Download A Guide to Your Modular Math Course Contents Joseph Lee Fall 2014

A Guide to Your Modular Math Course
Joseph Lee
Fall 2014
Contents
Introduction
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
To the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7
Module 1
Mod 1 Mod 1 Mod 1 Mod 1 -
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Chapter 1 Pre-Test
Sec 1.5 . . . . . . .
Sec 1.6 . . . . . . .
Sec 1.7 . . . . . . .
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Module 2
Mod 2 Mod 2 Mod 2 Mod 2 Mod 2 Mod 2 -
Sec 1.8 . . . . . . . .
Quiz - Sec 1.5 to 1.8
Exam - Chapter 1 .
Chapter 2 Pre-Test .
Sec 2.1 . . . . . . . .
Sec 2.2 . . . . . . . .
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10
Module 3
Mod 3 Mod 3 Mod 3 Mod 3 Mod 3 -
Sec 2.3 . .
Sec 2.4 . .
Sec 2.5 . .
Quiz - Sec
Sec 2.6 . .
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Module 4
Mod 4 Mod 4 Mod 4 Mod 4 Mod 4 -
Sec 2.7 . . . . . . . . .
Sec 2.8 . . . . . . . . .
Sec 2.9 . . . . . . . . .
Quiz - Sec 2.6, 2.7, 2.8,
Exam - Chapter 2 . .
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2.9
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12
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2.1
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to 2.5
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Module 5
14
Mod 5 - Chapter 3 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Mod 5 - Sec 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Mod 5 - Sec 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1
A Guide to Your Modular Math Course
Joseph Lee
Mod 5 - Sec 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mod 5 - Quiz - Sec 3.1, 3.2, 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mod 5 - Sec 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Module 6
Mod 6 Mod 6 Mod 6 Mod 6 Mod 6 -
Sec 3.5 . . . . . . . . .
Sec 3.6 . . . . . . . . .
Sec 3.7 . . . . . . . . .
Quiz - Sec 3.4, 3.5, 3.6,
Exam - Chapter 3 . .
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3.7
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14
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Module 7
Mod 7 Mod 7 Mod 7 Mod 7 Mod 7 Mod 7 -
Chapter 4 Pre-Test . .
Sec 4.1 . . . . . . . . .
Sec 4.2 . . . . . . . . .
Sec 4.3 . . . . . . . . .
Sec 4.4 . . . . . . . . .
Quiz - Sec 4.1, 4.2, 4.3,
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4.4
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Module 8
Mod 8 Mod 8 Mod 8 Mod 8 Mod 8 Mod 8 Mod 8 -
Exam - Chapter 4 . .
Chapter 5 Pre-Test . .
Sec 5.1 . . . . . . . . .
Sec 5.2 . . . . . . . . .
Sec 5.3A . . . . . . . .
Sec 5.3B . . . . . . . .
Quiz - Sec 5.1, 5.2, 5.3
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Module 9
Mod 9 Mod 9 Mod 9 Mod 9 Mod 9 -
Sec 5.4 . . . . . . .
Sec 5.5 . . . . . . .
Quiz - Sec 5.4, 5.5
Exam - Chapter 5
Chapter 9 Pre-Test
Module 10
Mod 10 Mod 10 Mod 10 Mod 10 Mod 10 Mod 10 Mod 10 Module 11
Mod 11 Mod 11 Mod 11 Mod 11 -
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Sec 9.1 . . . . . . . . . . .
Sec 9.2 . . . . . . . . . . .
Sec 9.3 . . . . . . . . . . .
Sec 9.4 . . . . . . . . . . .
Quiz - Sec 9.1, 9.2, 9.3, 9.4
Exam - Chapter 9 . . . . .
Comprehensive Quiz . . .
Chapter 1 Pre-Test
Sec 1.3 A . . . . . .
Sec 1.4 A . . . . . .
Quiz A - 1.3, 1.4 . .
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Page 2
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.
A Guide to Your Modular Math Course
Joseph Lee
Mod 11 - Sec 1.5 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mod 11 - Sec 1.6 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mod 11 - Quiz B - 1.5, 1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
25
25
Module 12
Mod 12 Mod 12 Mod 12 Mod 12 Mod 12 Mod 12 Mod 12 -
Sec 1.7 A . . . . .
Sec 1.8 A . . . . .
Quiz A - 1.7, 1.8 .
Sec 1.9 B . . . . .
Sec 1.10 B . . . .
Quiz B - 1.9, 1.10
Exam - Chapter 1
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Module 13
Mod 13 Mod 13 Mod 13 Mod 13 Mod 13 Mod 13 Mod 13 Mod 13 -
Chapter 2 Pre-Test .
Sec 2.1 A . . . . . . .
Sec 2.2 A . . . . . . .
Sec 2.3 A . . . . . . .
Quiz A - 2.1, 2.2, 2.3
Sec 2.4 B . . . . . . .
Sec 2.5 B . . . . . . .
Quiz B - 2.4, 2.5 . . .
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Module 14
Mod 14 Mod 14 Mod 14 Mod 14 Mod 14 Mod 14 Mod 14 Mod 14 Module 15
Mod 15 Mod 15 Mod 15 Mod 15 Mod 15 Module 16
Mod 16 Mod 16 Mod 16 Mod 16 Mod 16 Mod 16 Mod 16 -
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Sec 2.6 A . . . . . .
Sec 2.8 A . . . . . .
Quiz A - 2.6, 2.8 . .
Exam - Chapter 2 .
Chapter 3 Pre-Test
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Sec 3.2 B . . . . . .
Quiz B - 3.1, 3.2 . .
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Sec 3.3 A . . . . . . .
Sec 3.4 A . . . . . . .
Quiz A - 3.3, 3.4 . . .
Exam - Chapter 3 . .
Comprehensive Quiz
Chapter 4 Pre-Test
Sec 4.1 A . . . . . .
Sec 4.2 A . . . . . .
Quiz A - 4.1, 4.2 . .
Sec 4.3 B . . . . . .
Sec 4.4 B . . . . . .
Quiz B - 4.3, 4.4 . .
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A Guide to Your Modular Math Course
Module 17
Mod 17 Mod 17 Mod 17 Mod 17 Mod 17 Mod 17 Mod 17 Mod 17 Module 18
Mod 18 Mod 18 Mod 18 Mod 18 Mod 18 Mod 18 Mod 18 -
Joseph Lee
Sec 4.5 A . . . . . . . . . . . .
Sec 4.6 A . . . . . . . . . . . .
Quiz A - 4.5, 4.6 . . . . . . . .
Exam - Chapter 4 . . . . . . .
Chapter 5 & 6.1, 6.2 Pre-Test
Sec 5.1 B . . . . . . . . . . . .
Sec 5.2 B . . . . . . . . . . . .
Quiz B - 5.1, 5.2 . . . . . . . .
Sec 5.3 A . . . . . . .
Sec 5.4 A . . . . . . .
Quiz A - 5.3, 5.4 . . .
Sec 5.5 B . . . . . . .
Sec 5.6 B . . . . . . .
Sec 6.1 B . . . . . . .
Quiz B - 5.5, 5.6, 6.1
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Module 19
Mod 19 Mod 19 Mod 19 Mod 19 Mod 19 Mod 19 Mod 19 -
Sec 6.2 A . . . . . . . . . . .
Quiz A - 6.2 . . . . . . . . .
Exam - Chapter 5 & 6.1, 6.2
Chapter 7 Pre-Test . . . . .
Sec 7.1 B . . . . . . . . . . .
Sec 7.2 B . . . . . . . . . . .
Quiz B - 7.1, 7.2 . . . . . . .
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Module 20
Mod 20 Mod 20 Mod 20 Mod 20 Mod 20 Mod 20 Mod 20 -
Sec 7.3 A . . . . . . .
Sec 7.4 A . . . . . . .
Quiz A - 7.3, 7.4 . . .
Sec 7.5 B . . . . . . .
Quiz B - 7.5 . . . . .
Exam - Chapter 7 . .
Comprehensive Quiz
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Module 21
Mod 21 Mod 21 Mod 21 Mod 21 Mod 21 -
Chapter 3 & 4 Pre-Test
Sec 3.1 A . . . . . . . .
Sec 3.2 A . . . . . . . .
Sec 3.3 A . . . . . . . .
Quiz A - 3.1, 3.2, 3.3 .
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43
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Module 22
44
Mod 22 - Sec 3.5 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Mod 22 - Sec 4.1 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Mod 22 - Sec 4.2 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Page 4
A Guide to Your Modular Math Course
Joseph Lee
Mod 22 - Sec 4.3 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mod 22 - Quiz A - 3.5, 4.1, 4.2, 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
44
Module 23
45
Mod 23 - Exam - Chapters 3 & 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Mod 23 - Chapter 5 & 6 Pre-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Module 24
Mod 24 Mod 24 Mod 24 Mod 24 Mod 24 -
Sec 6.1 A . .
Sec 6.2 A . .
Sec 6.3 A . .
Sec 6.4 A . .
Quiz A - 6.1,
Module 25
Mod 25 Mod 25 Mod 25 Mod 25 Mod 25 -
Exam - Chapters 5 & 6
Chapter 7 Pre-Test . .
Sec 7.1 A . . . . . . . .
Sec 7.2 A . . . . . . . .
Quiz A - 7.1, 7.2 . . . .
Module 26
Mod 26 Mod 26 Mod 26 Mod 26 -
Sec 7.3 A . . . . .
Sec 7.4 A . . . . .
Quiz A - 7.3, 7.4 .
Exam - Chapter 7
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53
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6.4
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Module 27
Mod 27 Mod 27 Mod 27 Mod 27 Mod 27 Mod 27 -
Chapter 8 Pre-Test .
Sec 8.1 A . . . . . . .
Sec 8.2 A . . . . . . .
Sec 8.3 A . . . . . . .
Sec 8.4 A . . . . . . .
Quiz A - 8.1, 8.2, 8.3,
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54
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55
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56
Module 28
Mod 28 Mod 28 Mod 28 Mod 28 -
Sec 8.5 A . .
Sec 8.6 A . .
Sec 8.7 A . .
Quiz A - 8.5,
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57
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60
60
Module 29
Mod 29 Mod 29 Mod 29 Mod 29 Mod 29 -
. . . . .
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8.6, 8.7
Exam - Chapter 8 .
Chapter 9 Pre-Test
Sec 9.1 A . . . . . .
Sec 9.2 A . . . . . .
Quiz A - 9.1, 9.2 . .
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.
A Guide to Your Modular Math Course
Module 30
Mod 30 Mod 30 Mod 30 Mod 30 Mod 30 -
Sec 9.3 A . . . . . . .
Sec 9.4 A . . . . . . .
Quiz A - 9.3, 9.4 . . .
Exam - Chapter 9 . .
Comprehensive Quiz
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Joseph Lee
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Page 6
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61
61
61
61
61
61
A Guide to Your Modular Math Course
Joseph Lee
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 a
week of material.
About the Author
My name is Joseph Lee, and I am one of the many math instructors here at Metropolitan Community
College who teaches a modular math course. I have taught at this college since 2007, and the first modular
course I taught was in the Winter of 2012. During the Spring quarter of 2014, I tried to organize some
of the details of this course in this guide. This guide is by no means a complete work, but if any of this
information 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 may
be successful in any of these formats. The reverse is also true. Students who are absent from class, who
don’t spend time outside of class studying, and who don’t seek out their instructor for help will struggle
in any of our four formats. To be successful, students enrolled in modular math should understand being
successful in this course will require the same time and dedication that would be needed for any other
math course. Being on time and present for every class is expected. Spending time outside of class working
on homework every night is expected. Students who do not practice these habits in my traditional classes
find they are unlikely to succeed, as do students in my modular courses.
Page 7
A Guide to Your Modular Math Course
Joseph Lee
Module 1
Due: Thursday, September 4, 2014, 10:00 P.M.
Contents:
Mod 1 - Chapter 1 Pre-Test
Mod 1 - Sec 1.5
Mod 1 - Sec 1.6
Mod 1 - Sec 1.7
Mod 1 - Chapter 1 Pre-Test
Number of questions: 30
No 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 following
order:
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 :
Correct Order :
18 ÷ 2 × 9 6= 18 ÷ 18 = 1
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 first
digit in each number, and then perform any operations. For example,
265 + 1459 + 43, 921
≈ 300 + 1000 + 40, 000
≈ 41, 300.
1
2
not zero
according to the principal of estimation
Page 8
A Guide to Your Modular Math Course
Joseph Lee
Module 2
Due: Thursday, September 11, 2014, 10:00 P.M.
Contents:
Mod 2 - Sec 1.8
Mod 2 - Quiz - Sec 1.5 to 1.8
Mod 2 - Exam - Chapter 1
Mod 2 - Chapter 2 Pre-Test
Mod 2 - Sec 2.1
Mod 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 to
your answer.
Mod 2 - Quiz - Sec 1.5 to 1.8
Number of questions: 15
No calculator allowed.
Mod 2 - Exam - Chapter 1
Number of questions: 30
No calculator allowed.
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 - Chapter 2 Pre-Test
Number of questions: 33
No calculator allowed.
Mod 2 - Sec 2.1
Section 2.1 introduces fractions.
Page 9
A Guide to Your Modular Math Course
Joseph Lee
Mod 2 - Sec 2.2
Section 2.2 covers reducing fractions. The section offers two methods for reducing fractions. Here are the
54
two methods for the fraction
:
90
• You may reduce a fraction by finding a common factor in the numerator and in the denominator.
54
54 ÷ 18
3
=
=
90
90 ÷ 18
5
• You may reduce a fraction by factoring the numerator and denominator into prime numbers.
((
(×
(3
54
2×3×3×3 (
2×
3×3
3
=
=
=
((
(×
(3
90
2×3×3×5 (
5
2×
3×5
Page 10
A Guide to Your Modular Math Course
Joseph Lee
Module 3
Due: Thursday, September 18, 2014, 10:00 P.M.
Contents:
Mod 3 - Sec 2.3
Mod 3 - Sec 2.4
Mod 3 - Sec 2.5
Mod 3 - Quiz - Sec 2.1 to 2.5
Mod 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 fractions
before you multiply:
3 · 1 2 · 1
3 2
1
× =
×
= .
4 9
6
2 · 2 3 · 3
To multiply mixed numbers, change the mixed numbers to improper fractions, and then multiply:
1
2
7 17
119
14
2 ×3 = ×
=
=7 .
3
5
3
5
15
15
Mod 3 - Sec 2.5
Section 2.5 covers dividing fractions. To divide fractions, multiply by the reciprocal :
2 3
2 4
8
÷ = × = .
3 4
3 3
9
To divide mixed numbers, change the mixed numbers to improper fractions, and then divide:
2
7 17
7
5
35
1
= ×
=
2 ÷3 = ÷
3
5
3
5
3 17
51
Mod 3 - Quiz - Sec 2.1 to 2.5
Number of questions: 16
No calculator allowed.
Mod 3 - Sec 2.6
Section 2.6 covers finding a least common denominator and constructing equivalent fractions. The least
5
common denominator is the smallest number that all denominators divide into evenly. For the fractions
8
7
and
, the least common denominator is 24 because 24 is the smallest number that both 8 and 12 divide
12
into evenly. We may now write these fractions as equivalent fractions with the least common denominator:
5 3
15
× =
8 3
24
7
2
14
× =
12 2
24
Page 11
A Guide to Your Modular Math Course
Joseph Lee
Module 4
Due: Thursday, September 25, 2014, 10:00 P.M.
Contents:
Mod 4 - Sec 2.7
Mod 4 - Sec 2.8
Mod 4 - Sec 2.9
Mod 4 - Quiz - Sec 2.6, 2.7, 2.8, 2.9
Mod 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 as
equivalent fractions with the least common denominator, and then add or subtract:
2
7
2 3
7
+
= · +
5 15
5 3 15
=
6
7
+
15 15
=
13
15
Mod 4 - Sec 2.8
Section 2.8 covers adding and subtracting mixed numbers. To add mixed numbers, add the whole parts
together and add the fraction parts together.
2
3
2 4
3 3
6 +2 = 6 · +2 ·
3
4
3 4
4 3
= 6
8
9
+2
12
12
= 8+
17
12
= 8+1
= 9
5
12
5
12
Page 12
A Guide to Your Modular Math Course
Joseph Lee
To subtract mixed numbers, you may have to borrow to subtract your fraction parts:
2
3
2 4
3 3
6 −2 = 6 · −2 ·
3
4
3 4
4 3
= 6
8
9
−2
12
12
= 5+1
8
9
−2
12
12
= 5
20
9
−2
12
12
= 3
11
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
Number of questions: 16
No calculator allowed.
Mod 4 - Exam - Chapter 2
Number of questions: 33
No calculator allowed.
What are the most important things to know for this test?
• Be able to reduce a fraction.
• Know when you need a common denominator (adding, subtracting), and when you do not a common
denominator (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, subtracting, 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
A Guide to Your Modular Math Course
Joseph Lee
Module 5
Due: Thursday, October 2, 2014, 10:00 P.M.
Contents:
Mod 5 - Chapter 3 Pre-Test
Mod 5 - Sec 3.1
Mod 5 - Sec 3.2
Mod 5 - Sec 3.3
Mod 5 - Quiz - Sec 3.1, 3.2, 3.3
Mod 5 - Sec 3.4
Mod 5 - Chapter 3 Pre-Test
Number of questions: 30
No calculator allowed.
Mod 5 - Sec 3.1
Section 3.1 introduces decimals. It is important that you know the names of the place-values after the
decimal place. The place-values are as follows:
0.3 8 6 2 5
↑ ↑ ↑ ↑ ↑
| | | | ↓
| | | ↓ hundred-thousandths
| | ↓ ten-thousandths
| ↓ thousandths
↓ hundredths
tenths
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
309
=
100,000
800
Mod 5 - Sec 3.2
Section 3.2 covers ordering and rounding. To compare two decimals, keep in mind that you may add extra
0’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 decimals
is lining up the decimals.
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Joseph Lee
Mod 5 - Quiz - Sec 3.1, 3.2, 3.3
Number of questions: 15
No calculator allowed.
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 places
in the product.
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Joseph Lee
Module 6
Due: Thursday, October 9, 2014, 10:00 P.M.
Contents:
Mod 6 - Sec 3.5
Mod 6 - Sec 3.6
Mod 6 - Sec 3.7
Mod 6 - Quiz - Sec 3.4, 3.5, 3.6, 3.7
Mod 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 the
numerator 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
Number of questions: 15
No calculator allowed.
Mod 6 - Exam - Chapter 3
Number of questions: 30
No calculator allowed.
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Joseph Lee
Module 7
Due: Thursday, October 16, 2014, 10:00 P.M.
Contents:
Mod 7 - Chapter 4 Pre-Test
Mod 7 - Sec 4.1
Mod 7 - Sec 4.2
Mod 7 - Sec 4.3
Mod 7 - Sec 4.4
Mod 7 - Quiz - Sec 4.1, 4.2, 4.3, 4.4
Mod 7 - Chapter 4 Pre-Test
Number of questions: 28
Calculator 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 ? 34
=
39
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
28
=
7
7
n= 4
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Joseph Lee
Second, use cross multiplication to change the proportion to a simple equation.
n
3
=
5
7
n × 7 = 15
n×7
15
=
7
7
n= 2
1
7
Mod 7 - Sec 4.4
Section 4.4 covers application problems involving proportions. Make sure you start each problem by setting
up a proportion.
Mod 7 - Quiz - Sec 4.1, 4.2, 4.3, 4.4
Number of questions: 14
Calculator allowed.
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Joseph Lee
Module 8
Due: Thursday, October 23, 2014, 10:00 P.M.
Contents:
Mod 8 - Exam - Chapter 4
Mod 8 - Chapter 5 Pre-Test
Mod 8 - Sec 5.1
Mod 8 - Sec 5.2
Mod 8 - Sec 5.3A
Mod 8 - Sec 5.3B
Mod 8 - Quiz - Sec 5.1, 5.2, 5.3
Mod 8 - Exam - Chapter 4
Number of questions: 28
Calculator allowed.
Mod 8 - Chapter 5 Pre-Test
Number of questions: 26
Calculator allowed.
Mod 8 - Sec 5.1
Section 5.1 introduces percents. Any percent may be changed to a fraction or mixed number:
34% =
34
17
=
100
50
150
1
=1
100
2
or
150% =
or
150% = 1.5
Also, any percent may be changed to a decimal:
34% = 0.34
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 each
problem as an equation by using key words:
Problem
what
of
is
find
3
Equation
n
×
=
n=
Section 5.3B will present all the same problems but offer a different method for solving them.
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Joseph Lee
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 each
problem as a proportion involving the amount, a, the base, b, and the percent, p.
a
p
=
b
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
24
=
150
100
a × 100 = 150 × 24
n × 100
3600
=
100
100
a = 36
Mod 8 - Quiz - Sec 5.1, 5.2, 5.3
Number of questions: 17
Calculator allowed.
4
However, in 5.3B, you will not change the percent to a decimal.
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Joseph Lee
Module 9
Due: Thursday, October 30, 2014, 10:00 P.M.
Contents:
Mod 9 - Sec 5.4
Mod 9 - Sec 5.5
Mod 9 - Quiz - Sec 5.4, 5.5
Mod 9 - Exam - Chapter 5
Mod 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 of
increase 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
Number of questions: 9
Calculator allowed.
Mod 9 - Exam - Chapter 5
Number of questions: 26
Calculator allowed.
Mod 9 - Chapter 9 Pre-Test
Number of questions: 30
No calculator allowed.
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Joseph Lee
Module 10
Due: Thursday, November 6, 2014, 10:00 P.M.
Contents:
Mod 10 - Sec 9.1
Mod 10 - Sec 9.2
Mod 10 - Sec 9.3
Mod 10 - Sec 9.4
Mod 10 - Quiz - Sec 9.1, 9.2, 9.3, 9.4
Mod 10 - Exam - Chapter 9
Mod 10 - Comprehensive Quiz
Mod 10 - Sec 9.1
Section 9.1 covers adding positive and negative numbers. If you add two negatives together, add them
together, 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 the
subtraction 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 two
numbers with the same sign, the product or quotient is positive. If you multiply or divide two numbers
with 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.
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.
Mod 10 - Quiz - Sec 9.1, 9.2, 9.3, 9.4
Number of questions: 18
No calculator allowed.
5
absolute value
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A Guide to Your Modular Math Course
Joseph Lee
Mod 10 - Exam - Chapter 9
Number of questions: 30
No calculator allowed.
What are the most important things to know for this test?
• If you add a positive number and a negative number, find their difference (subtract) and take the
sign 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 you
multiply 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.
6
absolute value
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Joseph Lee
Module 11
Due: Thursday, September 11, 2014, 10:00 P.M.
Contents:
Mod 11 - Chapter 1 Pre-Test
Mod 11 - Sec 1.3 A
Mod 11 - Sec 1.4 A
Mod 11 - Quiz A - 1.3, 1.4
Mod 11 - Sec 1.5 B
Mod 11 - Sec 1.6 B
Mod 11 - Quiz B - 1.5, 1.6
Mod 11 - Chapter 1 Pre-Test
Number of questions: 39
No calculator allowed.
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
Number of questions: 15
No calculator allowed.
Mod 11 - Sec 1.5 B
Section 1.5 covers ordering and absolute value.
7
These are also called the counting numbers or the positive integers.
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A Guide to Your Modular Math Course
Joseph Lee
Mod 11 - Sec 1.6 B
Section 1.6 covers adding positive and negative numbers. If you add two negatives together, add them
together, 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
Number of questions: 19
No calculator allowed.
8
absolute value
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Joseph Lee
Module 12
Due: Thursday, September 25, 2014, 10:00 P.M.
Contents:
Mod 12 - Sec 1.7 A
Mod 12 - Sec 1.7 A
Mod 12 - Quiz A - 1.7, 1.8
Mod 12 - Sec 1.9 B
Mod 12 - Sec 1.10 B
Mod 12 - Quiz B - 1.9, 1.10
Mod 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 the
subtraction 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 two
numbers with the same sign, the product or quotient is positive. If you multiply or divide two numbers
with opposite signs, the product or quotient is negative.
Mod 12 - Quiz A - 1.7, 1.8
Number of questions: 15
No calculator allowed.
Mod 12 - Sec 1.9 B
Section 1.9 covers exponents and the order of operations:
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.
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Joseph Lee
Mod 12 - Sec 1.10 B
Section 1.10 introduces the properties of the real number system. You must memorize the following
properties:
Addition
Commutative Property
Associative Property
a+b=b+a
a + (b + c) = (a + b) + c
Distributive Property
Multiplication
a·b=b·a
a(bc) = (ab)c
a(b + c) = ab + ac
Identity Property
a+0=a
Inverse Property
a + (−a) = 0
Mod 12 - Quiz B - 1.9, 1.10
Number of questions: 15
No calculator allowed.
Mod 12 - Exam - Chapter 1
Number of questions: 40
No calculator allowed.
Page 27
a·1=a
a·
1
=1
a
A Guide to Your Modular Math Course
Joseph Lee
Module 13
Due: Thursday, October 9, 2014, 10:00 P.M.
Contents:
Mod 13 - Chapter 2 Pre-Test
Mod 13 - Sec 2.1 A
Mod 13 - Sec 2.2 A
Mod 13 - Sec 2.3 A
Mod 13 - Quiz A - 2.1, 2.2, 2.3
Mod 11 - Sec 2.4 B
Mod 11 - Sec 2.5 B
Mod 11 - Quiz B - 2.4, 2.5
Mod 13 - Chapter 2 Pre-Test
Number of questions: 31
No calculator allowed.
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 of
equality 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 property
of 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
Number of questions: 15
No calculator allowed.
Mod 13 - Sec 2.4 B
Section 2.4 covers solving equations with variables on one side of the equation.
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A Guide to Your Modular Math Course
Joseph Lee
Mod 13 - Sec 2.5 B
Section 2.5 covers solving equations with variables on both sides of an equation. When you have variables
on 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
Number of questions: 14
No calculator allowed.
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Joseph Lee
Module 14
Due: Thursday, October 23, 2014, 10:00 P.M.
Contents:
Mod 14 - Sec 2.6 A
Mod 14 - Sec 2.8 A
Mod 14 - Quiz A - 2.6, 2.8
Mod 14 - Exam - Chapter 2
Mod 14 - Chapter 3 Pre-Test
Mod 14 - Sec 3.1 B
Mod 14 - Sec 3.2 B
Mod 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
Number of questions: 13
No calculator allowed.
Mod 14 - Exam - Chapter 2
Number of questions: 31
No calculator allowed.
Mod 14 - Chapter 3 Pre-Test
Number of questions: 19
Calculator allowed.
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
Number of questions: 14
Calculator allowed.
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A Guide to Your Modular Math Course
Joseph Lee
Note: Question 13 on this quiz is from Section 3.39 .
9
Good luck :)
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A Guide to Your Modular Math Course
Joseph Lee
Module 15
Due: Thursday, November 6, 2014, 10:00 P.M.
Contents:
Mod 15 - Sec 3.3 A
Mod 15 - Sec 3.4 A
Mod 15 - Quiz A - 3.3, 3.4
Mod 15 - Exam - Chapter 3
Mod 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 the
perimeter of a rectangle (P = 2l + 2w), you will have to come up with your own formula (based on the
picture).
Mod 15 - Sec 3.4 A
Section 3.4 covers applications involving motion, money, and mixtures. You will need to complete one of
the following tables for each problem.
Motion Table:
rate
·
time
Person A
Person B
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=
distance
A Guide to Your Modular Math Course
Joseph Lee
Interest Table:
·
principal
rate
·
Rate
=
time
=
interest
Account A
Account B
Non-Interest Money Table:
Quantity
·
Amount
Mixture Table:
Quantity
·
Rate
Part A
Part B
Mix
Mod 15 - Quiz A - 3.3, 3.4
Number of questions: 15
Calculator allowed.
Mod 15 - Exam - Chapter 3
Number of questions: 17
Calculator allowed.
Mod 15 - Comprehensive Quiz
Number of questions: 16
No calculator allowed.
Page 33
=
Amount
A Guide to Your Modular Math Course
Joseph Lee
Module 16
Due: Thursday, September 11, 2014, 10:00 P.M.
Contents:
Mod 16 - Chapter 4 Pre-Test
Mod 16 - Sec 4.1 A
Mod 16 - Sec 4.2 A
Mod 16 - Quiz A - 4.1, 4.2
Mod 16 - Sec 4.3 B
Mod 16 - Sec 4.4 B
Mod 16 - Quiz B - 4.3, 4.4
Mod 16 - Chapter 4 Pre-Test
Number of questions: 37
No calculator allowed.
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
= xm−n
xn
Understand this rule in the following manner:
x4
xxxx xxxx
x
=
=
= = x.
3
x
xxx
1
xxx
• Zero Exponent Rule: x0 = 1
Understand this rule in the following manner:
x4
= x4−4 = x0 = 1.
x4
• Power Rule: (xm )n = xmn
3
Understand this rule in the following manner: x4 = x4 · x4 · x4 = (xxxx)(xxxx)(xxxx) = x12 .
m
ax
am xm
• Expanded Power Rule:
= m m
by
b y
3 2x
2x
2x
2x
8x3
Understand this rule in the following manner:
=
=
.
y2
y2
y2
y2
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
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A Guide to Your Modular Math Course
Joseph Lee
−n y n
x
=
• Fraction Raised to a Negative Exponent Rule:
y
x
Mod 16 - Quiz A - 4.1, 4.2
Number of questions: 16
No calculator allowed.
What are the most important things to know for this test?
• Product Rule: xm · xn = xm+n
• Quotient Rule:
xm
= xm−n
xn
• Zero Exponent Rule: x0 = 1
• Power Rule: (xm )n = xmn
m
am xm
ax
= m m
• Expanded Power Rule:
by
b y
• Negative Exponent Rule: x−m =
1
xm
−n x
y n
• Fraction Raised to a Negative Exponent Rule:
=
y
x
Mod 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
Number of questions: 19
No calculator allowed.
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Joseph Lee
Module 17
Due: Thursday, September 25, 2014, 10:00 P.M.
Contents:
Mod 17 - Sec 4.5 A
Mod 17 - Sec 4.6 A
Mod 17 - Quiz A - 4.5, 4.6
Mod 17 - Exam - Chapter 4
Mod 17 - Chapter 5 & 6.1, 6.2 Pre-Test
Mod 17 - Sec 5.1 B
Mod 17 - Sec 5.2 B
Mod 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
Number of questions: 15
No calculator allowed.
Despite the name of the quiz, this quiz only consists of questions for Section 4.5. Questions from Section 4.6 will appear on the Chapter 4 Exam, so make sure you review that section as well.
Mod 17 - Exam - Chapter 4
Number of questions: 30
No calculator allowed.
Mod 17 - Chapter 5 & 6.1, 6.2 Pre-Test
Number of questions: 39
No calculator allowed.
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.
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A Guide to Your Modular Math Course
Joseph Lee
Mod 17 - Quiz B - 5.1, 5.2
Number of questions: 17
No calculator allowed.
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A Guide to Your Modular Math Course
Joseph Lee
Module 18
Due: Thursday, October 9, 2014, 10:00 P.M.
Contents:
Mod 18 - Sec 5.3 A
Mod 18 - Sec 5.4 A
Mod 18 - Quiz A - 5.3, 5.4
Mod 18 - Sec 5.5 B
Mod 18 - Sec 5.6 B
Mod 18 - Sec 6.1 B
Mod 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
Number of questions: 15
No calculator allowed.
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
Number of questions: 20
No calculator allowed.
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Joseph Lee
Module 19
Due: Thursday, October 23, 2014, 10:00 P.M.
Contents:
Mod 19 - Sec 6.2 A
Mod 19 - Quiz A - 6.2
Mod 19 - Exam - Chapter 5 & 6.1, 6.2
Mod 19 - Chapter 7 Pre-Test
Mod 19 - Sec 7.1 B
Mod 19 - Sec 7.2 B
Mod 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
Number of questions: 8
No calculator allowed.
Note: The first 3 questions on this quiz are from Section 6.1. Review how to “determine the value or
values of the variable for which the expression is defined.”
Mod 19 - Exam - Chapter 5 & 6.1, 6.2
Number of questions: 25
No calculator allowed.
What are the most important things to know for this test?
• 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).
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Joseph Lee
• 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 - Chapter 7 Pre-Test
Number of questions: 30
No calculator allowed.
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 perfectly
good 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
Number of questions: 14
No calculator allowed.
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A Guide to Your Modular Math Course
Joseph Lee
Module 20
Due: Thursday, November 6, 2014, 10:00 P.M.
Contents:
Mod 20 - Sec 7.3 A
Mod 20 - Sec 7.4 A
Mod 20 - Quiz A - 7.3, 7.4
Mod 20 - Sec 7.5 A
Mod 20 - Quiz A - 7.5
Mod 20 - Exam - Chapter 7
Mod 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 − y1
.
x2 − 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
Number of questions: 16
No calculator allowed.
Mod 20 - Sec 7.5 B
Section 7.5 covers graphing linear inequalities.
Mod 20 - Quiz B - 7.5
Number of questions: 8
No calculator allowed.
Note: The last 2 quesions on this quiz are from Section 7.4.
Mod 20 - Exam - Chapter 7
Number of questions: 25
No calculator allowed.
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Joseph Lee
Mod 20 - Comprehensive Quiz
Number of questions: 20 (Multiple Choice)
No calculator allowed.
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A Guide to Your Modular Math Course
Joseph Lee
Module 21
Due: Thursday, September 4, 2014, 10:00 P.M.
Contents:
Mod 21 - Chapters 3 & 4 Pre-Test
Mod 21 - Sec 3.1 A
Mod 21 - Sec 3.2 A
Mod 21 - Sec 3.3 A
Mod 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
y2 − y1
m=
.
x2 − x1
It is also important to know that for a nonzero n,
0
n
= 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 opposite
reciprocals (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
Number of questions: 17
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A Guide to Your Modular Math Course
Joseph Lee
Module 22
Due: Thursday, September 11, 2014, 10:00 P.M.
Contents:
Mod 22 - Sec 3.5 A
Mod 22 - Sec 4.1 A
Mod 22 - Sec 4.2 A
Mod 22 - Sec 4.3 A
Mod 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 first
set are called the domain. Elements of the second set are called the range. A function is a specific type of
a 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
Number of questions: 17
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Joseph Lee
Module 23
Due: Thursday, September 18, 2014, 10:00 P.M.
Contents:
Mod 23 - Exam - Chapters 3 & 4
Mod 23 - Chapters 5 & 6 Pre-Test
Mod 23 - Exam - Chapters 3 & 4
Number of questions: 26
Mod 23 - Chapter 5 & 6 Pre-Test
Number of questions: 35
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A Guide to Your Modular Math Course
Joseph Lee
Module 24
Due: Thursday, September 25, 2014, 10:00 P.M.
Contents:
Mod 24 - Sec 6.1 A
Mod 24 - Sec 6.2 A
Mod 24 - Sec 6.3 A
Mod 24 - Sec 6.4 A
Mod 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 example
of 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 the
following special factoring formulas:
a2 + 2ab + b2 = (a + b)2
a2 − 2ab + b2 = (a − b)2
a2 − b2 = (a + b)(a − b)
Difference of Squares
Sum of Cubes
Difference of Cubes
a3 + b3 = (a + b)(a2 − ab + b2 )
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
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A Guide to Your Modular Math Course
Joseph Lee
write a polynomial with those solutions. Hence, you do everything in reverse:
x= −
2
3
x= 4
3x = −2
x−4= 0
3x + 2 = 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 the
square 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
Number of questions: 19
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A Guide to Your Modular Math Course
Joseph Lee
Module 25
Due: Thursday, October 2, 2014, 10:00 P.M.
Contents:
Mod 25 - Exam - Chapters 5 & 6
Mod 25 - Chapter 7 Pre-Test
Mod 25 - Sec 7.1 A
Mod 25 - Sec 7.2 A
Mod 25 - Quiz A - 7.1, 7.2
Mod 25 - Exam - Chapters 5 & 6
Number of questions: 28
Mod 25 - Chapter 7 Pre-Test
Number of questions: 22
Mod 25 - Sec 7.1 A
Section 7.1 covers simplifying, multiplying, and dividing rational expressions. To simplify, multiply, or
divide a rational expression, you must factor first.
Questions 18 and 19 ask you to find the domain of the rational expression. You will recall from
Chapter 3 that the domain represents the possible x-values of the function. Thus, we are looking for what
values of x would make the expression undefined. If the denominator of the rational expression is 0, then
the 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 least
common denominator as follows:
x−1
− 5x + 6
3x
4x − 12
x−1
(x − 2)(x − 3)
3x
4(x − 3)
x−1
4
·
(x − 2)(x − 3) 4
3x
x−2
·
4(x − 3) x − 2
4x − 4
4(x − 2)(x − 3)
3x2 − 6x
4(x − 2)(x − 3)
x2
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Joseph Lee
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
x−2
·
(2x − 1)(x + 4) x − 2
x−3
x+4
·
(2x − 1)(x − 2) 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+3
x−1
x+3
x−1
+
=
+
x2 + x − 20 x2 − 5x + 4
(x + 5)(x − 4) (x − 1)(x − 4)
=
x−1
x−1
x+3
x+5
·
+
·
(x + 5)(x − 4) x − 1 (x − 1)(x − 4) x + 5
=
x2 + 8x + 15
x2 − 2x + 1
+
(x + 5)(x − 4)(x − 1) (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
2
3
2
− 2
=
−
− 9 6x − 9x
(2x + 3)(2x − 3) 3x(2x − 3)
4x2
=
3x
2
2x + 3
3
·
−
·
(2x + 3)(2x − 3) 3x 3x(2x − 3) 2x + 3
=
4x + 6
9x
−
3x(2x + 3)(2x − 3) 3x(2x + 3)(2x − 3)
=
9x − (4x + 6)
3x(2x + 3)(2x − 3)
=
9x − 4x − 6
3x(2x + 3)(2x − 3)
=
5x − 6
3x(2x + 3)(2x − 3)
Mod 25 - Quiz A - 7.1, 7.2
Number of questions: 10
Page 49
(Be this careful while subtracting!)
A Guide to Your Modular Math Course
Joseph Lee
Module 26
Due: Thursday, October 9, 2014, 10:00 P.M.
Contents:
Mod 26 - Sec 7.3 A
Mod 26 - Sec 7.4 A
Mod 26 - Quiz A - 7.3, 7.4
Mod 26 - Exam - Chapter 7
Mod 26 - Sec 7.3 A
Section 7.3 covers simplifying complex rational expressions. To simplify complex rational expressions, the
preferred method is to multiply both the numerator and denominator by the least common denominator.
See the following three examples:
 2
x2
x
 y  y4
y

= 
 x3  · y 4
x3
y4
y4
=
x2 y 3
x3
=
y3
x
Example 2:

7 10
+ 2
1−
x x = 

2 15
1− − 2
1−
x x
1−

7 10
+ 2
x2
x x 
· 2

2 15
x
−
x x2
=
x2 − 7x + 10
x2 − 2x − 15
=
(x − 2)(x − 5)
(x − 5)(x + 3)
=
x−2
x+3
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A Guide to Your Modular Math Course
Joseph Lee
Example 3:

2 
2
x−
x−
x+3 = 
x+3· x+3

x
x  x+3
3+
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 rational
expressions, multiply both sides by the least common denominator:
4
5
=
x+1
x+1
4
5
(x + 1) · 6 −
=
· (x + 1)
x+1
x+1
6−
6(x + 1) − 4 = 5
6x + 6 − 4 = 5
6x + 2 = 5
6x = 3
x=
1
2
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Joseph Lee
Important: You must check your answers for extraneous solutions. If your answer makes the denominator equal to 0, your answer is an extraneous solution:
18
x
=
−3
−9
x+3
x
18
=
(x + 3)(x − 3) ·
− 3 · (x + 3)(x − 3)
(x + 3)(x − 3)
x+3
x2
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
2
x+3=0
x = −3
3
However, x = −3 is extraneous. Thus, the solution is x = .
2
Here is another example:
1
20
−1=
2
x
x
20
1
x2 ·
−1 =
· x2
2
x
x
20 − x2 = x
0 = x2 + x − 20
0 = (x + 5)(x − 4)
x−4= 0
x+5= 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
Number of questions: 10
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Joseph Lee
Mod 26 - Exam - Chapter 7
Number of questions: 18
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A Guide to Your Modular Math Course
Joseph Lee
Module 27
Due: Thursday, October 16, 2014, 10:00 P.M.
Contents:
Mod 27 - Chapter 8 Pre-Test
Mod 27 - Sec 8.1 A
Mod 27 - Sec 8.2 A
Mod 27 - Sec 8.3 A
Mod 27 - Sec 8.4 A
Mod 27 - Quiz A - 8.1, 8.2, 8.3, 8.4
Mod 27 - Chapter 8 Pre-Test
Number of questions: 46
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
√
√ m
xm/n = n x = n xm
You will also have to recall the rules of exponents10 :
• Product Rule: xm · xn = xm+n
• Quotient Rule:
xm
= xm−n
xn
• Power Rule: (xm )n = xmn
m
ax
am xm
• Expanded Power Rule:
= m m
by
b y
• Negative Exponent Rule: x−m =
1
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
10
from Section 5.1 or Beginning Algebra Part II
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A Guide to Your Modular Math Course
common denominator:
Joseph Lee
√
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,
Example 2:
√
72 =
√
36 ·
√
√
2=6 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:
p
p
32x5 y 4 = 25 · x5 · y 4
√
= 22 x2 y 2 2x
√
= 4x2 y 2 2x
Example 4:
p
p
3
54x8 y 7 = 3 2 · 33 · x8 · y 7
= 3x2 y 2
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3
2x2 y
A Guide to Your Modular Math Course
Joseph Lee
Mod 27 - Sec 8.4 A
Section 8.4 covers adding, subtracting, and multiplying radicals. You may only add or subtract radicals
with 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
Number of questions: 22
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Joseph Lee
Module 28
Due: Thursday, October 23, 2014, 10:00 P.M.
Contents:
Mod 28 - Sec 8.5 A
Mod 28 - Sec 8.6 A
Mod 28 - Sec 8.7 A
Mod 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 any
fractions in any radicals and any radicals in the denominators. You may get rid of a radical in a denominator
by multiplying both the numerator and denominator by the appropriate radical:
r
√
9
9
=√
5
5
3
=√
5
√
5
3
=√ ·√
5
5
√
3 5
=
5
Here is another example:
√
3 2
1
2
1
√
√
√
=
·
3 2
3
3
2
2
2
√
3
4
=
2
If the denominator has more than one term, you must multiply by the conjugate to rationalize the denominator:
√
3
4+ 2
3
√ =
√ ·
√
4− 2
4− 2 4+ 2
√
12 + 3 2
=
16 − 2
√
12 + 3 2
=
14
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Joseph Lee
Mod 28 - Sec 8.6 A
Section 8.6 covers solving equations involving radicals. To solve an equation with square roots, square
both 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−3=0
x+4=0
x = −4
However, x = −4 is extraneous, as
−4 + 1 6=
x=3
√
−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 throughout
the section:
i2 = −1
√
i = −1
The imaginary unit, i, allows us to simplify the square root of negative numbers:
√
√
√
−12 = −1 · 12
=
√
−1 ·
√
= 2i 3
Mod 28 - Quiz A - 8.5, 8.6, 8.7
Number of questions: 15
Page 58
√
4·3
A Guide to Your Modular Math Course
Joseph Lee
Module 29
Due: Thursday, October 30, 2014, 10:00 P.M.
Contents:
Mod 29 - Exam - Chapter 8
Mod 29 - Chapter 9 Pre-Test
Mod 29 - Sec 9.1 A
Mod 29 - Sec 9.2 A
Mod 29 - Quiz A - 9.1, 9.2
Mod 29 - Exam - Chapter 8
Number of questions: 30
Mod 29 - Chapter 9 Pre-Test
Number of questions: 21
Mod 29 - Sec 9.1 A
Section 9.1 covers the square root principle and completing the square. The square root principle states
that you may take the square root of both sides an equation, but only if you consider both positive
and 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
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Joseph Lee
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
p
√
(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
p
√
(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,
√
−b ± b2 − 4ac
x=
.
2a
Mod 29 - Quiz A - 9.1, 9.2
Number of questions: 11
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Joseph Lee
Module 30
Due: Thursday, November 6, 2014, 10:00 P.M.
Contents:
Mod 30 - Sec 9.3 A
Mod 30 - Sec 9.4 A
Mod 30 - Quiz A - 9.3, 9.4
Mod 30 - Exam - Chapter 9
Mod 30 - Comprehensive Quiz
Mod 30 - Sec 9.3 A
Section 9.3 covers solving equations with rational expressions and radicals and solving equations using a
substitution. 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
Number of questions: 11
Important: Show all your work for this test. Some questions on the test ask you to solve by completing 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 the
quadratic formula: you will need to show your work here as well.
Mod 30 - Exam - Chapter 9
Number of questions: 17
Mod 30 - Comprehensive Quiz
Number of questions: 10 (Multiple Choice)
Page 61