**Unformatted text preview: **Linear Relations
and Functions
Chapter Overview and Pacing
PACING (days)
Regular
Block LESSON OBJECTIVES Basic/
Average Advanced Basic/
Average Advanced Relations and Functions (pp. 56–62)
• Analyze and graph relations.
• Find functional values. 1 optional 0.5 optional Linear Equations (pp. 63–67)
• Identify linear equations and functions.
• Write linear equations in standard form and graph them. 1 optional 0.5 optional Slope (pp. 68–74)
• Find and use the slope of a line.
• Graph parallel and perpendicular lines. 1 optional 0.5 optional Writing Linear Equations (pp. 75–80)
• Write an equation of a line given the slope and a point on the line.
• Write an equation of a line parallel or perpendicular to a given line. 1 optional 0.5 optional 2
(with 2-5
Follow-Up) optional 1 optional Special Functions (pp. 89–95)
• Identify and graph step, constant, and identity functions.
• Identify and graph absolute value and piecewise functions. 1 optional 0.5 optional Graphing Inequalities (pp. 96–99)
• Graph linear inequalities.
• Graph absolute value inequalities. 1 optional 0.5 optional Study Guide and Practice Test (pp. 100–105)
Standardized Test Practice (pp. 106–107) 1 2 0.5 0.5 Chapter Assessment 1 1 0.5 0.5 10 3 5 1 Modeling Real-World Data: Using Scatter Plots (pp. 81–88)
• Draw scatter plots.
• Find and use prediction equations.
Follow-Up: Lines of Regression TOTAL Pacing suggestions for the entire year can be found on pages T20–T21. 54A Chapter 2 Linear Relations and Functions Timesaving Tools
™ All-In-One Planner
and Resource Center Chapter Resource Manager See pages T12–T13. 57–58 59–60 61 62 63–64 65–66 67 68 69–70 71–72 73 74 75–76 77–78 79 80 81–82 83–84 85 86 87–88 89–90 91 92 114 93–94 95–96 97 98 114 Materials 2-1 2-1 2-2 2-2 2-3 2-3 GCS 30,
SC 3 2-4 2-4 SC 4
SM 97–102 2-5 2-5 tape measure, graph paper
(Follow-Up: graphing calculator) GCS 29 2-6 2-6 graphing calculator, toothpicks 2-7 2-7 113 113, 115 5-M
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116–118 *Key to Abbreviations: GCS Graphing Calculator and Speadsheet Masters,
SC School-to-Career Masters,
SM Science and Mathematics Lab Manual Chapter 2 Linear Relations and Functions 54B Mathematical Connections
and Background
Continuity of Instruction
Prior Knowledge
In prior years students worked with coordinate systems, ordered pairs, and linear equations and functions. They manipulated and
solved linear equations and inequalities
algebraically. Also, they used graphs to
represent two-variable
data sets. This Chapter
Students explore algebraic descriptions of
linear functions, graphs of lines, and how to
go back and forth between linear equations
and graphed lines. They find the slope of a
line containing two given points, relate the
slope and y-intercept of a line to the values
m and b in the slope-intercept form of an
equation, and write equations for lines given
two points or given one point and the slope.
They find lines of fit for data and graph
inequalities and special functions such as the
greatest integer function and the
absolute value function. Future Connections
Students will continue to learn how algebraic
expressions and the coordinate plane are
related. At a simple level, they will learn how
the “horizontal line test” identifies a one-toone function. As activities become more complex, they will use graphs to explore quadratic
and other non-linear equations and inequalities and use graphs to represent and solve
systems of equations and inequalities. 54C Chapter 2 Linear Relations and Functions Relations and Functions
This lesson begins an exploration of two important themes of algebra. One theme is how algebraic
equations and the Cartesian coordinate system are
related. The other theme is the relationships between
equations that represent entire lines and numbers that
represent points on or properties of that line.
For this lesson the central idea is ordered pairs.
First, ordered pairs are explored as names for points in a
coordinate plane. Second, several ordered pairs are used
to describe how the set of the first elements (the domain)
can be related to the set of the second elements (the
range). Third, for equations that represent a line or a
curve, ordered pairs are used to determine the graph
that represents that equation in the coordinate plane.
In the lesson the relation between functions and
relations is explored in two ways. Mappings of domain
elements to range elements are used to identify functions
that are one to one, functions that are not one to one,
and relations that are not functions. In the coordinate
plane, the vertical line test is used to distinguish a relation from a function. Relationships between ordered
pairs and functions are explored in two ways. First,
students are given an equation (for a curve or for a line)
and make a table of ordered pairs for the equation.
Second, students are given a function and a domain
value, and evaluate the function to find the range value. Linear Equations
In this lesson students deal with linear functions
and intercepts. Linear functions and equations can be
written in slope-intercept form, f(x) mx b or
y mx + b, or in standard form, Ax By C. The
graph of a linear function or equation is always a line. Slope
Slope is a fundamental concept in algebra and
higher mathematics. In this lesson, students calculate
the slope of a line given two points on the line and
explore the slopes of families or pairs of lines that are
parallel and the slopes of pairs of lines that are
perpendicular.
Students graph a line given two points or given
one point and the slope. In the coordinate plane,
students associate lines with slopes that are positive,
negative, zero, or undefined. Writing Linear Equations
This lesson focuses on the slope and y-intercept of a linear equation. In the slope-intercept form
of a linear equation, y mx b, m represents the
slope of the line and b is the y-intercept.
Students use two forms of a linear equation,
the slope-intercept form and the point-slope form, to
write an equation given two points, given a point and
the slope, or given a point and the equation of a
parallel or a perpendicular line. Special Functions
In this lesson, students explore special functions. The identity and constant functions are special
linear functions. The graph of a step function is a
series of line segments. An absolute value function
has a V-shaped graph made up of portions of two
lines. A piecewise function is a function written using
two or more algebraic expressions. Graphing Inequalities
Modeling Real-World Data:
Using Scatter Plots
This lesson explores equations that approximate the relation between domain values and range
values, extending the idea of using an algebraic equation to represent a set of points in a plane. Starting
with a scatter plot of data, students mentally picture
a line through the data. After selecting two points on
that line, they calculate the slope and y-intercept of
that line. The equation, called a line of fit or a prediction equation, may be used to calculate the value of
one variable given a value of the other.
Activities in this lesson require three steps:
given a set of ordered pairs, students identify a line
that represents a set of ordered pairs; then they select
two ordered pairs that lie on the line; and finally they
calculate the slope and y-intercept for that line. In this lesson the graph of an equation is seen
as the boundary between two regions of the coordinate plane. An inequality is a description of one of
the two regions, and whether the boundary is part of
that region depends on the inequality symbol that is
used. Students explore how inequalities, including
absolute value inequalities, are modeled by points in
the coordinate plane, and vice versa.
Additional mathematical information and teaching notes
are available in Glencoe’s Algebra 2 Key Concepts:
Mathematical Background and Teaching Notes,
which is available at .
The lessons appropriate for this chapter are as follows.
• Linear Relations and Functions (Lesson 5)
• Graphing Linear Equations (Lessons 6 and 12)
• Slope (Lesson 7)
• Writing Linear Equations in Point-Slope and Standard
Forms (Lesson 8)
• Writing Linear Equations in Slope-Intercept Form
(Lesson 10)
• Integration: Geometry/Parallel and Perpendicular
Lines (Lesson 13)
• Statistics: Scatter Plots and Best-Line Fits (Lesson 9)
• Graphing Inequalities in Two Variables (Lesson 17)
Chapter 2 Linear Relations and Functions 54D and Assessment ASSESSMENT INTERVENTION Type Student Edition Teacher Resources Ongoing Prerequisite Skills, pp. 55, 62,
67, 74, 80, 86, 95
Practice Quiz 1, p. 74
Practice Quiz 2, p. 95 5-Minute Check Transparencies
Quizzes, CRM pp. 113–114
Mid-Chapter Test, CRM p. 115
Study Guide and Intervention, CRM pp. 57–58,
63–64, 69–70, 75–76, 81–82, 87–88, 93–94 Mixed
Review pp. 62, 67, 74, 80, 86, 95, 99 Cumulative Review, CRM p. 116 Error
Analysis Find the Error, pp. 60, 71 Find the Error, TWE pp. 60, 71
Unlocking Misconceptions, TWE p. 58
Tips for New Teachers, TWE pp. 62, 74, 90 Standardized
Test Practice pp. 62, 67, 74, 76, 78, 80, 86,
95, 99, 105, 106–107 TWE p. 76
Standardized Test Practice, CRM pp. 117–118 Open-Ended
Assessment Writing in Math, pp. 62, 67, 73,
80, 86, 94, 99
Open Ended, pp. 60, 65, 71, 78,
83, 92, 98 Modeling: TWE pp. 67, 74, 95
Speaking: TWE pp. 62, 98
Writing: TWE pp. 80, 86
Open-Ended Assessment, CRM p. 111 Chapter
Assessment Study Guide, pp. 100–104
Practice Test, p. 105 Multiple-Choice Tests (Forms 1, 2A, 2B),
CRM pp. 99–104
Free-Response Tests (Forms 2C, 2D, 3),
CRM pp. 105–110
Vocabulary Test/Review, CRM p. 112 Technology/Internet
Alge2PASS: Tutorial Plus
Standardized Test Practice
CD-ROM
standardized_test TestCheck and Worksheet Builder
(see below)
MindJogger Videoquizzes
vocabulary_review
Key to Abbreviations: TWE = Teacher Wraparound Edition; CRM = Chapter Resource Masters Additional Intervention Resources
The Princeton Review’s Cracking the SAT & PSAT
The Princeton Review’s Cracking the ACT
ALEKS TestCheck and Worksheet Builder
This networkable software has three modules for intervention
and assessment flexibility:
• Worksheet Builder to make worksheet and tests
• Student Module to take tests on screen (optional)
• Management System to keep student records (optional)
Special banks are included for SAT, ACT, TIMSS, NAEP, and
End-of-Course tests. 54E Chapter 2 Linear Relations and Functions Reading and Writing
in Mathematics
Intervention Technology
Alge2PASS: Tutorial Plus CD-ROM offers a
complete, self-paced algebra curriculum.
Algebra 2
Lesson Alge2PASS Lesson 2-2 3 Graphing Linear Equations on the
Coordinate Plane 2-7 4 Graphing Linear Inequalities on the
Coordinate Plane ALEKS is an online mathematics learning system that
adapts assessment and tutoring to the student’s needs.
Subscribe at . Glencoe Algebra 2 provides numerous opportunities to
incorporate reading and writing into the mathematics
classroom.
Student Edition
• Foldables Study Organizer, p. 55
• Concept Check questions require students to verbalize
and write about what they have learned in the lesson.
(pp. 60, 65, 71, 78, 83, 92, 98, 100)
• Writing in Math questions in every lesson, pp. 62, 67,
73, 80, 86, 94, 99
• Reading Study Tip, pp. 56, 59, 71, 82
• WebQuest, p. 84
Teacher Wraparound Edition Intervention at Home
Log on for student study help.
• For each lesson in the Student Edition, there are Extra
Examples and Self-Check Quizzes.
• For chapter review, there is vocabulary review, test
practice, and standardized test practice.
For more information on Intervention and
Assessment, see pp. T8–T11. • Foldables Study Organizer, pp. 55, 100
• Study Notebook suggestions, pp. 60, 65, 71, 78, 83,
93, 97
• Modeling activities, pp. 67, 74, 95
• Speaking activities, pp. 62, 98
• Writing activities, pp. 80, 86
• Differentiated Instruction, (Verbal/Linguistic), p. 92
• ELL Resources, pp. 54, 61, 66, 73, 79, 85, 92, 94,
99, 100
Additional Resources
• Vocabulary Builder worksheets require students to
define and give examples for key vocabulary terms as
they progress through the chapter. (Chapter 2 Resource
Masters, pp. vii-viii)
• Reading to Learn Mathematics master for each lesson
(Chapter 2 Resource Masters, pp. 61, 67, 73, 79, 85,
91, 97)
• Vocabulary PuzzleMaker software creates crossword,
jumble, and word search puzzles using vocabulary lists
that you can customize.
• Teaching Mathematics with Foldables provides
suggestions for promoting cognition and language.
• Reading and Writing in the Mathematics Classroom
• WebQuest and Project Resources For more information on Reading and Writing in
Mathematics, see pp. T6–T7.
Chapter 2 Linear Relations and Functions 54F Linear Relations
and Functions Notes
Have students read over the list
of objectives and make a list of
any words with which they are
not familiar. • Lesson 2-1 Analyze relations and functions.
• Lessons 2-2 and 2-4 Identify, graph, and write
linear equations.
• Lesson 2-3 Find the slope of a line.
• Lesson 2-5 Draw scatter plots and find prediction
equations.
• Lessons 2-6 and 2-7 Graph special functions, linear
inequalities, and absolute value inequalities. Point out to students that this is
only one of many reasons why
each objective is important.
Others are provided in the
introduction to each lesson. Key Vocabulary
•
•
•
•
• linear equation (p. 63)
linear function (p. 63)
slope (p. 68)
slope-intercept form (p. 75)
point-slope form (p. 76) Linear equations can be used to model relationships between many
real-world quantities. One of the most common uses of a linear model
is to make predictions.
Most hot springs are the result of groundwater passing through
or near recently formed, hot, igneous rocks. Iceland, Yellowstone
Park in the United States, and North Island of New Zealand are
noted for their hot springs. You will use a linear equation to find the
temperature of underground rocks in Lesson 2-2. Lesson
2-1
2-2
2-3
2-4
2-5
2-5
Follow-Up
2-6
2-7 NCTM
Standards Local
Objectives 1, 2, 7, 9, 10
1, 2, 4, 6, 8, 9
1, 2, 4, 6, 7, 8,
9, 10
1, 2, 6, 8, 9, 10
1, 2, 5, 6, 8, 9,
10
1, 2, 4, 5, 6, 9,
10
1, 2, 5, 6, 8, 9,
10
1, 2, 6, 8, 9, 10 Key to NCTM Standards:
1=Number & Operations, 2=Algebra,
3=Geometry, 4=Measurement,
5=Data Analysis & Probability, 6=Problem
Solving, 7=Reasoning & Proof,
8=Communication, 9=Connections,
10=Representation
54 54 Chapter 2 Linear Relations and Functions Vocabulary Builder ELL The Key Vocabulary list introduces students to some of the main vocabulary terms
included in this chapter. For a more thorough vocabulary list with pronunciations of
new words, give students the Vocabulary Builder worksheets found on pages vii and
viii of the Chapter 2 Resource Masters. Encourage them to complete the definition
of each term as they progress through the chapter. You may suggest that they add
these sheets to their study notebooks for future reference when studying for the
Chapter 2 test. Chapter 2 Linear Relations and Functions Prerequisite Skills To be successful in this chapter, you’ll need to master
these skills and be able to apply them in problem-solving situations. Review
these skills before beginning Chapter 2.
For Lesson 2-1 Identify Points on a Coordinate Plane Write the ordered pair for each point.
1. A (3, 3) 2. B (2, 3) 3. C (3, 1) 4. D (2, 0) 5. E (0, 4) 6. F (3, 2) y A B
D C x O F
E For Lesson 2-1 Evaluate Expressions Evaluate each expression if a 1, b 3, c 2, and d 0.
7. c d 2 8. 4c b 11 9. a2 5a 3 9
ac
12. 3 ab
11. 2 10. 2b2 b 7 28 (For review, see Lesson 1-1.) bc cd For Lesson 2-4 Simplify Expressions Simplify each expression. (For review, see Lesson 1-2.)
13. x (1) x 1 14. x (5) x 5
1
1
17. [x (4)] x 2
2
2 16. 4[x (2)] 4x 8 For Lessons 2-6 and 2-7 This section provides a review of
the basic concepts needed before
beginning Chapter 2. Page
references are included for
additional student help.
Prerequisite Skills in the Getting
Ready for the Next Lesson section
at the end of each exercise set
review a skill needed in the next
lesson.
For
Lesson
2-2
2-4
2-5
2-6
2-7 Prerequisite
Skill
Solving Equations (p. 62)
Solving Equations (p. 74)
Finding a Median (p. 80)
Absolute Value (p. 86)
Inequalities (p. 95) 15. 2[x (3)] 2x 6
1
1
18. [x (6)] x 2
3
3 Evaluate Expressions with Absolute Value Evaluate each expression if x = 3, y = 4, and z = 4.5. (For review, see Lesson 1-4.)
19. x 3 20. y 4 21. 5x 15 22. 2z 9 23. 5y z 2.5 24. 3x y x z 10.5 Make this Foldable to help you organize information about
relations and functions. Begin with two sheets of grid paper.
Fold Cut and Label Fold in half
along the width
and staple along
the fold. Cut the top
three sheets and
label as shown. Graphing Graphing
Linear
Linear
Relations Functions Reading and Writing As you read and study the chapter, write notes, examples,
and graphs under the tabs. Chapter 2 Linear Relations and Functions 55 TM For more information
about Foldables, see
Teaching Mathematics
with Foldables. Organization of Data: Annotating As students read and work
their way through the chapter, have them make annotations under
the appropriate tabs of their Foldable. Explain to them that
annotations are usually notes taken in the margins of books, which
we own, to organize the text for review or studying. Annotations
often include questions that arise, reader comments and reactions,
short summaries, steps or data numbered by the reader, and key
points highlighted or underlined.
Chapter 2 Linear Relations and Functions 55 Lesson
Notes 5-Minute Check
Transparency 2-1 Use as a
quiz or review of Chapter 1.
Mathematical Background notes
are available for this lesson on
p. 54C. do relations and functions apply to biology?
Ask students:
• What is the difference between
average lifetime and maximum
lifetime? The average lifetime is
a representative number of years
for any animal of that type, while
the maximum lifetime is the
greatest age ever attained by an
animal of that type.
• Why can you be sure that the
second number in the ordered
pairs for this data is always
greater than or equal to the
first? For each animal, the
maximum age will always equal or
exceed the average age. Vocabulary
• ordered pair
• Cartesian coordinate
plane
• quadrant
• relation
• domain
• range
• function
• mapping
• one-to-one function
• vertical line test
• independent variable
• dependent variable
• functional notation do relations and functions apply to biology?
The table shows the average lifetime
and maximum lifetime for some
animals. The data can also be
represented as ordered pairs . The
ordered pairs for the data are (12, 28),
(15, 30), (8, 20), (12, 20), and (20, 50).
The first number in each ordered pair
is the average lifetime, and the second
number is the maximum lifetime.
(12, 28)
average
lifetime ← In Chapter 1, students solved
equations and inequalities. In
this lesson, students relate
equations to functions and
relations, as well as to their
graphs. • Find functional values. ← Building on Prior
Knowledge • Analyze and graph relations. Average
Lifetime
(years) Maximum
Lifetime
(years) Cat 12 28 Cow 15 30 Deer 8 20 Dog 12 20 Horse 20 50 Source: The World Almanac GRAPH RELATIONS You can graph the
ordered pairs above by creating a coordinate
system with two axes. Each point represents
one of the ordered pairs above. Remember that
each point in the coordinate plane can be named
by exactly one ordered pair and that every
ordered pair names exactly one point in the
coordinate plane.
The graph of the animal lifetime data lies
in only one part of the Cartesian coordinate
plane—the part with all positive numbers. The
Cartes...

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