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7.2

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

MLR8: Discussion Supports

Materials

To Gather

Activity Narrative

The purpose of this activity is to develop students’ understanding of equivalent expressions. Students use tape diagrams to represent pairs of algebraic expressions when a one-digit whole number is given for the variable. Students see that expressions may be equal at some values for the variables, but not at others. Through repeated reasoning, they also begin to generalize that equivalent expressions are expressions that are equal at all values (MP8).

Launch

Arrange students in groups of 2. Ask students to work independently on each question and check in with their partner, discussing and resolving any disagreements, before moving on to the next question. Give students 10–12 minutes of work time, followed by a whole-class discussion. Tell students that they should draw two separate diagrams on each grid—one diagram to represent each expression.

While a grid is provided for each question that requires drawing, some students may wish to have additional drawing space. Provide access to graph paper if requested.

Action and Expression: Internalize Executive Functions. Begin with a small-group or whole-class demonstration and a think-aloud of the first question to remind students how to draw tape diagrams on grids. Keep the worked-out examples on display for students to reference as they work. Supports accessibility for: Memory, Conceptual Processing

Representing, Conversing, Listening: MLR8 Discussion Supports. Display sentence frames to support students when they share their answers and explanations for the questions, “When are and equal? When are they not equal?” Examples: “I know these expressions are equal (or not equal) when ______ because. . . .” This frame can help students use mathematical language as they connect the representations of equal and not equal values of expressions.
Advances: Speaking, Representing

Activity

None

Here are tape diagrams that represent and when is 4. Notice that the two diagrams are lined up on their left sides, so you can compare their lengths.

Two tape diagrams. Top diagram 2 parts, x, 2. Bottom diagram 3 parts x,x,x

On each grid, line up your two diagrams on one side.

Draw tape diagrams that represent and when is 3.
Draw tape diagrams that represent of and when is 2.
Draw tape diagrams the represent and when is 1.
Draw tape diagrams that represent and when is 0.
When are and equal? When are they not equal? Use your diagrams to explain.
1. Draw tape diagrams of and . Choose your own value for .
2. When are and equal? When are they not equal? Use your diagrams to explain.

Student Response

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Building on Student Thinking

Activity Synthesis

The goal of this discussion is to make sure students understand what it means for two expressions with a variable to be equivalent.

Invite several students to share the two diagrams they drew for and . Ask:

“How are the two diagrams in each pair alike?” (The two diagrams are always the same length regardless of the value of chosen.)
“How are the two diagrams in each pair different?” (The placement of and 3 are not the same.)

Display the equation . Point out that we can see that this equation is always true regardless of the value of . We call and equivalent expressions, because their values are equal no matter what the value of is.

Display the equation , and ask students to consider this equation as they review the diagrams they drew on the grids for the first four problems. Point out that we can see that this equation is true when is 1, but not for the other values of that we tried. So, we can say that is equal to when is 1. Ask:

“How do you know that and are not equivalent expressions?” (They are not equal for every value of .)

7.3

Activity

Standards Alignment

Building On

Addressing

Instructional Routines

None

Materials

None

Activity Narrative

In this activity, students apply what they know about the meaning and properties of operations to deepen their understanding of “equivalent expressions.” The focus is on looking for and making use of structure (MP7), rather than identifying all the types of equivalent expressions appropriate to grade 6.

Monitor for students who apply their knowledge of operations with numbers to reason about operations with variable expressions. For instance:

Having learned that is equivalent to , students then reason that .
Knowing that , students reason that .

Students may also use diagrams to reason or to explain their reasoning.

Launch

Give students 5 minutes of quiet work time, followed by a whole-class discussion.

Activity

None

Here is a list of expressions. Find any pairs of expressions that are equivalent. If you get stuck, consider drawing diagrams.

Activity Synthesis

Invite students to share their pairs and reasoning, including those who used diagrams, if any. To highlight the idea that expressions with variables are equivalent only if they have the same value for all values of the variable, ask questions such as:

“How can you be sure that the expressions on both sides will have the same value no matter what value is used for ?”
“Are there two expressions that you knew right away are equivalent, without needing to test values or draw diagrams? If so, which ones, and how did you know?”

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Break the class into small discussion groups and then invite a representative from each group to report back to the whole class. This will provide students with additional opportunities to compare strategies and hear from others.
Supports accessibility for: Language, Social-Emotional Functioning

Student Lesson Summary

We can use tape diagrams to see when expressions are equal. For example, the expressions and are equal when is 3, but they are not equal for other values of .

8 tape diagrams of various sizes and various labels on a grid. Each diagram is matched with an expression.

Sometimes two expressions are equal for only one particular value of their variable. Other times, they seem to be equal no matter what the value of the variable.

Expressions that are always equal for the same value of their variable are called equivalent expressions. However, it would be impossible to test every possible value of the variable. How can we know for sure that expressions are equivalent?

We can use the meaning of operations and properties of operations to know that expressions are equivalent. Here are some examples:

is equivalent to because of the commutative property of addition. The order of the values being added doesn’t affect the sum.
is equivalent to because of the commutative property of multiplication. The order of the factors doesn’t affect the product.
is equivalent to because adding 5 copies of something is the same as multiplying it by 5.
is equivalent to because dividing by a number is the same as multiplying by its reciprocal.

In the coming lessons, we will see how another property, the distributive property, can show that expressions are equivalent.