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Algebra Teacher's Activities Kit

150 Ready-to-Use Activities with Real-World Applications

150 Ready-to-Use Activities with Real-World Applications

By Judith A. Muschla Gary Robert Muschla

By Judith A. Muschla Gary Robert Muschla

John Wiley & Sons

John Wiley & Sons

Copyright © 2003

Copyright © 2003

Judith A. Muschla

All right reserved.

Judith A. Muschla

All right reserved.

ISBN: 0-7879-6598-7

ISBN: 0-7879-6598-7

Chapter One

**The Language of Algebra**

(Using Whole Numbers)

(Using Whole Numbers)

Every discipline has its own language. Soccer has its goals, figure

skating has its triple axels, and geometry has its polygons. Specialized

terminology makes it possible to understand the discipline.

How could someone describe baseball without mentioning the words

"innings," "strikes," and "outs"?

The fifteen activities of this section focus on the basic terminology

of algebra. Understanding this language will help your students to

understand numerical relationships, describe operations, and reason

algebraically. By becoming fluent in the language of algebra, your students

will be prepared to move on to more advanced skills and concepts.

**Teaching Suggestions for the Activities**

**1-1** Using the Order of Operations I

This activity requires your students to simplify an expression. The

expressions of the activity focus on the four basic operations-addition,

subtraction, multiplication, and division. No grouping symbols or exponents

are included. All numerical values are positive wholenumbers. The

symbols · and x are used to denote multiplication.

Introduce this activity by discussing the rules for the Order of Operations,

which are noted on the worksheet. Go over the example on the

worksheet together, noting the steps that are taken to arrive at the

numerical value.

Review the instructions on the worksheet with your students.

Explain that the first column lists the expression and the second column

the steps necessary to simplify an expression. Point out that each step

on the right can be paired with only one expression on the left and that

each step can be used only once. Remind your students to include the

numerical values of the expressions in their answers.

**1-2** Using the Order of Operations II

This activity builds upon the skills addressed in Activity 1-1 and pro-vides

more practice with the Order of Operations. It includes the use of

grouping symbols.

Begin the activity by reviewing the Order of Operations with your

students. Emphasize that parentheses, braces, brackets, and the fraction

bar are all considered to be grouping symbols. Operations within these

symbols must be done first. Remind your students that a number

directly to the left of a grouping symbol implies multiplication.

Go over the instructions for the activity with your students. Note

that, along with identifying the incorrect answer, they are to explain why

the answer is wrong.

**1-3** Using the Order of Operations III

For this activity your students are to simplify various expressions using

the Order of Operations. Grouping symbols, exponents, and fractions

appear in the expressions. Unless your students are experienced with

simplifying expressions, you should assign Activities 1-1 and 1-2 before

assigning this one.

Start the activity by reviewing the Order of Operations. You should

also explain the concept of a base and an exponent. For example, in the

expression [2.sup.3] , 2 is called the base and 3 is the exponent. The exponent

means that 2 is used as a factor three times. Thus, [2.sup.3] means 2 x 2 x 2 or

8. Point out that a common error students make when working with

exponents is to multiply the base times the exponent, for example, in this

case, 2 x 3, obtaining the incorrect answer of 6.

Depending on the abilities and backgrounds of your students, you

may also wish to introduce the concept of a square number, which is a

number raised to the second power. For example, [5.sup.2] = 25 shows that 25

is a square number. Some other examples of square numbers are 9, 16,

36, 49, 64, and 81.

Make sure your students understand the instructions for the activity.

Remind them to follow the directions closely when placing their

answers in the Code Box.

**1-4** Using Square Numbers

The purpose of this activity is to familiarize your students with square

numbers. Review the examples of square numbers included on the worksheet.

Depending on the needs of your students, you may wish to expand

the list.

Start this activity by drawing squares on the board or an overhead

projector. Draw: a 1 x 1-inch square, a 2 x 2-inch square, a 3 x 3-inch

square, and a 4 x 4-inch square. Emphasize to your students that the

number that represents the area in each square is a square number.

Next, explain the Square Number Theorem, referring to the examples

on the worksheet. Remind your students to find the square numbers

first, then add. You might mention that this procedure is the same as following

the Order of Operations.

Go over the instructions on the worksheet, and emphasize that the

last five problems require students to find the four square numbers that

add up to the number. This is the opposite of what they have to do in

problems 1 through 10. Suggest that guess and check is a good strategy

to use in solving the last four problems.

**1-5** Translating Algebraic Expressions into Phrases

This activity requires students to complete a crossword puzzle with a

word omitted from a phrase. It provides practice in writing expressions.

Introduce the activity by explaining that an algebraic expression is

a combination of a variable (or variables) and a number (or numbers).

Note the examples on the worksheet. Emphasize that order matters

when subtracting and dividing. For instance, n - 3 means 3 less than a

number and not 3 minus a number. Incorrect order is a common mistake

when writing expressions.

Review the instructions on the worksheet with your students.

Remind them that this is a crossword puzzle, and encourage them to

focus their attention on the clues.

**1-6** Writing Phrases as Algebraic Expressions I

This activity is a follow-up to Activity 1-5. For this activity, your students

are given a phrase containing an expression. They are to determine if

the expression is stated correctly. If it is incorrect, they are to correct the

expression.

Note the common errors that many students make with these types

of expressions. For example, they may overlook order. n - 8 is not the

same as 8 - n. They may also overlook grouping symbols. For example,

three times the sum of a number and 10 is 3(n + 10) or 3(10 + n), but not

3n + 10.

Review the instructions on the worksheet with your students.

Remind them that half of the problems on the worksheet are correct.

Students must correct the incorrect problems.

**1-7** Writing Phrases as Algebraic Expressions II

This activity provides your students with more practice using algebraic

expressions. In each problem of this activity, your students are to think of

a number, do a series of numerical operations, and obtain an answer that

the teacher can predict. Your students, with the aid of algebra, are to

explain these problems as well as create problems of their own. To complete

this activity successfully, your students must be able to translate

phrases into algebraic expressions.

Start this activity by reviewing the Distributive Property. Go over

the instructions on the worksheet, then do the first problem as a class

exercise. Instruct your students to write the number they begin with in

the blank in Column I, recording each successive number in the blanks

provided. (You may wish to suggest that for problems 1 through 3 students

choose a number between 1 and 9 to keep the math simple.) Complete Column

II (for the first problem) as a class. The steps are n, which represents

the first number the students record, [n.sup.2] , [n.sup.2] - 4, [2n.sup.2] - 8, [2n.sup.2] , n. Point out

that in this problem your students end with the number with which they

started.

Note that your students should use a variable to represent a number

in Column II. If they are asked to use another number, they should

choose a different variable.

**1-8** Simplifying Expressions by Combining Like Terms

This activity is designed to provide an introduction to combining similar

terms. Begin the activity by explaining basic notation: 3 x 4n can also be

written as 3 · 4n or 3(4n), all of which equal 12n. Also, n can be written

as 1n. 0 xn equals 0. Review the vocabulary on the worksheet and make

sure that your students understand the Distributive Property.

Go over the instructions on the worksheet. Remind your students to

simplify each expression completely.

**1-9** Simplifying and Evaluating Expressions

In this activity your students will simplify expressions by combining like

terms and then evaluate the expressions. The problems on the worksheet

do not contain negative numbers or exponents.

Begin the activity by reviewing how to simplify expressions.

Depending on the abilities of your students, you may also find it helpful

to discuss the Distributive Property. Remind your students that expressions

such as 2a can be expressed as 2 x a and that b can be expressed

as 1 x b. Note that to substitute a number for a variable, your students

should write the number in place of the variable. For example, if a = 3,

then 2a = 2 x 3 or 6.

Go over the instructions on the worksheet with your students. Caution

them to pay close attention to grouping symbols and remind them

to always multiply before adding or subtracting.

**1-10** Evaluating Expressions Using Exponents

This activity requires your students to evaluate expressions using exponents.

The answers will be positive whole numbers.

Begin this activity by discussing 0 and 1 as exponents. Note that [x.sup.1]

= x and that [x.sup.0] = 1. Depending on the background and abilities of your

students, you may also find it useful to review grouping symbols.

Go over the instructions with your students and discuss the examples

on the worksheet. Remind your students to be sure to follow the

Order of Operations where necessary.

**1-11** Writing Equations

In this activity your students are provided with information that they

are to express in terms of an equation. Although they are not required

to solve the equation, they may be curious to find the solution. The equations

and their solutions are provided in the Answer Key.

Begin this activity by writing some equations on the board or an

overhead projector. Two examples are P = 4s, for finding the perimeter

of a square, and A = l x w, for finding the area of a rectangle. Explain that

the equal sign means that the number on the left of the equation has the

same value as the number on the right. Encourage your students to volunteer

examples of other equations, which you may list on the board or

an overhead projector. As you do, emphasize the equality and the meaning

of the variables. Also review key words such as "variable," "more

than," and "product."

Go over the instructions on the worksheet with your students. Caution

them to be as accurate as possible in writing equations.

**1-12** Writing Equations and Inequalities I

For this activity your students must recall, find, research, and synthesize

various facts. (Most facts fall within the category of general knowledge.)

Your students are then required to compare numbers and write

an equation or inequality.

Since your students may need to conduct minor research to find

some of the information necessary to complete this activity successfully,

you may prefer to assign this activity as homework. If you have

access to the Internet from your classroom, the activity can easily be conducted

there. Another option is to reserve time in your school's library

so that students may use reference sources. This is a nice approach,

because it provides an example to students of how math is linked to

other areas.

Start this activity by reviewing the meaning of an equation. Note

that the values on both sides of an equal sign are the same, and emphasize

that in an inequality the numbers are not equal. If necessary,

explain the meanings of the symbols > and <, which appear on the worksheet.

Review the instructions on the worksheet with your students.

Encourage them to concentrate on the phrases as they work to complete

the activity.

**1-13** Writing Equations and Inequalities II

This activity builds on the skills covered in Activity 1-12 and provides

more practice with equations and inequalities. Along with the symbols

> and <, the symbols [Greater than or equal to] and [less than or equal to] are used in many of the equations.

Start the activity by discussing the symbols on the worksheet, then

go over the instructions. Depending on the abilities of your students, you

may find it helpful to do the first problem together. Remind students to

read each problem carefully before writing an equation.

**1-14** Identifying the Solution of an Equation

In this activity your students are required to match solutions to equations.

They are to substitute a given value and determine whether or not

it is a solution to the equation. Obtaining the correct answers will enable

them to complete a statement at the bottom of the worksheet.

Begin the activity by reviewing the Order of Operations. Because

there are no grouping symbols, your students should substitute, then

multiply and divide in order from left to right, then add and subtract.

Go over the instructions on the worksheet. Note that of the five possible

solutions above the four equations, only one is not a solution for any

of the equations. Your students are to write the letter of each solution in

the blank before the equation. After determining the correct solutions,

they are to write the letters in order to complete the statement at the

bottom of the worksheet.

**1-15** Determining the Solutions of Equations and Inequalities

This activity builds on the skills covered in Activity 1-14. For this activity

your students are given twenty equations and inequalities for which

they are to choose the solutions.

Begin the activity by reviewing the meaning of the equal sign and

the four inequality symbols: >, <, [greater than or equal to] and [less than or equal to] Go over the

instructions with your students. Emphasize that they are to record the letter of every solution.

Some problems have more than one. If no solution is given, they are

to record the letter that precedes "none."

**Answer Key for Section 1**

**1-1.** 1. 8 - 6 + 5 = 7 2. 3 + 8 + 4 = 15 3. 3 + 2 + 4 = 9 4. 21 - 14 + 6 = 13

5. 2 + 9 - 1 = 10 6. 48 + 2 - 7 = 43 7. 3 + 25 - 2 = 26 8. 4 + 12 - 16 =

0 9. 5 + 2 - 4 = 3 10. 60 - 60 + 2 = 2 11. 6 + 18 + 4 = 28 12. 1 + 6 -

2 = 5

**1-2.** 1. 15 - 2 x 3 = 39 is incorrect. Subtraction was done first, then multiplication

was done. 2. 16 + 8 ÷ 2 = 12 is incorrect. Addition was done first,

then division was done. 3. 3[2 + 4] = 10 is incorrect. Grouping symbols

were ignored; multiplication was done first, then 4 was added. 4. 50 ÷

(5 x10) = 100 is incorrect. Grouping symbols were ignored; division was

done, then multiplication was done. 5. *Continues...*

Excerpted fromAlgebra Teacher's Activities Kit

byJudith A. Muschla Gary Robert Muschla

Copyright © 2003 by Judith A. Muschla.

Excerpted by permission.

All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.

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