Lesson 7

Reasoning about Solving Equations (Part 1)

7.1: Hanger Diagrams (10 minutes)

Warm-up

Students encounter and reason about a concrete situation, hangers with equal and unequal weights on each side. They then see diagrams of balanced and unbalanced hangers and think about what must be true and false about the situations. In subsequent activities, students will use the hanger diagrams to develop general strategies for solving equations.

Launch

Display the photo of socks and ask students, “What do you notice? What do you wonder?”

Photo of socks on hangers.

Give students 1 minute to think about the picture. Record their responses for all to see.

Things students may notice:

  • There are two pink socks and two blue socks.
  • The socks are clipped to either ends of two clothes hangers. The hangers are hanging from a rod.
  • The hanger holding the pink socks is level; the hanger holding the blue socks is not level. 

Things students may wonder:

  • Why is the hanger holding the blue socks not level?
  • Is something inside one of the blue socks to make it heavier than the other sock?
  • What does this picture have to do with math?

Use the word “balanced” to describe the hanger on the left and “unbalanced” to describe the hanger on the right. Tell students that the hanger on the left is balanced because the two pink socks have an equal weight, and the hanger on the right is unbalanced because one blue sock is heavier than the other. Tell students that they will look at a diagram that is like the photo of socks, except with more abstract shapes, and they will reason about the weights of the shapes.

Give students 3 minutes of quiet work time followed by a whole-class discussion.

Student Facing

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 
Two hanger diagrams.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share some things that must be true, could be true, and cannot possibly be true about the diagrams. Ask them to explain their reasoning. The purpose of this discussion is to understand how the hanger diagrams work. When the diagram is balanced, there is equal weight on each side. For example, since diagram B is balanced, we know that one triangle weighs the same as three squares. When the diagram is unbalanced, one side is heavier than the other. For example, since diagram A is unbalanced, we know that one triangle is heavier than one square. 

7.2: Hanger and Equation Matching (15 minutes)

Activity

Students are presented with four hanger diagrams and are asked to match an equation to each hanger. They analyze relationships and find correspondences between the two representations. Then students use the diagrams and equations to find the unknown value in each diagram. This value is a solution of the equation.

Launch

Display the diagrams and explain that each square labeled with a 1 weighs 1 unit, and each shape labeled with a letter has an unknown weight. Shapes labeled with the same letter have the same weight.

Arrange students in groups of 2. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, use the same color to highlight the variables in the hanger with the same variables in its corresponding equation.
Supports accessibility for: Visual-spatial processing

Student Facing

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

  1. Match each hanger to an equation. Complete the equation by writing \(x\), \(y\), \(z\), or \(w\) in the empty box.
     
    • \(2 \boxed{\phantom{3}} + 3 = 5\)
    • \(3 \boxed{\phantom{3}} + 2 = 3\)
    • \(6 = 2 \boxed{\phantom{3}} + 3\)
    • \(7 = 3 \boxed{\phantom{3}} + 1\)
       
  2. Find the solution to each equation. Use the hanger to explain what the solution means.
Four balanced hanger diagrams, A, B, C, D.


 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Demonstrate one of the hangers alongside its equation, removing the same number from each side, and then dividing each side by the same thing. Show how these moves correspond to doing the same thing to each side of the equation. (See the student lesson summary for an example of this.)

Representing, Speaking: MLR7 Compare and Connect. After students have discussed what the solutions to the four equations mean, invite students to compare approaches to finding unknown values through different representations (e.g., visual hanger, equation). Help students make connections between the representations by asking questions such as, “Where do you see division in both the hanger diagram and the equation?” This will help students reason about the ways to find unknown values in balanced hangers and to explain the meaning of a solution to an equation.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

7.3: Use Hangers to Understand Equation Solving (15 minutes)

Activity

This activity continues the work of using a balanced hanger to develop strategies for solving equations. Students are presented with balanced hangers and are asked to write equations that represent them. They are then asked to explain how to use the diagrams, and then the equations, to reason about a solution. Students notice the structure of equations and diagrams and find correspondences between them and between solution strategies.

Launch

Draw students’ attention to the diagrams in the task statement. Ensure they notice that the hangers are balanced and that each object is labeled with its weight. Some weights are labeled with numbers but some are unknown, so they are labeled with a variable.

Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Student Facing

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.
Four balanced hanger diagrams, A, B, C, D.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Invite students to demonstrate, side by side, how they reasoned with both the diagram and the equation. For example, diagram A can be shown next to the equation \(7=3x+1\). Cross out a piece representing 1 from each side, and write \(7-1=3x+1-1\), followed by \(6=3x\). Encircle 3 equal groups on each side, and write \(6 \div 3 = 3x \div 3\), followed by \(2=x\). Repeat for as many diagrams as time allows. If diagrams A and B did not present much of a challenge for students, spend most of the time on diagrams C and D. 

We want students to walk away with two things:

  1. An instant recognition of the structure of equations of the form \(px+q=r\) where \(p\), \(q\), and \(r\) are specific, given numbers.
  2. A visual representation in their minds that can be used to support understanding of why for equations of this type, you can subtract \(q\) from each side and then divide each side by \(p\) to find the solution.
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Include the following terms and maintain the display for reference throughout the unit: hanger diagram. For example, display an example of a balanced hanger. With class participation, create step-by-step instructions on how to write and solve an equation based on the hanger.
Supports accessibility for: Memory; Language
Representing, Speaking: MLR8 Discussion Supports. To invite participation in the whole-class discussion, use sentence frames to support students’ explanations for how they found the weights by reasoning about the hangers and equations. For example, provide the frame “First, I ___ because . . .”, “Then I ___ because . . . ." Be sure to verbalize and amplify mathematical language in the students’ explanations (e.g., “subtracting the constant”, and “dividing by the coefficient”). This will help students explain their reasoning with the diagram and the equation.
Design Principle(s): Optimize output (for explanation); Cultivate conversation

Lesson Synthesis

Lesson Synthesis

Display the equation \(4x+6=9.2\). Ask students to work with their partner to draw a corresponding hanger diagram. Then, one partner solves by reasoning about the equation, the other solves by reasoning about the diagram. Ask students to compare the two strategies and discuss how they are alike and how they are different.

7.4: Cool-down - Solve the Equation (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

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 \(7=3x+1\).

Balanced hanger, left side, 7 squares, right side, 3 circles and 1 square.

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. 

Balanced hanger, and  to the side, an equation.

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

Balanced hanger, left side, 6 blue squares, right side, 3 green circles.  To the side, an equation says 6 = 3 x.

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

Balanced hanger.

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.

Balanced hanger, left side 2 squares, right side 1 circle.  To the side, an equation says 2 = x.

Here is a concise way to write the steps above:

\(\begin {align} 7&=3x+1 & \\ 6&=3x & \text{after subtracting 1 from each side} \\ 2 &= x & \text{after multiplying each side by } \tfrac13 \\ \end{align}\)