Exponents and Exponential Functions

Students extend their understanding of properties of exponents to include rational exponents and radicals, and investigate rates of change in linear and exponential sequences and functions.

Download Full Unit

Unit Summary

In Unit 6, Exponents and Exponential Functions, students compare linear and exponential functions in novel ways to reveal new information about and applications of each one. They also extend their understanding of properties of exponents from eighth grade to include rational exponents and radicals.

In Topic A, students recall the properties of exponents and operations that enable complex-looking expressions to be written more simply. Students are briefly introduced to polynomials, and they add, subtract, and multiply polynomials using properties. Students learn how to write rational exponential expressions from radicals, and vice versa, and they hone their skills in simplifying and computing with both forms.

In Topic B, students are introduced to sequences, specifically arithmetic and geometric sequences. They write recursive and explicit formulas for both types of sequences, focusing on precision in their notation and language. By identifying the common difference or ratio and looking at tables and graphs of sequences, students make connections between the rate of growth of a sequence and whether it is linear or exponential.

In Topic C, students make more explicit connections between increasing or decreasing rates of change and exponential functions. They graph exponential functions and transformations of exponential functions, and they write equations for those functions using key features from the graphs. Students look at several real-world applications of exponential growth and decay, in particular the concept of compound interest.

Pacing: 24 instructional days (22 lessons, 1 flex day, 1 assessment day)

Assessment

This assessment accompanies Unit 6 and should be given on the suggested assessment day or after completing the unit.

Unit Prep

Intellectual Prep

Internalization of Standards via the Unit Assessment

Take unit assessment. Annotate for:
- Standards that each question aligns to
- Purpose of each question: spiral, foundational, mastery, developing
- Strategies and representations used in daily lessons
- Relationship to Essential Understandings of unit
- Lesson(s) that assessment points to

Internalization of Trajectory of Unit

Read and annotate “Unit Summary.”
Notice the progression of concepts through the unit using “Unit at a Glance.”
Do all target tasks. Annotate the target tasks for:
- Essential understandings
- Connection to assessment questions

Essential Understandings

Exponential expressions with rational exponents follow the same properties of exponents as integer exponents. In addition, a rational exponent, $${a\over b}$$, defines a radical where $$a$$ is the exponent of the radicand and $$b$$ is the index of the radical. For example, $${10^{5\over6}}$$ can be written as $${\sqrt[6]{10^5}}$$.
An arithmetic sequence consists of terms that increase or decrease linearly by a constant value, called the common difference. Comparatively, a geometric sequence consists of terms that increase or decrease exponentially by a constant factor called the common ratio. Both arithmetic and geometric sequences can be represented by recursive and explicit formulas.
Exponential growth and decay functions can be used to model situations that involve a constant percent rate of growth or decay over time, such as compound interest. An exponentially increasing quantity will always eventually exceed a linearly increasing quantity.

Vocabulary

Properties of exponents	Polynomial, binomial, trinomial	Degree of term or polynomial	Leading coefficient
Exponential expression	Radical	Root	Radicand
Index	Recursive formula	Explicit formula	Sequence
Sequence notation, $${a_n}$$	Fibonacci sequence	Arithmetic sequence	Geometric sequence
Common difference	Common ratio	Exponential growth/decay function	Compound interest

Materials

Technology for graphing
Calculators

Lesson Map

Topic A: Exponent Rules, Expressions, and Radicals

1

8.EE.A.1

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

2

A.APR.A.1

Add and subtract polynomial expressions using properties of operations.

3

A.APR.A.1

Multiply polynomials using properties of exponents and properties of operations.

4

8.EE.A.1

Solve mathematical applications of exponential expressions.

5

8.EE.A.1

A.SSE.A.2

Use negative exponent rules to analyze and rewrite exponential expressions.

6

N.RN.A.1

N.RN.A.2

Define rational exponents and convert between rational exponents and roots.

7

N.RN.A.2

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

8

N.RN.A.2

Simplify radical expressions.

9

N.RN.A.2

N.RN.B.3

Multiply and divide rational exponent expressions and radical expressions.

10

N.RN.A.2

N.RN.B.3

Add and subtract rational exponent expressions and radical expressions.

Topic B: Arithmetic and Geometric Sequences

11

F.BF.A.2

F.IF.A.2

F.IF.A.3

Describe and analyze sequences given their recursive formulas.

12

F.BF.A.2

F.IF.A.2

F.IF.A.3

Write recursive formulas for sequences, including the Fibonacci sequence.

13

F.BF.A.2

F.LE.A.2

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

14

F.BF.A.2

F.IF.A.3

F.LE.A.2

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

15

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.

Topic C: Exponential Growth and Decay

16

A.SSE.A.1

F.IF.C.9

F.LE.A.1

F.LE.A.3

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

17

A.SSE.A.1

F.LE.A.1

F.LE.A.2

F.LE.B.5

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

18

F.BF.B.3

F.IF.C.7.E

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

19

F.IF.C.8.B

F.LE.A.2

F.LE.B.5

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

20

F.IF.C.8.B

F.LE.A.1.C

Write, graph, and evaluate exponential decay functions.

21

F.IF.C.8.B

F.LE.A.1.C

Identify features of exponential decay in real-world problems.

22

F.IF.C.8.B

F.LE.A.1

Solve exponential growth and exponential decay application problems.

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Arithmetic with Polynomials and Rational Expressions

A.APR.A.1 — Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Building Functions

F.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 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.

Building Functions

F.BF.A.2 — Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. 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.
F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Building Functions

F.BF.B.3 — Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3^-5 = 3^-3 = 1/3³ = 1/27.

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3^-5 = 3^-3 = 1/3³ = 1/27.

High School — Number and Quantity

N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.

High School — Number and Quantity

N.RN.A.1 — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^1/3 to be the cube root of 5 because we want (5^1/3)³ = 5(^1/3)³ to hold, so (5^1/3)³ must equal 5.
N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

High School — Number and Quantity

N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.
N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

High School — Number and Quantity

N.RN.B.3 — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Interpreting Functions

F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Interpreting Functions

F.IF.A.2 — Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.

Interpreting Functions

F.IF.A.3 — Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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.

Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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.
F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Interpreting Functions

F.IF.C.7.E — Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Interpreting Functions

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
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.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Interpreting Functions

F.IF.C.9 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Linear, Quadratic, and Exponential Models

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.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.
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).
F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

Linear, Quadratic, and Exponential Models

F.LE.A.3 — Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
F.LE.B.5 — Interpret the parameters in a linear or exponential function in terms of a context.

Linear, Quadratic, and Exponential Models

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

Seeing Structure in Expressions

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.

Seeing Structure in Expressions

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.
A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ — y⁴ as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

Foundational Standards

Expressions and Equations

6.EE.A.1

Expressions and Equations

6.EE.A.1 — Write and evaluate numerical expressions involving whole-number exponents.
7.EE.A.1

Expressions and Equations

7.EE.A.1 — Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
8.EE.A.1

Expressions and Equations

8.EE.A.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3^-5 = 3^-3 = 1/3³ = 1/27.
8.EE.A.2

Expressions and Equations

8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
8.EE.C.7

Expressions and Equations

8.EE.C.7 — Solve linear equations in one variable.

Functions

8.F.A.2

Functions

8.F.A.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3

Functions

8.F.A.3 — Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
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.

Ratios and Proportional Relationships

7.RP.A.3

Ratios and Proportional Relationships

7.RP.A.3 — Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

The Number System

8.NS.A.1

The Number System

8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Future Standards

Building Functions

F.BF.B.4

Building Functions

F.BF.B.4 — Find inverse functions.
F.BF.B.5

Building Functions

F.BF.B.5 — Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Interpreting Functions

F.IF.C.7

Interpreting Functions

F.IF.C.7 — Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 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.
F.IF.C.8

Interpreting Functions

F.IF.C.8 — Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Linear, Quadratic, and Exponential Models

F.LE.A.4

Linear, Quadratic, and Exponential Models

F.LE.A.4 — For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Seeing Structure in Expressions

A.SSE.B.4

Seeing Structure in Expressions

A.SSE.B.4 — Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments. 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.

Standards for Mathematical Practice

CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 — Model with mathematics.
CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 — Attend to precision.
CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.