Lesson 5
Reasoning about Equations and Tape Diagrams (Part 2)
Let’s use tape diagrams to help answer questions about situations where the equation has parentheses.
Problem 1
Here are two stories:
- A family buys 6 tickets to a show. They also each spend $3 on a snack. They spend $24 on the show.
- Diego has 24 ounces of juice. He pours equal amounts for each of his 3 friends, and then adds 6 more ounces for each.
Here are two equations:
- \(3(x+6)=24\)
- \(6(x+3)=24\)
- Which equation represents which story?
- What does \(x\) represent in each equation?
- Find the solution to each equation. Explain or show your reasoning.
- What does each solution tell you about its situation?
Problem 2
Here is a diagram and its corresponding equation. Find the solution to the equation and explain your reasoning.
![Tape diagram, 6 equal parts labeled x + 1, total 24](https://staging-cms-im.s3.amazonaws.com/d1W7yL8nt8LAtuMo9scB2t91?response-content-disposition=inline%3B%20filename%3D%227-7.6.A5.newPP.01.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.A5.newPP.01.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082953Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5be46615fd989862b04b87bf5a2ad3d4444dbe9a7106c5395d3435b3dcc4d7d2)
\(\displaystyle 6(x+1)=24\)
Problem 3
Find a sequence of rotations, reflections, translations, and dilations showing that one figure is similar to the other. Be specific: give the amount and direction of a translation, a line of reflection, the center and angle of a rotation, and the center and scale factor of a dilation.
![polar coordinate plane with center at A. quadrilateral BCDE and quadrilateral B prime prime, C prime prime, D prime prime, E prime prime graphed.](https://staging-cms-im.s3.amazonaws.com/CqjP4YzkLtXk6ubzVupuzjEK?response-content-disposition=inline%3B%20filename%3D%228-8.2.B.PP.Image.10.png%22%3B%20filename%2A%3DUTF-8%27%278-8.2.B.PP.Image.10.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF37H2AMFB%2F20240703%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240703T082953Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=21ab9f9fc03ee1949fc44f2b6f9c51a6c25a2342a32ee11d8fc8fcb0d2be4e99)
Problem 4
Suppose Quadrilaterals A and B are both squares. Are A and B necessarily scaled copies of one another? Explain.