Algebra 1 - Unit 1 Functions - Jed's Portfolio
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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