Unit 6, Lesson 8
Reasoning about Solving Equations (Part 2)
Grade 7
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In this activity, students identify expressions that are equivalent to a given expression, which involves applying the distributive property. In order to understand the two ways of solving an equation of the form
Arrange students in groups of 2. Give 3 minutes of quiet work time and then invite students to share their responses with their partner, followed by a whole-class discussion.
Select all the expressions equivalent to
The purpose of this discussion is to recall that
Possible discussion questions:
In this activity, students are presented with a balanced hanger diagram and are asked to explain why each of two different equations could represent it. They are then asked to find the unknown weight. Note that no particular solution method is prescribed. Give students a chance to come up with a reasonable approach, and then use the Activity Synthesis to draw connections between the diagram and each of the two equations. Monitor for one student who structures the diagram like
When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.
Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.
Explain why either of these equations could represent this hanger:
The purpose of this discussion is to understand viable alternatives for solving for an unknown weight by reasoning about a hanger diagram. Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to why either equation could represent this hanger. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help their partner clarify and strengthen their ideas and writing.
If time allows, display these prompts for feedback:
Close the partner conversations and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer. If time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
After Stronger and Clearer Each Time, ask one student to present who divided by 2 first, and another student to present who subtracted 6 first.
If no one mentions one of these approaches, demonstrate it. Show how the hanger supports either approach. The finished work might look like this for the first equation:
For the second equation, rearrange the right side of the hanger, first, so that 2
In this activity, students match hangers to equations, and then solve for an unknown weight first by reasoning about the diagram and then by reasoning about the equation. Monitor for students who:
Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.
Select work from students with different strategies, such as those described in the Activity Narrative, to share later.
Here are some balanced hanger diagrams. Each piece is labeled with its weight.
First diagram, left side, two groups, each group has circle x, square 5, right side, rectangle, 16.
Second diagram, left side, two groups, each group has triangle y, square 200, right side, rectangle, 3000.
Third diagram, left side, rectangle 20 point 8, right side 4 groups, each group has pentagon z, square 1 point 1.
Fourth diagram, left side, rectangle, 20 over 3, left side, two groups, each group has crown w and square 2 over 3.
For each diagram:
Identify which of the following equations matches the diagram. Then explain how to figure out the weight of a piece labeled with a variable by reasoning about the equation.
The purpose of this discussion is to understand different viable methods for solving an equation of the form
Select one hanger diagram for which one student divided by
The balanced hanger diagram shows the amounts on the left equal the amounts on the right. The left side has 3 pieces that each have unknown weight
Since there are 3 groups of
The two sides of the hanger balance with these weights: 3 groups of
Balanced hanger, left side, circle labeled x, square labeled 2, circle labeled x, square labeled 2, circle labeled x, square labeled 2, right side, rectangle not labeled. A dotted line is drawn around three groups, each group contains one circle and one square from the left side and a third of the rectangle on the right side. To the side, an equation says 1/3 * 3 ( x + 2 ) = 1/3 * 18.
The two sides of the hanger will balance with
We can remove 2 units of weight from each side, and the hanger will stay balanced. This is the same as subtracting 2 from each side of the equation.
Balanced hanger. Left side, circle labeled x and square labeled 2, the square appears to be loose from the hanger. Right side, rectangle labeled 4 and square labeled 2, the square appears to be loose from the hanger. To the side, an equation says x + 2 - 2= 6 - 2.
An equation for the new balanced hanger is
Here is a concise way to write the steps above:
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