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Equal and Equivalent

Lesson #8

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2019 Open Up Resources |

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Let’s use diagrams to figure out which expressions are equivalent and which are just sometimes equal.

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Today’s Goals

I can explain what it means for two expressions to be equivalent.
I can use a tape diagram to figure out when two expressions are equal.
I can use what I know about operations to decide whether two expressions are equivalent.

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Algebra Talk:

Solving Equations by Seeing Structure

Warm Up

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Find a solution to each equation mentally.

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Using Diagrams to Show That Expressions are equivalent

Activity 1

MLR8: Discussion Supports

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Draw diagrams that show:

2 + 3 = 3 + 2
2 + 3 does not equal 3 ᐧ 2

Launch

Provide access to graph paper. Ask students to draw diagrams that show:

2+ 3 = 3 + 2

2 ᐧ 3 does not equal 3 ᐧ 2

We can tell that 2+ 3 and 3+2 are equal because the length of the diagrams represent the value of each expression, and the diagrams are the same length. We can tell that 2 ᐧ 3 is not equal to these because this value is represented by the length of its diagram, and it's not the same length as the others. 2+3 and 3+2 are examples of expressions that are not identical, but are equal. Another example students have seen of this phenomenon are fractions like ½ and 3/6 , which are not identical but equal.

When we start talking about expressions that have letters in them, the language gets more complicated, because expressions can be equal or not equal depending on the value the letter represents.

Arrange students in groups of 2. Ask students to work independently on each question and then check in with their partner, discussing and resolving any disagreements. Allow 15 minutes to work and share responses with a partner, followed by a whole-class discussion.

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Work independently on each question. After each question check your answer with your partner.

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Equivalent Expressions

Equivalent expressions are always equal to each other. If the expressions have variables, they are equal whenever the same value is used for the variable in each expression.

For example, 3x + 4x is equivalent to 5x + 2x . No matter what value we use for x , these expressions are always equal. When x = 3, both expressions equal 21. When x = 10 , both expressions equal 70.

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Identifying Equivalent Expressions

Activity 2

MLR3: Clarify, Critique, Correct

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Work quietly on the task (5 min)

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Are you ready for more?

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Lesson Synthesis

Can you explain why these are equivalent without drawing diagrams?

x and x ᐧ 1
x+1 and 1+x
x ᐧ 3 and 3 ᐧ x
x and x + 0
x + x + x and 3x
x ÷4 and ¼ x

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Today’s Goals

I can explain what it means for two expressions to be equivalent.
I can use a tape diagram to figure out when two expressions are equal.
I can use what I know about operations to decide whether two expressions are equivalent.

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Decisions About Equivalence

Cool Down