Lesson 21 | Exponents and Exponential Functions | 9th Grade Mathematics

Objective

Identify features of exponential decay in real-world problems.

Common Core Standards

Core Standards

The core standards covered in this lesson

F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01 ^12t, y = (1.2)^t/10, and classify them as representing exponential growth or decay.

Interpreting Functions

F.IF.C.8.B — Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01 12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

F.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Linear, Quadratic, and Exponential Models

F.LE.A.1.C — Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Foundational Standards

The foundational standards covered in this lesson

8.F.B.4

Functions

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.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Determine if a real-world situation is exponential growth or exponential decay.
Identify the rate of decay in a real-world situation.
Write and evaluate exponential decay functions for applications.

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Malik bought a new car for $15,000. As he drove it off the lot, his best friend, Will, told him that the car’s value just dropped by 15% and that it would continue to depreciate 15% of its current value each year.

If the car’s value is now $12,750 (according to Will), what will its value be after 5 years? Complete the table below to determine the car’s value after each of the next five years. Round each value to the nearest cent.

# of years, t, since driving the car off the lot	Car value after t years	15% depreciation of current value	Car value minus 15% depreciation
0	$12,750.00	$1,912.50	$10,837.50
1
2
3
4
5

Write an exponential function to model the value of Malik’s car $$t$$ years after driving it off the lot.
Use the function from part (b) to determine the value of Malik’s car 5 years after its purchase. Round your answer to the nearest cent. Compare the value with the value in the table. Are they the same?
Use the function from part (b) to determine the value of Malik’s car 7 years after its purchase. Round your answer to the nearest cent.

Guiding Questions

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References

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 7 — Example 1

Algebra I > Module 3 > Topic A > 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 Fishtank Learning, Inc.

Problem 2

Every day Brian takes 20 mg of a drug that helps with his allergies. His doctor tells him that each hour the amount of drug in his bloodstream decreases by 20%.

Construct an exponential function in the form $${f(t)=ab^t}$$, for constants $$a$$ and $$b$$, that gives the quantity of the drug, in milligrams, that remains in Brian’s bloodstream $$t$$ hours after he takes the medication.
How much of the drug remains one day after taking it?
Do you expect the percentage of the dose that leaves the bloodstream in the first half hour to be more than or less than 20%? What percentage is it?
How much of the drug remains one minute after taking it?

Guiding Questions

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References

Illustrative Mathematics Allergy Medication

Allergy Medication, accessed on May 17, 2018, 1:35 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

A set of suggested resources or problem types that teachers can turn into a 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 task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

A huge ping-pong tournament is held in Beijing with 65,536 participants at the start of the tournament. Each round of the tournament eliminates half the participants.

If $${p(r)}$$ represents the number of participants remaining after $$r$$ rounds of play, write a function to model the number of participants remaining.
Use your model to determine how many participants remain after 10 rounds of play.
How many rounds of play will it take to determine the champion ping-pong player?

References

EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 7 — Exit Ticket

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.

Illustrative Mathematics Decaying Dice — (This can also be done using a simulator for the dice if physical dice are not available)
EngageNY Mathematics Algebra I > Module 3 > Topic A > Lesson 7 — Exercises 2-6 and Problem Set

Lesson 20

Lesson 22

Lesson Map

Topic A: Exponent Rules, Expressions, and Radicals

Use exponent rules to analyze and rewrite expressions with non-negative exponents.

8.EE.A.1

Add and subtract polynomial expressions using properties of operations.

A.APR.A.1

Multiply polynomials using properties of exponents and properties of operations.

A.APR.A.1

Solve mathematical applications of exponential expressions.

8.EE.A.1

Use negative exponent rules to analyze and rewrite exponential expressions.

8.EE.A.1 A.SSE.A.2

Define rational exponents and convert between rational exponents and roots.

N.RN.A.1 N.RN.A.2

Write equivalent radical and rational exponent expressions. Identify quantities as rational or irrational.

N.RN.B.3

Simplify radical expressions.

N.RN.A.2

Multiply and divide rational exponent expressions and radical expressions.

N.RN.A.2 N.RN.B.3

Add and subtract rational exponent expressions and radical expressions.

N.RN.A.2 N.RN.B.3

Topic B: Arithmetic and Geometric Sequences

Describe and analyze sequences given their recursive formulas.

F.BF.A.2 F.IF.A.2 F.IF.A.3

Write recursive formulas for sequences, including the Fibonacci sequence.

F.BF.A.2 F.IF.A.2 F.IF.A.3

Define arithmetic and geometric sequences, and identify common ratios and common differences in sequences.

F.BF.A.2 F.LE.A.2

Write explicit rules for arithmetic sequences and translate between explicit and recursive formulas.

F.BF.A.2 F.IF.A.3 F.LE.A.2

Write explicit rules for geometric sequences and translate between explicit and recursive formulas.

F.BF.A.2 F.IF.A.3 F.LE.A.2

Topic C: Exponential Growth and Decay

Compare rates of change in linear and exponential functions shown as equations, graphs, and situations.

A.SSE.A.1 F.IF.C.9 F.LE.A.1 F.LE.A.3

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

A.SSE.A.1 F.LE.A.1 F.LE.A.2 F.LE.B.5

Graph exponential growth functions and write exponential growth functions from graphs.

F.BF.B.3 F.IF.C.7.E

Write exponential growth functions to model financial applications, including compound interest.

F.IF.C.8.B F.LE.A.2 F.LE.B.5

Write, graph, and evaluate exponential decay functions.

F.BF.B.3 F.IF.C.7.E F.IF.C.8.B F.LE.A.1.C

Identify features of exponential decay in real-world problems.

F.IF.C.8.B F.LE.A.1.C

Solve exponential growth and exponential decay application problems.

F.IF.C.8.B F.LE.A.1 F.LE.A.2

Exponents and Exponential Functions

Objective

Common Core Standards

Core Standards

Interpreting Functions

Linear, Quadratic, and Exponential Models

Foundational Standards

Functions

Criteria for Success

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

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