Misc. Some prime and composite numbers<br>Misc. Sets. Functions lines and sequences<br>Statistics formula sheet Page 1<br>Statistics Page 2<br>Statistics Page 3<br><br>SD Installation support<br><br>If you wish you can show your appreciation and support future development by donating here! You can also donate via Bitcoin. My address is: 1GFAcD6piVbv4fN6Xw6tNe5yrgGmdG2kcz<br><br>Recent changes:<br>Fixed problem with permissions. App was requesting too many. Sorry for that. Should be fine now.C4droidGCC plugin forC4droid 19 Jul 2011 11:27:12 +0000WARNING! It's not a standalone application and it doesn't contain any activities to launch.<br>Make sure to use latest version of main app installed from Google Play.<br>Don't write a review if C4droid wasn't installed from Google Play - it will be ignored anyway.<br><br>This plugin allows to compile C and C++ using GNU GCC (C and C++ compiler).<br>It contains GCC 4.8.1 with Bionic (Android libc), email me to get the source code.<br><br>To download this app you must have C4droid installed (other usage is prohibited).<br>Android is a trademark of Google Inc.<br>Some executables in this app package are licensed under (L)GPL, email me to get the source code.<br><br>Recent changes:<br>Added SDL2 support.<br>Stdlibc++ is now 4.8, was fixed by me.Mountaineering 16 Jul 2011 22:11:38 +0000 <br><br>This <br><br>* This application is completely based on the United States MILITARY MOUNTAINEERING Field Manual FM 3-97.61. * <br><br>Features <br>****** <br>- Both Portrait and Landscape modes are supported. Landscape mode is preferred for better reading experience. <br>- Zooming is supported in order to see the text/images better. <br><br>List of Contents <br>************ <br><br>CHAPTER 1. MOUNTAIN TERRAIN, WEATHER, AND HAZARDS <br>CHAPTER 2. MOUNTAIN LIVING <br>CHAPTER 3. MOUNTAINEERING EQUIPMENT <br>CHAPTER 4. ROPE MANAGEMENT AND KNOTS <br>CHAPTER 5. ANCHORS <br>CHAPTER 6. CLIMBING <br>CHAPTER 7. ROPE INSTALLATIONS <br>CHAPTER 8. MOUNTAIN WALKING TECHNIQUES <br>CHAPTER 9 MOUNTAIN STREAM CROSSING <br>CHAPTER 10. MOVEMENT OVER SNOW AND ICE <br>CHAPTER 11. MOUNTAIN RESCUE AND EVACUATION <br>APPENDIX A. LEVELS OF MILITARY MOUNTAINEERING <br>APPENDIX B. MEASUREMENT CONVERSION FACTORS <br>APPENDIX C. AVALANCHE SEARCH AND RESCUE TECHNIQUES <br><br><br>Note: Ad supported application. For ad free version buy Mountaineering Pro<br><br>Recent changes:<br>v1.1 Fixed Force Close issue on Android v4 devices.Decimals 03 Mar 2013 03:12:05 +00001) Learn Decimals <br> - Converting fraction to a decimal, <br> - comparing decimals, <br> - decimal addition, <br> - decimal subtraction, <br> - decimal multiplication and<br> - decimal division<br> - 150 questions on above topics<br> - questions are presented randomly<br> - tutorial on all the above topics<br><br> 2) General Features - These are common features on all apps from sarmacs.com:<br> The screens are designed for simplicity and easy navigation.<br> No display of ads on any screen.<br> No location tracking.<br> No offline notifications or messages.<br><br>3) Our Privacy Policy:<br> We do not collect any information from the user.<br> We do not ask the user for any personal details like name, age etc.<br> We do not transmit any information on user.<br> We want to provide a very safe environment for kids to use our apps.<br> <br>4) Apps currently available from sarmacs.com<br>1 Simple Equations (3rd to 8th Grades)<br>2 Simple Math (1st to 6 th Grades)<br>3 PEMDAS ( 3rd to 8th Grades)<br>4 Mean, Median And Mode (3rd to 7th Grades)<br>5 General Knowledge (3rd to 5th Grades)<br>6 Divisibility Rules (3rd - 7th Grades)<br>7 Pre-Algebra (3rd to 7th Grades)<br>8 Basic Facts – Math Puzzles (2nd - 7th)<br>9 Geometry (3rd to 7th Grades)<br>10. Fractions (4th to 7th Grades)<br>11. Decimals ((5th to 8th Grades))<br>12. Kindergarten Math (Pre-K to 1st Grade)<br>13. 2nd Grade Math (2nd Grade)<br>14. 3rd Grade Math (3rd Grade)<br>15. 4th Grade Math (4th Grade)<br>16. 5th Grade Math (5th Grade)<br>17. English Grammar (2nd to 7th grades)<br><br>Recent changes:<br>Version 6 changes: <br>- Updated tutorial. <br>- Added Previous and Next buttons to tutorial and also Page No. User can now navigate back and forth using the Previous and Next buttons.Grade 6 Science by 24by7exams 09 Mar 2013 08:27:35 +0000Most comprehensive study material for Grade 6 Science in the form of multiple choice questions. Topics covered are:<br>- Sources of food<br>- Components of Food<br>- Fiber to Fabric<br>- Sorting Materials<br>- Separation of Substances<br>- Changes around us<br>- Living and non living things<br>- Plants: structure & functions<br>- Body and Movements<br>- Measurement and Motion<br>- Electricity<br>- Work & energy<br>- Nature of matter<br>- Light and shadow<br>- Magnets<br>- Fabric from fibre<br><br> If you would like to cover any additional topics, please send an email to support@24by7exams.com with your feedback. We will try our best to cover that topic in the next upgrade. ***<br><br>Other positive points:<br>- No distracting animations<br>- Focus is on exam but students also end up learning from their mistakes<br>- Pause after every questions by changing settings<br>- Confusing options to ensure students has really understood the topic<br>- Tap trash button to reset score and practice again<br><br>There is no shortcut to teaching and spending quality time with kids/students. We request you to observe your kids/students while they try to answer questions presented in this app. Help them where needed. Once you are confident that they have understood all concepts - encourage them to practice using this app every day till perfect score becomes a norm.<br><br>We are proud about the carefully selected questions on various topics. At the end of every test you will get clear picture about student's understanding of the topic.<br><br>This question bank helps students to get their basics right. They can build further expertise once they gain more confidence in getting perfect score.Factorization, GCF and LCM 26 Sep 2013 10:12:29 +0000•Type your number, Factorization Tool will do the factoring and prime factorization for your<br>•Practice Mode and Quiz Mode<br>•Lessons and Example for Factorization, GCF, LCM<br>•Scratch Pad <br>•Elegantly Designed and User Friendly Interface<br><br>Factorization, GCF and LCM is a cool way to learn, practice and quiz on how to factor a number, prime factorization, GCF and LCM.<br><br>This app is a wonderful tool for any student, parents and teachers. Type your number, out come a list of all factors, and a list of prime factors for the number you typed. Now prime factorization is as easy as simply typing your numbers.<br><br>Practice Mode and Quiz Mode offer a fun, innovative and challenging way for young learners to master the concepts and their uses. <br><br>Lessons and Examples section offers comprehensive lessons which introduce the concepts, techniques used to solve problems. Various types of examples are included to demonstrate how the problem solving techniques are used.Grammar Express : Nouns 04 Sep 2010 05:55:22 +00001. Over 48 Pages of Grammar Lessons.<br>2. Over 327 examples with Grammar Rules.<br>3. Over 450 Test Questions with Explanation. <br><br>Grammar Express: Nouns is the complete course in mastering the kinds of noun such as common noun, proper noun, collective noun, material noun and abstract noun. It contains over 48 pages of lessons explaining Noun-Number, Gender and Case with several examples. You can learn the grammar rules, study the examples and test their understanding by taking quiz. At the end of quiz the user is presented with test summery and explanation for each test question. Grammar Express can help you to improve your understanding of kinds of noun and their classification into concrete and abstract noun. Custom timer setting can assist you to improve response times under exam time constraints. Research suggest that kids and adults learn most quickly when playing learning games with real-time error-feedback. Grammar Express provides learners with an opportunity to improve strategies for grammar success. <br><br>~~~~~~~~~~~~~~~<br>PREPARE BY TOPICS:<br>~~~~~~~~~~~~~~~<br>You can study the grammar rules by topic. All questions are sorted by topic. It also tells you the areas that each topic covers. <br>1. Nouns <br>2. Noun-Case <br>3. Noun-Gender <br>4. Noun-Number <br><br>~~~~~~~~~~~~~~~<br>MOCK TEST MODE:<br>~~~~~~~~~~~~~~~<br>In mock test questions are presented randomly from all the topics. <br><br>~~~~~~~~~~~~~~~<br>DETAILED TEST RESULTS:<br>~~~~~~~~~~~~~~~<br>A summary of the practice test is presented at the end of each test. It shows you the time you took, the score, which questions you answered correctly and where you were wrong. <br><br>~~~~~~~~~~~~~~~<br>PROGRESS METER:<br>~~~~~~~~~~~~~~~<br>The app records your progress as you start giving practice tests. <br>It shows you a beautiful bar chart so that you can track your weak areas and give more focus on them. <br><br>~~~~~~~~~~~~~~~<br>VERY EASY TO USE:<br>~~~~~~~~~~~~~~~<br>The slick user interface allows you to choose from possible answers. <br>You don’t need to press too many buttons or encounter any alert messages. <br><br>~~~~~~~~~~~~~~~<br>FEATURE LIST:<br>~~~~~~~~~~~~~~~<br>• Over 48 pages of grammar lessons and rules with over 327 examples.<br>• Over 450 multiple -choice questions with explanation.<br>• Choose number of questions you would like in each test.<br>• A new module, “Pie chart” keeps track of how you are performing in a particular topic or mock test.<br>• Choose your own timer settings.<br>• Special algorithm that randomizes questions every time you take a test.<br><br>Keywords :- EFL, ESL, GRE, IELTS, TOEFL, TOIEC, basics, case, drills, english, gender, grammar, guide, language, learn, lessons, material, notes, nouns, number, quiz, teacher, test, tutor, vocabulary<br><br>Recent changes:<br>Fixed crash related bugs.<br>Fixed all data related bugs.<br>New theme.<br>Progress tracker with beautiful Pie chart.<br>Added support for both orientations.<br>Added beautiful Bar chart for Mock Test<br>Added support for tablets.Geometry Quest 19 Mar 2013 05:48:50 +0000<br><br>• Animated character gives hints on solving problems<br>• Passport stamps awarded for perfect quests<br>• Successfully solving a city unlocks the next city on the map<br>• Covers Common Core Standards: 3MD, 3G, 4MD, 4G, 5MD, 5G, 6G, 7G, 8G<br>• No ads<br>• No in-app purchases or links to the web<br><br>.<br><br>Recent changes:<br>Adds image for HD icon.Thu, 26 Sep 2013 10:12:29 +0000Thu, 26 Sep 2013 10:12:29 +0000
The Nuffield Advanced Mathematics reader provided articles as background or extensions to topics covered elsewhere in the course. The aim was to encourage students to make further study of the development and applications of the ideas about which they were learning. This was one of the ways by which the course team illustrated how… The Nuffield Advanced Mathematics Resources file provided supporting material for four types of calculator and for the spreadsheet, Excel. The program listings in the calculator sections were designed to be transparent and easy to understand. They followed the algorithms in the Nuffield texts as closely as possible, and they used… This Nuffield Advanced Mathematics option consisted of two units of work on different themes. A likely model for coursework was that students would develop an aspect of interest from one of the themes into a longer project. There are a variety of types of topic within the chapters of this book. In the 'Music and mathematics'The Nuffield Advanced Mathematics course took advantage of computer graphics programmes to introduce into the A-level course an option of studying surfaces, their gradients and other properties. This field had traditionally been the subject of first year university courses. Students who attempted this option were encouraged to use… A substantial part of this Nuffield Advanced Mathematics option was about algorithms. Students used calculators, or computers with a structured programming language, to turn algorithms into programs. They were introduced to various aspects of discrete mathematics and, in particular, they were shown how many of these relate to important… The two units in this Nuffield Advanced Mathematics option were independent of one another, and of unequal in length. 'Complex numbers' needed more time than 'Numerical methods'. The unit on complex numbers developed the arithmetic and geometry of complex numbers and led up to a section on fractals. The… The emphasis in this Nuffield Advanced Mathematics option was on non-parametric methods to help students to gain a good understanding of statistical processes. Students required access to a computer with a professional statistics package to gain the full benefit from this option. Contents 1. Introduction 2. Collecting real… As with Mechanics 1, the approach in this Nuffield Advanced Mathematics book was one of guided modelling supported by practical work with the same expectations that, in each investigation students would: • define the problem that you are going to investigate • set up a mathematical model of the situation • analyse… The approach in this Nuffield Advanced Mathematics books was one of guided modelling supported by practical work. In each investigation, students were expected to: • define the problem that you are going to investigate • set up a mathematical model of the situation • analyse the situation mathematically •… Book Five of Nuffield Advanced Mathematics contained the last four units of the core A2 course. The units in this book completed the treatment of calculus and statistics in the core. Contents Unit 21: More calculus Unit 22: Plane curves Unit 23: More differential equations Unit 24: Distributions Summaries and exercises Hints… … Book Three of Nuffield Advanced Mathematics contained the units that, with Books One and Two, made up the AS course. These units extended the students' knowledge of algebra, calculus, exponential functions and handling data. It also introduced vectors. Contents Unit 11: Equations and inequalities Unit 12: Correlation and… Contents Unit… The teacher's notes for Nuffield Advanced Mathematics were designed to help teachers organise and to support their work with students. The student materials were designed to support a flexible approach to teaching and learning mathematics. The teacher's notes made suggestions for varying teaching approaches. They encouraged… Book One of Nuffield Advanced Mathematics was the first of five core text books. It bridged the gap between GCSE and AS and A level Mathematics. It was designed to be accessible to students who had attained GCSE grade C and above, and to develop a strong foundation for further study. Contents Unit 1: Modelling and calculators Unit…
Contoh Latihan Matematik Darjah 1The Math Forum @ Drexel University. The Math Forum is the comprehensive resource for math education on the Internet. Some features include a K-12 math expert help service, an extensive database of math ....
In addition to teaching courses for math majors, Dr. Sutherland teaches math courses that act as service courses for many... see more In addition to teaching courses for math majors, Dr. Sutherland teaches math courses that act as service courses for many departments within the University. These courses are typically taught with multiple lecturers covering the same material in lectures sized between 35 and 260, with total enrollments of up to 1000 students. There are unique challenges to teaching these courses since students are often only present to fulfill general education requirements and mathematics is not a specific interest for them. Scott discusses the pros and cons of integrating discipline specific examples into his classes. He uses multiple digital video cameras to capture the development of math problems on the classroom blackboards and interactive software to show dynamically the effects of changing formulae and values on structures etc. Dr. Sutherland also uses a web based homework management system called WebAssign and has found that the use of clickers has had a significant effect of the performance of his students.Stony Brook University TLT website:This is a free online calculator for common statistics and data analysis for categorical data, continuous data, statistical... see more This is a free online calculator for common statistics and data analysis for categorical data, continuous data, statistical distributions and interpreting P values, random numbers and chemical and radiochemical data. SplashMath is a fun and innovative way to practice math. With 140+ adaptive worksheets with virtually infinite problems... see more SplashMath is a fun and innovative way to practice math. With 140+ adaptive worksheets with virtually infinite problems spanning across 10 chapters, Splash Math covers the math curriculum for Grades 1-4. It is by far the most comprehensive math workbook in the app store. All the included problems are aligned with the common core standards. Each is available on this website.There is a separate app for each grade. iPhone apps are $4.99 and iPad apps are $9.99 This website presents materials and results from NSF supported work on Diagnostic Question Clusters to Improve Student... see more This website presents materials and results from NSF supported work on Diagnostic Question Clusters to Improve Student Reasoning and Understanding in General Biology Courses. The website provides access to the Diagnostic Question Clusters and active learning strategies associated with the questions plus the pedagogical and research context for these materials.We have targeted two challenges to the teaching of General Biology:1) Most students do not address biological questions with the principles? and reasoning used by biologists and2) Most faculty do not teach students how to use the principles and to think like practicing biologists. The project centers on a set of interrelated Diagnostic Question Clusters (DQCs) designed to "hook" biology faculty to question and learn about their students' understanding of core biological concepts and ways of thinking about biology.The DQCs we focus on here concern tracing energy and matter through 3 levels biological complexity (atomic/molecular/cellular, organism and ecosystem). Dr. Gatteau discusses his experience
BEGINNING AND INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS, shows students how to apply traditional mathematical skills in real-world contexts. The emphasis on skill building and applications engages students as they master algebraic concepts, problem solving, and communication skills. Students develop sound mathematical skills by learning how to solve problems generated from realistic applications, instead of learning techniques without conceptual understanding. Authors Mark Clark and Cynthia Anfinson have developed several key ideas to make concepts real and vivid for students. First, the authors place an emphasis on developing strong algebra skills that support theMore... applications, enhancing student comprehension and developing their problem solving abilities. Second, applications are integrated throughout, drawing on realistic and numerically appropriate data to show students how to apply math and to understand why they need to know it. These applications require students to think critically and develop the skills needed to explain and think about the meaning of their answers. Third, important concepts are developed as students progress through the course and overlapping elementary and intermediate content in kept to a minimum. Chapter 8 sets the stage for the intermediate material where students explore the "eyeball best-fit" approach to modeling and understand the importance of graphs and graphing including graphing by hand. Fourth, Mark and Cynthia's approach prepares students for a range of courses including college algebra and statistics. In short, BEGINNING AND INTERMEDIATE ALGEBRA: CONNECTING CONCEPTS THROUGH APPLICATIONS develops strong mathematical skills using an engaging, application-driven and problem solving-focused approach to algebra
Discrete Mathematics and its Applications, Seventh and demonstrates the relevance and practicality of discrete mathematics to a wide a wide variety of real-world applications…from computer science to data networking, to psychology, to chemistry, to engineering, to linguistics, to biology, to business, and to many other important fields. Key features Improved Introduction and Organization - For the seventh edition the first part of the book has been restructured to present core topics in a more efficient, more effective, and more flexible way. Exercises and Worked Examples – There are over 3800 exercises and 750 examples in the text, from straightforward problems that develop basic skills to a large number of intermediate and challenging exercises. Exercise sets also contain special discussions that develop new concepts not covered in the text, enabling students to discover new ideas through their own work. *Answers to ODD numbered problems are in the back of the book. WORKED OUT SOLUTIONS for these ODD numbered problems are in the PRINTED Student's Solutions Guide (0-07-7353501). Complete SOLUTIONS for the EVEN NUMBERED PROBLEMS are available for the Instructor ONLY in the Instructor's Resource Guide link under the Instructor Resources. Historical Information, Biographies, and Updates on Latest Discoveries – The background of many topics are succinctly described in the text using historical footnotes and brief biographies of more than 65 mathematicians and computer scientists who were (and are) important contributors to discrete mathematics. Clarity and Precision – Rosen's writing style is direct and pragmatic. Precise mathematical language is used without excessive formalism and abstraction. Care has been taken to balance the mix of notation and words in mathematical statements. More Flexible Organization – The dependence of chapters on previous material has been minimized to allow instructors flexibility to pick and choose topics. Each chapter is divided into sections of approximately the same length, and each section is divided into subsections that form natural blocks of material for teaching. Instructors can easily pace their lectures using these blocks. Separate chapters on Algorithms and Number Theory and Cryptography. Accessibility – This text has proven to be easy to read and understand by beginning students. There are no mathematical prerequisites beyond college algebra for almost all of this text, and the few places in the book where calculus is referred to are explicitly noted. An updated Web Resources Guide containing new links to hundreds of external websites relevant to the text material. * An updated Exploring Discrete Mathematics with Maple guide featuring new material tied to the text and full compatibility with Maple 10 * An updated Applications of Discrete Mathematics supplement containing in-depth explorations of applications, with exercises and projects * Additional instructor resources for in-class use, such as printable tests, image banks, lecture notes, and materials donated by our community of users Expanded Coverage of Logic and Sets - helps students better understand these fundamental concepts using well known examples like Sudoku and Hilbert's Grand Hotel. New and Enhanced Features in the Text - The seventh edition offers improvements that make the text easier and more rewarding to use. Margin notes, added explanation and detail, revision of examples and problems, and Bourbaki's "dangerous bend" symbol to alert students to topics that require extra attention have been added to keep students on track. About the author Kenneth Rosen Kenneth H. Rosen is a Distinguished Member of the Technical Staff at AT&T Laboratories in Middletown, New Jersey. His current assignment involves the assessment of new technology and the creation of new services for AT&T. Dr. Rosen has written several leading textbooks and many articles. Rosen received his Ph.D. from MIT.
Description of Course Content This course will cover function notation, basic graphing techniques, polynomial, rational, exponential, and logarithmic functions.Applications of the mathematics will be stressed.A graphing calculator is required, preferably a Texas Instruments TI-83 Plus, TI-83 or a TI-82. Students will be taught: Chapter 2-Functions and their Graphs Definition of a relation, Domain of a function, Increasing/decreasing intervals for a function, Symmetry of functions, Graphs of basic function Law of sines, law of cosines, polar equations, graphs of polar equations, complex numbers in polar form Rating There will be 6 tests @ 75 points each and a comprehensive Final @ 100 points. The total possible points earnable will be 550.Grades will be assigned on a point scale format: A = 495 or more B = 440-494 C = 385-439 D = 330-384 F = 0-329 Attendance Policy Attendance is mandatory to do your best in the course.I will attempt to take the roll daily.If you miss more than 5 times (following the last day to drop the course) you will automatically be dropped from the course. Cheating or any other behavior that is disruptive in the class will not be TOLERATED.If you are caught cheating on a test you will be given an F. No make up exams will be given.If you miss a test you will wait until the final exam to get amake-up grade. THE DEADLINE TO DROP THIS CLASS WITHOUT PENALITY (YOU RECEIVE A W) IS February, 27th.
Mathcentre provide these resources which cover a selection of the mathematics used in the field of engineering and include working with Fractions, aspects of Algebra and Arithmetic, through to Differentiation, Integration, Matrices and Complex Numbers. They aim to offer support to students, in the early stages of their degree… Mathcentre provide these resources, which cover a selection of the mathematics which resources, which cover a selection of the mathematics that refresher packs covering Algebra, Numeracy, Differentiation and Calculus. They were designed as activities to support students by preparing them, prior to the commencement of their university course, for the mathematical demands of their programmes. The Algebra Refresher booklet is sent out to all honours…
Automatically Generating Algebra Problems Abstract We propose computer-assisted techniques for helping with pedagogy in Algebra. In particular, given a proof problem p (of the form "Left-hand-side-term = Right-hand-side-term"), we show how to automatically generate problems that are similar to p. We believe that such a tool can be used by teachers in making examinations where they need to test students on problems similar to what they taught in class, and students in generating practice problems tailored to their specific needs. Our first insight is that we can generalize p syntactically to a query Q that implicitly represents a set of problems [[Q]] (which includes p). Our second insight is that we can explore the space of problems [[Q]] automatically, use classical results from polynomial identity testing to generate only those problems in [[Q]] that are correct, and then use pruning techniques to generate only unique and interesting problems. Our third insight is that with a small amount of manual tuning on the query Q, the user can interactively guide the computer to generate problems of interest to her. We present the technical details of the above mentioned steps, and also describe a tool where these steps have been implemented. We also present an empirical evaluation on a wide variety of problems from various sub-fields of algebra including polynomials, trigonometry, calculus, determinants etc. Our tool is able to generate a rich corpus of similar problems from each given problem; while some of these similar problems were already present in the textbook, several were new!
Formats Trade in Oxford Primary Maths Dictionary (2008 edition) for an Amazon.co.uk gift card of up to £1.20, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more Book Description a graduate with 'A' level maths (and incidentally author of the crazy golf odyssey 'Nutters with Putters') and wanted a reference book for my 9 year old. We were discussing polygons and polyhedrons and I asked my son to look up 'polyhedron'. I was dismayed that a picture of an octahedron was described as a tetrahedron! So I decided to give the book a quick scan for other errors and it didn't take me long to find a host more. I wont detail them all here. The definition for 'diagonal' is wrong. The explanations for cube roots and particularly square roots were very poor. I wasn't aware that there was a symbol for 'indirect proportion' but there is one for direct proportion which wasn't listed. Also not listed under division is the '/' symbol. Where is the definition for Trapezoid? The 'link words' should have been more thorough, 'cube' should link to 'power' for instance. What is the audience in the UK for this publication? If, it is as the title suggests when was the last time the publishers viewed the relevant curricula? And I would have liked a little section at the back on common prefixes (poly, deci, equi etc) and suffixes. A 'family tree' of triangles and quadrilaterals would be nice too! I expected much more given that this is from the Oxford University Press. And back to the original question and the answer is I'm not sure! Not sure at all!! Fantastic book, clearly laid out and easy to use. Has been a great help with homework and would thoroughly recommend to help mums and dads to help understand the terminology now used in schools. Would not be without it. This book was purchased for my grandson to help him with his school homework. As this is the same book used at the school he attends I am hoping it will be of help to him in the coming years. Also for his younger brother when the time arrives.
The Core and More This resource has been compiled primarily for mathematics teachers of the 16-18 age range through support from the Mathematics Centre at the University of Chichester by four teachers who were central to the RAMP A level course from its inception. They were released from their teaching to analyse and evaluate the course in operation, identify strengths and weaknesses and consider the contribution of the course in improvement in teaching and learning mathematics at advanced level. The booklet covers some of the mathematical activities undertaken by students. They are representative of those introduced by teachers to enable students to master effectively a broad base of mathematical content and skills in the context of problems relevant to them. Activities include: Design a barrel, Finding which items in a fruit bowl can be mathematically modelled, Statistics using travel data, 3-dimensional puzzles and problems, Trigonometric graphs, and Vectors
Students will be able to correctly determine the area under a curve (on a graph of a function) using Riemann Sums and the JCM Applet available on Technical Notes: Each student needs access to a computer. If there are not enough computers per student then students may work in groups. Text of Learning Exercise: Students will be shown the JCM Applet and how to use it. There will be a couple examples completed with the class. (10 minutes) Each student will be given a worksheet with multiple examples that correspond with the applet. They will complete the worksheet in class and/or for homework, depending on time.
Summary Essential Resources for Teachers and Students Math On File™ is a three-volume set designed to supplement the classroom instruction of mathematics in accordance with the National Council of Teachers of Mathematics (NCTM) standards. The set includes three valuable resources—Math On File™: Algebra, Math On File™: Geometry, and Math On File™: Calculus. Each volume covers a wide range of topics in an accessible and attractive format. This comprehensive collection of approximately 150 engaging problem sets ranges from simple to challenging and emphasizes the systematic development of mathematical skills. All exercises are reproducible and easy to distribute for classroom use or take-home practice. Each exercise contains an introduction as well as text and diagrams that fully articulate the ideas and concepts being studied. Coverage spans arithmetic series and balancing equations to the angles and lengths of sides of polygons to derivatives, integrals, and beyond. An essential addition to any high school or college mathematics classroom, Math On File™ provides fast access to high-quality materials that teachers can assign for additional practice, provide as supplements to their lesson plans, or use as a basis for testing. It helps students achieve a better understanding of important principles in mathematics
Books Geometry & Topology Who is on the biggest square? How many circles are there? How many sides does a rectangle have? Toddlers will love learning about shapes with the cutest, friendliest dinosaur family ever! The colorful creatures put on a happy show as they play with everything from triangles to ovals, diamonds, and stars. Each spread features the labeled shape on the left side, and the dinos interacting with it on the right—along with a fun question about each one. This is a very successful textbook for undergraduate students of pure mathematics. Students often find the subject of complex analysis very difficult. Here the authors, who are experienced and well-known expositors, avoid many of such difficulties by using two principles: (1) generalising concepts familiar from real analysis; (2) adopting an approach which exhibits and makes use of the rich geometrical structure of the subject. An opening chapter provides a brief history of complex analysis which sets it in context and provides motivation. For the second edition of this very successful text, Professor Binmore has written two chapters on analysis in vector spaces. The discussion extends to the notion of the derivative of a vector function as a matrix and the use of second derivatives in classifying stationary points. Some necessary concepts from linear algebra are included where appropriate. The first edition contained numerous worked examples and an ample collection of exercises for all of which solutions were provided at the end of the book. The second edition retains this feature but in addition offers a set of problems for which no solutions are given. Teachers may find this a helpful innovation. The subject of space-filling curves has generated a great deal of interest in the 100 years since the first such curve was discovered by Peano. Cantor, Hilbert, Moore, Knopp, Lebesgue, and Polya are among the prominent mathematicians who have contributed to the field. However, there have been no comprehensive treatments of the subject since Siepinsky's in 1912. Cantor showed in 1878 that the number of points on an interval is the same as the number of points in a square (or cube, or whatever), and in 1890 Peano showed that there is indeed a continuous curve that continuously maps all points of a line onto all points of a square, though the curve exists only as a limit of very convoluted curves. This book discusses generalizations of Peano's solution and the properties that such curves must possess and discusses fractals in this context. The only prerequisite is a knowledge of advanced calculus. The first monograph to treat topological, group-theoretic, and geometric problems of ideal hydrodynamics and magnetohydrodynamics from a unified point of view. It describes the necessary preliminary notions both in hydrodynamics and pure mathematics with numerous examples and figures. The book is accessible to graduates as well as pure and applied mathematicians working in hydrodynamics, Lie groups, dynamical systems, and differential geometry. Fractal Geometry is a recent edition to the collection of mathematical tools for describing nature, and is the first to focus on roughness. Fractal geometry also appears in art, music and literature, most often without being consciously included by the artist. Consequently, through this we may uncover connections between the arts and sciences, uncommon for students to see in maths and science classes. This book will appeal to teachers who have wanted to include fractals in their mathematics and science classes, to scientists familiar with fractal geometry who want to teach a course on fractals, and to anyone who thinks general scientific literacy is an issue important enough to warrant new approaches. Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically of interest in classical analysis and mathematical physics. After developing the properties of these integrals, their use in determining the asymptotic behavior of special functions is detailed. Although such integrals have a long history, the book's account includes recent research results in analytic number theory and hyperasymptotics. The book also fills a gap in the literature on asymptotic analysis and special functions by providing a thorough account of the use of Mellin-Barnes integrals that is otherwise not available in standard references on asymptotics. This text is intended for a one-semester undergraduate course in topology. The fundamental concepts of general topology are covered rigorously but at a gentle pace and an elementary level. It is accessible to students with only an elementary calculus background. In particular, abstract algebra is not a prerequisite. The first chapter develops the elementary concepts of sets and functions, and in Chapter 2 the general topological space is introduced. Subspaces, continuity, and homeomorphisms are covered in Chapter 3. The remaining chapters cover product spaces, connected spaces, separation properties, and metric spaces. The late Professor G.N. Watson wrote his monumental treatise on the theory of Bessel functions in 1922 with two objects in view. The first was the development of applications of the fundamental processes of the theory of complex variables, and the second was compiling a collection of results of value for mathematicians and physicists who encounter Bessel functions in the course of their researches. The completeness of the theoretical account, combined with the wide scope of the collection of practical examples have resulted in a book that will be indispensable for pure mathematicians, applied mathematicians, and physicists.
Strategies for Success Strategies for Success: Study Skills for the College Math Student is a workbook of study skills activities specific to fostering success in college mathematics. These are true student activities, not just paragraphs talking about study skills. They enable the students to take an active approach to determine specifically what they can do to become successful math students. By using Strategies for Success, students develop effective study skills to help them succeed in college. The Strategies for Success student workbook is available from Pearson Higher Education. Check the PearsonHigherEd website for details. Strategies for Success can be ordered as a standalone workbook, or bundled together as a package with a Pearson mathematics textbook. The Strategies for Success Instructor's Manual accompanies the Strategies for Success Student Workbook. The Instructor's Manual provides guidance about implementing the Strategies for Success study skills activities in your college mathematics class.
Business Math (Complete) -With Study Guide - 8th edition Summary: For arithmetic-based Business Math courses at the undergraduate level. Will sometimes fit courses titled Consumer Math or Personal Finance.The focus of the 8th Edition of Business Mathematics is to provide students with the tools they need to solve mathematical problems they will encounter in both their personal and professional lives. Students are presented math in contexts that are familiar to them and that they care about: math needed for everyday business transactions, math neede...show mored to make important personal finance decisions, and math needed to start or run a small business.Now available with Business Math, 8/e:MathXL�and MyMathLab�for Business Math provide a powerful classroom management, homework, tutorial, and assessment tools.Students can take chapter quizzes or tests in MathXL and MyMathLaband receive personalized study plans based on their test results. The study plan diagnoses weaknesses and links students directly to tutorial exercises for the outcomes they need to study and retest. All student work can be tracked in MathXL's online gradebook. Three packaging options--MyMathLab, MathXL, or MathXL Tutorialson CD--provide flexible platforms to fit your course goals. For more information, visit our websitesat and ,or contact your sales representative.This text is also available in a brief version: Business Math, Brief Edition, 8e, Cleaves/Hobbs21 +$3.99 s/h Good One Stop Text Books Store Sherman Oaks, CA 2008-02-01 Paperback Good68 +$3.99 s/h Good Big Planet Books Burbank, CA 2008-02-01 Paperback Good Expedited shipping is available for this item! $99.39
Polynomials Worksheets You have reached the best page for polynomials worksheets which include identifying the type of polynomials, degree of polynomials, performing math operations on polynomials, factoring polynomials and more.
A First Course in Abstract Algebra Summary Considered a classic by many, A First Course in Abstract Algebra is an in-depth, introductory text which gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. The Sixth Edition continues its tradition of teaching in a classical manner, while integrating field theory and new exercises. Table of Contents 0 A FEW PRELIMINARIES 1 (30) 0.1 Mathematics and Proofs 1 (6) 0.2 Sets and Relations 7 (10) 0.3 Mathematical Induction 17 (4) 0.4 Complex and Matrix Algebra 21 (10) 1 GROUPS AND SUBGROUPS 31 (62) 1.1 Binary Operations 31 (12) *Finite-State Machines (Automata) 41 (2) 1.2 Isomorphic Binary Structures 43 (8) 1.3 Groups 51 (14) 1.4 Subgroups 65 (10) 1.5 Cyclic Groups and Generators 75 (18) Cayley Digraphs 87 (6) 2 MORE GROUPS AND COSETS 93 (68) 2.1 Groups of Permutations 93 (14) Automata 105 (2) 2.2 Orbits, Cycles, and the Alternating Groups 107 (13) Plane Isometries 117 (3) 2.3 Cosets and the Theorem of Lagrange 120 (8) 2.4 Direct Products and Finitely Generated Abelian Groups 128 (20) Periodic Functions 139 (2) Plane Isometries 141 (7) 2.5 Binary Linear Codes 148 (13) 3 HOMOMORPHISMS AND FACTOR GROUPS 161 (48) 3.1 Homomorphisms 161 (11) 3.2 Factor Groups 172 (7) 3.3 Factor-Group Computations and Simple Groups 179 (11) 3.4 Series of Groups 190 (7) 3.5 Group Action on a Set 197 (7) 3.6 Applications of G-Sets to Counting 204 (5) 4 ADVANCED GROUP THEORY 209 (44) 4.1 Isomorphism Theorems: Proof of the Jordan-Holder Theorem 209 (8) 4.2 Sylow Theorems 217 (7) 4.3 Applications of the Sylow Theory 224 (6) 4.4 Free Abelian Groups 230 (8) 4.5 Free Groups 238 (6) 4.6 Group Presentations 244 (9) 5 INTRODUCTION TO RINGS AND FIELDS 253 (72) 5.1 Rings and Fields 253 (11) 5.2 Integral Domains 264 (7) 5.3 Fermat's and Euler's Theorems 271 (6) 5.4 The Field of Quotients of an Integral Domain 277 (8) 5.5 Rings of Polynomials 285 (12) 5.6 Factorization of Polynomials over a Field 297 (11) 5.7 Noncommutative Examples 308 (8) 5.8 Ordered Rings and Fields 316 (9) 6 FACTOR RINGS AND IDEALS 325 (30) 6.1 Homomorphisms and Factor Rings 325 (9) 6.2 Prime and Maximal Ideals 334 (10) 6.3 Grobner Bases for Ideals 344 (11) 7 FACTORIZATION 355 (28) 7.1 Unique Factorization Domains 355 (13) 7.2 Euclidian Domains 368 (7) 7.3 Gaussian Integers and Norms 375 (8) 8 EXTENSION FIELDS 383 (48) 8.1 Introduction to Extension Fields 383 (10) 8.2 Vector Spaces 393 (9) 8.3 Algebraic Extensions 402 (10) 8.4 Geometric Constructions 412 (7) 8.5 Finite Fields 419 (5) 8.6 Additional Algebraic Structures 424 (7) 9 AUTOMORPHISMS AND GALOIS THEORY 431 (64) 9.1 Automorphisms of Fields 431 (10) 9.2 The Isomorphism Extension Theorem 441 (7) 9.3 Splitting Fields 448 (5) 9.4 Separable Extensions 453 (8) 9.5 Totally Inseparable Extensions 461 (4) 9.6 Galois Theory 465 (9) 9.7 Illustrations of Galois Theory 474 (7) 9.8 Cyclotomic Extensions 481 (7) 9.9 Insolvability of the Quintic 488 (7) BIBLIOGRAPHY 495 (4) NOTATIONS 499 (4) ANSWERS TO ODD-NUMBERED EXERCISES NOT ASKING FOR DEFINITIONS OR PROOFS
Glencoe/McGraw-Hill iv GlencoePre-Algebra Teacheru0027s Guide to Using the Florida Guide to Daily Intervention Today it is vital that students understand the mathematics that ... Copyright by The McGraw-Hill Companies, Inc. All rights reserved. Except as permitted under the United States Copyright Act, no part of this publication may be reproduced ... Glencoe/McGraw-Hill 282 GlencoeAlgebra 1 Rate of Change The rate of change tells, on average, how a quantity is changing over time. Slope describes a rate of change. Library/Chapter 5 Binder.pdf iv Teacheru0027s Guide to Using the Chapter 2 ResourceMasters The Fast File ChapterResource system allows you to conveniently file the resources you use most often.
Product Details: The two-line display scientific calculator combines statistics and advanced scientific functions and is a durable and affordable calculator for the classroom. The two-line display helps students explore math and science concepts in the classroom. Ideal for: General Math Algebra 1 & 2 Geometry Trigonometry Statistics Science Display Two-Line Shows entries on the top line and results on the bottom line. Scrolling Entry line (top) shows up to 11 characters and can scroll left and right up to 88. Result line (bottom) shows up to a 10-digit answer and 2-digit exponent. Key features for math and science Previous entry Lets you review previous entries and look for patterns. Description: ARP2438: Scientific calculator performs 335 scientific functions and features WriteView Technology. With WriteView, the four line, 12 digit LCD displays a formula exactly the way it is shown in textbooks. Other functions include three digit punctuation, two variable statistics, ...
Algebraic expressions and linear equations are applied throughout a thorough review of operations on integers, fractions, decimals, percents, and radicals. Students explore relations and functions using equations, tables, and graphs. Chapters on statistics and geometry extend foundational concepts in preparation for high school courses. Problem solving and real life uses of math are featured in each chapter. Dominion mathematics can be used to manage God's creation to His glory. Product: Pre-Algebra 8 Tests Answer Key (2nd Edition) Vendor: BJU Press Edition Number: 2 Binding Type: Paperback Minimum Grade: 8th Grade Maximum Grade: 8th Grade Weight: 1 pounds Length: 11 inches Width: 8.6 inches Height: 0.4 inches Vendor Part Number: 233106 Subject: Algebra, Calculus & Trig, Math Pre-Algebra 8 Tests Answer Key (2nd Edition).
I have a way out for you and it might just prove to be a better one than purchasing a new textbook. Try Algebrator, it covers a pretty comprehensive list of mathematical topics and is highly recommended. With it you can solve various types of problems and it'll also address all your enquiries as to how it came up with a particular answer. I tried it when I was having difficulty solving problems based on prentice hall mathematics course 3 answer key and I really enjoyed using it. I always use Algebrator to help me with my math projects . I have tried several other software programs but so far this is the best I have encountered . I guess it is the uncomplicated way of explaining the solution to problems that makes the whole process appear so easy. It is indeed a very good piece of software and I can vouch for it. Algebrator is a remarkable software and is certainly worth a try. You will find several exciting stuff there. I use it as reference software for my math problems and can say that it has made learning math much more fun .
Maths Plus is organised according to the three content strands of the NSW Syllabus Australian Curriculum: Mathematics Number and Algebra Measurement and Geometry Statistics and Probability Relevant NSW Syllabus outcome codes, Australian Curriculum: Mathematics content descriptions, and Working Mathematically and Learning Across the Curriculum references are included on every Student Book activity page and colour-coded Australian Curriculum: Mathematics cross-reference chart provided for each year level across all components. Maths Plus teaching components also align with student component by unit and page number. CLICK HERE to view Maths Plus NSW Syllabus Australian Curriculum Edition samples online.
Practical Problems in Mathematics series offers students and trainees of specific trades the opportunity to practice math principles in problems designed for each area of interest. Practical Problems in Mathematics for Electricians, 6th Edition contains 44 instructional units featuring updated material and the addition of new problems and examples related to electricity. The book offers an excellent opportunity to test and develop problem-solving skills, while at the same time providing a valuable review of electrical terminology. Stephen L. Herman has been both a teacher for industrial electricity and an industrial electrician for many years. He has worked as a maintenance electrician and as a class "A" electrician. Presently he is an instructor of industrial electricity at Lee College in Baytown, Texas
MIT News - Topic - Linear algebraMIT News is dedicated to communicating to the media and the public the news and achievements of the students, faculty, staff and the greater MIT community. Fri, 13 Dec 2013 15:05:11 +0000en-usExplained: Matrices Matrices arose originally as a way to describe systems of linear equations, a type of problem familiar to anyone who took grade-school algebra. "Linear" just means that the variables in the equations don't have any exponents, so their graphs will always be straight lines. The equation x - 2y = 0, for instance, has an infinite number of solutions for both x and y, which can be depicted as a straight line that passes through the points (0,0), (2,1), (4,2), and so on. But if you combine it with the equation x - y = 1, then there's only one solution: x = 2 and y = 1. The point (2,1) is also where the graphs of the two equations intersect. The matrix that depicts those two equations would be a two-by-two grid of numbers: The top row would be [1 -2], and the bottom row would be [1 -1], to correspond to the coefficients of the variables in the two equations. In a range of applications from image processing to genetic analysis, computers are often called upon to solve systems of linear equations — usually with many more than two variables. Even more frequently, they're called upon to multiply matrices. Matrix multiplication can be thought of as solving linear equations for particular variables. Suppose, for instance, that the expressions t + 2p + 3h; 4t + 5p + 6h; and 7t + 8p + 9h describe three different mathematical operations involving temperature, pressure, and humidity measurements. They could be represented as a matrix with three rows: [1 2 3], [4 5 6], and [7 8 9]. Now suppose that, at two different times, you take temperature, pressure, and humidity readings outside your home. Those readings could be represented as a matrix as well, with the first set of readings in one column and the second in the other. Multiplying these matrices together means matching up rows from the first matrix — the one describing the equations — and columns from the second — the one representing the measurements — multiplying the corresponding terms, adding them all up, and entering the results in a new matrix. The numbers in the final matrix might, for instance, predict the trajectory of a low-pressure system. Of course, reducing the complex dynamics of weather-system models to a system of linear equations is itself a difficult task. But that points to one of the reasons that matrices are so common in computer science: They allow computers to, in effect, do a lot of the computational heavy lifting in advance. Creating a matrix that yields useful computational results may be difficult, but performing matrix multiplication generally isn't. One of the areas of computer science in which matrix multiplication is particularly useful is graphics, since a digital image is basically a matrix to begin with: The rows and columns of the matrix correspond to rows and columns of pixels, and the numerical entries correspond to the pixels' color values. Decoding digital video, for instance, requires matrix multiplication; earlier this year, MIT researchers were able to build one of the first chips to implement the new high-efficiency video-coding standard for ultrahigh-definition TVs, in part because of patterns they discerned in the matrices it employs. In the same way that matrix multiplication can help process digital video, it can help process digital sound. A digital audio signal is basically a sequence of numbers, representing the variation over time of the air pressure of an acoustic audio signal. Many techniques for filtering or compressing digital audio signals, such as the Fourier transform, rely on matrix multiplication. Another reason that matrices are so useful in computer science is that graphs are. In this context, a graph is a mathematical construct consisting of nodes, usually depicted as circles, and edges, usually depicted as lines between them. Network diagrams and family trees are familiar examples of graphs, but in computer science they're used to represent everything from operations performed during the execution of a computer program to the relationships characteristic of logistics problems. Every graph can be represented as a matrix, however, where each column and each row represents a node, and the value at their intersection represents the strength of the connection between them (which might frequently be zero). Often, the most efficient way to analyze graphs is to convert them to matrices first, and the solutions to problems involving graphs are frequently solutions to systems of linear equations.]]>Larry Hardesty, MIT News OfficeConcepts familiar from grade-school algebra have broad ramifications in computer science.Electrical Engineering and Computer ScienceFourier transforms,graphs,Linear algebra,Matrices,matrix multiplicationLinear algebraFri, 06 Dec 2013 05:00:00 +0000Short algorithm, long-range consequences This animation shows two different "spanning trees" for a simple graph, a grid like those used in much scientific computing. The speedups promised by a new MIT algorithm require "low-stretch" spanning trees (green), in which the paths between neighboring nodes don't become excessively long (red).Images courtesy of the researchers — Kelner; Lorenzo Orecchia, an instructor in applied mathematics; and Kelner's students Aaron Sidford and Zeyuan Zhu — believe — a big grid of numbers — that — in time, money or energy — ofingParadigm shift Daniel Spielman, a professor of applied mathematics and computer science at Yale University, was Kelner's thesis advisor and one of two co-authors of the 2004 paper. According to Spielman, his algorithm solved Laplacians in nearly linear time "on problems of astronomical size that you will never ever encounter unless it's a much bigger universe than we know. Jon and colleagues' algorithm is actually a practical one." Spielman points out that in 2010, researchers at Carnegie Mellon University also presented a practical algorithm for solving Laplacians. Theoretical analysis shows that the MIT algorithm should be somewhat faster, but "the strange reality of all these things is, you do a lot of analysis to make sure that everything works, but you sometimes get unusually lucky, or unusually unlucky, when you implement them. So we'll have to wait to see which really is the case." The real value of the MIT paper, Spielman says, is in its innovative theoretical approach. "My work and the work of the folks at Carnegie Mellon, we're solving a problem in numeric linear algebra using techniques from the field of numerical linear algebra," he says. "Jon's paper is completely ignoring all of those techniques and really solving this problem using ideas from data structures and algorithm design. It's substituting one whole set of ideas for another set of ideas, and I think that's going to be a bit of a game-changer for the field. Because people will see there's this set of ideas out there that might have application no one had ever imagined." ]]>Larry Hardesty, MIT News OfficeA new technique for solving 'graph Laplacians' is drastically simpler than its predecessors, with implications for a huge range of practical problems.MathematicsAlgorithms,Graph Laplacian,Graph theory,Laplacians,Linear algebra,Mathematics,Theoretical computer scienceImage courtesy of the researchersLinear algebraFri, 01 Mar 2013 15:00:03 +0000Unraveling the Matrix In a paper published in the July 13 issue of Proceedings of the National Academy of Science, MIT math professor Gilbert Strang describes a new way to split certain types of matrices into simpler matrices. The result could have implications for software that processes video or audio data, for compression software that squeezes down digital files so that they take up less space, or even for systems that control mechanical devices. Strang's analysis applies to so-called banded matrices. Most of the numbers in a banded matrix are zeroes; the only exceptions fall along diagonal bands, at or near the central diagonal of the matrix. This may sound like an esoteric property, but it often has practical implications. Some applications that process video or audio signals, for instance, use banded matrices in which each band represents a different time slice of the signal. By analyzing local properties of the signal, the application could, for instance, sharpen frames of video, or look for redundant information that can be removed to save memory or bandwidth. Working backwards Since most of the entries in a banded matrix — maybe 99 percent, Strang says — are zero, multiplying it by another matrix is a very efficient procedure: You can ignore all the zero entries. After a signal has been processed, however, it has to be converted back into its original form. That requires multiplying it by the "inverse" of the processing matrix: If multiplying matrix A by matrix B yields matrix C, multiplying C by the inverse of B yields A. But the fact that a matrix is banded doesn't mean that its inverse is. In fact, Strang says, the inverse of a banded matrix is almost always "full," meaning that almost all of its entries are nonzero. In a signal-processing application, all the speed advantages offered by banded matrices would be lost if restoring the signal required multiplying it by a full matrix. So engineers are interested in banded matrices with banded inverses, but which matrices those are is by no means obvious. In his PNAS paper, Strang describes a new technique for breaking a banded matrix up into simpler matrices — matrices with fewer bands. It's easy to tell whether these simpler matrices have banded inverses, and if they do, their combination will, too. Strang's technique thus allows engineers to determine whether some promising new signal-processing techniques will, in fact, be practical. Faster than Fourier? One of the most common digital-signal-processing techniques is the discrete Fourier transform (DFT), which breaks a signal into its component frequencies and can be represented as a matrix. Although the matrix for the Fourier transform is full, Strang says, "the great fact about the Fourier transform is that it happens to be possible, even though it's full, to multiply fast and to invert it fast. That's part of what makes Fourier wonderful." Nonetheless, for some signal-processing applications, banded matrices could prove more efficient than the Fourier transform. If only parts of the signal are interesting, the bands provide a way to home in on them and ignore the rest. "Fourier transform looks at the whole signal at once," Strang says. "And that's not always great, because often the signal is boring for 99 percent of the time." Richard Brualdi, the emeritus UWF Beckwith Bascom Professor of Mathematics at the University of Wisconsin-Madison, points out that a mathematical conjecture that Strang presents in the paper has already been proven by three other groups of researchers. "It's a very interesting theorem," says Brualdi. "It's already generated a couple of papers, and it'll probably generate some more." Brualdi points out that large data sets, such as those generated by gene sequencing, medical imaging, or weather monitoring, often yield matrices with regular structures. Bandedness is one type of structure, but there are others, and Brualdi expects other mathematicians to apply techniques like Strang's to other types of structured matrices. "Whether or not those things will work, I really don't know," Brualdi says. "But Gil's already said that he's going to look at a different structure in a future paper."
Your step-by-step solution to mastering precalculus Understanding precalculus often opens the door to learning more advanced and practical math subjects, and can also help satisfy college requisites. Precalculus Demystified , Second Edition, is your key to mastering this sometimes tricky subject. This self-teaching guide presents general precalculus concepts first, so you'll ease into the basics. You'll gradually master functions, graphs of ... Everyone is a perfect size for something. PillHow can we solve the national debt crisis? Should you or your child take on a student loan? Is it safe to talk on a cell phone while driving? Are there viable energy alternatives to fossil fuels? Could simple policy changes reduce political polarization? These questions may all seem very different, but they share two things in common. First, they are all questions with important implications for either personal success or our ... ... 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As many college instructors already know, many undergraduates enrolling in their chosen discipline lack the necessary math skills to do well in the programs of their choice. Many of these students have nothing beyond basic math skills and struggle to keep up with the work. Many students are forced to take make up courses or pre-requisites before even entering their field of study. Other college candidates are so unfortunate with such below average skills in math that they score low on their college entrance exams and miss the opportunity to get into the college or university of their choice. These college students who manage to get into their programs struggle with their course curriculum and are at a high risk for failing their courses or dropping out. Those who do manage to pass their courses, because of their mediocre skill level in mathematics, tend to miss out on good paying jobs, in the actuarial, science, and high tech fields. In fact many employers find it difficult to find a skilled work force. For students who are having trouble, a simple solution would be to take college math courses online or college preparatory math courses online. There are several college math courses sites to choose from. Some of these college math courses online are free which is particularly important for students of meager means. College math courses online will start the student at the basics and take them to the level they need to succeed in their career. The basic skills that college math courses tend to begin with are fractions, decimals, percentages, pre algebra and the foundations of early algebra courses that will give the student the skill level they need for state high school proficiency. These courses will then provide the foundation needed for college level math requirements. These college math courses online will also provide problem-solving skills needed for finance, accounting and the field of economics. They then will provide more complex mathematical statistics, and can be found for whatever field and type of statistic needed at the college level. For example, business statistics, psychology statistics, and college math courses online for nursing statistics are available. College math courses online will prepare students for a degree in mathematics or a rewarding career in the highest paying jobs available today.
The Year 8 Interactive Maths software is compatible with both Windows® and Mac® computers. Discover more about Year 8 Interactive Maths by: Viewing the 17 chapter screen shots shown below that show step-by-step solutions in G S Rehill's proven writing style. Students use the solutions after answering questions from the interactive exercises to identify mistakes in their working and to reinforce concepts, reasoning or mathematical laws.
All About Math Bundle Grades 6-9 (Four Enhanced eBooksGive your students all the essential tools for a solid introduction to Algebra and Geometry. The All About Math Bundle contains four reproducible books. Retail Value $55.80 The zipped folder that accompanies this kit contains four enhanced eBooks. The enhanced eBooks in this kit give you the freedom to copy and paste the content of each page into the format that fits your needs. You can post lessons on your class website, make student copies, and more. For more information on enhanced eBooks, Click Here.
Math Courses 7 - 12 Math Practice Math Courses K - 6 Curriculum based Math Course for grades k-6. With these exercises the students can understand the concepts of Addition, Subtraction, Fact Families, Place Values, Number Concepts, Time, Money Geometry and More!! Lot of practice problems. Math Courses 7 - 12 Curriculum based Pre-Algrebra, Algebra and Calculus for grades 7-12. With these exercises the students can understand the fundamentals of Integers, Equations, Percents, Number Theory, Probability and Statistics and More!! Lot of practice problems
Top 5 Pre-Algebra Resources: Books An excellent selection of books and resources to assist you with the pre-algebraic concepts and skills which will ready you for Algebra. Good grounding in the fundamentals lead to academic success in higher learning skills. This Nelson Pre-Algebra Resource provides effective tutorials and helps you master core mathematical concepts in pre-algebra -- from fractions, decimals, and statistics to graphs, integers, and exponents -- and get the best possible grade. A great self-teaching paperback with a CD to assist you with all of the pre-albebra and introductory algebra skills needed to prepare you for algebra concepts. All the basics you need are included here. How can you go wrong with a resource that offers to take the mystery out of math? This book is one of my favorites - it provides an excellent approach to 'self teaching' Topics include:integers, fractions, decimals, ratios and proportions, percents, roots and exponents, etc. All the topics you need before moving into algebra. This resource is another one of my favorites. A comprehensive approach to the many pre-algebraic concepts including whole numbers, fractions, decimals, percents, linear equations, expressions and much more. Dr. Math from the famous math forum provides a question/answer approach based on the frequently asked questions from students.
students a step further in learning algebra Specially written for low-level learners, Algebra 2 covers several methods for solving quadratic equations, such as factoring, completing the square, and graphing. The text also introduces trigonometry and exponential functions—vital concepts for real world applications. Filled with full-color illustrations and examples throughout, Algebra 2 motivates students to learn. Overall, this high-interest, low-readability text makes it easy for you to engage students who struggle with reading, language, or a learning disability.Lexile Level790Reading Level3-4Interest Level6-12
This updated and refreshed version of CGP's bestselling Revision Guide is the ideal companion to Foundation Level GCSE Maths - it even includes a free online edition that can be used wherever you have internet access. Every topic is explained in a concise, friendly style, with a sprinkling of CGP humour to keep things interesting. Grade information is included to show the difficulty level of each topic, and there are summary questions at the bottom of each page to test you on the important skills. And finally, a unique code is printed in the book that gives you access to the free online digital version (which also includes fully worked answers to all the test questions in the book
Summary: This book gives a clear, practical and self-contained presentation of the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to differential and difference equations. These methods allow one to analyze problems that are encountered in physics and engineering that may not be solvable in closed form and for which brute-force numerical methods may not converge to useful solutions. The presentation is aimed at teaching the insig...show morehts that are most useful in approaching new problems; it avoids special methods and tricks that work only for particular problems, such as the traditional transcendental functions. Intended for graduate students and advanced undergraduates, the book assumes only a limited familiarity with differential equations and complex variables. The presentation begins with a review of differential and difference equations; develops local asymptotic methods for differential and difference equations; explains perturbation and summation theory; and concludes with an exposition of global asymptotic methods, including boundary-layer theory, WKB theory, and multiple-scale analysis. Emphasizing applications, the discussion stresses care rather than rigor and relies on many well-chosen examples to teach the reader how an applied mathematician tackles problems. There are 190 computer-generated plots and tables comparing approximate and exact solutions; over 600 problems, of varying levels of difficulty; and an appendix summarizing the properties of special functions.165
18.06 Linear Algebra, Fall 2013 Class Philosophy on Computing MATLAB and Julia In this course, we emphasize interactive exploration to supplement theory. Therefore, even though no experience with programming or computational software is required, some homework problems will require you to use a tool. Julia or Matlab will suffice for all such assignments, and so we use them as the "demonstration language" of 18.06 (though you may use any tool/language you want). We may occasionally demonstrate MATLAB and Mathematica, and we have incentives for those willing to try Julia. (Julia is newer and not as polished as the other tools, but it often has better performance, is freely available, and has more constructs expected in a real programming language. It also interfaces seamlessly to Python and C. It is becoming a key langauge for technical computing.)
sufficient score on UWB Math Assessment. Recognize and be comfortable using polynomial, exponential, logarithmic, and trig functions. Able to graph and manipulate functions symbolically. Apply functions and concepts to solve real world problems. Learn to become problem solvers. General method of instruction Discussion and group discovery – sometimes as a whole class and sometimes in small groups. Recommended preparation A 2.0 in BCUSP 122 or placement test. Desire to work hard. Class assignments and grading Daily HW collected and graded once a week. pop quizzes, 2 midterms and a final. At least 65% to receive a 2.0. The grades are not curved. Class median is calculated and used as a guideline. Participation and attendance is rewarded Barry Minai
Annotations for GED Mathematics Baker & Taylor Offers general information about what to expect on the GED test and how to prepare for it, as well as covering mathematics topics such as arithmetic, charts and graphs, probability, statistics, algebra, and geometry. ---------------------- Video Aided Instruction "Once you've mastered the algebra, geometry, and coordinate geometry topics covered in ""Pre-GED Mathematics"",." ---------------------- Video Aided Instruction Once you've mastered the algebra, geometry, and coordinate geometry topics covered in "Pre-GED Mathematics,".
To facilitate those instructors that use a more hands-on approach to teaching, an extensive Exploration Activities manual accompanies the text. The manual contains a variety of explorations for each section, which are referenced in the text by an icon in the margin. Some explorations deal directly with the content of the chapter, often making use of relevant manipulatives or other hands-on activities. Other explorations extend the content of the section either mathematically or by building a connection to the K - 8 classrooms. Most of the explorations can be done individually or with groups and should take about 30 - 45 minutes to complete
Sketchpad Math Software Goes Universal 11.06.2006—Geometer has released an update to Sketchpad, a software tool for teaching and learning math. The new 4.07 update adds native compatibility for Intel-based Macs and continues to support PowerPC-based Macs and Windows systems as well. Sketchpad is a suite designed for both students and educators, with separate editions for each. It allows students to explore mathematics by providing tools for them to create diagrams and figures. Educators can also use the software to generate teaching aids. It includes classroom activities, presentation and sketch samples, learning guides and reference materials. And it offers modules for various math courses. Specific curriculum modules include: Exploring Algebra 1; Exploring Algebra 2; Exploring Geometry; Pythagoras Plugged In: Proofs and Problems; Rethinking Proof; Exploring Conic Sections; Exploring Precalculus; Exploring Calculus; Geometry Activities for Middle School Students; Shape Makers: Developing Geometric Reasoning in Middle School; And Geometry in Action. The latest release, version 4.07, is now a Universal Binary for Macintosh systems, supporting both Intel and PowerPC hardware. It also adds Web links to the Sketchpad Resource Center and adds sample documents to the Help menu. Several bug fixes are also included in the update. The new 4.07 update is available free for current users. The student version of Sketchpad is available now for Mac OS X and Windows for $39.95. The full edition runs $129.95. Multi-license versions are also available, as is an evaluation version for instructors. See the company's Web site, below,
Syllabus Course Meeting Times Lectures: 2 sessions / week, 1 hour / session Course Description The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. Format Every week, the first session is for lectures on different techniques and results related to bijective proofs. Problems of varying levels of difficulty (some unsolved as the course progresses) related to the lecture will be passed out on the first session. In every second session, students will report on their work on the problems. All students must participate in this process. "Reasonable" collaboration on problem sets is permitted, but you shouldn't simply copy someone else's solution or a solution from an outside source. Text There will be no text required for the course. All material will be based on the handouts. There is currently one book in print devoted solely to bijective proofs: Stanton, D., and D. E. White. Constructive Combinatorics. This book is rather sophisticated. A more elementary book by A. T. Benjamin, and J. J. Quinn, Proofs That Really Count, will soon be published, but it is not yet available
Real World Math Course Description: Technology and 21st century learning skills advance the learning of mathematics.We must help students master the basic skills and then learn the meaning of the calculations in a life-related context
Finite Mathematics, CourseSmart eTextbook, 9th Edition Description®, a complete online course solution, a comprehensive series of video lectures is available for this text. CourseSmart textbooks do not include any media or print supplements that come packaged with the bound book. Table of Contents Chapter R: Algebra Reference R-1 Polynomials R-2 Factoring R-3 Rational Expressions R-4 Equations R-5 Inequalities R-6 Exponents R-7 Radicals Chapter 1: Linear Functions 1-1 Slopes and Equations of Lines 1-2 Linear Functions and Applications 1-3 The Least Squares Line Chapter Review Extended Application: Using Extrapolation to Predict Life Expectancy Chapter 2: Systems of Linear Equations and Matrices 2-1 Solution of Linear Systems by the Echelon Method 2-2 Solution of Linear Systems by the Gauss-Jordan Method 2-3 Addition and Subtraction of Matrices 2-4 Multiplication of Matrices 2-5 Matrix Inverses 2-6 Input-Output Models Chapter Review Extended Application: Contagion Chapter 3: Linear Programming: The Graphical Method 3-1 Graphing Linear Inequalities 3-2 Solving Linear Programming Problems Graphically 3-3 Applications of Linear Programming Chapter Review Chapter 4: Linear Programming: The Simplex Method 4-1 Slack Variables and the Pivot 4-2 Maximization Problems 4-3 Minimization Problems; Duality 4-4 Nonstandard Problems Chapter Review Extended Application: Using Integer Programming in the Stock-Cutting Problem Chapter 5: Mathematics of Finance 5-1 Simple and Compound Interest 5-2 Future Value of an Annuity 5-3 Present Value of an Annuity; Amortization Chapter Review Extended Application: Time, Money, and Polynomials Chapter 6: Logic 6-1 Statements 6-2 Truth Tables and Equivalent Statements 6-3 The Conditional and Circuits 6-4 More on the Conditional 6-5 Analyzing Arguments and Proofs 6-6 Analyzing Arguments with Quantifiers Chapter Review Extended Application: Logic Puzzles Chapter 7: Sets and Probability 7-1 Sets 7-2 Applications of Venn Diagrams 7-3 Introduction to Probability 7-4 Basic Concepts of Probability 7-5 Conditional Probability; Independent Events 7-6 Bayes' Theorem Chapter Review Extended Application: Medical Diagnosis Chapter 8: Counting Principles: Further Probability Topics 8-1 The Multiplication Principle; Permutations 8-2 Combinations 8-3 Probability Applications of Counting Principles 8-4 Binomial Probability 8-5 Probability Distributions; Expected Value Chapter Review Extended Application: Optimal Inventory for a Service Truck Chapter 9: Statistics 9-1 Frequency Distributions; Measures of Central Tendency 9-2 Measures of Variation 9-3 The Normal Distribution 9-4 Normal Approximation to the Binomial Distribution Chapter Review Extended Application: Statistics in the Law - The Castaneda Decision Chapter 10: Markov Chains 10-1 Basic Properties of Markov Chains 10-2 Regular Markov Chains 10-3 Absorbing Markov Chains Chapter Review Extended Application: A Markov Chain Model for Teacher Retention Chapter 11: Game Theory 11.1 Strictly Determined Games 11.2 Mixed Strategies 11.3 Game Theory and Linear Programming Chapter Review Extended Application: The Prisoner's Dilemma - Non-Zero-Sum Games in Economics Table Area under a Normal Curve Answers to Selected Exercises Photo Acknowledgements Index Special Topics to Accompany Finite Mathematics (TBC)* Digraphs and Networks Graphs and Digraphs Dominance Graphs Communication Graphs Networks Review Exercises *These supplements will be provided at no additional charge to adopters who request them.
Solving Equations Graphic Organizer with Example Word Document File Be sure that you have an application to open this file type before downloading and/or purchasing. 0.05 MB PRODUCT DESCRIPTION I use this with my students in Algebra. I make a foldable with them and instruct them to fill in the steps, and then we do an example to illustrate the steps with them. They have their own foldable on hand with them all the time. I typed it up so that I could turn it into a poster sized sheet, laminate it, and refer to the poster when doing examples. I love this tool, and it works well with Max Thompson's learning focused strategies, which my school has adopted. Comments & Ratings Product Questions & Answers Be the first to ask Vanessa Moon a question about this product. Please log in to do that. $1.25 Digital Download List Price: $1.50 You Save: $0.25
Math-Related Credit Crosswalk for Career Technical Education Classes in Macomb County Program Information District: L'Anse Creuse F. V. Pankow Center Program Name: Programming CIP Code Number: 11.0201 Career Pathway: Information Technology Instructor Name: Nick Paterni Date: May 2009 Strand STANDARDS CTE APPLICATION and PRACTICE L1 REASONING ABOUT NUMBERS, SYSTEMS AND QUANTITATIVE LITERACY L1.1 Number Systems and Number Sense L1.1.1 Know the different properties that hold in Integers, rational numbers and real numbers and all different number systems and recognize applicable properties are used throughout the course that the applicable properties change in the in a variety of programs. transition from the positive integers to all integers, to the rational numbers, and to the real numbers. L1.1.2 Explain why the multiplicative inverse of a Students understand that multiplying by ½ is the number has the same sign as the number, same as dividing by 2 therefore the sign stays the while the additive inverse has the opposite same. sign. Students understand that subtracting 3 is the same as adding -3. L1.1.3 Explain how the properties of associativity, All properties of real numbers including the order of commutativity, and distributivity, as well as operation must be followed for accuracy. identity and inverse elements, are used in Programming language is very specific and therefore arithmetic and algebraic calculations. all properties of arithmetic and algebraic operations must used. L1.1.4 Describe the reasons for the different When writing a compound interest program, students effects of multiplication by, or understand the effects of fractional exponents, exponentiation of, a positive number by a multiplying by a fraction and multiplying by a number number less than 0, a number between 0 larger than 1. and 1, and a number greater than 1. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 1 05/18/2011 L1.2 Representations and Relationships L1.2.1 Use mathematical symbols (e.g., interval Mathematical symbols of =, -, , /, ^, <. >, \|, { } ,( ), \ notation, set notation, summation notation) are used throughout the course in programming. to represent quantitative relationships and Ex. A statement in programming a game situations. // Ball lost? If (ballPosition.Y > 0.985f) { // Play sound L1.2.2 Interpret representations that reflect Students use and understand the absolute value absolute value relationships (e.g.,│x-a│< b, function and can use the function in appropriate or a± b) in such contexts as error tolerance. programs. Ex. System.math.ABS((-50.2)) L1.2.3 Use vectors to represent quantities that Positions and movement of images on the screen are have magnitude and direction, interpret designated by vectors. direction and magnitude of a vector Ex. lineVertices[ lineNum 2 + 0 ] numerically, and calculate the sum and = new VertexPositionColor(newVector3(10,20,5) difference of two vectors. L1.2.4 Organize and summarize a data set in a Students can create database information charts. table, plot, chart, or spreadsheet; find Ex. Employee Database Information Program patterns in a display of data; understand Morgan Industries needs an employee and critique data displays in the media. database to include name, address, date of hire and pay rate. L1.3 Counting and Probabilistic Reasoning L1.3.2 Define and interpret commonly used When discussing the creation of a Yahtzee program, expressions of probability (e.g., chances of students understand that the odds of getting a an event, likelihood, odds). number is 1:6 and that since these numbers are randomly generated, the likelihood of getting a 5 is 1:6 L1.3.3 Recognize and explain common probability When programming a Rock, Scissor, Paper game, misconceptions such as "hot streaks" and the outcomes are randomly generated and therefore "being due." the winner might believe he is "due" or is on a "hot streak" but knows the outcomes are random. Add and Subtract Integers and Rational Numbers N.ME.06.08 Understand integer subtraction as the Students write a program to create a mini calculator inverse of integer addition. Understand which includes all functions of integers. integer division as the inverse of integer Ex. lngAddFirst = txtInput.Text multiplication. strOutput = "0" Operation = 1 End Sub N.FL.06.10 Add, subtract, multiply and divide positive Students write a program to create a mini calculator rational numbers fluently. which includes all functions of rational numbers. Ex. lngDivideTotal = txtInput.Text txtInput.Text = (lngDivideFirst / lngDivideTotal) Operation = 0 Solve Decimal, Percentage and Rational Number Problems N.FL.06.12 Calculate part of a number given the Students can write a program to calculate the annual percentage and the number. payment on a loan. Ex. Calculate the annual payment for a loan of $9000 for 3 years are 5% interest rate. Finanicial.Pmt(.05,3,9000) C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 2 05/18/2011 N.MR.06.13 Solve contextual problems involving Sales Tax Program percentages such as sales taxes and tips. Ex. Create a program so that the user can input the amount and then calculate a 6% sales tax. Double.TryParse(salesTextbox.Text,sales) tax6 = sales * 0.06 taxLabel.Text = convert.ToString(tax6) N.FL.06.14 For applied situations, estimate the answers To Debug a program, students must estimate to calculations involving operations with answers to calculations for accuracy purposes. rational numbers N.FL.06.15 Solve applied problems that use the four Mini Calculator program operations with appropriate decimal Students write a program to simulate a calculator numbers. involving the four basic operations with decimal numbers. Use Exponents N.ME.06.16 Understand and use integer exponents, Data type excluding powers of negative bases, Ex. Single: a number with a decimal range -45 38 express numbers in scientific notation. + /- 1.401298 x 10 to +/- 3.402823 x 10 Understand Rational Numbers and Their Location on the Number Line N.ME.06.18 Understand that rational numbers are Students understand that all fractions are a quotient quotients of integers (non zero of two integers and are converted to decimals. denominators). N.ME.06.19 Understand that 0 is an integer that is 0 is used in programming to indicate a false neither negative nor positive. statement and has no value. N.ME.06.20 Know that the absolute value of a number is Students understand that the absolute value function the value of the number ignoring the sign; or in Visual Basic returns the number without the sign. is the distance of the number from 0. Ex. System.Math.Abs((-67)) Returns 67 Understand Derived Quantities N.MR.07.02 Solve problems involving derived quantities Students can create a program that allows the user such as density, velocity and weighted to enter the points a student earns on four projects averages. and two tests. Ex. Part of the program Select case totalPointsAccumulator Case is <= 360 Grade = "A" Case is >= 320 Grade = "B" Understand and Solve Problems Involving Rates, Ratios, and Proportions N.FL.07.03 Calculate rates of change including speed. Rate of Pay program Ex. Students create a program to allow users to input the hours worked and rate of pay per hour and then compute wages. N.MR.07.04 Convert ratio quantities between different Students create a program to convert Fahrenheit to systems of units, such as feet per second to Celsius temperature. miles per hour. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 3 05/18/2011 Compute with Rational Numbers N.FL.07.07 Solve problems involving operations with Every computer programmer uses operations with integers. integers. Ex. Age Program age = age + 1 adds the integer 1 to the contents of the Integer age variable, then assigns the result to the age variable. N.FL.07.08 Add, subtract, multiply and divide positive Every computer program uses positive and negative and negative rational numbers fluently. rational numbers. Ex. Mini Calculator program Students write a program to simulate a calculator involving the four operations of rational numbers. N.FL.07.09 Estimate results of computations with Every programmer estimates the results of rational numbers. computations with rational number for programming accuracy. Understand Real Number Concepts N.ME.08.02 Understand meanings for zero and negative Data Type integer exponents. Students understand the meaning of negative exponents in data type variables. Ex. Single: a number with a decimal place -45 38 Range = -1.401298 x 10 to = -3.402823x10 When converting from decimal to binary system, 0 students understand that 2 = 1 N.ME.08.03 Understand that in decimal form, rational In programming, all rational numbers are converted numbers either terminate or eventually to decimal forms and truncated to specified decimal repeat, and that calculators truncate or places as needed. round repeating decimals; locate rational numbers on the number line; know fraction forms of common repeating decimals. N.ME.08.04 Understand that irrational numbers are Students understand that Pi is an irrational number those that cannot be expressed as the that cannot be expressed as the quotient of two quotient of two integers, and cannot be integers. represented by terminating or repeating decimals; approximate the position of familiar irrational numbers. Solve Problems N.MR.08.07 Understand percent increase and percent Students can write a program to find percent decrease in both sum and product form. increase and therefore must know the formula. Ex. Payroll program Write a program to determine an employee's new hourly pay given the employee's current hourly pay and raise. Private Function GetNewpay(Byval current As Double,_ Byval rate As Double)AsDouble raise = current * rate newPay = current + raise. Return newPay End Function C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 4 05/18/2011 N.MR.08.08 Solve problems involving percent increases Ex. Calculate the new price of an item with a 5% and decreases. price increase. Private Function CalcNew(ByVal price as Double)as Double Return price + price * .05 End Function N.FL.08.09 Solve problems involving compounded Students write a program to find compound interest interest or multiple discounts. Ex. Private Sub btnCalculate_Click1 Amount = (principal (1+rate/periods))pow(periodsyears)) msgbox(amount.ToString(C"2")) N.MR.08.10 Calculate weighted averages such as Grade book program course grades, consumer price indices and Ex. Students can create an application that displays sports ratings. the total credit hours and GPA for a student in one semester using the following data: A = 4 points, B= 3 points, C = 2 points, D = 2 points and F = 1 point N.FL.08.11 Solve problems involving ratio units, such Students can create a program that calculates a as miles per hour, dollars per pound or customer's water bill. persons per square mile. Ex. Create an a program the calculates and displays the number of gallons of water used and the total charge. The charge for the water is $1.75 per gallon. L2 STANDARDS CTE APPLICATION and PRACTICE CALCULATION, ALGORITHMS, AND ESTIMATION L2.1 Calculation Using Real and Complex Numbers L2.1.1 Explain the meaning and uses of weighted Students can write programs for course grades. averages (e.g., GNP, consumer price index, Ex. Grade Book program grade point average). Grade = gradeTextBox.Text.ToUpper If grade = "A" then msgLabel.Text = "Excellent" L2.1.6 Recognize when exact answers aren't Students format numbers by using the ToString always possible or practical. Use function. appropriate algorithms to approximate Ex. commissionLabel.Text = solutions to equations (e.g., to approximate commission.ToString("C2") square roots). If the commission variable contains the number 1250, the statement assigns the string "$1250.00" to Text property of the commissionLabel L2.2 Sequences and Iteration L2.2.3 Use iterative processes in such examples Nested loops as computing compound interest or Ex. For each month in year applying approximation procedures. For each day in month msgbox(day) Next day Next month C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 5 05/18/2011 L3 STANDARDS CTE APPLICATION and PRACTICE MEASUREMENT AND PRECISION L3.1 Measurement Units, Calculations, and Scales L3.1.1 Convert units of measurement within and Students can convert from the decimal system to the between systems; explain how arithmetic binary system. operations on measurements affect units, Ex. 15 in decimal system = 11112 binary system and carry units through calculations correctly. L3.2 Understanding Error L3.2.1 Determine what degree of accuracy is Calculations involving decimal variables are not reasonable for measurements in a given subject to the small rounding errors that may occur situation; express accuracy through use of when using Double or Single variables. When the significant digits, error tolerance, or percent application contains money it is best to use the of error; describe how errors in Decimal data type. measurements are magnified by Ex. Formatting Decimals computation; recognize accumulated error Decimal.ToString("C2") = currency, 2 decimal in applied situations. Places L3.2.2 Describe and explain round-off error, Data type rounding, and truncating. Students understand rounding off and truncating when using ToDecimal function or INTEGER function. L3.2.3 Know the meaning of and interpret Students understand that an error in writing statistical significance, margin of error, and programs can lead to errors in outcomes. confidence level. Garbage in Garbage out L4.1 Mathematical Reasoning L4.1.1 Distinguish between inductive and Deductive Reasoning: deductive reasoning, identifying and Used throughout the programming course when providing examples of each. writing and developing programs and doing flow charts. Inductive Reasoning : Pseudocode Uses short phrases to describe the steps a procedure needs to accomplish its goal. L4.1.2 Differentiate between statistical arguments Logical Operator Unit (statements verified empirically using Students understand and use rules of logic and examples or data) and logical arguments logical arguments when writing programs. based on the rules of logic. Logical operators: & (and),! (not), \| (or) L4.2 Language and Laws of Logic L4.2.1 Know and use the terms of basic logic (e.g., Logical Operators Unit proposition, negation, truth and falsity, Students understand all truth tables for all logical implication, if and only if, contrapositive, and statements used in programming. converse). Ex. If NOT isinsured Then This condition evaluates to True when the Boolean isInsured variable contains the Boolean value False, otherwise, it evaluates to False. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 6 05/18/2011 L4.2.2 Use the connectives "not," "and," "or," and Logical Operators Unit "if…, then," in mathematical and everyday Students understand all truth tables for all logical settings. Know the truth table of each statements used in programming. connective and how to logically negate Ex. Math conditional formatting statements involving these connectives. A salesperson would get a raise if he gets an A rating and sells more than $10000. This would be true only if both conditions are true. rating = "A' AndAlso sales > 10000 L4.2.3 Use the quantifiers "there exists" and "all" in Logical Operators Unit mathematical and everyday settings and Students understand all truth tables for all logical know how to logically negate statements statements used in programming. involving them. Ex. IF THEN ELSE If the sales are greater than1500 commission = sales * .02 (true) else commission = sales .01 (false) End if L4.2.4 Write the converse, inverse, and Discuss cause and effect of studying and doing well. contrapositive of an "If…, then…" Ex. If I score well on all projects, then I statement. Use the fact, in mathematical will understand the concepts of and everyday settings, that the programming. contrapositive is logically equivalent to the Converse: If I understand all concepts of original while the inverse and converse are programming, then I will score well not. on all projects. Inverse: If I do not score well on the projects, then I do not understand the concepts of programming. Contrapositive; If I do not understand the concepts of programming, then I will not score well on the projects. L4.3 Proof L4.3.1 Know the basic structure for the proof of an Logical Operators Unit "If…, then…" statement (assuming the Ex. IF THEN ELSE hypothesis and ending with the conclusion) Calculate and display an employee's gross pay. and that proving the contrapositive is If hoursWorked >=0. AndAlso hoursWorked <= equivalent. 40.0, then gross pay = hoursWorked 10.65 L4.3.2 Construct proofs by contradiction. Use Logical Operators Unit counter examples, when appropriate, to Ex. IF THEN ELSE disprove a statement. Find the Mouse game If randomNumber = 1 Then PictureBox1.image = mousePictureBox.Image Else Picturebox1.Image = notHerePictureBox.Image End if Find Volume and Surface Area M.TE.06.03 Compute the volume and surface area of Students can create programs to find surface area cubes and rectangular prisms given the and volume of rectangular prisms. lengths of their sides, using formulas. Ex. Create a program that allows the user to enter the length, width and height of a swimming pool and then calculates the number of cubic feet of water the pool will contain. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 7 05/18/2011 A1 STANDARDS CTE APPLICATION and PRACTICE EXPRESSIONS, EQUATIONS, AND INEQUALITIES A1.1 Construction, Interpretation, and Manipulation of Expressions (linear, quadratic, polynomial, rational, power, exponential, logarithmic, and trigonometric) A1.1.1 Give a verbal description of an expression Students can write and interpret programming codes. that is presented in symbolic form, write an Ex. If "reno" = cityTextBox.Text.ToLowere Then algebraic expression from a verbal compares the lowercase letters "reno" to the description, and evaluate expressions given lowercase version of the string stored in the values of the variables. cityTextBox. A1.2 Solutions of Equations and Inequalities (linear, exponential, logarithmic, quadratic, power, polynomial, and rational) A1.2.1 Write and solve equations and inequalities Students use equations and inequalities for specific with one or two variables to represent programs. mathematical or applied situations. Ex. A company wants an application that the sales manager can use to display the average amount the company sold during the prior year. salesCounter = salesCounter +1 salesAccumulator = salesAccumulator + sales If sale Counter > 0 then salesAverage = salesAccumulator /Convert.ToDecimal (salesCounter) averageLabel.Text = saleAverage.ToString A1.2.4 Solve absolute value equations and Students use the absolute value function in various inequalities (e.g., solve │x - 3│ ≤ 6) and programs justify. Ex. System.Math.Abs((-34.8)) A1.2.9 Know common formulas (e.g., slope, Students know a variety of formulas to write distance between two points, quadratic programs given specific information. formula, compound interest, distance = rate Ex. Write a program to calculate monthly payments · time), and apply appropriately in and interest on a loan using interest rates of 5% contextual situations. through 10% and terms of 2,3,4,or 5 years. For rate As Double = 0.05 To 0.01 Step 0.01 monthly payment = Financial.PMT(rate/12,term12,principal) paymentsLabel.Text = paymentslabel.Text_ &rate.ToString("PO") & "->"&monthylyPayment. ToString ("C2")_ &ControlChars.NewLine Next rate A2 STANDARDS CTE APPLICATION and PRACTICE FUNCTIONS A2.1 Definitions, Representations, and Attributes of Functions A2.1.2 Read, interpret, and use function notation Students can create a program that will display the and evaluate a function at a value in its multiplication tables when a user inputs a number domain. and the output is the result of multiplying that number by the numbers 1 through 9 Ex. For count as integer = 1 to 9 msgbox("y=" & input count) Next count C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 8 05/18/2011 A2.1.5 Recognize that functions may be defined For…Next Statements (Loops) recursively. Compute values of and graph Ex. Dim x As Decimal simple recursively defined functions (e.g., For x = .05D To .1D Step .01D f(0) = 5, and f(n) = f(n-1) + 2). ratelabel.Text = rateLabel.Text & x.ToString("PO")_ & controlChars.NewLine Next x Displays 5%,6%,7%,8%,9%,10% A2.4 Lines and Linear Functions A2.4.3 Relate the coefficients in a linear function to Collision Testing in a Pong game the slope and x- and y-intercepts of its Ex. In a Pong game, when the ball collides with the graph. border or a paddle, the programmer must find the vector associated with the original movement and invert the angle by finding the perpendicular vector coordinates. A2.10 Trigonometric Functions A2.10.2 Use the relationship between degree and In programming, students convert all radian radian measures to solve problems. measurements to degree measurements. A3 STANDARDS CTE APPLICATION and PRACTICE MATHEMATICAL MODELING A3.1 Models of Real-world Situations Using Families of Functions Example: An initial population of 300 people grows at 2% per year. What will the population be in 10 years? A3.1.1 Identify the family of functions best suited Mathematical modeling for modeling a given real-world situation Students can write a program that takes user input [e.g., quadratic functions for motion of an and places it in the appropriate function to meet object under the force of gravity or desired outcome. exponential functions for compound interest. In the example above, recognize that the appropriate general function is t exponential (P = P0a )]. A3.1.2 Adapt the general symbolic form of a Mathematical modeling function to one that fits the specifications of Students can write a program that takes user input a given situation by using the information to and places it in the appropriate function to meet replace arbitrary constants with numbers. desired outcome. In the example above, substitute the given Ex. A sales manager wants an application that values P0 = 300 and a = 1.02 to obtain P = determines the number of salespeople selling t 300(1.02) . above a specified amount. To accomplish this: For Each salesAmount As Integer In sales If salesAmount > search For Then Counter =m Counter + 1 End if Next Sales Amount. A3.1.3 Using the adapted general symbolic form, Mathematical modeling draw reasonable conclusions about the Students can write a program that takes user input situation being modeled. In the example and places it in the appropriate function to meet above, the exact solution is 365.698, but for desired outcome. this problem, an appropriate approximation Ex. In the above example, the sales manager can is 365. input the specified amounts. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 9 05/18/2011 A.PA.06.01 Solve applied problems involving rates, Rate of Pay program including speed. Ex. Students write a program that allows the user to input hours worked and rate of pay, then computes wages earned including overtime pay. Understand the Coordinate Plane A.RP.06.02 Plot ordered pairs of integers and use All positioning of images on the screen is indicated ordered pairs of integers to identify points in by vector ordered pairs. all four quadrants of the coordinate plane. Ex. A triangle with three vertices Vector 1 (0,1,5), Vertex 2 (-0.5,0,0.7), Vector 3 (1,1, 0.2) Use Variables, Write Expressions and Equations, and Combine Like Terms A.FO.06.03 Use letters with units, to represent Choosing variables that make sense quantities in a variety of contexts. When writing programs and choosing variables to represent data, students must choose a variable that makes sense for the program. Ex. When writing a program to calculate age, a = age is an appropriate variable. A.FO.06.04 Distinguish between an algebraic Expression: Aspect Ratio expression and an equation. aspectRatio = (float)width/(float)height Equation : Any algebraic equation use in programming bonus = sales * .05 A.FO.06.05 Use standard conventions for writing All algebraic expressions are written in standard algebraic expressions. algebraic order and all expressions are solved using the standard order of operation. A.FO.06.06 Represent information given in words using Visual Basic Programming algebraic expressions and equations. Ex. Convert the contents of an Integer variable named testScore to String, and then assign the result to the totalLabel's Text property. totalLabel.Text = Convert.ToString(testScore) Represent Linear Functions Using Tables, Equations, and Graphs A.RP.06.08 Understand that relationships between Students can write programs to display data in a quantities can be suggested by graphs and table. tables. Ex. The president of the Harvey Company wants an application that performs the payroll calculations including employee's weekly gross pay, Social Security and Medicare tax, federal withholding tax and net pay to be displayed in a table. A.RP.06.10 Represent simple relationships between Students can write programs, using appropriate quantities using verbal descriptions, equations, and display the information in a table formulas or equations, tables and graphs. spreadsheet. Ex. Students use the TOE (Task, Object, Event) chart to create order forms including name, address, phone numbers of the customer, price and total number of products ordered and total amount of order including sales tax. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 10 05/18/2011 Apply Basic Properties of Real Numbers in Algebraic Contexts A.PA.07.11 Understand and use basic properties of real Students understand all basic properties of real numbers: additive and multiplicative numbers and the order of operation for programming identities, additive and multiplicative tasks. inverses commutativity, associativity, and the distributive property of multiplication over addition. Understand the Concept of Non-linear Functions Using Basic Examples A.PA.08.02 For basic functions, describe how changes Any program involving user input changes the value in one variable affect the others. of the output. Understand Solutions and Solve Equations, Simultaneous Equations and Linear Inequalities A.FO.08.10 Understand that to solve the equation f(x) Pay Raise program allows user input of rate of raise means to find all values of x for which the and then computes the wages for all employees. equation is true. Ex. newHourPay = GetNewPay(pay,raise) newPaylabel.Text = newHourPay.ToString("C2") End Sub G1 STANDARDS CTE APPLICATION and PRACTICE FIGURES AND THEIR PROPERTIES G1.2 Triangles and Their Properties G1.2.2 Construct and justify arguments and solve Collision Testing in a Pong game multi-step problems involving angle Ex. In a Pong game, when the ball collides with the measure, side length, perimeter, and area border or a paddle, the programmer must find the of all types of triangles. vector associated with the original movement and invert the angle by finding the perpendicular vector coordinates. G1.4 Quadrilaterals and Their Properties G1.4.1 Solve multi-step problems and construct Students can create programs to find the area and proofs involving angle measure, side length, perimeter of various polygons. diagonal length, perimeter, and area of Ex. Create a program that allows a user to input the squares, rectangles, parallelograms, kites, length and width of a rectangle and the price of a and trapezoids. square foot of tile, then calculate and display the total area and the total price of the tile. G1.5 Other Polygons and Their Properties G1.5.2 Know, justify, and use formulas for the Students write programs to find perimeter and area perimeter and area of a regular n-gon and of various polygons and therefore must know the formulas to find interior and exterior angles formulas. of a regular n-gon and their sums. Ex. Area of a Square Public Function CalculateArea () As Integer Return _side * _side End Function C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 11 05/18/2011 G1.6 Circles and Their Properties G1.6.1 Solve multi-step problems involving Students create a program to find the area and circumference and area of circles. circumference of circles. Ex. Create a program to calculate area of a circle. Double.Tryparse(radiusTextBox.Text,radius) area =Pi * radiusradius AreaLabel.Text = Convert.ToString(area) G2 STANDARDS CTE APPLICATION and PRACTICE RELATIONSHIPS BETWEEN FIGURES G2.2 Relationships Between Two-dimensional and Three-dimensional Representations G2.2.1 Identify or sketch a possible three- 3D programming dimensional figure, given two-dimensional Students can import 3D data in a 2D screen using views (e.g., nets, multiple views). Create a WorldMatrix. They can also rotate the image for two-dimensional representation of a three- multiple views. dimensional figure. G3 STANDARDS CTE APPLICATION and PRACTICES TRANSFORMATIONS OF FIGURES IN THE PLANE G3.1 Distance-preserving Transformations: Isometries G3.1.1 Define reflection, rotation, translation, and Importing 3-D data from a model file on your 2D glide reflection and find the image of a screen. figure under a given isometryG3.1.2 Given two figures that are images of each Students can describe the transformation used in other under an isometry, find the isometry image position and placement. and describe it completely. G3.1.3 Find the image of a figure under the When programming, more than one isometry is composition of two or more isometries and usually used when positioning and placement of an determine whether the resulting figure is a image on a screen. reflection, rotation, translation, or glide reflection image of the original figure. G3.2 Shape-preserving Transformations: Isometries G3.2.1 Know the definition of dilation and find the Converting 3D data to a 2D screen is called image of a figure under a given dilation. projection (dilation). The ProjectionMatrix converts the matrix values to a 2D screen and specifies how deep one can look into the screen. G3.2.2 Given two figures that are images of each Converting 3D images to a 2D screen other under some dilation, identify the Ex. Use the Matrix CreateScale(2.5f) center and magnitude of the dilation. C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 12 05/18/2011 Understand the Concept of Congruence and Basic Transformations G.TR.06.03 Understand the basic rigid motions in the Importing 3-D data from a model file on a 2D screen. plane (reflections, rotations, translations). Ex. To rotate, scale and position a rocket in3D Relate these to congruence, and apply studio Max use WorldMatrix them to solve problems. Matrix.CreateRotationX(MathHelper.Pi/2* Matrix.CreateScale(2.5f)* Matrix.CreateTranslation(rocketPositiion); G.TR.06.04 Understand and use simple compositions of Importing 3-D data from a model file on a 2D screen. basic rigid transformationsUnderstand the Concept of Similar Polygons and Solve Related Problems G.TR.07.03 Understand that in similar polygons, 3D programming corresponding angles are congruent and Students can write a code to scale images from 3D the ratios of corresponding sides are equal; data to a 2D screen and understand that the resulting understand the concepts of similar figures image is similar to the original image with sides in and scale factor. proportion and equal angles. Solve Problems about Geometric Figures G.SR.08.03 Understand the definition of a circle; know Students create a program to find the area and when to use the formulas for circumference circumference of circles. and area of a circle to solve problems. Ex. Create a program to calculate area of a circle. Double.Tryparse(radiusTextBox.Text,radius) area =Pi * radiusradius AreaLabel.Text = Convert.ToString(area) G.SR.08.05 Solve applied problems involving areas of Students create a program to find the area and triangles, quadrilaterals and circles. circumference of circles. Ex. Create a program to calculate area of a circle. Double.Tryparse(radiusTextBox.Text,radius) area =Pi radiusradius AreaLabel.Text = Convert.ToString(area) Understand and Apply Concepts of Transformation and Symmetry G.TR.08.09 Understand the definition of a dilation from Converting 3D data to a 2D screen is called a point in the plane and relate it to the projection (dilation). The ProjectionMatrix converts definition of similar polygons. the matrix values to a screen and specifies how deep one can look into the screen. G.TR.08.10 Understand and use reflective and Importing 3-D data from a model file on your 2D rotational symmetries of two-dimensional screen. shapes and relate them to transformations Ex. To rotate, scale and position a rocket in3D to solve problems. studio Max use WorldMatrix Matrix.CreateRotationX(MathHelper.Pi/2 Matrix.CreateScale(2.5f)* Matrix.CreateTranslation(rocketPositiion); C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 13 05/18/2011 S2 STANDARDS CTE APPLICATION and PRACTICE BIVARIATE DATA - EXAMINING RELATIONSHIPS S2.1 Scatterplots and Correlation S2.1.4 Differentiate between correlation and Students understand that there is a strong correlation causation. Know that a strong correlation between time spent on creating a program and the does not imply a cause-and-effect quality of the program. Waiting until the last minute relationship. Recognize the role of lurking to finish a program can put undo pressure on the variables in correlation. programmer and can result in a poor quality program. S3 STANDARDS CTE APPLICATION and PRACTICE SAMPLES, SURVEYS, AND EXPERIMENTS S3.1 Data Collection and Analysis S3.1.1 Know the meanings of a sample from a Customer Satisfaction population and a census of a population, Ex. Students understand that the needs and and distinguish between sample statistics requirements of a customer must be met for and population parameters. customer satisfaction, therefore, using a TOE (Task, Object, Event) chart is useful in planning the application. S3.1.2 Identify possible sources of bias in data Customer Satisfaction collection and sampling methods and Ex. There is a self bias involved in programming that simple experiments; describe how such bias must be overcome to meet the customer's needs can be reduced and controlled by random and requirements when planning a program. sampling; explain the impact of such bias A client might have different requirements than on conclusions made from analysis of the what the programmer would like to do. data; and know the effect of replication on the precision of estimates. S4 STANDARDS CTE APPLICATION and PRACTICE PROBABILITY MODELS AND PROBABILITY CALCULATION Understand Probability Concepts for Simple and Compound Events D.PR.08.06 Understand the difference between Lottery Program independent and dependent events and Students understand that the numbers generated recognize common misconceptions randomly are mutually exclusive events and that the involving probability. output must not contain any duplicate numbers. References: XNA Programming – Benjamin Nitschke Microsoft: Visual basic 2008 – Diane Zak C:\Docstoc\Working\pdf\3bdbe5a8-f6d8-4000-a21c-6c13587e45f7.doc 14 05/18
Objective: MATLAB is a very powerful computational program for computers which has its roots in Linear Algebra. We hope that by reading through the help files and trying some of the exercises on your own, MATLAB can become a helpful tool for both your current assignments and studying in Linear Algebra, and also for any future mathematics or engineering classes you may take. Please click on your desired chapter to scroll down and see a list of topics in that chapter:
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Buy Used Textbook eTextbook 180 day subscription $7124*ACP FUND. OF ALGEBRAIC MODELING 2E Fundamentals of Algebraic Modeling Fundamentals of Algebraic Modeling : An Introduction to Mathematical Modeling with Algebra and Statistics Fundamentals of Algebraic Modeling: An Introduction to Mathematical Modeling with Algebra and Statistics Student Solutions Manual for Timmons/Johnson/McCook's Fundamentals of Algebraic Modeling: An Introduction to Mathematical Modeling with Algebra and Statistics, 5th Summary FUNDAMENTALS OF ALGEBRAIC MODELING 5e presents Algebraic concepts in non-threatening, easy-to-understand language and numerous step-by-step examples to illustrate ideas. This text aims to help you relate math skills to your daily as well as a variety of professions including music, art, history, criminal justice, engineering, accounting, welding and many others. Table of Contents A Review of Algebra Fundamentals Mathematical Models Real Numbers and Mathematical Equations Solving Linear Equations Formulas Ratio and Proportion Percents Word Problem Strategies Graphing Rectangular Coordinate System Graphing Linear Equations Slope Writing Equations of Lines Applications and Uses of Graphs Functions Functions Using Function Notation Linear Functions as Models Direct and Inverse Variation Quadratic Functions and Power Functions as Models Exponential Functions as Models Mathematical Models in Consumer Math Mathematical Models in the Business World Mathematical Models in Banking Mathematical Models in Consumer Credit Mathematical Models in Purchasing an Automobile Mathematical Models in Purchasing a Home Mathematical Models in Insurance Options and Rates Mathematical Models in Stocks, Mutual Funds, and Bonds Mathematical Models in Personal Income Additional Applications of Algebraic Modeling Models and Patterns in Plane Geometry Models and Patterns in Right Triangles Models and Patterns in Art and Architecture: Perspective and Symmetry Models and Patterns in Art, Architecture, and Nature: Scale and Proportion
Basic Maths - Using A Calculator Ex Overall rating: 1 out of 5 from 9 members Award winning mathematics teacher Dr Murray Siegel helps those who struggle to understand mathematics, teaching not only the practical methods used in mathematics but the theory behind why these methods work. This edition presents all you need to know about using a calculator.
Professional Commentary: Students use matrices and technology to solve the Meadows or Malls problem, a linear programming problem with six variables. Students who have not done linear programming problems before are advised to begin with The Busing Problem before attempting Meadows or Malls.... Professional Commentary: This activity focuses on having students create and solve systems of linear equations in real-world settings. By solving a system of two equations in two unknowns, students find the equilibrium point for supply and demand.... Professional Commentary: This site shows how to represent rotations, reflections, expansions, compressions, and shears using standard matrices. An applet allows students to explore the connection between entries in the matrix and the effect of the transformation geometrically.... Professional Commentary: This tutorial reviews some of the basic definitions relative to matrices, such as trace, transpose, identity matrix, and inverse of an invertible matrix. Elementary algebraic operations on matrices are also reviewed, including sum, difference, scalar multiple, product, and determinant.... Professional Commentary: This site illustrates in general, and then with detailed examples, how to solve a system of equations using Gaussian elimination. Applying elementary row operations to the augmented matrix for the system of equations, the student can reduce the matrix to row-echelon form, from which the solution of the system, or the nonexistence of a solution, can... Professional Commentary: This lesson on cryptology from the National Security Agency, "America's Codemakers and Codebreakers," is accessible to a wide range of students. The 34-page lesson material begins with a discussion of shift transformations and letter frequencies, techniques of code analysis that middle school students can easily understand and enjoy.... Professional Commentary: Students read an antifreeze chart to determine the necessary mix of antifreeze and water for protection against various temperatures. Students then numerically analyze a particular cooling system to determine how much fluid to drain and replace with antifreeze to get the desired percentage concentration of antifreeze.... Professional Commentary: Land has been donated to River City. Students must decide how to split the land use between development and recreation in a way that will minimize the cost to the city of the necessary improvements, while adhering to restrictions agreed on by the River City Council....
SimulEquationsSoftware for teaching and learning of linear equations with two unknowns. Solutions by the elimination and substitution methods. Worked examples of solutions with each method shown side by side for easy comparison. Explanations provided for important steps in the solution to help understanding. It can generate hundreds of sums (up to 200 per session) for practice, but users may also choose to key in their own equations. The interactive guide can help the user do his homework if he keys in the sums. Marks automatically and can print a detailed report after each session. Version 3 includes New feature that allows two player to compete with each other on one computer Synchronised start for both players Each player can do the sums at his own pace Prints two separate detailed reports, each showing the sums done by the individual player Each report includes the scores of both players and the winners name Other minor improvements
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"I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!" There are some good suggestions in the comments, and some not so good ones. Surely our wise and mathematically sophisticated readers will be able to help. Add what you can there, and in the comments here if you like. I'd suggest "Introduction to Mathematical Reasoning" by Peter Ecceles. It's a textbook that helps bridge the gap between high school and college level mathematics. It's a great book that teaches student how to write up arguments in a logically rigorous manner; another book in the same vein would be "Tools of the Trade" by Paul Sally Jr. Also, I'd suggest anything by HMS Coexter. Another book, also focusing on geometry/topology, would be "Intuitive Topology" by Prasolov. It's a book used for students at Moscow Math School 57, and the first couple of sections are pretty readable–just playing around with knots and links. I think it's a nice book, but it might be a little "unstructured" for students. Also, there are two classics I'd recommend: The Moscow Puzzles (Kordemsky) and Mathematical Circles: Russian Experience (Fomin). Both books are excellent. They cover a wide array of mathematical topics disguised as brain teasers, puzzles and riddles. Brian 1. After algebra II, the high school math track moves toward calculus. Fine. You need calculus for any kind of engineering or science. But during the Fifteenth Century, mathematics moved in the direction of more advanced algebra. People figured out how to solve cubic and then quartic equations. When no one could crack general fifth degree equations, Abel and Galois investigated the roots of such equations and determined the impossibility of solution by radicals. I don't know of any general book, but I bet a good student could trace this development by Googling. 2. Diophantine Equations. 3. There are some good books of hard-to-prove geometry theorems. 4. Fermat's Last Theorem – among other things this book makes students aware that there are unsolved problems. Oh yeah. The Code Book, by Simon Singh. Maybe it's not all strictly math, but it is a good read about codes and ciphers. ts The list may vary depending on the goal. The age group (16-18) doesn't really mean much without knowing their goals and at what level they are being taught. Are they the kind who would be satisfied by merely passing remedial type math courses? Or are they future engineers and scientists? Few high school students get intrigued by pure maths, though they actually get intrigued by something like physics when the right buttons get pushed. For many it's easier to "get it" when there are intuitive contexts, just like dinosaurs, astronomy, and such are a great tool to lure the scientifically challenged into learn some science. jay George Polya's "How to Solve It" remains one of my all time favorites. Accessible to students with a rudimentary knowledge of geometry, it nevertheless retains its power for undergraduate and graduates students alike. Also, Imre Lakatos's "Proofs and Refutations" was a "must have" according to one of my CS professors.. Theo I highly recommend The Symmetry of Things, by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. The first section does lots of real math (classifies compact surfaces with boundary, and therefore orbifolds with nonnegative curvature), but is completely accessible to a math-inclined reader who doesn't know anything. These were classics that I found very helpful as a high-school student. Classics that are over a century old and still in publication. There was one more book for calculus that engineering students in the US use in their freshman years, but I forget which one – had a russian author. Ryan Dickherber Wikipedia. I know that sounds ridiculous, but I learned a lot about set theory on Wikipedia that helped immensely in the courses I later took. There are many great books, but Wikipedia is an excellent way to browse a ton of mathematics freely and easily. Brian In my post at 8:33 PM, I mistakenly referred to the book Fermat's Enigma, by Simon Singh, as "Fermat's Last Theorem." james I have to agree with "Flatland" – it's an interesting blend of philosophy and mathematics. Regardless of the subtext of the book, it's a mind opening story. Patrick Dennis "Fundamentals of Mathematics" by Moses Richardson (MacMillan, Various editions 1939-1966, now out of print. There is also a 1973 edition co-authored with – I think – his son.) I have the 1966 edition. It is a survey, yet one not only of astounding breadth, but also of great depth. Richardson maintains throughout the book the spirit of mathematical rigor. He begins with logical systems, then moves through the customary progression from counting numbers through to complex numbers, arithmetic, algebra (including group theory), functions,calculus, probability, even non-euclidian geometry and transfinites. All this in about 550 very well-written pages! It is a book that for forty years I have been able to pick up, ever confident that I would come upon an interesting passage or chapter. Soto I always like learning the history of the mathematical concept. I find that this helps with understanding. A book that does this well is "Zero: The Biography of a Dangerous Idea" by Charles Seife. Numbers, series, and integrals are among the many mathematical topics covered in the history of zero. Jonathan Moore Sticking to just one, I'll recommend "Forever Undecided: A Puzzle Guide to Godel", by Ray Smullyan. It's fun, challenging, and introduces some serious but sexy mathematics. Also, all of Joshua's suggestions sound good. Respectfully, I am going to recommend against some of the others: (i) Polya; better to try solving some problems (ii) the Singh books ; basically pop science (iii) Gardner; generally good, but the excellent Moscow Puzzles is very much better. Mark Most useful mathematics book: Advanced Mathematics for Engineers and Scientists. Yes, it's a Schaum's Outline, it's easily the most useful mathematics textbook I ever had and I wish I had a copy earlier in life. Most inspiring sciences book: A Short History of Nearly Everything by Bill Bryson, this book puts the sciences into perspective. It tells a story of the knowledge of the earth and the way that knowledge was attained. Angus Bohanon Coincidences, Chaos, and All That Math Jazz. Fantastic book, very accessible and actually fun to read. Won't actually teach anything, it's more a way to grasp concepts like infinity, dimensions beyond the third, fractals, etc. It's perfect for high school. Mr B I have a manuscript posted on my math blog that some of my students are reading to get better acquainted with basic calculus concepts. It's written for people who have a passing acquaintance with algebra and a dash of trig. Folks are welcome to take a look. "Differential and Integral Calculus" – Richard Courant Most of my undergraduate peers (in my country) understand calculus as a series of derivative and integral formula for standard functions. They can't appreciate what a limit or divergence means. A good book on Calculus must build stuff from the ground up. That apart, "One, Two Three . . . Infinity" – by George Gamov was a fun general read. And as someone already mentioned, Feynman's Lectures are a must. Eugene They could have a lot of fun reading Neal Stephenson's Cryptonomicon. senderista Why not linear algebra? It's not exactly traditional high school material, but it doesn't really have any prerequisites (beyond high school algebra and complex numbers), and it's full of easy-to-visualize examples. Elementary group theory would also be appropriate, I would think. I recommend "Linear Algebra Done Right" by Sheldon Axler, although it's clearly intended for math rather than science students. moshe David Foster Wallace on set theory and the foundations of mathematics: Morris Kline was a math professor and a critic of how math was taught. jester What about the following books? J. Weeks, The Shape of Space T. Needham, Visual Complex Analysis Bruce Rout Flatland is excellent but there is also a set of four books called from one to infinity that is a collection of all the major papers of mathematicians throughout history. A 15-year-old girl who hated math said it was the best book she had ever read. Ever! Tristram Brelstaff From Here to Infinity by Ian Stewart. The Knot Book by Colin C Adams. Fermi-Walker Public Transport How about "Number Theory in Science and Communication: With Applications to Cryptography, Physics, Digital Information, Computing and Self-Similarity" by Manfred Schroeder. Minimal background needed and as the title suggests, it has an applied orientation. trailblazer I think that some basic differential geometry/vector calculus/complex analysis will be suitable-for tensor calculus I think the best book is Borisenko and Tarapov-old russian book-one of the best in the subject(a good book is also Introduction to Geometry by Coxeter). Probability theory is also a very good add in to the curriculum-for instance the book by Kolmogorov-not the classic Foundations of …, but a small, really interesting, high school level book , which goes all the way from dice to the central limit theorem-it is really interesting to learn the subject from the master's book. randomeda Flatland -Edwin Abbot Pieter Kok Chaos by James Gleich. David Derbes I'm a high school physics teacher. There has been an explosion of extremely enticing math books for the general reader (I really think Hawking's "Brief History" was the start of this.) Many are published by Princeton, some by Johns Hopkins. The three big names are Ian Stewart, Paul J. Nahin and Eli Maor. Another reader above suggested abstract algebra; Stewart's "Why Beauty is Truth" goes a long way toward addressing group theory for high school students. Barry Mazur and Simon Singh have also published good books for high school students and the general public. For logic, it's very hard to beat Raymond Smullyan's books. My favorite is his first, "What is the Name of This Book?" which gets to Gödel's theorem via jokes and riddles. For classics, there are these: "Calculus Made Easy" by Silvanus P. Thompson, or the revised version with Martin Gardner, and Michael Spivak's largely unknown "The Hitchhiker's Guide to Calculus". Courant and Robbins, "What is Mathematics?" is another classic, as is Hilbert and Cohn-Vossen's "Geometry and the Imagination". These are a little dry, as is Weyl's "Symmetry". I think the suggestion of the Feynman Lectures is not really a good one for most high school students. On the other hand, Feynman's "Character of Physical Law", though not strictly mathematical, is an excellent choice. Better yet in my opinion is "Feynman's Lost Lecture" which is actually mathematics (that the inverse square law leads to elliptical orbits, done with very little more than Euclidean geometry.) Finally, some of the quirky books of Lillian and Hugh Lieber have recently been reprinted. These are "The Education of T. C. Mits", "Infinity" and "The Einstein Theory of Relativity". (Disclaimer: I LaTeXed and helped to edit the last, though I have no financial interest in the book's success.) Harold Abelson & Gerald Jay Sussman – "Structure and Interpretation of Computer Programs" ingenious introduction to, well, I'm not really sure… "programming" would be a vast understatement. Its arc spans from Ackermann's function, to symbolic computation of polynomials, to an interpreter for the Scheme programming language (which the book is written in, by the way) Richard Feynman – the lectures, volume 1 no introduction needed ollie I'd suggest NO EXTRA CREDIT at all. What happens is that students get used to this option and then show up at college and expect a "way out" of learning the assigned material. But as far as reading, Abott's "flatland" and (forget the author) "How to Lie with Statistics" are both readable and good. The Shape of Space by Weeks will probably be too much for them, but has lots of cool pictures. Nova Terata Neil Stephenson's Anathem would be particularly relevant to students Kyle G For an introduction to advanced mathematical ideas that should interest, rather than frighten, high schoolers, I think Burger & Starbird's Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas is a good choice. It's heavy on the concepts but light on the complicated math. I've had Starbird as a professor before, and he's extremely talented at teaching non-math majors about weighty mathematical concepts. I've seen him teach topology to run-of-the-mill liberal arts majors with success. Count Iblis I have yet to find a suitable math book for high school students. The problem is that in math teaching, little is done to present the material so that it looks very exciting. What you need to do is to present some easy to understand spectacular stuff that will motivate the students to spend a lot of time studying math. E.g., you could teach modular arithmetic and then immediately show the power of this method. You can give an elementary proof of Fermat's little theorem (a^(p-1) = 1 Mod p), Chinese Remainder theorem, Euler's generalization of Fermat's little theorem etc. Calculus: Why not explain Taylor's expansion intuitively (just fit a polynomial by requiring that the derivatives match). Then a spectacular result would be to use it to give Euler's intuitive nonrigorous argument that zeta(2) = pi^2/6. Also, use that to prove that the probability that two large integers have no common factor is 6/pi^2. If students are exposed to math in this way, then they will like it a lot more and be prepared to go through the tedious rigorous proofs. Bryan Richard Courant's What is Mathematics?. This is the mathematician's classic. It's a serious introduction to the field. It's also readable in highschool. and has the amazing propensity to make you want to do graduate work in math, before you even get to college. (That's what happened to me, anyway.) Mike I teach mathematics at a highly selective liberal arts college in the Midwest, and I have assigned as supplementary readings a number of excellent books written for a general audience that do a wonderful job of conveying the joy of mathematical discovery. I've seen two of them in the preceding comments, but I'll repeat them here to add another vote for their selection. Chaos, by James Gleick The Shape of Space, by Jeffrey Weeks Sync, by Steven Strogatz Godel, Escher, and Bach, by Douglas Hofstadter Beckyj I haven't read a ton of math books, but I did like the History of Pi by Petr Beckman. It describes the development (or estimation) of the value of pi through the centuries. Gavin Polhemus Four Colors Suffice: How the Map Problem Was Solved, by Robin Wilson This is one I haven't seen among the many great suggestions. Four Colors Suffice is great because it connects an ancient problem to it recent solution, addresses a problem that may people wouldn't even think of as math (map coloring), and actually does a great job of explaining the actual proof, in addition to giving the story of the people who did it. Early in the book readers learn how to prove that any map can be colored with six colors. This is pretty easy. Later they learn how to prove that any map can be colored with five, which is more difficult and includes all of the essential elements of the four color proof. The four color proof was eventually done with the assistance of a computer, which is an interesting twist to the story. It talks about many related problems as well as giving the stories of the many interesting people involved. Also, it's not very long. It has all of the equations that are needed, but they are very simple formulas because it is a mapping problem. There are far more pictures than equations. I teach high school too, I know these will be a primary consideration for some kids. However, even with a Ph.D. in physics, I found this book fascinating. robert61 When I was that age, I read A History of Pi in one sitting after picking it off a friend's shelf. efp You might want to check out The Mathematical Experience by Hersh & Davis. I recall reading this around high school age. It consists of short, self-contained essays, so you could pick and choose. And I second the notion that there should be no such thing as extra credit. Ever. Blake Stacey I read a fair number of math-y books when I was that age, but in thinking back, it's easier to come up with glitz and glamour than it is to recall books which actually helped develop the mathematical skills I use as a science person. Knuth's Surreal Numbers was fascinating, and I probably got some indirect benefits from encountering proof techniques (induction and such), but how often do physicists use surreal numbers? Moreover, books on more recent developments — like Singh and Aczel's books on Fermat's Last Theorem, or Keith Devlin's various attempts to popularize the Riemann hypothesis — have too many gaps, too many places where the abstruse sophistication of the mathematical arguments are glossed over with a bit of narrative. It's fun, yes, and it keeps the enthusiasm stoked, but you can't actually solve problems in group theory by applying the life story of Evariste Galois. And, if you can't actually use the mathematics, it's not really part of your life, is it? Having issued all these caveats, then, here are some books I've enjoyed, ranked in increasing order of "I could do stuff after having read this": Surreal Numbers, by Donald Knuth The Book of Numbers, by John Conway and Richard Guy QED: The Strange Theory of Light and Matter, by Richard Feynman Chaos, by James Gleick The Cartoon Guide to Statistics, by Larry Gonick and Woollcott Smith The Manga Guide to Statistics, by Shin Takahashi Also, I'm midway through Douglas Hofstadter's I Am a Strange Loop and Marcus du Sautoy's Symmetry: A Journey Into the Patterns of Nature (originally published as Finding Moonshine in the UK), and I bet I would have liked both of them when I was a ninth-grader. John T. Scott I very much enjoyed John Derbyshire's book on the Riemann Hypothesis. Blake Stacey, above, seems to downplay an apparently similar book by Keith Devlin, which I don't know, but Derbyshire held my interest throughout, and I suspect the book might help a bright highschooler appreciate the power of mathematics. Another one would be Morris Kline's Mathematical Thought from Ancient to Modern Times. Sure, it's a history book, but Kline teaches a lot of mathematics along the way. I continue to find that the amount of mathematics known to the Babylonians, Greeks, Egyptians et al. quite staggering, and I hope that your high-school students would feel the same. Eunoia Innumeracy. Beyond Numeracy. G.E.B lcjohnson I read from a great text in a college course on the history of Math, but would be great for high school because it is split up into short stories – a sketch for each element explained. Berlinghoff and Gouvêa, Math Through The Ages. Published by Mathematical Association of America. hegemonicon Another vote for Godel, Escher, Bach. Tom S. For a more personal touch: "Men of Mathematics" by E. T. Bell. Although Bell doesn't aways get the facts right, the book is a very entertaining look at the lives and work of famous mathematicians of the past. JK I was the sort who read lots of the books mentioned above at that age (at least the ones that were published then…). The most outstanding were: Weeks, The Shape of Space Hilbert and Cohn-Vossen, Geometry and the Imagination Hoftstader, Godel Escher Bach You can give them Penrose, either The Road to Reality or, maybe better, The Emporer's New Mind. If they're like me, they will only understand a fraction, but the writing will set them on the right path. I don't think anyone has mentioned Stewart and Tall, The Foundations of Mathematics Stewart's popularisations have been mentioned, but this is his text book specially designed to make the transition between school mathematics and the rigour of university study. I leant how to appreciate 'real' mathematics by self-study of that book and would highly recommend it. Scott E I am a high school math teacher as well and there are some interesting books on the list. Here a few not mentioned: Math Devil (an odd little book), Conned Again, Watson (Sherlock Holmes and Dr. Watson take on cases with mathematical solutions) and Journey Through Genious (a very well written -with lots of real math – history book). casey jane THE UNIVERSE IN A TEACUP by K.C. Cole Igor Another vote for "Godel, Escher, Bach" – this can be, no hyperbole, a life changing book for the mathematically inclined. I'd also suggest "When Least is Best" by Paul Nahin, which is a fantastic book about mathematical optimization. It's a very accessible but also very rigorous introduction to some of the most interesting and important applied mathematics out there. Nick One I wish I had read in high school is "How to prove it" by Velleman (sp?). Very useful for someone who intents to take maths in university but has not taken a class on how to write proofs. Jimbo Frankly, most of these professionals have lost sight of the reality of HS, when you have hot blood flowing thru your veins, its hard to concentrate on anything, much less the king of abstraction, mathematics. Nonetheless, IAll the above esoteric math refs might be OK for gifted students, but the average Jane/Joe Schmoe will ultimately use math as a tool to facilitate earning a living, not an end to itself. Do not short change them. Equip them with practical algebra, trig, and elementary calculus so they leave with a diploma that empowers them, not retards or intimidates them. I have taught college physics in 4 states, at 4-yr universities, community colleges, & tech schools. Without questtion, math is the primary hangup. Conquer that, and all walls fall. It explains proof ideas for the mathematical novice alongside the actual material. And it's probably the next course the student will study in college anyway — except he'll probably be forced to learn it the "wrong way" — through matrices, row reduction, systems of equations, etc. Rien I agree with Simon Singh's Fermat book and James Gleick's Chaos. Both make math seem exciting and you will probably get at least a little bit of math with you from reading them. Courant's book is very nice, but I doubt that anyone but the already quite interested will want to read it. If you are a math geek it will be perfect, though. a grad student Calculus by Spivak get it done right Blake Stacey John T. Scott, I was thinking of the chapter "Hard Problems About Complex Numbers" in Keith Devlin's Mathematics: The New Golden Age. It's a good book, and if you want an introduction to Mersenne primes or Cantorian higher orders of infinity, it's probably great for your purposes. (I believe there's since been a revised edition which updates the chapter on Fermat's Last Theorem and such.) My only issue is that I'd have to rank it only middling on the "I could actually do math and science after reading this" scale. (Yes, there are all sorts of factors influencing how much practical competence one gains from a book or a class, including one's self-discipline, but holding all else constant, there's still a gradation of books in this regard, I believe.) It might be heresy to mention a TV show in a thread about books, but I have to plug the Caltech production Project Mathematics!, which is an all-around nifty treatment of geometry and trigonometry. Paul Murray I Well in that case, boolean algebra and math that relates to computing is a must. I don't know what gets taught in computing classes these days: probably "how to use microsoft word to prepare a job application". If you are interested in exposing the students to joys of abstract math, then LISP and Prolog might be the go. There are free interpreters out there. Or: why not teach boolean algebra by having them assemble logic circuits on breadboards? The real rudiments of computing: AND, OR and NOT gates, and making the LEDs flash. The chips are reasonably cheap, I think. You can go onto groups and modular arithmetic and whatnot from there. Barbara One, Two, Three . . . Infinity, by Gamow. Old, but as good as ever. It fascinated me in high school and decades later I startled a nephew by giving him a copy. Methalos The Archimedes Codex By Reviel Netz, William Noel Adan.Mike.Selene Who Is Fourier?: A Mathematical Adventure (Paperback) by Transnational College of LEX (Author) This book was written by members of a Japanese educational commune who devote themselves to innovative learning styles. They learn languages by immersing themselves in language — they say expose themselves to 11 languages simultaneously, and it works! They were interested in Fourier series, as they related it to understanding sound, which was related to their interest in language, and they set out to understand it in creatively. This book has cartoonish pictures, but don't get put off. It sets out to derive the concepts it needs from the ground up. I love math books that lead me along a chain of logic that makes everything fall into place. This book starts by explaining graphs and trig functions, and goes through Fourier transforms. I love Courant's calculus, because it explains everything (although Courant keeps making comments that his work is simplified and not truly rigorous!), and I love this book for the same quality, or though it is the opposite end of the spectrum from Courant, which is about as formal as one will be exposed to these days. This book is unique. Check it out, I cannot do it justice. estraven anon at 11:36 said: "While in school, someone had gifted me Courant & Robbins 'What is Mathematics?' That book was a revelation." Same happened to me. That book made me a mathematician. I also strongly second Feynman's Physical Law. It takes away the guilt from wanting to be a mathematician . To Jimbo, who claimed hormones make study difficult: I disagree. Mathematics is the only thing gripping enough to take your thoughts away from sex. Or so it seemd to me as a teenager. coolstar Jimbo has it EXACTLY right. It's hard to fathom how truly BAD most of these suggestions are (no, I don't have any better ones other than seconding Mark's very valuable and pragmatic suggestion). The fraction of HS students interested in pure mathematics is probably one or two orders of magnitude LOWER than those interested in a career in physics or astronomy. I'm very happy when I get students out of HS who are not 1) totally innumerate 2) totally turned off by math and science (almost always by bad secondary teaching). Oh, want to change these numbers? Go volunteer at a high school or your local community college (and not just to teach the "smart" kids!). (my own teaching career mirrors Jimbo's pretty well, by the way). RDeen "Concise Introduction to Pure Mathematics", Martin Liebeck – great book that I read a couple of months ago, and I'm in high school too. David Park John Stillwell, _____Numbers and Geometry ____ Mathematics And Its History Jimbo CoolStar, Comrade in Arms…Merci beau coup, Monsieur ! Paul Murray…Ask them to count in base 2 to 64. Me thinks you will get a sobering lesson in math reality, and soon forget about logic circuits…Better to focus only on soldering ! The state of American HS math is worse than the American economy… And that's sayin A-Lot. I have to agree with coolstar & Jimbo. I can see that most people here don't deal with real high school students, outside of perhaps those occasional "feel good" science outreach programs. Looking at the book suggestions here, I can clearly see overenthusiastic math teachers who cannot communicate at all with students. So sad. Scott If these students are already into calculus at all, I'd suggest "Calculus Made Easy" by Silvanus Thompson, or (even better) Martin Gardner's annotated reprint of the same book. As for the concept of infinity, Rudy Rucker's "Infinity and the Mind" is a good one – very entertaining! Lastly, if any of your students enjoyed Edwin Abbott's "Flatland", I'd also suggest "Flatterland" by Ian Stewart, and "Spaceland – A Novel of the Fourth Dimension" by Rudy Rucker. Anil Maybe "What is mathematics" – by courant, robbins and stewart And "Does god play dice" by stewart. Ginger Yellow "Another vote for "Godel, Escher, Bach" – this can be, no hyperbole, a life changing book for the mathematically inclined. " Absolutely. I only have A-level maths, but it's still the best non-fiction book I've read by some margin (sorry, Phil).
More About This Textbook Overview Stahl's Second Edition continues to provide students with the elementary and constructive development of modern geometry that brings them closer to current geometric research. At the same time, repeated use is made of high school geometry, algebra, trigonometry, and calculus, thus reinforcing the students' understanding of these disciplines as well as enhancing their perception of mathematics as a unified endeavor. This distinct approach makes these advanced geometry principles accessible to undergraduates and graduates
Item code: SAX-3281 The unique approach of these 7th grade homeschool lessons and teacher books has helped make Saxon Math a best seller. The solutions to Saxon Math 87 3rd edition problems usually involve many distinct and important steps. The Solutions Manual is the only place that provides the step by step solutions to every problem in the student textbook. Saxon Math 87 homeschool curriculum 3rd edition student textbook as well as the Tests and Worksheets are sold separately. The student textbook presents and explains concepts and provides practice exercises. The Tests and Worksheets provide test questions for review after every fiver or ten lessons in the student textbook as well as consumable problem solving exercises, investigations, facts practice, and activities. Typically used in the seventh grade, Math 87 is a transition program for students who have completed Math 76 but are not ready to enroll in pre-algebra. Basic mathematical concepts and skills are reviewed and reinforced. Concepts, procedures, and vocabulary needed to succeed in upper-level mathematics courses are introduced and developed incrementally with continual practice. Math 87 includes the study of fractions, decimals, percents, and ratios; perimeter, circumference, area, and volume; exponents; scientific notation; and signed numbers. Students continually practice problem-solving techniques through word problems. 3rd Edition, 2004 (135 lessons).
including algebra 1 and algebra 2including algebra 1
...I have been using it professionally for years, because it is one of the math subjects that connects fully to the physical world we live in. That makes calculus very intuitive and easy to explain. What makes calculus difficult is that it requires strong math skills in foundational math up to the level of trigonometry, and any holes or weaknesses in those areas need to be addressed. ...Elementary Math explores pi (3.14 or 22/7) and its relationship to the formulas for the circumference and area of a circle. It shows what is to come in Algebra by introducing the variable in formulas involving geometric shapes and ratios and solving one-step linear equations. Elementary Math de
Modern Engineering Mathematics Building on the foundations laid in the companion text Modern Engineering Mathematics, this book gives an extensive treatment of some of the advanced ...Show synopsisBuilding on the foundations laid in the companion text Modern Engineering Mathematics, this book gives an extensive treatment of some of the advanced areas of mathematics that have applications in various fields of engineering, particularly as tools for computer-based system modelling, analysis and design. The philosophy of learning by doing helps students develop the ability to use mathematics with understanding to solve engineering problems. A wealth of engineering examples and the integration of MATLAB and MAPLE further support students.Hide synopsis Modern Engineering Mathematics The series make learning engineering maths simpler. Unfortunately there are little questions with answers within this book. Its the "new teaching method," where nothing can be known, unless your the lecturer with the answers booklet. Still it is a great series, worth it as it covers many fields
Science and Math Products Selection of Tables of Squares, Cubes, Square Roots, Cube Roosts, Fifth Roots and Powers, Circumferencecs and Areas of Circles, Common Logarithms of Numbers and of the Trigonometric Functions, the Natural Trigonometric Functions, Natural Logarithms, Exponential and Hyperbolic Functions, and Integrals. Explanatory Notes by Edward S Allen, Assoc Professor of Mathematics, Iowa State College. Second Edition Fourth Impression. Published by McGraw-Hill Book Company, Inc. 144 Pages are age yellowed, with no tears or bends, but a little soil and penciled notes here and there. Leather Bound cover has very light edge wear, light soil and tiny spots. Inside front and back cover pages have owner name, street address and phone number in ink, plus lots of penciled math figures. Great addition to a vintage math library. Elementary Algebra Structure and Use Raymond A. Barnett This is an introductory course in algebra. It is written for students with no background in algebra and for those who need a review of algebra from a contemporary point of view. McGraw-Hill Book Company, 1968. ISBN 07-003785-X. Hard cover, no dust jacket, 6-3/4 x 9-1/4 inches, 402 pages. Used label on spine. Scar at top of front cover. may be a few pencil notes. Firmly bound. Under 3 lbs. Our World in Space and Time International Pictorial Treasury of Knowledge, Vol. 1 The most absorbing story on earth is the story of the world itself. . . Its mysterious beginnings and countless ages without life! Mighty changes that made molehills into mountains! Nature's experiments with the first creatures and giant reptiles! Man's evolution, epic discoveries, inventions and explorations! Today's challenge of the unknown in outer space! All this is excitingly and accurately depicted in this pictorial treasure. A basic guide to knowledge. International Graphic Society, 1960. Hard cover, with dust jacket, 7-3/4 x 9-3/4 inches, 192 pages. Dust jacket has slight edge and corner wear. Book is in excellent condition. Under 3 lbs. Calculus By Discovery Douglas Downing, ill by Susan Detrich This book tells of adventures that took place in the land of Carmorra. The story is told here because, by following these adventures, you can learn differential and integral calculus. This book includes material suitable for a first-year calculus course. It is designed to be used in a classroom, but it can also be used by someone wishing to learn calculus on his or her own, or as a supplement to a course. This book is unlike regular math books, though. You are invited to read the book as you would read a fantasy novel. Barron's Educational Series, Inc., 1981. ISBN 0-8120-5451-2. Hard pictorial cover, 8-1/4 x 10-1/2 inches, 273 pages. School stamp on title page and back fly. Back cover has chewed corner and small nick along edge. Number on top and bottom page edge. Text is good, clean and firmly bound. Under 3 lbs. These flash cards are for grade level 3 - 5. I believe that these particular cards are no longer in print. This box contains 42 cards and one instruction card. The cards are in excellent condition, are still sealed in a cellophane package and have never been opened. This is a great time of year to purchase so your student can be practicing this Summer!!! Mathematics in Action Reteaching Activities Grade 7 The Reteaching Activities book is designed to reinforce the learning of specific concepts or skills in Macmillan / McGraw-Hill Mathematics in Action. These activities provide brief instruction opportunities for practice and application that match the Practice exercises in the text. The activities in the reteaching activities book can be used as students work through a chapter or after they have taken the chapter test. Answers not provided. Macmillan / McGraw-Hill, 1994. ISBN 0-02-109358-X. Soft cover, 4to - over 9¾ inches - 12 inches tall, about 149 pages. Punched for three-hole binder. Slight cover creasing, otherwise near new. Another impressive volume from the 'Mysteries of the Unknown' series, which examined the history and nature of seemingly paranormal phenomena, THE UFO PHENOMENON was published in 1988 by the Editors of Time-Life. In the 160-page hardbound book are chapters on 'Elusive Visitors,' 'Fear and Hope on Film,' 'Into the Saucer Era,' 'Kidnappers from Space,' A Time of Close Encounters,' 'Project Blue Book,' 'A Deepening Controversy,' 'The Enduring Enigma,' and 'A Universe of Possibilities.' As is usually the case with books from Time-Life, this is filled with wonderful color artwork and the articles are highly detailed. People, places and names are specific and make the information extremely interesting! The book is in Very Good condition, with just some very minor edge/corner/spine end wear and a couple of minor scuffs to the covers Inside pages have no apparent defects. If you're interested in UFOs, don't miss this one! It'll keep you wondering (5169) Addison-Wesley Mathematics Text book Teacher's notes written in about this book. One says 'Good home practice for fourth grade.' Another says 'Good practice. Use several times,' and another 'This could be fun..' Illustrations are colorful, activities are very interesting for the fourth grade student. An excellent math text book for home study. Addison Wesley Pub. Co., 1987. ISBN 0-201-26400-5. Hard pictorial cover, 8-1/4 x 10-1/4 inches, 442 pages. Besides the notes, neatly written in ink, and other ink marks, the corners and spine ends are worn. There is a much used 'Property of' stamp inside front, and inside front and back have crayon dust soil. Fly is creased. Page 377 has a long rip that needs to be taped. Most of the book is in very good condition. Under 3 lb. Standard Mathematical Tables, Student Edition Samuel M. Selby, Ph. D., Ed. This book contains important mathematical and scientific information. An aid to the teaching professional, the mathematician, the physicist, the engineer, and to many others who require a mathematical technique, a fact, or a table, for investigating and creating the answers to challenging problems in the academic and scientific fields. The Chemical Rubber Co., Cleveland, OH, 1967, 15th ed.. Hard blue cloth cover, no dust jacket, 6-1/8 x 9-1/4 inches, 664 pages. Very good condition. price stamped on front fly. Dent on top of back cover. Index pages have creased corners. Under 3 lb. Under the Sea Science Adventures Helen H. Carey, & Judeth E. Greenberg Illustrated by Andrea Z. Tachiera, This book takes you on an underwater adventure! First, you will explore the surface of the ocean, or sea. Then you will go deeper underwater to explore and to live for a time. You will become par of a new world, one that is almost as strange as another planet. You will also do two experiments that will help you understand water pressure and the darkness of the sea floor. Raintree Publishers, 1990, first edition. Hard pictorial cover, 8 x 9-1/2 inches, 32 pages. Ex-library, with date page inside front, and library and discard stamps on copyright page. Clean, well bound, no tears. Under 2 lb. Safe & Simple Projects with Electricity Charles D. Neal A book of practical, easy projects that help readers discover, and this understand, some of the principles of electricity and how it works. Children's Press, 1965. Hard cover with dust jacket, 8-1/2 x 11 inches, 157 pages. Ex-library, with pocket and date page in front. Library markings throughout. Dust jacket is covered in mylar, taped to book cover. Extensive shelf wear to bottom edge. Slight loosening of spine and small tears at bottom of some pages. Taped tear in back fly. Under 3 lb. The Artificial World Around Us Lucy Kavaler drawings by Gloria McKeown Have you ever gone to a supermarket and been lured straight to the shelves of packaged cake by the artificial smell of freshly baked cinnamon buns? Have you ever been to a used car dealer and been surprised that the second-hand cars smell new? Do you know that hardware, automotive parts and machinery can be made out of plastic instead of metal? Do you know that diamonds, rubies and emeralds need not come from a mine? Do you know that America's favorite ice cream is flavored with wood pulp and many soft drinks owe their sweetness to coal? Do you know that you can satisfy your hunger without food? The Artificial World Around Us introduces the fascinating world born in a test tube. Some of the discoveries made in the laboratory provide only pleasure or comfort. But most of them serve a real purpose in these times of rapidly expanding population and rapidly diminishing natural resources. Published by The John Day Company, 1963. Hard cover, no dust jacket. 125 pages. Stamped inside front cover, library pocket inside back cover, a bit of wear to corners. A little discoloration at bottom of spine. But very clean and neat for an ex-library book. Under 1 lb. Mechanics of Materials - by Archie Higdon, Edward H. Ohlsen, and William B. Stiles. 1960 by John Wiley & Sons, Inc. 502 pages with hardback cover, in very good condition. Book category: Technical. PDP42 The Golden Picture Book of Science by Rose Wyler Pictures by Marjorie Hartwell and Valerie Swenson This book lists subjects as Animals, Plants, Rocks, Gravity, Day and Night, Rain and Snow, the Sky and the Ocean. With 45 experiments and activities. Published by Golden Press in large format, 1957. Hard pictorial cover, glossy finish, 8-1/4 x 11 inches, 57 pages. Spine has been taped to repair tear halfway down back side of spine, which does not affect binding. Corners are worn. Owner's name stamped into inside front cover and fly. Clean. Tightly bound. Under 2 lb. Engineering Projects for Young Scientists by Peter H. Goodwin Each book in the Projects for Young Scientists series tells you what a science project is and how to do one; discusses science fairs; and presents numerous ideas for science projects suitable for independent study, classroom requirements, or science fairs. Presents practical problems & science fair projects related to engineering & physics, covering such subjects as force, friction, motion, sound waves, light waves, and mechanics. Publilshed by Projects for Young Scientists, a Grolier Co., 1987. 2nd printing. ISBN 0-531-10339-0. Hard back, DJ covered with mylar. Ex-library with date due sheet on front fly, in very good condition. Under 2 lb. Energy Projects for Young Scientists by Robert Gardner Each book in the Projects for Young Scientists series tells you what a science project is and how to do one; discusses science fairs; and presents numerous ideas for science projects suitable for independent study, classroom requirements, or science fairs. Instructions for a variety of projects and experiments related to solar, thermal, electrical, kinetic, and potential energy. Publilshed by Projects for Young Scientists, a Grolier Co., 1987. 2nd printing. Hard back, DJ covered with mylar. Ex-library with date due sheet on front fly, library stamp on edge. DJ has 1-3/4 inch tear in lower front. Othewise in very good condition. Under 2 lb.
from a wide variety of mathematical subjects, this book aims to show how mathematics is realised in practice in the everyday world. Dozens of applications are used to show that applied mathematics is much more than a series of academic calculations. Mathematical topics covered include distributions, ordinary and partial differential equations, and asymptotic methods as well as basics of modelling. The range of applications is similarly varied, from the modelling of hair to piano tuning, egg incubation and traffic flow. The style is informal but not superficial. In addition, the text is supplemented by a large number of exercises and sideline discussions, assisting the reader's grasp of the material. Used either in the classroom by upper-undergraduate students, or as extra reading for any applied mathematician, this book illustrates how the reader's knowledge can be used to describe the world around them. less
Covering the main fields of mathematics, this handbook focuses on the methods used for obtaining solutions of various classes of mathematical equations that underlie the mathematical modeling of numerous phenomena and processes in science and technology. The authors describe formulas, methods, equations, and solutions that are frequently used in scientific and engineering applications and present classical as well as newer solution methods for various mathematical equations. The book supplies numerous examples, graphs, figures, and diagrams and contains many results in tabular form, including finite sums and series and exact solutions of differential, integral, and functional equations.
Diploma in Mathematics Diploma in Mathematics The free online Diploma in Mathematics course from ALISON gives you a comprehensive knowledge and understanding of key subjects in mathematics. This course covers calculus, geometry, algebra, trigonometry, functions, vectors, data distributions, probability and probability and statistics. Math qualifications are in great demand from employers and this Diploma will greatly enhance your career prospects. Greatly improve your geometry skills Learning Outcome Learning outcome This course will greatly enhance your skills in many areas of mathematics, giving you a greater understanding of core mathematics components such as geometry, trigonometry, calculus and more as well as expanding your knowledge base in areas such as chance, data distributions, statistics, probability, correlations and regression. You will learn about using binomial expansions for problem solving and will understand the relationship between the graphs of functions and their antiderivatives. You will be able to confidently create graphs and make advanced calculations such as straight-line calculations, kinematics, motion, vectors, algebra, binomial expressions, and quadratic functions.
Self-instructional Mathematics Tutorials The following mathematics tutorials development as part of the project, Increasing Students Success: Addressing Prerequisite Mathematics Assumptions in Introductory Non-mathematics Courses, funded by The Fund for the Improvement of Postsecondary Education. (project #P116B60125) Various introductory courses at six universities have been selected for this project. One goal is to provide self-instructional mathematics tutorials for individuals who may need review of certain topics. This self instructional approach will: • Let you move at your own pace. • Provide you with additional review (if necessary). • Let you know how well you are doing. Currently the non-interactive versions have been developed. While some do not have a lot of graphics, the review materials 3, 4, and 5 are fairly graphic intensive and may take a few minutes to load. Interactive versions are currently being developed and will be added to this site at a later date. \|
Accessible Mathematics is Steven Leinwand?s latest important book for math teachers. He focuses on the crucial issue of classroom instruction. He scours the research and visits highly effective classrooms for practical examples of small adjustments to teaching that lead to deeper student learning in math. Some of his 10 classroom-tested teaching shifts... more... Advanced numerical simulations that use adaptive mesh refinement (AMR) methods have now become routine in engineering and science. Originally developed for computational fluid dynamics applications these methods have propagated to fields as diverse as astrophysics, climate modeling, combustion, biophysics and many others. The underlying physical models... more... A compendium of fundamental computer science topics and techniques. It illustrates how the topics and techniques come together to deliver efficient solutions to important practical problems. It contains four chapters that cover external memory and parameterized algorithms as well as computational number theory and algorithmic coding theory. more... A compendium of fundamental computer science topics, techniques, and applications. It covers self-stabilizing and pricing algorithms as well as the theories of privacy and anonymity, databases, computational games, and communication networks. It also discusses computational topology, natural language processing, and grid computing. more... The field of multiscale problems occurs in many fields of science, such as microstructures in materials, sharp-interface models, and others. Reporting on the mathematical developments in the DFG Priority programme, this book provides the state-of-the-art on the mathematical foundations of the modeling and the numerical treatment of such problems. book collects up-to-date papers from world experts in a broad variety of relevant applications of approximation theory, including dynamical systems, multiscale modelling of fluid flow, metrology, and geometric modelling to mention a few. The 14 papers in this volume document modern trends in approximation through recent theoretical developments,... more... Approximation theory and numerical analysis are central to the creation of accurate computer simulations and mathematical models. Research in these areas can influence the computational techniques used in a variety of mathematical and computational sciences. This collection of contributed chapters, dedicated to renowned mathematician Gradimir V. Milovanovia,... more... Contains a cross-section of the papers that are presented at the BAIL 2008 conference, which was held from 28 July to 1 August 2008 at the University of Limerick, Ireland. This book gives an overview of the research into many engineering and mathematical problems of a singularly perturbed character. more...

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