Lesson 6

Equality Diagrams

Let's use hanger diagrams to understand equivalent equations. 

6.1: Notice and Wonder: Solving Equations

What do you notice? What do you wonder?

Hanger diagram. 2 circles and 2 triangles on left, 6 triangles on right.
Hanger diagram. 1 circle and 1 triangle on left, 3 triangles on right.
Hanger diagram. 1 circle on left, 2 triangles on right.

 

6.2: Hanger Diagrams

  1. Hanger diagram.

    The hanger with 1 square and 2 circles is in balance.
    Which of these should also be in balance? Explain your reasoning.

    1. Hanger diagram. 2 squares on left, 4 circles on right.
    2. Hanger diagram. 2 squares on left, 1 square and 2 circles on right.
    3. Hanger diagram. 3 squares on left, 1 square and 5 circles on right.
    4. Hanger diagram. 4 squares on left, 2 squares and 4 circles on right.
    5. Hanger diagram. 4 squares on left, 1 square and 3 circles on right.
    6. Hanger diagram. 2 circles on left, 1 square on right.
  2. This hanger containing 2 pentagons and 6 circles is in balance. Use the hanger diagram to create two additional hangers that would be in balance.

    Hanger diagram. 2 pentagons on left, 6 circles on right.

6.3: Diagrams and Equations

In the previous activity, each square weighs 10 pounds and each circle weighs \(x\) pounds.

Hanger diagram.

So, this diagram could be represented by the equation. \(10 = 2x\)

  1. Use each of the 6 hanger diagrams containing squares and circles from the previous activity to write an equation that represents the weights on the hanger.

    1.  
    2.  
    3.  
    4.  
    5.  
    6.  
  2. Solve each equation.

    1.  
    2.  
    3.  
    4.  
    5.  
    6.  
  3. Compare the solutions to the equations with the answers from the previous activity. What do you notice?

Summary