Verify the distributive property of division over addition using
a = 1/3 b = -2/3 & c = 1/2.
Answers
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: