unit 6 lesson 11 using equations to solve problems

Using Equations to Solve Problems

Lesson Narrative

This lesson brings together the skills and concepts that have been studied in the unit so far. Students solve problems that can be represented by equations of the form \(p(x+q) =r\) and \(px+q = r\) . A bit of scaffolding is offered in the first activity to reactivate their understanding of tape diagrams, but after that no scaffolding is offered so that students can make sense of problems (MP1) and choose representations to use (MP5).

Learning Goals

Teacher Facing

• Interpret and coordinate tape diagrams, equations, and verbal descriptions for situations involving signed numbers.
• Solve an equation of the form $px+q=r$ or $p(x+q)=r$ to determine an unknown quantity in a situation, and present the solution method (orally, in writing, and through other representations).
• Write an equation of the form $px+q=r$ or $p(x+q)=r$ to represent a situation involving signed numbers.

Student Facing

Let’s use tape diagrams, equations, and reasoning to solve problems.

Required Materials

• Sticky notes
• Tools for creating a visual display

Required Preparation

Decide if students will conduct group presentations or a gallery walk for the last activity. If so, prepare tools for creating a visual display and around 3 sticky notes per student. If not, these materials are not necessary.

Learning Targets

• I can solve story problems by drawing and reasoning about a tape diagram or by writing and solving an equation.

CCSS Standards

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Additional Resources

Using Equations to Solve Problems

11.1: Remember Tape Diagrams (5 minutes)

CCSS Standards

Routines and Materials

Instructional Routines

• Think Pair Share

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.

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

• Write a story that could be represented by this tape diagram.
• Write an equation that could be represented by this tape diagram.

Student Response

For access, consult 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)

• Anticipate, Monitor, Select, Sequence, Connect
• MLR7: Compare and Connect

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.

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

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.

• How many invitations should Tyler make each day to finish his goal within a week? Explain or show your reasoning.
• Use Tyler’s diagram to write an equation that represents the situation. Explain how each part of the situation is represented in your equation.
• Show how to solve your equation.

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.

• Explain how the diagram represents the situation.
• Noah writes the equation \(3(x+7) = 57\) to represent the situation. Do you agree with him? Explain your reasoning.
• How many stickers is Noah's sister putting in each prize bag? Explain or show your reasoning.
• 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.

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.

11.3: Running Around (15 minutes)

• Group Presentations

Required Materials

• Sticky notes
• Tools for creating a visual display

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.

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.

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

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.

• How many miles did Priya run last week?
• 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?

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?

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?

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?

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?

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. 

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

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)

Student lesson summary.

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.

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

Curriculum  /  Math  /  6th Grade  /  Unit 6: Equations and Inequalities  /  Lesson 6

Equations and Inequalities

Lesson 6 of 14

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

Solve percent problems using equations.

Common Core Standards

Core standards.

The core standards covered in this lesson

Expressions and Equations

6.EE.B.7 — Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

Ratios and Proportional Relationships

6.RP.A.3.C — Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Foundational Standards

The foundational standards covered in this lesson

6.RP.A.1 — Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."

6.RP.A.2 — Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. Expectations for unit rates in this grade are limited to non-complex fractions. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger."

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Determine, through repeated reasoning, an equation to represent the relationship between percent, whole, and part:   $$percent\times{whole}=part$$   (MP.8).
• Write an equation to represent a percent situation when given a part and a percent.
• Write and solve equations to find the whole, given the part and percent.

Suggestions for teachers to help them teach this lesson

In Unit 2, students solved percent problems by reasoning about diagrams, double number lines, and tables. Now having learned about equations in the form  $${px=q}$$ , students revisit percent problems to see how they can be modeled and solved efficiently using an equation (MP.4).

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

• 30% of 120?

In general, if you’re given a percent and a whole, how can you find the part? Write this as an equation.

Guiding Questions

30% of what number is 12?

Solve this problem first by drawing a diagram. Then write and solve an equation to verify your solution. 

For each situation below, write and solve an equation to answer the question.

a.   There are 6 liters of water in a bucket, which is 20% of the maximum number of liters the bucket can hold. What is the maximum number of liters the bucket can hold?

b.   A softball team won 18 games, which was 60% of the games they played this season. How many games did the softball team play this season?

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Paula is saving money to buy a tablet. So far, she has saved $54, which is 45% of what she needs to buy the tablet. 

Write and solve an equation to find the price of the tablet. 

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Challenge: In a choir, there are 28 female singers which is 40% of the choir. How many female singers would have to be added to the group so exactly 50% of the choir were females?
• EngageNY Mathematics Grade 6 Mathematics > Module 1 — Lessons 27–29 (Revisit these lessons from Unit 2 and have students write equations to solve.)
• Open Up Resources Grade 6 Unit 6 Practice Problems — Lesson 7 #1–3

Topic A: Reasoning About and Solving Equations

Represent equations in the form  $${ x+p=q }$$ and  $${px=q}$$ using tape diagrams and balances.

6.EE.B.6 6.EE.B.7

Define and identify solutions to equations.

Write equations for real-world situations.

Solve one-step equations with addition and subtraction.

Solve one-step equations with multiplication and division.

6.EE.B.7 6.RP.A.3.C

Solve multi-part equations leading to the form  $${x+p=q }$$  and $${px=q}$$ .

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Topic B: Reasoning About and Solving Inequalities

Define and identify solutions to inequalities.

6.EE.B.5 6.EE.B.8

Write and graph inequalities for real-world conditions. (Part 1)

Write and graph inequalities for real-world conditions. (Part 2)

Solve one-step inequalities.

6.EE.B.6 6.EE.B.8

Topic C: Representing and Analyzing Quantitative Relationships

Write equations for and graph ratio situations. Define independent and dependent variables.

6.EE.C.9 6.RP.A.3.A

Represent the relationship between two quantities in graphs, equations, and tables. (Part 1)

Represent the relationship between two quantities in graphs, equations, and tables. (Part 2)

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6th grade (Illustrative Mathematics)

Unit 1: area and surface area, unit 2: introducing ratios, unit 3: unit rates and percentages, unit 4: dividing fractions, unit 5: arithmetic in base ten, unit 6: expressions and equations, unit 7: rational numbers, unit 8: data sets and distribution.