A Guide to Your Modular Math Course - faculty.mccneb.edufaculty.mccneb.edu/jdlee3/modular_guide.pdf · A Guide to Your Modular Math Course Joseph Lee Mod 5 - Sec 3.3 . . . . .

A Guide to Your Modular Math Course

Joseph Lee

Spring 2015

Contents

Introduction 7About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Module 1 8Mod 1 - Chapter 1 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Mod 1 - Sec 1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Mod 1 - Sec 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Mod 1 - Sec 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Module 2 9Mod 2 - Sec 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mod 2 - Quiz - Sec 1.5 to 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mod 2 - Exam - Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mod 2 - Chapter 2 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mod 2 - Sec 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Mod 2 - Sec 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Module 3 11Mod 3 - Sec 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Mod 3 - Sec 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Mod 3 - Sec 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Mod 3 - Quiz - Sec 2.1 to 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Mod 3 - Sec 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Module 4 12Mod 4 - Sec 2.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Mod 4 - Sec 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Mod 4 - Sec 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Mod 4 - Quiz - Sec 2.6, 2.7, 2.8, 2.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Mod 4 - Exam - Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Module 5 14Mod 5 - Chapter 3 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Mod 5 - Sec 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Mod 5 - Sec 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

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A Guide to Your Modular Math Course Joseph Lee

Mod 5 - Sec 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Mod 5 - Quiz - Sec 3.1, 3.2, 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Mod 5 - Sec 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Module 6 16Mod 6 - Sec 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Mod 6 - Sec 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Mod 6 - Sec 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Mod 6 - Quiz - Sec 3.4, 3.5, 3.6, 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Mod 6 - Exam - Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Module 7 17Mod 7 - Chapter 4 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Mod 7 - Sec 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Mod 7 - Sec 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Mod 7 - Sec 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Mod 7 - Sec 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Mod 7 - Quiz - Sec 4.1, 4.2, 4.3, 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Module 8 19Mod 8 - Exam - Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mod 8 - Chapter 5 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mod 8 - Sec 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mod 8 - Sec 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mod 8 - Sec 5.3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Mod 8 - Sec 5.3B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Mod 8 - Quiz - Sec 5.1, 5.2, 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Module 9 21Mod 9 - Sec 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Mod 9 - Sec 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Mod 9 - Quiz - Sec 5.4, 5.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Mod 9 - Exam - Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Mod 9 - Chapter 9 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Module 10 22Mod 10 - Sec 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Mod 10 - Sec 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Mod 10 - Sec 9.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Mod 10 - Sec 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Mod 10 - Quiz - Sec 9.1, 9.2, 9.3, 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Mod 10 - Exam - Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Mod 10 - Comprehensive Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Module 11 24Mod 11 - Chapter 1 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Mod 11 - Sec 1.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Mod 11 - Sec 1.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Mod 11 - Quiz A - 1.3, 1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

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Mod 11 - Sec 1.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Mod 11 - Sec 1.6 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Mod 11 - Quiz B - 1.5, 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Module 12 26Mod 12 - Sec 1.7 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mod 12 - Sec 1.8 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mod 12 - Quiz A - 1.7, 1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mod 12 - Sec 1.9 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Mod 12 - Sec 1.10 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Mod 12 - Quiz B - 1.9, 1.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Mod 12 - Exam - Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Module 13 28Mod 13 - Chapter 2 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Sec 2.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Sec 2.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Sec 2.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Quiz A - 2.1, 2.2, 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Sec 2.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Mod 13 - Sec 2.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Mod 13 - Quiz B - 2.4, 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Module 14 30Mod 14 - Sec 2.6 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Sec 2.8 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Quiz A - 2.6, 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Exam - Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Chapter 3 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Sec 3.1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Sec 3.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Mod 14 - Quiz B - 3.1, 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Module 15 32Mod 15 - Sec 3.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Mod 15 - Sec 3.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Mod 15 - Quiz A - 3.3, 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Mod 15 - Exam - Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Mod 15 - Comprehensive Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Module 16 34Mod 16 - Chapter 4 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Mod 16 - Sec 4.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Mod 16 - Sec 4.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Mod 16 - Quiz A - 4.1, 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mod 16 - Sec 4.3 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mod 16 - Sec 4.4 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35Mod 16 - Quiz B - 4.3, 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

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Module 17 36Mod 17 - Sec 4.5 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Sec 4.6 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Quiz A - 4.5, 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Exam - Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Chapter 5 & 6.1, 6.2 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Sec 5.1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Sec 5.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Mod 17 - Quiz B - 5.1, 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Module 18 38Mod 18 - Sec 5.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Sec 5.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Quiz A - 5.3, 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Sec 5.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Sec 5.6 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Sec 6.1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mod 18 - Quiz B - 5.5, 5.6, 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Module 19 39Mod 19 - Sec 6.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Mod 19 - Quiz A - 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Mod 19 - Exam - Chapter 5 & 6.1, 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Mod 19 - Chapter 7 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Mod 19 - Sec 7.1 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Mod 19 - Sec 7.2 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Mod 19 - Quiz B - 7.1, 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Module 20 41Mod 20 - Sec 7.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Sec 7.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Quiz A - 7.3, 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Sec 7.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Quiz B - 7.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Exam - Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Mod 20 - Comprehensive Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Module 21 43Mod 21 - Chapter 3 & 4 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mod 21 - Sec 3.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mod 21 - Sec 3.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mod 21 - Sec 3.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Mod 21 - Quiz A - 3.1, 3.2, 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Module 22 44Mod 22 - Sec 3.5 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Mod 22 - Sec 4.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Mod 22 - Sec 4.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

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Mod 22 - Sec 4.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Mod 22 - Quiz A - 3.5, 4.1, 4.2, 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Module 23 45Mod 23 - Exam - Chapters 3 & 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Mod 23 - Chapter 5 & 6 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Module 24 46Mod 24 - Sec 6.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Mod 24 - Sec 6.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Mod 24 - Sec 6.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Mod 24 - Sec 6.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46Mod 24 - Quiz A - 6.1, 6.2, 6.3, 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Module 25 48Mod 25 - Exam - Chapters 5 & 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Mod 25 - Chapter 7 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Mod 25 - Sec 7.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Mod 25 - Sec 7.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Mod 25 - Quiz A - 7.1, 7.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Module 26 50Mod 26 - Sec 7.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Mod 26 - Sec 7.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Mod 26 - Quiz A - 7.3, 7.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Mod 26 - Exam - Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Module 27 54Mod 27 - Chapter 8 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Mod 27 - Sec 8.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Mod 27 - Sec 8.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Mod 27 - Sec 8.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Mod 27 - Sec 8.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Mod 27 - Quiz A - 8.1, 8.2, 8.3, 8.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Module 28 57Mod 28 - Sec 8.5 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Mod 28 - Sec 8.6 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Mod 28 - Sec 8.7 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Mod 28 - Quiz A - 8.5, 8.6, 8.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Module 29 59Mod 29 - Exam - Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Mod 29 - Chapter 9 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Mod 29 - Sec 9.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Mod 29 - Sec 9.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Mod 29 - Quiz A - 9.1, 9.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

Page 5


Module 30 61Mod 30 - Sec 9.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Mod 30 - Sec 9.4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Mod 30 - Quiz A - 9.3, 9.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Mod 30 - Exam - Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Mod 30 - Comprehensive Quiz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Page 6


Introduction

Welcome to modular math. Modular math is designed to get students through Developmental Math,Beginning Algebra, and Intermediate Algebra. The course is divided into “modules” which represent aweek of material.

About the Author

My name is Joseph Lee, and I am one of the many math instructors here at Metropolitan CommunityCollege who teaches a modular math course. I have taught at this college since 2007, and the first modularcourse I taught was in the Winter of 2012. During the Spring quarter of 2014, I tried to organize someof the details of this course in this guide. This guide is by no means a complete work, but if any of thisinformation is helpful in your course, please use it at your convenience.

If there are any questions or comments, I can be reached by phone (402-289-1356) or email ([email protected]).Other infomation about my classes may be found on my faculty webpage (faculty.mccneb.edu/jdlee3).

To the Student

Here at Metropolitan Community College, students may take math courses in one of four different formats:

• traditional,

• online,

• computer enhanced, and

• modular.

While students may prefer one mode of instruction to another, it has been my experience that students maybe successful in any of these formats. The reverse is also true. Students who are absent from class, whodon’t spend time outside of class studying, and who don’t seek out their instructor for help will strugglein any of our four formats. To be successful, students enrolled in modular math should understand beingsuccessful in this course will require the same time and dedication that would be needed for any othermath course. Being on time and present for every class is expected. Spending time outside of class workingon homework every night is expected. Students who do not practice these habits in my traditional classesfind they are unlikely to succeed, as do students in my modular courses.

Page 7


Module 1

Due: Tuesday, March 17, 2015

Contents: Mod 1 - Chapter 1 Pre-TestMod 1 - Sec 1.5Mod 1 - Sec 1.6Mod 1 - Sec 1.7

Mod 1 - Chapter 1 Pre-Test

Number of questions: 30No calculator allowed.

Chapter 1 covers division, exponents, the order of operations, and application problems.

Mod 1 - Sec 1.5

Section 1.5 covers long division. Understand the following difference:

• 0 divided by a number1 is 0.

• a number divided by 0 is undefined.

Mod 1 - Sec 1.6

Section 1.6 covers exponents and the order of operations. Operations must be performed in the followingorder:

1. Perform any operations inside parenthesis.

2. Evaluate any exponents.

3. Perform any multiplication or division, from left to right.

4. Perform any addition or subtraction, from left to right.

A common mistake:Incorrect Order : 18÷ 2× 9 6= 18÷ 18 = 1Correct Order : 18÷ 2× 9 = 9× 9 = 81

Mod 1 - Sec 1.7

Section 1.7 covers the rounding and the principal of estimation. To estimate2, always round the very firstdigit in each number, and then perform any operations. For example,

265 + 1459 + 43, 921≈ 300 + 1000 + 40, 000≈ 41, 300.

1not zero2according to the principal of estimation

Page 8


Module 2


Contents: Mod 2 - Sec 1.8Mod 2 - Quiz - Sec 1.5 to 1.8Mod 2 - Exam - Chapter 1Mod 2 - Chapter 2 Pre-TestMod 2 - Sec 2.1Mod 2 - Sec 2.2

Mod 2 - Sec 1.8

Section 1.8 covers solving application problems involving basic operations.

Hint (Questions 16, 17): The figures are given in millions. Thus, you will have to add 6 zeros toyour answer.

Mod 2 - Quiz - Sec 1.5 to 1.8


Mod 2 - Exam - Chapter 1


What are the most important things to know for this test?

• Be able to perform any long division.

• 0 divided by a number is 0; a number divided by 0 is undefined.

• Understand the principle of estimation – always round to the very first digit.

• Understand the order of operations.

• Be able to identify the correct operation (add, subtract, multiply, or divide) in an application problem.



Mod 2 - Sec 2.1

Section 2.1 introduces fractions.

Page 9


Mod 2 - Sec 2.2

Section 2.2 covers reducing fractions. The section offers two methods for reducing fractions. Here are the

two methods for the fraction54

90:

• You may reduce a fraction by finding a common factor in the numerator and in the denominator.

54

90=

54÷ 18

90÷ 18=

3

5

• You may reduce a fraction by factoring the numerator and denominator into prime numbers.

54

90=

2× 3× 3× 3

2× 3× 3× 5=(((

((2× 3× 3× 3

(((((2× 3× 3× 5

=3

5

Page 10


Module 3


Contents: Mod 3 - Sec 2.3Mod 3 - Sec 2.4Mod 3 - Sec 2.5Mod 3 - Quiz - Sec 2.1 to 2.5Mod 3 - Sec 2.6

Mod 3 - Sec 2.3

Section 2.3 covers changing between mixed numbers and improper fractions.

Mod 3 - Sec 2.4

Section 2.4 covers multiplying fractions. When multiplying fractions, make sure to reduce the fractionsbefore you multiply:

3

4× 2

9=�3 · 1�2 · 2

× �2 · 1�3 · 3

=1

6.

To multiply mixed numbers, change the mixed numbers to improper fractions, and then multiply:

21

3× 3

2

5=

7

3× 17

5=

119

15= 7

14

15.

Mod 3 - Sec 2.5

Section 2.5 covers dividing fractions. To divide fractions, multiply by the reciprocal :

2

3÷ 3

4=

2

3× 4

3=

8

9.

To divide mixed numbers, change the mixed numbers to improper fractions, and then divide:

21

3÷ 3

2

5=

7

3÷ 17

5=

7

3× 5

17=

35

51

Mod 3 - Quiz - Sec 2.1 to 2.5


Mod 3 - Sec 2.6

Section 2.6 covers finding a least common denominator and constructing equivalent fractions. The least

common denominator is the smallest number that all denominators divide into evenly. For the fractions5

8

and7

12, the least common denominator is 24 because 24 is the smallest number that both 8 and 12 divide

into evenly. We may now write these fractions as equivalent fractions with the least common denominator:

5

8× 3

3=

15

247

12× 2

2=

14

24

Page 11


Module 4

Due: Tuesday, April 7, 2015

Contents: Mod 4 - Sec 2.7Mod 4 - Sec 2.8Mod 4 - Sec 2.9Mod 4 - Quiz - Sec 2.6, 2.7, 2.8, 2.9Mod 4 - Exam - Chapter 2

Mod 4 - Sec 2.7

Section 2.7 covers adding and subtracting fractions. To add or subtract fractions, write each fraction asequivalent fractions with the least common denominator, and then add or subtract:

2

5+

7

15=

2

5· 3

3+

7

15

=6

15+

7

15

=13

15

Mod 4 - Sec 2.8

Section 2.8 covers adding and subtracting mixed numbers. To add mixed numbers, add the whole partstogether and add the fraction parts together.

62

3+ 2

3

4= 6

2

3· 4

4+ 2

3

4· 3

3

= 68

12+ 2

9

12

= 8 +17

12

= 8 + 15

12

= 95

12

Page 12


To subtract mixed numbers, you may have to borrow to subtract your fraction parts:

62

3− 2

3

4= 6

2

3· 4

4− 2

3

4· 3

3

= 68

12− 2

9

12

= 5 + 18

12− 2

9

12

= 520

12− 2

9

12

= 311

12

Mod 4 - Sec 2.9

Section 2.9 covers application problems involving fractions.

Mod 4 - Quiz - Sec 2.6, 2.7, 2.8, 2.9





• Be able to reduce a fraction.

• Know when you need a common denominator (adding, subtracting), and when you do not a commondenominator (multiplying, dividing).

• Change mixed numbers to improper fractions when you are multiplying or dividing.

• Reduce first when multiplying fractions. You can only reduce when multiplying (not adding, sub-tracting, or dividing).

• Multiply by the reciprocal when you are dividing fractions.

• Be able to identify the correct operation (add, subtract, multiply, or divide) in an application problem.

Page 13


Module 5


Contents: Mod 5 - Chapter 3 Pre-TestMod 5 - Sec 3.1Mod 5 - Sec 3.2Mod 5 - Sec 3.3Mod 5 - Quiz - Sec 3.1, 3.2, 3.3Mod 5 - Sec 3.4



Mod 5 - Sec 3.1

Section 3.1 introduces decimals. It is important that you know the names of the place-values after thedecimal place. The place-values are as follows:

0 . 3 8 6 2 5↑ ↑ ↑ ↑ ↑| | | | ↓| | | ↓ hundred-thousandths| | ↓ ten-thousandths| ↓ thousandths↓ hundredthstenths

The previous number is read: thirty-eight thousand, six hundred twenty-five hundred-thousandths.Moreover, now we can write this decimal as a fraction:

0.38625 =38,625

100,000=

309

800

Mod 5 - Sec 3.2

Section 3.2 covers ordering and rounding. To compare two decimals, keep in mind that you may add extra0’s to right of the decimal. For example, to compare

2.47 2.476,

write 2.47 as 2.470. Thus, we may see that

2.470 < 2.476.

Mod 5 - Sec 3.3

Section 3.3 covers adding and subtracting decimals. The important step to adding and subtracting decimalsis lining up the decimals.

Page 14


Mod 5 - Quiz - Sec 3.1, 3.2, 3.3


Mod 5 - Sec 3.4

Section 3.4 covers multiplying decimals. To multiply decimals, you do not have to line up the decimals.Instead, add number of decimal places in both factors, and this number is the number of decimal placesin the product.

Page 15


Module 6


Contents: Mod 6 - Sec 3.5Mod 6 - Sec 3.6Mod 6 - Sec 3.7Mod 6 - Quiz - Sec 3.4, 3.5, 3.6, 3.7Mod 6 - Exam - Chapter 3

Mod 6 - Sec 3.5

Section 3.5 covers dividing decimals.

Mod 6 - Sec 3.6

Section 3.6 covers converting from fractions to decimals. To change a fraction to a decimal, divide thenumerator by the denominator.

Mod 6 - Sec 3.7

Section 3.7 covers application problems involving decimals.

Mod 6 - Quiz - Sec 3.4, 3.5, 3.6, 3.7




Page 16


Module 7


Contents: Mod 7 - Chapter 4 Pre-TestMod 7 - Sec 4.1Mod 7 - Sec 4.2Mod 7 - Sec 4.3Mod 7 - Sec 4.4Mod 7 - Quiz - Sec 4.1, 4.2, 4.3, 4.4


Number of questions: 28Calculator allowed.

Mod 7 - Sec 4.1

Section 4.1 introduces ratios.

Mod 7 - Sec 4.2

Section 4.2 intoduces proportions. To identify if an equation is a proportion, use cross multiplication:

26

39

?=

34

51

26× 51?= 39× 34

1326 = 1326

If the cross multiplication results in a true equation, then the equation is a proportion.

Mod 7 - Sec 4.3

Section 4.3 covers solving proportions. First, be able to solve a simple equation:

7× n = 28 (divide both sides by 7)

7× n

7=

28

7

n = 4

Page 17


Second, use cross multiplication to change the proportion to a simple equation.

n

5=

3

7

n× 7 = 15

n× 7

7=

15

7

n = 21

7

Mod 7 - Sec 4.4

Section 4.4 covers application problems involving proportions. Make sure you start each problem by settingup a proportion.

Mod 7 - Quiz - Sec 4.1, 4.2, 4.3, 4.4


Page 18


Module 8

Due: Tuesday, May 5, 2015

Contents: Mod 8 - Exam - Chapter 4Mod 8 - Chapter 5 Pre-TestMod 8 - Sec 5.1Mod 8 - Sec 5.2Mod 8 - Sec 5.3AMod 8 - Sec 5.3BMod 8 - Quiz - Sec 5.1, 5.2, 5.3





Mod 8 - Sec 5.1

Section 5.1 introduces percents. Any percent may be changed to a fraction or mixed number:

34% =34

100=

17

50or 150% =

150

100= 1

1

2

Also, any percent may be changed to a decimal:

34% = 0.34 or 150% = 1.5

Mod 8 - Sec 5.2

Section 5.2 covers changing between percents, decimals, and fractions.

Mod 8 - Sec 5.3A

Section 5.3A offers one method for solving percent problems3. Section 5.3A asks you to rewrite eachproblem as an equation by using key words:

Problem Equation

what n

of ×is =

find n =

3Section 5.3B will present all the same problems but offer a different method for solving them.

Page 19


For example,What is 24% of 150?↓ ↓ ↓ ↓ ↓n = 0.24 × 150

n = 36

It is important that you change the percent into a decimal while solving these problems4.

Mod 8 - Sec 5.3B

Section 5.3B offers a second method for solving percent problems. Section 5.3B asks you to set up eachproblem as a proportion involving the amount, a, the base, b, and the percent, p.

a

b=

p

100

The amount always proceeds or follows “is,” and the base always follows “of.” Unlike the previous section,we will write p as a percent instead of a decimal. For example,

What is 24% of 150?

a

150=

24

100

a× 100 = 150× 24

n× 100

100=

3600

100

a = 36

Mod 8 - Quiz - Sec 5.1, 5.2, 5.3


4However, in 5.3B, you will not change the percent to a decimal.

Page 20


Module 9


Contents: Mod 9 - Sec 5.4Mod 9 - Sec 5.5Mod 9 - Quiz - Sec 5.4, 5.5Mod 9 - Exam - Chapter 5Mod 9 - Chapter 9 Pre-Test

Mod 9 - Sec 5.4

Section 5.4 covers solving application problems involving percents.

Mod 9 - Sec 5.5

Section 5.5 covers more percent applications and the percent of increase and decrease. The percent ofincrease or decrease is calcuclated as follows:

Percent of Increase (or Decrease) =Amount of Increase (or Decrease)

Original Amount

Mod 9 - Quiz - Sec 5.4, 5.5






Page 21


Module 10


Contents: Mod 10 - Sec 9.1Mod 10 - Sec 9.2Mod 10 - Sec 9.3Mod 10 - Sec 9.4Mod 10 - Quiz - Sec 9.1, 9.2, 9.3, 9.4Mod 10 - Exam - Chapter 9Mod 10 - Comprehensive Quiz

Mod 10 - Sec 9.1

Section 9.1 covers adding positive and negative numbers. If you add two negatives together, add themtogether, and the sum is negative. If you add a positive number and a negative number, find their difference(subtract) and take the sign of the bigger number5.

Mod 10 - Sec 9.2

Section 9.2 covers subtracting positive and negative numbers. To subtract any two numbers, switch thesubtraction to addition by adding the opposite. For example,

13− 7 = 13 + (−7) = 6

Mod 10 - Sec 9.3

Section 9.3 covers multiplying and dividing positive and negative numbers. If you multiply or divide twonumbers with the same sign, the product or quotient is positive. If you multiply or divide two numberswith opposite signs, the product or quotient is negative.

Mod 10 - Sec 9.4

Section 9.4 covers the order of operations with positive and negative numbers.





Mod 10 - Quiz - Sec 9.1, 9.2, 9.3, 9.4


5absolute value

Page 22





• If you add a positive number and a negative number, find their difference (subtract) and take thesign of the bigger number6.

• If you add two negatives together, add them together, and the sum is negative.

• To subtract any two numbers, switch the subtraction to addition by adding the opposite. For example,

13− 7 = 13 + (−7) = 6

• If you multiply or divide two numbers with the same sign, the product or quotient is positive. If youmultiply or divide two numbers with opposite signs, the product or quotient is negative.

Mod 10 - Comprehensive Quiz

Number of questions: 15 (Multiple Choice)No calculator allowed.

6absolute value

Page 23


Module 11


Contents: Mod 11 - Chapter 1 Pre-TestMod 11 - Sec 1.3 AMod 11 - Sec 1.4 AMod 11 - Quiz A - 1.3, 1.4Mod 11 - Sec 1.5 BMod 11 - Sec 1.6 BMod 11 - Quiz B - 1.5, 1.6



Mod 11 - Sec 1.3 A

Section 1.3 reviews fractions.

Mod 11 - Sec 1.4 A

Section 1.4 introduces the real number system. You must memorize the following sets of numbers:

Natural numbers7: {1, 2, 3, . . .}

Whole numbers: {0, 1, 2, 3, . . .}

Integers: {. . . ,−3,−2,−1, 0, 1, 2, 3, . . .}

Irrational numbers include: {√

2,√

3,√

5,√

6,√

7,√

8,√

10, . . .}

Rational numbers are all numbers that are not irrational.

Real numbers are all numbers.

Mod 11 - Quiz A - 1.3, 1.4


Mod 11 - Sec 1.5 B

Section 1.5 covers ordering and absolute value.

7These are also called the counting numbers or the positive integers.

Page 24


Mod 11 - Sec 1.6 B

Section 1.6 covers adding positive and negative numbers. If you add two negatives together, add themtogether, and the sum is negative. If you add a positive number and a negative number, find their difference(subtract) and take the sign of the bigger number8.

Mod 11 - Quiz B - 1.5, 1.6


8absolute value

Page 25


Module 12


Contents: Mod 12 - Sec 1.7 AMod 12 - Sec 1.7 AMod 12 - Quiz A - 1.7, 1.8Mod 12 - Sec 1.9 BMod 12 - Sec 1.10 BMod 12 - Quiz B - 1.9, 1.10Mod 12 - Exam - Chapter 1

Mod 12 - Sec 1.7 A

Section 1.7 covers subtracting positive and negative numbers. To subtract any two numbers, switch thesubtraction to addition by adding the opposite. For example,

13− 7 = 13 + (−7) = 6

Mod 12 - Sec 1.8 A

Section 1.8 covers multiplying and dividing positive and negative numbers. If you multiply or divide twonumbers with the same sign, the product or quotient is positive. If you multiply or divide two numberswith opposite signs, the product or quotient is negative.

Mod 12 - Quiz A - 1.7, 1.8


Mod 12 - Sec 1.9 B

Section 1.9 covers exponents and the order of operations:





Page 26


Mod 12 - Sec 1.10 B

Section 1.10 introduces the properties of the real number system. You must memorize the followingproperties:

Addition Multiplication

Commutative Property a + b = b + a a · b = b · a

Associative Property a + (b + c) = (a + b) + c a(bc) = (ab)c

Distributive Property a(b + c) = ab + ac

Identity Property a + 0 = a a · 1 = a

Inverse Property a + (−a) = 0 a · 1

a= 1

Mod 12 - Quiz B - 1.9, 1.10




Page 27


Module 13


Contents: Mod 13 - Chapter 2 Pre-TestMod 13 - Sec 2.1 AMod 13 - Sec 2.2 AMod 13 - Sec 2.3 AMod 13 - Quiz A - 2.1, 2.2, 2.3Mod 11 - Sec 2.4 BMod 11 - Sec 2.5 BMod 11 - Quiz B - 2.4, 2.5



Mod 13 - Sec 2.1 A

Section 2.1 covers simplifying algebraic expressions.

Mod 13 - Sec 2.2 A

Section 2.2 covers solving equation using the addition property of equality. The addition property ofequality states that you may add or subtract the same number on both sides of an equation.

Mod 13 - Sec 2.3 A

Section 2.3 covers solving equation using the multiplicationn property of equality. The addition propertyof equality states that you may multiply or divide the same number on both sides of an equation.

Mod 13 - Quiz A - 2.1, 2.2, 2.3


Mod 13 - Sec 2.4 B

Section 2.4 covers solving equations with variables on one side of the equation.

Page 28


Mod 13 - Sec 2.5 B

Section 2.5 covers solving equations with variables on both sides of an equation. When you have variableson both sides of the equation, you may end up with a solution of all real numbers or have no solution:

8x− (3 + 4x) = 2(2x− 2)

8x− 3− 4x = 4x− 4

4x− 3 = 4x− 4

−3 = −4

The equation represents a contradiction, as −3 6= −4. Therefore, there is no solution to this equation.

Here is another example:−5 + 3x + 9 = 4x− (x− 4)

3x + 4 = 4x− x + 4

3x + 4 = 3x + 4

4 = 4

The equation represents an identity, as 4 = 4. Therefore, the solution to this equation is all real numbers.

Mod 13 - Quiz B - 2.4, 2.5


Page 29


Module 14


Contents: Mod 14 - Sec 2.6 AMod 14 - Sec 2.8 AMod 14 - Quiz A - 2.6, 2.8Mod 14 - Exam - Chapter 2Mod 14 - Chapter 3 Pre-TestMod 14 - Sec 3.1 BMod 14 - Sec 3.2 BMod 14 - Quiz B - 3.1, 3.2

Mod 14 - Sec 2.6 A

Section 2.6 covers formulas.

Mod 14 - Sec 2.8 A

Section 2.8 covers linear inequalities.

Mod 14 - Quiz A - 2.6, 2.8






Mod 14 - Sec 3.1 B

Section 3.1 covers writing expressions and equations to represent applications.

Mod 14 - Sec 3.2 B

Section 3.2 covers solving applications.

Mod 14 - Quiz B - 3.1, 3.2


Page 30


Note: Question 13 on this quiz is from Section 3.39.

9Good luck :)

Page 31


Module 15


Contents: Mod 15 - Sec 3.3 AMod 15 - Sec 3.4 AMod 15 - Quiz A - 3.3, 3.4Mod 15 - Exam - Chapter 3Mod 15 - Comprehensive Quiz

Mod 15 - Sec 3.3 A

Section 3.3 covers solving application involving geometry. You will have to memorize the following formulas:

The angles of a triangle add up to 180◦: ∠1 + ∠2 + ∠3 = 180.

The angles of a quadrilateral add up to 360◦: ∠1 + ∠2 + ∠3 + ∠4 = 360.

Complementary angles add up to 90◦: A + B = 90.

Supplementary angles add up to 180◦: A + B = 180.

Vertical angles are equal: A = B.

The perimeter of a triangle is given by: P = s1 + s2 + s3.

The perimeter of a rectangle is given by: P = 2l + 2w.

Hint (Question 12): Refer to the picture in the upper right. Instead of using the formula for theperimeter of a rectangle (P = 2l + 2w), you will have to come up with your own formula (based on thepicture).

Mod 15 - Sec 3.4 A

Section 3.4 covers applications involving motion, money, and mixtures. You will need to complete one ofthe following tables for each problem.

Motion Table:

rate · time = distance

Person A

Person B

Page 32


Interest Table:

principal · rate · time = interest

Account A

Account B

Non-Interest Money Table:

Quantity · Rate = Amount

Mixture Table:

Quantity · Rate = Amount

Part A

Part B

Mix

Mod 15 - Quiz A - 3.3, 3.4






Page 33


Module 16


Contents: Mod 16 - Chapter 4 Pre-TestMod 16 - Sec 4.1 AMod 16 - Sec 4.2 AMod 16 - Quiz A - 4.1, 4.2Mod 16 - Sec 4.3 BMod 16 - Sec 4.4 BMod 16 - Quiz B - 4.3, 4.4



Mod 16 - Sec 4.1 A

You will need to use the following rules throughout this section:

• Product Rule: xm · xn = xm+n

Understand this rule in the following manner: x4 · x3 = (xxxx)(xxx) = xxxxxxx = x7.

• Quotient Rule:xm

xn= xm−n

Understand this rule in the following manner:x4

x3=

xxxx

xxx=��xxxx

��xxx=

x

1= x.

• Zero Exponent Rule: x0 = 1

Understand this rule in the following manner:x4

x4= x4−4 = x0 = 1.

• Power Rule: (xm)n = xmn

Understand this rule in the following manner:(x4)3

= x4 · x4 · x4 = (xxxx)(xxxx)(xxxx) = x12.

• Expanded Power Rule:

(ax

by

)m

=amxm

bmym

Understand this rule in the following manner:

(2x

y2

)3

=

(2x

y2

)(2x

y2

)(2x

y2

)=

8x3

y6.

Mod 16 - Sec 4.2 A

In addition to all the rules you learned in the previous section, you will also use the following rules:

• Negative Exponent Rule: x−m =1

xm

Page 34


• Fraction Raised to a Negative Exponent Rule:

(x

y

)−n

=(yx

)nMod 16 - Quiz A - 4.1, 4.2





xn= xm−n

• Zero Exponent Rule: x0 = 1



(ax

by

)m

=amxm

bmym


xm

• Fraction Raised to a Negative Exponent Rule:

(x

y

)−n

=(yx

)nMod 16 - Sec 4.3 B

Section 4.3 covers scientific notation.

Mod 16 - Sec 4.4 B

Section 4.4 introduces polynomials and covers the addition and subtraction of polynomials.

Mod 16 - Quiz B - 4.3, 4.4


Page 35


Module 17


Contents: Mod 17 - Sec 4.5 AMod 17 - Sec 4.6 AMod 17 - Quiz A - 4.5, 4.6Mod 17 - Exam - Chapter 4Mod 17 - Chapter 5 & 6.1, 6.2 Pre-TestMod 17 - Sec 5.1 BMod 17 - Sec 5.2 BMod 17 - Quiz B - 5.1, 5.2

Mod 17 - Sec 4.5 A

Section 4.5 covers the multiplication of polynomials.

Mod 17 - Sec 4.6 A

Section 4.6 covers the division of polynomials by a monomial.

Mod 17 - Quiz A - 4.5, 4.6


Despite the name of the quiz, this quiz only consists of questions for Section 4.5. Questions from Sec-tion 4.6 will appear on the Chapter 4 Exam, so make sure you review that section as well.



Mod 17 - Chapter 5 & 6.1, 6.2 Pre-Test


Mod 17 - Sec 5.1 B

Section 5.1 covers factoring out the greatest common factor (GCF) from a polynomial.

Mod 17 - Sec 5.2 B

Section 5.2 covers factoring by grouping.

Page 36


Mod 17 - Quiz B - 5.1, 5.2


Page 37


Module 18


Contents: Mod 18 - Sec 5.3 AMod 18 - Sec 5.4 AMod 18 - Quiz A - 5.3, 5.4Mod 18 - Sec 5.5 BMod 18 - Sec 5.6 BMod 18 - Sec 6.1 BMod 18 - Quiz B - 5.5, 5.6, 6.1

Mod 18 - Sec 5.3 A

Section 5.3 covers factoring trinomials with a leading coefficient of 1.

Mod 18 - Sec 5.4 A

Section 5.4 covers factoring trinomials with a leading coefficient greater than 1.

Mod 18 - Quiz A - 5.3, 5.4


Mod 18 - Sec 5.5 B

Section 5.5 covers factoring using special formulas and provides a general review of factoring.

Mod 18 - Sec 5.6 B

Section 5.6 uses factoring to solve equations.

Mod 18 - Sec 6.1 B

Section 6.1 introduces rational expressions and covers simplify rational expressions.

Mod 18 - Quiz B - 5.5, 5.6, 6.1


Page 38


Module 19


Contents: Mod 19 - Sec 6.2 AMod 19 - Quiz A - 6.2Mod 19 - Exam - Chapter 5 & 6.1, 6.2Mod 19 - Chapter 7 Pre-TestMod 19 - Sec 7.1 BMod 19 - Sec 7.2 BMod 19 - Quiz B - 7.1, 7.2

Mod 19 - Sec 6.2 A

Section 6.2 covers multiplying and dividing rational expressions.

Mod 19 - Quiz A - 6.2


Note: The first 3 questions on this quiz are from Section 6.1. Review how to “determine the value orvalues of the variable for which the expression is defined.”

Mod 19 - Exam - Chapter 5 & 6.1, 6.2



• Always factor out the GCF first. For example,

12x2 + 75 = 3(4x2 + 25).

• Be able to factor any trinomial. For example,

6x2 + 11x− 10 = (3x− 2)(2x + 5).

• Be able to factor the difference of squares, a2 − b2 = (a + b)(a− b). For example,

4x2 − 25 = (2x)2 − (5)2

= (2x + 5)(2x− 5)

• Understand that the sum of squares, a2 + b2, is prime (cannot be factored).

Page 39


• Be able to factor the sum and difference of cubes, a3 + b3 = (a + b)(a2 − ab + b2 and a3 − b3 =(a− b)(a2 + ab + b2). For example,

27x3 + 8 = (3x)3 + (2)3

= (3x + 2)(9x2 − 6x + 4)

• When asked to “determine the value or values of the variable for which the expression is defined,”understand that you must identify what would make the denominator equal to zero.

• To simplify, multiply, or divide rational expressions, you must factor everything first.



Mod 19 - Sec 7.1 B

Section 7.1 introduces the coordinate plane and linear equations in two variables.

Note: The last 2 questions on this homework assignment are from Chapter 2. While they are perfectlygood review, I recommend that you skip these two questions.

Mod 19 - Sec 7.2 B

Section 7.2 covers graphing linear equations in two variables.

Mod 19 - Quiz B - 7.1, 7.2


Page 40


Module 20


Contents: Mod 20 - Sec 7.3 AMod 20 - Sec 7.4 AMod 20 - Quiz A - 7.3, 7.4Mod 20 - Sec 7.5 AMod 20 - Quiz A - 7.5Mod 20 - Exam - Chapter 7Mod 20 - Comprehensive Quiz

Mod 20 - Sec 7.3 A

Section 7.3 introduces slope. The slope of the line passing through points (x1, y1) and (x2, y2) is given by

m =y2 − y1x2 − x1

.

Mod 20 - Sec 7.4 A

Section 7.4 introduces slope-intercept form and point-slope form. Slope-intecept form is given by

y = mx + b,

where m is the slope and (0, b) is the y-intercept. Point-slope form is given by

y − y1 = m(x− x1),

where m is the slope and (x1, y1) is any point the line passes through.

Mod 20 - Quiz A - 7.3, 7.4


Mod 20 - Sec 7.5 B

Section 7.5 covers graphing linear inequalities.

Mod 20 - Quiz B - 7.5


Note: The last 2 quesions on this quiz are from Section 7.4.



Page 41



Number of questions: 20 (Multiple Choice)No calculator allowed.

Page 42


Module 21


Contents: Mod 21 - Chapters 3 & 4 Pre-TestMod 21 - Sec 3.1 AMod 21 - Sec 3.2 AMod 21 - Sec 3.3 AMod 21 - Quiz A - 3.1, 3.2, 3.3

Mod 21 - Chapter 3 & 4 Pre-Test

Number of questions: 33

Mod 21 - Sec 3.1 A

Section 3.1 covers graphing linear equations.

Mod 21 - Sec 3.2 A

Section 3.2 covers find slope and y-intercept. Slope-intecept form is

y = mx + b,

where m is the slope and (0, b) is the y-intercept. The slope of the line passing through points (x1, y1) and(x2, y2) is given by

m =y2 − y1x2 − x1

.

It is also important to know that for a nonzero n, 0n = 0, while n

0 is undefined.

Mod 21 - Sec 3.3 A

Section 3.3 covers finding the equation of a line.

To find the equation of a line, use point-slope form:

y − y1 = m(x− x1),

where m is the slope and (x1, y1) is any point the line passes through.

Two lines are parallel if they have the same slope. Two lines are perpendicular if the slopes are oppositereciprocals (if their product is −1). Moreover:

• Any two horizontal lines are parallel.

• Any two vertical lines are parallel.

• A horizontal line and a vertical line are perpendicular.

Mod 21 - Quiz A - 3.1, 3.2, 3.3


Page 43


Module 22


Contents: Mod 22 - Sec 3.5 AMod 22 - Sec 4.1 AMod 22 - Sec 4.2 AMod 22 - Sec 4.3 AMod 22 - Quiz A - 3.5, 4.1, 4.2, 4.3

Mod 22 - Sec 3.5 A

Section 3.5 introduces functions. A relation is a correspondence between two sets. Elements of the firstset are called the domain. Elements of the second set are called the range. A function is a specific type ofa relation where each element in the the domain corresponds to exactly one element in the range.

Mod 22 - Sec 4.1 A

Section 4.1 covers solving systems of equations in two variables.

Mod 22 - Sec 4.2 A

Section 4.2 covers solving systems of equations in three variables.

Mod 22 - Sec 4.3 A

Section 4.3 covers solving application problems using systems of equations.

Mod 22 - Quiz A - 3.5, 4.1, 4.2, 4.3


Page 44


Module 23


Contents: Mod 23 - Exam - Chapters 3 & 4Mod 23 - Chapters 5 & 6 Pre-Test

Mod 23 - Exam - Chapters 3 & 4


Mod 23 - Chapter 5 & 6 Pre-Test


Page 45


Module 24


Contents: Mod 24 - Sec 6.1 AMod 24 - Sec 6.2 AMod 24 - Sec 6.3 AMod 24 - Sec 6.4 AMod 24 - Quiz A - 6.1, 6.2, 6.3, 6.4

Mod 24 - Sec 6.1 A

Section 6.1 covers factoring out a common factor and factoring by grouping.

Mod 24 - Sec 6.2 A

Section 6.2 covers factoring trinomials and introduces the idea of using a substitution. Here is an exampleof using a substitution to factor a polynomial:

(x + 7)2 − 6(x + 7)− 16 = u2 − 6u− 16 Substitute: u = x + 7

= (u− 8)(u + 2)

= [(x + 7)− 8][(x + 7) + 2]

= (x− 1)(x + 9)

Mod 24 - Sec 6.3 A

Section 6.3 covers factoring using special factoring formulas. Students are expected to recognize thefollowing special factoring formulas:

a2 + 2ab + b2 = (a + b)2

a2 − 2ab + b2 = (a− b)2

Difference of Squares a2 − b2 = (a + b)(a− b)

Sum of Cubes a3 + b3 = (a + b)(a2 − ab + b2)

Difference of Cubes a3 − b3 = (a− b)(a2 + ab + b2)

Mod 24 - Sec 6.4 A

Section 6.4 covers solving equations using factoring.

Hint (Questions 11, 12): These questions give you the solutions of a polynomial and asks you to

Page 46


write a polynomial with those solutions. Hence, you do everything in reverse:

x = −2

3x = 4

3x = −2

3x + 2 = 0 x− 4 = 0

(3x + 2)(x− 4) = 0

3x2 − 10x− 8 = 0

Hint (Question 13): This question asks you to apply the Pythagorean Theorem to the given triangle.The Pythagorean Theorem states the sum of the squares of the legs of a right triangle is equal to thesquare of its hyptonuse. Here’s an example:

x2 + (x + 1)2 = (x + 2)2

x2 + x2 + 2x + 1 = x2 + 4x + 4

2x2 + 2x + 1 = x2 + 4x + 4

x2 − 2x− 3 = 0

(x− 3)(x + 1) = 0

The solution set is {−1, 3}. Since x represents the length of a leg of the triangle, we know x = 3. Thus,the sides of the triangle are 3, 4, and 5.

Mod 24 - Quiz A - 6.1, 6.2, 6.3, 6.4


Page 47


Module 25


Contents: Mod 25 - Exam - Chapters 5 & 6Mod 25 - Chapter 7 Pre-TestMod 25 - Sec 7.1 AMod 25 - Sec 7.2 AMod 25 - Quiz A - 7.1, 7.2

Mod 25 - Exam - Chapters 5 & 6




Mod 25 - Sec 7.1 A

Section 7.1 covers simplifying, multiplying, and dividing rational expressions. To simplify, multiply, ordivide a rational expression, you must factor first.

Questions 18 and 19 ask you to find the domain of the rational expression. You will recall fromChapter 3 that the domain represents the possible x-values of the function. Thus, we are looking for whatvalues of x would make the expression undefined. If the denominator of the rational expression is 0, thenthe function is undefined, so that x-value is not in the domain.

Mod 25 - Sec 7.2 A

Section 7.2 covers adding and subtracting rational expressions. To add and subtract rational expressions,you need to write both fractions with a common denominator. You may write two fractions with the leastcommon denominator as follows:

x− 1

x2 − 5x + 6

3x

4x− 12

x− 1

(x− 2)(x− 3)

3x

4(x− 3)

x− 1

(x− 2)(x− 3)· 4

4

3x

4(x− 3)· x− 2

x− 2

4x− 4

4(x− 2)(x− 3)

3x2 − 6x

4(x− 2)(x− 3)

Page 48


Here is another example:

x + 2

2x2 + 7x− 4

x− 3

2x2 − 5x + 2

x + 2

(2x− 1)(x + 4)

x− 3

(2x− 1)(x− 2)

x + 2

(2x− 1)(x + 4)· x− 2

x− 2

x− 3

(2x− 1)(x− 2)· x + 4

x + 4

x2 − 4

(2x− 1)(x + 4)(x− 2)

x2 + x− 12

(2x− 1)(x + 4)(x− 2)

Once you have a common denominator, you may add or subtract as needed:

x− 1

x2 + x− 20+

x + 3

x2 − 5x + 4=

x− 1

(x + 5)(x− 4)+

x + 3

(x− 1)(x− 4)

=x− 1

(x + 5)(x− 4)· x− 1

x− 1+

x + 3

(x− 1)(x− 4)· x + 5

x + 5

=x2 − 2x + 1

(x + 5)(x− 4)(x− 1)+

x2 + 8x + 15

(x + 5)(x− 4)(x− 1)

=2x2 + 6x + 16

(x + 5)(x− 4)(x− 1)

While subtracting, be careful to distribute the negative sign correctly:

3

4x2 − 9− 2

6x2 − 9x=

3

(2x + 3)(2x− 3)− 2

3x(2x− 3)

=3

(2x + 3)(2x− 3)· 3x

3x− 2

3x(2x− 3)· 2x + 3

2x + 3

=9x

3x(2x + 3)(2x− 3)− 4x + 6

3x(2x + 3)(2x− 3)

=9x− (4x + 6)

3x(2x + 3)(2x− 3)(Be this careful while subtracting!)

=9x− 4x− 6

3x(2x + 3)(2x− 3)

=5x− 6

3x(2x + 3)(2x− 3)

Mod 25 - Quiz A - 7.1, 7.2


Page 49


Module 26


Contents: Mod 26 - Sec 7.3 AMod 26 - Sec 7.4 AMod 26 - Quiz A - 7.3, 7.4Mod 26 - Exam - Chapter 7

Mod 26 - Sec 7.3 A

Section 7.3 covers simplifying complex rational expressions. To simplify complex rational expressions, thepreferred method is to multiply both the numerator and denominator by the least common denominator.See the following three examples:

x2

y

x3

y4

=

x2

y

x3

y4

· y4y4=

x2y3

x3

=y3

x

Example 2:

1− 7

x+

10

x2

1− 2

x− 15

x2

=

1− 7

x+

10

x2

1− 2

x− 15

x2

· x2x2

=x2 − 7x + 10

x2 − 2x− 15

=(x− 2)(x− 5)

(x− 5)(x + 3)

=x− 2

x + 3

Page 50


Example 3:

x− 2

x + 3

3 +x

x + 3

=

x− 2

x + 3

3 +x

x + 3

· x + 3

x + 3

=x(x + 3)− 2

3(x + 3) + x

=x2 + 3x− 2

3x + 9 + x

=x2 + 3x− 2

4x + 9

Mod 26 - Sec 7.4 A

Section 7.4 covers solving equations involving rational expressions. To solve an equation with rationalexpressions, multiply both sides by the least common denominator:

6− 4

x + 1=

5

x + 1

(x + 1) ·(

6− 4

x + 1

)=

(5

x + 1

)· (x + 1)

6(x + 1)− 4 = 5

6x + 6− 4 = 5

6x + 2 = 5

6x = 3

x =1

2

Page 51


Important: You must check your answers for extraneous solutions. If your answer makes the denom-inator equal to 0, your answer is an extraneous solution:

18

x2 − 9=

x

x + 3− 3

(x + 3)(x− 3) ·(

18

(x + 3)(x− 3)

)=

(x

x + 3− 3

)· (x + 3)(x− 3)

18 = x(x− 3)− 3(x + 3)(x− 3)

18 = x2 − 3x− 3(x2 − 9)

18 = x2 − 3x− 3x2 + 27

2x2 + 3x− 9 = 0

(2x− 3)(x + 3) = 0

2x− 3 = 0 x + 3 = 0

x =3

2x = −3

However, x = −3 is extraneous. Thus, the solution is x =3

2.

Here is another example:20

x2− 1 =

1

x

x2 ·(

20

x2− 1

)=

(1

x

)· x2

20− x2 = x

0 = x2 + x− 20

0 = (x + 5)(x− 4)

x + 5 = 0 x− 4 = 0

x = −5 x = 4

Hint (Questions 1, 2): These questions only want you to identify which values would be extraneous.Do not solve these equations.

Mod 26 - Quiz A - 7.3, 7.4


Page 52




Page 53


Module 27


Contents: Mod 27 - Chapter 8 Pre-TestMod 27 - Sec 8.1 AMod 27 - Sec 8.2 AMod 27 - Sec 8.3 AMod 27 - Sec 8.4 AMod 27 - Quiz A - 8.1, 8.2, 8.3, 8.4



Mod 27 - Sec 8.1 A

Section 8.1 introduces radicals.

Mod 27 - Sec 8.2 A

Section 8.2 covers rational exponents. The following two definitions are used throughout the section:

x1/n = n√x

xm/n =(

n√x)m

= n√xm

You will also have to recall the rules of exponents10:



xn= xm−n



(ax

by

)m

=amxm

bmym


xm

Hint (Questions 17-20): First, write all radicals with rational exponents. Then simplify.

Hint (Question 21): After writing the radicals with rational exponents, write the fractions with a

10from Section 5.1 or Beginning Algebra Part II

Page 54


common denominator: √5 · 3√

7 = 51/2 · 71/3

= 53/6 · 72/6

=(53 · 72

)1/6= (6125)1/6

= 6√

6125

Mod 27 - Sec 8.3 A

Section 8.3 covers simplifying radicals. See the following four examples:

√72 =

√23 · 32

=√

22 · 2 · 32

= 2 ·√

2 · 3

= 6√

2

Alternatively, √72 =

√36 ·√

2 = 6√

2

Example 2:

−3√

54 = −3√

2 · 33

= −3√

2 · 32 · 3

= −3 · 3 ·√

2 · 3

= −9√

6

Alternatively,−3√

54 = −3 ·√

9 ·√

6 = −9√

6

Example 3: √32x5y4 =

√25 · x5 · y4

= 22x2y2√

2x

= 4x2y2√

2x

Example 4:3√

54x8y7 = 3√

2 · 33 · x8 · y7

= 3x2y2 3√

2x2y

Page 55


Mod 27 - Sec 8.4 A

Section 8.4 covers adding, subtracting, and multiplying radicals. You may only add or subtract radicalswith like radicals. To add or subtract radicals, first simplify all radicals:

√18 +

√32 =

√9 · 2 +

√16 · 2

= 3√

2 + 4√

2

= 7√

2

Multiplying radicals is no different than multiplying polynomials. Just make sure you simplify all radicals:(2√

6− 3√

5) (

5√

6− 9√

5)

= 10√

36− 18√

30− 15√

30 + 27√

25

= 10 · 6− 18√

30− 15√

30 + 27 · 5

= 60− 18√

30− 15√

30 + 135

= 195− 33√

30

Mod 27 - Quiz A - 8.1, 8.2, 8.3, 8.4


Page 56


Module 28


Contents: Mod 28 - Sec 8.5 AMod 28 - Sec 8.6 AMod 28 - Sec 8.7 AMod 28 - Quiz A - 8.5, 8.6, 8.7

Mod 28 - Sec 8.5 A

Section 8.5 covers rationalizing denominators. To rationalize a denominator, you must get rid of anyfractions in any radicals and any radicals in the denominators. You may get rid of a radical in a denominatorby multiplying both the numerator and denominator by the appropriate radical:√

9

5=

√9√5

=3√5

=3√5·√

5√5

=3√

5

5

Here is another example:

13√

2=

13√

2·

3√

22

3√

22

=3√

4

2

If the denominator has more than one term, you must multiply by the conjugate to rationalize the denom-inator:

3

4−√

2=

3

4−√

2· 4 +

√2

4 +√

2

=12 + 3

√2

16− 2

=12 + 3

√2

14

Page 57


Mod 28 - Sec 8.6 A

Section 8.6 covers solving equations involving radicals. To solve an equation with square roots, squareboth sides. As you did in Section 7.4, you must also check that your answers are not extraneous:

x + 1 =√x + 13

(x + 1)2 =(√

x + 13)2

x2 + 2x + 1 = x + 13

x2 + x− 12 = 0

(x + 4)(x− 3) = 0

x + 4 = 0 x− 3 = 0

x = −4 x = 3

However, x = −4 is extraneous, as−4 + 1 6=

√−4 + 13.

Thus, our solution is x = 3.

Mod 28 - Sec 8.7 A

Section 8.7 introduces complex numbers. The following definitions are identical and are used throughoutthe section:

i2 = −1

i =√−1

The imaginary unit, i, allows us to simplify the square root of negative numbers:

√−12 =

√−1 ·√

12

=√−1 ·√

4 · 3

= 2i√

3

Mod 28 - Quiz A - 8.5, 8.6, 8.7


Page 58


Module 29


Contents: Mod 29 - Exam - Chapter 8Mod 29 - Chapter 9 Pre-TestMod 29 - Sec 9.1 AMod 29 - Sec 9.2 AMod 29 - Quiz A - 9.1, 9.2





Mod 29 - Sec 9.1 A

Section 9.1 covers the square root principle and completing the square. The square root principle statesthat you may take the square root of both sides an equation, but only if you consider both positiveand negative solutions:

x2 = 9

√x2 = ±

√9

x = ±3

Completing the square involves finding a constant to add to a polynomial to make it a square:

x2 + 8x +

(x + )2 = (x + 4)2

= x2 + 8x + 16

Page 59


This procedure allows you to solve any quadratic equation:

x2 + 8x + 2 = −5

x2 + 8x = −7

x2 + 8x + 16 = −7 + 16

(x + 4)2 = 9√(x + 4)2 = ±

√9

x + 4 = ±3

x = −4± 3

x = −1 x = −7

Here is another example:x2 − 6x + 1 = 0

x2 − 6x = −1

x2 − 6x + 9 = −1 + 9

(x− 3)2 = 8√(x− 3)2 = ±

√8

x− 3 = ±2√

2

x = 3± 2√

2

x = 3 + 2√

2 x = 3− 2√

2

Mod 29 - Sec 9.2 A

Section 9.2 covers the quadratic formula. For any equation ax2 + bx + c = 0,

x =−b±

√b2 − 4ac

2a.

Mod 29 - Quiz A - 9.1, 9.2


Page 60


Module 30


Contents: Mod 30 - Sec 9.3 AMod 30 - Sec 9.4 AMod 30 - Quiz A - 9.3, 9.4Mod 30 - Exam - Chapter 9Mod 30 - Comprehensive Quiz

Mod 30 - Sec 9.3 A

Section 9.3 covers solving equations with rational expressions and radicals and solving equations using asubstitution. These ideas were first presented in Chapters 6, 7, and 8.

Mod 30 - Sec 9.4 A

Section 9.4 covers graphing quadratic functions.

Mod 30 - Quiz A - 9.3, 9.4


Important: Show all your work for this test. Some questions on the test ask you to solve by com-pleting the square. If you do not solve by completing the square, you will not get credit for the problem(whether MyLabsPlus gives you credit or not). Similarly, some questions ask you to solve by using thequadratic formula: you will need to show your work here as well.




Number of questions: 10 (Multiple Choice)

Page 61

Documents

A Guide to Your Modular Math Course - faculty.mccneb.edufaculty.mccneb.edu/jdlee3/modular_guide.pdf · A Guide to Your Modular Math Course Joseph Lee Mod 5 - Sec 3.3 . . . . .