ANSWER:

Are They

Saying the

Same Thing?

Verifying Equivalent Expressions

4

Lesson Overview

Students begin by reviewing the properties of arithmetic and algebra that they have formally

or informally studied in the past. This allow students to use properties as they rewrite algebraic

expressions in equivalent forms. Students analyze pairs of expressions. They use properties, tables,

and graphs to show that the expressions are or are not equivalent. Students compare the algebraic

expressions and are asked to use tables and graphs to determine if they equivalent. This opens the

discussion that one non-example is necessary to disprove a claim, while an infinite number of examples

are necessary to prove a claim.

Grade 6 Expressions and Equations

Apply and extend previous understandings of arithmetic to algebraic expressions.

4. Identify when two expressions are equivalent (i.e., when the two expressions name the same

number regardless of which value is substituted into them).

Essential Ideas

• The Commutative Properties of Addition and Multiplication state that the order in which you

add or multiply two or more numbers does not affect the sum or the product.

• The Associative Properties of Addition and Multiplication state that changing the grouping of

the terms in an addition or multiplication problem does not change the sum or product.

• The Distributive Property states that if a, b, and c are any real numbers, then a(b 1 c) 5 ab 1 ac.

Because subtraction is a special form of addition and division is a special form of

multiplication, the Distributive Property can also be expressed as a(b 2 c) 5 ab 2 ac, _____

a 1 b

c 5

__

a

c 1

__

b

c, and _____

a 2 b

c 5

__

a

c 2

__

b

c.

• Two algebraic expressions are equivalent expressions if, when any values are substituted for

the variables, the results are equal.

• Two algebraic expressions can be proven to be equivalent by: (1) using algebraic properties to

simplify them until they are written the EXACT same way; and (2) graphing each expression on

the same graph to determine if their graphs are the same.

MATERIALS

Scissors

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M3-53B • TOPIC 1: Expressions

Lesson Structure and Pacing: 1 Day

Engage

Getting Started: Property Sort

Students cut out cards that display the names of properties, a numeric example of each property,

an algebraic statement of each property, and a visual of each property. Then they match the

representations of the properties and discuss their reasoning.

Develop

Activity 4.1: Determining Whether Expressions Are Equivalent

Students compare two equivalent expressions and two non-equivalent expressions. They use

properties to verify that the two expressions are or are not equivalent, complete a table of

values for each expression, and use graphs to show when two expressions are equivalent. Then

students determine if two expressions are equivalent using tables and graphs. This problem also

addresses the concept of proof, namely, what is enough to prove a claim and what is enough to

disprove a claim.

Demonstrate

Talk the Talk: Property Management

Students identify the properties used to simplify expressions. They rewrite an expression in

simplest form and explain how to verify that their simplification is correct.

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LESSON 4: Are They Saying the Same Thing? • M3-53C

Getting Started: Property Sort

Facilitation Notes

In this activity, students cut out cards that display the names of properties, a

numeric example of each property, an algebraic statement of each property,

and a visual of each property. Then they match the representations of the

properties and discuss their reasoning.

Direct students to cut out the cards at the end of the lesson. Have students

sort the cards according to the property NAMED or illustrated. Have

students work with a partner or in groups to complete Questions 1 and 2.

Share responses as a class.

Questions to ask

• How do the PICTURES represent addition and/or multiplication?

• How can you tell the difference between the Commutative Property

and the Associative Property?

• Why is the commutative property useful in arithmetic?

• Why is the associative property useful in arithmetic?

• Why is the distributive property useful in arithmetic?

• What properties do you use when you combine like terms?

Differentiation strategies

To assist all students,

• MAKE other connections using words. For example, commute means

to move. In commutative property of addition, the numbers “move”.

• Make other connections using symbols. Write each algebraic notation

under its picture to emphasize its meaning.

To support students who struggle, create 2 smaller sorts, one with

Commutative Property and the Associative Property and the other sort

with the remaining properties.

Summary

ion: