Math 1:
Unit 9: Representations of Exponential Functions
Approximate Time Frame: 4-5 weeks
Connections to Previous Learning:
Students learn the rules of exponents and compute exponential expressions arithmetically. Students solve basic exponential equations when the problem
involved perfect squares or cubes and the answer resulted in a whole number. Students began to compare two linear functions and to analyze these and nonlinear graphs. Students do extensive work with linear equations and functions. Students have found patterns and developed structure in expressions, terms,
units and statistical analysis. Students have specifically looked at arithmetic sequences, recognizing these as linear functions with domains in the integers.
Focus within Grade:
Using the understanding of arithmetic sequences, students see how the recursive and explicit formulas are different for geometric sequences. Students
compare what happens when the growth of a sequence is a common ratio rather than a common difference. Students contrast exponential and linear functions
as they explore exponential models using tables, graphs and symbols. In Math 1, exponential functions are limited to a domain of the integers, providing a
connection between geometric sequences and exponential functions in multiple representations.
From the Grade 8, High School Functions Progression Document p. 15:
Linear and Exponential Models
Construct and compare linear and exponential models and solve problems. Distinguishing between situations that can be modeled with linear functions and
with exponential functions turns on understanding their rates of growth and looking for indications of these types of growth rates (MP7). One indicator of these
growth rates is differences over equal intervals, given, for example, in a table of values drawn from the situation – with the understanding that such a table may
only approximate the situation.
To prove that a linear function grows by equal differences over equal intervals, students draw on the understanding developed in Grade 8 that the ratio of the
rise and run for any two distinct points on a line is the same (see Expressions and Equations Progression). An interval can be seen as determining two points on
the line whose x-coordinates occur at the boundaries of the intervals. The equal intervals can be seen as the runs for two pairs of points. Because these runs
have equal length and ratio between rise and run is the same for any pairs of points, the consequence that
the corresponding rises are the same. These rises are the growth of the function over each interval.
Students note the correspondence between rises and run on a graph and differences of inputs and outputs in
a symbolic form of the proof (MP1). A symbolic proof has the advantage that the analogous proof showing
that exponential functions grow by the equal factors over equal intervals begins in an analogues way.
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Math 1:
Unit 9: Representations of Exponential Functions
From the High School, Algebra Progression Document, p. 10
Reasoning with Equations and Inequalities
Equations in one variable A naked equations, such as 𝑥 2 = 4, without any surrounding text, is merely a sentence fragment, neither true nor false, since it
contains a variable x about which nothing is said. A written sequence of steps to solve an equation, such as in the margin, is coded for a narrative line of
reasoning using words like “if”, “then”, “for all”, and “there exists.” In the process of learning to solve equations, students learn certain standard “if-then”
moves, for example “if x = y then x + 2 = y + 2.” The danger in learning algebra is that students emerge with nothing but the moves, which may make it difficult to
detect incorrect or made-up moves later on. Thus the first requirement in the standards in this domain is that students understand that solving equations is a
process of reasoning. This does not necessarily mean that they always write out the full text; part of the advantage of algebraic notation is its compactness. Once
students know what the code stands for, they can start writing in code. Thus, eventually students might make 𝑥 2 = 4 ⇒ 𝑥 = ±2 on step.
Connections to Subsequent Learning:
In the following unit, students expand their knowledge by comparing exponential and linear functions and deciding which most closely represents a situation. In
Math 2, students expand their knowledge of functions to include quadratics and compare these to the linear and exponential functions they learned in Math 1.
In Math 3, this extension continues through polynomial, rational, logarithmic and trigonometric functions.
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Math 1:
Unit 9: Representations of Exponential Functions
Desired Outcomes
Standard(s):
Interpret the structure of expressions.
 A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
a) Interpret parts of an expression, such as terms, factors, and coefficients.
b) Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P (1+r)n as the product of P and a factor
not depending on P.
Create equations that describe numbers or relationships.
 A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and
simple rational and exponential functions.
Represent and solve equations and inequalities graphically.
 A.REI.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted on the coordinate plane, often forming a curve
(which could be a line).
Functions understand the concept of a function and use function notation.
 F.IF.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), f(n+1) = f(n) + f(n-1) for n ≥ 1.
Analyze functions using different representation.
 F.IF.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.
Build a function that models relationship between two quantities.
 F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situation and translate between the two
forms.
Construct and compare linear, quadratic, and exponential models and solve problems.
 F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
a) Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
b) Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c) Recognize situations in which one quantity grows or decays by a constant percent rate per unit interval relative to another.
 F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table.)
Seeing Structure in Expressions.
 A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.
c) Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as
(1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
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Math 1:
Unit 9: Representations of Exponential Functions
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
 explicit vocabulary instruction with regard to the parts of expressions, equations and inequalities as they represent contexts.
 guidance for using visual and tabular representations to support proofs regarding linear and exponential functions and their growth factors.
Understandings: Students will understand …
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Arithmetic and geometric sequences both have a domain of the integers, but arithmetic sequences have equal intervals (common difference) and
geometric have equal factors (constant ratio).
Geometric sequences can be represented by both recursive and explicit formulas.
Exponential functions can be represented by a table, graph, verbal description or equation. Each representation can be transferred to another
representation.
Discrete and continuous functions have properties that appear differently when graphed.
Exponential expressions represent a quantity in terms of its context.
Exponential expressions have equivalent forms that can reveal new information to aid in solving problems.
Essential Questions:
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How can you decide what type of sequence or function is represented?
What are the different ways you can represent an exponential function?
How do you create an appropriate function to model data or situations given within context?
What new information will be revealed if this equation is written in a different but equivalent form?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
*7. Look for and make use of structure. Students write equations using the structure of the terms within context. Students discover the differences in
arithmetic and geometric sequences and use this structure to write equations and create exponential and linear functions. Students use properties of
operations to rewrite exponential functions to reveal new information.
*8. Look for and express regularity in repeated reasoning. Students look at multiple sequences to determine rules to define them as geometric or arithmetic.
Students use the repetition of a sequence to write an explicit or recursive equation.
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Math 1:
Unit 9: Representations of Exponential Functions
Prerequisite Skills/Concepts:
Students should already be able to:
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Advanced Skills/Concepts:
Some students may be ready to:
Knowledge: Students will know…
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Solve one variable equations using the properties of
equivalency.
Model real-world context with one- or two-variable equations.
Define a function and the parts of expressions, equations and
functions.
Represent and interpret functions on a graph and in a table
using scale and labels appropriately.
Write arithmetic sequences and linear functions and represent
them in multiple ways.
Recognize sequences that are neither exponential nor linear and describe their
growth.
Skills: Students will be able to do…
The distinction between discrete and continuous functions and
their representations.
The definition of terms and constant ratio.
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Write recursive and explicit equations for arithmetic and geometric functions.
Create tables or other representations given recursive or explicit equations for
sequences.
Recognize patterns in exponential functions and geometric sequences.
Model situations using explicit and recursive equations. Translate among
representations of exponential functions including tables, graphs, equations and
real-life situations.
Distinguish between linear and exponential functions from multiple
representations.
Distinguish between arithmetic and geometric sequences.
Academic Vocabulary:
Critical Terms:
Supplemental Terms:
Geometric sequence
Recursive formula
Explicit formula
Exponential function
Equal factor
Equal interval
Domain
Continuous
Discrete
Arithmetic sequence
Linear equation
Term
Initial value
Common difference
Constant ratio
2/9/2016 4:30:07 AM
Adapted from UbD framework
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Math 1:
2/9/2016 4:30:07 AM
Unit 9: Representations of Exponential Functions
Adapted from UbD framework
Page 6