Lesson 3
Staying in Balance
Problem 1
Select all the equations that represent the hanger.
![Balanced hanger. Left side, 3 identical circles labeled, x. Right side, 6 identical squares.](https://cms-im.s3.amazonaws.com/hgFo7pWL1wAtGVdF9qTXy373?response-content-disposition=inline%3B%20filename%3D%226-6.6.A3.PP.Rev.Image.0101.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A3.PP.Rev.Image.0101.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T201956Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b8ecde17de1c0e2cc7ae13554b9c29ef02c1bcdeb2553b3f83c42b8af952ba5c)
\(x+x+x = 1+1+1+1+1+1\)
\(x \boldcdot x \boldcdot x = 6\)
\(3x = 6\)
\(x + 3 = 6\)
\(x \boldcdot x \boldcdot x = 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1\)
Solution
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Problem 2
Write an equation to represent each hanger.
![Four balanced hangers, A, B, C, and D.](https://cms-im.s3.amazonaws.com/4oh2N5hoWqXjKx3HBPPExgYa?response-content-disposition=inline%3B%20filename%3D%226-6.6.A3.PP.Rev.Image.0606.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A3.PP.Rev.Image.0606.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T201956Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6c016207aa6653669f4314782cdead943a2d22cf1c3630d6f6dbd9de631f71e3)
Solution
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Problem 3
- Write an equation to represent the hanger.
- Explain how to reason with the hanger to find the value of \(x\).
- Explain how to reason with the equation to find the value of \(x\).
![Balanced hanger. Left side, 2 identical circles, x, right side, 1 rectangle, 14 point 6 2.](https://cms-im.s3.amazonaws.com/MFZv37dmdfjqRYSejuPwjLXg?response-content-disposition=inline%3B%20filename%3D%226-6.6.A3.PP.Rev.Image.0303.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A3.PP.Rev.Image.0303.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T201956Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b7ac6980f6555eb438a1c523ce20e757e255533f76e821cfa8aa5a89c3e8f443)
Solution
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Problem 4
Andre says that \(x\) is 7 because he can move the two 1s with the \(x\) to the other side.
![Balanced hanger. Left side, 1 circle, x, 2 identical squares, 1, right side, five identical squares, 1.](https://cms-im.s3.amazonaws.com/hNSg67HAhpa1t6s37zkyEohc?response-content-disposition=inline%3B%20filename%3D%226-6.6.A3.PP.Rev.Image.0404.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.A3.PP.Rev.Image.0404.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T201956Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=dbc678144f970c975c8dd95aeba5e100eab56e688d6eac4fec8f1cb0283d2f64)
Do you agree with Andre? Explain your reasoning.
Solution
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Problem 5
Match each equation to one of the diagrams.
- \(12-m=4\)
- \(12=4\boldcdot m\)
- \(m-4=12\)
- \(\frac{m}{4}=12\)
![Four tape diagrams labeled A, B, C, and D.](https://cms-im.s3.amazonaws.com/xWZvhUQNKrVhtZMoBnWoRW76?response-content-disposition=inline%3B%20filename%3D%226-6.6.PP.msand12s.png%22%3B%20filename%2A%3DUTF-8%27%276-6.6.PP.msand12s.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240630%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240630T201956Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=e843025ae45074694a474f79af6c42c35f63d3ac0e300d97e8dcb5c5a7751915)
Solution
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(From Unit 4, Lesson 1.)Problem 6
The area of a rectangle is 14 square units. It has side lengths \(x\) and \(y\). Given each value for \(x\), find \(y\).
- \(x=2\frac13\)
- \(x=4\frac15\)
- \(x=\frac76\)
Solution
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(From Unit 3, Lesson 10.)Problem 7
Lin needs to save up $20 for a new game. How much money does she have if she has saved each percentage of her goal. Explain your reasoning.
- 25%
- 75%
- 125%
Solution
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(From Unit 2, Lesson 20.)