Equations and Inequalities

Lesson 3

Objective

Solve equations in the forms $${px+q=r}$$  and $${p(x+q)=r}$$  using tape diagrams.

Common Core Standards

Core Standards

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  • 7.EE.B.3 — Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

  • 7.EE.B.4.A — Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

Foundational Standards

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  • 6.EE.B.7

Criteria for Success

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  1. Model situations with tape diagrams and equations.
  2. Understand operations needed to solve for a variable using tape diagrams.
  3. Compare the solution strategy for equations in the form $${px+q=r}$$  to those in the form $${p(x+q)=r}$$.

Tips for Teachers

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Anchor Problem #1 is designed like a jigsaw puzzle problem, where groups first work collaboratively on one problem and then mix to form new groups where students share their work from their first group. Each group works on a different scenario of the same common context, using a tape diagram and equation to model the situation and determine a solution (MP.4).

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Anchor Problems

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Problem 1

Divide students into five to seven groups, and give each group the introduction and one of the scenarios below. 
In each group, students should:

  • Represent the scenario with a tape diagram and equation.
  • Collaborate on a sequence of operations to find the solution.
  • If time, model how to find the answer using the equation.

Once all groups are done, mix up the groups so that each group has one student from each scenario. In these new groups, students should:

  • Present their scenario.
  • Share their model and solution.
  • Complete the summary chart and questions.

Introduction:

The Sanchez family just got back from a family vacation. Jon and Ava are summarizing some of the expenses from their family vacation for themselves and their three children, Louie, Missy, and Bonnie:

Car and insurance fees: $400

Airfare and insurance fees: $875

Motel and tax: $400

Baseball game and hats: $103.83

Movies for one day: $75

Soda and pizza: $37.95

Sandals and t-shirts: $120

Scenario 1: 

During one rainy day on the vacation, the entire family decided to go watch a matinee movie in the morning and a drive-in movie in the evening. The price for a matinee movie in the morning is different than the cost of a drive-in movie in the evening. The tickets for the matinee movie cost $6 each. How much did each person spend that day on movie tickets if the ticket cost for each family member was the same? What was the cost for a ticket for the drive-in movie in the evening? 

Scenario 2:
For dinner one night, the family went to the local pizza parlor. The cost of a soda was $3. If each member of the family had a soda and one slice of pizza, how much did one slice of pizza cost? 

Scenario 3:
One night, Jon, Louie, and Bonnie went to see the local baseball team play a game. They each bought a game ticket and a hat that cost $10. How much was each ticket to enter the ballpark?

Scenario 4: 
While Jon, Louie, and Bonnie went to see the baseball game, Ava and Missy went shopping. They bought a t-shirt for each member of the family and bought two pairs of sandals that cost $10 a pair. How much was each t-shirt? 

Scenario 5: 
The family flew in an airplane to the vacation destination. Each person had their own ticket for the plane and also paid $25 in insurance fees per person. What was the cost of one plane ticket? 

Scenario 6:
While on vacation, the family rented a car to get them to all the places they wanted to see for 5 days. The car cost a certain amount each day, plus a one-time insurance fee of $50. How much was the daily cost of the car (not including the insurance fees)?

Scenario 7:
The family decided to stay in a motel for 4 nights. The motel charged a nightly fee plus $60 in state taxes. What was the nightly charge with no taxes included?

Summary Chart & Questions:

Cost of 1 evening movie      
Cost of 1 slice of pizza  
Cost of admission ticket to baseball game  
Cost of 1 t-shirt  
Cost of 1 airplane ticket  
Daily cost for car rental  
Nightly charge for motel  
  1. Determine the cost of 1 airplane ticket, 2 nights at the motel, and 1 evening movie.
  2. Determine the cost of 1 t-shirt, 1 ticket to a baseball game, and 2 days of the rental car.

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic C > Lesson 17Exploratory Challenge

Grade 7 Mathematics > Module 2 > Topic C > Lesson 17 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by The Match Foundation, Inc.

Problem 2

The cost of a babysitting service is $10 for the first hour and $12 for each additional hour. If the total cost of babysitting baby Aaron was $58, how many hours was Aaron with the babysitter?


Draw a tape diagram and write an equation to represent the situation. Then find the solution. 

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 2 > Topic C > Lesson 17Exercise

Grade 7 Mathematics > Module 2 > Topic C > Lesson 17 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Problem Set

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Target Task

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Eric’s father works two part-time jobs, one in the morning and one in the afternoon, and works a total of 40 hours each 5-day work week. If his schedule is the same each day and he works 3 hours each morning, how many hours does Eric’s father work each afternoon? 

Draw a tape diagram and write an equation to represent the situation. Use either model to solve. 

References

Mastery Response

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