Lesson 8
Reasoning about Solving Equations (Part 2)
Let’s use hangers to understand two different ways of solving equations with parentheses.
Problem 1
Here is a hanger:
- Write an equation to represent the hanger.
- Solve the equation by reasoning about the equation or the hanger. Explain your reasoning.
![Hanger has 5 groups of a circle + rectangle labeled with 2 on the left, rectangle labeled with 11 on the right](https://staging-cms-im.s3.amazonaws.com/hcD9fpjbHF3H5E9JwMupwPZC?response-content-disposition=inline%3B%20filename%3D%227-7.6.B2.newPP.04.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B2.newPP.04.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T133245Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80b8765935d7cd1f8b972f1bf120f1d6d232d5053337f02a073c991b4653de2e)
Problem 2
Explain how each part of the equation \(9=3(x+2)\) is represented in the hanger.
- \(x\)
- 9
- 3
- \(x+2\)
- \(3(x+2)\)
- the equal sign
![Balanced hanger, left side, 9 squares each labeled 1. Right side, three groups, each group contains one circle labled x and 2 squares labeled 1.](https://staging-cms-im.s3.amazonaws.com/K979vo5T8aUYQvDrLDShnCB6?response-content-disposition=inline%3B%20filename%3D%227-7.6.B2.newPP.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B2.newPP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T133245Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=83eecb30da7cf99b0deb7c845dfe32160eed9e4ca6a9e4e2c64ee580392e4344)
Problem 3
Select the word from the following list that best describes each situation.
Problem 4
Clare drew this diagram to match the equation \(2x+16=50\), but she got the wrong solution as a result of using this diagram.
![A tape diagram partitioned into three different sized rectangles, labeled 2, x and 16. The total length of the bar is labeled 50.](https://staging-cms-im.s3.amazonaws.com/56WU7UHC7TvuswRthhztpiBy?response-content-disposition=inline%3B%20filename%3D%227-7.6.B.PP.Image.07.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B.PP.Image.07.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T133245Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5492be1cc5e90b3af99c2370c3cf3a6e7469cd7a276094382f4eb3af0917b4c3)
- What value for \(x\) can be found using the diagram?
- Show how to fix Clare’s diagram to correctly match the equation.
- Use the new diagram to find a correct value for \(x\).
- Explain the mistake Clare made when she drew her diagram.