Linear Expressions & Single-Variable Equations/Inequalities

Lesson 7

Math

Unit 3

9th Grade

Lesson 7 of 12

Objective


Define variables; write and solve equations to represent a contextual situation.

Common Core Standards


Core Standards

  • F.BF.A.1 — Write a function that describes a relationship between two quantities Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
  • A.CED.A.1 — Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
  • A.CED.A.2 — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A.CED.A.4 — Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
  • N.Q.A.1 — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
  • F.IF.B.5 — Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.

Foundational Standards

  • 8.F.B.4

Criteria for Success


  1. Define variables and units from a contextual situation. 
  2. Write simple expressions and equations to model contextual situations. 
  3. Use known quantities to try the algebraic model developed and determine reasonability. 
  4. Assign domain restrictions based on context and algebraic relationships. 
  5. Rearrange equation used to model a relationship to highlight a quantity of interest. 
  6. Use nested expressions to describe a larger model. 
  7. Solve contextual problems and interpret solution in the context of a problem.

Tips for Teachers


The criteria for success are largely the same as in lesson 6 with the caveat that the focus of this lesson is on identifying any errors or changes in the model and making revisions so that the model serves the purpose.

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Anchor Problems


Problem 1

Joshua works for the post office and drives a mail truck. At the end of each day, he records his mileage. Since his route never changes, he will drive the same $$r$$ miles every day. He is the only mail carrier that drives his truck, so it is not used on the days he has off. At the end of Joshua’s nth day of work, the mail truck shows a mileage of $$m$$ miles.

  1. Assuming that Joshua’s mail truck was driven by another mail carrier before Joshua started working at the post office, what variable is missing from the context? 
  2. Write an equation that models the mileage of the mail truck, $$T$$, as a function of the number of days that Joshua works. 
  3. Use your equation to determine the variable you defined in part (a), assuming that: 
  • Joshua’s route is 14 miles long. 
  • At the end of Joshua’s 100th day of work, the mail truck shows a mileage of 76,762 miles. 

Guiding Questions

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References

Illustrative Mathematics Delivering the Mail, Assessment Variation

Delivering the Mail, Assessment Variation, accessed on March 15, 2017, 12:59 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.

Problem 2

The taxi fare in Gotham city is determined by a set amount for the first fixed portion of a mile and then charged at a dollar per mile rate. A tip is generally expected at an approximate rate of 20% of the total fare. 

  1. Write an equation that models the cost of a fare as a function of the number of miles you travel. 
  2. Write an equation that models the number of miles you can travel as a function of how much money you have to spend. 
  3. Using the following information about one particular ride in Gotham City, determine whether your model makes sense. 

Guiding Questions

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References

Illustrative Mathematics Gotham City Taxis

Gotham City Taxis, accessed on Sept. 14, 2017, 1:18 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 Fishtank Learning, Inc.

Problem Set


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

Target Task


A student has had a collection of baseball cards for several years. Suppose that $$B$$, the number of cards in the collection, can be described as a function of $$t$$, time in years since the collection was started. 

a.    $$B=200+100t$$

b.    $$B=100+200t$$

c.    $$B=2000-100t$$

d.    $$B=100-200t$$

 

  1. Explain what each of the following equations would tell us about the number of cards in the collection over time.
  2. How would you change one equation to represent a problem where he has 500 baseball cards and gives away 50 per year? 
  3. Why doesn’t (d) make sense? How would you change it so it does make sense? 

References

Illustrative Mathematics Baseball Cards

Baseball Cards, accessed on Dec. 6, 2016, 3:22 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 Fishtank Learning, Inc.

Additional Practice


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.

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Lesson 6

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Lesson 8

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Properties and Solutions of Single-Variable Linear Expressions and Equations

Topic B: Modeling with Single-Variable Linear Equations

Topic C: Properties and Solutions of Single-Variable Linear Inequalities

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