Carnegie Learning
Algebra I
© 2012 Carnegie Learning
Teacher’s
Implementation Guide
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Copyright © 2012 by Carnegie Learning, Inc. All rights reserved. Carnegie Learning, Cognitive
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Inc. All other company and product names mentioned are used for identification purposes only
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expressed consent of the publisher.
© 2012 Carnegie Learning
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ISBN: 978-1-60972-155-8
Teacher’s Implementation Guide, Volume 2
Printed in the United States of America
1-05/2012 HPS
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Dear Teacher,
It is our goal to provide you with instructional materials to support your implementation of the
Common Core State Standards for Mathematics (CCSS) and the Standards for Mathematical
Practice (SMP). At Carnegie Learning, we analyzed the CCSS and coupled it with the best
academic research on teaching and learning practices. The results: a text that introduces and
develops mathematical concepts with coherence, combats common student misconceptions, and
accommodates all 9th Grade students as well as a variety of classroom implementations.
The CCSS and SMP are a call for change. It is our responsibility as educators to create a safe
environment to learn, provide appropriate instructional materials, and believe that all students
can achieve academic excellence and become productive mathematical thinkers. To produce
successful learners, we must support students’ effective communication
These
through dialogue and discussion of different strategies. This textbook
resources are
encourages active engagement through a student-centered classroom
designed to align
teaching to
environment, which inspires students to learn from each other. It is our
learning.
intent that students become knowledgeable and independent learners.
© 2012 Carnegie Learning
We realize that students enter your classroom with varying degrees of
mathematical experience and success. Prior knowledge that is fragmented or based
on memorization rather than a deep conceptual understanding is an unstable
foundation for developing mathematical relationships and concepts. This text is
intentionally designed to help students make connections, develop a
conceptual understanding of mathematics, and Learn by Doing™. Key
formative assessment questions geared toward student comprehension are
embedded throughout each lesson.
It is our recommendation that you take the time at the beginning of each
chapter to do the math yourself. This will provide you the first-hand experience
necessary to make informed instructional decisions about which parts of the lesson
will drive your mathematical goals.
Yours in Education,
The Carnegie Learning ®Curriculum Development Team
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Acknowledgments
Carnegie Learning
Authoring Team
• Sandy Bartle
• David Dengler
Senior Academic Officer Director, Curriculum Development
• Joshua Fisher
• Jen Dilla
Math Editor Editorial Assistant
• David “Augie” Rivera
• Lezlee Ross
Math Editor Curriculum Developer
Acknowledgments
Contributing Authors
• Jaclyn Snyder
• Dr. Mary Lou Metz
Vendors
• Cenveo Publisher Services
• Mathematical Expressions
• Hess Print Solutions
• Bradford & Bigelow
• Mind Over Media
• Lapiz
• eInstruction
Special Thanks
and content.
• Teacher reviewers and students for their input and review of lesson content.
• Carnegie Learning Software Development Team for their contributions to
research and content.
• William S. Hadley for being a mentor to the development team, his leadership,
and his pedagogical pioneering in mathematics education.
• Amy Jones Lewis for her review of content.
FM-4 © 2012 Carnegie Learning
• Carnegie Learning Managers of School Partnerships for their review of design
Acknowledgments
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© 2012 Carnegie Learning
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Table of Contents
8
Analyzing Data
Sets for One Variable
8.1
453
Start Your Day the Right Way
Graphically Representing Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
8.2
Which Measure Is Better?
Determining the Best Measure of Center for a Data Set . . . . . . . . . . . . . . . . . . . 469
8.3
You Are Too Far Away!
Acknowledgments
Table of Contents
Calculating IQR and Identifying Outliers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
8.4
Whose Scores Are Better?
Calculating and Interpreting Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . 489
8.5
Putting the Pieces Together
Analyzing and Interpreting Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
Chapter 8 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
9
Correlation and Residuals
9.1
521
Like a Glove
9.2
Gotta Keep It Correlatin’
Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
9.3
The Residual Effect
Creating Residual Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
9.4
To Fit or Not To Fit? That Is The Question!
Using Residual Plots. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
9.5
Who Are You? Who? Who?
© 2012 Carnegie Learning
Least Squares Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
Causation vs. Correlation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563
Chapter 9 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
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10
Analyzing Data Sets for
Two Categorical Variables
577
10.1 Could You Participate in Our Survey?
Interpreting Frequency Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
10.2 It’s So Hot Outside!
Relative Frequency Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
10.3 She Blinded Me with Science!
Relative Frequency Conditional Distribution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
10.4 Oh! Switch the Station!
Drawing Conclusions from Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601
Chapter 10 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
Introduction to
Quadratic Functions
Table of Contents
11 615
11.1 Up and Down or Down and Up
Exploring Quadratic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
11.2 Just U and I
Comparing Linear and Quadratic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625
11.3 Walking the . . . Curve?
Domain, Range, Zeros, and Intercepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 637
11.4 Are You Afraid of Ghosts?
Factored Form of a Quadratic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645
© 2012 Carnegie Learning
11.5 Just Watch that Pumpkin Fly!
Investigating the Vertex of a Quadratic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . 653
11.6 The Form Is “Key”
Vertex Form of a Quadratic Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
11.7 More Than Meets the Eye
Transformations of Quadratic Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675
Chapter 11 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
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12
Polynomials and Quadratics
701
12.1 Controlling the Population
Adding and Subtracting Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
12.2 They’re Multiplying—Like Polynomials!
Multiplying Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 717
12.3 What Factored Into It?
Factoring Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 731
12.4 Zeroing In
Solving Quadratics by Factoring. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743
12.5 What Makes You So Special?
Special Products. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 751
12.6 Could It Be Groovy to Be a Square?
Approximating and Rewriting Radicals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 761
Table of Contents
12.7 Another Method
Completing the Square. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 771
Chapter 12 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 779
13
Solving Quadratic
Equations and Inequalities
787
13.1 Ladies and Gentlemen: Please Welcome
the Quadratic Formula!
The Quadratic Formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 789
Using a Calculator-Based Ranger to Model Quadratic Motion . . . . . . . . . . . . . . 803
13.3 They’re a Lot More Than Just Sparklers!
Solving Quadratic Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 813
13.4 You Must Have a System
Systems of Quadratic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 821
Chapter 13 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 829
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13.2 It’s Watching and Tracking!
Table of Contents
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14
Real Number Systems
835
14.1 The Real Numbers . . . For Realsies!
The Numbers of the Real Number System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837
14.2 Getting Real, and Knowing How . . .
Real Number Properties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845
14.3 Imagine the Possibilities
Imaginary and Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 851
14.4 It’s Not Complex—Just Its Solutions Are Complex!
Solving Quadratics with Complex Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 859
Chapter 14 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 869
Other Functions and Inverses
875
Table of Contents
15
15.1 I Graph in Pieces
Linear Piecewise Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 877
15.2 Step By Step
Step Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885
15.3 The Inverse Undoes What a Function Does
Inverses of Linear Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895
15.4 Taking the Egg Plunge!
Inverses of Non-Linear Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 909
© 2012 Carnegie Learning
Chapter 15 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 921
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16 Mathematical Modeling
929
16.1 People, Tea, and Carbon Dioxide
Modeling Using Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931
16.2 Stop! What Is Your Reaction?
Modeling Stopping Distances and Reaction Times. . . . . . . . . . . . . . . . . . . . . . . 939
16.3 Modeling Data Helps Us Make Predictions
Using Quadratic Functions to Model Data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 947
16.4 BAC Is BAD News
Choosing a Function to Model BAC. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 957
16.5 Cell Phone Batteries, Gas Prices,
and Single Family Homes
Modeling with Piecewise Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963
© 2012 Carnegie Learning
Table of Contents
Chapter 16 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971
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© 2012 Carnegie Learning
Table of Contents
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The Research
Dr. Ritter Introduces the Research Behind the
Carnegie Learning Products
Both our textbooks and Cognitive Tutor Software have been strongly influenced by
research into how students learn and how to best motivate students to succeed
academically. The research addressed in the solution includes:
The Research
Motivational Research
Dr. Steve Ritter, Co-Founder
and Chief Cognitive Scientist,
Carnegie Learning
Some students may lose interest in academic success, which can be attributed to
students’ alienation, resulting from a feeling that the school environment is unwelcoming
to them. Recent research identifies several elements of this alienation, as well as
practices that can re-focus students on academic achievement.
Students who believe that they can get smarter will work harder.
Teaching students about the way that the brain changes as they learn has been
shown to encourage students to believe that they have the capability to learn.
Those students who consider long-term learning as their goal may learn more
flexibly. In the Software we provide feedback focused on success and use badges
to reward effective learning behaviors.
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Students who approach a task with the intention of succeeding (rather
than avoiding failure) are more likely to excel.
The Research
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Learning Research
Carnegie Learning does not just read the research on how students learn; we are active
participants in this research and frequently publish results in peer-reviewed journals and
conferences. The Carnegie Learning products incorporate a wide variety of activities and
approaches that represent the result of decades of this research, including:
Active Learning
The research makes clear that students need to actively engage with content if they
are to benefit from it. All of our activities within the text and Software encourage
students to consider hypotheses and conclusions from different perspectives, and
build deep understandings of mathematics.
Worked Examples
The Research
Research shows that learning is best supported with a mix of problem solving and
worked examples. Algebra I includes extensive worked examples as well as multiple
examples of student work to encourage comparison and self-explanation.
Fluency Tasks
Success in mathematics builds upon the ability to fluently recognize mathematical
relationships. The Software includes game-like fluency tasks to help students build
representational and procedural fluency.
© 2012 Carnegie Learning
Software
Cognitive Tutor Tasks have long been the heart of the research-based approach
behind Carnegie Learning’s software. These tasks emphasize problem solving and
build a cognitive model of each student’s abilities in order to provide them with
appropriate pacing and tasks.
The Research 8068_TIG_FM_00i-liv_Vol2.indd 13
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Product Introduction
Introducing
Each course includes a textbook and Cognitive Tutor Software. Each course provides a complete set
of research-based educational materials to inspire all high school students to master mathematical
concepts and skills. The instructional materials align to the Common Core State Standards
(CCSS) for high school mathematics, which define what students should know and be able to do at
each grade level. Our pedagogical approach focuses on how students think, learn, and apply new
knowledge in mathematics and empowers them to take ownership of their learning. This approach is
Product Introduction
consistent with the Common Core Standards for Mathematical Practice.
The primary goal of the texts is to get students to think! We recognize the responsibility of providing
instruction that respects the research on how students learn mathematics and believe in a continuous
improvement model. Research can be difficult to implement in a practical way. Research shows that
there is no magic bullet—there are no shortcuts—learning math requires mental effort. Our materials
are designed to provide students with the appropriate tools to think deeply about mathematics and
Our goal
is to support
your team of teachers,
coaches, and leaders to
obtain the results your
students deserve.
FM-14 © 2012 Carnegie Learning
fluently execute the procedures.
Product Introduction
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3
Big Ideas
explain why we’re different . . .
The Carnegie Learning solutions inspire students to succeed in high school mathematics
through a combination of collaborative classroom activities, adaptive software, and
teacher professional development. The instructional materials help students gain a deep
understanding of the mathematics that they will need to succeed in school and in life.
© 2012 Carnegie Learning
2
3
students’ beliefs about the nature of intelligence, their goals within a learning task, and their
perception of academic expectations have strong effects on their academic performance.
Each course includes elements designed to guide your students toward appropriate and
effective attitudes about learning.
Promote Deep Conceptual Understanding. Deep understanding means that concepts are
well represented and well connected to other concepts. Each course makes extensive use of
models­­—real-world situations, manipulatives, graphs, and diagrams, among others—to help
students see the connections between different topics. Your students will view mathematics as
a set of related topics as opposed to a set of discrete topics.
Powerful, On-Going Formative Assessment. Formative assessment is a reflective
process that promotes student learning. It is the part of instruction designed to provide
crucial feedback for you and your students—to diagnose, not to assign a grade. Each
course provides ongoing opportunities for students to be active participants in the learning
process by expressing their knowledge and ideas to you, to their peers, and themselves.
Product Introduction 8068_TIG_FM_00i-liv_Vol2.indd 15
Product Introduction
1
Engage and Motivate. Recent research regarding academic achievement shows that
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Carnegie Learning courses include:
Product Introduction
Texts
Cognitive Tutor Software
Carnegie Learning Math Texts help
Carnegie Learning offers math
students make connections among
software solutions powered by
different mathematical concepts and
Cognitive Tutor. This software features
understand relationships. Students
the most precise method of
build on prior knowledge and obtain
differentiated instruction available.
new knowledge by solving real-world
Our adaptive solutions individualize
problems. This Learning By Doing®
instruction based on how students
approach helps them develop a deep
learn. Our engagement features, such
understanding of mathematics.
as interest areas, characters, and
Students will construct and interpret
choice options, help hold students’
mathematical models, and explain
interest while they are using it. Our
their reasoning as they build a
motivational aspects like badges,
solid foundation for success in high
message-of-the-day, and dynamic
school mathematics.
map keep them coming back for more.
Professional Development
Carnegie Learning offers professional development solutions for both our text and
Cognitive Tutor Software implementations. Carnegie Learning is working side-by-side
with schools and districts implementing our curricula. We are dedicated to partnering
with you to increase teacher effectiveness and student achievement in mathematics.
As you work with our professional services team to build a standards-based,
student-centered classroom, and effectively integrate technology to inform
sustain student achievement.
FM-16 © 2012 Carnegie Learning
data-driven instruction, your district will build the capacity you need to raise and
Product Introduction
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Student Text Materials
Student Edition
The student edition is the primary classroom resource. This consumable text is a record of the
students’ thinking, reasoning, and problem solving. The student lessons interleave questions,
instruction, and worked examples to engage students as they develop their own mathematical
understanding. The lessons are structured to provide students with various opportunities to reason,
to model, and to explain mathematical ideas.
Student Assignments
The student assignments provide opportunities for students to practice and apply their understanding
of the mathematical objectives addressed in the corresponding student lesson. Assignments mirror
the types of problems presented in the student lesson and focus on further developing students’
ability to make sense of problems, reason abstractly, and persevere in problem solving.
Product Introduction
Student Skills Practice
The skills practice worksheets are a supplemental resource to provide targeted practice of discrete skills
within each student lesson. Each skills practice worksheet contains two sections—vocabulary and
problem sets. The vocabulary section provides additional practice with the key terminology of the lesson
through a variety of tasks such as matching, fill-in-the-blank, and identifying similarities and differences.
The problem sets should be assigned as needed based on formative assessment. The solution for the first
question in each problem set is provided as a worked example to help students. The answers for the odd
questions are provided in the back of their workbook.
Student Resource Center
The student edition, student assignments, and student skills practice are available to students and
© 2012 Carnegie Learning
parents online for viewing and printing.
Each student
is given a
consumable textbook
that they can write in, take
notes, highlight key
information, and solve
a problem.
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Teacher Text Materials
Teacher’s Implementation Guide
9. Identify the
slope
_______ ________ ______
Lesson 4.3
Name
guiding, and facilitating student learning. Additional questions
are provided for the teacher to ask during the discuss and share
term
on the preceding
a sequence based
ses each term of
recursive formula expres
You may have
1. A(n)
noticed that when
you transformed
to isolate
ce. the a or
of the sequence.
the equations
s variables, the
a term in a sequen
is the position of
position $3000 goal “disap
Where did the
index
term’s
the
ce using money go? Let’s
peared .”
2. The
a sequen
perform unit analys
each part of one
tes each term of
is on
explicit formula calcula
isolated a equat
3. A(n)
ion to see just
$3000 went .
where that
ce.
sequen
in the
12. Identify the
units of measu
explicit formulare. for each part of the
5susing
the
1
Problem Set
10a
ce
5
3000 .
equation,
tic sequen
the given arithme
unknown term in
of the sequence
Determine each
ine5the 30th term
2. Determa.
sequence
...
20th term of the
210, 215, 220,
dollars
1. Determine the
b. s
1)
studen
2
d(n
.
.
t
1
.
tickets
a
7,
5
4,
an
1,
1
student tickets
2 1)
1)
an 5 a1 1 d(n 2
a30 5 210 1 (25)(30
)
1)
a20 5 1 1 3(20 2
a30 5 210 1 (25)(29
a20 5 1 1 3(19)
a30 5 210 1 (2145)
c. 10
a20 5 1 1 57
a30 5 2155
dollars
d. a
a20 5 58
adult tickets
phases of each student lesson. A lesson map provides a lesson
overview, pacing suggestions, learning goals, key terms, essential
4
ideas, Common Core State Standards covered, and a warm-up
3
remember, when
you’re doing
analysis, you have tounit
identify
the units for each part
the equation! of
ie Learning
______________
© 2012 Carnegie
Learning
© 2012 Carneg
to assist with implementation. A “Check for Understanding”
can help teachers quickly ascertain which students comprehend
Skills Practice
10. Compare
the x-intercepts
and the y-inte
What do you Date
rcepts of the
notice?
two graphs you
just created .
The intercepts
are opposites
of each other.
y-intercept of
The x-intercept
the second graph
of the first graph
, and the y-inte
x-intercept of
is the
rcept of the first
the second graph
graph is the
.
Curious Thing
a
Is
bra
Alge
The Power of
11. Is there a
s of a Sequence
Term
mine
way
to
Deter
determine the
to
total amount
Explain why or
Using Formulas
of money collec
why not .
ted from either
graph?
I can determ
ine when the
group raises
cannot determ
$3000 as where
ine an exact
Vocabulary
the graph lies.
amount of mone
on the graph
However, I
tes each statement.
y for any point
.
that best comple
that does not
Choose the term
lie directly
recursive formula
explicit formula
index
The Teacher’s Implementation Guide is a resource for planning,
question is provided at the end of each lesson. These questions
of the graph .
Interpret its meani
y2 2 y
ng in terms of
1
the problem situati
600 2 0
600
x2 2 x 5
on .
1
0 2 300 5 2300 5 22
The slope is
22. This mean
s that for every
adult ticket needs
two more stude
to be sold.
nt tickets sold,
1 fewer
____________
ce
sequen
25th term of the
3. Determine the
3.3, 4.4, 5.5, . . .
1)
an 5 a1 1 d(n 2
2 1)
a25 5 3.3 1 1.1(25
a 5 3.3 1 1.1(24)
the concepts and instruction, and which may need more time to
25
a25 5 3.3 1 26.4
a25 5 29.7
sequence
50th term of the
4. Determine the
100, 92, 84, . . .
1)
an 5 a1 1 d(n 2
e.
3000
2 1)
a50 5 100 1 (28)(50
dollars
)
a50 5 100 1 (28)(49
a 5 100 1 (2392)
adult tickets
50
a50 5 2292
3.2 Standard
Form of Linear
8043_Ch03.indd
Practice
Chapter 4 Skills
master the mathematical concepts or skills.
Equations
181
181
113
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AM
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113
Product Introduction
Teacher’s Resources and Assessments
The Teacher’s Resource and Assessments contains a suite of assessments for each chapter, including
a Pre-Test, a Post-Test, a Mid-Chapter test, an End of Chapter Test, and Standardized Test Practice.
Answers to the student assignments, student skills practice, and assessments are provided.
ExamView
All student assignments, student skills practice, and assessment
questions are available through ExamView, which allows you to
alter individual items and construct customized tests.
Teacher Resource Center
The Carnegie Learning Resource Center provides instructors with a variety of implementations tools. All
student and teacher materials are available online for viewing and printing. Dynamic lessons, available in
The Teacher,s
Implementation
Guide contains useful
sections to help you plan and
prepare to facilitate classroom
activities. An image of each
student text page, including
answers, is provided in the
Teacher’s Implementation
Guide.
FM-18 © 2012 Carnegie Learning
the Cognitive Tutor Software, can also be accessed via the Resource Center for use in your classrooms.
Product Introduction
8068_TIG_FM_00i-liv_Vol2.indd 18
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Teacher and Student Resource Center
The Carnegie Learning Resource Center is a dynamic community providing instructors and students
with access to Cognitive Tutor Software, textbook files, support, and valuable implementation tools
twenty-four hours a day, seven days a week. All access is determined by the licensing of Carnegie
Learning products and services.
You can access all of your print materials in electronic form on the resource center in the Textbooks
section. You will find student and teacher versions of texts, student assignments, student skills
practice, and assessments there. You can access these materials anywhere you have an internet
connection through your Carnegie Learning Resource Center account.
© 2012 Carnegie Learning
Product Introduction
Product Introduction 8068_TIG_FM_00i-liv_Vol2.indd 19
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Modules and Standards
1
Module
2
N.Q.1
A.REI.1
F.IF.7.a
N.RN.1
A.REI.10
F.BF.1.a
N.Q.2
A.REI.3
F.IF.9
N.RN.2
A.REI.11
F.BF.2
N.Q.3
A.REI.10
F.BF.1.b
N.Q.2
A.REI.12
F.BF.3
A.SSE.1.a
F.IF.1
F.LE.1.b
A.SSE.1.a
F.IF.1
F.LE.1
A.SSE.1.b
F.IF.2
F.LE.1.c
A.SSE.1.b
F.IF.2
F.LE.1.a
Core State
A.CED.1
F.IF.4
F.LE.2
A.CED.1
F.IF.3
F.LE.1.b
Standards
A.CED.2
F.IF.5
S.ID.6
A.CED.2
F.IF.4
F.LE.1.c
A.CED.3
F.IF.6
S.ID.7
A.CED.3
F.IF.6
F.LE.2
A.REI.3
F.IF.7.e
F.LE.3
A.REI.5
F.BF.1
F.LE.5
Modules and Standards
Common
A.CED.4
A.REI.6
Software
Ch. 4 Sequences
Ch. 2 Graphs, Equations, and Inequalities
Ch. 5 Graphs of Linear and Exponential Functions
Ch. 3 Linear Functions
Ch. 6 Systems of Equations
Ch. 7 Systems of Inequalities
Unit 1 Relations and Functions
Unit 13 Sequences
Unit 2 Graphs of Functions
Unit 14 Sequences and Functions
Unit 3 Linear Models and Four Quadrant Graphs
Unit 15 Exponential Models
Unit 4 Linear Equation and Inequality Solving with
Graphs
Unit 16 Graphs of Exponential Functions
Unit 5 Linear Inequalities
Unit 18 Properties of Exponents
Unit 6 Absolute Value Equations and Inequalities
Unit 19 Systems Linear Modeling
Unit 7 Non-Linear Equation and Inequality Solving
with Graphs
Unit 20 Linear System Solving using Substitution
Unit 8 Linear Models in General Form
Unit 9 Literal Equations
Unit 21 Linear System Solving using Linear
Combinations
Unit 10 Linear Models and the Distributive Property
Unit 22 Graphs of Linear Inequalities in Two
Variables
Unit 11 Equations of a Line
Unit 23 Systems of Linear Inequalities
Unit 12 Linear Function Operations and
Composition
FM-20 Unit 17 Linear and Exponential Transformations
© 2012 Carnegie Learning
Textbook
Ch. 1 Quantities and Relationship
Modules and Standards
8068_TIG_FM_00i-liv_Vol2.indd 20
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3
4
5
N.RN.2
A.REI.4.a
N.RN.1
F.IF.2
S.ID.2
A.SSE.1
A.REI.4.b
N.RN.2
F.IF.4
S.ID.3
A.SSE.1.a
A.REI.7
N.RN.3
F.IF.5
S.ID.5
A.SSE.2
F.IF.4
N.CN.1
F.IF.7
S.ID.6.a
A.SSE.3.a
F.IF.5
N.CN.7
F.IF.7b
S.ID.6.b
A.SSE.3.b
F.IF.6
A.APR.1
F.BF.1
S.ID.6.c
A.APR.1
F.IF.7a
A.CED.1
F.BF.1.a
S.ID.7
A.CED.1
F.BF.1.b
A.CED.4
F.BF.4
S.ID.8
A.CED.2
F.BF.3
A.REI.4.b
F.BF.4.a
F.LE.1.a
F.IF.1
F.BF.4.b
S.ID.9
Modules and Standards
S.ID.1
F.LE.1
F.LE.2
Ch. 8 Analyzing Data Sets for One
Variable
Ch. 11 Graphs of Quadratic Functions
Ch. 14 Real Number Systems
Ch. 12 Polynomial Expressions
Ch. 15 Other Functions and Inverses
Ch. 9 Analyzing Data Sets for Two
Quantitative Variables
Ch. 13 Solving Quadratic Equations
Ch. 16 Mathematical Modeling
Unit 32 Quadratic Models in Factored
Form
Unit 44 Rational and Irrational Numbers
Ch. 10 Analyzing Data Sets for Two
Categorical Variables
Unit 24 Measure of Central Tendency
Unit 25 Categorical Data Display
Comparisons
Unit 26 Numerical Data Display
Comparisons
© 2012 Carnegie Learning
Unit 27 Mean Absolute Deviation
Unit 33 Equivalent Forms of Quadratic
Functions and Graphs
Unit 34 Linear and Quadratic
Transformations
Unit 45 Operations with Complex
Numbers
Unit 46 Quadratic Equation Solving with
Complex Roots
Unit 47 Piecewise Linear Functions
Unit 35 Polynomial Operations
Unit 48 Inverse of Functions
Unit 49 Curve of Best Fit Analysis
Unit 29 Lines of Best Fit
Unit 36 Like Terms and Order of
Operations
Unit 30 Lines of Best Fit Analysis
Unit 37 Quadratic Expression Factoring
Unit 31 Frequency and Relative
Frequency
Unit 38 Quadratic Equation Solving
Using Factoring
Unit 28 Variance and Standard
Deviation
Unit 50 Linear and Non-Linear
Regression Analysis
Unit 39 Simplification and Operations
with Radicals
Unit 40 Forms of Quadratics
Unit 41 Quadratic Equation Solving
Unit 42 Quadratic Models in General
Form
Unit 43 Systems of Linear Quadratic
Equations
Modules and Standards 8068_TIG_FM_00i-liv_Vol2.indd 21
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Aligning Teaching to Learning
The primary goal of Carnegie Learning instructional materials is to get students to think! We recognize
the responsibility to provide instruction that respects the research on how students learn mathematics,
and believe in a continuous improvement model. Research can be difficult to implement in a practical
way. This same research shows that there is no magic bullet—there are no shortcuts—learning math
requires mental effort. Our materials are designed to provide students with the appropriate tools to
think deeply about mathematics and fluently execute the procedures.
Instructional Design
Within each student lesson, questions, instruction, and worked examples are interleaved to engage
Instructional Design
students as they develop their own mathematical understanding. The lessons are structured to
provide students with various opportunities to reason, to model, and to expand on explanations about
mathematical ideas. The over-arching questioning strategy throughout the text promotes analysis and
higher order thinking skills beyond simple “yes” or “no” responses. By explaining problem-solving
steps or the rationale for a solution, students will internalize the processes and reasoning behind the
mathematics. To achieve the learning goals of each lesson, students will respond to questions that
• Look for patterns
• Compare and contrast
• Estimate
• Calculate
• Predict
• Solve
• Describe
• Write a rule
• Determine
• Generalize
• Represent
• Explain their reasoning
Lessons will include a variety of problem types for students. These instructional features include lesson
openers, worked examples, pre-written student methods, error analysis, sorting activities, and more.
These instructional materials thoughtfully lead and support students to develop an understanding of
mathematical ideas. The materials were designed to teach students that math is relevant not because it
comes with a rule book that must be followed in a rote manner, but because it provides a common and
useful language for discussing and solving complex problems in everyday life.
FM-22 © 2012 Carnegie Learning
ask them to:
Aligning Teaching to Learning
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Planning
Each chapter is written to accommodate a variety of learners. With every instructional decision you
make, keep in mind your mathematical objectives for the chapter and the course. Connecting to the
3 Big Ideas of Carnegie Learning, plan each lesson by thinking about how you will create access
for your particular group of students, maintain access and pace throughout the lesson, and assess
their understanding along the way. We recommend that you do the math in each chapter before
implementing the activities with your specific group of students.
Engage and Motivate:
Create Access
Promote Deep Conceptual
Understanding:
Maintain Access
• What accommodations will
you need to make for your
• Which problem(s) or parts of
student population?
problems will accomplish your
• How will you use the warm up
• How will you orchestrate
• How will you access and
connect students’ prior
knowledge to the mathematical
concepts of this lesson?
• How will you group students?
Planning
mathematical goals?
and lesson opener?
opportunities for student-to-
Aligning Teaching
to Learning
student discourse?
Providing students with learning
opportunities to deepen their
© 2012 Carnegie Learning
mathematical understanding
Formative Assessment:
Assess Student Knowledge
• How will students demonstrate their
understanding?
• How will you summarize the
mathematical concepts of
the lesson?
• How will you use the student
assignments, student skills practice,
and Cognitive Tutor Software?
Planning 8068_TIG_FM_00i-liv_Vol2.indd 23
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Common Core Standards
for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators
at all levels should seek to develop in their students. These practices rest on important “processes
and proficiencies” with longstanding importance in mathematics education. The first of these are the
NCTM process standards of problem solving, reasoning and proof, communication, representation,
and connections. The second are the strands of mathematical proficiency specified in the National
Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual
understanding (comprehension of mathematical concepts, operations and relations), procedural fluency
(skill in carrying out procedures flexibly, accurately, efficiently, and appropriately), and productive
belief in diligence and one’s own efficacy).
© Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved.
Each lesson provides
opportunities for
students to think, reason, and
communicate their mathematical
understanding. However, it is our
responsibility as teachers to recognize
these opportunities and incorporate these
practices into our daily rituals. Expertise
is a long-term goal and students must
be encouraged to apply these
practices to new content
throughout their school
career.
Effective
communication
and collaboration are
essential skills of the
successful learner. It is through
dialogue and discussion of
different strategies that
students become
knowledgeable, independent
learners.
© 2012 Carnegie Learning
Mathemaical Practice
disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a
FM-24 Mathemaical Practice
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Standards for Mathematical Practice
Supporting the Practices
Mathematically proficient students start by explaining to
themselves the meaning of a problem and looking for entry
points to its solution. They analyze givens, constraints,
relationships, and goals. They make conjectures about
the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt.
They consider analogous problems, and try special cases
and simpler forms of the original problem in order to gain
insight into its solution. They monitor and evaluate their
progress and change course if necessary. Older students
might, depending on the context of the problem, transform
algebraic expressions or change the viewing window
on their graphing calculator to get the information they
need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions,
tables, and graphs or draw diagrams of important features
and relationships, graph data, and search for regularity
or trends. Younger students might rely on using concrete
objects or pictures to help conceptualize and solve a
problem. Mathematically proficient students check their
answers to problems using a different method, and they
continually ask themselves, “Does this make
sense?” They can understand the approaches
of others to solving complex problems
and identify correspondences between
different approaches.
The Carnegie Learning implementation model (discuss,
think, pair, share) provides a classroom environment for
students to make sense of problems, develop strategies,
persevere in implementing the strategy, and analyze
the results.
As students work collaboratively through problems,
they will plan and execute a solution strategy. It is
the responsibility of each group member to monitor
and evaluate the progress of the group, and to make
suggestions for changing course, if necessary. As
a facilitator, circulate through the room monitoring
students’ work, assessing progress, and redirecting
with guided questions.
Mathemaical Practice
To bring closure and provide a summary for each
problem, ask thought-provoking questions that require
students to explain their thinking and process. Allow
multiple groups to present their solutions with the class
discussion centered around alternate solution paths,
connections to prior concepts, and generalizations.
1
Make sense of
problems and
persevere in
solving them.
Example
Problem 1
Gearing For Success
© 2012 Carnegie Learning
Gwen has a part-time job working at Reliable Robots (RR) which sells electronics and
hardware parts for robot creators. One of her tasks is to analyze RR’s finances in terms of
cost and income. Her boss, Mr. Robo, asks her to determine the break-even point for the
cost and the income. The break-even point is the point when the cost and the income are
equal. Gwen begins with the income and costs for gearboxes.
1. Let the function I( g) represent the income (I ) from selling gearboxes ( g) and the function
C( g) represent the cost (C ) of purchasing gearboxes ( g).
a. Describe the relationship between the income function and the cost function that will
show the break-even point. Explain your reasoning.
I( g) 5 C( g)
Because I know that the break-even point is where the cost and income are equal to each other, I
can write the functions as equal to each other.
b. Describe the relationship between the income function and the cost function that will
show a profit from selling gearboxes. Explain your reasoning.
I( g) . C( g)
I know that RR will earn a profit when the income from selling gearboxes is greater than the cost
of purchasing gearboxes.
2. RR purchases gearboxes from The Metalists for $5.77 per gearbox plus a one-time
credit check fee of $45.00. RR sells each gearbox for $8.50.
a. Write the function for the income generated from selling gearboxes.
I(g) 5 8.5g
b. Write the function for the cost of purchasing gearboxes from The Metalists.
C(g) 5 5.77g 1 45
3. Sketch a graph of each function on the coordinate plane to predict the break-even point
of the income from RR selling the gearboxes and the cost of purchasing the gearboxes.
450
6
350
Dollars
8068_TIG_FM_00i-liv_Vol2.indd 25
I(g) 5 8.5g
400
300
250
200
C(g) 5 5.77g 1 45
Be sure
to label each graph
so you know which graph
represents cost and which
represents income.
© 2012 Carnegie Learning
Mathematical Practice y
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Supporting the Practices
The scenarios throughout the text help students
recognize and understand that quantitative relationships
seen in the real-world are no different than quantitative
relationships in mathematics. Some problems
begin with a real-world context to remind students
that the quantitative relationships they already use
can be formalized mathematically. Other problems
will use real-world situations as an application of
mathematical concepts.
Mathemaical Practice
Standards for Mathematical Practice
Mathematically proficient students make sense of
quantities and their relationships in problem situations.
They bring two complementary abilities to bear on
problems involving quantitative relationships: the
ability to decontextualize—to abstract a given situation
and represent it symbolically and manipulate the
representing symbols as if they have a life of their own,
without necessarily attending to their referents—and the
ability to contextualize, to pause as needed during the
manipulation process in order to probe into the referents
for the symbols involved. Quantitative reasoning
entails habits of creating a coherent representation of
the problem at hand; considering the units involved;
attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different
properties of operations and objects.
2
Reason
abstractly and
quantitatively.
Example
Problem 2
Analyzing Equations and Graphs
1. Complete the table shown for the problem situation described in Problem 1, Analyzing
Tables. First, determine the unit of measure for each expression. Then, describe the
contextual meaning of each part of the function. Finally, choose a term from the word
box to describe the mathematical meaning of each part of the function.
output value
2
input value
rate of change
What It Means
Unit
t
minutes
the time, in minutes, that the plane
has been in the air
input value
1800
feet
_____
the number of feet that the plane
climbs each minute
rate of change
1800t
feet
minute
Contextual Meaning
Mathematical Meaning
the height, in feet, of the plane
output value
2. Write a function, h(t), to describe the plane’s
height over time, t.
Why do you
think h(t) is
used to name this
function?
© 2012 Carnegie Learning
h ( t ) 5 1800t
© 2012 Carnegie Learning
Expression
FM-26 Mathematical Practice
78
Chapter 2
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Standards for Mathematical Practice
Supporting the Practices
b. |x 2 9| 5 12
2
(x 2 9) 5 12
x 2 9 5 12
x 5 21
3
Construct viable
arguments and
critique the
reasoning of
others.
2(x 2 9) 5 12
x 2 9 5 212
x 2 9 1 9 5 12 1 9
c. |3x 1 7| 5 28
Mathemaical Practice
Mathematically proficient students understand and use
In a Carnegie Learning classroom, students are active
stated assumptions, definitions, and previously established
participants in their learning; they are doing the work,
results in constructing arguments. They make conjectures
presenting solutions, and critiquing each other. Your role
and build a logical progression of statements to explore
is to facilitate the discussion and highlight important
the truth of their conjectures. They are able to analyze
connections, strategies, and conclusions.
situations by breaking them into cases, and can recognize
Each lesson ends with the statement “Be prepared
and use counterexamples. They justify their conclusions,
to share your solutions and methods.” Students are
communicate them to others, and respond to the arguments
expected to be able to communicate their reasoning and
of others. They reason inductively about data, making
critique the explanation of others. As students explain
plausible arguments that take into account the context
problem-solving steps or the rationale for a solution,
from which the data arose. Mathematically proficient
they will internalize the processes and reasoning behind
students are also able to compare the effectiveness of two
the mathematics.
plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an
argument—explain why it is flawed. Elementary students
can construct arguments using concrete referents such as
objects, drawings, diagrams, and actions. Such arguments
can make sense and be correct, even though they are not
generalized or made formal until later
grades.
Later,
students
6. Solve
each linear
absolute
value equation. Show your work.
Before you
1 7| 5 3
learn to determine domains to which a.an|xargument
applies.
start solving each
(x 1arguments
7) 5 3
equation, think about the
Students at all grades can listen or read the
of 2(x 1 7) 5 3
number of solutions each equation
x1753
x 1 7 5 23
others, decide whether they make sense, and
ask useful
may have. You may be able to
x17275327
x 1 7 2 7 5 23 2 7
questions to clarify or improve the arguments.
save yourself some
x 5 24
x 5 210
work—and time!.
x 2 9 1 9 5 212 1 9
x 5 23
No solution. Linear absolute value can never
be a negative number.
d. |2x 1 3| 5 0
Example
2x 1 3 5 0
2x 1 3 2 3 5 0 2 3
2x 5 23
23
2x 5 ___
___
2
2
3
x 5 2__
2
7. Cho, Steve, Artie, and Donald each solved the equation |x| 2 4 5 5.
Artie
Donald
(x) = 9
|x| – 4 = 5
|x| = 9
–(x) – 4 = 5
–x = 9
(x) = 9
–(x) = 9
x = –9
x = –9
Cho
Steve
|x| – 4 = 5
|x| – 4 = 5
(x) – 4 = 5
–[(x) – 4] = 5
x–4=5
–x + 4 = 5
x=9
–x = 1
x = –1
128
Chapter 2
8043_Ch02_78-128.indd 128
(x) – 4 = +5
x=9
–(x) – 4 = –5
© 2012 Carnegie Learning
© 2012 Carnegie Learning
|x| – 4 = 5
(x) – 4 = 5
–x – 4 = –5
–x = –1
x=1
Graphs, Equations, and Inequalities
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Mathemaical Practice
Standards for Mathematical Practice
Supporting the Practices
Mathematically proficient students can apply the
mathematics they know to solve problems arising
in everyday life, society, and the workplace. In early
grades, this might be as simple as writing an addition
equation to describe a situation. In middle grades, a
student might apply proportional reasoning to plan a
school event or analyze a problem in the community. By
high school, a student might use geometry to solve a
design problem or use a function to describe how one
quantity of interest depends on another. Mathematically
proficient students who can apply what they know are
comfortable making assumptions and approximations
to simplify a complicated situation, realizing that
these may need revision later. They are able to identify
important quantities in a practical situation and map
their relationships using such tools as diagrams, twoway tables, graphs, flowcharts, and formulas. They can
analyze those relationships mathematically to draw
conclusions. They routinely interpret their mathematical
results in the context of the situation and
reflect on whether the results make sense,
possibly improving the model if it has not
served its purpose.
Activities throughout the text provide opportunities for
students to create and use multiple representations
(words, tables, graphs, and symbolic statements) to
organize, record, and communicate mathematical ideas.
Manipulatives and various models are incorporated
throughout to develop a conceptual understanding
of mathematical concepts. These activities provide
opportunities for students to develop strategies and
reasoning that will serve as the foundation for learning
more abstract mathematics. To foster the transfer of
student understanding from concrete manipulatives
to the abstract procedures, a variety of instructional
prompts are used.
4
Model with
mathematics.
Example
Problem 1
Analyzing Tables
A 747 airliner has an initial climb rate of 1800 feet per minute until it reaches a height
of 10,000 feet.
1. Identify the independent and dependent quantities in this problem situation.
Explain your reasoning.
The height of the airplane depends on the time, so height is the dependent quantity
and time is the independent quantity.
2
2. Describe the units of measure for:
a. the independent quantity (the input values).
The independent quantity of time is measured in minutes.
The dependent quantity of height is measured in feet.
3. Which function family do you think best represents this situation? Explain your reasoning.
Answers will vary.
The situation shows a linear function because the rate the plane ascends is constant. So, this situation
belongs to the linear function family.
y
Height (feet)
When you
sketch a graph,
include the axes’ labels
and the general graphical
behavior. Be sure to
consider any
intercepts.
Time (minutes)
74
8043_Ch02.indd 74
FM-28 x
© 2012 Carnegie Learning
4. Draw and label two axes with the independent and dependent quantities and their units
of measure. Then sketch a simple graph of the function represented by the situation.
© 2012 Carnegie Learning
b. the dependent quantity (the output values).
Chapter 2 Graphs, Equations, and Inequalities
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Mathematical Practice
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Standards for Mathematical Practice
Supporting the Practices
Mathematically proficient students consider the available
tools when solving a mathematical problem. These tools
might include pencil and paper, concrete models, a ruler,
a protractor, a calculator, a spreadsheet, a computer
algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar
with tools appropriate for their grade or course to make
sound decisions about when each of these tools might
be helpful, recognizing both the insight to be gained and
their limitations. For example, mathematically proficient
high school students analyze graphs of functions and
solutions generated using a graphing calculator. They
detect possible errors by strategically using estimation
and other mathematical knowledge. When making
mathematical models, they know that technology can
enable them to visualize the results of varying assumptions,
explore consequences, and compare predictions with
data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical
resources, such as digital content located on a website,
and use them to pose or solve problems. They are able
to use technological tools to explore and deepen their
understanding of concepts.
Activities throughout the text facilitate the appropriate use
of tools including graphing calculators, rulers, protractors,
compasses, and manipulatives. Tools are used in a variety
of ways—to build conceptual understanding, to explore
concepts, and to verify solutions. Worked examples are
provided as appropriate within lessons to demonstrate
how to use the various tools.
The Cognitive Tutor Software is a dynamic instructional
element that offers opportunities for students to use
tools appropriately within each section.
Mathemaical Practice
5
Use appropriate
tools
strategically.
Example
You can input equations written in function notation into your graphing calculator.
Your graphing calculator will list different functions as Y1, Y2, Y3, etc.
1
Let’s graph the function f(x) 5 8x 1 15 on a calculator by following the steps shown.
You can use a graphing calculator to graph
a function.
Step 1: Press Y=. Your cursor should be
blinking on the line \Y1=. Enter
© 2012 Carnegie Learning
the equation. To enter a variable
like x, press the key with
X, T, Ø, n once.
The way
you set the
window will vary each
time depending on the
equation you are graphing.
Step 2: Press WINDOW to set the
bounds and intervals you
want displayed.
Step 3: Press GRAPH to view the graph.
The Xmin represents the least point on the x-axis that will be seen on the screen. The Xmax
represents the greatest point that will be seen on the x-axis. Lastly, the Xscl represents the
intervals. Similar names are used for the y-axis (Ymin, Ymax, and Yscl).
A convention to communicate the viewing WINDOW on a graphing calculator is shown.
Xmin: 210
Xmax: 10
Ymin: 220
Ymax: 20
}
}
[210, 10]
[220, 20]
}
[210, 10] 3 [220, 20]
8068_TIG_FM_00i-liv_Vol2.indd 29
2012 Carnegie Learning
Mathematical Practice FM-29
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Supporting the Practices
Each lesson throughout the text provides opportunities
for students to communicate precisely when writing
in their consumable books, and then sharing their
solutions with their peers. It is your responsibility to
ensure that students label units of measure and explain
their reasoning using appropriate definitions and
mathematical language.
Mathemaical Practice
Standards for Mathematical Practice
Mathematically proficient students try to communicate
precisely to others. They try to use clear definitions in
discussion with others and in their own reasoning. They
state the meaning of the symbols they choose, including
using the equal sign consistently and appropriately.
They are careful about specifying units of measure,
and labeling axes to clarify the correspondence with
quantities in a problem. They calculate accurately and
efficiently, express numerical answers with a degree of
precision appropriate for the problem context. In the
elementary grades, students give carefully formulated
explanations to each other. By the time they reach high
school they have learned to examine claims and make
explicit use of definitions.
The answers provided in the Teacher’s Implementation
Guide are exemplars of student responses and model
precision appropriately. A Vocabulary section is included
in the Skills Practice for each lesson that provides
additional opportunities to use appropriate definitions
and mathematical language.
6
Attend to
precision.
Example
?
4. Dawson would like to exchange $70 more.
Jonathon thinks Dawson should have a total of
£343.54707. Erin says he should have a total of
£343.55, and Tre says he should have a total of £342.
Who’s correct? Who’s reasoning is correct? Why
are the other students not correct? Explain your reasoning.
2
The pound
(£) is made up of
100 pence (p), just like
the dollar is made up of
100 cents.
Jonathon
=
=
=
=
300 + 0.622101d
300 + 0.622101(70)
300 + 43.54707
343.54707
Erin
f(d) = 300 + 0.622101d
f(d) = 300 + 0.622101(70)
f(d) = 300 + 43.54707
© 2012 Carnegie Learning
f(d)
f(d)
f(d)
f(d)
Tre
f (d ) = 300 + 0.6d
f (d ) = 300 + 0.6 (70)
f(d) = 343.54707
f (d ) = 300 + 42
f(d) ¯ 343.55
f (d ) = 342
5. How many total pounds will Dawson have if he only exchanges an additional $50?
Show your work.
© 2012 Carnegie Learning
Erin is correct. Since Dawson is exchanging money, he must round to the
hundredths place. Jonathon did not round his answer to reflect pence. Tre rounded
the conversion rate and short-changed the actual amount of British pounds that
Dawson should receive.
f(d ) 5 300 1 0.622101d
FM-30 Mathematical Practice
f(d ) 5 300 1 0.622101(50)
f(d ) 5 331.10505
If Dawson exchanges an additional $50, he will have a total of £331.11.
94
Chapter 2
Graphs, Equations, and Inequalities
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Standards for Mathematical Practice
Mathematically proficient students look closely to discern
a pattern or structure. Young students, for example, might
notice that three and seven more is the same amount as
seven and three more, or they may sort a collection of
shapes according to how many sides the shapes have.
Later, students will see 7 3 8 equals the well remembered
7 3 5 1 7 3 3, in preparation for learning about the
distributive property. In the expression x2 1 9x 1 14,
older students can see the 14 as 2 3 7 and the 9 as
2 1 7. They recognize the significance of an existing line
in a geometric figure and can use the strategy of drawing
an auxiliary line for solving problems. They also can step
back for an overview and shift perspective. They can see
complicated things, such as some algebraic expressions, as
single objects or as being composed of several objects. For
example, they can see 5 2 3(x 2 y)2 as 5 minus a positive
number times a square and use that to realize
that its value cannot be more than 5 for any
real numbers x and y.
Supporting the Practices
Activities throughout the text provide opportunities for
students to analyze numeric, geometric, and algebraic
patterns. Accompanying questions help students
notice relationships for themselves as opposed to
memorization of facts.
Mathemaical Practice
7
Look for and
make use of
structure.
Example
3. Ax 1 By 5 C
a. slope-intercept form:
Ax 1 By 5 C
By 5 2Ax 1 C
By ____
C
___
5 2Ax 1 __
B
B
B
A
C
y 5 2__ x 1 __
B
B
c. y-intercept:
A
C
y 5 2__ x 1 __
B
B
A
C
y 5 2__ (0) 1 __
B
B
C
y 5 __
B
3
b. x-intercept:
Ax 1 By 5 C
Ax 1 B (0) 5 C
Ax 5 __
C
___
A
A
C
x 5 __
A
d. slope:
A
C
y 5 2__ x 1 __
B
B
A
m 5 __
B
4. If you want to determine the y-intercept of an equation, which form is more efficient?
Explain your reasoning.
© 2012 Carnegie Learning
The slope-intercept form is more efficient for determining the y-intercept because
I do not have to solve to determine it.
Answers may vary.
The standard form is more efficient for determining the x-intercept. In order to
determine the x-intercept in standard form, I only have to perform one operation,
division. To determine the x-intercept in slope-intercept form, I have to perform at
least two operations.
© 2012 Carnegie Learning
5. If you want to determine the x-intercept of an equation, which form is more efficient?
Explain your reasoning.
6. If you wanted to graph an equation on your calculator, which form is more efficient?
Explain your reasoning.
Slope-intercept form is more efficient for graphing on my calculator. The equation
must be in slope-intercept form to enter it into the calculator. There is no way to
enter an equation in standard form.
192
Chapter 3
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Linear Functions
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Standards for Mathematical Practice
Supporting the Practices
Mathemaical Practice
Mathematically proficient students notice if calculations
are repeated, and look both for general methods and
for shortcuts. Upper elementary students might notice
when dividing 25 by 11 that they are repeating the same
calculations over and over again, and conclude they
have a repeating decimal. By paying attention to the
calculation of slope as they repeatedly check whether
points are on the line through (1, 2) with slope 3,
students might abstract the equation (y 2 2)/(x 2 1) 5 3.
Noticing the regularity in the way terms cancel when
expanding (x 2 1)(x 1 1), (x 2 1)(x2 1 x 1 1), and (x 2 1)
(x3 1 x2 1 x 1 1) might lead them to the general formula
for the sum of a geometric series. As they work to solve
a problem, mathematically proficient students maintain
oversight of the process, while attending to the details.
They continually evaluate the reasonableness of their
intermediate results.
Activities throughout the text provide opportunities
for students to make observations, notice patterns,
and make generalizations. Students are required
to communicate their generalizations verbally and
symbolically. This understanding will lead to greater
transfer and ability to solve non-routine problems.
As you facilitate discussions be sure to highlight
important connections, efficient strategies, and
conclusions.
8
Look for and
express
regularity in
repeated
reasoning.
Example
4. Write the x-value of each ordered pair for the three given functions. You can use your
graphing calculator to determine the x-values.
h(x) 5 2x
v(x) 5 2(x 1 3)
1)
( 22 , __
4
1)
( 25 , __
4
(
1
1)
( 21 , __
2
1)
( 24 , __
2
(
1)
2 , __
0 , 1)
( 23 , 1)
(
3 , 1)
(
1
, 2)
( 22 , 2)
(
4 , 2)
(
2 , 4)
( 21 , 4)
(
5 , 4)
1)
, __
4
Why are
there no
negative y-values
given in this table?
HINT: You learned about
it in the previous
lesson!
2
5. Use the table to compare the ordered pairs of the graphs of v(x) and w(x) to the ordered
pairs of the graph of the basic function h(x). What do you notice?
For the same y-coordinate, the x-coordinate of v (x ) is 3 less than the
x-coordinate of h(x ). For the same y-coordinate, the x-coordinate of w (x ) is
3 more than the x-coordinate of h(x ).
A horizontal translation of a graph is a shift of the entire graph left or right. A horizontal
translation affects the x-coordinate of each point on the graph.
You can use the coordinate notation shown to indicate a horizontal translation.
(x, y) → (x 1 a, y), where a is a real number.
6. Use coordinate notation to represent the horizontal translation of each function.
5
• h(x) 5 2
x
(x, y) → (x 2 3, y )
• w(x) 5 2(x 2 3)
(x, y) → (x 1 3, y )
So, if
a constant is
added or subtracted
OUTSIDE a function, like
g(x) + 3 or g(x) – 3,
then only the y-values
change, resulting in a
vertical translation.
320
Chapter 5
8043_Ch05_320.indd 320
FM-32 And, if
a constant is added
or subtracted INSIDE a
function, like g(x + 3) or g(x – 3),
then only the x-values
change, resulting in a horizontal
translation.
© 2012 Carnegie Learning
(x, y)
• v(x) 5 2(x 1 3)
© 2012 Carnegie Learning
(
w(x) 5 2(x 2 3)
Exponential Functions
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Mathematical Practice
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Home Connection
Encourage your students to
share the Dear Student letter
with their parents/guardians
to inform them about the
instructional materials and the
approach of this text.
Dear Student,
You are about to begin an exciting endeavor using mathematics! To be successful, you
will need the right tools. This book is one of the most important tools you will use this
year. Throughout this book there is space for note-taking, sketching, and calculating.
You will be given opportunities to think and reason about various mathematical concepts
and use tools such as tables, graphs, and graphing calculators.
This year you will face many new challenges both in and outside of the classroom. While
some challenges may seem difficult, it is important to remember that effort maters. You
must realize that it may take hard work and perseverance to succeed—and your hard work
will pay off!
© 2012 Carnegie Learning
I bet the
folks at home
would like to know what
we’re going to do
this year!
Implementation
© 2012 Carnegie Learning
Connections in mathematics are important. Throughout this
text, you will build new knowledge based upon your prior
knowledge. It is our goal that you see mathematics as relevant
because it provides a common and useful language for discussing
and solving real-world problems.
Don’t wory—you will not be working alone. Working with others is a skill
that you will need throughout your life. When you begin your career, you
will most likely work with all sorts of people, from shy to outgoing, from
leaders to supporters, from innovators to problem solvers—and many
more types of people! Throughout this book, you will have many
opportunities to work with your classmates. You will be able to discuss
your ideas and predictions to different problem situations; present your
calculations and solutions to questions; and analyze, critique and
sugest, or support your classmates’ answers to problem situations.
Today’s workplace demands teamwork and self-confidence. At Carnegie
Learning, our goal is to provide you with opportunities to be successful in
your math course. Enjoy the year and have fun Learning by Doing(TM)!
© 2012 Carnegie Learning
—The Carnegie Learning Curriculum Development Team
Student Page 8068_TIG_FM_00i-liv_Vol2.indd 33
FM-33
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Note
Characters are embedded
throughout the text to remind
students to stop and think in
order to promote productive
reflection. The characters are
used in a variety of ways: they
may remind students to recall a
previous mathematical concept,
help students develop expertise
to think through problems, and
occasionally, present a fun fact.
The Crew
Implementation
The Crew
The Crew is here to help you throughout this text. Sometimes they will remind you about
things you have already learned. Sometimes they will ask you questions to help you think
about different strategies. Sometimes they will share fun facts. They are members of your
group—someone you can rely on!
x FM-34 The Crew
© 2012 Carnegie Learning
© 2012 Carnegie Learning
Teacher aides will guide you along your way. They will help you make connections and
remind you to think about the details.
Student Page
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Implementation
Recommendation
© 2012 Carnegie Learning
© 2012 Carnegie Learning
If you expect students to work
well together, they need to
understand what it means and
how it will benefit them. It is
our recommendation that you
establish classroom guidelines
and structure groups to create
a community of learners. When
you are facilitating groups listen
carefully and value diversity
of thought, redirect student’s
questions with guiding questions,
provide additional support with
those struggling with a task,
Introduction
During this course, you will solve problems and work with many different representations of
mathematical concepts, ideas, and processes to better understand the world. Each lesson
will provide you with opportunities to discuss your ideas, work within groups, and share your
solutions and methods with your class. These process icons are placed throughout the text.
Discuss to Understand
•
•
•
•
Read the problem carefully.
What is the context of the problem? Do we understand it?
What is the question that we are being asked? Does it make sense?
Is this problem similar to some other problem we know?
Think for Yourself
• Do I need any additional information to answer the question?
• Is this problem similar to some other problem that I know?
• How can I represent the problem using a picture, a diagram, symbols, or some
other representation?
Work with Your Partner
•
•
•
•
How did you do the problem?
Show me your representation.
This is the way I thought about the problem—how did you think about it?
What else do we need to solve the problem?
• Does our reasoning and our answer make sense to each other?
• How will we explain our solution to the class?
Share with the Class
• Here is our solution and the methods we used.
• Are we communicating our strategies clearly?
• We could only get this far with our solution. How can we finish?
• Could we have used a different strategy to solve the problem?
Representations and hold groups accountable for an end product. During the share phase make your
expectations clear, require that students defend and talk about their solutions, and
monitor student progress by checking for understanding.
xi
Mathematical Practice Standard 1 Make sense of problems and persevere in solving them.
This type of implementation model provides a classroom environment that
allows students to make sense of problems, develop strategies, persevere in
implementing the strategy, and analyze results.
Student Page 8068_TIG_FM_00i-liv_Vol2.indd 35
Implementation
Process icons appear throughout
each lesson as helpful prompts for
students. The notes throughout
your Teachers Implementation
Guide provide recommendations
for grouping students and provide
additional guiding questions for
the discuss and share phases.
Mathematical Representations
Represeentations
A key goal with instruction is
to get students to think about
and discuss math. Research
shows that different classroom
environments may require
different methods to engage
students. Our recommendations
offer a balance of whole group,
small group, partner pairs,
and individual instructional
approaches. Ideally, students are
provided multiple opportunities to
discuss mathematics and share
their thinking and ideas daily.
The dynamics for teaching and
learning in your classroom may
vary from one year to the next. As
a teaching professional, you have
choices about how to implement
these instructional materials in the
way you deem best to engage
your students to think about and
understand the content.
FM-35
5/11/12 12:18 PM
Implementation
Recommendation
Academic Glossary
Key Terms of the Course
There are important terms you will encounter throughout this book. It is important that you
have an understanding of these words as you get started through the mathematical
concepts. Knowing what is meant by these terms and using these terms will help you think,
reason, and communicate your ideas. The Graphic Organizers shown display a definition for
a key term, related words, sample questions, and examples.
You will
create graphic
organizers like these as your
own references of key
mathematical ideas.
Academic Glossary
It is our recommendation to be
explicit about your expectations
of language use and the way
students write responses
throughout the text. Encourage
students to answer questions
with complete sentences.
Complete sentences help
students reflect on how they
arrived at a solution, make
connections between topics,
and consider what a solution
means both mathematically as
well as in context. Answers
are provided in the Teacher’s
Implementation Guide in the
form of complete sentences
whenever appropriate.
Encourage your students to
share these pages with their
parents/guardians to inform
them of the types of questions
that appear throughout this text.
FM-36 xii Academic Glossary
© 2012 Carnegie Learning
Home Connection
© 2012 Carnegie Learning
My folks
are always
trying to get me to
be organized!
© 2012 Carnegie Learning
Implementation
It is critical for students to
possess an understanding
of the language of their text.
Students must learn to read for
different purposes and write
about what they are learning.
Use the graphic organizers
included in the Academic
Glossary to help students
become familiar with the key
words and the questions they
can ask themselves when they
encounter these words.
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Mathematical Practice
Standard 1 Make sense of problems and
persevere in solving them.
Ask Yourself
Definition
To study or look closely for patterns.
Analyzing can involve examining or breaking
a concept down into smaller parts to gain a
better understanding of it.
The Ask Yourself questions
help students develop the
proficiency to explain to
themselves the meaning
of problems.
determine
observe
consider
what do you notice?
• What is the context?
• What does the solution mean in terms
of this problem situation?
what do you think?
sort and match
identify
Analyze
b. At least how many boxes would Alan have to sell to be able to choose his own prize?
3.75b 1 25 $ 1500
3.75b 1 25 2 25 $ 1500 2 25
Example
3.75b $ 1475
3.75b $ _____
1475
______
3.75
3.75
b $ 393.33. . .
2
Alan would need to sell at least 394 boxes to be able to choose his own prize.
Problem 3
Reversing the Sign
Alan’s camping troop hikes down from their campsite at an elevation of 4800 feet
to the bottom of the mountain. They hike down at a rate of 20 feet per minute.
© 2012 Carnegie Learning
h(m) 5 220m 1 4800
2. Analyze the function.
a. Identify the independent and dependent quantities and their units.
The independent quantity is the number of minutes hiked, and the
dependent quantity is the elevation in feet.
b. Identify the rate of change and explain what it means in terms of this
problem situation.
The rate of change is 220. This represents a decrease of 20 feet every minute.
c. Identify the y-intercept and explain what it means in terms of this problem situation.
© 2012 Carnegie Learning
Students use similar
graphic organizers
throughout the text to
create their own references
of key mathematical
concepts. The graphic
organizers require students
to clarify definitions
and communicate their
understanding and
reasoning clearly.
1. Write a function, h(m), to show the troop’s elevation as a function of time in minutes.
© 2012 Carnegie Learning
© 2012 Carnegie Learning
Mathematical Practice
Standard 6 Attend to precision.
The y-intercept is 4800. This shows that the troop started their hike
at an elevation of 4800 feet.
d. What is the x-intercept and explain what it means in terms of this problem situation?
0 5 220m 1 4800
24800 5 220m
220m
5 ______
220
220m
240 5 m
24800
_______
The x-intercept is (240, 0). The hikers will be at the bottom of the mountain
in 240 minutes, or 4 hours.
2.3
8043_Ch02.indd 107
Modeling Linear Inequalities
107
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Academic Glossary Student Page 8068_TIG_FM_00i-liv_Vol2.indd 37
Implementation
The matching and
sorting activities provide
opportunities for students
to recognize and search
for patterns.
evaluate
to accomplish?
investigate
Academic Glossary
Mathematical Practice
Standard 7 Look for and make use
of structure.
•
•
•
•
•
examine
representation, or numbers change?
• What is the question asking me
Related Words
•
•
•
•
•
• Do I see any patterns?
• Have I seen something like this before?
• What happens if the shape,
xiii
FM-37
5/11/12 12:18 PM
Mathematical Practice
Standard 3 Construct viable arguments
and critique the reasoning
of others.
Ask Yourself
Definition
To give details or describe how to
determine an answer or solution.
•
•
•
•
•
Explaining your reasoning helps
justify conclusions.
Each lesson provides
opportunities for students
to justify their conclusions,
communicate them to
others, and respond to
feedback.
Related Words
•
•
•
•
How should I organize my thoughts?
Is my explanation logical?
Does my reasoning make sense?
How can I justify my answer to others?
Did I use complete sentences in
my answer?
show your work
Don’t
forget to
check your
answers!
explain your calculation
justify
why or why not?
Real-world contexts confirm
concrete examples of
mathematics. The scenarios
in the lessons help students
recognize and understand that
quantitative relationships seen
in the real world are no different
than quantitative relationships in
mathematics. Some problems
begin with a real-world context
to remind students that the
quantitative relationships they
already use can be formalized
mathematically. Other problems
will use real-world situations as
an application of mathematical
concepts.
Academic Glossary
Explain
YourReasoning
Example
Problem 1
Analyzing Tables
A 747 airliner has an initial climb rate of 1800 feet per minute until it reaches a height
of 10,000 feet.
1. Identify the independent and dependent quantities in this problem situation.
Explain your reasoning.
2. Describe the units of measure for:
a. the independent quantity (the input values).
The independent quantity of time is measured in minutes.
b. the dependent quantity (the output values).
The dependent quantity of height is measured in feet.
3. Which function family do you think best represents this situation? Explain your reasoning.
© 2012 Carnegie Learning
The height of the airplane depends on the time, so height is the dependent quantity
and time is the independent quantity.
2
© 2012 Carnegie Learning
Implementation
Note
Answers will vary.
y
xiv Academic Glossary
Solving problems from
real-world contexts
helps students make
sense of quantities and
relationships.
Time (minutes)
74
8043_Ch02.indd 74
FM-38 When you
sketch a graph,
include the axes’ labels
and the general graphical
behavior. Be sure to
consider any
intercepts.
Chapter 2
x
© 2012 Carnegie Learning
4. Draw and label two axes with the independent and dependent quantities and their units
of measure. Then sketch a simple graph of the function represented by the situation.
Height (feet)
Mathematical Practice
Standard 2 Reason abstractly and
quantitatively.
© 2012 Carnegie Learning
The situation shows a linear function because the rate the plane ascends is constant. So, this situation
belongs to the linear function family.
Graphs, Equations, and Inequalities
03/05/12 11:08 AM
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Mathematical Practice
Standard 2 Reason abstractly and
quantitatively.
Ask Yourself
Definition
To display information in various ways.
• How should I organize my thoughts?
• How do I use this model to show a
Representing mathematics can be done
using words, tables, graphs, or symbols.
Questions provide
opportunities for students
to represent relationships is
a variety of ways.
concept or idea?
•
•
•
•
Related Words
•
•
•
•
•
•
•
•
show
sketch
draw
create
plot
graph
write an equation
What does this representation tell me?
Is my representation accurate?
What units or labels should I include?
Are there other ways to model
this concept?
complete the table
Mathematical Practice
Standard 4 Model with mathematics.
Represent
Example
3. Label the function on the coordinate plane.
y
Campsite Elevation (feet)
4500
© 2012 Carnegie Learning
© 2012 Carnegie Learning
2
h(m) 5 220m 1 4800
4000
3500
y 5 3200
3000
2500
2000
1500
1000
500
0
60
120
180
240
Time (minutes)
x
Implementation
Academic Glossary
Questions provide
opportunities for students
to use models to represent
mathematical concepts.
4. Use the graph to determine how many minutes passed if the troop is below 3200 feet.
Draw an oval on the graph to represent this part of the function and write the
corresponding inequality statement.
More than 80 minutes has passed if the troop is below 3200 feet.
m . 80
5. Write and solve an inequality to verify the solution set you interpreted from the graph.
220m 1 4800 , 3200
220m 1 4800 2 4800 , 3200 2 4800
220m , 21600
© 2012 Carnegie Learning
21600m
220m , ________
______
6. Compare and contrast your solution sets using the graph and the function.
What do you notice?
The solution sets are the same.
108
8043_Ch02.indd 108
Chapter 2
© 2012 Carnegie Learning
220
220m
m . 80
Academic Glossary Graphs, Equations, and Inequalities
03/05/12 11:42 AM
Student Page 8068_TIG_FM_00i-liv_Vol2.indd 39
xv
FM-39
5/11/12 12:18 PM
Mathematical Practice
Standard 1 Making sense of problems
and persevere in solving
them.
problem situation?
Related Words
• predict
• approximate
Academic Glossary
• What predictions can I make from this
problem situation?
• expect
• about how much?
Estimating
gets you in the
neighborhood,
calculating gets you
the address.
Estimate
Example
3. Use the data from the table to create a graph of the problem situation on the
coordinate plane.
y
18
2
y 5 13.45
12
10
8
6
4
2
0
1.0
2.0
3.0
4.0
Amount of Ground Meat (pounds)
x
4. Consider a total bill of $13.45.
a. Estimate the amount of ground beef purchased.
The graph of y 5 13.45 crosses the original graph at about 2, so I predict that
2 pounds of ground meat were purchased.
© 2012 Carnegie Learning
14
© 2012 Carnegie Learning
16
Total Cost (dollars)
Implementation
Questions provide
opportunities for students
to make sense of quantities
and their relationships in
problems.
• Does my reasoning make sense?
• Is my solution close to my estimation?
• What do I know about this
Estimating first helps inform reasoning.
Questions provide
opportunities for students
to make conjectures before
simply jumping into a
solution path.
Mathematical Practice
Standard 2 Reason abstractly and
quantitatively.
Ask Yourself
Definition
To make an educated guess based on the
analysis of given data.
b. Determine the exact amount of ground meat purchased.
© 2012 Carnegie Learning
xvi © 2012 Carnegie Learning
Using the intersection function on my graphing calculator, I determined the exact
amount to be 2.2 pounds of ground meat.
5. Based on the number of people coming to the cookout, you decide to buy 6 pounds of
ground meat for the hamburgers.
a. If your budget for the food is $25.00, do you have enough money? Why or why not?
The total cost of the bill will be $24.81, so $25.00 is enough money.
Academic Glossary
b. If you have enough money, how much money do you have left over? If you do not
have enough money, how much more will you need?
The bill will be $24.81, so I will have $0.19 left over.
2.6
8043_Ch02.indd 139
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139
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Mathematical Practice
Standard 2 Reason abstractly and
quantitatively.
Ask Yourself
Definition
To represent or give an account of in words.
Describing communicates mathematical
ideas to others.
Questions provide
opportunities for students
to make sense of quantities
and their relationships in
problems.
• How should I organize my thoughts?
• Is my explanation logical?
• Did I consider the context of
this situation?
Related Words
•
•
•
•
•
•
Mathematical Practice
Standard 6 Attend to precision.
• what are the
demonstrate
my answer?
advantages?
label
•
display
compare
• Did I include appropriate units and labels?
• Will my classmates understand
what are the
disadvantages?
my reasoning?
• what is similar?
• what is different?
define
determine
Describe
Example
Talk the Talk
You just worked with different representations of a linear function.
1. Describe how a linear function is represented:
a. in a table.
b. in a graph.
© 2012 Carnegie Learning
Questions provide
opportunities for students
to make observations,
notice patterns, and make
generalizations.
2
A linear function is represented in a graph by a straight line.
c. in an equation.
A linear function is represented by a function in the form f (x) 5 ax 1 b.
2. Name some advantages and disadvantages of the graphing method and the algebraic
method when determining solutions for linear functions.
Answers will vary.
Graphs provide visual representations of functions, and they can provide a wide
range of values, depending on the intervals. A disadvantage is that I have to estimate
values if points do not fall exactly on grid line intersections. The algebraic method
provides an exact solution for every input, but I may be unable to solve more difficult
equations correctly.
3. Do you think the graphing method for determining solutions will work for any function?
Answers will vary.
No. The graphing method would probably be too difficult to use for complicated
functions.
84
8043_Ch02.indd 84
Chapter 2
© 2012 Carnegie Learning
© 2012 Carnegie Learning
© 2012 Carnegie Learning
Mathematical Practice
Standard 8 Look for and express
regularity in repeated
reasoning.
When the input values in a table are in successive order and the first differences
of the output values are constant, the table represents a linear function.
Academic Glossary xvii
Graphs, Equations, and Inequalities
03/05/12 11:09 AM
Student Page 8068_TIG_FM_00i-liv_Vol2.indd 41
Implementation
Academic Glossary
Questions provide
opportunities for students
to communicate their
mathematical ideas.
Always encourage students
to include the correct unit
of measure when they write
and speak.
• Does my reasoning make sense?
• Did I use complete sentences in
FM-41
5/11/12 12:18 PM
Problem Types You Will See
Worked Example
WHEN YOU SEE A WORKED EXAMPLE
• Take your time to read through it,
• What is the main idea?
• Question your Aown
understanding,
geometric
sequence isand
a sequence of numbers in• How would this work if I changed
Think of
• Think about the
connections
between
steps.
which
the ratio between
any two
consecutive terms is a the numbers?
the constant you
constant. In other words, it is a sequence of numbers•in Have I used these strategies before?
multiply each term by to
which you multiply each term by a constant to determine
produce the next term. This will
the next term. This integer or fraction constant is called
tell you whether r is an
the common ratio. The common ratio is represented by
integer or a fraction.
the variable r.
Consider the sequence shown.
FM-42 1, 2, 4, 8, . . .
The pattern is to multiply each term by the same number, 2, to determine the
next term.
multiply
by 2
Sequence:
1
,
multiply
by 2
2
multiply
by 2
4
,
,
8
,...
This sequence is geometric and the common ratio r is 2.
4
3. Suppose a sequence has the same starting number as the sequence in the worked
example, but its common ratio is 3.
The sequence would still increase, but the terms would be different.
The sequence would increase more rapidly.
Yes. The sequence is still geometric because the ratio between any two
consecutive terms is constant.
c. If possible, write the first 5 terms for the new sequence.
1, 3, 9, 27, 81
xviii Problem Types
4.2
Arithmetic and Geometric Sequences
231
develop the desired habits of mind for being conscientious about the importance
of steps and their order.
8043_Ch04.indd 231
03/05/12 11:17 AM
© 2012 Carnegie Learning
b. Is the sequence still geometric? Explain your reasoning.
© 2012 Carnegie Learning
a. How would the pattern change?
© 2012 Carnegie Learning
Students will learn to recognize
the notebook call-out of a
worked example as they appear
throughout the lessons. Worked
examples provide a means
for students to view the flow
of each step taken to solve
the example problem. In most
cases, students will answer
questions posed about the
worked example. The questions
are designed to serve as a
model for self-questioning
and self-explanations which is
useful because some students
don’t do this naturally. The
questions represent and mimic
an internal dialog about the
mathematics and the strategies.
This approach doesn’t allow
students to skip over the
example without interacting
with it, thinking about it, and
responding to the questions.
This approach will help students
ASK YOURSELF
© 2012 Carnegie Learning
Research shows students learn
best when they are actively
engaged with a task. Often
students only focus or mentally
engage with a problem when
they are required to produce a
“product” or “answer”. Carnegie
Learning texts offer a different
approach to worked examples
to help students benefit from
this mode of instruction. Many
students need a model to know
how to engage effectively with
worked examples. Students
need to be able to question
their understanding, make
connections with the steps,
and ultimately self-explain the
progression of the steps and
the final outcome.
Acknowledgments
Problem Types
Implementation
Note
Mathematical Practice Standard 8 Look for and express regularity in repeated reasoning.
The corresponding questions associated with worked examples help
students to look for and express regularity in repeated reasoning.
Student Page
8068_TIG_FM_00i-liv_Vol2.indd 42
5/11/12 12:18 PM
c. Identify three different solutions of the system of linear inequalities you graphed.
What do the solutions represent in terms of the problem situation?
Answers will vary.
Three possible solutions are (3, 2), (2, 4), and (4, 0).
The solution (3, 2) represents that Chase, 2 other adults, and 2 children weigh at
most 800 pounds and pay at least $150.
The solution (2, 4) represents that Chase, 1 other adult, and 4 children weigh at
most 800 pounds and pay at least $150.
Note
The solution (4, 0) represents that Chase and 3 other adults weigh at most
800 pounds and pay at least $150.
Research shows that only
providing positive examples
does not eliminate some of the
things students may think; it is
also efficient to show negative
examples. From the incorrect
responses, students learn to
determine where the error in
calculation is, why the method
is an error, and also how to
correct the method to correctly
calculate the solution. These
types of problems will help
students analyze their own work
for errors and correctness.
d. Determine one combination of adults and children that is not a solution for this
Thumbs Down
system of linear inequalities. Explain your reasoning.
Answers
will vary.
WHEN YOU SEE
A THUMBS
DOWN ICON
200(2) 1 100(1) # 800
• Take your time
to read through the
•
b. Choose a point in each shaded region of the graph. Determine whether each point
is a solution of the system. Then describe how the shaded region represents
5. Analyze the solution
set of the system of linear inequalities shown.
the solution.
x1y.1
 2x 1 y # 3
a. Graph the system
Pointof linear xinequalities.
1y.1
(28, 2)
y
8
6
(2, 8)
2(28) 1 2 # 3
10 # 3 ✗
218.1
10 . 1 ✓
4
0
2
4
6
8
22 1 8 # 3
6#3✗
The point is a solution of the
first inequality, but not the
second. It is located in the
region shaded by the first
inequality.
28 1 2 # 3
26 # 3 ✓
The point is a solution for both
inequalities and it is located in
the region shaded by both
inequalities.
x
812.1
10 . 1 ✓
24
28
7
c. Alan makes the statement shown.
422
© 2012 Carnegie Learning
The point is a solution of the
22 1 (28) . 1
2(22) 1 (28) # 3
210 . 1 ✗
26 # 3 ✓ second inequality, but not the
first. It is located in the region
shaded by the second
inequality.
Chapter 7
Systems of Inequalities
The intersection point is always an algebraic
solution to a system of inequalities because that
is where the two lines meet.
8043_Ch07.indd 422
© 2012 Carnegie Learning
Alan
03/05/12 11:40 AM
Explain why Alan’s statement is incorrect. Use the intersection point of this system
to explain your reasoning.
(21, 2)
21 1 2 . 1
2(21) 1 2 # 3
1.1
3#3
Alan is incorrect because the intersection point is not always a solution to the
system of linear inequalities. The intersection point for this system only works for
one of the inequalities, not both which means it is not a solution. If the inequality
symbols are not both “or equal to” then the intersection point is not a solution.
7
7.2 Systems of Linear Inequalities
423
Problem Types 8043_Ch07.indd 423
xix
03/05/12 3:44 PM
Student Page 8068_TIG_FM_00i-liv_Vol2.indd 43
Implementation
26
(22, 28)
The point is not a solution to
either inequality and it is located
in the region that is not shaded
by either inequality.
Problem Types
Mathematical Practice
Standard 3 Construct viable arguments
and critique the reasoning
of others.
2x 1 y # 3
28 1 2 . 1
26 . 1 ✗
Notice the
inequality
symbols. How do you
think this ofwilllocation
affect
Description
your graph?
© 2012 Carnegie Learning
Points chosen will vary.
22
© 2012 Carnegie Learning
• Where is the error?
• Why is it an error?
125 $ 150 ✗
• How can I correct it?
75 1 50 $ 150
Although Chase, 1 other adult, and 1 child are within the weight limit for the raft,
the money earned is less than $150. Because this ordered pair does not produce
true statements in both inequalities, it is not a solution.
(8, 2)
© 2012 Carnegie Learning
75(2 2 1) 1 50(1) $ 150
400 1 100 # 800
incorrect solution.
500was
# 800
✓
Think about what error
made.
28 26 24 22
Thumbs down problem
types provide opportunities
for students to analyze
incorrect logic and explain
the flaw in the reasoning.
ASK YOURSELF
The point (2, 1) does not represent a solution.
FM-43
5/11/12 12:18 PM
Note
Thumbs Up
WHEN YOU SEE A THUMBS UP ICON
•
• Why is this method correct?
• Have I used this method before?
correct solution.
Think about the connections between steps.
8. Pat and George each wrote a function to represent the number of rice grains for any
square number using different methods.
Pat
George
I compared the exponents of the power
to the square number in the table. Each
exponent is 1 less than the square number.
f (s) = 2s –1
This problem type is designed
to foster flexibility and a
student’s internal dialog about
the mathematics and strategies
used to solve problems.
I know this is an exponential function with a
common base of 2. If I extend the pattern
back on the graph I get the y-intercept of
(0, __1 ), so a = __1 .
2
2
f (s) = __1 (2)s
2
1 (2)s are equivalent.
Use properties of exponents to verify that 2s21 and __
2
2s21 5 (2s)(221)
5 (2 )( __1 )
2
5 ( __1 )(2 )
s
2
Mathematical Practice
Standard 1 Make sense of problems
and persevere in solving
them.
s
__1 (2) 5 (2)
2
s
(2)s
21
5 2211s
5 2s21
s 5 19
On the 19th square, there are 262,144 rice grains.
I determined the intersection point of the graphs
of f(s) 5 2s21 and g(s) 5 262,144. The intersection
point is at (19, 262,144).
5
Make sure
you adjust the
settings for your graph
window so that you can answer
each question!
Thumbs up problem types
provide opportunities for
students to analyze work
for correctness, learn new
strategies, or confirm a
method they have invented.
FM-44 © 2012 Carnegie Learning
a. Which square on the chessboard contains 262,144 rice grains?
s 5 16
I determined the intersection point of the graphs of
f(s) 5 2s21 and g(s) 5 32,768. The intersection point is at
(16, 32,768).
xx Problem Types
350
8043_Ch05.indd 350
Chapter 5
© 2012 Carnegie Learning
b. Which square on the chessboard contains 32,768 rice grains?
On the 16th square, there are 32,768 rice grains.
Mathematical Practice
Standard 3 Construct viable arguments
and critique the reasoning
of others.
© 2012 Carnegie Learning
9. Use the intersection feature of a graphing calculator to answer each question.
Write each answer as an equation or compound inequality. Explain how you determined
your answer.
© 2012 Carnegie Learning
Questions provide
opportunities for students
to analyze the approaches
of others.
ASK YOURSELF
• Take your time to read through the
Problem Types
Implementation
Thumbs Up problems provide
a framework that allows
students the opportunity to
analyze viable methods and
problem-solving strategies.
Questions are presented along
with the student work to help
students think deeper about
the various strategies, and to
focus on an analysis of correct
responses. One goal of these
problems is to help students
make inferences about correct
responses. These types of
problems will help students
analyze their own work for
errors and correctness.
Exponential Functions
03/05/12 11:23 AM
Student Page
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5/11/12 12:18 PM
Note
Research shows that only
providing positive examples
does not eliminate some of the
things students may think; it is
also efficient to show negative
examples. From the incorrect
responses, students learn to
determine where the error in
calculation is, why the method
is an error, and also how to
correct the method to correctly
calculate the solution. These
types of problems will help
students analyze their own work
for errors and correctness.
• Take your time to read through the situation.
• Question the strategy or reason given.
• Determine which solution is correct and which is
• Does the reasoning make sense?
• If the reasoning makes sense, what is
the justification?
• If the reasoning does not make sense,
not correct.
what error was made?
?
8. Carlos and Mikala do not like working with fractions. They rewrite their equation so
that it does not have fractions. Their work is shown.
Carlos
F 5 _59 C 1 32
(5)F 5 5 _59 C 1 32
5F 5 9C 1 160
5F 2 9C 5 160
(
)
Mikala
5 (F 2 32)
C 5 __
9
5 (F 2 32)
(9)C 5 (9)__
9
9C 5 5(F 2 32)
9C 5 5F 2 160
9C 2 5F 5 2160
Carlos and Mikala got two different equations. Who is correct?
3
Both Carlos and Mikala are correct. If they divide either equation by 21 they will
get the other equation.
© 2012 Carnegie Learning
© 2012 Carnegie Learning
ASK YOURSELF
Implementation
Who’s Correct problem
types provide opportunities
for students to analyze
incorrect logic and explain
the flaw in the reasoning.
WHEN YOU SEE A WHO’S CORRECT? ICON
Problem Types
Mathematical Practice
Standard 3 Construct viable arguments
and critique the reasoning
of others.
Who’s Correct?
9 and __
5 as well as the constant 32 had
9. In the original equations, the coefficients __
5
9
meaning based on temperature. What do the coefficients, 9 and 5 and the constant
160 represent in Carlos’s and Mikala’s equations?
Problem Types 190
Chapter 3
xxi
Linear Functions
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© 2012 Carnegie Learning
© 2012 Carnegie Learning
The values 9, 5, and 160 represent nothing in terms of temperature.
FM-45
5/11/12 12:18 PM
Note
The Standards for
Mathematical Practice
describe the expertise that we
seek to develop in our students.
Effective communication and
collaboration are essential skills
of the successful learner. It is
critical for students to have daily
opportunities to develop these
habits of mind. Use the “I can”
statements included to help
students become productive
mathematical thinkers.
TheStandardsforMathematicalPractice
Effective communication and collaboration are essential skills of a successful learner. With practice,
you can develop the habits of mind of a productive mathematical thinker.
Make sense of problems and
persevere in solving them.
explain what a problem “means”
in my own words.
•
•
analyze and organize information.
•
always ask myself, “does this
make sense?”
keep track of my plan and
change it if necessary
•
•
calculate accurately and efficiently.
•
specify units of measure and
label diagrams and other
figures appropriately to clarify
the meaning of different
representations.
use clear definitions when I talk
with my classmates, my teacher,
and others.
ReasoningandExplaining
Reason abstractly and
quantitatively.
Construct viable arguments
and critique the reasoning
of others.
I can:
create an understandable
representation of a
problem situation.
•
consider the units of measure
involved in a problem.
•
understand and use properties
of operations.
Habits of Mind
I can:
•
use definitions and previously
established results in
constructing arguments.
•
communicate and defend my
own mathematical reasoning
using examples, drawings, or
diagrams.
•
distinguish correct reasoning
from reasoning that is flawed.
•
listen to or read the conclusions
of others and decide whether
they make sense.
•
ask useful questions in an
attempt to understand other
ideas and conclusions.
© 2012 Carnegie Learning
•
© 2012 Carnegie Learning
xxii FM-46 •
I can:
© 2012 Carnegie Learning
Implementation
Habits of Mind
I can:
Attend to precision.
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ModelingandUsingTools
Use appropriate tools
strategically.
Model with mathematics.
I can:
identify important relationships
in a problem situation and
represent them using tools such
as, diagrams, tables, graphs, and
formulas.
I can:
•
apply mathematics to solve
problems that occur in
everyday life.
•
interpret mathematical results
in the contexts of a variety of
problem situations.
•
reflect on whether my results
make sense, improving the model
I used if it is not appropriate for
the situation.
•
•
use a variety of different tools
that I have to solve problems.
•
use a graphing calculator to
explore mathematical concepts.
•
recognize when a tool that I
have to solve problems might
be helpful and also when it
has limitations.
© 2012 Carnegie Learning
© 2012 Carnegie Learning
© 2012 Carnegie Learning
Look for and make
use of structure.
Look for and express regularity
in repeated reasoning.
I can:
I can:
•
look closely to see a pattern or
a structure in a mathematical
argument.
•
notice if calculations
are repeated.
•
•
can see complicated things
as single objects or as being
composed of several objects.
look for general methods
and more efficient methods to
solve problems.
•
•
can step back for an overview
and can shift my perspective.
evaluate the reasonableness of
intermediate results.
•
make generalizations based
on results.
Habits of Mind Student Page 8068_TIG_FM_00i-liv_Vol2.indd 47
Implementation
Habits of Mind
SeeingStructureandGeneralizing
xxiii
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Note
Each lesson provides opportunities for you to think, reason, and communicate mathematical understanding.
Here are a few examples of how you will develop expertise using the Standards for Mathematical Practice
throughout this text.
Problem 1
1. Compare this problem situation to the problem situation in
Lesson 2.1, The Plane! How are the situations the same?
How are they different?
2. Complete the table to represent this problem situation.
Independent Quantity
Reason abstractly
and quantitatively.
You will move from a real-life
context to the mathematics and
back to the context
throughout problems.
Dependent Quantity
Quantity
Units
0
Model with
mathematics.
You will identify
relationships and
represent them using
diagrams, tables,
graphs, and
formulas.
2
4
6
18,000
6000
Expression
t
Look for
and make use
of structure.
You will look for paterns
in your calculations and
use those to write formal
expressions and
equations.
© 2012 Carnegie Learning
Implementation
As We Make Our Final Descent
At 36,000 feet, the crew aboard the 747 airplane begins making preparations
to land. The plane descends at a rate of 1500 feet per minute until it lands.
Habits of Mind
The Standards for
Mathematical Practice
describe the expertise
that we seek to develop in
our students. Each lesson
provides opportunities for
students to think, reason,
and communicate their
mathematical understanding.
However, it is the responsibility
of teachers to recognize these
opportunities and incorporate
these practices into daily
classroom routines. Expertise is
a long-term goal and students
must be encouraged to apply
these practices to new content
throughout their school career.
8043_Ch02.indd 88
xxiv FM-48 Habits of Mind
© 2012 Carnegie Learning
Chapter 2 Graphs, Equations, and Inequalities
07/05/12 2:12 PM
© 2012 Carnegie Learning
88
© 2012 Carnegie Learning
3. Write a function, g(t), to represent this problem situation.
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4. Complete the table shown. First, determine the unit of measure for each expression.
Then, describe the contextual meaning of each part of the function. Finally, choose
a term from the word box to describe the mathematical meaning of each part of
the function.
input value
output value
y-intercept
rate of change
x-intercept
Description
Expression
Atend to
precision.
You will specify
units of measure to
clarify meaning.
Units
Contextual
Meaning
Mathematical
Meaning
t
21500
21500t
21500t 1 36,000
5. Graph g(t) on the coordinate plane shown.
Use appropriate
tools strategically.
You will use multiple
representations
throughout the
text.
y
© 2012 Carnegie Learning
© 2012 Carnegie Learning
Construct viable
arguments and critique
the reasoning of others.
You will share your answers with
your classmates and listen to their
responses to decide whether
they make sense.
28,000
24,000
20,000
16,000
12,000
8000
4000
0
4
8 12 16 20 24 28 32 36
Time (minutes)
2.2
8043_Ch02.indd 89
© 2012 Carnegie Learning
32,000
Height (feet)
© 2012 Carnegie Learning
36,000
x
Analyzing Linear Functions
89
07/05/12 2:12 PM
Habits of Mind Student Page 8068_TIG_FM_00i-liv_Vol2.indd 49
Implementation
Habits of Mind
36,000
xxv
FM-49
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