Lesson 5
Reasoning about Equations and Tape Diagrams (Part 2)
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?
Solution
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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%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T133659Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=2fff4c15686c0ed3ad828bf83aad56c4a38b833d034a7e065d8499b8a035146a)
\(\displaystyle 6(x+1)=24\)
Solution
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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%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T133659Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ba356f8c591868954165b7b382900dca3994d789e670f13628bee6f617e1d4ac)
Solution
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(From Unit 2, Lesson 11.)Problem 4
Suppose Quadrilaterals A and B are both squares. Are A and B necessarily scaled copies of one another? Explain.
Solution
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(From Unit 2, Lesson 2.)