Functions

Lesson 4

Objective

Represent functions with equations.

Common Core Standards

Core Standards

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  • 8.F.A.1 — Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Function notation is not required in Grade 8.

  • 8.F.B.4 — Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

Foundational Standards

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  • 7.RP.A.2.B

  • 7.RP.A.2.C

Criteria for Success

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  1. Write an equation to represent a rule for a function when the initial value is 0.
  2. Write an equation to represent a rule for a function when there is a non-zero initial value. 
  3. Evaluate a function equation for given values.

Tips for Teachers

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  • This lesson builds off of Lesson 3, where students identified rates of change and initial values of functions. In this lesson, students use these properties to construct equations for real-world functions (MP.4). 
  • The Anchor Problems in this lesson build on one another. In Anchor Problem 1, students look at a situation where the initial value is 0 with a single rate of change. In Anchor Problem 2, students use the same rate of change, but must determine how a non-zero initial value changes the equation. In Anchor Problem 3, students find the rate of change and interpret the initial value from a table in order to write the equation. 

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Anchor Problems

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Problem 1

Water flows from a faucet into a bathtub at a constant rate of 7 gallons of water pouring out every 2 minutes. The bathtub is initially empty and the plug is in. 

  1. Determine a rule that describes the volume of water in the tub as a function of time. Write your rule as an equation. 
  2. If the tub can hold 50 gallons of water, how long will it take to fill the tub?

Guiding Questions

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References

EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 3Example 4

Grade 8 Mathematics > Module 5 > Topic A > Lesson 3 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

You are filling the same tub (from Anchor Problem #1) with water flowing in at the same constant rate, but there were initially 8 gallons of water in the tub. 

  1. Determine a new rule that describes the volume of water in the tub as a function of time. Write your rule as an equation. 
  2. The tub can still hold 50 gallons of water. How long will it take now to fill the tub?

Guiding Questions

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References

EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 3Example 4

Grade 8 Mathematics > Module 5 > Topic A > Lesson 3 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

Water flows from a faucet at a constant rate. Assume that 6 gallons of water are already in a tub by the time we notice the faucet is on. This information is recorded in the first column of the table below. The other columns show how many gallons of water are in the tub at different numbers of minutes since we noticed the running faucet. 

Time in minutes (x) 0 3 5 9
Total volume in tub in gallons (y) 6 9.6 12 16.8
  1. What is the rate of change? Explain what this means in context of the situation.
  2. Write an equation that describes the volume of water in the tub as a function of time.
  3. How long was the faucet on before it was noticed? 

Guiding Questions

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References

EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 3Example 5

Grade 8 Mathematics > Module 5 > Topic A > Lesson 3 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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Target Task

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The information in the table shows the number of pages a student can read in a certain book as a function of time in minutes spent reading. Assume a constant rate of reading. 

Time in minutes (x) 2 6 11 20
Total number of pages read in a certain book (y) 7 21 38.5 70
  1. Write the equation that describes the total number of pages read, $$y$$, as a function of the number of minutes, $$x$$, spent reading.
  2. How many pages can be read in 45 minutes?
  3. A certain book has 396 pages. The student has already read $${\frac{3}{8}}$$ of the pages and now picks up the book again at time $$x = 0$$ minutes. Write the equation that describes the total number of pages of the book read as a function of the number of minutes of further reading.
  4. Approximately how much time, in minutes, will it take to finish reading the book?

References

EngageNY Mathematics Grade 8 Mathematics > Module 5 > Topic A > Lesson 3Exit Ticket

Grade 8 Mathematics > Module 5 > Topic A > Lesson 3 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..

Mastery Response

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