Lesson 8Reasoning About Solving Equations (Part 2)

Learning Goal

Let’s use hangers to understand two different ways of solving equations with parentheses.

Learning Targets

  • I can explain how a balanced hanger and an equation represent the same situation.

  • I can explain why some balanced hangers can be described by two different equations, one with parentheses and one without.

  • 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: Equivalent to

Problem 1

Select all the expressions equivalent to .

Activity 1: Either Or

Problem 1

  1. Explain why either of these equations could represent this hanger:

    A balanced hanger diagram with 14 on the left and 2 groups of circle x and square 3 on the right

  2. Find the weight of one circle. Be prepared to explain your reasoning.

Activity 2: Use Hangers to Understand Equation Solving, Again

Problem 1

Here are some balanced hangers. Each piece is labeled with its weight.

  1. Assign one of these equations to each hanger:

    Hanger A:

    Balanced hanger diagram, left side, two groups, each group has circle x, square 5, right side, rectangle, 16.

    Hanger B:

    Balanced hanger diagram, eft side, two groups, each group has triangle y, square 200, right side, rectangle, 3000. T

    Hanger C:

    Balanced hanger diagram, left side, rectangle 20 point 8, right side 4 groups, each group has pentagon z, square 1 point 1.

    Hanger D:

    Balanced hanger diagram, left side, rectangle, 20 over 3, left side, two groups, each group has crown w and square 2 over 3.

    1. Hanger A

    2. Hanger B

    3. Hanger C

    4. Hanger D

  2. For each diagram, explain how to figure out the weight of a piece labeled with a letter by reasoning about the diagram.

  3. For each diagram, explain how to figure out the weight of a piece labeled with a letter by reasoning about the equation.

Lesson Summary

The balanced hanger shows 3 equal, unknown weights and 3 2-unit weights on the left and an 18-unit weight on the right.

There are 3 unknown weights plus 6 units of weight on the left. We could represent this balanced hanger with an equation and solve the equation the same way we did before.

A balanced hanger diagram with 3 groups of circle, x and square 2 on the left and 18 on the right.

Since there are 3 groups of on the left, we could represent this hanger with a different equation: .

A balanced hanger diagram with 3 groups of circle, x and square 2 on the left with dotted lines around them and 18 on the right, with equation 3(x + 2) = 18.

The two sides of the hanger balance with these weights: 3 groups of on one side, and 18, or 3 groups of 6, on the other side.

A balanced hanger diagram with 3 groups of circle, x and square 2 on the left with dotted lines around them and 3 blank squares on the right, with equation 3(x + 2) = 18.

The two sides of the hanger will balance with of the weight on each side: .

A balanced hanger with circle, x and square, 2 on the left and 6 on the right with equation x + 2 = 6.

We can remove 2 units of weight from each side, and the hanger will stay balanced. This is the same as subtracting 2 from each side of the equation.

A balanced hanger with circle, x on left and square, 4 on right with a 2 coming off both sides and equation x + 2 = 4 + 2

An equation for the new balanced hanger is . This gives the solution to the original equation.

A balanced hanger with circle, x on left and square, 4 on right and equation x = 4

Here is a concise way to write the steps above: