Students are presented with three hanger diagrams and are asked to match an equation to each diagram. They analyze relationships and find correspondences between the two representations (diagrams and equations), and use them to find the unknown value in each diagram. This value is a solution of the equation.

Draw on the diagrams to help students reason about the “weights” of the shapes. In Diagram A, circle groups of shapes on the two sides that show equal weights. In Diagram B, remove shapes on either side that show equal weights. When you are done drawing, your diagrams might look like this:

Two balanced hangers, A and B. A, left side, 3 identical circles, x, right side, 6 identical squares, 1. 3 groups are circled, each containing 1 circle and 2 squares. B, left side, 1 pentagon, y, and 3 identical squares, 1, right side, 6 identical squares, 1. Three squares are crossed off from each side.

Emphasize that, in each case, the result shows the weight of one lettered shape, which represents the value of the variable in the matching equation. In Diagram A, we see that each weighs the same as 2 squares. So, the solution to the matching equation is 2. Verify that 2 is the solution by substituting 2 for in the diagram and equation.

Repeat for Diagram B showing is equal to 3. Consider asking the following questions:

• “Why does the equation represent Diagram A?”
• “How does the grouping we drew on Diagram A show the solution to that equation?”
• “For Diagram B, how is crossing off 3 squares on each side of the diagram like subtracting 3 from each side of its matching equation?”
• “What are some moves that ensure that a balanced hanger diagram stays balanced?”

This activity continues the work of using hanger diagrams to develop strategies for solving equations. Students write equations that represent given diagrams explain how to use the diagrams and equations to reason about a solution. As students complete this activity, they notice correspondences in the structure of equations and diagrams and in the solution strategies (MP7).

Draw Diagram A next to the equation . If needed, show dividing the “8” piece into 8 equal parts and then cross out 3 of them. Under the equation, write , followed by . Ask:

• “Why does the diagram stay balanced when we remove 3 from both sides?” (The weight of both sides are still equal.)
• “Why does the equation stay balanced when we subtract 3 from both sides?” (The value of both sides are still equal.)
• “Is a balanced equation true? How do you know?” (Yes. An equation is balanced when both sides have the same value. This is also what it means for an equation to be true.)

Invite students to demonstrate, side by side, how they reasoned with both the diagram and the equation for Diagrams B, C, and D. For the diagrams represented by a multiplication equation, prompt students to show dividing each side into the same number of equal-size groups.