Module Overview: In this module students will explore different situations that can be modeled with exponential functions and equations. Students use tables and graphs to contrast the repeated multiplication of exponential patterns with the repeated addition of linear patterns. This unit also deepens students’ understanding of functions and their notation. Students will investigate key features, domains and ranges of exponential functions; write exponential functions to model relationships between two quantities as in; and compare properties of exponential functions.
Essential Question:
- How can very large and very small numbers be represented?
- How can expressions with exponents be simplified?
- What is an example of a real world situation that is represented by an exponential function?
- What are the characteristics of exponential functions?
Prerequisite Skills and Knowledge: simplifying expressions with exponents using laws of exponents
Tier III Vocabulary:
Scientific Notation, exponential functions, exponential growth, growth factor, compound interest, exponential decay, decay factor
Common Core Standards for Mathematical Practices
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning. / Key
- ★ Modeling Standard
- *indicates a standard that appears in multiple modules
- + indicates a standard included to increase coherence
- PH Prentice Hall Algebra One textbook 2011
- BOLD indicates TN Common Core focus standards
Common Core State Standards for Math Content / Students will be able to / Activities/Resources
N-RN Number-Real Numbers
N-RN.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 (5 1/3)3 = 5(1/3)3 to hold, so (5 1/3)3 must equal 5. /
- Use laws of exponents learned in middle school to simplify basic expressions that contain exponents.
- For school year 2013-14, include instruction that reviews the eighth grade content: Scientific Notation and Operations with Scientific Notation.
A-CED Creating Equations
A. Create equations that describe numbers or relationships.
A-CED.A.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.*
A-CED.A.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.* /
- Create and solve exponential functions from context or data.
- Determine viable and non-viable solutions in the context of the problem.
F-LE Functions-Linear, Quadratic, and Exponential Models
A. Construct and compare linear, quadratic, and
exponential models and solve problems.
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 (including these from a table). / Create exponential functions given geometric sequences. /
F-LE Linear, Quadratic, and Exponential Models
A. Construct and compare linear, quadratic, and exponential models and solve problems.
F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
- Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
- Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
B. Interpret expressions for functions in terms of the situation they model.
F-LE.B.5 Interpret the parameters in a linear or exponential function in terms of context. /
- Show that linear functions change at the same rate over time and that exponential functions change by equal factors over time.
- Describe situations where a quantity grows or decays at a constant percent rate per unit interval as compared to another.
- Use graphs and tables to show that a quantity that is increasing exponentially will eventually exceed the same quantity increasing linearly, quadratically, or by any other polynomial function.
- Based on the context of a situation, explain the meaning of the coefficients, factors, exponents, and/or intercepts of an exponential function.
A-SSE Algebra- Seeing Structure in Expressions
B. Write expressions in equivalent forms to solve problems.
A-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. ★
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12tto reveal the approximate equivalent monthly interest rate if the annual rate is 15%. /
- Use properties of exponents (such as power of a power, product of powers, power of a product, etc.) to write an equivalent form of an exponential function to reveal and explain specific information about its approximate rate of growth or decay.
PF-IF Functions-Interpreting Functions
C. Analyze functions using different representations.
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. ★
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 exponential function and an algebraic expression for another say which one will reach a certain value first. /
- Graph exponential functions that arise from conceptual situations.
- Describe intercepts and end behavior of exponential functions.
- Determine domain and range of exponential functions.
- Compare the key features of two exponential functions each represented a different way.
S-ID Interpreting Categorical and Quantitative Data
B. Summarize, represent, and interpret data on two categorical and quantitative variables.
S-ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
- Fit a function to the data: use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.*
- Create a scatter plot from two quantitative variables.
- Categorize data as exponential or not. Use algebraic methods and technology to fit an exponential function to the data. Use the function to predict values.
- Explain the meaning of the growth rate and y-intercept in context.
- Explain the meaning of the constant and coefficients in context.
Cumberland County Algebra One Curriculum Guide 1