Equations and Inequalities

Lesson 6

Objective

Solve word problems leading to equations in the forms $${px+q=r}$$ and $${p(x+q)=r }$$ (Part 2).

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. Write an equation to represent an arithmetic sequence.

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.

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Anchor Problems

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Problem 1

Julia, Keller, and Isreal are volunteer firefighters. On Saturday, the volunteer fire department held its annual coin drop fundraiser at a streetlight. After one hour, Keller had collected $42.50 more than Julia, and Isreal had collected $15 less than Keller. The three firefighters collected $125.95 in total. How much did each person collect?

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 3 > Topic B > Lesson 8Example 1

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.

Problem 2

Your parents are redecorating the dining room and want to place two rectangular pictures, each measuring 25 inches across, on the wall, which measures $${10 {2\over3}}$$ feet long. They want to place the pictures so that the distance between the pictures and the distances from each picture to the end of the wall are the same, as shown in the diagram below. 

Determine the distance each picture should be placed from the ends of the wall. Explain the strategy that you used. 

Guiding Questions

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References

EngageNY Mathematics Grade 7 Mathematics > Module 3 > Topic B > Lesson 7Opening Exercise

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 3

 Three toothpick configurations are shown in the chart below. 

 

  1. Extend the pattern to complete the chart.
  2. How many toothpicks will be in the 10th design? 
  3. Write an expression to represent the number of toothpicks in the $$n$$th design. 
  4. Which design will have 175 toothpicks? Write and solve an equation.  

Guiding Questions

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References

SERP Poster Problems Toothpick Patterns

Toothpick Patterns from Poster Problems is made available by SERP under the CC BY-NC-SA 4.0 license. Accessed Nov. 9, 2017, 12:50 p.m..

Modified by The Match Foundation, Inc.

Problem Set

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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).
  • Challenge: Perla worked 4 hours at her job on Saturday and then spent $12.75 on her way home. She worked 5 hours on Sunday and earned 1.5 times her normal hourly rate because it was a holiday. She spent $8.50 on her way home on Sunday. If Perla had $70.75 after her spending, determine her normal hourly rate. 

Target Task

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A museum sells three types of tickets, including those for seniors, adults, and children. On a Tuesday morning, the museum sells 164 tickets. The number of adult tickets sold was three times as many as senior tickets sold, and the number of child tickets sold was 10 more than the number of adult tickets sold. 

How many of each type of ticket did the museum sell on Tuesday morning? Use either a tape diagram or an equation to represent the situation. 

Mastery Response

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