Chapter 3 Solving Equations and Inequalities

Chapter 3 Solving Equations and Inequalities

Mathematical Overview

In this chapter, students use algebraic expressions as the building blocks for writing equations and inequalities. Arrow diagrams are introduced as a way to represent expressions and then later as a way to model the steps used when solving equations. The order of operations guidelines and the TABLE feature on a calculator, as well as the Properties of Equality and the Properties of Inequality, are all explored in the context of finding solutions for real-world problems. Students note that whereas a single number solution is graphed as a point on a number line, the graph of an inequality is an interval on the line. The concept of an inequality is extended to include compound inequalities. Students also transform statements into inequalities and graph the solution sets on number lines.

Lesson Summaries

Lesson 3.1 Activity: Tree Growth

In this Activity, students measure the heights of drawings of a tree in six different years of growth. They then use a proportion to compute the actual height of the tree for each of those six years. Using their tables of data, students write expressions that represent the height of the tree for any year of growth and use them to make predictions about the height of the tree. Students are introduced to arrow diagrams and use them to represent/evaluate expressions. They also review the order of operations guidelines to help them evaluate expressions that contain many mathematical operations.

Lesson 3.2 Investigation: From Expression to Equation

In this Investigation, students build on their work with simple expressions from Lesson 3.1 and construct equations that model a second stage of tree growth. They learn that an equation is a statement that two expressions are equal. Students become aware that solving some problems may require different strategies than they normally use. The process of using equations to model realworld problems that are unlike problems students have solved in the past is the main focus of this lesson.

Lesson 3.3 Using Arrow Diagrams and Tables to Solve Equations

In this lesson, the use of arrow diagrams is expanded to represent two-step equations. Students solve equations by using a completed diagram to tell them which operations to perform and in which order. Students also learn to use the calculator TABLE feature to solve equations. The methods used in this lesson illustrate important algebraic properties, the Properties of Equality, which are used to solve many types of equations.

Lesson 3.4 R.A.P. In this lesson, students Review And Practice solving problems that require the use of skills and concepts taught in previous math levels. The skills reviewed in this lesson are skills that are needed as a basis for solving problems throughout this course.

Lesson 3.5 Investigation: Solving Multi-Step Equations and 3.6 Investigation: Solving Inequalities In Lesson 3.5, students explore using expressions and equations to model, evaluate, and solve real-world problems. Emphasis is placed on finding and combining like terms and isolating the variable. In Lesson 3.6 students apply this same knowledge to solving and graphing inequalities. The main purpose of the Investigation in this lesson is to provide students with a review of the Properties of Inequality.

Lesson 3.7 Compound Inequalities In this lesson, students use what they learned in Lessons 3.5 and 3.6 to write, solve, and graph compound inequalities. Students learn that the intersection of two graphs consists of all points that are common to both graphs, and the union of two graphs consists of all the points in one or the other or in both. Absolute value inequalities are included in some of the practice exercises.

55a

Lesson Guide

Lesson/Objectives

Chapter 3 Opener: How Can We Care for Our Forests? ? recognize that equations can be used to help maintain and

protect our country's forests.

3.1 Activity: Tree Growth ? use arrow diagrams to represent expressions. ? evaluate expressions. ? write expressions to model real-world situations.

3.2 Investigation: From Expression to Equation ? write an equation that models a real-world situation.

Per pair: ? inch ruler

3.3 Using Arrow Diagrams and Tables to Solve Equations ? use arrow diagrams to solve two-step equations. ? use tables to solve equations.

3.4 R.A.P. ? solve problems that require previously learned concepts and

skills.

3.5 Investigation: Solving Multi-Step Equations ? combine like terms in an expression. ? solve equations. ? use equations to solve real-world problems.

3.6 Investigation: Solving Inequalities ? solve inequalities. ? graph inequalities.

3.7 Compound Inequalities ? write compound inequalities. ? graph compound inequalities on a number line. ? solve compound inequalities.

Materials

Optional: ? TRM table shell for Question 2. ? TRM table shell for Exercises 6 and 9. Optional: ? TRM table shells for Exercises 1 and 18. ? grid paper

Optional: ? algebra tiles

Pacing Guide

Day 1

Basic

p. 56, 3.1

Day 2

3.2

Day 3

3.3

Day 4

3.4

Day 5

3.5

Day 6

3.5

Day 7

3.6

Day 8

3.7

Day 9

project

Day 10

review

Standard p. 56, 3.1 3.2

3.3

3.4

3.5

3.5

3.6

3.7

project review

Block

p. 56, 3.1, 3.2, 3.3 3.3, 3.4 3.5 3.2

3.6, 3.7

3.7, project

review

Supplement Support See the Book Companion Website at highschool.ModelingwithMathematics and the Teacher's Resource Materials (TRM) for additional resources.

55b

31CCCHHhAAPPaTTEEpRR ter 2 Solving

Direct VariatiMEoqnauthaetimonatsicaanld MInoedqeulalities

CHAPTER 3

Solving Equations and

Inequalities

Comap2e_Modeling_Ch03.indd 55

CONTENTS

CCOhaNpTteEr NOTpeSner:

HowHoIswMCaatnheWmeaCtiacrseRfeolratOeudrtFoorests? 56

LeBssuonng3e.e1Jumping?

35

LessAoCnTI2V.I1TY: Tree Growth

57

LeAssctoivnit3y:.2Bungee Jumping

37

LessINoVnE2S.T2IGATION:

InFvroesmtigEaxtiporne: ssion to Equation

61

LePssroopno3r.t3ional Relationships

39

LessUosnin2g.3Arrow Diagrams and

DTiarbecletsVtaoriSatoilovne FEuqnucattioionnss

4367

LLeessssoonn23.4.4

RRA.AP.P.

4773

LLeessssoonn23.5.5

SIlNoVpEeSTIGATION:

53

Solving Multi-Step Equations

75

Modeling Project:

LeIsts'soOnn3l.y6Water Weight

57

I N V E S T I G AT I O N :

ChaSpotlevrinRgeIvnieqwualities

6082

ELxetsesnosnio3n.7:

ICnvoemrspeoVuanrdiaItnioenqualities

6787

Modeling Project:

Algebra's Next Top Model

91

Chapter Review

92

03/02/12 7:32 PM

55

CHAPTER 3 OPENER

5e Engage

Lesson Objective

? recognize that equations can be used to help maintain and protect our country's forests.

Vocabulary

none

Description

This reading helps students see that sustainable forest management involves planting large numbers of trees, and that mathematics can be used to predict the rate at which they will grow.

TEACHING TIP

After students have read the Chapter Opener, lead a class discussion on the importance of forests to their state and/or community. Ask students to name products that they use that come from trees. Examples could include furniture, paper, and books.

How Can We Care for Our Forests?

One-third of the United States is covered with trees. In some states, the majority of land consists of forests. For example, in Virginia, 62% or 16 million acres is forested. Altogether, the United States contains about 230 billion trees or about 1,000 trees per person.

We often hear about problems related to the loss of forests, such as rain forests, throughout the world. In the United States, efforts are made to maintain a balance between the removal of trees for commercial and other purposes, and the natural or controlled regrowth of trees. One company that supplies wood for paper

products, J. D. Irving Limited, has planted over 800 million trees over the past fifty years. A nonprofit organization called Sustainable Forestry Initiative, Inc., certifies forest

lands in the United States and Canada to help ensure that responsible forest management is practiced.

How tall will a tree be after ten years of growth? How long will it take a tree to reach 50 feet in height? Mathematical equations can be used to answer

questions like these.

56

Chapter 3

SOLVING EQUATIONS AND INEQUALITIES

Comap2e_Modeling_Ch03.indd 56

03/02/12 7:32 PM

56

Lesson 3.1 ACTIVITY: Tree Growth

Algebraic expressions are some of the basic building blocks of mathematics. They allow us to describe real-world situations in such a way that the power of mathematics can be applied to them. In this Activity, you will write expressions to describe the growth of trees.

New trees arise naturally in a forest, from seeds that come from existing trees. In the United States, over 2 billion trees are planted each year by the forest products industry, tree farmers, and government agencies. Many of these plantings consist of small seedlings or saplings that are already more than a foot tall.

The figure below shows several stages in the growth of a tree. The first drawing on the left represents a newly planted sapling. This sequence of drawings shows successive years in the tree's growth.

Year 0 Year 1 Year 2

Year 3

Year 4

Year 5

Year Height on Actual Drawing Height

0 1 2 3 4 5

1.

The

scale

of

the

drawings

is

_1_ 4

inch

1

foot.

Measure

the

height

of

the drawing of the newly planted sapling on the left to the nearest

eighth of an inch. Then use a proportion to find the actual height

of the tree.

2. Complete the table to the left by measuring each of the drawings and using a proportion to find the actual height of each tree.

3. Describe any pattern you see in the third column of your table.

4. If the pattern in the table continues, what do you expect the height of the tree to be after 6 years? After 10 years?

5. Let the variable n represent any number of years of tree growth. Assuming the pattern continues, what do you expect the height of the tree to be after n years?

TREE GROWTH

Lesson 3.1

57

Comap2e_Modeling_Ch03.indd 57

Lesson 3.1 Activity Answers

1.

_1_ 2

inch;

2

feet

2. Sample answer:

Height on

Actual

Year

Drawing

Height

0

_ 1 2

inch

2 feet

1

_ 7 8

inch

3

_ 1 2

feet

2

1

_ 1 4

inches

5 feet

3

1

_ 5 8

inches

6

_ 1 2

feet

4

2 inches

8 feet

5

2

_ 3 8

inches

9

_ 1 2

feet

03/02/12 7:32 PM

3. Sample answer: The heights

increase by a constant amount of

1

_1_ 2

feet

each

year.

4. 11 feet; 17 feet

5. Sample answer: 2 1.5n feet

LESSON 3.1

5e Engage

Lesson Objectives

? use arrow diagrams to represent expressions.

? evaluate expressions. ? write expressions to model real-

world situations.

Vocabulary

? algebraic expression ? coefficient ? evaluate an expression ? order of operations

Materials List

? rulers

Description

Preparation:

Have students work in pairs. Provide each pair with a ruler.

During the Activity:

Students should measure the heights

of the trees in the drawing to the

nearest eighth of an inch and then

use

the

_1_ 4

inch

1

foot

scale

to

compute the corresponding tree

heights.

Closing the Activity:

Make sure students see that the construction of an arrow diagram is based on the Order of Operations Guidelines. Write an expression, such as _x_7__4_, on the board. Ask students to represent the expression with an arrow diagram. Then have them evaluate the expression for different values of x.

TEACHING TIP

Ask students how they converted measurements from the drawing to actual tree heights.

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