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://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=20240630T182025Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=64d52fb5cedc6a2c1cb362ee71b9c07f4575238a4035ae20f9fe09a7a8f292f7)
\(x+x+x = 1+1+1+1+1+1\)
\(x \boldcdot x \boldcdot x = 6\)
\(3x = 6\)
\(x + 3 = 6\)
\(x \boldcdot x \boldcdot x = 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1 \boldcdot 1\)
Problem 2
Write an equation to represent each hanger.
![Four balanced hangers, A, B, C, and D.](https://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=20240630T182025Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=38dbe4381615fe4acb5139fc2234509d19c595c9adcd61966722038b21882d38)
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=20240630T182025Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d0bc924224e2fba858398618294aa28b1deff8aebbdfea995753d165a2507788)
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=20240630T182025Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=fb25d931967308d462528ebc804e71c3940b1783e2485c777e455d699cb3ba4e)
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://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=20240630T182025Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=6ca1012b6f2b862124a22ade150e8398b028d79978f1e1b58ea7d0fc144852d8)
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%