Equations and Inequalities

Lesson 5

Objective

Solve word problems leading to equations in the forms $${px+q=r}$$  and $${p(x+q)=r}$$ (Part 1).

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. Write equations in the form $${px+q=r }$$ or $${{p(x+q)=r}}$$ to represent word problems.
  2. Solve equations using different approaches, including arithmetic approach and algebraic approach.
  3. Understand that $${{p(x+q)=r}}$$ can be solved in two different ways.

Tips for Teachers

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  • Lessons 5 and 6 engage students in solving various types of word problems using arithmetic, tape diagrams, and equations. Ensure students get exposure to a variety of types of problems in different classroom settings (collaboratively, independently, as a class, etc.) for them to see how different strategies can be used to solve these problems. In addition, include fluency practice with solving equations without context throughout the problem set. 
  • Students engage in MP.2 in two different ways in this lesson. First by manipulating equations to solve for variables and then re-contextualizing the solutions to make sense in context, and second by reasoning abstractly and using the properties of operations to understand two different ways to solve $$ p(x+q)=r$$.

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Anchor Problems

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Problem 1

At the candy store, M&Ms and Skittles are sold for $0.50 per ounce. Kevita puts some M&Ms in a bag and then added 8 ounces of Skittles. The total cost for her bag of candy is $6.50.

Kevita and Mary write the equation $${0.5(x+8)=6.50}$$ to represent the situation, where x represents the number of ounces of M&Ms. 

  • Kevita says that to solve this equation, you first distribute the 0.5 through the parentheses to get $${0.5x+4=6.50}$$.
  • Mary says that to solve this equation, you first divide by 0.5 on both sides to get $${x+8=13}$$.

Do you agree with either Kevita or Mary? Why? Finish solving the problem to find out how many ounces of M&Ms Kevita put in her bag of candy. 

Guiding Questions

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Problem 2

The taxi fare in Gotham City is $2.40 for the first $${1 \over 2}$$ mile and additional mileage is charged at the rate of $0.20 for each additional 0.1 mile. You plan to give the driver a $2 tip. How many miles can you ride for $10?

What are two different ways you can solve this problem? 

Guiding Questions

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References

Illustrative Mathematics Gotham City Taxis

Gotham City Taxis, accessed on Nov. 6, 2017, 2:24 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Modified by The Match Foundation, Inc.

Problem 3

The ages of three cousins are consecutive integers. The sum of their ages is 45. Calculate their ages.

  1. Draw and use a tape diagram to find the ages of the cousins. 
  2. If the youngest cousin is $$x$$ years old, describe the ages of the other cousins in terms of $$x $$. Write and solve an equation to find the ages of the cousins.

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 3 > Topic B > Lesson 7Example parts a and b

Grade 7 Mathematics > Module 3 > Topic B > Lesson 7 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 Set

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With Fishtank Plus, you can download a complete problem set and answer key for this lesson. Download Sample

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

  • Include additional fluency practice of solving equations (without context).

Target Task

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Mrs. Canale’s class is selling frozen pizzas to earn money for a field trip. For every pizza sold, the class makes $5.35. They have already earned $182.90, but they need $750. How many more pizzas must they sell to earn $750?

References

EngageNY Mathematics Grade 7 Mathematics > Module 3 > Topic B > Lesson 8Exit Ticket

Grade 7 Mathematics > Module 3 > Topic B > Lesson 8 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.

Mastery Response

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