Using Algebra Tiles Effectively

MATHEMATICS

USING ALGEBRA TILES

EFFECTIVELY

TOOLS FOR UNDERSTANDING

Bettye C. Hall

Reviewers

James Gates, Ed.D.

Yvonne S. Gentzler, Ph.D.

AUTHOR

Bettye C. Hall is the former Director of Mathematics in the Houston Unified

School District in Houston, Texas. Ms. Hall, admired as a “teacher’s teacher”

because of her practical perspective on students’ and teachers’ needs, is active

as a mathematics consultant, speaker, and workshop leader throughout the

United States.

REVIEWERS

JAMES GATES, ED.D.

Executive Director Emeritus, National Council of Teachers of Mathematics

YVONNE S. GENTZLER, PH.D.

Associate Professor, College of Education, University of Idaho

Worksheets and tests may be duplicated for classroom use, the number not to exceed the number of

students in each class. Notice of copyright must appear on all copies. No other part of this book may be

reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying,

recording, or by any information storage or retrieval systems, without permission in writing from the

publisher. Printed in the United States of America.

PRENTICE HALL MATH

USING ALGEBRA TILES

EFFECTIVELY

TOOLS FOR UNDERSTANDING

Contents

Professional Development Overview------------------------------------------------------iv

Introduction --------------------------------------------------------------------------------------vi

Using Algebra Tiles Effectively Workshop

Pre-Workshop Preparation--------------------------------------------------------------1

Workshop Topics--------------------------------------------------------------------------2

Workshop Opening: Examining the Algebra Tiles----------------------------------3

Activity 1: Adding Integers-------------------------------------------------------------5

Activity 2: Subtracting Integers--------------------------------------------------------7

Activity 3: Simplifying Algebraic Expressions--------------------------------------8

Activity 4: Solving Linear Equations------------------------------------------------ 10

Activity 5: Solving Word Problems with Algebra Tiles-------------------------- 12

Activity 6: Modeling, Adding, and Subtracting Polynomials-------------------- 13

Activity 7: Multiplying Polynomials------------------------------------------------- 15

Activity 8: Factoring Polynomials--------------------------------------------------- 17

Workshop Closing ---------------------------------------------------------------------- 19

Visual Aids------------------------------------------------------------------------------- 20

Introducing a new,

hands-on Professional

Development package from

Prentice Hall.

Now, all the algebra tiles resources you need—in one

place—to effectively train your trainers.

With this new package, all educators can better utilize algebra tiles to reach today’s

students. New users are given the support they need to start using algebra tiles in the

classroom—right away. Experienced users are provided with a host of new

applications.

Comprehensive, train-the-trainer package . . .

• Fully-developed algebra tiles workshop

• Written-out scripts that are easy to follow

• Activities for various levels

• Visual Aid masters for transparencies

• Ideas for customizing the workshop for middle grades and

high school teachers

• Suggestions for additional resources, including Internet site

. . . written by an expert in the field.

• Workshop crafted by a leading educator, with expertise in using

algebra tiles, professional development, and curriculum support

• Meets a range of classroom needs

• Can be customized to your middle grades or high school curriculum

• Allows you to effectively and easily train your trainers

Two more professional development exclusives from Prentice Hall:

Annual Math and Technology Summer Institutes

Two-day interactive institutes for middle grades and high school mathematics

educators

• Providing new ideas on how to utilize technology to reach today’s

students

• Highlighting strategies for improving student performance on

standardized tests

• Held each June in cities across the country

• Led by Dr. Jim Gates, Director Emeritus of NCTM, and other leading

educators

Prentice Hall Internet Home Page

• Faculty Forum: an on-line place to dialogue about topics of interest

and to share ideas

• Professional Library, containing a number of handbooks for educators

• Links to sites about grants and fellowships, professional

organizations, departments of education, school sites, and more

• Teacher support for projects, activities, etc., related to Prentice

Hall’s new series of math textbooks for Grades 6–12

• Visit us at: www.phschool.com

PRENTICE HALL

Reaching today’s students

For more information, call 1-800-848-9500

iv Using Algebra Tiles Effectively

Introducing a new,

hands-on Professional

Development package from

Prentice Hall.

Now, all the algebra tiles resources you need—in one

place—to effectively train your trainers.

With this new package, all educators can better utilize algebra tiles to reach today’s

students. New users are given the support they need to start using algebra tiles in the

classroom—right away. Experienced users are provided with a host of new

applications.

Comprehensive, train-the-trainer package . . .

• Fully-developed algebra tiles workshop

• Written-out scripts that are easy to follow

• Activities for various levels

• Visual Aid masters for transparencies

• Ideas for customizing the workshop for middle grades and

high school teachers

• Suggestions for additional resources, including Internet site

. . . written by an expert in the field.

• Workshop crafted by a leading educator, with expertise in using

algebra tiles, professional development, and curriculum support

• Meets a range of classroom needs

• Can be customized to your middle grades or high school curriculum

• Allows you to effectively and easily train your trainers

Using Algebra Tiles Effectively v

Two more professional development exclusives from Prentice Hall:

Annual Math and Technology Summer Institutes

Two-day interactive institutes for middle grades and high school mathematics

educators

• Providing new ideas on how to utilize technology to reach today’s

students

• Highlighting strategies for improving student performance on

standardized tests

• Held each June in cities across the country

• Led by Dr. Jim Gates, Director Emeritus of NCTM, and other leading

educators

Prentice Hall Internet Home Page

• Faculty Forum: an on-line place to dialogue about topics of interest

and to share ideas

• Professional Library, containing a number of handbooks for educators

• Links to sites about grants and fellowships, professional

organizations, departments of education, school sites, and more

• Teacher support for projects, activities, etc., related to Prentice

Hall’s new series of math textbooks for Grades 6–12

• Visit us at: www.phschool.com

PRENTICE HALL

Reaching today’s students

For more information, call 1-800-848-9500

vi Using Algebra Tiles Effectively

Algebra Tiles: Tools for

Understanding

Purposes of the Professional Development Workshop

Today’s mathematics teachers face tremendous challenges, including changes in

curricula, teaching techniques, and assessment practices. To help teachers rise to

these teaching challenges and opportunities, Prentice Hall presents this self-contained

“train the trainer” workshop, developed by nationally-recognized and respected

mathematics educators. This workshop is designed to aid educators in meeting the

diverse needs and expectations of their students.

The introduction of algebra tiles and other manipulatives into the classroom provides

mathematics teachers with exciting opportunities to empower students of all learning

styles. Through hands-on activities, this workshop will help teachers become familiar

with the uses and applications of algebra tiles. New users will become comfortable

using algebra tiles in their classrooms. Experienced users will learn new applications.

All users will more fully appreciate the ways that these manipulative tools can help

them reach a greater percentage of students.

Components of This Workshop

This workshop contains activities and visual aid masters that are suitable for use by

staff developers, department chairpersons, curriculum specialists, or lead teachers in

conducting a workshop on using algebra tiles effectively. This workshop also includes

a statement regarding the purpose of the workshop, the intended audience, a list of

necessary pre-workshop materials and audio-visual requirements, as well as

suggestions for room preparation.

Because of the wide variety of activities provided, middle school, high school, first-

year, and veteran teachers will all benefit from this program. The activities can be

customized to accommodate different needs and levels of understanding. Trainers can

select the activities that best meet their specific time constraints and professional

requirements.

You can introduce the session using any of a variety of ideas that are presented at the

beginning. Next, there is a variety of activities designed to address the workshop

goals and objectives. At the end of the workshop are summary ideas and suggestions

for closing activities.

The workshop contains masters for making transparencies or black line copies of the

visual aids.

Using Algebra Tiles Effectively vii

Facilitator’s Roles and Responsibilities

As facilitator, you should read through the workshop to become familiar with the

objectives and activities. If any of the material is new to you, explore it thoroughly.

Select the activities that best suit the needs of the educators attending your

workshop. Each section and activity includes the suggested amount of time deemed

necessary to accomplish the goal of the activity. Create a reasonable schedule for your

workshop, being sure to allow time for exploration and questions.

Suggestions for Getting Started

Introduce yourself. Workshop participants may or may not know each other. If the

latter is true, set aside a few minutes for members of the group to get acquainted with

one another. Encourage them to talk about their own experiences and their students’

experiences using algebra tiles. This should ease participant anxiety.

Customizing the Workshop

As is always the case, knowledge is best understood and interpreted when it meets

the needs of the learner. Knowing the needs of the group to be served, the facilitator

will be in the best position to customize the workshop. This workshop, Using Algebra

Tiles Effectively, contains a sufficient variety of activities so that you can use the

workshop with middle grade teachers or high school teachers, educators new to using

algebra tiles or educators experienced at teaching with manipulatives and algebra tiles

or any combination of these groups. On page 2 of the workshop is a table containing

the activities and suggested amount of time necessary to accomplish each activity.

Use this table to determine which activities you will use in this workshop.

USING ALGEBRA TILES EFFECTIVELY 1

Using

Algebra Tiles

Effectively

(x + 3)(x – 2) = x

+ x – 6

When I listen, I hear.

When I see, I remember.

But when I do, then I understand.

This workshop helps participants use algebra tiles effectively in

their classrooms.

Purpose

Middle School and/or High School Teachers

Participants

Materials

• Algebra tiles (1 set for every two participants)

• Flip charts or large pieces of newsprint for recording

drawings and number sentences.

• Crayons or colored markers (for each group)

Pre-Workshop

Preparation

Audio-Visual Requirements

• Overhead projector and screen

• Algebra tiles for the overhead projector

• Blank transparency film and pens for transparencies

• Transparencies of Visual Aids 1, 2, 3, 4, 5A, 5B, 6A, 6B, 7A,

7B, 8A, 8B, 9A, 9B

Room Preparation

1. Select desks or tables with flat tops as workspaces.

2. Arrange tables to accommodate groups of two or four.

3. Provide sufficient wall space for posting summaries of group

discussions on flip charts or newsprint.

2 USING ALGEBRA TILES EFFECTIVELY

Because many teachers were not exposed to manipulative activities

as students, they do not use hands-on activities with their classes.

In this workshop, the Facilitator acts as the teacher and models the

lessons on an overhead projector. The participants act as students

and experience the benefits of hands-on activities.

Workshop

Approach

Activity Level Materials Time

Workshop Topics

WORKSHOP OPENING

Examining the

Algebra Tiles

All Participants Visual Aid 1 10 minutes

Note:

In addition to the

Opening and Closing

components,

choose at least two

activities that best suit

your group and your time

constraints.

ACTIVITY 1

Adding Integers

Middle School

Teachers

Tiles: x, +1, –1

Visual Aid 2

10 minutes

ACTIVITY 2

Subtracting Integers

Middle School

Teachers

Tiles: x, +1, –1

Visual Aid 3

10 minutes

ACTIVITY 3

Simplifying Algebraic

Expressions

Middle School

and Algebra I

Teachers

Tiles: x, +1, –1

Visual Aid 4

10 minutes

ACTIVITY 4

Solving Linear Equations

Middle School

and Algebra I

Teachers

Tiles: x, +1, –1

Visual Aids 5A,

15 minutes

ACTIVITY 5

Solving Word Problems

with Algebra Tiles

Middle School

and Algebra I

Teachers

Tiles: x, +1, –1

Visual Aids 6A,

10 minutes

ACTIVITY 6

Adding and Subtracting

Polynomials

Algebra I and

Algebra II

Teachers

Tiles: x

, x, –x, +1,

–1

Visual Aids 7A,

15 minutes

ACTIVITY 7

Multiplying Polynomials

Algebra I and

Algebra II

Teachers

Tiles: x

, – x

, x,

– x, +1, –1

Visual Aids 8A,

15 minutes

ACTIVITY 8

Factoring Polynomials

Algebra I and

Algebra II

Teachers

Tiles: x

, – x

, x,

– x, +1, –1

Visual Aids 9A,

15 minutes

WORKSHOP CLOSING All Participants Recordings from

each group

10 minutes

USING ALGEBRA TILES EFFECTIVELY 3

In this workshop, the Facilitator acts as the teacher and models

the lessons on an overhead projector. The participants act as

students and experience the benefits of hands-on activities in

math classrooms.

Workshop Opening:

Examining the

Algebra Tiles

Welcome Participants (5 minutes)

1. After you introduce yourself, discuss the purpose of the

workshop:

♦ Say: The purpose of this workshop is to help you help

your students learn how to use algebra tiles effectively.

♦ Say: Algebra tiles help to develop concepts related to

integers, algebraic expressions, equations, and

polynomials.

2. Explain the approach you will use in the workshop.

♦ Say: In this workshop, I, as the Facilitator, will act as

the teacher. I’ll use algebra tiles and Visual Aid

transparencies to model the activities on an overhead

projector. You, the participants, will act as students and

experience the benefits of hands-on activities in math

classrooms.

Note:

The Facilitator works with

Visual Aid 1 and algebra tiles

on the overhead projector

while participants work in

pairs, or in groups of 4, at

their desks.

Materials:

• Visual Aid 1

• Algebra tiles for the

overhead projector

• Algebra tiles for each group

• Newsprint or large paper

sheets

Note:

When appropriate, pictures of

models will appear in this

column.

3. Tell participants which activities you will use in the

workshop.

Note:

Choose at least two activities

in addition to the workshop

opening and closing.

4 USING ALGEBRA TILES EFFECTIVELY

Examine the Algebra Tiles (5 minutes)

4. Have participants spread the algebra tiles on their

worktables and examine them.

Note

Use a transparency of

Visual Aid 1.

Place algebra tiles on the

transparency.

5. Discuss the colors and shapes of the different algebra tiles.

Ask questions such as:

♦ Ask: What do you notice about all the negative tiles,

–1, –x, and –x

–1

–x

Answer:

All the negative tiles are

red.

6. Have participants store the tiles in a corner of their

worktables so they will have room to make models in the

center of their tables.

7. Inform participants that they will use the newsprint to record

their findings by drawing the models and writing the number

sentences and equations they will create in the workshop

activities.

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for classroom activities related

to algebra tiles.

USING ALGEBRA TILES EFFECTIVELY 5

Algebra tiles help to develop concepts related to integers,

algebraic expressions, equations, and polynomials. Point out

that small yellow tiles (+1 tiles) are used for positive integers

and small red tiles (–1 tiles) are used for negative integers.

Activity 1: Adding

Integers

Adding Integers (10 minutes)

Note:

The Facilitator works with

algebra tiles on the overhead

projector while participants

work at their desks in pairs or

in groups of 4.

Materials:

• +1 tile

• –1 tile

• Visual Aid 2

• Newsprint

1. Use your algebra tiles on the overhead projector in the steps

below to illustrate the addition of two positive numbers:

5 + 3

♦ Have participants show two groups of positive tiles. In

one group, model +5. In the other group, model +3.

♦ Ask How can we model 5 + 3 with these tiles?

Elicit the fact that to add the two groups, they should be

moved together.

♦ Ask: What number sentence describes the model?

Answer: 5 + 3 = 8

2. Since the sum of a number and its opposite is zero, together,

a positive tile and a negative tile represent zero and are

called a zero pair. Use your algebra tiles to model a zero

pair on the overhead projector.

♦ Ask: What number sentence describes this zero pair?

Answer: 1 + (–1) = 0

3. Have participants model 3 + (–3) at their desks while you

model the expression on the overhead projector. Model 3

with 3 yellow tiles and model –3 with 3 red tiles.

♦ Ask: What number sentence describes the model?

Answer: 3 + (–3) = 0

6 USING ALGEBRA TILES EFFECTIVELY

4. Have participants model 2 + (– 7) at their desks while you

model the expression on the overhead projector.

Model 2 with 2 positive tiles and –7 with 7 negative.

♦ Ask: What expression does this model represent?

Answer: 2 + (–7)

Have participants join the two groups of tiles. Match

pairs of positive and negative tiles and remove them.

Elicit the fact that you can remove zero pairs because

their value is zero. Elicit the fact that you cannot form

any more zero pairs because all yellow tiles have been

used.

♦ Ask: After removing the zero pairs, what tiles are left?

Answer: 5 red tiles,

representing –5

♦ Ask: What number sentence describes the model?

Answer: 2 + –7 = –5

5. Challenge participants to find another sum.

♦ Ask: What is the sum of – 6 + 4?

♦ Have participants model the expression and find the

sum. Their models should show 4 zero pairs and 2 red

tiles, similar to what is shown at the right. On a piece of

paper, have teams record their findings by drawing and

coloring the algebra tiles and writing the number

sentence.

Answer: –6 + 4 = –2

6. Place a transparency of Visual Aid 2 on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to find the following sums.

After each group has recorded its models and number

sentences, uncover the “Model/Answer” column and have

participants compare their models to those on the

transparency.

a. 4 + (– 9)

b. – 3 + (– 8)

c. 9 + (– 3)

Note:

Use a transparency of

Visual Aid 2

(5 minutes)

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for more activities related to

integers.

USING ALGEBRA TILES EFFECTIVELY 7

Algebra tiles help to develop concepts related to integers,

algebraic expressions, equations, and polynomials. Point out

that small yellow tiles (+1 tiles) are used for positive integers

and small red tiles (–1 tiles) are used for negative integers.

Activity 2:

Subtracting Integers

Subtracting Integers (10 minutes)

Note:

The Facilitator uses algebra

tiles on the overhead projector

while participants work at their

desks in pairs or in groups of 4.

Materials:

• +1 tile

• –1 tile

• Visual Aid 3

• Newsprint

1. Use algebra tiles on the overhead projector in the steps

below to illustrate the subtraction of two integers.

♦ Say: We can model – 6 – (–2).

Start with 6 negative tiles. To subtract –2, remove 2

negative tiles.

♦ Ask: What number sentence describes the model?

Answer: –6 – (–2) = –4

2. Say: We can model 4 – 7.

♦ Have each group place four positive tiles in a group. Ask

them to add zero pairs until they have seven positive

tiles in the set.

♦ Have the participants remove seven positive tiles.

♦ Say: This is one form of subtraction.

♦ Ask: How many tiles are left?

Answer: Three red tiles,

representing – 3.

♦ Ask: What number sentence describes the model?

Answer: 4 – 7 = – 3

3. Place a transparency of Visual Aid 3 on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles for the following subtractions.

After each group has recorded its models and number

sentences, uncover the “Model/Answer” column.

a. 2 – 6 b. 2 + (–6) c. –3 – 8 d. –3 + (–8)

Note:

Use a transparency of

Visual Aid 3

(5 minutes)

4. Elicit discussion about the differences and sums in Visual

Aid 3.

Say: Complete the following statement. To subtract an

integer, you can add —.

Answer: its opposite

8 USING ALGEBRA TILES EFFECTIVELY

One pre-service teacher commented about the class in which he

was a substitute teacher:

“They just don’t get it unless they use the tiles. I’ve tried

without them, and it just doesn’t work.”

Activity 3:

Simplifying Algebraic

Expressions

Simplifying Algebraic Expressions

(15 minutes)

1. Describe the algebra tiles the participants will use. Point out

that negative tiles are red.

Note:

The Facilitator works with

algebra tiles on the overhead

projector while participants

work at their desks in pairs or

in groups of 4.

Materials:

• +1 tile

• –1 tile

• x tile

• –x tile

• Visual Aid 4

• Newsprint

2. Write this expression on the chalkboard or on a

transparency:

♦ Write: 5 + 4x

♦ Ask: How can we model this expression?

♦ Give participants an opportunity to respond. Then model

the expression with the group.

5 + 4x

3. Write this expression on the chalkboard or on a

transparency:

♦ Write: 3x + 2 – 4x – 5

♦ Say: Before we model this expression, remember that

subtracting is the same as adding the opposite, so we

can write the expression as 3x + 2 + (–4x) + (–5).

♦ Ask: How can we model this expression?

♦ Give participants an opportunity to respond. Then model

the expression with the group.

3x + 2 – 4x – 5

USING ALGEBRA TILES EFFECTIVELY 9

4. ♦ Ask: How can we simplify the expression?

♦ Elicit the fact that simplifying means collecting like

terms (like tiles) by using zero pairs.

♦ Remove zero pairs of x tiles and zero pairs of integer

tiles.

♦ Say: After we move aside the zero pairs, the simplified

expression is left.

Answer:

3x + 2 – 4x – 5 = –x – 3

5. Write this expression on the chalkboard or on a

transparency:

♦ Write: –2x + 5 – 4x – 5

♦ Ask: How can we model this expression?

♦ Give participants an opportunity to respond. Then model

the expression with the group.

–2x + 5 – 4x – 5

♦ Ask: How can we simplify the expression?

♦ Repeat that simplifying means collecting like terms (like

tiles) by using zero pairs.

♦ Remove the integer zero pairs.

♦ Ask: What is the simplified expression?

Answer: –6x

6. Place a transparency of Visual Aid 4 on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to model and simplify the

following expressions. After each group has recorded its

models and expressions, uncover the “Model/Answer”

column.

a. 4x + 8 – 3x

b. 5x – 9 – 2 – 3x

c. –3x + 7 + x – 6

Note:

Use a transparency of

Visual Aid 4

(5 minutes)

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for further classroom activities

related to integers.

10 USING ALGEBRA TILES EFFECTIVELY

Algebra tiles can help students develop concepts related to

integers, algebraic expressions, equations, and polynomials.

Activity 4:

Solving Linear

Equations

Solving Linear Equations (15 minutes)

1. Describe the algebra tiles, shown at the right, that the

participants will use. Point out that negative tiles are red.

Note:

The Facilitator works with

algebra tiles on the overhead

projector while participants

work at their desks in pairs or

in groups of 4.

Materials:

• +1 tile

• –1 tile

• x tile

• –x tile

• Visual Aid 5

• Newsprint

2. Write this expression on the chalkboard or on a transparency:

♦ Write: x – 2 = 7

♦ Ask: How can we model this equation?

♦ Give participants an opportunity to respond. Then model

the equation. Point out that just as the two sides of an

equation are separated by an equals sign, the two parts

of the equation are separated by the bar in the model.

x – 2 = 7

♦ Ask: How can we use algebra tiles to solve this

equation?

♦ Give participants an opportunity to respond.

♦ Say: To isolate the variable, that is, to get the x tile

alone, add 2 positive tiles to each side of the equation.

♦ Remove the zero pairs to show the solution.

♦ Ask: What is the solution?

Answer: x = 9

USING ALGEBRA TILES EFFECTIVELY 11

3. Write this expression on the chalkboard or on a

transparency:

♦ Write: 2x + 3 = –9

♦ Ask: How can we model and solve this equation?

♦ Give participants an opportunity to respond. Then

manipulate the tiles while you explain each step.

2x + 3 = –9

♦ Add three negative tiles to each side to create zero pairs

on the side with the x-tiles.

♦ Remove zero pairs to show 2x = –12.

2x = –12

♦ Say: We want to get x alone for a solution. First, we

can make two groups of equal numbers of tiles on each side

of the bar. Then we can remove one set of the tiles from

each side of the bar. Remember that whatever we do to one

side of an equation, we must do to the other side.

2x ÷ 2 = –12 ÷ 2

♦ Ask: What is the solution?

Answer: x = –6

4. Place a transparency of Visual Aid 5A on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to solve the following

equation. After each group has recorded its model and

solution, uncover the “Model/Answer” column.

♦ Say: Solve this equation: 3x – 2 = 4

Note:

Use a transparency of

Visual Aid 5A.

Answer: x = 2

5. Place a transparency of Visual Aid 5B on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to solve the following

equations. After each group has recorded its models and

solutions, uncover the “Model/Answer” column.

♦ Say: Solve this equation: 5x + 6 = –4

Note:

Use a transparency of

Visual Aid 5B.

Answer: x = –2

12 USING ALGEBRA TILES EFFECTIVELY

In this activity you will challenge participants to work in

cooperative groups to analyze a word problem, determine how

they can use algebra tiles to model the problem, and how they

can use algebra tiles to solve the problem.

Activity 5: Solving

Word Problems with

Algebra Tiles

Using Algebra Tiles to Solve a Word

Problem (10 minutes)

1. Read this problem to participants or display a transparency

of Visual Aid 6A.

♦ A total of 38 students from two schools attended a

mathematics tournament. Six students from

Jefferson High School rode bicycles. The rest were

transported in three cars. Seven students from

Lincoln Middle School rode bicycles. The rest were

transported in two cars. If all the cars transported

the same number of students, how many students

rode in each car?

Materials:

• Each group of 4 participants

needs enough algebra tiles to

represent 51 unit tiles and

5 x-tiles.

• Visual Aids 6A, 6B

Note:

Use a transparency of Visual Aid

6A to present the problem.

2. Tell participants they will work together in their groups for 5

minutes to:

♦ Write an equation for the problem.

♦ Model the problem with algebra tiles.

♦ Solve the problem.

Note:

Allow 5 minutes.

Remind students to record their

models and equations.

Equation:

3x + 6 + 2x + 7 = 38

3. After 5 minutes display a transparency of Visual Aid 6B on

the overhead projector. Discuss the equation, the models,

and the solution with participants.

Note:

Uncover the transparency for

Visual Aid 6B row by row.

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for further classroom activities

related to problem solving.

USING ALGEBRA TILES EFFECTIVELY 13

There are many more activities that can be done with algebra

tiles beyond representing algebraic expressions and linear

equations. Algebra tiles can be used to represent polynomials

and operations with polynomials. The tiles can also be used to

factor trinomials and to solve quadratic equations.

Activity 6: Adding

and Subtracting

Polynomials

Modeling Polynomials (5 minutes)

1. Review the meaning of each algebra tile with participants.

Materials:

+1 –1

x –x

–x

• Visual Aids 1, 7A, 7B

• Newsprint

2. Place the tiles to the right on the overhead projector.

♦ Ask: What expression does this model represent?

Answer: 2x

– 3x – 4

3. Challenge participants to build a model of the expression:

♦ Say: Build a model for this expression:

– 2x – 3.

– 2x – 3

Adding Polynomials (5 minutes)

1. Remind participants that they can model addition of

polynomials by modeling the two polynomials, joining them,

and removing zero pairs.

♦ Ask: How can we model this addition?

+ 3x + 5 and x

– 2x – 3

♦ Accept all reasonable answers. Then model the addition

with tiles on the overhead projector. Call attention to the

zero pairs.

Ask: What is the sum?

Answer: 3x

+ x + 2

14 USING ALGEBRA TILES EFFECTIVELY

2. Place a transparency of Visual Aid 7A on the overhead

projector. Cover the “Model/Answer” section. Have

participants use algebra tiles to add the following

polynomials. After each group has recorded its models and

solutions, uncover the “Model/Answer” section.

♦ Say: Find the sum:

(3x

+ 2x – 4) + (–2x

+ x – 3)

Note:

Use a transparency of Visual

Aid 7A.

Answer:

+ 3x – 7

Subtracting Polynomials (5 minutes)

1. Remind participants that they can model subtraction of a

polynomial by adding the model of the inverse of the

polynomial to be subtracted to the model of the first

polynomial.

♦ Ask: How can we model this subtraction?

+ 4x – 5 – (x

+ 2x – 3)

♦ Accept all reasonable answers. Then model the addition

with tiles on the overhead projector.

Note:

By the time students get to

polynomials, they should know

that subtraction is adding the

inverse, or opposite, number.

2. Build a model for 2x

+ 4x – 5.

3. Subtract x

+ 2x – 3 by adding its opposite, that is

–x

– 2x + 3. Model this expression.

♦ Combine models and remove zero pairs to model the

result.

♦ Ask: What expression does the resulting model

represent?

Answer: x

+ 2x – 2

4. Place a transparency of Visual Aid 7B on the overhead

projector. Cover the “Model/Answer” section. Have

participants use algebra tiles to subtract the following

polynomials. After each group has recorded its models and

solutions, uncover the “Model/Answer” section.

♦ Say: Find the difference:

(2x

+ 2x – 1) – (x

– x + 3)

Note:

Use a transparency of Visual

Aid 7B.

Answer: x

+ 3x – 4

USING ALGEBRA TILES EFFECTIVELY 15

In this activity participants will use rectangular arrays to model

the product of two polynomials. You can relate the multiplication

of polynomials to a basic whole-number multiplication table.

Place the first factor vertically under the multiplication dot. Place

the second factor horizontally to the right of the multiplication

dot.

Activity 7:

Multiplying

Polynomials

Multiplying Polynomials (15 minutes)

1. Review the meaning of each algebra tile with participants.

Materials:

+1 –1

x –x

–x

• Visual Aids 1, 8A, 8B

• Newsprint

2. Ask participants how the table on Visual Aid 8A shows

basic multiplication facts.

♦ Ask: How can we find the product 3 • 4 in this table?

♦ Ask: What is the product of 3 • 4?

Note:

Use Visual Aid 8A.

Answer:

Find the entry to the right of 3

and under 4.

3 • 4 = 12

3. Help participants relate a basic multiplication table to a

rectangular array to model the multiplication of polynomials.

♦ Ask: How can we model the multiplication of

(x + 3)(x – 2)?

♦ Elicit the fact that algebra tiles that represent the first

factor (x + 3) are placed on the vertical axis and algebra

tiles that represent the second factor (x – 2) are placed

on the horizontal axis.

(x – 2)

(x + 3)

16 USING ALGEBRA TILES EFFECTIVELY

♦ Ask: How can we model (x + 3)(x – 2)?

♦ Give participants a chance to respond and model the

multiplication with algebra tiles on the overhead

projector.

(x + 3)(x – 2)

♦ Ask: What is the product of (x + 3)(x – 2)?

Answer: x

+ 3x – 2x – 6 =

+ x – 6

4. Place a transparency of Visual Aid 8A on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to model the following

multiplications of polynomials. After each group has

recorded its models and solutions, uncover the

“Model/Answer” column and discuss the groups’ findings.

(x – 1)(x – 4)

Note:

Use a transparency of Visual

Aid 8A.

Answer: x

– 5x + 4

5. Place a transparency of Visual Aid 8B on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to model the following

multiplications of polynomials. After each group has

recorded its models and solutions, uncover the

“Model/Answer” column and discuss the groups’ findings.

(–2x + 2)(x – 3)

Note:

Use a transparency of Visual

Aid 8B.

Answer: –2x

+ 8x – 6

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for further classroom activities

related to polynomials.

USING ALGEBRA TILES EFFECTIVELY 17

In this activity participants will use rectangular arrays to model

factoring polynomials. Relate factoring polynomials to Activity

7, Multiplying Polynomials. The first factor is placed vertically

under the multiplication dot. The second factor is placed

horizontally to the right of the multiplication dot.

Activity 8: Factoring

Polynomials

Factoring Polynomials (15 minutes)

1. Review the meaning of each algebra tile with participants.

Materials:

+1 –1

x –x

–x

• Visual Aids 1, 9A, 9B

• Newsprint

2. Briefly review Multiplying Polynomials.

♦ Ask: How can we model the multiplication of

(x + 3)(x – 2)?

♦ Elicit the fact that algebra tiles that represent the first

factor (x + 3) are placed on the vertical axis and algebra

tiles that represent the second factor (x – 2) are placed

on the horizontal axis.

♦ Give participants a chance to respond. Then model the

multiplication with algebra tiles on the overhead

projector.

(x + 3)(x – 2) = x

+ x – 6

3. Show the model at the right on the overhead projector.

♦ Call attention to the fact that the model shows a

rectangular array with the tiles arranged in descending

order.

♦ Ask: What expression does this model represent?

Note: Use a transparency of

Visual Aid 8A or model the

expression with algebra tiles on

the overhead projector.

Answer: x

+ 4x – 3x – 12

18 USING ALGEBRA TILES EFFECTIVELY

4. Challenge participants to suggest how the expression can

be factored by using algebra tiles. Give participants a chance

to respond and then demonstrate on the overhead projector

how to factor a polynomial.

♦ Build an axis around the rectangle.

♦ To factor the polynomial, find the dimensions of the

rectangle. Determine which tiles should be placed on the

horizontal axis and which tiles should be placed on the

vertical axis. Call attention to the fact that all negative

tiles should be placed on the same axis.

♦ Ask: What expression does the resulting model

represent?

Answer: (x – 3)(x + 4)

5. Place a transparency of Visual Aid 9A on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to model the factoring of the

following trinomial. After each group has recorded its model

and solution, uncover the “Model/Answer” column and

discuss the groups’ findings.

♦ Factor x

+ 5x + 6.

Note:

Use a transparency of Visual

Aid 9A.

Answer:

+ 5x + 6 =

(x + 2)(x + 3)

6. Place a transparency of Visual Aid 9B on the overhead

projector. Cover the “Model/Answer” column. Have

participants use algebra tiles to model the factoring of the

following trinomial. After each group has recorded its model

and solution, uncover the “Model/Answer” column and

discuss the groups’ findings.

♦ Factor x

– 7x + 12.

Note:

Use a transparency of Visual

Aid 9B.

Answer:

– 7x + 12 =

(x – 4)(x + 3)

Classroom Applications

Encourage participants to consult the Teacher’s Editions for

Prentice Hall Middle Grades Math, Prentice Hall Algebra, or

Prentice Hall Advanced Algebra for further classroom activities

related to factoring polynomials.

USING ALGEBRA TILES EFFECTIVELY 19

While a representative from each group displays the pages with

the group’s recordings, you may wish to collect the algebra tiles

and other materials that must be returned.

Workshop Closing:

Recapping the

Activities

Closing Activities (10 minutes)

1. Have a representative from each group display around the

room the recordings (drawings, number sentences,

equations) from that group.

2. Recap the activities that have been presented and end by

having participants discuss what they have learned and how

they can use it.

♦ Ask: How can the activities that we worked on today

have an impact on students in your classroom from now

on?

3. Stress with participants the importance of continuing to

experiment with algebra tiles in their classrooms.

♦ Say: Remember, algebra tiles can help students

develop concepts related to integers, algebraic

expressions, equations, and polynomials.

4. You may wish to refer your group to any worthwhile articles,

books, or Web sites with which you are familiar.

5. Take any questions that remain.

6. Collect all algebra tiles and other materials that have been

distributed.

7. You may wish to close with the quotation that introduced

this professional development workshop:

♦ Say: Remember:

When I listen, I hear.

When I see, I remember.

But when I do, then I understand.

Note:

For more professional

development information and

training materials, visit the

Prentice Hall site:

www.phschool.com

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 1

WORKING WITH ALGEBRA TILES

+1 –1

–x

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 2

ADDING INTEGERS

Use algebra tiles to find the following sums.

Write a number sentence for each.

Record your models and number sentences.

Expression Model/Answer

4 + (– 9)

4 + (–9) = –5

– 3 + (– 8)

–3 + (–8) = –11

9 + (– 3)

9 + (–3) = 6

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 3

SUBTRACTING INTEGERS

Model these differences and sums with

algebra tiles. Write a number sentence for each.

Record your models and number sentences.

Expression Model/Answer

2 – 6

2 – 6 = –4

2 + (–6)

2 + (– 6) = –4

–3 – 8

–3 – 8 = –11

–3 + (–8)

–3 + (–8) = –11

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 4

SIMPLIFYING ALGEBRAIC EXPRESSIONS

Simplify these expressions.

Record your findings.

Expression Model/Answer

4x + 8 – 3x

4x + 8 – 3x = x + 8

5x – 9 – 2 – 3x

5x – 9 – 2 – 3x = 2x – 11

–3x + 7 + x – 6

–3x + 7 + x – 6 = –2x + 1

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 5A

SOLVING LINEAR EQUATIONS

Model this equation with algebra tiles.

Use the model to solve the equation.

Record your findings.

Equation Model/Answer

3x – 2 = 4

Add two positive tiles

to each side. Then remove

zero pairs.

3x = 6

x = 2

Divide.

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 5B

SOLVING LINEAR EQUATIONS

Model this equation with algebra tiles.

Use the model to solve the equation.

Record your findings.

5x + 6 = –4

Add six negative tiles to

each side. Then remove

zero pairs.

5x = –10

x = –2

Divide.

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 6A

SOLVING WORD PROBLEMS

Model the word problem with algebra tiles.

Use the model to solve the problem.

Record your findings.

A total of 38 students from two

schools attended a mathematics

tournament. Six students from

Jefferson High School rode bicycles.

The rest were transported in three

cars. Seven students from Lincoln

Middle School rode bicycles. The

rest were transported in two cars. If

all the cars transported the same

number of students, how many

students rode in each car?

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 6B

SOLVING WORD PROBLEMS

Use the model for Visual Aid 6A.

Solve the word problem. Record your findings.

3x + 6 + 2x + 7 = 38

Simplify the left side

of the equation.

Add –13 to each side

of the equation to

form zero pairs.

Remove zero pairs.

5x = 25

x = 5

Five students rode in

each car.

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 7A

ADDING POLYNOMIALS

Use algebra tiles to find the following sum.

Write an equation for the addition.

Record your model and equation.

Expression

(3x

+ 2x – 4) + (–2x

+ x – 3)

Model/Answer

(3x

+ 2x – 4) + (–2x

+ x – 3) = x

+ 3x – 7

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 7B

SUBTRACTING POLYNOMIALS

Use algebra tiles to subtract.

Write an equation for the subtraction.

Record your model and equation.

Remember: Subtraction is adding the inverse number.

Expression

(2x

+ 2x – 1) – (x

– x + 3)

Model/Answer

(2x

+ 2x – 1) – (x

– x + 3) = x

+ 3x – 4

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 8A

MULTIPLYING POLYNOMIALS

• 2 3 4 5

2 4 6 8 10

3 5 9 12 15

4 8 12 16 20

5 10 15 20 25

3 • 4 = 12

Model this multiplication with algebra tiles.

Record your findings.

Multiply Model/Answer

(x – 1)(x – 4)

– 5x + 4

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 8B

MULTIPLYING POLYNOMIALS

Model this multiplication with algebra tiles.

Record your findings.

(–2x + 2)(x – 3)

Answer:

–2x

+ 8x – 6

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 9A

FACTORING POLYNOMIALS

Use algebra tiles to factor this trinomial.

Record your findings.

Factor Model/Answer

+ 5x + 6

Answer:

+ 5x + 6 = (x + 2)(x + 3)

USING ALGEBRA TILES EFFECTIVELY

VISUAL AID 9B

Use algebra tiles to factor this trinomial.

Record your findings.

Factor Model/Answer

– 7x + 12

Answer:

– 7x + 12 = (x – 4)(x – 3)