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Lesson 7Reasoning 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.

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.

7.1 Hanger Diagrams

2 hangers representing the equations

In the two diagrams, all the triangles weigh the same and all the squares weigh the same.

For each diagram, come up with . . .

  1. One thing that must be true 
  2. One thing that could be true
  3. One thing that cannot possibly be true 

7.2 Hanger and Equation Matching

On each balanced hanger, figures with the same letter have the same weight.

4 hangers representing the equations
  • 2 \boxed{\phantom{3}} + 3 = 5
  • 3 \boxed{\phantom{3}} + 2 = 3
  • 6 = 2 \boxed{\phantom{3}} + 3
  • 7 = 3 \boxed{\phantom{3}} + 1
  1. Match each hanger to an equation. Complete the equation by writing x , y , z , or w in the empty box.
  2. Find the solution to each equation. Use the hanger to explain what the solution means.

7.3 Use Hangers to Understand Equation Solving

3 hangers representing the equations

Here are some balanced hangers where each piece is labeled with its weight. For each diagram:

  1. Write an equation.
  2. Explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.
  3. Explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.

Are you ready for more?

When you have the time, visit the site https://solveme.edc.org/Mobiles.html to solve some trickier puzzles that use hanger diagrams like the ones in this lesson. You can even build new ones. (If you want to do this during class, check with your teacher first!)

Lesson 7 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 .

An image of a pair of red socks and a pair of blue socks on hangers

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 hanger representing the equation

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

a hanger representing the equation

So the hanger will balance with \frac13  of the weight on each side: \frac13 \boldcdot 6 = \frac13 \boldcdot 3x .

a hanger representing the equation

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 hanger representing the equation

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}

Lesson 7 Practice Problems

  1. There is a proportional relationship between the volume of a sample of helium in liters and the mass of that sample in grams. If the mass of a sample is 5 grams, its volume is 28 liters. (5, 28) is shown on the graph below.

    A coordinate plane with the origin labeled “O”. The horizontal axis is labeled “mass in grams” and the numbers 0 through 22, in increments of 2, are indicated. The vertical axis is labeled “volume in liters” and the numbers 0 through 180, in increments of 10, are indicated. The point with coordinates 5 comma 28 is indicated.
    1. What is the constant of proportionality in this relationship?
    2. In this situation, what is the meaning of the number you found in part a?
    3. Add at least three more points to the graph above, and label with their coordinates.
    4. Write an equation that shows the relationship between the mass of a sample of helium and its volume. Use m for mass and v for volume.

  2. Explain how the parts of the balanced hanger compare to the parts of the equation.

    7=2x+3

    a hanger representing the equation
  3. Here is a hanger:
    Hanger has 5 circles and rectangle labeled 2 on the left, rectangle labeled 17 on the right
    1. Write an equation to represent the hanger.
    2. Draw more hangers to show each step you would take to find x . Explain your reasoning.
    3. Write an equation to describe each hanger you drew. Describe how each equation matches its hanger.