For the last CAMI meeting before the 2022 summer break, we explored and created hanger diagrams. The session began with a notice and wonder of this hanger.
Participants noticed that even though the shapes on each side of the hanger were different colors, shapes, and the strands were of different lengths, the hanger seemed to be balanced. We wondered if 2 purple diamonds were equal to 1 green square, and if we could use substitution to swap out a certain number of purple shapes and replace them with green shapes.
Then it was time for new information. We went to SolveMe Mobiles and found out that the purple triangle was equivalent to a value of 2 pounds (or some other weight). We used the annotate tools at the bottom of the screen to find the value of the green squares, as illustrated below.
After that we compared the original hanger diagram with another one and we thought about how they were same and how they were different.
The similarities we noticed:
- They have the same number of squares and diamonds on each side.
- They have the same values so both hangers are balanced.
- They might both be represented with the equation: 5x + 4 = 3x + 12 or 5x + 2 x 2 = 3x + 6 x 2
How we thought they are different:
- One has a middle square (but it’s still balanced).
- One has been expanded
- There may be different equations to represent each hanger.
Next, we went into breakout rooms to solve and talk about the differences and similarities between various types of hanger diagrams. We discovered that some hanger diagrams have a total value that helps to solve the problem. Other hanger diagrams seemed to have more than one solution but we came back to the main room before we had a chance to explore that possibility.
As a whole group we generated some questions we have about hanger diagrams:
- Can you put in multiple values in the same hanger?
- Are there conditions where there are more than one solution but other conditions where there is not more than one solution?
- Is it possible to have a solution to a hanger diagram that is not a whole number?
- Am I still doing algebra if I solve the hangers without doing equations?
To explore the answers to our questions, we broke off into groups and spent some time creating our own hanger diagrams and shared them in the CAMI Hanger Diagram Presentation.
Each group then presented their hanger diagram along with their takeaways. Some of our conclusions:
- Some hanger diagrams have only one solution while others allow for multiple solutions.
- Solving and creating hanger diagrams requires algebraic thinking even if no equations are generated.
- There is more than one way to approach creating a hanger.
- Start with an equation and create a hanger that represents the equation.
- Start with relationships between shapes and then add values after the relationships are established.
- Add and subtract values from each side of the hanger until it is balanced.
- Are there other ways?
- Hanger diagrams are a visualization of the rule “what you do to one side of the equation you have to do to the other side.”
- When creating a hanger diagram it’s important to be mindful of what values to make public and what values to keep hidden. We realized that we need to provide enough information so the hanger can be solved algebraically but not more information than is needed to solve the puzzle.
Our Hanger Diagrams:
Khom, Vera, and Maya
Peg, Edward, and Eric
Raynell, Ruben and Sarah
After presenting our hangers, we all realized there were changes we wanted to make to our hangers and thought about how we might approach creating hangers differently the next time.
We encourage you to solve the hanger diagrams we created and to create hanger diagrams of your own at SolveMe Mobiles. Enjoy!
