Change the way
students see math
Prentice Hall
Algebra 1 Geometry Algebra 2
A New
Perspective on Math Prentice Hall Algebra 1, Geometry, Algebra 2 ©2011 is changing the way students see math! By delivering instruction through a blended medium of digital and print components, we are helping you reach today’s digital natives.
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Hi, my name is Darius. I’m one of the six peer coaches you’ll find throughout the program— in print and online.
Make Math Meaningful For many students who struggle, math shows up as a collection of rules, formulas, and properties that they learn temporarily, forget quickly, and never use again. Students find mathematics meaningless if they don’t see the connections. Prentice Hall Algebra 1, Geometry, Algebra 2 teaches for understanding by incorporating an interwoven strand of thinking and reasoning into problem solving. This connects the math that students learn, from the first lesson to the last. By focusing on thinking, reasoning, and problem solving, students will become more prepared for success in school, in their careers, and in life.
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Distinguished Authorship
Custom Solutions for Your Classroom
Series Authors Randall I. Charles, Ph.D., is Professor Emeritus in the Department of Mathematics and Computer Science at San Jose State University in California. He was a high school mathematics teacher and supervisor for five years. Dr. Charles has been a member of several NCTM committees and is the former Vice President of the National Council of Supervisors of Mathematics. Much of his writing and research has been in the area of problem solving. He has authored more than 75 mathematics textbooks for kindergarten through college.
Dan Kennedy, Ph.D., is a classroom teacher and the Lupton Distinguished Professor of Mathematics at the Baylor School in Chattanooga, Tennessee. Dr. Kennedy is a frequent speaker on the subject of mathematics education reform. He is coauthor of textbooks in calculus and precalculus, and he chaired the College Board’s AP* Calculus Development Committee. He is a 1992 Tandy Technology Scholar and a 1995 Presidential Award winner.
Basia Hall currently serves as Manager of Instructional Programs for the Houston Independent School District. With 33 years of teaching experience, Ms. Hall has served as a department chair, instructional supervisor, school improvement facilitator, and professional development trainer. She has developed curriculum for Algebra 1, Geometry, and Algebra 2 and codeveloped the Texas state mathematics standards. A 1992 Presidential Award winner, Ms. Hall is past president of the Texas Association of Supervisors of Mathematics, and a state representative for the National Council of Supervisors of Mathematics (NCSM).
Comprehensive On-Level Program
Struggling Learners Program
Prentice Hall Algebra 1, Geometry, Algebra 2 incorporates a blend of print and digital components to tap into the power of mathematics and mathematical reasoning. The wealth and flexibility of resources will enable you to easily adapt to the changing needs of your classroom.
Prentice Hall Algebra 1, Geometry, Algebra 2, Foundations Series is a great option for low-level and inclusion classrooms, delivering comprehensive content in an accessible manner to struggling students. With less difficult content, shorter chapters and lessons include more support and scaffolding for students, as well as more frequent assessments.
Consulting Authors Stuart J. Murphy is a visual learning author and consultant. He is a champion of developing visual learning skills to help children become more successful students. He is the author of Math Start, a series of children's books that present mathematical concepts in the context of stories. A graduate of the Rhode Island School of Design, he has worked in educational publishing and has been on the authorship teams of a number of elementary and high school mathematics programs. He presents at meetings of the National Council of Teachers of Mathematics, and the International Reading Association.
Grant Wiggins, Ed.D., is the President of Authentic Education in Hopewell, New Jersey. He earned his Ed.D. from Harvard University and his B. A. from St. John's College in Annapolis, Maryland. Dr. Wiggins consults with schools, districts, and state education departments on a variety of reform matters. He is perhaps best known for being the co-author, with Jay McTighe, of Understanding By Design and The Understanding By Design Handbook, the award-winning and highly successful materials on curriculum reform published by ASCD. His work has been supported by the Pew Charitable Trusts, the Geraldine R. Dodge Foundation, and the National Science Foundation.
Complete Online Program PowerAlgebra.com and PowerGeometry.com is the digital component for the series that can be used as part of the blended model with print or as a stand-alone digital course. The digital component includes a wealth of assets, including the Student and Teacher’s Editions, instruction and presentation tools, student-generated videos, classroom management tools and editable resources, and online assessment with remediation.
Table of Contents* Algebra 1
Geometry
Algebra 2
1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6. Systems of Equations and Inequalities 7. Exponents and Exponential Functions 8. Polynomials and Factoring 9. Quadratic Functions and Equations 10. Radical Expressions and Equations 11. Rational Expressions and Functions 12. Data Analysis and Probability
1. Tools of Geometry 2. Reasoning and Proof 3. Parallel and Perpendicular Lines 4. Congruent Triangles 5. Relationships Within Triangles 6. Polygons and Quadrilaterals 7. Similarity 8. Right Triangles and Trigonometry 9. Transformations 10. Area 11. Surface Area and Volume 12. Circles
1. Expressions, Equations, and Inequalities 2. Functions, Equations, and Graphs 3. Linear Systems 4. Quadratic Functions and Equations 5. Polynomials and Polynomial Functions 6. Radical Functions and Rational Exponents 7. Exponential and Logarithmic Functions 8. Rational Functions 9. Sequences and Series 10. Quadratic Relations and Conic Sections 11. Probability and Statistics 12. Matrices 13. Periodic Functions and Trigonometry 14. Trigonometric Identities and Equations
*
4
Advanced Placement, Advanced Placement Program, AP, and Pre-AP are registered trademarks of The College Board, which was not involved in the production of, and does not endorse, these products.
*
Foundations Series Table of Contents varies slightly.
5
7-6
Engage Today’s Students
Objective
D
AC
ES
AM YN I
TIVITI
To evaluate and graph
ns
exponential functio
wants to practice Your soccer team in amount of a drill for a certa plan will give time each day. Which practice time total your team more 8 days? Explain over 4 days? Over your reasoning.
Family feud! These functions don’t belong in the same family of functions.
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Introducing PowerAlgebra.com and PowerGeometry.com—the gateway for students and teachers to all the digital components available for the series. This includes access to the online Student Edition with audio, Teacher’s Edition, student-generated videos, animations, presentation tools, online assessment with remediation, as well as lesson planning, editable worksheets, and a sophisticated classroom management system.
Exponential Functions
ic Activity D Dynam Exponential Functions
LLesson Vocabulary V ential e • expon function
Solve It have different The two plans in the n. erent type of functio of growth with a diff
Plan 1
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patterns of growth.
Some functions model
You can model each
an initial amount that
Learning by Doing Dynamic Activities provide an interactive way for students to explore lesson concepts. Additionally, math tools enable you and your students to utilize the functionality of tools such as a graphing calculator, algebra tiles, and geometry software.
type
is
the ding for these functions, Essential Understan e number. In the rules lied by the same positiv repeatedly multip le is an exponent. independent variab
ential Function Key Concept Expon x, where a 2 0, b . 0, Definition the form y 5 a ? b on is a function of An exponential functi number. b 2 1, and x is a real
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Lights, Camera…Math! My Math Videos, found at the beginning of each chapter, are student-produced videos that engage students in math concepts that are relevant to their lives. Through the Pearson Video Challenge, students can demonstrate their own understanding and creativity by generating their own videos to be included on PowerAlgebra.com and PowerGeometry.com.
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ential Functio Lesson 7-6 Expon
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Practice Makes Perfect MathXL® for School tutorial exercises provide interactive practice at the midpoint and end of each chapter. Each exercise provides learning aids—including an interactive guided solution— sample problems, and similar problems that refresh with new numbers so students can retry exercises.
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“Today’s students are digital natives. These students are not merely technology-savvy; they are approaching their lives differently as they integrate digital technologies seamlessly throughout their daily activities. Let’s not have them power-down when they get to math class.”
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–Laurie Bass, program author
7
Build Conceptual Understanding Math proficiency is developed through fluency, reasoning, and application. By teaching for understanding, you are enabling your students to demonstrate their ability to transfer their knowledge from one situation or problem to another, particularly on high-stakes exams. Students are also able to make critical connections between concepts, making math meaningful.
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“The visual models in the Student Edition allow students to interact with mathematical concepts, process the information, observe change, reflect on their experiences, modify their thinking, and draw conclusions. They learn.”
–Stuart J. Murphy, visual instruction consulting author
Big Ideas This program incorporates the groundbreaking Understanding by Design framework. Co-developed by consulting author Grant Wiggins, UbD changes the way students approach math by introducing them first to the Big Idea of each chapter. Within each chapter, students will develop answers to the Essential Questions posed and make connections around the Big Ideas. Chapter Preview 1
Equivalence Essential Question: How can you represent very large and very small numbers?
7
2
Properties Essential Question: How can you simplify expressions involving exponents? To solve these
3
will pull together Function many concepts Essential Question: What are the and skills that you have studied about similarity. characteristics of exponential functions?
problems, you
All Together
Task 1
are proportional.
In the diagram below, AC 6 DF 6 BH
and CB 6 FE. F C
A
H G E
B
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413
Chapter 7 Exponents and Exponential Functions all similar. Explain how you know that they are
a. Find four similar triangles. extended proportion. found in part (a), complete the following b. Using the similar triangles you AB 5 DE 5 j 5 j j DG j AC
Similarity the sides and angles of a triangle determine Lines with special relationships to know the lengths of some of the segments, proportional segments. When you unknown length. you can use a proportion to find an
Task 2
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”A Big Idea is a way of seeing better and working smarter, not just another piece of knowledge.”
Visual Instruction is about acquiring and communicating information. Visuals support students as they analyze complex word problems. They clarify important concepts, and they engage students and encourage them to make connections with real-life situations. Visual learning strategies are a powerful teaching tool for a student’s depth of understanding about mathematics.
Pull It All Together, at the end of each chapter, enables students to demonstrate their understanding of the concepts and skills they studied in the chapter lessons and in previous chapters. In doing so, they apply their reasoning strategies and growth as independent problem solvers.
7-1 Zero and Negative Exponents 7-2 Scientific Notation 7-3 Multiplying Powers With the Same Base 7-4 More It Multiplication Properties of Pull Exponents 7-5 Division Properties of Exponents and Proof, and Similarity 7-6 Exponential Functions Visualization, Reasoning exist between similar when certain relationships You can show that two triangles are triangles are similar, then know two 7-7 Exponential Growth Decay g parts. If youand two or three pairs of correspondin you know their corresponding sides
Visual Learning
–Grant Wiggins, consulting author
2-3 Connect to What You Know Visual Instruction increases the learning potential of all students. The Solve It! at the start of each lesson makes use of engaging visuals and real-world examples to help students tap into their prior knowledge and connect it to important concepts in the lesson.
Solving Multi-Step Equations Objective
This problem has a twist. The unknown amount occurs twice.
tions in one variable To solve multi-step equa
ds. You a group of your frien e tickets online for You are buying movi on the screen shown. er of tickets you want total of a have have to enter the numb and tickets t card to pay for the ain your answer. You are using a debi ts can you buy? Expl ticke y man How d. $45 to spen : wou ld like to purc hase Num ber of ticke ts you Ticket price
$9.00 ×
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number of tickets
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Total
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$5.00
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multi-step equations.
s of equations, you form a serie ing To solve multi-step ations, Essential Understand s of equality, inverse oper ertie prop the use this, do tions. To ble. simpler equivalent equa s until you isolate the varia bers. You use the propertie and properties of real num
9
2-3
Objective
To solve multi-step
equations in one var
Develop Problem Solving iable
“Research shows that understanding develops during the process of solving problems in which important math concepts and skills are embedded…”
You are buying mo vie tickets online for a group of you have to enter the r friends. You number of tickets you want on the You are using a deb screen shown. it card to pay for the tickets and hav $45 to spend. Ho e a total of w many tickets can you buy? Explain your answer.
Problem-solving strategies are an integral part of the program and are embedded throughout each lesson. The worked-out problems model effective thinking and reasoning strategies and can help foster students’ mathematical reasoning.
This problem has a twist. The unknown amount occurs twice.
Nu mb er of tick ets you wo uld like to pu rch ase : Ticket price
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of tickets
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ns.
“Think” and “Plan” call-outs model mathematical reasoning and problem solving for every problem in a lesson. Some reveal “Step Zero,” or the reasoning that goes on before the first step of the solution. Some worked-out problems provide even more support as they model the thinking behind each step of a problem-solving plan.
To solve multi-step simpler equivalent equ equations, you form ations. To do this, use a series of the properties of equ and properties of rea ality, inverse operations l numbers. You use , the properties until you isolate the variab le.
Problem 1 Comb ining Like Terms What is the solution of 5 5 5m 2 23 1 2m ? 5 5 5m 2 23 1 2m 5 5 5m 1 2m 2 23
How is this equatio n different from equations you’ve seen before? The variable occurs in two terms. You can simplify the equation by grouping like terms and combining them.
Commutative Property of Addition Combine like terms.
5 5 7m 2 23 5 1 23 5 7m 2 23 1 23 28 5 7m
Add 23 to each side. Simplify.
28 7m 7 5 7
45m
Check
Divide each side by 7. Simplify.
5 5 5m 2 23 1 2m 5 0 5(4) 2 23 1 2(4 ) 555 ✓
–Randy Charles, program author
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Reasoning Call-outs
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Essential Underst anding
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Chapter 2 Solving Equations
Online Problems Each problem in the Student Edition is also modeled online at PowerAlgebra.com and PowerGeometry.com. Step-by-step instruction, with guided support from one of six avatars, enables students to follow along at their own pace.
er. equation? Check each answ b. 22y 1 5 1 5y 5 14
ation Solving a Multi-Step Equ nephew to a concert. and ha takes her niece Concert Merchandise Mart them. The bumper stickers for ers stick per bum and She buys T-shirts stickers, and her e wants 1 shirt and 4 bumper cost $1 each. Martha’s niec ers. If Martha’s total is $67, stick per bum no but s nephew wants 2 shirt ? what is the cost of one shirt
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2s
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cost of niece’s items (1 shirt and 4 stickers)
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Commutative Property of Addit Combine like terms. Subtract 4 from each side.
Now It’s Your Turn Got It? following each problem, checks for understanding. Students can work through each Got It? in the Student Companion worktext.
Simplify. Divide each side by 3. Simplify.
One shirt costs $21.
ic store. for new guitar strings in a mus and a Kate buys 2 packs of strings gs. strin of s pack 2 buys Noah much $16. Their total cost is $72. How music book. The book costs gs? strin of pack is one
shopping Got It? 2. Noah and Kate are
10
11
Differentiate Instruction Students learn in different ways and at different paces. Unique, built-in resources differentiate instruction to support all levels of learners in becoming successful problem solvers. Differentiating instruction helps all students develop conceptual understanding, fosters mathematical reasoning, and refines problem-solving strategies. Options are available to differentiate instruction at the start of each chapter and throughout the lessons.
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“Differentiated instruction does not change what is taught; it changes how it is taught.”
–Basia Hall, program author
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