Inside Algebra Overview
with Sample Pages
Strategic Intervention
for Algebra
Grades 8–12
NEW!
INTERACTI
VE TEXT
Larry Bradsby
Helps students who
“just don’t get it”
Innovative, proven program
designed to help struggling
students at risk of not
passing algebra
Voyager—Your RtI Partner
Dedicated to the success of every student, Voyager
provides strategic and intensive interventions designed
to accelerate skill acquisition for all struggling students.
Voyager is a member of Cambium Learning® Group, the leading
educational company focused primarily on serving the needs of
at-risk and special student populations. The company operates three core
divisions: Voyager, which provides comprehensive interventions; Sopris, which is known for
supplemental and behavioral interventions; and Cambium Learning Technologies (CLT) which
includes IntelliTools®, Kurzweil Educational Systems®, Learning A–Z, and ExploreLearning.
Partnering for RtI Solutions
As your intervention partner, we will work with you to develop a customized and integrated
solution to meet your Response to Intervention (RtI) needs.
• Our math and literacy interventions are
•
research based and research validated
Experienced consultants and practitioners will work with you to
develop a customized intervention plan to meet your unique
systemwide needs and goals
• Our unparalleled
implementation support team will provide onsite and
online staff development to ensure fidelity of implementation
Voyager’s powerful and effective support interventions, services and educational technology
help accelerate
all struggling students to grade-level proficiency.
English
language learners (ELLs) and students with disabilities derive particular benefits from the
interventions and make dramatic gains.
2
Effective RtI Key Features and Benefits
There is no one-size-fits-all solution for struggling learners. Each system and student has specific needs—some
only require occasional additional instruction, while others require more comprehensive, long-term support.
What Does Your School Look Like?
Few
Students
Some
Students
Many
Students
Many
Students
Many
Students
Our interventions and support services are designed to meet the needs of all struggling students by providing
multitiered instructional interventions aligned to content standards and benchmarks, including the
Common Standards.
Key Features of Voyager Interventions
Benefit to Your School/District
Multitiered, systematic, scalable approach with
supports and tools for differentiated instruction
Implement an effective and comprehensive
intervention plan
Universal screening and embedded progress
monitoring along with a comprehensive
Web-based data-management system
Easily access RtI documentation online
and regularly assess and monitor every
student’s progress
Research-based and -validated
Deploy interventions that are proven to work and
to turn around low-performing schools
Onsite and online professional development to
increase fidelity of implementation
Build teacher capacity to ensure interventions are
implemented as intended and increase struggling
students’ academic achievement
3
Voyager—Your RtI Partner
A Continuum of Interventions for Your Struggling Students
Voyager’s interventions are intended to identify struggling students early—before they fall behind—and
provide the support they need to be successful.
We provide a continuum of academic interventions in reading and math designed to address the needs of your
struggling learners—from strategic interventions that support core curricula to intensive interventions for
students who need a completely different approach.
Intensive Interventions (Tier III)
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Strategic Interventions (Tier II)
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Core Programs (Tier I)
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4
More Intense,
More Comprehensive
Grades
Intensive
Literacy
Strategic
K–2
Strategic
3–12
Strategic
K–3
Intensive
K–5
6–9
Grades
5–10
2–8
Small group reading intervention designed to wrap around core curriculum and
accelerate students to grade level using a blended approach of teacher-led instruction
and online interactive practice
• 30–45 minutes daily
Small group Spanish reading intervention designed to build a strong foundation in
students’ native language to help strengthen their transition to English
• 30 or 40 minutes daily
Mastery-based, intensive reading and language arts intervention that targets the
needs of nonreaders, struggling readers, and English learners
• 90 or 120 minutes daily
High-interest reading intervention for middle and high school students that builds
academic vocabulary, comprehension, and fluency through motivating topics,
teacher-led instruction, and student-centered technology
• 50 minutes daily
Description
Mastery-based, intensive intervention that focuses on the foundational concepts and
problem-solving strategies needed for successful entry into algebra
• 50–60 minutes daily
Strategic intervention with a modular approach for targeted skill intervention to reach
grade-level expectations
• 40–45 minutes daily
Strategic
®
Mastery-based, intensive reading and language arts intervention with focus on
decoding, comprehension, spelling, and writing
• 60–90 minutes daily
8–12
Mastery-based, strategic intervention that provides additional strategies for
algebra success
• 50–60 minutes daily
Core
Strategic
Intensive
Mathematics
Description
K–5
Flexible elementary curriculum, organized by grade-level content and broken into two
components: Anchors and Excursions
• 50–60 minutes daily
5
What is Inside Algebra?
A balanced approach to teaching algebra
re
og
Pr
The four-step lesson design is a powerful tool that weaves:
Concept Development Activities that build conceptual
understanding through concrete modeling experiences
•
Problem-Solving Activities that build problem-solving
skills through relevant, real-world connections
•
Mastery
of Algebra
Concepts
Problem
Solving
Progress-Monitoring Activities that help build
computational fluency and monitor student understanding
or
Practice
in
g
• Aligns with all algebra curricula
Larry Bradsby
E TEXT
INTERACTIV
STUDENT
ASSESSME
NT
Larry Bradsby
• Flexible implementations to
address credit recovery needs
Visit www.voyagerlearning.com for
more information on Inside Algebra.
6
Mo
nito
ring
Practice Activities that support new learning through
games and small group activities
it
on
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Progres
•
Concept Development
es
s
•
ss Monito
rin
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Inside Algebra is an engaging, mastery-based algebra program to help at-risk
students through a multisensory approach to achieving algebra success.
These three aspects of research support every Inside Algebra objective.
Students develop conceptual understanding
Concept Development Activities
The National Mathematics
• Utilize manipulatives (provided with program)
Advisory Panel (NMP)
• Provide concrete modeling experiences
recommends that teachers
Practice Activities
employ a balanced approach
• Strengthen understanding of newly learned concepts
• Provide peer interaction through small group activities
and games
to teaching algebra focused
on conceptual understanding,
understanding
developing fluency in
Students increase computational fluency
Progress-Monitoring Activities
• Develop automaticity in basic skills
procedures and number
operations, and building strong
problem-solving skills.
• Provide information to adjust instruction
• Build fluency with one-minute drills
Algebra Skill Builders (online resource)
Students build problem-solving skills
Problem-Solving Activities
• Allow synthesis of a variety of skills
• Reinforce problem-solving strategies
• Provide real-world relevance
7
A
At
At-A-Glance
Who Is It for?
Inside Algebra is a strategic
intervention for all students
in grades 8–12 who “just
don’t get it” and need a
mastery-based, multisensory
approach to develop the
skills necessary for
algebra success.
What Makes
Inside Algebra Work?
The program promotes
flexible grouping and
provides multiple modeling
activities. It also provides
enhanced instruction for
English language learners.
• Pictorial representation
to help students visualize
concepts
• Hands-on manipulatives
to make concepts more
complete
• ExploreLearning Gizmos
for interactive learning
Research Base
The Inside Algebra
Components
Inside Algebra is written
with a focus on the mastery
of objectives and provides
a scaffold instructional
approached that is built
around 60 objectives that
support standards set forth
by the National Council of
Teachers of Mathematics
(NCTM).
The program contains a
wide array of components
to support student learning
and teacher implementation,
such as ExploreLearning
Gizmos, VPORT® online
data management system,
student texts, teacher guides,
assessments, blackline
masters, and much more.
Special Education: 2007–2008
60%
Classrooms Using High Implementation
59%
50%
40%
30%
20%
28%
10%
0%
Students Averaged a 110% Gain
Pretest
Posttest
Larry Bradsby
TEAC
HER PLAC
EMEN
T
Larry Bradsby
Pages 10–11
8
Pages 12–13
Pages 14–15
Pages 16–17
Larry Brad
sby
Cambium Learning® Group is the leading educational company focused
exclusively on at-risk and special student populations.
How Inside
Algebra Works
Inside Algebra
in the Classroom
Professional
Development
Inside Algebra supports
students through explicit
instruction organized in a
clear, consistent manner.
Explicit instruction is
supported by clearly defined
concepts and skills. A
variety of activities help
students learn and recognize
the relationships between
those concepts and skills.
This support is integrated
into each of the 12 chapters
in Inside Algebra, which are
organized into objectives
and activities. This includes:
• 60 objectives
• More than 500 activities
With the implementation
of higher math standards
nationwide, and algebra
becoming a prerequisite for
graduation in many states,
Inside Algebra is a muchneeded addition to any
classroom. This effective
and flexible program
allows teachers to design
instruction according to
individual student needs,
choose from a multitude
of activities that promote
mastery, and carefully
and accurately monitor
student progress.
At Cambium Learning
Group, we understand
that intervention solutions
don’t come from programs
alone. Voyager’s professional
development partnership
provides ongoing training
and implementation
support to maximize the
effectiveness of instruction.
Scope and Sequence
Algebra, according to the
NMP, is the gateway to
higher learning success.
A student’s performance
in algebra has a strong
correlation to their success
in upper-level mathematics.
A solid algebraic foundation
also correlates strongly with
access to college, graduation
from college, and
earning potential.
Pages 44–45
Sample Lesson
Chapter 9
CHAPTER
9
Objective 3
Instructional Plans
5-Day Instructional Plan
Use the 5-Day Instructional Plan when pretest results indicate that students would benefit
from a slower pace. This plan is used when the majority of students need more time or did
not demonstrate mastery on the pretest. This plan does not include all activities.
STUDENT PLACEMENT
Larry Bradsby
CD 1 Using Algebra Tiles
Day 1
)PA 1 Sharing the Factors
Pretest/Posttest 2006–2007
80%
PM 1 A
Apply
l Skills 1
Day 2
)CD 2 Making Area Rugs
(September–March)
PM 2 A
Apply
l Skills 2
Day 3
Objective 3
68%
60%
CD 3 Solving
S l i the
h TTrinomial
i
i l Equation
E
i
PM 3
Concept Development
PA 2
Activities
Day 4
) CD
2
PM 4
Making Area Rugs
ACCELERATE
ENTRY POINT 1
ENTRY POINT 2
Chapter 1: Variables and Expressions
Chapter 3: Solving
g Linear Equations
40%
20%
In this chapter, students develop an understanding
of expressions by comparing verbal and mathematical
expressions. They discover and apply the order of
operations to evaluate and simplify expressions and
determine whether an expression is true, false, or
open. Variables are introduced through substitution
and in general representations of basic identities
and properties.
Objective 1
Translate verbal expressions into mathematical
expressions and vice versa.
Objective 2
Evaluate expressions using the order of operations.
Objective 3
Solve open sentences by performing arithmetic
operations.
Solving Linear
Equations
In this chapter, students begin to solve basic linear
equations using addition, subtraction, multiplication,
division, or a combination of these operations. They
use a variety of tools to solve equations by keeping the
equation balanced. Students also explore and solve
proportions, and apply their learning to solve word
problems involving linear equations and proportions.
Chapter 1
VOCABULARY
3
35%
Solve linear equations with addition and subtraction.
Objective 2
Solve linear equations with multiplication and division.
power, page 9
Objective 3
square, page 9
Students Averaged a 94% Gain
Solve linear equations using one or more operations.
false, page 38
acute triangle, page 229
Pretest
isosceles triangle, page 229
Objective 4
obtuse triangle, page 229
right triangle, page 229
Objective 5
equivalent, page 235
Solve proportions that have a missing part.
percent, page 242
Objective 6
proportion, page 242
Use proportions to solve percent problems.
ratio, page 242
Chapter 9 t Object
rectangle, called an area rugg here.
3. Point out that although a trinomial has only three
elements, the area rug has four rectangles. Note
that the area rug diagram is similar to the algebra
tile concept.
4. Tell students we will start with trinomials that
have no leading coefficient for the x 2 term. In other
words, it is just like having the coefficient 1 in front
of it.
6. Write x 2 + 5x + 6 on the
x2
board. Have students place
the x 2 term in the upper left
rectangle and the constant
number, 6, in the lowest right rectangle.
equilateral triangle, page 229
Solve problems that can be represented as equations.
factor A monomial that evenly divides a value
5. Have students draw a blank area rug made up of
four rectangles, as shown on the board.
equation, page 174
multiplicative inverse, page 190
linear equation, page 200
open, page 38
PM 5
1. Review the following terms with students:
quadratic trinomial
A polynomial of the
Posttest
form ax 2 + bxx + c
806
0%
Chapter 3
true, page 38
Objective 4
Use mathematical properties to evaluate expressions.
DIRECTIONS
2. Draw a rectangularPretest
area rug
diagram. Explain to students
that a quadratic trinomial can
VOCABULARY
Objective 1
algebraic expression, page 8
variable, page 8
cube, page 9
Day 5
DIFFERENTIATE
Use with 5-Day or 4-Day Instructional Plan. In this activity,
students factor quadratic trinomials using area rugs.
8. Explain to students that they can use the area rug
to find the factors of x 2 + 5x (VJEFTUVEFOUT
as they label the outside lengths and widths of the
large rectangle. Make sure students recognize that
an x is written as both the length and width for the
upper left rectangle.
9. Tell students to look at the 3x in the upper right
rectangle. Point out that we already labeled the
width for this rectangle with an x . Make sure
students recognize that the length for this rectangle
is 3, making the overall length for the rectangle
x + 3. Have students find the overall width, x + 2.
Have a volunteer identify the factors of the original
trinomial by multiplying the length by the width.
x
x
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
represent
the total
area of)=
a Includes Problem Solving
PA = Practice
Activity
CHAPTER
1
CHAPTER
Variables
and Expressions
PM 5
)PS 1
Posttest
10. List more quadratic trinomials on the board, one at a
time. Have students factor the quadratic trinomials
CZNBLJOHBOBSFBSVHGPSFBDI$IPPTFTUVEFOUT
to present the area rugs by drawing them on the
board for all to see. Make sure they label the overall
length and width for the large rectangle. Also, ask
them to prove, by multiplying the factors, that the
length times the width equals the original trinomial.
Sample problems:
x 2 + 2x + 1 x
x
x 2 + 5x + 4 x
x
x 2 + 7x + 10 x
x
6
7. Tell students to list all
x2
3xx
combinations of factors
for the constant number.
2x
6
Point out that only one
combination of factors from the list will add up
OPUTVCUSBDU
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UFSNJOUIFPSJHJOBMUSJOPNJBM
&YQMBJOUIBUUIJT
combination will be the two coefficients that are
used inside the remaining two rectangles, the upper
right and lower left, in the area rug. 2x + 3x
x 2 + 7x + 12 x
x
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 2, page 815—All students, to assess progress
4-Day Instructional Plan:
CD 3, page 811—All students, for additional
concept development
)= Includes Problem Solving
Chapter 1
Pages 18–19
1
Pages 20–25
Chapter 3
169
810
Pages 26–43
Chapter 9 t Objective 3
Pages 46–85
9
Who Is It for?
Addresses All Major Topics of Algebra Recommended by
the NMP
Algebra, according to the NMP, is the gateway to higher learning
success. A student’s performance in algebra has a strong correlation to
their success in upper-level mathematics. A solid algebraic foundation
also correlates strongly with access to college, graduation from college,
and earning potential.
In fact, a student who completes Algebra II is
more than twice as likely to graduate from college.
Foundational Skills
The NMP says: “The coherence and sequential nature
of mathematics dictate the foundational skills that are
necessary for the learning of algebra. The most
important foundational skill not presently developed
appears to be proficiency with fractions (including
decimals, percents, and negative fractions).”
Chapters 1–2
Prealgebra Skills
Develops Mastery of These
Algebra Concepts
• Using operations with rational numbers
• Locating, comparing, and ordering
real numbers
• Finding the square root of a number
• Using variables to represent
specific values
• Using variables to write general
statements
• Using mathematical properties and
order of operations
• Using proportions to solve problems
• Graphing ordered pairs and relations
• Finding the domain and range of a relation
10
Who is it for?
Inside Algebra is for struggling students who need a multisensory, hands-on approach to achieve algebra success.
It can be used as:
• A supplement to a core program
• An intervention for students at risk of not passing algebra
• An intervention for students who need Algebra I credit recovery
• A two-year algebra course
Skills build sequentially to mastery.
Basic Algebra
The NMP says: “The Panel recommends that school
algebra be consistently understood in terms of
the major topics of school algebra.” These include
symbols and expressions, linear equations,
quadratic equations, functions, algebra of polynomials,
and combinatorics and finite probability.
Chapters 3–10
Advanced Algebra
The NMP says: “…research shows that completion
of Algebra II correlates significantly with success
in college and earnings from employment. In fact,
students who complete Algebra II are more than
twice as likely to graduate from college compared to
students with less mathematical preparation.”
Chapters 11–12
Basic Algebra Skills
Additional Algebra Skills
• Writing and solving equations in one
and two variables
• Using operations with rational
expressions
• Graphing linear equations
• Simplifying rational expressions
• Identifying functions from a graph,
table, or equation
• Solving rational equations
• Writing equations in standard, pointslope, and slope-intercept form, and
converting between forms
• Simplifying radical expressions
• Solving and graphing linear inequalities
• Solving equations involving radicals
• Solving systems of linear equations
and inequalities
• Using the Pythagorean theorem to
solve problems
• Adding, subtracting, and multiplying
polynomials
• Using coordinate geometry to
solve problems
• Dividing polynomials
• Using operations with radical expressions
• Factoring polynomials
• Solving quadratic polynomial equations
using a variety of methods
11
What Makes Inside Algebra Work?
Promotes flexible grouping and provides
multiple modeling activities
Whole class, small group, pairs, and individual models are recommended
d
and may be teacher-led or student-centered.
Teachers’ use of modeling is an essential element in promoting
understanding. Inside Algebra uses:
•
Pictorial representation to help students visualize concepts
•
Hands-on manipulatives to make concepts more complete
for interactive learning
•
The activities of Inside Algebra include tools that help students connect
their hands-on experiences with the pictorial representations, then relate
these to the symbolic representations of algebra.
12
Provides enhanced instruction for English language learners (ELLs)
Strategies include:
• Explicit vocabulary instruction
• Open-ended and short response prompts brief constructed responses
• Small group activities and games
• Formal and informal assessments
Teaches the language of math
Throughout Inside Algebra, students are increasingly responsible for understanding and applying math vocabulary as they:
• Collaborate with peers
• Justify their thinking on mathematical tasks and problem solving
• Demonstrate their understanding and proficiency in math on high-stakes assessments
13
Research Base
Concept Development: Students are often taught that algebra is a sequence of steps rather than a set of concepts that can be
meaningful and useful to them. As a consequence, students tend to memorize algorithms and miss the underlying concept. Equipped only
with fragmented understanding, students often become confused when the problems they encounter become increasingly complex.
A more effective approach to teaching algebra is to teach concepts, emphasizing the “how” and “why” of what students are doing: how to
solve problems, why they are using certain techniques, how and why the concept they have learned relates to new problems, and various
procedures for solving them (Witzel et al., 2001).
Practice Activities: In order to gain a true understanding of concepts and mastery of skills, students need focused practice
time (Marzano, Pickering & Polluck, 2001). Practice gives students the chance to shape the skills they have learned into a conceptual
understanding. If this does not occur, students may develop gaps in their learning and superficial understandings that will interfere
with their ability to apply what they have learned to more complex problems, causing frustration and failure (Mathematical Sciences
Education Board, 1990; Witzel et al., 2001).
Focus on Objectives: Most students develop a deeper understanding of algebraic content when their learning is structured
around an objective-based approach. In a well-designed, objective-based approach, students master objectives in a logical sequence in
which new learning builds on the foundation of prior learning.
Objective-based teaching has been shown to be especially useful for teaching students with learning disabilities, because these students often
have substantial difficulty retaining information (Ysseldyke, Thurlow, Langenfeld, Nelson, Teelucksingh & Seyfarth, 1998). Teaching to
objectives and using a variety of activities until mastery is achieved is more likely to ensure student success than a lesson-based approach.
Explicit Instruction: When curricular design clearly defines concepts and skills and identifies the relationships between those
concepts and skills, it is considered explicit (Woodward, 1991). Research findings show that more explicit math instruction improves
student achievement (Montague, 1997). In a study involving secondary students with learning disabilities, students made greater
improvement in the acquisition, application, maintenance, and generalization of information through explicit instruction than through
traditional instruction (Montague, 1997).
Hands-on and Manipulative-based Activities: Hands-on activities can make math relevant and interesting to students.
Giving students the opportunity to work through abstract algebraic concepts with manipulatives and hands-on activities helps them
see how concepts can be translated into real life (Devlin, 2000; Maccini & Gagnon, 2000). As a result, students are more interested in
the concepts being taught—they can see how they make sense in concrete terms and they can grasp the content more easily.
Cooperative Learning: Cooperative learning, or group investigation, denotes an instructional arrangement for teaching
academic and collaborative skills to small, heterogeneous groups of students (Rich, 1993). Cooperative learning is supported by the
NCTM (1991) and has been shown to benefit mathematics achievement in students with and without learning disabilities (Slavin,
Leavy & Madden, 1984).
Problem Solving: Problem-solving activities can foster deep and lasting learning in algebra. Problem solving also provides students
with opportunities for self-evaluation through reflection on their strategies. When using a problem-solving approach, teachers should
guide students through the activities with step-by-step directions—identifying problem areas for various students, helping students
understand the significance of each step, and drawing attention to relationships between abstract and concrete concepts (Witzel et al.,
2001; Miller & Mercer, 1997).
Ongoing Assessment: Frequent assessment of student learning, with reference to student performance on specific tasks, is
essential to effective instruction. Curriculum-based assessment, in which each student’s progress is measured as she or he moves
through the curriculum, generates data that can inform teachers’ evaluations of student learning and progress. Teachers can use
these data to develop an accurate assessment of each student’s progress and design appropriate instructional interventions for
students who are falling behind. Thus, the collection of data on student progress improves both instruction and student learning
(Jones, Wilson & Bhojwani, 1997).
14
Inside Algebra is Proven Effective
Implementation results
In a large urban New Mexico school district during the 2007–2008 school year, students with learning
disabilities used Inside Algebra as the primary algebra program. In the implementation, students completed
at least four lessons per week and teachers consistently used all the lesson components. Students showed an
average improvement of 110 percent.
Special Education: 2007–2008
60%
Classrooms Using High Implementation
59%
50%
Pretest
40%
Posttest
30%
20%
28%
10%
0%
Students Averaged a 110% Gain
In another implementation in the same district, students identified as low-achieving based
on state assessment scores and at risk of failing algebra used Inside Algebra during the
2006–2007 school year. Of the more than 100 students, some used Inside Algebra as their core
algebra program and others first attended their Inside Algebra classroom and then attended
an additional session utilizing a traditional algebra program. The pretest and posttest scores
for this group indicate that students showed remarkable gains.
Pretest/Posttest 2006–2007
80%
(September–March)
68%
60%
Posttest
40%
20%
Pretest
35%
0%
Students Averaged a 94% Gain
15
The Inside Algebra Components
Teacher materials
• Teacher Guides: two-volume set
• Teacher Placement Guide—guides teachers in
administering and scoring the placement test
• VPORT® Online Data Management System
• Online Resources
TEA
ACH
H
Larry Bradsby
TEAC
HER PLAC
EMEN
T
• Access to Selected ExploreLearning Gizmos
Larry Bradsby
Larry Brad
sby
• Algebra Skill Builders Blackline Masters
Student materials
• Student Interactive Text
• Assessment Book
STUD
ENT PLAC
EMEN
T
Larry Brad
sby
STUDENT ASSESSMENT
T
16
• Student Placement Test
• Access to Selected ExploreLearning Gizmos
Larryy Bradsbyy
Larry Bradsby
Inside Algebra supports teachers and
students with relevant technology
Enhance concept development
ExploreLearning Gizmos take advantage of research-proven
instructional strategies that provide fun, interactive simulations
to help students visualize and understand important concepts.
Teachers can supplement and enhance instruction with
powerful interactive visualizations of mathematics concepts.
Students can use Gizmos to manipulate key variables,
Screenshot reprinted with permission of ExploreLearning.
generate and test hypotheses, and engage in extensive
“what-if ” experimentation. These end-of-chapter
differentiation activities help make connections
between algebra and the real world.
VPORT tracks student progress
Inside Algebra includes the VPORT online data
management system that allows educators to collect and
report student results. Using VPORT, teachers, schools, and
districts can:
• Input scores from all assessments
• View individual student scores
• Print a variety of reports for multiple uses
Online resources provide daily teacher support
• Complete Online Teacher Guide
• Blackline Masters
• Student Extension Activity Pages
• Student Reinforcement Activity Pages
• Alternate Form B Chapter Tests
• Algebra Skill Builders Blackline Masters
• Additional Activity Resources
17
Scope and Sequence
Chapter 1—Variables and Expressions
Objective 1: Translate verbal expressions into mathematical expressions and vice versa.
Objective 2: Evaluate expressions using the order of operations.
Objective 3: Solve open sentences by performing arithmetic operations.
Objective 4: Use mathematical properties to evaluate expressions.
Chapter 2—Exploring Rational Numbers
Objective 1: Graph rational numbers on the number line.
Objective 2: Add and subtract rational numbers.
Objective 3: Compare and order rational numbers.
Objective 4: Multiply and divide rational numbers.
Objective 5: Find the principal square root of a number.
Chapter 3—Solving Linear Equations
Objective 1: Solve linear equations with addition and subtraction.
Objective 2: Solve linear equations with multiplication and division.
Objective 3: Solve linear equations using one or more operations.
Objective 4: Solve problems that can be represented as equations.
Objective 5: Solve proportions that have a missing part.
Objective 6: Use proportions to solve percent problems.
Chapter 4—Graphing Relations and Functions
Objective 1: Graph ordered pairs and relations.
Objective 2: Identify the domain, range, and the inverse of a relation.
Objective 3: Determine the range for a given domain of a relation.
Objective 4: Graph linear equations.
Objective 5: Determine whether a relation is a function, and find a value for a given function.
Chapter 5—Analyzing Linear Equations
Objective 1: Determine the slope given a line on a graph or two points on the line.
Objective 2: Write the equation of a line in standard form given two points on the line.
Objective 3: Draw a best-fit line, and find the equation of the best-fit line for a scatter plot.
Objective 4: Write linear equations in slope-intercept form to find the slope, x-intercept, and y-intercept, and
sketch the graph.
Objective 5: Use the slope of lines to determine if two lines are parallel or perpendicular.
Chapter 6—Solving Linear Inequalities
Objective 1: Solve and graph the solution set of inequalities with addition and subtraction.
Objective 2: Solve and graph the solution set of inequalities with multiplication and division.
Objective 3: Solve and graph the solution set of inequalities using more than one operation.
Objective 4: Solve and graph the solution set of compound inequalities and inequalities involving absolute value.
Objective 5: Graph inequalities in the coordinate plane.
18
Chapter 7—Solving Systems of Linear Equations and Inequalities
Objective 1: Solve systems of equations by graphing.
Objective 2: Determine whether a system of equations has one solution, no solutions, or infinitely many solutions.
Objective 3: Solve systems of equations using the substitution method.
Objective 4: Solve systems of equations by eliminating one variable.
Objective 5: Solve systems of inequalities by graphing.
Chapter 8—Exploring Polynomials
Objective 1: Multiply and divide monomials and simplify expressions.
Objective 2: Write numbers in scientific notation and find products and quotients of these numbers.
Objective 3: Add and subtract polynomials and express the answer so the powers of the terms are in
descending order.
Objective 4: Multiply a polynomial by a monomial and arrange the terms in descending order by powers.
Objective 5: Multiply two binomials and simplify the expressions, including special products of (a + b)(a + b) and
(a + b)(a – b).
Chapter 9—Using Factoring
Objective 1: Find the greatest common factor through prime factorization for integers and sets of monomials.
Objective 2: Use the greatest common factor and the Distributive Property to factor polynomials with the
grouping technique, and use these techniques to solve equations.
Objective 3: Factor quadratic trinomials of the form ax2 + bx + c, and solve equations by factoring.
Objective 4: Factor quadratic polynomials that are perfect squares or differences of squares, and solve equations
by factoring.
Objective 5: Solve quadratic equations by completing the square.
Chapter 10—Exploring Quadratic and Exponential Functions
Objective 1:
Objective 2:
Objective 3:
Objective 4:
Graph parabolas, and find the coordinates of the vertex and axis of symmetry.
Estimate the roots of a quadratic equation by graphing the associated function.
Solve quadratic equations by factoring or using the quadratic formula.
Graph exponential functions, and solve problems using the graphs.
Chapter 11—Exploring Rational Expressions and Equations
Objective 1: Simplify rational expressions.
Objective 2: Multiply and divide rational expressions.
Objective 3: Divide a polynomial by a binomial.
Objective 4: Add and subtract rational expressions.
Objective 5: Solve equations involving rational expressions.
Chapter 12—Exploring Radical Expressions and Equations
Objective 1: Simplify and perform operations with radical expressions.
Objective 2: Solve equations with radical expressions.
Objective 3: Use the Pythagorean theorem to solve problems.
Objective 4: Find the distance between two points in the coordinate plane.
Objective 5: Find the unknown measures of the sides of similar triangles.
19
How Inside Algebra Works
Delivers Content Through Explicit Instruction
as Recommended by the NMP
Inside Algebra supports students through explicit instruction organized in a
clear, consistent manner
Explicit instruction is supported by clearly defined concepts and skills. A variety of activities help students learn and
recognize the relationships between those concepts and skills.
This support is integrated into each of the 12 chapters in Inside Algebra, which are organized into objectives and activities.
This includes:
• 60 objectives
• More than 500 activities
20
Instructional design of each objective
Every objective in Inside Algebra begins with a pretest. Students complete a combination of Concept Development, Practice,
and Progress-Monitoring activities before completing one or more Problem-Solving activities that synthesize student
learning and provide relevant applications. A posttest measures student mastery of the objective.
Consistent lesson format provides explicit direction
for teachers to present instruction to support student mastery
Pretest
Concept
Development
Progress
Monitoring
Practice
Problem Solving
Posttest
21
How Inside Algebra Works
Placement Test Pinpoints Skill Levels
Inside Algebra placement is based on students’ skill levels. Before instruction begins, the Placement Test is administered to all
students being considered for the Inside Algebra program. Student results determine placement into one of two entry points:
STUDENT PLACEMENT
Chapter 1: Variables and Expressions
Chapter 3: Solving Linear Equations
In this chapter, students develop an understanding
of expressions by comparing verbal and mathematical
expressions. They discover and apply the order of
operations to evaluate and simplify expressions and
determine whether an expression is true, false, or
open. Variables are introduced through substitution
and in general representations of basic identities
and properties.
Objective 1
Translate verbal expressions into mathematical
expressions and vice versa.
Objective 2
Evaluate expressions using the order of operations.
Objective 3
Solve open sentences by performing arithmetic
operations.
CHAPTER
ENTRY POINT 2
CHAPTER
ENTRY POINT 1
Variables
and Expressions
Solving Linear
Equations
In this chapter, students begin to solve basic linear
equations using addition, subtraction, multiplication,
division, or a combination of these operations. They
use a variety of tools to solve equations by keeping the
equation balanced. Students also explore and solve
proportions, and apply their learning to solve word
problems involving linear equations and proportions.
Chapter 1
VOCABULARY
Chapter 3
VOCABULARY
Objective 1
algebraic expression, page 8
Solve linear equations with addition and subtraction.
variable, page 8
Objective 2
cube, page 9
Solve linear equations with multiplication and division.
power, page 9
equation, page 174
multiplicative inverse, page 190
linear equation, page 200
acute triangle, page 229
square, page 9
Objective 3
equilateral triangle, page 229
false, page 38
Solve linear equations using one or more operations.
isosceles triangle, page 229
open, page 38
Objective 4
obtuse triangle, page 229
true, page 38
Solve problems that can be represented as equations.
right triangle, page 229
Objective 5
equivalent, page 235
Solve proportions that have a missing part.
percent, page 242
Objective 6
proportion, page 242
Use proportions to solve percent problems.
ratio, page 242
Objective 4
Use mathematical properties to evaluate expressions.
Chapter 1
Guide,
Chapter 1 O
Opener
TTeacher
h G
id Ch
22
Larry Bradsby
1
Chapter 3
Guide,
Chapter 3 O
Opener
TTeacher
h G
id Ch
169
The Comprehensive Assessment System Tracks and Monitors
Student Growth from Placement to Mastery
This user-friendly assessment system provides teachers with the measures they need to accurately place students and
monitor their progress though the curriculum. It furnishes the teacher with the data necessary to inform instruction to
ensure each student meets his or her goals.
Placement
Test
Larry Bradsby
VPORT
ins
t
ce
ct
ru
reinfo
r
STUDENT PLACEMENT
Ongoing
Assessment
assess
•D
Daily
il application
li i
• Objective pretests
• Objective posttests
• Chapter tests
• Extension activities
• Reinforcement activities
Placement
Based on students’ demonstrated understanding of key mathematics concepts and skills, data from the Inside Algebra
Placement Test accurately place students at one of the two entry points of the program.
Ongoing assessments
Regular assessment of student mastery of the concepts and skills taught in the program ensures that teachers can adjust
pacing or instruction to meet the needs of individual students.
VPORT
This user-friendly data management system allows teachers and administrators to record, track, and report student test
results. Reports can be generated at the individual, class, building, and district levels.
23
How Inside Algebra Works
Easy-to-access Data Informs Differentiation
During Instruction …
Inside Algebra offers multiple opportunities to assess, reinforce,
and differentiate instruction to promote mastery of each objective
CD 1 Using Algebra Tiles
After each Objective
Pretest teachers use
data to select an
appropriate instructional
plan for the class.
ACCELERATE
Day 1
PM 2 Apply Sk
Skills 2
CD 3 Solving the
Trinomial Equation
PM 4 Apply Sk
Skills 4
Throughout each
instructional plan
teachers use informal
assessment data to identify
groups for acceleration or
differentiation, providing
a second layer of
differentiation to support a
range of learners.
DIFFERENTIATE
PM 1 Apply S
Skills 1
)PA 1 Sharing the Factors
PM 2 Apply S
Skills 2
DIFFERENTIATE
Day 2
PA 2 Finding the
Solution Bingo
CD 3 Solving
S l i the
t
Trinomial Equation
PM 5 Apply Skills 5
PM 3 Apply Skills 3
PA 2 Fi
Finding
di the
t
Solution Bingo
Day 3
)CD 2 M
Making
ki Area
Rugs
CD 3 Solving the
Trinomial Equation
PM 3 Apply
A l Sk
Skills 3
)PS 1 Paving the Yard
PM 4 Apply Skills 4
After each Objective
Posttest teachers use
data to identify students
who may need additional
instruction, either one-onone or in small groups.
DIFFERENTIATE
)PS 2 Finding
Dimensions
)PS 1 P
Paving
i th
the Yard
PA 2 Finding the
Solution Bingo
PM 4 Apply
A l Skills
Sk 4
PM 5 Apply Skills 5
Day 4
Posttest
estt Obj
Objective 3
Pretest Objective 4
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
24
…
And After Assessment
Extensions and reinforcements are provided in every chapter
After administering the Chapter Test, teachers determine differentiation using student data by:
• Scoring the Chapter Test by objective and reviewing student scores
• Identifying differentiation needs
• Establishing small groups for extension or reinforcement
Differentiation Planner (Chapter 9)
Students who demonstrated mastery and
score at or above 80 percent on the
Chapter Test complete extension activities
individually or in pairs.
Students who demonstrated mastery on every objective
posttest and scored 80% or above on the Chapter Test
Extend learning using:
•
Students who demonstrated mastery but
score below 80 percent on the Chapter
Test complete independent reinforcement
activities in pairs or small groups.
Gizmos Use the Quadratics in Polynomial Form—
Activity A Gizmo with the Extension Activity. Have students
work in pairs or individually to complete the activity
Students who demonstrated mastery on every objective
posttest but scored below 80% on the Chapter Test
Reinforce learning using:
ExploreLearning Gizmos provide
relevant, real-world activities.
•
Gizmos Use the Modeling the Factorization of
x2 + bx + c Gizmo with the Reinforcement Activity.
Have students work in pairs or small groups to
complete the activity
• Additional Activities from the online resources
Students who did not demonstrate
mastery on any or all the objectives
on the objective posttests or the
Chapter Test complete teacher-guided
reinforcement activities in a small group.
• Algebra Skill Builders for Chapter 9 from the
online resources
Students who did not demonstrate mastery on any or all of
the objective posttests or the chapter test
Reinforce learning using:
•
Gizmos Present the Modeling the Factorization of
x2 + bx + c Gizmo to students in small groups using
the instruction on page 861
• Additional Activities from the online resources
Retest students who scored below 80
percent using the Chapter Test, Form B,
from the online resources.
• Algebra Skill Builders for Chapter 9 from the
online resources
Students who scored below 80%
Retest—Administer Chapter 9 Test, Form B, from the online
resources to students who scored below 80 percent on
Form A when time allows.
25
Inside Algebra in the Classroom
Progress Monitoring and Reporting
Teachers and administrators use VPORT to inform ongoing decision-making
so that every child is successful. The VPORT online data management system
incorporates benchmark and progress-monitoring assessments with real-time data
management to:
• Identify individual instructional needs and goals
• Adjust instruction based on skill need
• Monitor progress against goals
• Communicate progress to the instructional team
• Generate parent reports in English and Spanish
Real-time reporting
The key to effective instruction is real-time data that track student progress
throughout the year. The Inside Algebra Assessment System uses VPORT data
to provide multiple reports that help identify student needs, adjust instruction,
monitor progress, and evaluate instructional effectiveness.
26
Introducing the Chapter
Each chapter of Inside Algebra is focused on helping students master the concepts and skills necessary for future
success. Chapters are organized into Objectives, which students master through Concept Development, Practice,
CHAPTER
Progress-Monitoring, and Problem-Solving Activities.
Using Factoring
In this chapter, students explore and gain an
understanding of polynomials, including quadratic
trinomials. They apply concepts of factoring to
monomials and use a variety of factoring strategies
with polynomials. Students use factoring and models
as tools for solving quadratic polynomials.
Objective 1
Find the greatest common factor through prime
factorization for integers and sets of monomials.
Objective 2
Chapter 9
VOCABULARY
factor, page 772
Use the greatest common factor and the Distributive
Property to factor polynomials with the grouping
technique, and use these techniques to solve equations.
greatest common factor (GCF), page 772
Objective 3
quadratic formula, page 820
Factor quadratic trinomials of the form ax 2 + bx + c,
and solve equations by factoring.
perfect square, page 826
Objective 4
difference of squares, page 829
Factor quadratic polynomials that are perfect squares or
differences of squares, and solve equations by factoring.
quadratic polynomial, page 829
prime factorization, page 777
quadratic trinomial, page 808
perfect square trinomial, page 826
completing the square, page 848
Objective 5
Solve quadratic equations by completing the square.
Key vocabulary listed at the beginning of
each chapter facilitates the preteaching of
important math ideas.
Clearly defined
Objectives for
each chapter
present concepts
and skills in a
logical sequence.
Using Factoring, Teacher Guide, Chapter 9
27
Inside Algebra in the Classroom
Administer the Pretest for Each Objective
9
CHAPTER
Each Objective
Pretest provides
baseline data to
determine the
instructional path.
Objective 3
Factor quadratic trinomials of the form
ax 2 + bx + c, and solve equations by factoring.
Objective 3 Pretest
Students complete the Objective 3 Pretest on the same day
as the Objective 2 Posttest.
Using the Results
t4DPSFUIFQSFUFTUBOEVQEBUFUIFDMBTTSFDPSEDBSE
t*GUIFNBKPSJUZPGTUVEFOUTEPOPUEFNPOTUSBUFNBTUFSZ
PGUIFDPODFQUTVTFUIF%BZ*OTUSVDUJPOBM1MBOGPS
Objective 3.
CHAPTER 9 t Objective 3
9
KWKVW
t*GUIFNBKPSJUZPGTUVEFOUTEFNPOTUSBUFNBTUFSZPGUIF
DPODFQUTVTFUIF%BZ*OTUSVDUJPOBM1MBOGPS0CKFDUJWF
Name __________________________________________ Date ____________________________
Factor the quadratic polynomials.
1.
x 2 + 5x + 6
2.
(x + 2)(x + 3)
3.
x 2 – 4x – 45
(x + 3)(x + 5)
4.
(x + 5)(x – 9)
5.
x 2 + 8x + 15
3x 2 – 19x + 6
(3x – 1)(x – 6)
x 2 – 5x – 24
(x + 3)(x – 8)
Solve the quadratic equations by factoring.
6.
x2 + x – 6 = 0
7.
(x + 3)(x – 2) = 0
x = –3, 2
8.
x 2 – 5x – 14 = 0
(x + 2)(x – 7) = 0
x = –2, 7
(x + 6)(x – 4) = 0
x = –6, 4
9.
6x 2 + x – 15 = 0
(3x + 5)(2x – 3) = 0
x = –5, 3
3 2
9x 2 + 12x – 5 = 0
(3x + 5)(3x – 1) = 0
x = –5, 1
3 3
128
Chapter 9 t Objective 3
Using Factoring, Teacher Guide, Chapter 9
28
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
10.
x 2 + 2x – 24 = 0
Objective 3
Goals and Activities
Objective 3 Goals
The following activities, when used with the instructional plans on
pages 806 and 807, enable students to:
t'BDUPSUIFRVBESBUJDQPMZOPNJBMx 2 + 6x – 16 to get
(x − 2)(x + 8)
t4PMWFUIFRVBESBUJDFRVBUJPOx 2 + 5x – 14 = 0 to get
x = 2, −7
Objective Goals
provide specific
examples of the skills
and concepts students
are expected to
learn through the
Objective Activities.
Objective 3 Activities
Concept Development Activities
CD 1 Using Algebra
Tiles, page 808
)CD 2 Making Area
CD 34PMWJOHUIF
5SJOPNJBM&RVBUJPO
page 811
Rugs, page 810
Practice Activities
PA 14IBSJOHUIF'BDUPSTQBHF
)
PA 2'JOEJOHUIF4PMVUJPO#JOHP
page 813
Progress-Monitoring Activities
PM 1
"QQMZ4LJMMT
1, page 814
PM 2
"QQMZ4LJMMT
2, page 815
PM 3
"QQMZ4LJMMT
3, page 816
PM 4
"QQMZ4LJMMT
4, page 817
A color-coded
objective overview
outlines the different
types of activities
provided to meet the
Objective Goals.
PM 5
"QQMZ4LJMMT
5, page 818
)
Problem-Solving Activities
)PS 1 Paving the Yard, page 819
)PS 2'JOEJOH%JNFOTJPOTQBHF
Ongoing Assessment
Posttest Objective 3, page 821
Pretest Objective 4, page 822
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
29
Inside Algebra in the Classroom
Determine the Appropriate Instructional Path Based on
Pretest Results
CHAPTER
9
When the majority
of students do not
demonstrate
mastery on the
Objective Pretest,
an intensified
Instructional Plan
provides additional
activities.
Two distinct Instructional
Plans provide explicit
guidance in the selection
of appropriate activities for
differentiation.
Objective 3
Instructional Plans
5-Day Instructional Plan
Use the 5-Day Instructional Plan when pretest results indicate
di t that
th t students
t d t would
ld benefit
b fit
from a slower pace. This plan is used when the majority of students need more time or did
not demonstrate mastery on the pretest. This plan does not include all activities.
CD 1 Using Algebra Tiles
Day 1
)PA 1 Sharing the Factors
PM 1 A
Apply
l Skills 1
Day 2
)CD 2 Making Area Rugs
PM 2 A
Apply
l Skills 2
Day 3
CD 3 Solving the Trinomial Equation
PM 3 Apply Skills 3
Differentiation
occurs through
alternate activities
based on whether
students demonstrate
understanding of the
concept or need
additional support.
PA 2 Finding the
h S
Solution Bingo
Day 4
PM 4 Apply Skills 4
ACCELERATE
DIFFERENTIATE
PM 5 Apply Skills
Skill 5
PM 5 Apply Skills 5
)PS 1 Paving the Yard
Day 5
Posttest
estt Obj
Objective 3
Pretest Objective 4
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
Using Factoring, Teacher Guide, Chapter 9
30
When the majority of students
demonstrate mastery on the
Objective Pretest, a streamlined
Instructional Plan provides an
alternate pathway for when the
class can move through the
activities at a faster pace.
4-Day Instructional Plan
Use the 4-Day Instructional Plan when pretest results indicate that students can move
through the activities at a faster pace. This plan is ideal when the majority of students
demonstrate mastery on the pretest.
CD 1 Using Algebra Tiles
ACCELERATE
Day 1
PM 2 Apply Skills
Sk 2
CD 3 Solving the
Trinomial Equation
PM 4 Apply Skills
Sk 4
DIFFERENTIATE
PM 1 Apply S
Skills 1
)PA 1 Sharing the Factors
PM 2 Apply S
Skills 2
DIFFERENTIATE
Day 2
PA 2 Finding the
Solution Bingo
CD 3 Solving
S l i the
t
Trinomial Equation
PM 5 Apply Skills 5
PM 3 Apply Skills 3
PA 2 Finding
Fi di the
t
Solution Bingo
Day 3
DIFFERENTIATE
)CD 2 Making
M ki Area
Rugs
CD 3 Solving the
Trinomial Equation
Differentiation
occurs through
alternate activities
based on whether
students demonstrate
understanding of the
concept or need
additional support.
PM 3 Apply
A l Skills
Sk 3
)PS 1 Paving the Yard
PM 4 Apply Skills 4
)PS 2 Finding
Dimensions
)PS 1 Paving
P i the
th Yard
PA 2 Finding the
Solution Bingo
PM 4 Apply
A l Skills
Sk 4
PM 5 Apply Skills 5
Day 4
Posttest
estt Objective
Obj
3
Pretest Objective 4
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
31
Inside Algebra in the Classroom
Provide Instruction for Content Mastery
Concept
Development
Activities use
manipulatives to
develop algebraic
thinking and
provide concrete
representations of
abstract concepts.
Objective 3
Concept Development
Activities
CD 1
Using Algebra Tiles
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students factor quadratic trinomials using
algebra tiles.
ExploreLearning Gizmos provide
alternate presentations of concepts
using interactive simulations and
virtual manipulatives.
Variation: Gizmos For this activity, use the tiles
in the Gizmo Modeling the Factorization of
x 2 + bx + c to model the factoring of these
quadratic expressions.
t Gizmos
MATERIALS
t "MHFCSB UJMFT POF TFU GPS FWFSZ UXP TUVEFOUT
t Variation: Gizmos
Modeling the Factorization of x 2 + bx + c
Consistent lesson
format provides
explicit direction for
teachers to present
instruction to support
student mastery.
DIRECTIONS
1. Review the following term with students:
factor A monomial that evenly divides a value
2. Review how to find the product of two binomials using
algebra tiles; for example, write (x
x + 1)(x + 2) on the
board and use the following rectangle to discuss:
x+2
x
+
1
x2
x
x
1 1
5. Write several polynomials on the board, and have
students use algebra tiles to find the factors. Call
on students to give you the factors they found and
write them under the appropriate polynomials.
x
Sample problems:
Be sure students see that
(x
x + 1)(x + 2) = x 2 + 3x
x + 2.
3. Discuss the following term with students:
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
4. Next, show students that to find factors of a
trinomial, they should make a rectangle out of the
given trinomial. In other words, work backward
from what is shown in Step 2. Write x 2 + 4x
x+3
on the board, and use algebra tiles to factor the
trinomial. Show students how to determine the
dimensions of the overall rectangle. (x + 1)(x + 3)
x+3
x
+
1
x2
x
x
1 1 1
x
x
Using Factoring, Teacher Guide, Chapter 9
32
Modeling the Factori
Factorization
ation of x 2 + bbxx + c
x 2 + 5x + 6 (x + 2)(x + 3)
x 2 + 4x + 4 (x + 2)2
x 2 + x − 6 (x − 2)(x + 3)
x 2 + 6x + 5 (x + 1)(x + 5)
6. Demonstrate how to factor x 2 + 5x
x + 6. (x + 2)(x + 3)
Discuss the relationship between the numbers (5 and
6) and the factors (2 and 3). Make sure students
recognize that 2 + 3 = 5 and 2 t 3 = 6. Use the model
to show why the relationship exists. Repeat this
process for all polynomials on the board.
7. Ask students to find the factors of x 2 + 7x
x + 10
and x 2 + x − 12 . Allow students to use the algebra
tiles if they need the model to find the factors.
x 2 + 7x + 10 = (x + 2)(x + 5), x 2 + x + 12 = (x – 3)(x + 4)
Note:: If students need more practice multiplying
binomials, refer to Chapter 8, Objective 5.
Demonstrate Conceptual Understanding through
Concept Development and Practice Activities
Practice Activities use games and
small group interaction to strengthen
conceptual understanding.
Objective 3
Name _______________________________________________________ Date __________________
Practice
Activities
PA 2
4
×
38
4 BINGO CARD
Finding the Solution Bingo
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students factor quadratic trinomials.
MATERIALS
t #MBDLMJOF .BTUFS t (BNF NBSLFST UP DPWFS TRVBSFT
Important vocabulary
is highlighted and
reviewed at point of
use to promote math
language development.
DIRECTIONS
1. Review the following terms with students:
factor A monomial that evenly divides a value
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
2. %JTUSJCVUF POF DPQZ PG #MBDLMJOF .BTUFS ¨ Bingo Card, to each student. Have each student put
the numbers −3, −2, −1, 0, 1, 2, 3 at random in the
squares of the bingo card. Point out that they will
have to repeat some numbers to fill the 16 squares.
3. Write an equation on the board, selected at
random from the list below. Tell students to solve
the equation and cover the squares that have the
solution(s) with their markers. Have students write
the equations and solutions on a piece of paper to
hand in at the end of the activity.
Equations to Use Solutions
Equations to Use Solutions
1. x 2 xx + 2 = 0
–2, –1
14. x 2 – 2x o o
2. x 2 o x 15. x 2 – x – 2 = 0
2, –1
3. x 2 o x 2
16. x 2 – 5xx + 6 = 0
4. x 2 + x – 6 = 0
o
17. x 2 + 2x o o
5. x 2 + x – 2 = 0
–2, 1
18. x 2 x oo
6. x 2 + 2xx + 1 = 0
–1
19. x 2 + 5xx + 6 = 0
oo
7. x 2 + 6xx + 9 = 0
o
20. x 2 + 2xx = 0
–2, 0
8. x 2 – x – 6 = 0
o
21. x 2 o –2, 2
9. x – 2xx = 0
0, 2
22. x xx = 0
o
10. x 2 x –2
23. x 2 – 2xx + 1 = 0
1
11. x 2 + x = 0
0, –1
24. x 2 o xx + 2 = 0
1, 2
12. x 2 – 6xx + 9 = 0
25. x 2 o x 2
13. x 2 o xx = 0
2
2
4. Continue with other equations. The first student to
get four markers in a row should call out, “Bingo!”
If the student’s answers are correct, that student
is the winner.
5. Alternatively, continue play until a student covers
all the squares on his or her card.
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 4, page 817—All students, to assess progress
4-Day Instructional Plan:
PM 5, page 818—Students who are on the
accelerated path, to assess progress
Next Steps provide
guidance based on student
performance along the
instructional path.
PM 4, page 817—Students who are on the
differentiated path, to assess progress
33
Inside Algebra in the Classroom
Progress-Monitoring
Activities determine
differentiation through
alternate activities as
they build fluency with
basic algebra skills.
Objective 3
Progress-Monitoring
Activities
PM 1
Apply Skills 1
Use with 5-Day or 4-Day Instructional Plan.
MATERIALS
t Interactive Text,
t page 346
DIRECTIONS
1. Have students turn to Interactive Text,
t page 346,
Apply Skills 1.
3. Monitor student work, and provide feedback as
necessary.
Watch for:
t %P TUVEFOUT GBDUPS UIF USJOPNJBMT VTJOH BMHFCSB
tiles to complete the rectangle?
Name __________________________________________ Date __________________________
A P P LY S K I L L S 1
Factor each of the quadratic trinomials.
Example:
x + 2)(x + 4)
x 2 + 6x + 8 = (x
1.
x 2 + 9x + 20 =
(x + 4)(x + 5)
2.
x 2 + 12x + 20 =
3.
x 2 – 4x – 32 =
(x + 4)(x – 8)
4.
x 2 + 4x + 3 =
5.
x2 + x – 6 =
6.
x 2 + 8x + 12 =
7.
x 2 + 6x + 5 =
(x + 1)(x + 5)
9.
x – 6x + 8 =
(x – 2)(x – 4)
10. x – 3x – 18 =
11.
x 2 – 4x + 3 =
(x – 1)(x – 3)
12.
x 2 + 10x + 21 =
13.
x 2 + x – 12 =
(x – 3)(x + 4)
14.
x 2 – 7x + 12 =
15.
x 2 + 9x – 10 =
(x + 10)(x – 1)
2
17. x – x – 30 =
2
19.
346
(x – 2)(x + 3)
(x + 5)(x – 6)
2x + 11x + 12 =
2
Chapter 9
t
(2x + 3)(x + 4)
8. x 2 + x – 2 =
(x + 1)(x + 3)
(x + 2)(x + 6)
(x – 1)(x + 2)
2
16. x 2 – 12x + 32 =
18.
(x + 2)(x + 10)
x – 8x – 9 =
2
20. 3x 2 + 16x + 5 =
(x + 3)(x – 6)
(x + 3)(x + 7)
(x – 3)(x – 4)
(x – 4)(x – 8)
(x + 1)(x – 9)
(3x + 1)(x + 5)
Objective 3 t PM 1
t %P BOZ TUVEFOUT USZ BO BMHFCSBJD NFUIPE
NEXT STEPS t Differentiate
5-Day Instructional Plan:
CD 2, page 810—All students, for additional
concept development and problem solving
4-Day Instructional Plan:
PA 1, page 812—All students, for additional
practice and problem solving
Using Factoring, Teacher Guide, Chapter 9
34
Modified wraparound
Teachers Guide includes
answer keys.
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
Informal assessment
strategies such as ask
for, watch for, and
listen for provide
further insight into
student progress.
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
T
M
KVV7OW
O7M
Monitor Progress Toward Mastering the Objective
Build Problem-Solving Skills and Demonstrate Relevance
Problem-Solving Activities reinforce problem-solving
strategies and reflective thinking as students synthesize
cumulative skills.
Objective 3
Problem-Solving
Activities
) PS
2
Finding Dimensions
Use with 4-Day Instructional Plan. In this activity,
students apply what they know about quadratic
equations to solve word problems.
DIRECTIONS
1. Discuss the following term with students:
√b 2 – 4ac
quadratic formula x = –bb ± 2a
where
ax 2 + bxx + c = 0
2. Read the following scenario to students:
A small calf needs to be kept
280
away from the herd of cattle
square feet
because of an infection. The
rancher has fences made of
tubing that can be put up quickly. The calf will
need 280 square feet of grazing land. The tube
frame will be six feet longer than it is wide. Find
the dimensions of the fence.
3. Guide students as they write an equation based
on the information they know. Remind students
to solve the equation to find the actual dimensions
of the area.
x(x + 6) = 280 sq. ft.
x 2 + 6x = 280
x 2 + 6x – 280 = 0
(x − 14)(x + 20) = 0
x = 14, −20; dimensions cannot be negative so the
fence is 14 ft. by 20 ft.
5. Read the following scenario to students:
A rectangular garden
(16 feet by 21 feet) has a
uniform rock path around
it. If the total area of the
garden and path is 500
square feet, what is the
width of the path?
Total area =
500 square feet
21 ft.
16 ft.
6. Guide students as they write an equation based
on the information they know. Remind students
to solve the equation to find the actual dimensions
of the area.
l t w = 500 sq. ft.
(21 + x + x )(16 + x + x ) = 500
(21 + 2x )(16 + 2x ) = 500
4x 2 + 74x + 336 = 500
4x 2 + 74x – 164 = 0
2x 2 + 37x – 82 = 0
(2x + 41)(x − 2) = 0
41
x = – 2 or 2; measurement must be positive so the
width of the path is 2 ft.
Examples of student
solutions showcase
one possible strategy
that students may use
to solve the problem.
NEXT STEPS t Differentiate
4-Day Instructional Plan:
Objective 3 Posttest, page 821—All students
4. Tell students to find the dimensions if the calf only
needs 160 square feet of grazing land.
x(x + 6) = 160 sq. ft.
x 2 + 6x = 160
x 2 + 6x – 160 = 0
(x − 10)(x + 16) = 0
x = 10, −16; dimensions cannot be negative so the
fence is 10 ft. by 16 ft.
)= Includes Problem Solving
35
Inside Algebra in the Classroom
Administer the Posttest for Each Objective
CHAPTER
9
Each Objective
Posttest measures
student growth in
mastering the
objective and
identifies concepts
that may need
reinforcement.
Objective 3
Ongoing Assessment
Objective 3 Posttest
Discuss with students the key concepts in Objective 3.
Following the discussion, administer the Objective 3
Posttest to all students.
Using the Results
t4DPSFUIFQPTUUFTUBOEVQEBUFUIFDMBTTSFDPSEDBSE
t1SPWJEFSFJOGPSDFNFOUGPSTUVEFOUTXIPEPOPU
EFNPOTUSBUFNBTUFSZPGUIFDPODFQUTUISPVHIJOEJWJEVBM
PSTNBMMHSPVQSFUFBDIJOHPGLFZDPODFQUT
1.
x 2 + 7x
x+6
2.
(x + 1)(x + 6)
3.
x 2 – 6x
x – 27
(x – 5)(x + 7)
4.
(x + 3)(x – 9)
5.
x 2 + 2x
x – 35
3x 2 – 19x
x – 14
(3x + 2)(x – 7)
CHAPTER 9 t Objective 3
Factor the quadratic polynomials.
9VWWKVW
Name __________________________________________ Date____________________________
4x 2 + 7x
x–2
(4x – 1)(x + 2)
Solve the quadratic equations by factoring.
6.
x – 10 = 0
x 2 + 3x
7.
(x + 5)(x – 2) = 0
x = –5, 2
8.
x 2 + x – 30 = 0
(x + 6)(x – 5) = 0
x = –6, 5
10.
x 2 + 3x
x – 28 = 0
(x + 7)(x – 4) = 0
x = –7, 4
9.
2x 2 – 3x
x – 14 = 0
(x + 2)(2x – 7) = 0
x = –2, 7
2
3x 2 + 14x
x+8=0
(x + 4)(3x + 2) = 0
x = –4, –2
3
Inside Algebra
Chapter 9 t Objective 3
Using Factoring, Teacher Guide, Chapter 9
36
129
Complete All Objectives in the Chapter
Flexible pacing meets the needs of a variety of learners
For each objective, teachers select the appropriate instructional plan to meet the needs of students.
• Teachers are encouraged to choose instructional plans based on student need and not time considerations
• Each objective is focused on mastery of the concepts and not a specific time frame
• Taking the time necessary to reach mastery is beneficial for students
The chart below outlines the minimum amount of time required to complete each chapter using the shorter Instructional
Plan for each objective.
Chapter
1
2
3
4
5
6
7
8
9
10
11
12
14
days
18
days
20
days
17
days
17
days
18
days
17
days
17
days
19
days
14
days
17
days
18
days
Total: 206 Days
The chart below outlines an alternative pacing for students who would benefit from an additional 15 to 20 days of
instruction beyond the minimum time required.
Summer School
School Year
Chapter 1
Chapters 2–12
Total: 221 Days
Using only the longer instructional plans, it will take students two full years to complete Inside Algebra.
37
Inside Algebra in the Classroom
After Completing the Chapter, Review Chapter Objectives
CHAPTER
9
.NTWK
%
K>OKZ
The Chapter Review
consolidates key
concepts to reinforce
objectives and
provides the
opportunity to monitor
student learning.
Name __________________________________________ Date __________________________
OBJECTIVE 1
Chapter
Review
Find the greatest common factor (GCF) for each pair.
1. 30 and 105
2. 42 and 54
30: 2 t 3 t 5
105: 3 t 5 t 7
GCF: 3 t 5 = 15
Chapter 9 Review
Use with 3-Day Instructional Plan A or 3-Day
Instructional Plan B. In this activity, students
review key chapter concepts prior to taking
the Chapter Test.
42: 2 t 3 t 7
54: 2 t 3 t 3 t 3
GCF: 2 t 3 = 6
3. 5a 3b 4 and 12a 3b
4. 12x 3y 2 and 9xy 3
5a 3b 4: 5 t a t a t a t b t b t b t b
12a 3b: 2 t 2 t 3 t a t a t a t b
GCF: a t a t a t b = a 3b
12x 3y 2: 2 t 2 t 3 t x t x t x t y t y
9xy 3: 3 t 3 t x t y t y t y
GCF: 3 t x t y t y = 3xy 2
OBJECTIVE 2
Factor the polynomials using the greatest common factor (GCF) and the
Distributive Property.
MATERIALS
5. 6x 2 + 3x
6. a 2b 3c 2 + ab 2c 3 + a 2b 2c 2
3x(2x + 1)
ab 2c 2(ab + c + a)
t Interactive Text,
t pages 363–364
DIRECTIONS
3. Monitor student work, and provide
feedback when necessary. If students
complete the review quickly, pair them
with other students or groups to discuss
their answers.
Review problems
organized by
objective facilitate
reteaching when
necessary.
7. x 2 + 4x = 0
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2. Have students complete the review
individually or in small groups. If the
activity is completed individually,
provide time for students to discuss their
solutions as a class or in small groups.
Solve the equations.
8. 5x 2 − 10x = 0
x(x + 4) = 0
x = 0, −4
5x(x − 2) = 0
x = 0, 2
Inside Algebra
.NTWK
%
K>OKZ
1. Have students turn to Interactive Text,
t
pages 363–364, Chapter 9 Review.
Chapter 9 t CR 9
363
Name __________________________________________ Date __________________________
OBJECTIVE 3
Factor the quadratic polynomials.
9. x 2 + 5x + 4
10. x 2 − 3x − 10
(x + 1)(x + 4)
(x − 5)(x + 2)
Solve the quadratic equations by factoring.
11. x 2 + 2x − 48 = 0
(x + 8)(x − 6)
x = −8, 6
12. x 2 + 2x − 3 = 0
(x + 3)(x − 1)
x = −3, 1
OBJECTIVE 4
Factor the quadratic polynomials.
13. x 2 + 4x + 4
14. x 2 − 25
(x + 2)2
(x + 5)(x − 5)
Solve the quadratic equations.
15. x 2 − 8x + 16 = 0
364
Chapter Review, Teacher Guide, Chapter 9
38
Chapter 9
t
CR 9
16. x 2 − 1 = 0
(x + 1)(x − 1) = 0
x = ±1
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
(x − 4)2 = 0
x=4
Administer the Chapter Test
CHAPTER
Ongoing
Assessment
Chapter 9 Test,
Form A
Objective 1
Find the greatest common factor (GCF) of these pairs using prime factorization.
1.
12 and 18
2.
6
3.
36 and 32
4
a 2b 2 and a 2b 5
4.
a 2b 2
6x 3y 4 and 9xy 2
3xy 2
Objective 2
MATERIALS
t Assessment Book, pages 135–136
DIRECTIONS
1. Have students turn to Assessment Book,
pages 135–136, Chapter 9 Test, Form A.
Administer the Chapter Test to all students.
Factor the polynomials using the greatest common factor (GCF) and the
distributive property.
5.
6x 2 + 9x
6.
3x(2x + 3)
12a 2b 2 + 8a 2b + 10ab 2
2ab(6ab + 4a + 5b)
Solve the equations.
7.
x=0
x 2 + 6x
8.
x(x + 6) = 0
x = 0, –6
3a 2 – 12a
a=0
3a(a – 4) = 0
a = 0, 4
2. Score the test by objective and update
the class record card.
NTWK
%WKVW0
-
9
Name __________________________________________ Date____________________________
The Chapter Test
measures student
mastery of skills
taught in the chapter
and forms the basis
for differentiation
using the Extension
or Reinforcement
Activities.
3. Use the test data to determine
differentiation needs.
.NTWK
%WKVW0
-
Inside Algebra
136
Chapter 9 t Test
135
Name __________________________________________ Date____________________________
Objective 3
Factor the quadratic polynomials.
9.
x 2 – x – 12
10.
(x + 3)(x – 4)
x 2 – 8x
x + 15
(x – 3)(x – 5)
Solve the quadratic equations by factoring.
11.
x+8=0
x 2 – 6x
12.
(x – 2)(x – 4) = 0
x = 2, 4
x 2 – 4x
x – 12 = 0
(x + 2)(x – 6) = 0
x = –2, 6
Objective 4
Factor the quadratic polynomials.
13.
9x 2 – 16
14.
(3x + 4)(3x – 4)
x 2 + 2x
x+1
(x + 1)2
Solve the quadratic equations.
15.
x + 49 = 0
x 2 + 14x
(x + 7)2 = 0
x = –7
Chapter 9 t Test
16.
x 2 – 36 = 0
(x + 6)(x – 6) = 0
x = ±6
Inside Algebra
Chapter Test, Teacher Guide, Chapter 9
39
Inside Algebra in the Classroom
Use Chapter Test Results to Identify Students for Differentiation
9
CHAPTER
For students who have
mastered objectives,
Extension Activities
use ExploreLearning
Gizmos in real-life
applications to
engage and extend
their knowledge of
chapter objectives.
Ongoing
Assessment
Differentiation
MATERIALS
t Gizmos Quadratics in Polynomial
Form—Activity A Gizmo
t Gizmos Extension Activity pages
t Gizmos Modeling the Factorization
of x 2 + bx + c Gizmo
Gizmos in either
student-centered or
teacher-led activities
that scaffold
instruction for
chapter objectives.
Students who demonstrated mastery on every
objective posttest and scored 80% or above
on the chapter test
Extend learning using:
t
Gizmos Use the Quadratics in Polynomial
Form—Activity A Gizmo with the Extension
Activity. Have students work in pairs or
individually to complete the activity.
t Additional Activities
Students who demonstrated mastery on every
objective posttest but scored below 80% on
the chapter test
t Algebra Skill Builders for Chapter 9
Reinforce learning using:
t Gizmos Reinforcement Activity page
For students who
have not completely
mastered objectives,
Reinforcement
Activities use
ExploreLearning
Differentiation Planner
t Chapter Test, Form B
DIRECTIONS
1. Review Chapter 9 Test, Form A, with
the class.
2. Use the results from Chapter 9 Test,
Form A, to identify students for
reinforcement or extension.
3. After students have been identified
for extension or reinforcement, break
students into appropriate groups. See
pages 859–861 for detailed differentiated
instruction.
t Gizmos Use the Modeling the
Factorization of x 2 + bx + c Gizmo with the
Reinforcement Activity. Have students
work in pairs or small groups to complete
the activity.
t "EEJUJPOBM"DUJWJUJFTGSPNUIFPOMJOF
resources.
t "MHFCSB4LJMM#VJMEFSTGPS$IBQUFSGSPN
the online resources.
Students who did not demonstrate mastery
on any or all of the objective posttests or the
chapter test
Reinforce learning using:
t Gizmos Present the Mo
odeling the
Factorization of x 2 + bx + c Gizmo to
students in small groups
s using the
instruction on page 861.
The color-coded
Differentiation
Planner quickly
t "EEJUJPOBM"DUJWJUJFTGSPN
NUIFPOMJOF
resources.
identifies whether
t "MHFCSB4LJMM#VJMEFSTGPS$IBQUFSGSPN
students
need extension
the online resources.
Retestt—Administer Chapter 9 Tesst, Form
B, from
the
or
reinforcement
based
online resources to students who scored
s
below 80 percent
on
the
results
of
the
on Form A when time allows.
Chapter Test.
NEXT STEPS t Pretest
t Administer Chapter 10, Objective 1
Pretest, page 864, to allll students.
Differentiation, Teacher Guide, Chapter 9
40
Launch Extension Activity to Differentiate
CHAPTER
9
Extension Activity
Ongoing
Assessment
Cassie is designing a large circular fountain. The distance from the
center of the fountain to the edge is 8 feet. Water will come from
many jets placed in the fountain. The path of the water from the jet
back into the fountain can be modeled by a quadratic polynomial.
The graph of a quadratic polynomial is called a parabola.
1.
Students who demonstrated mastery on every
objective posttest and scored 80% or above
on the chapter test
The path of the water from one of the jets can be modeled by the quadratic function y = −4x 2 + 8x,
x
where x is the distance in feet from the center of the fountain and y is the height of the water in
x = 0, 2
feet. Solve the equation 0 = −4x 2 + 8x.
Start the Quadratics in Polynomial Form—Activity A Gizmo.
Use the sliders to graph the function y = −4x 2 + 8x. What are
x = 0, 2
the x
x-values where the graph crosses the x-axis?
x
These are the x
x-intercepts of the graph. What is the y-value
y
0
that corresponds to an x-intercept?
x
Sketch a
graph of the function on the grid.
1. Divide students into pairs or allow them
to work individually for this activity.
2. Distribute one copy of the Extension
Activity from the online resources to
each student.
Name ___________________________________ Date _____________________
Q U A D R AT I C S I N P O LY N O M I A L F O R M — A C T I V I T Y A
y
5
4
3
2
1
x
How can you fi
find the x
x-intercepts by using the function
without looking at the graph?
1
2
3
4
5
6
7
Replace y with 0 and solve for x.
2.
The water leaves the jet in Problem 1 at (0, 0). This means that the jet is located at the center
of the fountain.
(2, 0)
At what point on the graph does the water return to the fountain?
2 feet
How many feet from the center does the water return to the fountain?
3. Direct students to the Gizmo
Quadratics in Polynomial Form—
Activity A through the Inside Algebra
Student Web site, http://insidealgebra.
voyagerlearning.com.
Does the water stay inside the fountain?
yes
How do you know?
The distance from the center to the edge is 8 feet and 2 feet is less than 8 feet.
4. Have students complete the Extension
Activity.
5. Peer Review. If there is time, have
students exchange papers with a peer.
They should review and discuss each
response and be prepared to explain
their thinking.
Inside Algebra % Chapter 9 % Extension
Extension Activity
The Extension Activity
engages students who
demonstrated mastery
of all objectives and
scored 80% or above
on the Chapter Test.
Students work
individually or in
pairs to complete
the activity.
1
Name ___________________________________ Date _____________________
Q U A D R A T I C S I N P O L Y N O M I A L F O R M — A C T I V I T Y A (continued )
Variation: If students do not have
access to the Gizmo, provide them
with graphs of the functions in
Problems 1–4.
3.
The path from another jet can be modeled by y = −2x 2 + 8x
x − 6. Use factoring to solve the equation
0 = −2x 2 + 8x
x − 6. x = 1, 3
According to this function and the x
x-intercepts, describe where the jet could be placed in relation
to the center of the fountain.
The jet could either be 1 foot or 3 feet from the center.
The water will return either 1 foot or 3 feet
Where will the water return to the fountain?
from the center.
t Gizmos
y
yes
The distance is 1 foot or 3 feet,
Does the water stay inside the fountain?
How do you know?
which is less than 8 feet.
4
3
2
1
Use the Gizmo to graph y = −2x 2 + 8x
x − 6 to verify your
answer. Sketch a graph on the grid.
4.
5
x
1
2
3
4
5
6
7
The path from another jet can be modeled by y = −0.5x 2 + 4x
x − 7.5. Use factoring to solve
0 = −0.5x 2 + 4x
x − 7.5. x = 3, 5 Hint:: The number −0.5 is a common factor of each term.
According to this function and the x
x-intercepts, describe where the jet could be placed in relation
to the center of the fountain.
The jet could either be 3 feet or 5 feet from the center.
Where will the water return to the fountain?
y
The water will return either 3 feet or 5 feet from
the center.
Does the water stay inside the fountain?
5
4
3
yes
2
1
Use the Gizmo to graph the function. Sketch a graph
on the grid.
5.
What do these values have in common?
Quadratics
Q
d ti iin Polynomial
P l
i l FForm—Activity
A ti it A
x
1
2
3
4
5
6
7
Each quadratic function is in the form y = ax 2 + bx
x + c. Look at the value of a in each function.
What do the shapes of the graphs have in common?
They are all negative.
They all open downward.
How do you think the value of a affects the graph?
Answers will vary, but students should recognize that when |a| < 1, the graph
is wider and when |a| > 1, the graph is narrower.
Inside Algebra % Chapter 9 % Extension
The Extension Activity
uses an ExploreLearning
Gizmo, a fun and
easy-to-use interactive
simulation that supports
many different
learning styles.
2
41
Inside Algebra in the Classroom
Launch Student-centered Reinforcement Activity to Differentiate
CHAPTER
9
Reinforcement Activity
Ongoing
Assessment
1.
6
6x
The GCF of 30x 2 and 24x
x is
C
What is the GCF of x 2 and x?
x
x
.
Use the GCF from Problem 1 to help you factor 30x 2 + 24x.
30x 2 + 24x
x=
3.
+ bX +
1, 2, 3, 5, 6, 10, 15, 30
1, 2, 3, 4, 6, 8, 12, 24
What are the factors of 24?
2.
X2
Find the greatest common factor (GCF) of 30x 2 and 24x.
What are the factors of 30?
What is the GCF of 30 and 24?
Students who demonstrated mastery on every
objective posttest but scored below 80% on
the chapter test
6x
5x
(
4
+
)
Use the factorization from Problem 2 to help you solve 30x 2 + 24x
x = 0.
30x 2 + 24x
x=0
1. Divide students into pairs or small
groups.
Peer Review promotes
discussion among
students. They use math
vocabulary to explain
their thinking which is a
key factor in solidifying
concepts.
Name ___________________________________ Date______________________
M O D EL IN G T HE FA C T O RI Z AT I O N O F
6x
(
6x
=0
x=
0
5x
4
+
or
or
)=0
5x + 4
–4
x=
=0
5
Start the Modeling the Factorization off x2 + bx + c Gizmo. Follow the instructions on the screen.
x + 5. Then arrange the tiles into a rectangle.
First use algebra tiles to model x 2 + 6x
2. Distribute one copy of the Individual
Reinforcement Activity from the online
resources to each student.
4.
3. Direct students to the Gizmo Modeling
the Factorization of x 2 + bx + c through
the Inside Algebra Student Web site,
http://insidealgebra.voyagerlearning.com.
5.
Write each quadratic polynomial and its factorization as shown by the algebra tiles.
6.
Each person in your group should use the Gizmo to factor as many polynomials as he or she can in
two minutes. One person should keep time and record the number of correct answers. The person
with the greatest number of correct answers wins.
4. Have students complete the
Reinforcement Activity.
5. Peer Review. If time permits, have
students exchange papers with a peer
to review and discuss each other’s
responses. Remind students to be
prepared to explain the reasoning
behind their responses.
What is the width of the rectangle?
x+1
What is the height?
x+1
Write the area of the rectangle as the width times the height: (
So, x 2 + 6x
x+5=(
2x 2 + 3x + 1
=(
x+1
x+1
)(
)(
x+5
2x + 1 )
x+5
)(
x+5
)
).
x 2 − 4x + 4
=(
x−2
)(
x−2
)
Inside Algebra t Chapter 9 t Reinforcement
1
t Gizmos
Variation: If students do not have
access to the Gizmo, provide them
with algebra tiles to use to model
and factor the polynomial in Problem
4. For Problem 6, provide students
with a list of quadratic polynomials
to factor.
Modeling the Factorization
Modeli
Factorizati of x 2 + bxx + c
Differentiation, Teacher Guide, Chapter 9
42
The student-centered Reinforcement Activity
uses ExploreLearning Gizmos to strengthen
understanding of chapter objectives for
students who demonstrated mastery of all
objectives and scored below 80% but at
or above 60%. Students work in pairs or in
small groups to complete the activity.
Use the Teacher-led Reinforcement Activity to Differentiate
CHAPTER
9
Ongoing
Assessment
Students who did not demonstrate mastery
on any or all of the objective posttests or the
chapter test
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TUVEFOUT VTF UIFJS BOTXFS UP 1SPCMFN t "TL TUVEFOUT XIBU x JT NVMUJQMJFE CZ
UP HFU x 5x "TL TUVEFOUT XIBU x
JT NVMUJQMJFE CZ UP HFU x 4
t "TL TUVEFOUT UP VTF UIF %JTUSJCVUJWF
1SPQFSUZ BOE TUBUF UIF GBDUPSFE
FYQSFTTJPO x xx(5x + 4)
t "TL TUVEFOUT UP IFMQ ZPV BSSBOHF UIF
UJMFT JOUP B SFDUBOHMF UP GBDUPS UIF
FYQSFTTJPO "T ZPV ESBH UJMFT JOUP
UIF SJHIU CPY QPJOU PVU UIF XJEUI BOE
IFJHIU XSJUUFO BMPOH UIF UPQ BOE MFGU
TJEFT PG UIF CPY
The teacher-led Reinforcement
Activity uses ExploreLearning
Gizmos and accompanying
teacher directions to reteach
key objectives for students
who did not demonstrate
mastery of any or
all objectives.
t "GUFS UIF SFDUBOHMF JT GPSNFE BTL
TUVEFOUT GPS UIF GBDUPST (x BOE
(x + 1)
5. "SSBOHF BMHFCSB UJMFT JOUP B SFDUBOHMF UP
TIPX x + 3x + 1 = (x x 'JSTU
BTL TUVEFOUT UP OBNF UIF FYQSFTTJPO
TIPXO CZ UIF UJMFT x + 3x + 1 /FYU
BTL UIFN UP OBNF GBDUPST HJWFO CZ UIF
MFOHUI BOE XJEUI PG UIF SFDUBOHMF (x + 1)
BOEx + 1)
6. 3FQFBU UIF QSPDFTT JO 1SPCMFN XJUI
x – 4x + 4 = (x o x o Variation: *G TUVEFOUT EP OPU IBWF
BDDFTT UP UIF (J[NP VTF B CMBDLCPBSE
PS PWFSIFBE QSPKFDUPS BOE BMHFCSB UJMFT
UP DPNQMFUF UIF BDUJWJUZ
The Variation describes
how to complete the
differentiation activities
if the teacher or the
students cannot access
ExploreLearning Gizmos.
t Gizmos
3. "TL TUVEFOUT UP TPMWF x x 6TF UIF BOTXFS UP 1SPCMFN 3FNJOE
TUVEFOUT PG UIF ;FSP 1SPEVDU 1SPQFSUZ
t "TL TUVEFOUT XIBU UXP FRVBUJPOT UIFZ
OFFE UP TPMWF UIF QSPCMFN xBOE
5x t "TL TUVEFOUT UP TPMWF UIF FRVBUJPOT
4
x BOEx = – 5
4. 4UBSU UIF Modeling the Factorization of
x 2 + CY + D (J[NP
t "TL TUVEFOUT UP OBNF UIF UJMFT OFFEFE
UP NPEFM x x %SBH UJMFT JOUP UIF
MFGU CPY BT UIFZ BOTXFS one x UJMFTJY
xUJMFTmWFVOJUUJMFT
.PEFMJOH UIF 'BDUPSJ
'BDUPSJ[BUJPO
BUJPO PG x + bbxx + c
43
Professional Development
At Cambium Learning® Group, we understand that intervention
solutions don’t come from programs alone. Voyager’s professional development
partnership provides ongoing training and implementation support to maximize the effectiveness
of instruction.
Focus on Fidelity
Voyager provides award-winning professional development to support effective teaching practices. The
hands-on, interactive design can be used in structured environments or in self-paced individual settings to
help teachers be successful from the start. Participants learn to:
• Use the VPORT data management system to assess students and differentiate instruction
• Apply new research and best practices
• Implement the program with ease and fidelity
The Voyager professional development partnership extends throughout the school year and integrates
continuous training and support services with detailed reporting on student achievement for teachers and
administrators. Our services embody the five keys to success.
• Reviews successes
and areas of focus
INITIAL
PLANNING
• Provides overview of
program and tools
LEADERSHIP
ORIENTATION
• Plans for the next
implementation
(summer or fall)
• Assists leaders in
setting goals and
expectations
AMOUNT OF
INSTRUCTION
YEAR-END
REVIEW AND
PLANNING
CLASSROOM
MANAGEMENT
FIVE KEYS
TO
SUCCESS
• Demonstrates handson application of
VPORT data
INSTRUCTIONAL
EFFECTIVENESS
TEACHER
TRAINING
• Facilitates consultative
analysis of student
benchmark data
DIFFERENTIATION
ASSESSMENT
• Prepares participants
for successful
implementation
• Provides tools to develop
action plans based on
student assessments
• Accesses real-time data
via VPORT
44
CONSULTATIVE
SUPPORT
• Available in both online
and face-to-face format
Initial Planning, Leadership Orientation, and Teacher Training
The professional development partnership begins with collaborative planning between district leadership
and Voyager’s support staff. This initial planning involves customizing program training and support to
align with district expectations and goals.
Voyager’s leadership orientation provides an opportunity for school leaders to review program
components and VPORT, Voyager’s online data management system. Leaders establish implementation
goals and expectations as well as an implementation plan and timeline for their school.
Student success depends on the strength of the teacher, and Voyager’s training focuses on improving the
quality of instruction by increasing teacher knowledge. Voyager’s professional development is unmatched in
the industry, offering teacher training through face-to-face sessions and an online course.
Consultative Support and Year-End Review/Planning
VPORT provides educators with immediate and transparent real-time data to track student progress
throughout the year. With Voyager’s consultative support, educators learn to:
• Identify student needs
• Monitor student progress against goals
• Evaluate student learning
• Adjust instruction based on skills and needs
One of the most important
benefits of the Voyager partnership
occurs during the year-end
review and planning stage.
Administrators and Voyager
support personnel review student
progress made during the year
and examine areas of focus for
the following year. Working
collaboratively, they analyze
benchmark data and set goals for
summer and fall implementations.
45
CHAPTER
Sample Lesson
Chapter 9
Using Factoring
In this chapter, students explore and gain an
understanding of polynomials, including quadratic
trinomials. They apply concepts of factoring to
monomials and use a variety of factoring strategies
with polynomials. Students use factoring and models
as tools for solving quadratic polynomials.
Objective 1
Find the greatest common factor through prime
factorization for integers and sets of monomials.
Chapter 9
VOCABULARY
factor, page 772
Objective 2
Use the greatest common factor and the Distributive
Property to factor polynomials with the grouping
technique, and use these techniques to solve equations.
greatest common fac
ctor (GCF)
g , page 772
Objective 3
quadratic formula, pa
age 820
prime factorization, p
page 777
quadratic trinomial, p
page 808
Factor quadratic trinomials of the form ax + bx + c,
and solve equations by factoring.
perfect square, pagee 826
Objective 4
difference of square
ess, page 829
Factor quadratic polynomials that are perfect squares or
differences of squares, and solve equations by factoring.
quadratic polynomiaal, page 829
2
perfect square trinomial
m , page 826
completing the squa
are
r , page 848
Objective 5
Solve quadratic equations by completing the square.
Chapter 9
46
767
CHAPTER
9
Objective 3
Factor quadratic trinomials of the form
ax 2 + bx + c, and solve equations by factoring.
Objective 3 Pretest
Students complete the Objective 3 Pretest on the same day
as the Objective 2 Posttest.
Using the Results
t4DPSFUIFQSFUFTUBOEVQEBUFUIFDMBTTSFDPSEDBSE
t*GUIFNBKPSJUZPGTUVEFOUTEPOPUEFNPOTUSBUFNBTUFSZ
of the concepts, use the 5-Day Instructional Plan for
Objective 3.
CHAPTER 9 t Objective 3
9
KWKVW
t*GUIFNBKPSJUZPGTUVEFOUTEFNPOTUSBUFNBTUFSZPGUIF
concepts, use the 4-Day Instructional Plan for Objective 3.
Name __________________________________________ Date____________________________
Factor the quadratic polynomials.
1.
x 2 + 5x
x+6
2.
(x + 2)(x + 3)
3.
x 2 – 4x – 45
(x + 3)(x + 5)
4.
(x + 5)(x – 9)
5.
x 2 + 8x
x + 15
3x 2 – 19x + 6
(3x – 1)(x – 6)
x 2 – 5x – 24
(x + 3)(x – 8)
Solve the quadratic equations by factoring.
6.
x2 + x – 6 = 0
7.
(x + 3)(x – 2) = 0
x = –3, 2
8.
x 2 – 5x
x – 14 = 0
(x + 2)(x – 7) = 0
x = –2, 7
10.
x 2 + 2x
x – 24 = 0
(x + 6)(x – 4) = 0
x = –6, 4
9.
6x 2 + x – 15 = 0
(3x + 5)(2x – 3) = 0
x = –5, 3
3 2
9x 2 + 12x
x–5=0
(3x + 5)(3x – 1) = 0
x = –5, 1
3 3
128
804
Chapter 9 t Objective 3
Inside Algebra
Chapter 9 t Objective 3
47
Objective 3
Goals and Activities
Objective 3 Goals
The following activities, when used with the instructional plans on
pages 806 and 807, enable students to:
t'BDUPSUIFRVBESBUJDQPMZOPNJBMx 2 + 6x – 16 to get
(x − 2)(x + 8)
t4PMWFUIFRVBESBUJDFRVBUJPOx 2 + 5x – 14 = 0 to get
x = 2, −7
Objective 3 Activities
Concept Development Activities
CD 1
Tiles, page 808
)CD 2 Making Area
CD 3 Solving the
Trinomial Equation,
page 811
Rugs, page 810
Practice Activities
PA 1 Sharing the Factors, page 812
)
PA 2'JOEJOHUIF4PMVUJPO#JOHP
page 813
Progress-Monitoring Activities
PM 1
Apply Skills
1, page 814
PM 2
Apply Skills
2, page 815
PM 3
Apply Skills
3, page 816
PM 4
Apply Skills
4, page 817
PM 5
Apply Skills
5, page 818
)
Problem-Solving Activities
)PS 1 Paving the Yard, page 819
)PS 2 Finding Dimensions, page 820
Ongoing Assessment
Posttest Objective 3, page 821
Pretest Objective 4, page 822
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
Chapter 9 t Objective 3
48
805
Sample Teacher Pages
CHAPTER
9
Objective 3
Instructional Plans
5-Day Instructional Plan
Use the 5-Day Instructional Plan when pretest results indicate that students would benefit
from a slower pace. This plan is used when the majority of students need more time or did
not demonstrate mastery on the pretest. This plan does not include all activities.
CD 1 Using Algebra Tiles
Day 1
)PA 1 Sharing the Factors
PM 1 A
Apply
l Skills 1
Day 2
)CD 2 Making Area Rugs
PM 2 A
Apply
l Skills 2
Day 3
CD 3 Solving the Trinomial Equation
PM 3 Apply Skills 3
PA 2'JOEJOHUIF4PMVUJPO#JOHP
UI 4
Day 4
PM 4 Apply Skills 4
ACCELERATE
DIFFERENTIATE
PM 5 Apply Skill
Skills 5
PM 5 Apply Skills 5
)PS 1 Paving the Yard
Day 5
Posttest
estt Objective
Obj
3
Pretest Objective 4
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
806
Chapter 9 t Objective 3
49
4-Day Instructional Plan
Use the 4-Day Instructional Plan when pretest results indicate that students can move
through the activities at a faster pace. This plan is ideal when the majority of students
demonstrate mastery on the pretest.
CD 1 Using Algebra Tiles
ACCELERATE
Day 1
PM 2 Apply Skills
Sk 2
CD 3 Solving the
Trinomial Equation
PM 4 Apply Skills
Sk 4
DIFFERENTIATE
PM 1 Apply S
Skills 1
)PA 1 Sharing the Factors
PM 2 Apply S
Skills 2
DIFFERENTIATE
Day 2
PA 2 Finding the
4PMVUJPO#JOHP
CD 3 Solving
S l i tthe
Trinomial Equation
PM 5 Apply Skills 5
PM 3 Apply Skills 3
PA 2 Fi
Finding
di tthe
4PMVUJPO#JOHP
Day 3
DIFFERENTIATE
)CD 2 M
Making
ki Area
A
Rugs
CD 3 Solving the
Trinomial Equation
PM 3 Apply
A l Sk
Skills 3
)PS 1 Paving the Yard
PM 4 Apply Skills 4
)PS 2 Finding
Dimensions
)PS 1 P
Paving
i th
the Yard
PA 2 Finding the
4PMVUJPO#JOHP
PM 4 Apply
A l Sk
Skills 4
PM 5 Apply Skills 5
Day 4
Posttest
estt Obj
Objective 3
Pretest Objective 4
CD = Concept Development PM = Progress Monitoring PS = Problem Solving
PA = Practice Activity )= Includes Problem Solving
Chapter 9 t Objective 3
50
807
Sample Teacher Pages
Objective 3
Concept Development
Activities
CD 1
Using Algebra Tiles
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students factor quadratic trinomials using
algebra tiles.
Variation: Gizmos For this activity, use the tiles
JOUIF(J[NP Modeling the Factorization of
x 2 + bx + c to model the factoring of these
quadratic expressions.
t Gizmos
MATERIALS
t "MHFCSBUJMFTPOFTFUGPSFWFSZUXPTUVEFOUT
t Variation: Gizmos
Modeling the Factorization of x 2 + bx + c
DIRECTIONS
1. Review the following term with students:
factor A monomial that evenly divides a value
2. Review how to find the product of two binomials using
algebra tiles; for example, write (x
x + 1)(x + 2) on the
board and use the following rectangle to discuss:
x+2
x
+
1
x2
x
x
1 1
5. Write several polynomials on the board, and have
TUVEFOUTVTFBMHFCSBUJMFTUPmOEUIFGBDUPST$BMM
on students to give you the factors they found and
write them under the appropriate polynomials.
x
Sample problems:
#FTVSFTUVEFOUTTFFUIBU
x + 2.
(x
x + 1)(x + 2) = x 2 + 3x
3. Discuss the following term with students:
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
4. Next, show students that to find factors of a
trinomial, they should make a rectangle out of the
given trinomial. In other words, work backward
x+3
from what is shown in Step 2. Write x 2 + 4x
on the board, and use algebra tiles to factor the
trinomial. Show students how to determine the
dimensions of the overall rectangle. x
x
x+3
x
+
1
808
x2
x
x
1 1 1
x
Modeling the Factori
Factorization
ation of x 2 + bbxx + c
x
x 2 + 5x + 6 x
x
x 2 + 4x + 4 x
2
x 2 + x − 6 x¦
x
x 2 + 6x + 5 x
x
x + 6. x
x
6. Demonstrate how to factor x 2 + 5x
%JTDVTTUIFSFMBUJPOTIJQCFUXFFOUIFOVNCFSTBOE
BOEUIFGBDUPSTBOE
.BLFTVSFTUVEFOUT
recognize that 2 + 3 = 5 and 2 t 3 = 6. Use the model
to show why the relationship exists. Repeat this
process for all polynomials on the board.
x + 10
7. Ask students to find the factors of x 2 + 7x
and x 2 + x − 12 . Allow students to use the algebra
tiles if they need the model to find the factors.
x 2 + 7xx
x
x 2 + xxo
x
Note:: If students need more practice multiplying
CJOPNJBMTSFGFSUP$IBQUFS0CKFDUJWF
Chapter 9 t Objective 3
51
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PA 1, page 812—All students, for additional
practice and problem solving
4-Day Instructional Plan:
PM 2, page 815—Students who demonstrate
understanding of the concept, to assess progress
PM 1, page 814—Students who need additional
support, to assess progress
Chapter 9 t Objective 3
52
809
Sample Teacher Pages
Objective 3
Concept Development
Activities
) CD
2
Making Area Rugs
Use with 5-Day or 4-Day Instructional Plan. In this activity,
students factor quadratic trinomials using area rugs.
DIRECTIONS
1. Review the following terms with students:
factor A monomial that evenly divides a value
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
2. Draw a rectangular area rug
diagram. Explain to students
that a quadratic trinomial can
represent the total area of a
rectangle, called an area rugg here.
3. Point out that although a trinomial has only three
elements, the area rug has four rectangles. Note
that the area rug diagram is similar to the algebra
tile concept.
4. Tell students we will start with trinomials that
have no leading coefficient for the x 2 term. In other
words, it is just like having the coefficient 1 in front
of it.
5. Have students draw a blank area rug made up of
four rectangles, as shown on the board.
6. Write x 2 + 5x + 6 on the
x2
board. Have students place
the x 2 term in the upper left
rectangle and the constant
number, 6, in the lowest right rectangle.
8. Explain to students that they can use the area rug
to find the factors of x 2 + 5x (VJEFTUVEFOUT
as they label the outside lengths and widths of the
large rectangle. Make sure students recognize that
an x is written as both the length and width for the
upper left rectangle.
9. Tell students to look at the 3x in the upper right
rectangle. Point out that we already labeled the
width for this rectangle with an x. Make sure
students recognize that the length for this rectangle
is 3, making the overall length for the rectangle
x + 3. Have students find the overall width, x + 2.
Have a volunteer identify the factors of the original
trinomial by multiplying the length by the width.
x
x
10. List more quadratic trinomials on the board, one at a
time. Have students factor the quadratic trinomials
CZNBLJOHBOBSFBSVHGPSFBDI$IPPTFTUVEFOUT
to present the area rugs by drawing them on the
board for all to see. Make sure they label the overall
length and width for the large rectangle. Also, ask
them to prove, by multiplying the factors, that the
length times the width equals the original trinomial.
Sample problems:
x 2 + 2x + 1 x
x
x 2 + 5x + 4 x
x
x 2 + 7x + 10 x
x
6
7. Tell students to list all
x2
3x
x
combinations of factors
for the constant number.
2x
6
Point out that only one
combination of factors from the list will add up
OPUTVCUSBDU
UPFRVBMUIFDPFGmDJFOUPGUIFNJEEMF
UFSNJOUIFPSJHJOBMUSJOPNJBM
&YQMBJOUIBUUIJT
combination will be the two coefficients that are
used inside the remaining two rectangles, the upper
right and lower left, in the area rug. 2x + 3x
x 2 + 7x + 12 x
x
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 2, page 815—All students, to assess progress
4-Day Instructional Plan:
CD 3, page 811—All students, for additional
concept development
)= Includes Problem Solving
810
Chapter 9 t Objective 3
53
Objective 3
Concept Development
Activities
CD 3
Solving the Trinomial Equation
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students solve quadratic trinomials by factoring.
DIRECTIONS
1. Review the following terms with students:
factor A monomial that evenly divides a value
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 3, page 816—All students, to assess progress
4-Day Instructional Plan:
PM 4, page 817—Students who are on the
accelerated path, to assess progress
PM 3, page 816—Students who are on the
differentiated path, to assess progress
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
2. Write x 2 + 5x + 4 = 0 on the board. Ask students
to think about how they would solve this equation.
Have volunteers try various methods by working the
problem on the board; for example, students may
use subtraction or they may divide by 5 or x.
3. If students do not suggest factoring, review
factoring and show that the problem can be written
as (x + 4)(x + 1) = 0.
4. Review the Zero Product Property:
If a t b = 0, then a = 0 or b = 0.
5. Demonstrate how to solve the factors.
(x + 4) = 0 or (x + 1) = 0
x = −4
or x = −1
x = −4, −1
6. Have students substitute the solutions into the
original equation to show that they work.
o
2o
o
2o
7. (JWFTUVEFOUTNPSFFRVBUJPOTBOEIBWFUIFNVTF
factoring to solve the equations.
Sample problems:
x + 8 = 0 x = −2, −4
x 2 + 6x
x 2 − 2x
x − 15 = 0 x = 5, −3
2x 2 + 11x
x + 12 = 0 x = − 3 , −4
2
Chapter 9 t Objective 3
54
811
Sample Teacher Pages
Objective 3
Practice
Activities
) PA
1
Sharing the Factors
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students factor quadratic trinomials.
DIRECTIONS
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 1, page 814—All students, to assess progress
4-Day Instructional Plan:
PM 2, page 815—All students, to assess progress
1. Review the following terms with students:
factor A monomial that evenly divides a value
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
2. Write (x
x ± a) and (x
x ± b), where −10 ≤ a ≤ 10 and
−10 ≤ b ≤ 10 on the board. Have the class come
VQXJUIUXPCJOPNJBMTJOUIJTGPSN(VJEFTUVEFOUT
as they multiply the binomials to get a trinomial, for
FYBNQMFxx xo
x
x 2 – 3xx – 28.
3. Divide the class into groups of four.
4. Have each group design three similar problems
using the guidelines on the board. Have them write
these problems on a piece of paper. On a new sheet
of paper, have students write the three trinomials
they get by multiplying their binomial pairs.
5. Have the groups exchange their trinomials with
another group in the class. Make sure students
hold onto the matching binomials they wrote. Tell
students to work in their groups to factor the three
trinomials they received.
6. After students finish, have each group pick one
problem to put on an overhead transparency and
present to the class. Tell groups to show how they
found the factors to the problem. This will allow
the class to see different ways to find the factors.
Students need to find a method they understand
and can use.
Variation: Writing Have each student write an
explanation of how to factor a trinomial, such as
x 2 + x – 6. Review the written explanations.
7. Repeat Steps 4–6 using two binomials of the form
(ax
x ± b) and (x
x ± c)
c . In this case, students practice
factoring trinomials with a coefficient for the x 2 term.
)= Includes Problem Solving
812
Chapter 9 t Objective 3
55
Objective 3
Name _______________________________________________________ Date __________________
Practice
Activities
4
×
38
4 BINGO CARD
PA 2 'JOEJOHUIF4PMVUJPO#JOHP
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students factor quadratic trinomials.
MATERIALS
t #MBDLMJOF.BTUFS
t (BNFNBSLFSTUPDPWFSTRVBSFT
DIRECTIONS
1. Review the following terms with students:
factor A monomial that evenly divides a value
quadratic trinomial A polynomial of the
form ax 2 + bxx + c
2. %JTUSJCVUFPOFDPQZPG#MBDLMJOF.BTUFS¨
#JOHP$BSEUPFBDITUVEFOU)BWFFBDITUVEFOUQVU
the numbers −3, −2, −1, 0, 1, 2, 3 at random in the
squares of the bingo card. Point out that they will
have to repeat some numbers to fill the 16 squares.
3. Write an equation on the board, selected at
random from the list below. Tell students to solve
the equation and cover the squares that have the
TPMVUJPOT
XJUIUIFJSNBSLFST)BWFTUVEFOUTXSJUF
the equations and solutions on a piece of paper to
hand in at the end of the activity.
Equations to Use Solutions
Equations to Use Solutions
1. x + 3xx + 2 = 0
–2, –1
14. x 2 – 2xx – 3 = 0
–1, 3
2. x – 4xx + 3 = 0
3, 1
15. x 2 – x – 2 = 0
2, –1
3. x 2 – 4xx + 4 = 0
2
16. x 2 – 5xx + 6 = 0
3, 2
4. x 2 + x – 6 = 0
–3, 2
17. x 2 + 2xx – 3 = 0
–3, 1
5. x + x – 2 = 0
–2, 1
18. x + 4xx + 3 = 0
–3, –1
6. x 2 + 2xx + 1 = 0
–1
19. x 2 + 5xx + 6 = 0
–3, –2
7. x + 6xx + 9 = 0
–3
20. x + 2xx = 0
–2, 0
8. x 2 – x – 6 = 0
3, –2
21. x 2 – 4 = 0
–2, 2
9. x 2 – 2xx = 0
0, 2
22. x 2 + 3xx = 0
0, –3
10. x + 4xx + 4 = 0
–2
23. x – 2xx + 1 = 0
1
11. x 2 + x = 0
0, –1
24. x 2 – 3xx + 2 = 0
1, 2
12. x – 6xx + 9 = 0
3
25. x – 4xx + 4 = 0
2
13. x 2 – 3xx = 0
0, 3
2
2
2
2
2
2
2
2
2
2
4. $POUJOVFXJUIPUIFSFRVBUJPOT5IFmSTUTUVEFOUUP
HFUGPVSNBSLFSTJOBSPXTIPVMEDBMMPVUi#JOHPw
*GUIFTUVEFOUTBOTXFSTBSFDPSSFDUUIBUTUVEFOU
is the winner.
5. Alternatively, continue play until a student covers
all the squares on his or her card.
NEXT STEPS t Differentiate
5-Day Instructional Plan:
PM 4, page 817—All students, to assess progress
4-Day Instructional Plan:
PM 5, page 818—Students who are on the
accelerated path, to assess progress
PM 4, page 817—Students who are on the
differentiated path, to assess progress
Chapter 9 t Objective 3
56
813
Objective 3
Progress-Monitoring
Activities
PM 1
Apply Skills 1
Use with 5-Day or 4-Day Instructional Plan.
MATERIALS
t Interactive Text,
t page 346
DIRECTIONS
1. Have students turn to Interactive Text,
t page 346,
Apply Skills 1.
3. Monitor student work, and provide feedback as
necessary.
Watch for:
t %PTUVEFOUTGBDUPSUIFUSJOPNJBMTVTJOHBMHFCSB
tiles to complete the rectangle?
Name __________________________________________ Date __________________________
A P P LY S K I L L S 1
Factor each of the quadratic trinomials.
Example:
x + 2)(x + 4)
x 2 + 6x + 8 = (x
1.
x 2 + 9x + 20 =
(x + 4)(x + 5)
2.
x 2 + 12x + 20 =
3.
x 2 – 4x – 32 =
(x + 4)(x – 8)
4.
x 2 + 4x + 3 =
5.
x2 + x – 6 =
6.
x 2 + 8x + 12 =
8.
x2 + x – 2 =
(x – 2)(x + 3)
(x + 1)(x + 3)
(x + 2)(x + 6)
7. x 2 + 6x + 5 =
(x + 1)(x + 5)
9.
x 2 – 6x + 8 =
(x – 2)(x – 4)
10. x 2 – 3x – 18 =
11.
x 2 – 4x + 3 =
(x – 1)(x – 3)
12.
x 2 + 10x + 21 =
13.
x 2 + x – 12 =
(x – 3)(x + 4)
14.
x 2 – 7x + 12 =
15.
x 2 + 9x – 10 =
(x + 10)(x – 1)
16. x 2 – 12x + 32 =
(x + 5)(x – 6)
18.
17. x 2 – x – 30 =
19.
346
2x 2 + 11x + 12 =
Chapter 9
t
(2x + 3)(x + 4)
Objective 3 t PM 1
(x + 2)(x + 10)
(x – 1)(x + 2)
x 2 – 8x – 9 =
20. 3x 2 + 16x + 5 =
(x + 3)(x – 6)
(x + 3)(x + 7)
(x – 3)(x – 4)
(x – 4)(x – 8)
(x + 1)(x – 9)
(3x + 1)(x + 5)
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
T
M
KVV7OW
O7M
Sample Teacher Pages
t %PBOZTUVEFOUTUSZBOBMHFCSBJDNFUIPE
NEXT STEPS t Differentiate
5-Day Instructional Plan:
CD 2, page 810—All students, for additional
concept development and problem solving
4-Day Instructional Plan:
PA 1, page 812—All students, for additional
practice and problem solving
814
Chapter 9 t Objective 3
57
A P P LY S K I L L S 2
Progress-Monitoring
Activities
Factor each of the quadratic trinomials.
Example:
x + 3)(x – 2)
2x 2 – x – 6 = (2x
1.
x 2 + 3x + 2 =
3.
7x 2 + 11x – 6 =
Use with 5-Day or 4-Day Instructional Plan.
5.
14x 2 – x – 4 =
MATERIALS
7. 2x 2 + 3x – 5 =
PM 2
T
M
KVV7OW
O7M
Name __________________________________________ Date __________________________
Objective 3
(x + 1)(x + 2)
(5x + 2)(x – 7)
2.
5x 2 – 33x – 14 =
(7x – 3)(x + 2)
4.
8x 2 – 19x + 6 =
(8x – 3)(x – 2)
(7x – 4)(2x + 1)
6.
x 2 + 9x + 20 =
(x + 4)(x + 5)
(2x + 5)(x – 1)
8.
3x 2 – 10x – 8 =
Apply Skills 2
(3x + 2)(x – 4)
t Interactive Text,
t page 347
DIRECTIONS
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
3. Monitor student work, and provide feedback as
necessary.
Watch for:
t %PTUVEFOUTGBDUPSUIFUSJOPNJBMTVTJOHBSFBSVHT
t %PTUVEFOUTDIFDLUIFJSBOTXFSTCZNVMUJQMZJOH
the resulting binomials?
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
1. Have students turn to Interactive Text,
t page 347,
Apply Skills 2.
9.
6x 2 + 17x + 10 =
(6x + 5)(x + 2)
11.
16x 2 – 8x – 3 =
(4x – 3)(4x + 1)
13.
12x 2 – 16x + 5 =
15.
2x 2 – x – 3 =
(2x – 3)(x + 1)
17. 5x 2 – 22x – 15 =
19.
6x 2 – 7x – 3 =
Inside Algebra
(2x – 1)(6x – 5)
(5x + 3)(x – 5)
(3x + 1)(2x – 3)
(2x + 1)(4x – 3)
10. 8x 2 – 2x – 3 =
12.
12x 2 – 29x + 15 =
14.
32x 2 – 4x – 1 =
16. 20x 2 + 12x + 1 =
18.
30x 2 + 1x – 3 =
20. 3x 2 – x – 2 =
(3x – 5)(4x – 3)
(4x – 1)(8x + 1)
(2x + 1)(10x + 1)
(3x + 1)(10x – 3)
(3x + 2)(x – 1)
Chapter 9 t Objective 3 t PM 2
347
NEXT STEPS t Differentiate
5-Day Instructional Plan:
CD 3, page 811—All students, for additional
concept development
3-Day Instructional Plan:
CD 3, page 811—Students who are on the
accelerated path, for additional concept
development
CD 3, page 811—Students on the differentiated
path who demonstrate understanding of the
concept, to extend understanding
CD 2, page 810—All other students, for additional
concept development
Chapter 9 t Objective 3
58
815
Sample Teacher Pages
Progress-Monitoring
Activities
PM 3
Name __________________________________________ Date __________________________
T
M
KVV7OW
O7M
Objective 3
Apply Skills 3
Use with 5-Day or 4-Day Instructional Plan.
MATERIALS
t Interactive Text,
t pages 348–349
DIRECTIONS
1. Have students turn to Interactive Text,
t pages
348–349, Apply Skills 3.
A P P LY S K I L L S 3
Solve the quadratic trinomials by factoring.
1. 3x 2 – 10x – 8 = 0
Example:
(3x + 2)(x – 4) = 0
3x + 2 = 0 or x – 4 = 0
3x = –2 or x = 4
x = –2 or x = 4
2x 2 – x – 6 = 0
(2x + 3)(x – 2) = 0
2x + 3 = 0 or x – 2 = 0
x = – 32 , 2
3
3. 6x 2 – 7x – 3 = 0
2. 2x 2 + 3x + 1 = 0
(3x + 1)(2x – 3) = 0
3x + 1 = 0 or 2x – 3 = 0
3x = –1 or 2x = 3
x = –1 or x = 3
(2x + 1)(x + 1) = 0
2x + 1 = 0 or x + 1 = 0
2x = –1 or x = –1
x = –1 or x = –1
3
2
4. 4x 2 + 4x – 15 = 0
5. x 2 + 12x + 20 = 0
(2x + 5)(2x – 3) = 0
2x + 5 = 0 or 2x – 3 = 0
2x = –5 or 2x = 3
x = –5 or x = 3
2
(x + 2)(x + 10) = 0
x + 2 = 0 or x + 10 = 0
x = –2 or x = –10
2
7. 12x 2 – 16x + 5 = 0
6. 2x 2 – x – 3 = 0
2
2
3. Monitor student work, and provide feedback as
necessary.
8. 2x 2 + 3x – 5 = 0
9. 6x 2 + 17x + 10 = 0
(2x + 5)(x – 1) = 0
2x + 5 = 0 or x – 1 = 0
2x = –5 or x = 1
x = –5 or x = 1
Watch for:
t %PTUVEFOUTTPMWFUIFUSJOPNJBMTCZGBDUPSJOH
(6x + 5)(x + 2) = 0
6x + 5 = 0 or x + 2 = 0
6x = –5 or x = –2
x = –5 or x = –2
2
t %PTUVEFOUTSFNFNCFSUPBDDPVOUGPSBMFBEJOH
coefficient?
348
Chapter 9
t
6
6
Inside Algebra
Objective 3 t PM 3
NEXT STEPS t Differentiate
A P P LY S K I L L S 3
5-Day and 4-Day Instructional Plans:
PA 2, page 813—All students, for additional
practice
(continued )
10. 8x 2 – 2x – 3 = 0
11. 16x 2 – 8x – 3 = 0
(2x + 1)(4x – 3) = 0
2x + 1 = 0 or 4x – 3 = 0
2x = –1 or 4x = 3
x = –1 or x = 3
2
T
M
KVV7OW
O7M
Name __________________________________________ Date __________________________
4
12. 2x 2 + 5x – 12 = 0
(2x – 3)(x + 4) = 0
2x – 3 = 0 or x + 4 = 0
2x = 3 or x = –4
x = 3 or x = –4
(4x – 3)(4x + 1) = 0
4x – 3 = 0 or 4x + 1 = 0
4x = 3 or 4x = –1
x = 3 or x = –1
4
4
13. x 2 + 3x + 2 = 0
(x + 1)(x + 2) = 0
x + 1 = 0 or x + 2 = 0
x = –1 or x = –2
2
14. x 2 – 4x – 32 = 0
15. x 2 + 9x + 20 = 0
(x + 4)(x – 8) = 0
x + 4 = 0 or x – 8 = 0
x = –4 or x = 8
(2x + 1)(7x – 4) = 0
2x + 1 = 0 or 7x – 4 = 0
2x = –1 or 7x = 4
x = –1 or x = 4
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2
816
7
18. 7x 2 + 11x – 6 = 0
(7x – 3)(x + 2) = 0
7x – 3 = 0 or x + 2 = 0
7x = 3 or x = –2
x = 3 or x = –2
7
Inside Algebra
(x + 4)(x + 5) = 0
x + 4 = 0 or x + 5 = 0
x = –4 or x = –5
17. 5x 2 – 3x – 2 = 0
16. 14x 2 – x – 4 = 0
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
(2x – 1)(6x – 5) = 0
2x – 1 = 0 or 6x – 5 = 0
2x = 1 or 6x = 5
x = 1 or x = 5
(2x – 3)(x + 1) = 0
2x – 3 = 0 or x + 1 = 0
2x = 3 or x = –1
x = 3 or x = –1
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
2
(5x + 2)(x – 1) = 0
5x + 2 = 0 or x – 1 = 0
5x = –2 or x = 1
x = –2 or x = 1
5
19. 5x 2 – 33x – 14 = 0
(5x + 2)(x – 7) = 0
5x + 2 = 0 or x – 7 = 0
5x = –2 or x = 7
x = –2 or x = 7
5
Chapter 9 t Objective 3 t PM 3
349
Chapter 9 t Objective 3
59
T
M
KVV7OW
O7M
Objective 3
Progress-Monitoring
Activities
PM 4
Apply Skills 4
Use with 5-Day or 4-Day Instructional Plan.
MATERIALS
t Interactive Text,
t pages 350–351
DIRECTIONS
1. Have students turn to Interactive Text,
t pages
350–351, Apply Skills 4.
Name __________________________________________ Date __________________________
A P P LY S K I L L S 4
Solve the quadratic trinomials by factoring.
1. x 2 + 12x + 20 = 0
Example:
(x + 2)(x + 10) = 0
x + 2 = 0 or x + 10 = 0
x = –2 or x = –10
x 2 + 6x + 5 = 0
(x + 1)(x + 5) = 0
x + 1 = 0 or x + 5 = 0
x = –1, –5
2. x 2 + 3x + 2 = 0
3. x 2 – 4x – 32 = 0
(x + 1)(x + 2) = 0
x + 1 = 0 or x + 2 = 0
x = –1 or x = –2
(x + 4)(x – 8) = 0
x + 4 = 0 or x – 8 = 0
x = –4 or x = 8
4. x 2 + 9x + 20 = 0
5. x 2 – 9x + 14 = 0
(x + 4)(x + 5) = 0
x + 4 = 0 or x + 5 = 0
x = –4 or x = –5
(x – 2)(x – 7) = 0
x – 2 = 0 or x – 7 = 0
x = 2 or x = 7
7. x 2 – 6x + 9 = 0
6. x 2 – 2x – 15 = 0
(x – 3)2 = 0
x–3=0
x=3
(x + 3)(x – 5) = 0
x + 3 = 0 or x – 5 = 0
x = –3 or x = 5
3. Monitor student work, and provide feedback as
necessary.
8. x 2 + 5x – 6 = 0
9. x 2 + 5x + 6 = 0
(x – 1)(x + 6) = 0
x – 1 = 0 or x + 6 = 0
x = 1 or x = –6
Watch for:
t %PTUVEFOUTSFNFNCFSUIF;FSP1SPQFSUZ1SPEVDU
of multiplication?
350
t %PTUVEFOUTVOEFSTUBOEIPXUPmOEBWBMVF
for x that makes a factor equal to zero?
Chapter 9
t
(x + 2)(x + 3) = 0
x + 2 = 0 or x + 3 = 0
x = –2 or x = –3
Inside Algebra
Objective 3 t PM 4
A P P LY S K I L L S 4
NEXT STEPS t Differentiate
(continued )
10. x 2 – 2x – 24 = 0
(x + 4)(x – 6) = 0
x + 4 = 0 or x – 6 = 0
x = –4 or x = 6
5-Day Instructional Plan:
PM 5, page 818—All students, for additional
progress assessment
12. x 2 – 3x – 18 = 0
4-Day Instructional Plan:
PA 2, page 813—Students who are on the
accelerated path, for additional practice
(x + 3)(x – 6) = 0
x + 3 = 0 or x – 6 = 0
x = –3 or x = 6
PS 1, page 819—Students on the differentiated
path who demonstrated understanding on PM 2,
to develop problem-solving skills
T
M
KVV7OW
O7M
Name __________________________________________ Date __________________________
14. x 2 + 2x – 3 = 0
(x – 1)(x + 3) = 0
x – 1 = 0 or x + 3 = 0
x = 1 or x = –3
11. x 2 – 8x – 9 = 0
(x + 1)(x – 9) = 0
x + 1 = 0 or x – 9 = 0
x = –1 or x = 9
13. x 2 – 7x + 10 = 0
(x – 2)(x – 5) = 0
x – 2 = 0 or x – 5 = 0
x = 2 or x = 5
15. x 2 + x – 12 = 0
(x – 3)(x + 4) = 0
x – 3 = 0 or x + 4 = 0
x = 3 or x = –4
PM 5, page 818—All other students, for additional
progress assessment
16. x 2 – 12x + 32 = 0
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
(x – 4)(x – 8) = 0
x – 4 = 0 or x – 8 = 0
x = 4 or x = 8
18. x 2 + 10x – 24 = 0
(x – 2)(x + 12) = 0
x – 2 = 0 or x + 12 = 0
x = 2 or x = –12
Inside Algebra
17. x 2 + 3x – 40 = 0
(x – 5)(x + 8) = 0
x – 5 = 0 or x + 8 = 0
x = 5 or x = –8
19. x 2 – 3x = 0
x(x – 3) = 0
x = 0 or x – 3 = 0
x = 0 or x = 3
Chapter 9 t Objective 3 t PM 4
351
Chapter 9 t Objective 3
60
817
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
Sample Teacher Pages
Progress-Monitoring
Activities
PM 5
Name __________________________________________ Date __________________________
T
M
KVV7OW
O7M
Objective 3
Apply Skills 5
Use with 5-Day or 4-Day Instructional Plan.
MATERIALS
t Interactive Text,
t pages 352–353
DIRECTIONS
1. Have students turn to Interactive Text,
t pages
352–353, Apply Skills 5.
A P P LY S K I L L S 5
Solve the quadratic trinomials by factoring.
1. a 2 – 6a – 16 = 0
Example:
(x – 4)(x + 2) = 0
x – 4 = 0 or x + 2 = 0
x = 4, –2
2. b 2 – 9b + 14 = 0
3. c 2 – 4c – 21 = 0
(b – 2)(b – 7) = 0
b – 2 = 0 or b – 7 = 0
b = 2 or b = 7
(c + 3)(c – 7) = 0
c + 3 = 0 or c – 7 = 0
c = –3 or c = 7
4. d 2 + 8d – 9 = 0
5. x 2 – 9x + 8 = 0
(d – 1)(d + 9) = 0
d – 1 = 0 or d + 9 = 0
d = 1 or d = –9
(x – 1)(x – 8) = 0
x – 1 = 0 or x – 8 = 0
x = 1 or x = 8
7. m 2 + 11m + 28 = 0
6. y 2 – 7y – 30 = 0
(m + 4)(m + 7) = 0
m + 4 = 0 or m + 7 = 0
m = –4 or m = –7
(y + 3)(y – 10) = 0
y + 3 = 0 or y – 10 = 0
y = –3 or y = 10
3. Monitor student work, and provide feedback as
necessary.
8. c 2 – 20c + 64 = 0
9. a 2 + 6a – 27 = 0
(c – 4)(c – 16) = 0
c – 4 = 0 or c – 16 = 0
c = 4 or c = 16
Watch for:
t "SFTUVEFOUTBCMFUPBQQMZUIFJSLOPXMFEHF
of factoring to solve trinomials?
352
t %PTUVEFOUTSFBMJ[FUIBUUIFOBNFPGUIF
variable is not important?
Chapter 9
t
(a – 3)(a + 9) = 0
a – 3 = 0 or a + 9 = 0
a = 3 or a = –9
Objective 3 t PM 5
Inside Algebra
A P P LY S K I L L S 5
NEXT STEPS t Differentiate
(continued )
10. x – x – 30 = 0
2
(x + 5)(x – 6) = 0
x + 5 = 0 or x – 6 = 0
x = –5 or x = 6
5-Day and 4-Day Instructional Plans:
PS 1, page 819—Students who are on the
accelerated path, to develop problem-solving skills
Objective 3 Posttest, page 821—Students who are
on the differentiated path
12. c 2 + 6c – 40 = 0
(c – 4)(c + 10) = 0
c – 4 = 0 or c + 10 = 0
c = 4 or c = –10
14. g 2 – 9g + 18 = 0
(g – 3)(g – 6) = 0
g – 3 = 0 or g – 6 = 0
g = 3 or g = 6
16. x 2 + 15x + 54 = 0
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
(x + 6)(x + 9) = 0
x + 6 = 0 or x + 9 = 0
x = –6 or x = –9
18. a 2 + 32a + 60 = 0
(a + 2)(a + 30) = 0
a + 2 = 0 or a + 30 = 0
a = –2 or a = –30
Inside Algebra
T
M
KVV7OW
O7M
Name __________________________________________ Date __________________________
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
2. Remind students of the key terms: quadratic
trinomiall and factor.
r
818
(a + 2)(a – 8) = 0
a + 2 = 0 or a – 8 = 0
a = –2 or a = 8
x 2 – 2x – 8 = 0
11. d 2 – 15d + 36 = 0
(d – 3)(d – 12) = 0
d – 3 = 0 or d – 12 = 0
d = 3 or d = 12
13. e 2 + e – 20 = 0
(e – 4)(e + 5) = 0
e – 4 = 0 or e + 5 = 0
e = 4 or e = –5
15. h 2 – 14h + 33 = 0
(h – 3)(h – 11) = 0
h – 3 = 0 or h – 11 = 0
h = 3 or h = 11
17. m 2 – m – 72 = 0
(m + 8)(m – 9) = 0
m + 8 = 0 or m – 9 = 0
m = –8 or m = 9
19. p 2 – 21p – 100 = 0
(p + 4)(p – 25) = 0
p + 4 = 0 or p – 25 = 0
p = –4 or p = 25
Chapter 9 t Objective 3 t PM 5
353
Chapter 9 t Objective 3
61
Objective 3
Problem-Solving
Activities
) PS
1
Paving the Yard
Use with 5-Day or 4-Day Instructional Plan. In this
activity, students calculate the area of a rectangle.
DIRECTIONS
1. Read the following scenario to students:
A homeowner wants to pave
a square area in his backyard
that is 9x 2 square feet in area.
He will use square pavers that
measure one foot on each side.
9x 2
He is considering extending the paving to two
rectangular areas adjacent to the original area.
The first rectangular area is to the east and is
6 feet long and as wide as the original square.
The second rectangular area is to the south and
is 4 feet wide and as long as his original square
plus the 6-foot extension.
6
2. Tell students to write an expression in terms
of x that would indicate how many pavers the
homeowner would need. x
x
3. Ask students to think about how large the original
square area is that the homeowner wanted to
pave if x = 3. 81 square feet Make sure students
recognize that the homeowner would need 81
pavers for the original square area if x = 3 because
he uses pavers that are one square foot.
4. Ask students to determine how many more pavers
he would need to pave the two rectangular areas if
x = 3. <
><
>oQBWFST
NEXT STEPS t Differentiate
5-Day Instructional Plan:
Objective 3 Posttest, page 821—All students
4-Day Instructional Plan:
PS 2, page 820—Students who are on the
accelerated path, for additional problem solving
PM 5, page 818—Students who are on the
differentiated path, to assess progress
9x 2
4
)= Includes Problem Solving
Chapter 9 t Objective 3
62
819
Sample Teacher Pages
63
64
Sample Teacher Pages
65
66
Sample Teacher Pages
67
68
Sample Teacher Pages
69
70
Sample Teacher Pages
71
72
Sample Student Pages
73
74
Sample Student Pages
75
76
Sample Student Pages
77
78
Sample Student Pages
79
80
CHAPTER 9 t Objective 3
r
Sample Student Pages
Name __________________________________________ Date____________________________
Factor the quadratic polynomials.
1.
x 2 + 5x
x+6
2.
x 2 + 8x
x + 15
3.
x 2ox o
4.
3x 2 ox + 6
5.
x 2ox o Solve the quadratic equations by factoring.
6.
x
x 2 + xo
7.
x 2 + 2x
x o
8.
x 2ox
x o
9.
6x 2 + xo
x
128
9x 2 + 12x
x o
Chapter 9 t 0CKFDUJWF
Inside Algebra
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
10.
81
1.
x 2 + 7x
x+6
2.
x 2 + 2xo
x
3.
x 2 ox
x o 4.
3x 2 ox
x o 5.
4x 2 + 7xo
x
CHAPTER 9 t Objective 3
Factor the quadratic polynomials.
o
Name __________________________________________ Date____________________________
Copyright 2011 Cambium Learning Sopris West.® All rights reserved.
Solve the quadratic equations by factoring.
82
6.
x o
x 2 + 3x
7.
x 2 + 3xo
x
8.
x 2 + x o
9.
2x 2 oxo
x
10.
3x 2 + 14x
x
Inside Algebra
Chapter 9 t 0CKFDUJWF
129
Notes ...
83
Notes ...
84
Notes ...
85
Notes ...
86
Notes ...
87
Z351_PO/191994/09-10/RRD/15K/.68
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