Lesson 7
Reasoning about Solving Equations (Part 1)
Let’s see how a balanced hanger is like an equation and how moving its weights is like solving the equation.
7.1: Hanger Diagrams
In the two diagrams, all the triangles weigh the same and all the squares weigh the same.
For each diagram, come up with . . .
- One thing that must be true
- One thing that could be true
- One thing that cannot possibly be true
![Two hanger diagrams.](https://cms-im.s3.amazonaws.com/tXCLd95NM4rLjKBduc8n5VuG?response-content-disposition=inline%3B%20filename%3D%227-7.6.Revision.Image.k8.06.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.Revision.Image.k8.06.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=9844b222051367b0256715d50ad24fb20a57880a94b69a9b5928e97410e07ace)
7.2: Hanger and Equation Matching
On each balanced hanger, figures with the same letter have the same weight.
- Match each hanger to an equation. Complete the equation by writing \(x\), \(y\), \(z\), or \(w\) in the empty box.
- \(2 \boxed{\phantom{3}} + 3 = 5\)
- \(3 \boxed{\phantom{3}} + 2 = 3\)
- \(6 = 2 \boxed{\phantom{3}} + 3\)
-
\(7 = 3 \boxed{\phantom{3}} + 1\)
- Find the solution to each equation. Use the hanger to explain what the solution means.
![Four balanced hanger diagrams, A, B, C, D.](https://cms-im.s3.amazonaws.com/2ufTkCE48RJvkWprEY9Zs7fS?response-content-disposition=inline%3B%20filename%3D%227-7.6.Revision.Image.k8.07.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.Revision.Image.k8.07.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=a1b87f7d3c74e5613242f31174bf2033f258476eff29f100a10d69079b2a9859)
7.3: Use Hangers to Understand Equation Solving
Here are some balanced hangers where each piece is labeled with its weight. For each diagram:
- Write an equation.
- Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
- Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.
![Four balanced hanger diagrams, A, B, C, D.](https://cms-im.s3.amazonaws.com/F6vBcaAdb9f2spfyKohheT1G?response-content-disposition=inline%3B%20filename%3D%227-7.6.Revision.Image.k8.08.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.Revision.Image.k8.08.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=15fb6faa14676af23f534cb5287c2efe67dbeb4b277e88f5eaa697305b47cacc)
Summary
In this lesson, we worked with two ways to show that two amounts are equal: a balanced hanger and an equation. We can use a balanced hanger to think about steps to finding an unknown amount in an associated equation.
The hanger shows a total weight of 7 units on one side that is balanced with 3 equal, unknown weights and a 1-unit weight on the other. An equation that represents the relationship is \(7=3x+1\).
![Balanced hanger, left side, 7 squares, right side, 3 circles and 1 square.](https://cms-im.s3.amazonaws.com/SSQnrz1omkAgj5m44v1aj495?response-content-disposition=inline%3B%20filename%3D%227-7.6.B7.Summary1.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B7.Summary1.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=17e75ec71a7d567d5c40aff1ef6a3fbe531f9c231b303e53b88b19ddd1a0d3dd)
We can remove a weight of 1 unit from each side and the hanger will stay balanced. This is the same as subtracting 1 from each side of the equation.
![Balanced hanger, and to the side, an equation.](https://cms-im.s3.amazonaws.com/ybfMWZ6xxZ5FRipdD8875kyn?response-content-disposition=inline%3B%20filename%3D%227-7.6.B7.Summary2.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B7.Summary2.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=beed1d7d4831da2579e686b605b2e3a9f3df1f6336aa94985665c5443fc3ad32)
An equation for the new balanced hanger is \(6=3x\).
![Balanced hanger, left side, 6 blue squares, right side, 3 green circles. To the side, an equation says 6 = 3 x.](https://cms-im.s3.amazonaws.com/9awskjWu64iq9oAiUQaiygwe?response-content-disposition=inline%3B%20filename%3D%227-7.6.B7.Summaryxyz.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B7.Summaryxyz.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=4f49777762ca25d041277d71d1ab8b4eb068268cfcb0999334a1913e226ad4ec)
So the hanger will balance with \(\frac13\) of the weight on each side: \(\frac13 \boldcdot 6 = \frac13 \boldcdot 3x\).
![Balanced hanger.](https://cms-im.s3.amazonaws.com/KSpfnvy5cKL8MppjGL4KjZ9S?response-content-disposition=inline%3B%20filename%3D%227-7.6.B7.Summary3.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B7.Summary3.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=d367b1afbe8312b9909e778799e6abda79b7d2c0b731ad959f4995402648fcce)
The two sides of the hanger balance with these weights: 6 1-unit weights on one side and 3 weights of unknown size on the other side.
![Balanced hanger, left side 2 squares, right side 1 circle. To the side, an equation says 2 = x.](https://cms-im.s3.amazonaws.com/1Q6o1jr43CjSUUGrJyVAAxo7?response-content-disposition=inline%3B%20filename%3D%227-7.6.B7.Summarypdq.png%22%3B%20filename%2A%3DUTF-8%27%277-7.6.B7.Summarypdq.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=20240630T162111Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=eb3920ec5f309b8e5d9d1735d996d9ceadfda0c29f830361ab9c8e7b07803e85)
Here is a concise way to write the steps above:
\(\begin {align} 7&=3x+1 & \\ 6&=3x & \text{after subtracting 1 from each side} \\ 2 &= x & \text{after multiplying each side by } \tfrac13 \\ \end{align}\)