Lesson 7Reasoning About Solving Equations (Part 1)
Learning Goal
Let’s see how a balanced hanger is like an equation and how moving its weights is like solving the equation.
Learning Targets
I can explain how a balanced hanger and an equation represent the same situation.
I can find an unknown weight on a hanger diagram and solve an equation that represents the diagram.
I can write an equation that describes the weights on a balanced hanger.
Warm Up: Hanger Diagrams
Problem 1
In the two diagrams, all the triangles weigh the same and all the squares weigh the same.
![2 hanger diagrams, The left one has a triangle on the left hanging lower than square on the right. The other is balanced with 1 triangle on left and 3 squares and 1 circle on right](../../../../../../embeds/16cd2ce3--7.6.Revision.Image.k8.06.png)
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
Activity 1: Hanger and Equation Matching
On each balanced hanger, figures with the same letter have the same weight.
Problem 1
Match each hanger to an equation.
![4 balanced hanger diagrams A-D. There are squares with 1's, circles with w's, triangles with z's, pentagons with x's, and crowns with y's.](../../../../../../embeds/cbed0433--7.6.Revision.Image.k8.07.png)
Hanger A
Hanger B
Hanger C
Hanger D
Problem 2
Complete the equation by writing
Problem 3
Find the solution to each equation. Use the hanger to explain what the solution means.
Activity 2: Use Hangers to Understand Equation Solving
Here are some balanced hangers where each piece is labeled with its weight. For each diagram:
Problem 1
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.
Problem 2
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.
Problem 3
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.
Problem 4
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.
Lesson 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
![A balanced hanger diagram with 7 squares on left and 3 circles and 1 square on right with the equation 7 = 3x + 1](../../../../../../embeds/83bc3bfc--7.6.B7.Summary1.png)
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.
![A balanced hanger diagram with 6 squares on left and 3 circles on right with the equation 7 - 1 = 3x + 1 - 1. Both diagrams have a red square coming off.](../../../../../../embeds/fe78ccd5--7.6.B7.Summary2.png)
An equation for the new balanced hanger is
![A balanced hanger diagram with 6 squares on the left and 3 circles on the right with the equation 6 = 3x](../../../../../../embeds/27ec390b--7.6.B7.Summaryxyz.png)
So the hanger will balance with
![A balanced hanger diagram with 6 squares on the left and 3 circles on the right with the equation 6 = 3x. Two squares and one circle are grouped together 3 times.](../../../../../../embeds/0739af7a--7.6.B7.Summary3.png)
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.
![A balanced hanger diagram with 2 squares on the left and 1 circle on the right and the equation 2 = x](../../../../../../embeds/86d2984c--7.6.B7.Summarypdq.png)
Here is a concise way to write the steps above: