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.

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.

Assessment

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

Unit Prep

Essential Understandings

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  • 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

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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

Unit Materials, Representations and Tools

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  • Technology for graphing
  • Calculators

Intellectual Prep

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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

Lesson Map

Topic A: Exponent Rules, Expressions, and Radicals

Topic B: Arithmetic and Geometric Sequences

Topic C: Exponential Growth and Decay

Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

Core Standards

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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.

  • 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.

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 51/3 to be the cube root of 5 because we want (51/3)³ = 5(1/3)³ to hold, so (51/3)³ must equal 5.

  • 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.

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.

  • 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.

  • 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.

  • 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.

  • 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).

  • 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.

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).

Foundational Standards

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Expressions and Equations
  • 6.EE.A.1

  • 7.EE.A.1

  • 8.EE.A.1

  • 8.EE.A.2

  • 8.EE.C.7

Functions
  • 8.F.A.2

  • 8.F.A.3

  • 8.F.B.4

Ratios and Proportional Relationships
  • 7.RP.A.3

The Number System
  • 8.NS.A.1

Future Standards

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Building Functions
  • F.BF.B.4

  • F.BF.B.5

Interpreting Functions
  • F.IF.C.7

  • F.IF.C.8

Linear, Quadratic, and Exponential Models
  • F.LE.A.4

Seeing Structure in Expressions
  • A.SSE.B.4

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.