Lesson 3
Staying in Balance
Let's use balanced hangers to help us solve equations.
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142336Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=1e92959904eea904f9b211bb4c4a0d7a4c334c8ee69c862eb970e9cd20e24c69)
\(x+x+x = 1+1+1+1+1+1\)
\(x \boldcdot x \boldcdot x = 6\)
\(3x = 6\)
\(x + 3 = 6\)
\(x \boldcdot x \boldcdot x = 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1\)
Problem 2
Write an equation to represent each hanger.
![Four balanced hangers, A, B, C, and D.](https://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142336Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=aaddae52cb2d6ff27f48c05f2ca388d06be3366c89ef5101900d35b2d38cdaba)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142336Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=037b7a598ed58a294c2f99b49fd267330d3ed00333f0857bfaed0fccaa50cb49)
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142336Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=02309daa393bfbb071c99d53dcf1495ec07cad90c74ff756f17602f62946e09d)
Do you agree with Andre? Explain your reasoning.
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://staging-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=AKIAXQCCIHWF37H2AMFB%2F20240722%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240722T142336Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b0196c013b8fc8b124f68bef6ccb6bf6fc634de9c7bcf6dfbf9a7f0da50656ea)
Problem 6
The area of a rectangle is 14 square units. It has side lengths \(x\) and \(y\). Given each value for \(x\), find \(y\).
- \(x=2\frac13\)
- \(x=4\frac15\)
- \(x=\frac76\)
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%