Curriculum  /  Math  /  9th Grade  /  Unit 5: Functions and Transformations  /  Lesson 4

Functions and Transformations

Lesson 4

Math

Unit 5

9th Grade

Lesson 4 of 16

Objective

Write and graph piecewise functions from contextual situations.

Common Core Standards

Core Standards

  • 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.3 — Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
  • 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. Given a contextual situation, identify the rate(s) of change, domain, range, and other important points necessary to represent the function graphically or algebraically.
  2. Evaluate a contextual situation from the algebraic and graphical representations.
  3. Interpret the situation to determine when the domain is inclusive or exclusive of values.
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Anchor Problems

25-30 minutes

Problem 1

You have a summer job that pays time and a half for working overtime. Overtime is determined as the number of hours over $${{40}}$$ hours you work in one week. Your regular pay is $${ $8 }$$ per hour for the first $${{40}}$$ hours you work in one week.

  1. Write a piecewise function to represent the amount of your weekly paycheck, $${P(h)}$$, for having worked $$h$$ hours in a week.
  2. Graph the function.

Guiding Questions

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Problem 2

There was no snow on the ground when it started falling at midnight at a constant rate of 1.5 inches per hour. At 4:00 a.m., it started falling at a constant rate of 3 inches per hour, and then from 7:00 a.m. to 9:00 a.m., snow was falling at a constant rate of 2 inches per hour. It stopped snowing at 9:00 a.m.

  1. Write a piecewise function that models the depth of snow as a function of time since midnight. 
  2. Create the graph of the function.
  3. When was the depth of the snow on the ground 8 inches? 
  4. How deep was the snow at 9:00 a.m.?

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 3 > Topic D > Lesson 24 — Problem Set, Question #2

Algebra I > Module 3 > Topic D > Lesson 24 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

15-20 minutes

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

Target Task

5-10 minutes

When Janaya started working, she researched how income tax was determined. She read that income up to and including $10,000 is taxed at a rate of 10%. Any income over $10,000 and up to and including $40,000 is taxed at a rate of 15%. For example, if Janaya earned $11,000 in one year, the first $10,000 would be taxed at a rate of 10% and the last $1,000 would be taxed at a rate of 15%. 

  1. How much income tax would Janaya pay if she earned $15,000 in one year?
  2. Write a piecewise function to represent the situation.
  3. How much more income tax would a person who earns $30,000 pay than a person who earns $10,000 in one year?

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.

  • Include problems involving context where students write a piecewise function, graph it, and then answer questions. Some contexts to include are:
    • Admission rates
    • Tax rates
    • Overtime pay
    • Bulk purchasing 
    • Changing rates (such as filling a pool with different rates of water flow from a hose)

Next

Graph, evaluate, and write step functions, with and without context.

Lesson 5
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Lesson Map

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

Topic A: Piecewise Functions

Topic B: Absolute Value Functions

Topic C: Function Transformations

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