Algebra I, Teacher’s Edition

2011·0.65 MB·English

by  Mary Fay-Zenk

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CK-12 Algebra I Teacher’s
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Printed: August 9, 2011
Authors
Mary Fay-Zenk, Andrew McFarland
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Contents
1 TE Equations and Functions 1
1.1 Variable Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Patterns and Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Equations and Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Functions as Rules and Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Functions as Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Problem-Solving Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.8 Problem-Solving Strategies: Make a Table; Look for a Pattern. . . . . . . . . . . . . . . . . 12
2 TE Real Numbers 14
2.1 Integers and Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Addition of Rational Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.3 Subtraction of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Multiplication of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 The Distributive Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Division of Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Square Roots and Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8 Problem-Solving Strategies: Guess and Check; Work Backward . . . . . . . . . . . . . . . . 28
3 TE Equations of Lines 29
3.1 One-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Two-Step Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Multi-Step Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Equations with Variables on Both Sides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5 Ratios and Proportions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 Scale and Indirect Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Percent Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.8 Problem-Solving Strategies: Use a Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
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4 TE Graphs of Equations and
Functions 44
4.1 The Coordinate Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.2 Graphs of Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Graphing Using Intercepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Slope and Rate of Change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.5 Graphs Using Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Direct Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.7 Linear Function Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.8 Problem-Solving Strategies - Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5 TE Writing Linear Equations 56
5.1 Linear Equations in Slope-Intercept Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Linear Equations in Point-Slope Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Linear Equations in Standard Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.4 Equations of Parallel and Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.5 Fitting a Line to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.6 Predicting with Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.7 Problem Solving Strategies: Use a Linear Model . . . . . . . . . . . . . . . . . . . . . . . . 64
6 TE Graphing Linear
Inequalities 66
6.1 Inequalities Using Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
6.2 Inequalities Using Multiplication and Division . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3 Multi-Step Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.4 Compound Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.5 Absolute Value Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.6 Absolute Value Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.7 Linear Inequalities in Two Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 TE Solving Systems of
Equations and Inequalities 79
7.1 Linear Systems by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.2 Solving Linear Systems by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.3 Solving Linear Systems by Elimination through Addition or Subtraction . . . . . . . . . . . 84
7.4 Solving Systems of Equations by Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 85
7.5 Special Types of Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
7.6 Systems of Linear Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
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8 TE Exponential Functions 90
8.1 Exponent Properties Involving Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2 Exponent Properties Involving Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.3 Zero, Negative, and Fractional Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8.4 Scientific Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
8.5 Exponential Growth Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
8.6 Exponential Decay Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.7 Geometric Sequences and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 100
8.8 Problem-Solving Strategies (reprise Make a Table; Look for a Pattern) . . . . . . . . . . . . 102
9 TE Factoring Polynomials 103
9.1 Addition and Subtraction of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
9.2 Multiplication of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.3 Special Products of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
9.4 Polynomial Equations in Factored Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
9.5 Factoring Quadratic Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
9.6 Factoring Special Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
9.7 Factoring Polynomials Completely . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
10 TE Quadratic Equations and
Quadratic Functions 116
10.1 Graphs of Quadratic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
10.2 Quadratic Equations by Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
10.3 Quadratic Equations by Square Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10.4 Quadratic Equations by Completing the Square . . . . . . . . . . . . . . . . . . . . . . . . . 122
10.5 Quadratic Equations by the Quadratic Formula . . . . . . . . . . . . . . . . . . . . . . . . . 125
10.6 The Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
10.7 Linear, Exponential and Quadratic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
10.8 Problem Solving Strategies: Choose a Function Model . . . . . . . . . . . . . . . . . . . . . 128
11 TE Algebra and Geometry
Connections; Working with Data 130
11.1 Graphs of Square Root Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.2 Radical Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
11.3 Radical Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.4 The Pythagorean Theorem and Its Converse . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
11.5 Distance and Midpoint Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.6 Measures of Central Tendency and Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 140
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11.7 Stem-and-Leaf Plots and Histograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
11.8 Box-and-Whisker Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
12 TE Rational Equations and
Functions; Topics in
Statistics 146
12.1 Inverse Variation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
12.2 Graphs of Rational Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
12.3 Division of Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
12.4 Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
12.5 Multiplication and Division of Rational Expressions . . . . . . . . . . . . . . . . . . . . . . 152
12.6 Addition and Subtraction of Rational Expressions . . . . . . . . . . . . . . . . . . . . . . . 153
12.7 Solutions of Rational Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
12.8 Surveys and Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
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Chapter 1
TE Equations and Functions
Overview
Equations and Functions consists of eight lessons that introduce students to the language of algebra.
Suggested Pacing:
Variable Expressions - 1 hr
Order of Operations - 1 hr
Patterns and Equations - 1−2 hrs
Equations and Inequalities - 1−2 hrs
Functions as Rules and Tables - 0:5 hrs
Functions as Graphs - 1 hr
Problem-Solving Plan - 0:5 hr
Problem-Solving Strategies: - 2 hrs
Make a Table; Look for a Pattern
If you would like access to the Solution Key FlexBook for even-numbered exercises, the Assessment Flex-
Book and the Assessment Answers FlexBook please contact us at [email protected].
Problem-Solving Strand for Mathematics
The problem-solving strategies presented in this chapter, Make a Table and Look for Patterns, are foun-
dational techniques. Making a Table can help structure a student’s ability to organize and clarify the data
presented and teach the student to communicate more clearly. When teaching Look for a Pattern, ask
questions such as:
• Can you identify a pattern in the given examples that would let you extend the data?
• Do you observe any pattern that applies to all the given examples?
• Can you find a relationship or operation(s) that would allow you to predict another term?
Alignment with the NCTM Process Standards
Two promising practices, focused on the communication standards, can be effectively used with these
strategies. The first is setting aside a time on a regular basis (i.e. once a week) to have students write
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about the problem solving they have been doing (COM.2). Present a daily warm-up problem which is well
suited to the strategy being learned such as Look for a Pattern. Students work on one problem a day,
first individually and then as a group with teacher leadership and summation. Each day the problem is
discussed and solved, and many different points of view are shared in the process (COM.1, COM.3). At
the end of the week, students are asked to write about any one of the problems they did earlier in the
week. This practice pushes students to develop logical thinking skills, to benefit from the classroom work
that was shared earlier in the week, and to learn to communicate mathematical ideas clearly (COM.4). It
also gives students a choice; they only have to write about one of the problems, and it can be the problem
that made the most sense to them. Oftentimes, unfortunately, we do not honor enough what makes sense
to students; we expect them only to be able to follow the logic presented to them. We must give them
experiences of “sense-making” as well (RP.1, RP.2).
A second practice is posting student work, such as:
• gallery walks where work in progress can be viewed
• posters that highlight exceptionally well done solutions
• student essays posted on the classroom wall
What matters is that students’ work is valued and shared (RP.3). If these pieces are large enough to be
seen from a distance, this practice helps students to recall various approaches to solving problems (RP.4)
and keeps their thinking “alive.”
• COM.1 - Organize and consolidate their mathematical thinking through communication.
• COM.2 - Communicate their mathematical thinking coherently and clearly to peers, teachers, and
others.
• COM.3 - Analyze and evaluate the mathematical thinking and strategies of others.
• COM.4 - Use the language of mathematics to express mathematical ideas precisely.
• RP.1 - Recognize reasoning and proof as fundamental aspects of mathematics.
• RP.2 - Make and investigate mathematical conjectures.
• RP.3 - Develop and evaluate mathematical arguments and proofs.
• RP.4 - Select and use various types of reasoning and methods of proof.
1.1 Variable Expressions
Learning Objectives
At the end of this lesson, students will be able to:
• Evaluate algebraic expressions.
• Evaluate algebraic expressions with exponents.
Vocabulary
Terms introduced in this lesson:
algebra
generalize
variables
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