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 , and another 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.

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:

• “The part I understood best was . . . .”
• “Can you say more about . . . ?”
• “How do you know . . . ? What else do you know is true?”

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 ’s are on the top and 6 units of weight are on the bottom. Then cross off 6 from each side and divide each side by 2. Show this side by side with “doing the same thing to each side” of the equation.

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:

• Rewrite as first.
• Divide both sides by first.

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 first and another student distributed first. Display the two solution methods side by side, along with the hanger diagram. Invite students to briefly describe their solution, then use Compare and Connect to help students compare, contrast, and connect the different solutions. Here are some questions for discussion:

• “What do the solutions have in common? How are they different?”
• “Did anyone solve the problem the same way but would explain it differently?”
• “Why do the different approaches lead to the same solutions?”