The purpose of this task is for students to understand and explain why they can add or subtract expressions from each side of an equation and still maintain the equality, even if the value of those expressions are not known. Both problems have shapes with unknown weight on each side to promote students' thinking about unknown values in this way before the transition to equations.

While the focus of this activity is on the relationship between both sides of the hanger and not equations, some students may start the second problem by writing and solving an equation to find the weight of a square. While students are working, identify those using equations and those not using equations. Ask them to answer the second problem during the whole-class discussion.

The purpose of this discussion is to examine moves that keep the hanger in balance and how those moves can help in solving for the weight of an unknown object.

Invite students to share their response and reasoning for the weight of a square. Ask if any groups used a different strategy to find the answer. If possible, record and display their work for all to see.

If it does not come up, ask students to consider how the hangers could be turned into equations.

To connect the hangers to equations, ask:

• “How could the the weight of a square be represented in an equation?” (I could use a variable, like .)
• “How do you account for there being 2 squares on the left side when finding the weight with the hanger diagram? How does it fit into the equation?” (I knew I had to divide the weight on the right by 2. In the equation, I used .)
• “You found the weight of a square using the last hanger. How do you know that is the same weight of a square in the original hanger?” (Because the hanger is in balance from each move, the weight of a square for any one of them should be the weight of the square in all of them.)
• “We didn’t know the weight of a square. How do we know the hanger will stay balanced when we remove 1 square from each side?” (Because all of the squares have the same weight, removing an equal number from each side should keep the hanger balanced.)
• “What other moves could we do to the hanger to keep it in balance?” (Add the same number of a shape to each side, double or halve the weight on each side by adding more shapes or cutting them in half.)
   

The goal of this activity is for students to transition their reasoning about solving hanger problems by maintaining the equality of each side, to solving equations using the same logic.

Students solve 2 more hanger problems and write equations to represent each hanger. In the first problem, the solution is not an integer, which will challenge any student who has been using guess-and-check in the previous activities and will encourage that student to look for a more efficient method.

The purpose of the discussion is to make connections between moves done to hangers that keep them balanced and moves done to equations to write equivalent equations. Prepare for all to see a display with 2 columns with hanger diagrams on the one side and equations on the other. Start the discussion with the original hanger diagram. 

Ask students “What is an equation you wrote for the hanger?” ( ). Record the student response next to the original hanger in the right column .

Next, invite a student to suggest a move they could do to the circles that would keep the hanger balanced (remove 1 circle from each side). Add a new hanger diagram to the hanger column showing the suggested change. Then ask, “What is an equation we can write for this new hanger diagram?” ( ) and record the new equation in the equation column.

Referring back to earlier work using arrows with rows of equivalent equations, in each column draw arrows on each side of the first hanger and equation to the second hanger and equation. Ask, “What should we write next to the arrows to describe what happened to the equations?” (remove 1 circle and subtract 6).

If time allows, add more rows to each column, each time inviting students to suggest moves that keep the hanger in balance and the equations equivalent.

Here are some questions for discussion:

• “Could we subtract from each side of the equation to create an equivalent equation? What does that look like on the hanger diagram?” (Yes, it is the same as removing 1 square from each side.)
• "How is dealing with removing the triangle different when using the hanger diagrams from using the equation?” (In the equation, we can subtract 3 from each side. For the original hanger diagram, we could remove a triangle from the left side but there is no like item on the right side, so we could substitute two triangles for the circle and then remove one of the triangles.