Exponents and Exponential Functions

Lesson 17

Objective

Write linear and exponential models for real-world and mathematical problems.

Common Core Standards

Core Standards

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  • F.LE.A.1 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

  • F.LE.A.2 — Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

  • F.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.

  • A.SSE.A.1 — Interpret expressions that represent a quantity in terms of its context 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

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  • 8.F.A.3

  • 8.F.B.4

Criteria for Success

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  1. Identify features of linear and exponential functions in contextual situations. 
  2. Write linear and exponential functions for real-world situations. 
  3. Write linear and exponential functions from two input-output pairs.

Anchor Problems

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Problem 1

For each of the scenarios below, decide whether the situation can be modeled by a linear function, an exponential function, or neither. For those with a linear or exponential model, create a function that accurately describes the situation.

  1. From 1910 until 2010, the growth rate of the United States has been steady at about 1.5% per year. The population in 1910 was about 92,000,000.  
  2. The circumference of a circle as a function of the radius
  3. According to an old legend, an Indian King played a game of chess with a traveling sage on a beautiful hand-made chessboard. The sage requested, as reward for winning the game, one grain of rice for the first square, two grains for the second, four grains for the third, and so on for the whole chess board. How many grains of rice would the sage win for the $${n^{th}}$$ square?
  4. The volume of a cube as a function of its side length  

Guiding Questions

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References

Illustrative Mathematics Finding Linear and Exponential Models

Finding Linear and Exponential Models, accessed on May 17, 2018, 12:49 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 population of a country is initially 2 million people and is increasing at 4% per year. The country's annual food supply is initially adequate for 4 million people and is increasing at a constant rate adequate for an additional 0.5 million people per year.

  1. Based on these assumptions, in approximately what year will this country first experience shortages of food?
  2. If the country doubled its initial food supply and maintained a constant rate of increase in the supply adequate for an additional 0.5 million people per year, would shortages still occur? In approximately which year?
  3. If the country doubled the rate at which its food supply increases, in addition to doubling its initial food supply, would shortages still occur?

Guiding Questions

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References

Illustrative Mathematics Population and Food Supply

Population and Food Supply, accessed on May 17, 2018, 12:50 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 3

Two points are shown below.

$${A (1,6)}$$

$${B (0,2)}$$

  1. Write a linear function that contains these two points. 
  2. Write an exponential function that contains these two points.

Guiding Questions

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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.

Target Task

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Chain emails are emails with a message suggesting you will have good luck if you forward the email on to others. Suppose a student started a chain email by sending the message to 3 friends and asking those friends to each send the same email to 3 more friends exactly 1 day after receiving it. 

  1. Write a function to represent the number of people who receive the email on the $${n^{th}}$$ day. Assume everyone who receives the email follows the directions. 
  2. Did you write an exponential function or a linear function? Explain your reasoning behind your decision. 
  3. Which day is the first day that the number of people receiving the email exceeds 100?

References