Lesson 22 | Exponents and Exponential Functions | 9th Grade Mathematics

Objective

Solve exponential growth and exponential decay application 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 — Distinguish between situations that can be modeled with linear functions and with exponential functions.

Linear, Quadratic, and Exponential Models

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).

Linear, Quadratic, and Exponential Models

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).

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

Identify when a situation has a constant rate of change, an increasing rate of change, or a decreasing rate of change.
Write exponential growth and decay functions for real-world situations.
Interpret exponential functions in context of their situations.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

In regard to pacing, this lesson may be extended over two days, especially if the Desmos modeling activity “iPhone 6s Opening Weekend Sales,” included in the Problem Set Guidance, is used.
This lesson only includes one Anchor Problem in order to allow for adequate time for independent practice or other targeted instruction based on the needs of your classroom.
The Anchor Problem and the Target Task both address exponential growth; ensure that students also encounter problems that target exponential decay.

Fishtank Plus

Unlock features to optimize your prep time, plan engaging lessons, and monitor student progress.

Anchor Problems

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

Leo leaves his house one morning and notices a small plant growing in the yard that he has never noticed before. Out of curiosity, he grabs a ruler and measures it; the plant is 3 cm tall. Exactly one week later, Leo notices the plant has grown quite a bit, so he measures it again; now it is 9 cm tall. And one week after that, it measures 27 cm.

Leo knows that it is unlikely that the plant will continue to triple in height each week indefinitely, but he starts to wonder about the height of the plant before he started to measure it and how he could model its growth mathematically. Suppose that the plant follows a rule, “triples in height each week.”

Read the information contained in the table to understand what Leo has written so far, and then complete the table. Write any heights that are less than 1 cm as fractions.

Week	$${-4}$$	$${-3}$$	$${-2}$$	$${-1 }$$	$$0$$	$$1$$	$$2$$	$$3$$	$$4$$	$$5$$	$$w$$
Height (cm)						$$3$$	$$9$$	$$27$$
Height Expression							$$3^2$$	$$3^3$$

Express the height of the plant, $$h$$, as a function of the week it was measured, $$w$$.
Explain in words the meaning of $$h(0)$$.
Use your function to find the height of the plant on Week {-4}. Write this value as a fraction. Does the result of the function agree with what you wrote in the table?

Guiding Questions

Create a free account or sign in to access the Guiding Questions for this Anchor Problem.

References

Illustrative Mathematics Predicting the Past — Part A

Predicting the Past, accessed on May 16, 2018, 4:51 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 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

Mr. Miller starts working for a technology company this year. His salary the first year is $40,000. According to the company’s employee handbook, each following year Mr. Miller works at the company, he is eligible for a raise equal to 2–5% of his previous year’s salary.

Mr. Miller calculates the range of his raise on his first year’s salary. He adds that amount as his raise for each following year. Mr. Miller thinks that:

in his second year working at the company, he would be earning a salary between $40,800 and $42,000, and
in his third year, he would be earning a salary between $41,600 and $44,000.

Based on this reasoning, what salary range would Mr. Miller expect to earn in his tenth year at the company?
Mr. Miller’s reasoning is incorrect. Show with diagrams, equations, expressions, or words why his reasoning is incorrect.

References

Smarter Balanced Assessment Consortium: Item and Task Specifications MAT.HS.ER.4.00FLE.E.566

MAT.HS.ER.4.00FLE.E.566 from Development and Design: Item and Task Specifications made available by Smarter Balanced Assessment Consortium. © The Regents of the University of California – Smarter Balanced Assessment Consortium. Accessed May 18, 2018, 3:31 p.m..

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 Predicting the Past — Part B
Desmos iPhone 6s Opening Weekend Sales
Illustrative Mathematics Solving Problems with Linear and Exponential Models

Lesson 21

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

Linear, Quadratic, and Exponential Models

Foundational Standards

Functions

Criteria for Success

Tips for Teachers

Anchor Problems

Guiding Questions

References

Problem Set

Target Task

References

Additional Practice

Lesson Map

Request a Demo

Contact Information

School Information

What courses are you interested in?

ELA

Math

Are you interested in onboarding professional learning for your teachers and instructional leaders?

Any other information you would like to provide about your school?

Effective Instruction Made Easy