Lesson 6

Equality Diagrams

These materials, when encountered before Algebra 1, Unit 2, Lesson 6 support success in that lesson.

6.1: Notice and Wonder: Solving Equations (5 minutes)

Warm-up

The purpose of this warm-up is to elicit what students recall about solving equations and moves which preserve the equality of both sides and the solution to the equation. This will be useful when students justify the steps of solving equations and begin to connect solving equations to solving systems of equations in later activities in their Algebra 1 class. While students may notice and wonder many things about these images, the moves and how they preserve the equality and solutions to the equations are the important discussion points.

Through articulating things they notice and things they wonder about the equations and diagrams, students have an opportunity to attend to precision in the language they use to describe the equation-solving moves they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Launch

Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the equations for all to see. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

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.

 

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the images or equations. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If the moves that were done at each step—and whether they preserve equality and the solutions to the equations—do not come up during the conversation, ask students to discuss this idea.

Encourage students to use their intuition about the hanger diagram, and why each move keeps it balanced and helps you know the weight of the red circle, to justify why each move keeps both sides balanced and keeps the weight of both sides equal.

Remind students of the meaning of the equals sign as indicating that both sides have the same value—the same weight in the case of balances.

6.2: Hanger Diagrams (15 minutes)

Activity

The purpose of this activity is for students to understand how hanger diagrams work in preparation for the activity that follows, where a connection is made to equations. This will be useful when students must determine acceptable moves in creating equivalent equations in associated Algebra 1 lessons. It also helps prepare them to create their own equivalent equations by building their conceptual understanding of what happens at each new step when solving an equation.

Launch

Display the original image of the hanger diagram for all to see. 

Hanger diagram.

Allow students to engage in a class discussion about how to interpret the hangar diagram. Ask students, "What do you notice?", "What do you think would happen if we take one circle away from one side?", "What might happen if you add a square to one side?". The goal is for students to understand the relationship between the square and circle to be that the weight of a square is double the weight of a circle. 

Student Facing

  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.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

The goal of this discussion is to highlight students' understanding of the relationship between the weight of each unique shape. Here are sample questions to promote class discussion:

  • "How did you determine if each hanger diagram was balanced?" (I know that the weight of one square is equal to the weight of two circles, so I applied this relationship to each new hanger diagram. If the diagram did not maintain the original ratio, then it was not balanced. Also, if the diagram had more weight or extra shapes on one side, I know that the same weight or extra shapes should be present on the other side of the diagram as well.)

  • "How can hanger diagrams be used to understand equivalent equations?" (The shapes can symbolize quantities in the same way that variables and numbers represent quantities.)

6.3: Diagrams and Equations (20 minutes)

Activity

The purpose of this activity is for students to recognize that equivalent equations share the same solutions. Students do this by drawing a connection between the hanger diagrams and expressions that use variables (MP2). The work in this activity will be useful when students determine what was done to an original equation to get an equivalent one, because here they build their understanding that each new step in solving an equation creates equations that are equivalent to the original. 

Student Facing

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?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Discuss how students use the hanger diagrams to create equations. Here are sample questions to promote class discussion:

  • "What information is important in creating an equation to represent the hanger diagram?" (Which shape is considered the unknown quantity, the weight of the other shape, how many of each shape there are.)

  • "What is significant about many of the equations having the same solution?" (Equivalent equations have the same solution. Equivalent equations represent hangers that were related to the original by either adding the same weight to each side or doubling to weight on each side.)

  • "What difference did you notice between the equations whose solutions were not the same as all of the other equations and those whose solutions were. (The equations representing balanced hangers that were equivalent to the original all had the same solution. The equations representing the other hangers had a different solution.)