Algebra 1 - Unit 1 Functions



ALGEBRA 1 UNIT 3 MAP – Exponential Functions – 4 Weeks | |

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|ESSENTIAL QUESTIONS |VOCABULARY |

|Describe the similarities and differences between exponential and linear |Asymptote |

|functions. |Base |

|What similarities do all tables of exponential functions share? |Exponent |

|How does the value of a in the equation f(x) =a(bx) affect the graph and table |Growth/Decay |

|of an exponential function? |Multiplicative rate of change |

|How do you recognize exponential growth or decay and initial value from a |Power |

|graph, function rule, table, or real-world situation? |Starting/Initial Value |

|How do you simplify expressions using properties of exponents? | |

|COMMON CORE MATHEMATICAL PRACTICES |

|Make sense of problems and persevere in solving them |Use appropriate tools strategically |

|Reason abstractly and quantitatively |Attend to precision |

|Construct viable arguments & critique the reasoning of others |Look for and make use of structure |

|Model with mathematics |Look for and express regularity in repeated reasoning |

|OVERVIEW OF GRAPHIC ORGANIZER |

|Many relationships between variables are non-linear.  Exponential functions are commonly used in a variety of real-world situations.  For example, exponential functions are used to solve problems related to population|

|change, interest on investments, half-lives of drugs, spread of and cleanup of pollution, and radioactive decay.  The two main topics in studying exponential functions are the areas of exponential growth and |

|exponential decay.   |

| Students learn that exponential functions can be written using a function rule as f(x) = abx.  In this rule, a represents the starting value when x=0, and b represents the rate of change.  Students identify that in a|

|recursive relationship for exponential functions, to move from one value to the next, a common multiplier is used.  For example tn+1= tn x b or NEXT = NOW x b, including initial value for t0 or NOW.  Moving between |

|the multiple representations of tables, graphs, real-world contexts and function rules (both explicit and recursive) and observing the connections deepens students' understanding of the rate of change modeled in each |

|case. |

|Students begin studying properties of exponents in 8th grade (starting in 2012/2013). Students should have prior knowledge of the product rule, quotient rule, negative exponent rule, and zero exponent rule using |

|basic expressions. |

|COMMON CORE STATE STANDARDS |UNPACKED STANDARDS AS RELATED TO EXPONETIAL FUNCTIONS |

|Seeing Structure in Expression |Seeing Structure in Expression |

|Interpret the structure of expressions |Interpret the structure of expressions |

|A-SSE.1a. Interpret parts of an expression, such as terms, factors, and coefficients. |A.SSE.1a. Students will be able to identify the initial value and growth/decay factor given an expression. |

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|Write expressions in equivalent forms to solve problems |Write expressions in equivalent forms to solve problems |

|A-SSE.3c. Use the properties of exponents to transform expressions for exponential functions. |A.SSE.3c Students will be able to use properties of exponents to write an equivalent form of an exponential |

| |expression. |

|Creating Equations |Creating Equations |

|Create equations that describe numbers or relationships |Create equations that describe numbers or relationships |

|A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph |A.CED.2 Students will be able to create and graph equations in two variables to represent relationships |

|equations on coordinate axes with labels and scales. |between quantities that describe exponential situations. Students will be able to graph exponential |

| |equations, label the axes appropriately (if given context), and scale the axes to properly represent a |

| |graph. For example, a student may use a scale of 1: 10 to graph the function f(x) = 4x on a standard 10 by |

| |10 graph |

|Represent and solve equations and inequalities graphically |Represent and solve equations and inequalities graphically |

|A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = |A.REI.11 Students will be able to use a graph of an exponential equation to approximate the solution to a |

|g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using |problem. Students will be able to use a graphing calculator to graph an exponential equations and create a |

|technology to graph the functions, make tables of values, or find successive approximations. Include cases |table of values |

|where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic | |

|functions. ★ | |

|Interpreting Functions |Interpreting Functions |

|Understand the concept of a function and use function notation |Understand the concept of a function and use function notation |

|F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) |F.IF.1 Students will be able to identify the domain and range of an exponential function using an equation, |

|assigns to each element of the domain exactly one element of the range. If f is a function and x is an |table, and graph. |

|element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the| |

|graph of the equation y = f(x). | |

|F-IF.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.2 Students will be able to use function notation and evaluate exponential functions for inputs in their|

| |domain. Interpret exponential functions that use function notation in terms of the context in which they are|

| |used. |

|Interpret functions that arise in applications in terms of the context | |

|F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs |Interpret functions that arise in applications in terms of the context |

|and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of |F.IF.4 Given an exponential function in context, students will be able to identify key features in graphs |

|the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, |and tables including: initial value, rate of growth/decay (constant ratio), horizontal asymptotes. Given the|

|positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ |initial value and rate of change of an exponential function, sketch the graph. |

|F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship| |

|it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n | |

|engines in a factory, then the positive integers would be an appropriate domain for the function.★ |F.IF.5 Students will be able to determine the practical domain of an exponential function as it relates to |

| |the context it describes. |

|Analyze functions using different representations | |

|F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases | |

|and using technology for more complicated cases.★ | |

| |Analyze functions using different representations |

| |F.IF.7 Students will be able to graph exponential functions expressed symbolically and show the initial |

|F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain |value and rate of growth/decay (constant ratio). Students should be able to graph by hand and by using a |

|different properties of the function. |graphing calculator. |

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|F-IF.8b. Use the properties of exponents to interpret expressions for exponential functions. |F.IF.8 Write a function in equivalent forms to show different properties of the function. Explain the |

|F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically,|different properties of a function that are revealed by writing a function in equivalent forms. |

|numerically in tables, or by verbal descriptions) |F.IF.8b Use the properties of exponents to interpret expressions for percent rate of change, and classify |

| |them as growth or decay |

| |F.IF.9 Students will be able to compare the initial value and rate of growth/decay (constant ratio) of two |

| |functions represented using a rule, a table, a graph, or context. For example, compare the initial value of |

| |two exponential functions; one of which is represented graphically and the other is represented using a |

| |table. |

|Building Functions |Building Functions |

|Build a function that models a relationship between two quantities |Build a function that models a relationship between two quantities |

|F-BF.1. Write a function that describes a relationship between two quantities.★ |F.BF.1 Students will be able to write an exponential function that describes a relationship between two |

| |quantities |

|F-BF.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context. |F.BF.1a From context, students will be able to write an explicit expression, define a recursive process, and|

| |describe the calculations needed to model an exponential function between two quantities. |

|Linear, Quadratic, and Exponential Models |Linear, Quadratic, and Exponential Models |

|Construct and compare linear, quadratic, and exponential models and solve problems |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 |F.LE.1 Given a contextual situation, students will be able to describe whether the situation has a linear |

|functions. |pattern of change or an exponential pattern of change. |

|F-LE.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential |F.LE.1a Students will be able to show that exponential functions change by equal factors over equal |

|functions grow by equal factors over equal intervals. |intervals. |

| F-LE.1c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit | |

|interval relative to another. |F.LE.1c Students will be able to describe situations where a quantity grows or decays at a constant percent |

|F-LE.2. Construct linear and exponential functions, including arithmetic and geometric sequences, given a |rate per unit interval as compared to another. |

|graph, a description of a relationship, or two input-output pairs (include reading these from a table). |F.LE.2 Students will be able to create exponential functions given the following situations: |

| |- a graph |

| |- a description of a relationship |

|F-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a |- two points, which can be read from a table |

|quantity increasing linearly, quadratically, or (more generally) as a polynomial function. |F.LE.3 Students will be able to make the connection, using graphs and tables that a quantity increasing |

| |exponentially eventually exceeds a quantity increasing linearly. In future units, Students will be able to |

| |make this connection as compared to quadratic functions. |

|Interpret expressions for functions in terms of the situation they model | |

|F-LE.5 Interpret the parameters in a linear or exponential function in terms of a context. |Interpret expressions for functions in terms of the situation they model |

| |F.LE.5 Based on the context of a situation, students will be able to explain the meaning of the coefficients|

|★-Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. |in an exponential function. |

|Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear | |

|throughout the high school standards indicated by a star symbol | |

|SUGGESTED PACING |

|A Pacing Guide will not be created; however, tasks that the group decided to implement will be listed below as well as what sections from the textbook align |

|with this unit. |

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|Prentice Hall Text – |

|7.1, 7.3-7.5: Properties of Exponents |

|7.6: Identifying exponential functions using a table and rule. Evaluating and graphing exponential functions |

|7.7: Exponential functions in context; growth, decay, and compound interest. |

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|Tasks – |

|M&Ms Data Collection (Essential Question 3 Folder) |

|Lets Make a Deal with Your Teacher (Essential Question 1 Folder) |

|Paper Folding (Essential Question 4 Folder) |

|Drug Filtering (Essential Question 4 Folder, answer sheet also available) |

|Exponential Situations (Essential Question 4 Folder) |

|ASSESSMENT RESOURCES |

|Mathematics Assessment Project Tasks: |

|​ Linear and Exponential Models:  A set of three short questions asking students to distinguish between linear and exponential using multiple representations.  |

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|"​Ponzi" Pyramid Schemes:  ​ Do you want to get rich quick?  Your task is to find the fatal catch in this sure-fire money making scheme. |

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|OTHER RESOURCES |

|NCTM Illuminations ([pic] ) |

|Trout Pond: This investigation illustrates the use of iteration, recursion and algebra to model and analyze a changing fish population. Graphs, equations, |

|tables, and technological tools are used to investigate the effect of varying parameters on the long-term population. |

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|National Debt and Wars: Students collect information about the National Debt, plot the data by decade, and determine whether an exponential curve is a good fit|

|for the data. Then student groups will determine and compare common traits and differences in changes in the National debt in three major eras: the Civil War, |

|World War I, and World War II. |

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|​Predicting Your Financial Future: Students often ask, “When are we ever going to use this?” Compound interest is a topic that provides an inherent answer to |

|this question. In this activity, students use their knowledge of exponents to compute an investment’s worth using a formula and a compound interest simulator. |

|Students also use the simulator to analyze credit card payments and debt.   |

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|Computer Lab – Exponential Functions: This activity explores the parameter changes of an exponential function of the form f(x) = abcx. |

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|Which Is More?: Which is more: being given one million dollars, or one penny the first day, double that penny the next day, then double the previous day's |

|pennies and so on for a month? |

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| Desert Bighorn Sheep: Use exponential functions to model the population of Desert Bighorn Sheep in North America. |

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| Exponential Decay: Collect data modeling exponential decay through an experiment measuring the temperature change is a cup of hot water over time. |

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| ​Bounce Time for a Bouncing Ball:  ​This applet simulates dropping a ball from a height at a given velocity.  These are two variables that can be changed |

|through the use of sliders. |

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|Interest Rates: These two activities model exponential growth using compound interest rates. |

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|​Student Explorations in Mathematics:  Are You Interested in Stretching Your Dollars? by Sandy Powers, November 2001.  This activity compares the amount of |

|money a student could earn by investing money in interest bearing accounts versus the amount of money the student could earn by purchasing lottery tickets. |

|Such concepts as simple interest, compound interest, exponents, exponential growth, and functions are introduced.  |

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| ​Graphing Exponential Functions:  This worksheet allows you to move the exponential function around by adjusting the "h" and "k" sliders. The "a" slider will |

|change the direction of the graph and the "b" slider will change the base. |

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|Texas Instruments ​([pic] ) |

| ​Chill Out: How Hot Objects Cool (TI-84+):  Students will use a temperature probe to collect data as the warmed probe cool. Students will also investigate |

|Newton's Law of Cooling and model cooling data with an exponential function. They will fit the data to a mathematical model after analysis. |

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| ​Exploring the Exponential Function (TI-84+):  Students study the exponential function and differentiate between exponential growth or decay from an equation. |

|They identify the coefficient in an equation that represents the rate of growth/decay. Students also explain the effect of changes in the values of A and B.​ |

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| ​Bounce Back​(TI-84+):  In this activity, students will explore the rebound height of a ball and develop a function that will model the rebound heights for a |

|particular bounce. The model can then be used to predict the height of the ball for any bounce. |

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| NUMB3RS Activity: Chains and Pyramids Episode: “Backscatter”:  In "Backscatter," Charlie uses the mathematics of "Backscatter Analysis" to trace an Internet |

|attack back to its source. While few home computer users are likely to become victims of an outright attack by Internet hackers, nearly everyone with an e-mail|

|address is familiar with the problems that can be caused by "message flooding," the mathematically inevitable consequence of designing messages so that they |

|will replicate themselves exponentially. |

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| ​Comparing Linear and Exponential Functions (TI-Nspire):  Compare data from two different scenarios -- linear and exponential growth. |

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| The Science of Racing: Fuel Strategy (Exponential Decay) (TI-Nspire):  NASCAR cars run on gas and a team’s fuel strategy on race day can be a huge factor on |

|winning versus losing. The T80SRC car, on the other hand, is an electric car; so as the fuel drains, its performance worsens. T80SRC teams also have to devise |

|a fuel strategy. For example, when should you do a pit stop to replace the battery or should you simply add two batteries to the car? This lesson will explore |

|the drain on the battery of a car through various means. |

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Drug Filtering and Exponential Situations are similar, choose one

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