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

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

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.

Hanger A
Hanger B
Hanger C
Hanger D

$2 + 3 = 5$
$3 + 2 = 3$
$6 = 2 + 3$
$7 = 3 + 1$

Problem 2

Complete the equation by writing $x$ , $y$ , $z$ , or $w$ in the empty box.

$2$ $+ 3 = 5$
$3$ $+ 2 = 3$
$6 = 2$ $+ 3$
$7 = 3$ $+ 1$

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

Balanced hanger, left side, rectangle, 7, right side, 3 circles, x, square, 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

Balanced hanger, left side, 2 diamonds, y, square, 10, right side, rectangle, 31.

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

Balanced hanger, left side, rectangle 6.8, right side 2 pentagons, z, square 2.2.

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 4

Balanced hanger, left side, 4 triangles, w, square, 3 over 2, right side, rectangle, 17 over 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.

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 $7 = 3 x + 1$ .

A balanced hanger diagram with 7 squares on left and 3 circles and 1 square on right with the equation 7 = 3x + 1

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.

An equation for the new balanced hanger is $6 = 3 x$ .

A balanced hanger diagram with 6 squares on the left and 3 circles on the right with the equation 6 = 3x

So the hanger will balance with $\frac{1}{3}$ of the weight on each side: $\frac{1}{3} \cdot 6 = \frac{1}{3} \cdot 3 x$ .

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.

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

Here is a concise way to write the steps above:

$\begin{aligned} 7 & = 3 x + 1 \\ 6 & = 3 x & after subtracting 1 from each side \\ 2 & = x & after multiplying each side by \frac{1}{3} \end{aligned}$