Lesson 11

Using Equations to Solve Problems

11.1: Remember Tape Diagrams (5 minutes)

Warm-up

The purpose of this warm-up is to reactivate students’ understanding of tape diagrams to make it more likely that tape diagrams are accessible as a tool for them to choose in this lesson. The diagram was deliberately constructed to encourage some students to write an equation like \(24=3(a+2)\) and others like \(24=3a+6\). Monitor for one student who writes each type of equation.

Launch

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

Student Facing

Tape diagram, 6 parts, a, 2, a, 2, a, 2, total 24.  Across the top, brackets above an a and 2 indicate a + 2.
  1. Write a story that could be represented by this tape diagram.
  2. Write an equation that could be represented by this tape diagram.

Student Response

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

Activity Synthesis

After students have had a chance to share their work with their partner, select a few students to share their stories. Then, select one student to share each type of equation and explain its structure: \(3(a+2)=24\) and \(3a+6=24\).

11.2: At the Fair (15 minutes)

Activity

In this activity, students use a tape diagram to help them reason about a situation, write an equation that represents it, and solve the equation. Students can use both the diagram and the solution strategy of doing the same to each side and undoing that they saw in the past few lessons. The first two questions provide more scaffolding and the last question provides none.

When students work on the last question, monitor for students who

  • reason numerically without any diagrams or representations.
  • create a tape diagram and use it to reason numerically.
  • write an equation like \(6(x-1.5)=46.5\) and solve it by using the distributive property to find the total amount saved, \(6\boldcdot 1.50\).
  • write an equation and solve it by first dividing by 6 to find the cost of each discounted ticket.

Launch

Keep students in the same groups. Give students 5–10 minutes of quiet work time and partner discussions followed by a whole-class discussion. 

Action and Expression: Internalize Executive Functions. Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, present one question at a time and monitor students to ensure they are making progress throughout the activity.
Supports accessibility for: Organization; Attention

Student Facing

  1. Tyler is making invitations to the fair. He has already made some of the invitations, and he wants to finish the rest of them within a week. He is trying to spread out the remaining work, to make the same number of invitations each day. Tyler draws a diagram to represent the situation.
    A tape diagram partitioned into 8 parts labeled x, x, x, x, x, x, x, and 66. A brace is drawn indicating the length of the diagram and is labeled 122.
    1. Explain how each part of the situation is represented in Tyler’s diagram:

      How many total invitations Tyler is trying to make.

      How many invitations he has made already.

      How many days he has to finish the invitations.

    2. How many invitations should Tyler make each day to finish his goal within a week? Explain or show your reasoning.
    3. Use Tyler’s diagram to write an equation that represents the situation. Explain how each part of the situation is represented in your equation.
    4. Show how to solve your equation.
  2. Noah and his sister are making prize bags for a game at the fair. Noah is putting 7 pencil erasers in each bag. His sister is putting in some number of stickers. After filling 3 of the bags, they have used a total of 57 items.

    Tape diagram, 6 parts, x, 7, x, 7, x, 7, total 57.
    1. Explain how the diagram represents the situation.
    2. Noah writes the equation \(3(x+7) = 57\) to represent the situation. Do you agree with him? Explain your reasoning.
    3. How many stickers is Noah's sister putting in each prize bag? Explain or show your reasoning.
  3. A family of 6 is going to the fair. They have a coupon for \$1.50 off each ticket. If they pay \$46.50 for all their tickets, how much does a ticket cost without the coupon? Explain or show your reasoning. If you get stuck, consider drawing a diagram or writing an equation.

Student Response

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

Activity Synthesis

Invite selected students to share their strategies for the last problem, following the sequence of approaches in the Activity Narrative. As students present, display the different approaches side by side, and ask students to explain the meaning of the numbers they find.

Speaking, Representing: MLR7 Compare and Connect. Use this routine during the whole-group discussion as students compare approaches for solving the last problem. Comparisons should focus on different representations of the situation. Listen for phrases like: “I used the tape diagram because….” to highlight students’ justifications for the representations they choose. Then ask students, “What is the same and what is different?” between the approaches. Amplify language that connect quantities between representations (e.g., “How is the ‘family of 6’ represented in your equation? In your diagram?”). This will help students produce language that describes their thinking about the connections between representations and approaches.
Design Principle(s): Maximize meta-awareness; Support sense-making

11.3: Running Around (15 minutes)

Activity

This activity offers four word problems. Depending on time constraints, you may have all students complete all four problems or assign a different problem to each group. The problems increase in difficulty. It is suggested that students create a visual display of one of the problems and do a gallery walk or presentation, but if time is short, you may choose to just have students work in their workbooks or devices.

Launch

Keep students in the same groups. Either instruct students to complete all four problems or assign one problem to each group. If opting to have students do presentations or a gallery walk, distribute tools for making a visual display.

Give students 5–6 minutes quiet work time and a partner discussion followed by a whole-class discussion or gallery walk.

Representation: Internalize Comprehension. Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems, and other text-based content.
Supports accessibility for: Language; Conceptual processing

Student Facing

Priya, Han, and Elena, are members of the running club at school.

  1. Priya was busy studying this week and ran 7 fewer miles than last week. She ran 9 times as far as Elena ran this week. Elena only had time to run 4 miles this week.

    1. How many miles did Priya run last week?
    2. Elena wrote the equation \(\frac19 (x-7) = 4\) to describe the situation. She solved the equation by multiplying each side by 9 and then adding 7 to each side. How does her solution compare to the way you found Priya's miles?
  2. One day last week, 6 teachers joined \(\frac57\) of the members of the running club in an after-school run. Priya counted a total of 31 people running that day. How many members does the running club have?

  3. Priya and Han plan a fundraiser for the running club. They begin with a balance of -80 because of expenses. In the first hour of the fundraiser they collect equal donations from 9 family members, which brings their balance to -44. How much did each parent give?

  4. The running club uses the money they raised to pay for a trip to a canyon. At one point during a run in the canyon, the students are at an elevation of 128 feet. After descending at a rate of 50 feet per minute, they reach an elevation of -472 feet. How long did the descent take?

Student Response

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

Student Facing

Are you ready for more?

A musician performed at three local fairs. At the first he doubled his money and spent \$30. At the second he tripled his money and spent \$54. At the third, he quadrupled his money and spent \$72. In the end he had \$48 left. How much did he have before performing at the fairs?

Student Response

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

Anticipated Misconceptions

The phrases “9 times as far” and “9 times as many” may lead students to think about multiplying by 9 instead of dividing (or multiplying by \(\frac19\)). Encourage students to act out the situations or draw diagrams to help reason about the relationship between the quantities. Remind them to pay careful attention to what or who a comparison refers to. 

Activity Synthesis

If students created a visual display and you opt to conduct a gallery walk, ask students to post their solutions. Distribute sticky notes and ask students to read others’ solutions, using the sticky notes to leave questions or comments. Give students a moment to review any questions or comments left on their display.

Invite any students who chose to draw a diagram to share; have the class agree or disagree with their diagrams and suggest any revisions. Next, invite students who did not try to draw a diagram to share strategies. Ask students about any difficulties they had creating the expressions and equations. Did the phrase “9 times as many” suggest an incorrect expression? If yes, how did they catch and correct for this error?

Lesson Synthesis

Lesson Synthesis

Ask students to reflect on the work done in this unit so far. What strategies have they learned? What kinds of problems can they solve that they weren’t able to, previously? Ask them to write down or share with a partner one new thing they have learned and one thing they still have questions or confusion about.

11.4: Cool-down - The Basketball Game (5 minutes)

Cool-Down

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

Student Lesson Summary

Student Facing

Many problems can be solved by writing and solving an equation. Here is an example:

Clare ran 4 miles on Monday. Then for the next six days, she ran the same distance each day. She ran a total of 22 miles during the week. How many miles did she run on each of the 6 days?

One way to solve the problem is to represent the situation with an equation, \(4+6x = 22\), where \(x\) represents the distance, in miles, she ran on each of the 6 days. Solving the equation gives the solution to this problem.

\(\begin{align} 4+6x &= 22 \\ 6x &= 18 \\ x &= 3 \\ \end{align}\)