# What Is School Mathematics?

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 What Is School Mathematics? Lisbon, Portugal January 30, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document.

2 The German conductor Herbert von Karajan once said that there is no vulgar music, only vulgar performance. I recall this statement by Karajan because mathematicians love to say that school mathematics is trivial. Let me therefore paraphrase Karajan by declaring that there is no trivial mathematics, only trivial mathematical exposition. I will argue my case by discussing a few basic topics from school mathematics from an American perspective, and let you decide if they are trivial or not.

3 Let us start with the simple long division of 587 by 3. You know the algorithm: Two questions: (1) What does this mean? (2) Why is it correct? The answer to (1) is that if 587 is divided by 3, then the quotient is 195 and the remainder is 2. The answer to (2) will occupy us in the next several minutes.

4 We cannot answer (2) without a clear understanding of the answer to (1). The usual expression, = 195 R 2 carries no information and also does not make sense, because: The left side is not a number, and the right side is not a number either. So what does it mean to say the left side is equal to the right side? The correct meaning of (1) is: we have a division-with-remainder, 587 = (195 3) + 2, where 2 < 3. Now we have to prove that the above algorithm leads to this equation.

5 We analyze the long division algorithm for each digit of the quotient 195: 5 = (1 3) Then: 28 = (9 3)

6 And finally: 17 = (5 3) Thus the long division algorithm is a compact summary of three divisions-with-remainder: 5 = (1 3) = (9 3) = (5 3) + 2

7 Here is the proof that 587 = (195 3) + 2: Using 5 = (1 3) + 2, we have: 587 = (5 100) + (8 10) + 7 = ( ((1 3) + 2 ) 100) + (8 10) + 7 = ( (1 100) 3 ) + (28 10) + 7 Using 28 = (9 3) + 1, we get: 587 = ( (1 100) 3 ) + ( ((9 3) + 1) 10) + 7 = ( (1 100) 3 ) + ((9 10) 3) + 17 Finally, using 17 = (5 3) + 2, we have: 587 = ( (1 100) 3 ) + ((9 10) 3) + (5 3) + 2 = ( (1 100) + (9 10) + 5 ) = (195 3) + 2

8 This proof has the virtue of bringing out the fundamental idea underlying all whole number algorithms: each step of the algorithms involves only one digit, when one digit is suitably interpreted. So far so good. Unfortunately, this is sophisticated mathematics. Teachers should know it, of course, but even fifth graders would generally find this too difficult. To make this proof usable as school mathematics, we have to simplify it as much as possible but without sacrificing the mathematical validity of the explanation or the central idea that it is one digit at a time. The following is one possible compromise.

9 We want whole numbers q and r so that 587 = (q 3) + r where r < 3 The number q cannot have 4 digits (because right side would exceed the left side), so it is at most a 3-digit number. Its hundreds digit cannot be 2 (again because right side would exceed the left side), so its hundreds digit is 1. So q = T, where T is a 2-digit number. Thus so that 587 = (100 + T ) 3 + r = (T 3) + r = (T 3) + r

10 Now 287 = (T 3) + r. The tens digit of T can be as big as 9 without any contradiction (if we want r < 3, then we need T to be as large as possible so long as T 3 does not exceed 287). So T = 90 + S, where S is a single digit number, and therefore so that 287 = (90 + S) 3 + r = (S 3) + r = (S 3) + r

11 Finally, we have 17 = (S 3) + r. Clearly we should let S = 5 and r = 2: 17 = (5 3) Recall: 587 = (q 3) + r, and we have determined that the hundreds digit, the tens digit, and the ones digit of q are 1, 9, 5 respectively, and r = 2. Thus 587 = (195 3) + r

12 Everything we have done thus far about long division is part of what we call school mathematics. We observe that: (a) These considerations do not belong to the university mathematics curriculum. They are too elementary. (b) The mathematics is not trivial. (c) Part of this discussion about bringing the mathematics down to the level of fifth graders goes beyond mathematics per se. (d) The discussion in (c) cannot take place without a complete understanding of the mathematics underlying the long division algorithm.

13 The phenomenon exhibited in (a) (d) is not special to long division, but is shared by most topics in school mathematics: (i) Conversion of a fraction m n of m by n. to a decimal by the long division (ii) The concept of a fraction, and everything related to fractions, including ratio and percent. (iii) The concept of constant speed or constant rate. (iv) Axioms and Euclidean geometry.

14 (v) The concepts of congruence and similarity. (vi) The concept of length and area, especially the computation of the circumference and area of a circle. (vii) The concept of a negative number, and everything related to negative numbers. (viii) Finding the maximum or minimum of a quadratic function. (ix) The concept of a polynomial and the algebra of polynomials.

15 Let me give one more illustration of the phenomenon exhibited in (a) (d) using negative numbers. In America, the Most Frequently Asked question in school mathematics is Why is negative negative equal to positive? Mathematicians consider this to be obvious. They prove something more general: For any numbers x, y, ( x)( y) = xy. Proof: We first prove that ( x)z = (xz) for any x and z. Observe that if a number A satisfies xy+a = 0, then A = (xy). But by the distributive law, xy + {( x)y} = (x + ( x))y = 0 y = 0, so ( x)y = (xy). Now let z = ( y), then we have ( x)( y) = (x( y)), which by the commutative law is equal to (( y)x) = ( (yx)) = yx = xy. So ( x)( y) = xy.

16 University mathematicians usually do not recognize how sophisticated this simple argument really is. It is not suitable for the consumption of school students. The basic resistance to accepting negative negative = positive is a psychological one. If we can explain this phenomenon for integers (rather than fractions), most of the battle is already won. We will give a relative simple explanation of why ( 2)( 3) = 2 3. The key step lies in the proof of ( 1)( 1) = 1

17 If we want to show a number is equal to 1, the most desirable way is to get it through a computation, e.g., if then A = 1 because A = (12 13) (6 25) 5, A = = 1 But sometimes, such a direct computation is not available. Then we have to settle for an indirect method of verification. (Think of dipping a ph strip into a solution to test for acidity.) So to test if a number A is equal to 1, we ask: is it true that A + ( 1) = 0? If so, then we are done.

18 Now let A = ( 1)( 1). We have A + ( 1) = ( 1)( 1) + ( 1) = ( 1)( 1) + 1 ( 1) By the distributive law, ( 1)( 1) + 1 ( 1) = (( 1) + 1) ( 1) = 0 ( 1) = 0 So A + ( 1) = 0, and we conclude that A = 1, i.e., ( 1)( 1) = 1

19 Now we can prove ( 2)( 3) = 2 3. We first show ( 1)( 3) = 3. We have ( 1)( 3) = ( 1) (( 1) + ( 1) + ( 1)) which, by the distributive law, is equal to ( 1)( 1) + ( 1)( 1) + ( 1)( 1) = = 3 Thus ( 1)( 3) = 3. Then, ( 2)( 3) = (( 1)+( 1)) ( 3) = ( 1)( 3)+( 1)( 3). By what we just proved, the latter is = 2 3. So ( 2)( 3) = 2 3. Proof of ( m)( n) = mn for whole numbers m, n is similar.

20 So far, I have given you bits and pieces of what School Mathematics is about. Let me go a step further and give you a more comprehensive view. We know, at least, what it is not: School Mathematics is not University Mathematics. There is more, however. We have seen that, in order to make a mathematical topic usable for school students, we need to take an extra step to ensure that the mathematical substance is not lost. This step involves more than mathematical knowledge; it involves a knowledge of the school classroom.

21 What we are saying is that School Mathematics is the product of Mathematical Engineering. What does this mean? Engineering: The discipline of customizing abstract scientific principles into processes and products that safely realize a human objective or function. Mathematical engineering: The discipline of customizing abstract mathematics into a form that can be correctly taught, and learned, in the K 12 classroom. Mathematical engineering is also known as K 12 mathematics education.

22 Chemical engineering: Chemistry the plexi-glass tanks in aquariums, the gas you pump into your car, shampoo, Lysol,... Electrical engineering: Electromagnetism computers, power point, ipod, lighting in this hall, motors,... Mathematical engineering: Abstract mathematics school mathematics

23 The recognition that school mathematics is an engineering product lends clarity to the current debate in mathematics education. There is no controversy in stating that engineering should not attempt to produce anything that caters to human wishes but defies scientific principles, e.g., perpetual motion machines, machines that extract oxygen from water without use of energy. There is also no controversy in stating that engineering should not waste time producing anything that is irrelevant to human needs no matter how scientifically sound.

24 Yet school mathematics was once mathematically sound but was decidedly not relevant to the school classroom: the New Math of the 1960 s. Injection of set theory into elementary school. Over-emphasis of precision (e.g., distinction between the number three and the numeral 3 that represents the number three ). Emphasis of abstractions (e.g., modular arithmetic, numbers in arbitrary bases, symbolic logic) at the expense of basic skills (computations in base 10).

25 Currently, there have been too many attempts in school mathematics to make mathematics easy to learn by ignoring basic mathematical principles. Consider the subject of fractions in school mathematics. Education researchers and textbook writers seem to believe that no engineering process can bring the mathematics of fractions to the school classroom. Their decision is therefore to teach fractions, not as mathematics, but as a language, which can conceivably be learned by osmosis, by listening to stories, by using analogies and metaphors, and by engaging in hands-on activities. The resulting fear of fractions is there for all to see.

26 Here are some of the problems in the way fractions is taught in America. No definition. There is no definition of a fraction (other than as a piece of pizza). The statement fractions have multiple representations is meaningless, because if we don t know what it is, what is there to represent? There is also no definition of any of the arithmetic operations on fractions: what does it mean to multiply two pieces of pizza? No reasoning. Analogies and metaphors replace reasoning. Why use the Least Common Denominator to add fractions? Why not add fractions the same way we multiply fractions: add the numerators and add the denominators? Why compute division by invert-and-multiply?

27 No coherence. This is perhaps the most serious of the three problems. Fractions are taught as different numbers from whole numbers. We are told that Children must adopt new rules for fractions that often conflict with well-established ideas about whole numbers. More is true: Decimals are taught as different numbers from fractions. There is also no logical connection between various concepts and skills within the subject of fractions. How can students cope with such fragmented knowledge?

28 Mathematical engineering, when competently done, does produce a presentation of fractions that is fully consistent with basic mathematical principles and suitable for students of grades 5 7. This engineering effort requires a knowledge of the school classroom as well as a deep mathematical knowledge.

29 Physicists and chemists recognized long ago that real-world applications of their theoretical knowledge are important and should be pursued as a separate discipline. Schools of engineering were born. However, neither mathematicians nor educators seem to recognize school mathematics as an engineering product. Each group end up producing mathematics not relevant to the school classroom, or materials usable in the classroom but mathematically flawed. We need good school mathematics. We need good mathematical engineering.

30 For a more thorough discussion of mathematics education as mathematical engineering, see wu/icmtalk.pdf

### The Mathematics School Teachers Should Know

The Mathematics School Teachers Should Know Lisbon, Portugal January 29, 2010 H. Wu *I am grateful to Alexandra Alves-Rodrigues for her many contributions that helped shape this document. Do school math

### The program also provides supplemental modules on topics in geometry and probability and statistics.

Algebra 1 Course Overview Students develop algebraic fluency by learning the skills needed to solve equations and perform important manipulations with numbers, variables, equations, and inequalities. Students

### Prentice Hall Connected Mathematics 2, 7th Grade Units 2009

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems

### WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT?

WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly

### MTH124: Honors Algebra I

MTH124: Honors Algebra I This course prepares students for more advanced courses while they develop algebraic fluency, learn the skills needed to solve equations, and perform manipulations with numbers,

### Math at a Glance for April

Audience: School Leaders, Regional Teams Math at a Glance for April The Math at a Glance tool has been developed to support school leaders and region teams as they look for evidence of alignment to Common

### Division with Whole Numbers and Decimals

Grade 5 Mathematics, Quarter 2, Unit 2.1 Division with Whole Numbers and Decimals Overview Number of Instructional Days: 15 (1 day = 45 60 minutes) Content to be Learned Divide multidigit whole numbers

### 3.4 Complex Zeros and the Fundamental Theorem of Algebra

86 Polynomial Functions.4 Complex Zeros and the Fundamental Theorem of Algebra In Section., we were focused on finding the real zeros of a polynomial function. In this section, we expand our horizons and

### Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

### Elementary School Mathematics Priorities

Elementary School Mathematics Priorities By W. Stephen Wilson Professor of Mathematics Johns Hopkins University and Former Senior Advisor for Mathematics Office of Elementary and Secondary Education U.S.

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### PERCENTS. Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% =

PERCENTS Percent means per hundred. Writing a number as a percent is a way of comparing the number with 100. For example: 42% = Percents are really fractions (or ratios) with a denominator of 100. Any

### Absolute Value of Reasoning

About Illustrations: Illustrations of the Standards for Mathematical Practice (SMP) consist of several pieces, including a mathematics task, student dialogue, mathematical overview, teacher reflection

### Elementary Number Theory We begin with a bit of elementary number theory, which is concerned

CONSTRUCTION OF THE FINITE FIELDS Z p S. R. DOTY Elementary Number Theory We begin with a bit of elementary number theory, which is concerned solely with questions about the set of integers Z = {0, ±1,

### MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

### The impact of Common Core Standards on the mathematics education of teachers

The impact of Common Core Standards on the mathematics education of teachers MAA Section Meeting Menonomie, WI, April 29, 2011 H. Wu [This is a slightly revised version of the lecture given on April 29,

### Unit 1 Number Sense. In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions.

Unit 1 Number Sense In this unit, students will study repeating decimals, percents, fractions, decimals, and proportions. BLM Three Types of Percent Problems (p L-34) is a summary BLM for the material

### Algebra I. In this technological age, mathematics is more important than ever. When students

In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

### 3.6. The factor theorem

3.6. The factor theorem Example 1. At the right we have drawn the graph of the polynomial y = x 4 9x 3 + 8x 36x + 16. Your problem is to write the polynomial in factored form. Does the geometry of the

### a = bq + r where 0 r < b.

Lecture 5: Euclid s algorithm Introduction The fundamental arithmetic operations are addition, subtraction, multiplication and division. But there is a fifth operation which I would argue is just as fundamental

### Integer Operations. Overview. Grade 7 Mathematics, Quarter 1, Unit 1.1. Number of Instructional Days: 15 (1 day = 45 minutes) Essential Questions

Grade 7 Mathematics, Quarter 1, Unit 1.1 Integer Operations Overview Number of Instructional Days: 15 (1 day = 45 minutes) Content to Be Learned Describe situations in which opposites combine to make zero.

### Continued Fractions and the Euclidean Algorithm

Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction

### LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

### Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm.

Chapter 4, Arithmetic in F [x] Polynomial arithmetic and the division algorithm. We begin by defining the ring of polynomials with coefficients in a ring R. After some preliminary results, we specialize

### Grade Level Year Total Points Core Points % At Standard 9 2003 10 5 7 %

Performance Assessment Task Number Towers Grade 9 The task challenges a student to demonstrate understanding of the concepts of algebraic properties and representations. A student must make sense of the

### Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

### CORE Assessment Module Module Overview

CORE Assessment Module Module Overview Content Area Mathematics Title Speedy Texting Grade Level Grade 7 Problem Type Performance Task Learning Goal Students will solve real-life and mathematical problems

### Phoenix Rising Bringing the Common Core State Mathematics Standards to Life

Phoenix Rising Bringing the Common Core State Mathematics Standards to Life By Hung-Hsi Wu Many sets of state and national mathematics standards have come and gone in the past two decades. The Common Core

### ECS CURRICULUM GUIDE

ECS CURRICULUM GUIDE MIDDLE SCHOOL OVERVIEW Classroom Structure: 7 class periods of 45 55 minutes daily (Bible, Language, History, Math, Science, plus two Elective/Exploration classes) Student/Teacher

### 0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

### History of Development of CCSS

Implications of the Common Core State Standards for Mathematics Jere Confrey Joseph D. Moore University Distinguished Professor of Mathematics Education Friday Institute for Educational Innovation, College

### A can of Coke leads to a piece of pi

theme / MATHEMATICS AND SCIENCE A can of Coke leads to a piece of pi A professional development exercise for educators is an adaptable math lesson for many grades BY MARILYN BURNS During a professional

### Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

### Mathematics Georgia Performance Standards

Mathematics Georgia Performance Standards K-12 Mathematics Introduction The Georgia Mathematics Curriculum focuses on actively engaging the students in the development of mathematical understanding by

### Copyrighted Material. Chapter 1 DEGREE OF A CURVE

Chapter 1 DEGREE OF A CURVE Road Map The idea of degree is a fundamental concept, which will take us several chapters to explore in depth. We begin by explaining what an algebraic curve is, and offer two

### Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

### MODULAR ARITHMETIC. a smallest member. It is equivalent to the Principle of Mathematical Induction.

MODULAR ARITHMETIC 1 Working With Integers The usual arithmetic operations of addition, subtraction and multiplication can be performed on integers, and the result is always another integer Division, on

### Chapter 11 Number Theory

Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

### Quotient Rings and Field Extensions

Chapter 5 Quotient Rings and Field Extensions In this chapter we describe a method for producing field extension of a given field. If F is a field, then a field extension is a field K that contains F.

### Math Circle Beginners Group October 18, 2015

Math Circle Beginners Group October 18, 2015 Warm-up problem 1. Let n be a (positive) integer. Prove that if n 2 is odd, then n is also odd. (Hint: Use a proof by contradiction.) Suppose that n 2 is odd

### Teaching Fractions According to the Common Core Standards

Teaching Fractions According to the Common Core Standards H. Wu c Hung-Hsi Wu 0 August, 0 (revised February 8, 04) Contents Preface Grade Grade 4 7 Grade Grade 6 9 Grade 7 80 I am very grateful to David

### Answer Key for California State Standards: Algebra I

Algebra I: Symbolic reasoning and calculations with symbols are central in algebra. Through the study of algebra, a student develops an understanding of the symbolic language of mathematics and the sciences.

### Content Standards 5.0 Numbers and Operations Standard Statement... 17 5.1 Number Sense. 18 5.2 Operations on Numbers... 26 5.3 Estimation...

TABLE OF CONTENTS Introduction.... 4 Process Standards....... 8 1.0 Problem Solving....... 9 2.0 Reasoning and Proof..... 12 3.0 Communication......... 13 4.0 Connections.... 15 Content Standards 5.0 Numbers

### Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

### Stephanie A. Mungle TEACHING PHILOSOPHY STATEMENT

Stephanie A. Mungle TEACHING PHILOSOPHY STATEMENT I am a self-directed, enthusiastic college mathematics educator with a strong commitment to student learning and excellence in teaching. I bring my passion

### parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL

parent ROADMAP MATHEMATICS SUPPORTING YOUR CHILD IN HIGH SCHOOL HS America s schools are working to provide higher quality instruction than ever before. The way we taught students in the past simply does

### Maths Progressions Number and algebra

This document was created by Clevedon School staff using the NZC, Maths Standards and Numeracy Framework, with support from Cognition Education consultants. It is indicative of the maths knowledge and

### The Most Widely Used. Mathematics Textbook Series in Japan is Now in English! Introducing Tokyo Shoseki s. and

The Most Widely Used Mathematics Textbook Series in Japan is Now in English! Introducing Tokyo Shoseki s Mathematics International (Elementary School, s 1 to 6) and Mathematics International (Lower Secondary

### Common Core State Standards General Brief

Common Core State Standards General Brief The Common Core State Standards Initiative is a state-led effort to establish consistent and clear education standards for English-language arts and mathematics

### If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

### TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

### Mathematics. Curriculum Content for Elementary School Mathematics. Fulton County Schools Curriculum Guide for Elementary Schools

Mathematics Philosophy Mathematics permeates all sectors of life and occupies a well-established position in curriculum and instruction. Schools must assume responsibility for empowering students with

### Master of Arts in Mathematics

Master of Arts in Mathematics Administrative Unit The program is administered by the Office of Graduate Studies and Research through the Faculty of Mathematics and Mathematics Education, Department of

### Math Opportunities Valuable Experiences Innovative Teaching. MOVE IT with manipulatives! MOVE IT through the curriculum! MOVE IT in the 21st century!

Math Opportunities Valuable Experiences Innovative Teaching MOVE IT with manipulatives! MOVE IT through the curriculum! MOVE IT in the 21st century! BIG REWARDS! FILLS GAPS, ENRICHES, ACCELERATES any Grade

### Academic Standards for Mathematics

Academic Standards for Grades Pre K High School Pennsylvania Department of Education INTRODUCTION The Pennsylvania Core Standards in in grades PreK 5 lay a solid foundation in whole numbers, addition,

### PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey)

PoW-TER Problem Packet A Phone-y Deal? (Author: Peggy McCloskey) 1. The Problem: A Phone-y Deal? [Problem #3280] With cell phones being so common these days, the phone companies are all competing to earn

### Mathematics of Cryptography

CHAPTER 2 Mathematics of Cryptography Part I: Modular Arithmetic, Congruence, and Matrices Objectives This chapter is intended to prepare the reader for the next few chapters in cryptography. The chapter

### Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

### Interpretation of Test Scores for the ACCUPLACER Tests

Interpretation of Test Scores for the ACCUPLACER Tests ACCUPLACER is a trademark owned by the College Entrance Examination Board. Visit The College Board on the Web at: www.collegeboard.com/accuplacer

### Whitnall School District Report Card Content Area Domains

Whitnall School District Report Card Content Area Domains In order to align curricula kindergarten through twelfth grade, Whitnall teachers have designed a system of reporting to reflect a seamless K 12

### 2. Methods of Proof Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try.

2. METHODS OF PROOF 69 2. Methods of Proof 2.1. Types of Proofs. Suppose we wish to prove an implication p q. Here are some strategies we have available to try. Trivial Proof: If we know q is true then

### Understanding Place Value of Whole Numbers and Decimals Including Rounding

Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned

### ACCUPLACER Arithmetic & Elementary Algebra Study Guide

ACCUPLACER Arithmetic & Elementary Algebra Study Guide Acknowledgments We would like to thank Aims Community College for allowing us to use their ACCUPLACER Study Guides as well as Aims Community College

### 9. POLYNOMIALS. Example 1: The expression a(x) = x 3 4x 2 + 7x 11 is a polynomial in x. The coefficients of a(x) are the numbers 1, 4, 7, 11.

9. POLYNOMIALS 9.1. Definition of a Polynomial A polynomial is an expression of the form: a(x) = a n x n + a n-1 x n-1 +... + a 1 x + a 0. The symbol x is called an indeterminate and simply plays the role

### Notes on Factoring. MA 206 Kurt Bryan

The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor

### Course Syllabus. MATH 1350-Mathematics for Teachers I. Revision Date: 8/15/2016

Course Syllabus MATH 1350-Mathematics for Teachers I Revision Date: 8/15/2016 Catalog Description: This course is intended to build or reinforce a foundation in fundamental mathematics concepts and skills.

### U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra

U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory

### Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

### Number Sense and Operations

Number Sense and Operations representing as they: 6.N.1 6.N.2 6.N.3 6.N.4 6.N.5 6.N.6 6.N.7 6.N.8 6.N.9 6.N.10 6.N.11 6.N.12 6.N.13. 6.N.14 6.N.15 Demonstrate an understanding of positive integer exponents

### COLLEGE ALGEBRA. Paul Dawkins

COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... Introduction... Integer Exponents... Rational Exponents... 9 Real Exponents...5 Radicals...6 Polynomials...5

### Van de Walle (2007) describes divid ing one fraction by another in this way: Invert the divisor and multiply

6 Rich and Engaging Mathematical Tasks: Grades 5 9 Measurement and Fair-Sharing Models for Dividing Fractions Jeff Gregg and Diana Underwood Gregg Van de Walle (007) describes divid ing one fraction by

### Mathematics Curricular Guide SIXTH GRADE SCHOOL YEAR. Updated: Thursday, December 02, 2010

Mathematics Curricular Guide SIXTH GRADE 2010-2011 SCHOOL YEAR 1 MATHEMATICS SCOPE & SEQUENCE Unit Title Dates Page 1. CMP 2 Bits and Pieces I... 3 2. How Likely Is It? (Probability)... 6 3. CMP 2 Bits

### 6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives

6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise

### With a Brief. For Students Entering McMaster University

With a Brief For Students Entering McMaster University Department of Mathematics and Statistics McMaster University 2009 Published by: Miroslav Lovric Department of Mathematics and Statistics McMaster

### FACTORISATION YEARS. A guide for teachers - Years 9 10 June 2011. The Improving Mathematics Education in Schools (TIMES) Project

9 10 YEARS The Improving Mathematics Education in Schools (TIMES) Project FACTORISATION NUMBER AND ALGEBRA Module 33 A guide for teachers - Years 9 10 June 2011 Factorisation (Number and Algebra : Module

### Useful Number Systems

Useful Number Systems Decimal Base = 10 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Binary Base = 2 Digit Set = {0, 1} Octal Base = 8 = 2 3 Digit Set = {0, 1, 2, 3, 4, 5, 6, 7} Hexadecimal Base = 16 = 2

### 8 Divisibility and prime numbers

8 Divisibility and prime numbers 8.1 Divisibility In this short section we extend the concept of a multiple from the natural numbers to the integers. We also summarize several other terms that express

### Mathematics Review for MS Finance Students

Mathematics Review for MS Finance Students Anthony M. Marino Department of Finance and Business Economics Marshall School of Business Lecture 1: Introductory Material Sets The Real Number System Functions,

### A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

### Multiplying and Dividing Algebraic Fractions

. Multiplying and Dividing Algebraic Fractions. OBJECTIVES. Write the product of two algebraic fractions in simplest form. Write the quotient of two algebraic fractions in simplest form. Simplify a comple

### WritePlacer Sample Topic. WritePlacer. Arithmetic

. Knowledge of another language fosters greater awareness of cultural diversity among the peoples of the world. Individuals who have foreign language skills can appreciate more readily other peoples values

### Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. Solve word problems that call for addition of three whole numbers

### 5 th Grade Common Core State Standards. Flip Book

5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at

### Mathematics Content Courses for Elementary Teachers

Mathematics Content Courses for Elementary Teachers Sybilla Beckmann Department of Mathematics University of Georgia Massachusetts, March 2008 Sybilla Beckmann (University of Georgia) Mathematics for Elementary

### Mathematics (MAT) MAT 061 Basic Euclidean Geometry 3 Hours. MAT 051 Pre-Algebra 4 Hours

MAT 051 Pre-Algebra Mathematics (MAT) MAT 051 is designed as a review of the basic operations of arithmetic and an introduction to algebra. The student must earn a grade of C or in order to enroll in MAT

### Florida Math Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper

Florida Math 0022 Correlation of the ALEKS course Florida Math 0022 to the Florida Mathematics Competencies - Lower and Upper Whole Numbers MDECL1: Perform operations on whole numbers (with applications,

### Florida Math 0018. Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower

Florida Math 0018 Correlation of the ALEKS course Florida Math 0018 to the Florida Mathematics Competencies - Lower Whole Numbers MDECL1: Perform operations on whole numbers (with applications, including

### The Number Line: An Overview

The Number Line: An Overview Learning to Think Mathematically with the Number Line One of the most overlooked tools of the elementary and middle school classroom is the number line. Typically displayed

### Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

### 1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]

1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not

### Polynomial Operations and Factoring

Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = 45 60 minutes) Content to be learned Identify terms, coefficients, and degree of polynomials.

### This Performance Standards include four major components. They are

Eighth Grade Science Curriculum Approved July 12, 2004 The Georgia Performance Standards are designed to provide students with the knowledge and skills for proficiency in science at the eighth grade level.

### Division of whole numbers is defined in terms of multiplication using the idea of a missing factor.

32 CHAPTER 1. PLACE VALUE AND MODELS FOR ARITHMETIC 1.6 Division Division of whole numbers is defined in terms of multiplication using the idea of a missing factor. Definition 6.1. Division is defined

### Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

### Florida Math 0028. Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper

Florida Math 0028 Correlation of the ALEKS course Florida Math 0028 to the Florida Mathematics Competencies - Upper Exponents & Polynomials MDECU1: Applies the order of operations to evaluate algebraic

### Problem of the Month Through the Grapevine

The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

### Multiplying and Factoring Notes

Multiplying/Factoring 3 Multiplying and Factoring Notes I. Content: This lesson is going to focus on wrapping up and solidifying concepts that we have been discovering and working with. The students have

### A GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate

A GENERAL CURRICULUM IN MATHEMATICS FOR COLLEGES W. L. DUREN, JR., Chairmnan, CUPM 1. A report to the Association. The Committee on the Undergraduate Program in Mathematics (CUPM) hereby presents to the