ALEX Lesson Plan

Exponential Growth and Decay

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This lesson provided by:
 Author: Priscilla Godwin-Harden System: Elmore County School: Stanhope Elmore High School
General Lesson Information
 Lesson Plan ID: 24092 Title: Exponential Growth and Decay Overview/Annotation: This lesson on exponential growth and decay involves a variety of teaching resources. There are a variety of websites used to teach and reinforce how to identify exponential growth or decay and how to solve problems relating to growth and decay. There is a lab provided that will help model these concepts being taught and computer based practice on these concepts. Videos are provided that give a picture image of how exponential growth and decay works. This lesson plan was created as a result of the Girls Engaged in Math and Science, GEMS Project funded by the Malone Family Foundation.
Associated Standards and Objectives
Content Standard(s):
 MA2015 (9-12) Algebra 7. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] a. Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] b. Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b] Example: Interpret P(1+r)n as the product of P and a factor not depending on P. MA2015 (9-12) Algebra 9. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.* [A-SSE3] a. Factor a quadratic expression to reveal the zeros of the function it defines. [A-SSE3a] b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. [A-SSE3b] c. Determine a quadratic equation when given its graph or roots. (Alabama) d. Use the properties of exponents to transform expressions for exponential functions. [A-SSE3c] 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%. MA2015 (9-12) Algebra 28. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.* [F-IF4] MA2015 (9-12) Algebra 31. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7] a. Graph linear and quadratic functions, and show intercepts, maxima, and minima. [F-IF7a] b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b] MA2015 (9-12) Algebra 32. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. [F-IF8] a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. [F-IF8a] b. Use the properties of exponents to interpret expressions for exponential functions. [F-IF8b] Example: Identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, and y = (1.2)t/10, and classify them as representing exponential growth and decay. MA2015 (9-12) Algebra 34. Write a function that describes a relationship between two quantities.* [F-BF1] a. Determine an explicit expression, a recursive process, or steps for calculation from a context. [F-BF1a] b. Combine standard function types using arithmetic operations. [F-BF1b] Example: Build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. MA2015 (9-12) Algebra 36. 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. [F-BF3] MA2015 (9-12) Algebra 37. Distinguish between situations that can be modeled with linear functions and with exponential functions. [F-LE1] a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. [F-LE1a] b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. [F-LE1b] c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. [F-LE1c] MA2015 (9-12) Algebra 38. 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-LE2] MA2015 (9-12) Algebra 39. 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-LE3] MA2015 (9-12) Algebraic Connections 3. Use formulas or equations of functions to calculate outcomes of exponential growth or decay. (Alabama) Example: Solve problems involving compound interest, bacterial growth, carbon-14 dating, and depreciation. MA2015 (9-12) Algebra II 30. Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7] a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b] Example f(x) = 2x3 or f(x) = (x+1)/(x-1) for x ≠ 1. b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c] c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e] MA2015 (9-12) Algebra II 34. 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. [F-BF3] MA2015 (9-12) Algebra II with Trigonometry 12. Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] a. Interpret parts of an expression such as terms, factors, and coefficients. [A-SSE1a] b. Interpret complicated expressions by viewing one or more of their parts as a single entity. [A-SSE1b] Example: Interpret P(1+r)n as the product of P and a factor not depending on P. MA2015 (9-12) Algebra II with Trigonometry 21. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. [A-CED2] MA2015 (9-12) Precalculus 16. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine odd, even, neither.)* [F-IF4] (Alabama) MA2015 (9-12) Algebra II with Trigonometry 30. Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [F-IF7] a. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. [F-IF7b] b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [F-IF7c] c. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [F-IF7e] MA2015 (9-12) Algebra II with Trigonometry 34. 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. [F-BF3] MA2015 (9-12) Precalculus 25. Compare effects of parameter changes on graphs of transcendental functions. (Alabama) Example: Explain the relationship of the graph y = ex-2 to the graph y = ex.

Local/National Standards:

Primary Learning Objective(s):

The students will use the lesson that is taught to make applications and then solve problems relating to exponential growth and decay.

Students can use the concepts taught here to draw conclusions about the population growth and whether the planet can support such a population growth. Click here

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Preparation Information
 Total Duration: 61 to 90 Minutes Materials and Resources: 1. M&M's for experiment 2. Small cups for experiment Technology Resources Needed: Computer lab for working problems cooperatively Background/Preparation: It would be beneficial for students to have a thorough understanding of graphing on the coordinate plane.
Procedures/Activities:
 1.)Provide an introduction to graphing exponential functions using the attached video from the Teacher Tube website.(Intro to Exponential Growth and Decay)The video introduces step by step how to graph an exponential function. 2.)Provide to the students a copy of the formulas for exponential growth and decay. Attachments are provided with the formulas on them. One attachment shows formulas for how the graphs of exponential functions change based upon the values of a and b. The other attachment shows formulas for when a quantity grows by a fixed percent at regular intervals. Introduce the concept of exponential growth and decay by using the website listed (regentsprep). This website provides students with everyday uses of exponential growth and decay.(Exponential Growth and Decay)This website provides examples of exponential growth and decay to model and explain to students. All examples are great to work through with students. 3.)Show model of exponential decay. (Radioactive Decay)This sight models the number of nuclei remaining at a given time (Exponential decay). 4.)Model of exponential growth of a bacteria colony.(Exponential Growth)This is a real demonstration of a colony of bateria growing. 5.)Students will explore the growth or decline of a trout population in a restocked pond by doing this activity on exponential growth in Thinkfinity. (Trout Population Calculator)Fishing and natural causes contribute to reduced numbers in a trout pond, but wildlife managers restock the pond with fish every year. See how these factors affect the longâ€‘term population in the pond. 6.)Have students work cooperatively in pairs for practice on this website which provides practice on exponential graphs and equations of growth and decay.(Practice for Exponential Growth an Decay)This site has 11 multiple choice questions that can be used for extra practice. 7.)Have students work cooperatively in pairs for practice on this website which provides practice on APPLIED exponential growth and decay. (Applied Exponential Growth and Decay)This site consists of four application type problems. 8.)Students can do a Growth and Decay Lab which is an excellent method of verifying the exponential growth we usually see in population problems and the exponential decay we experience in half-life problems. The directions are found at Click here for lab directions. A student handout can be found at Click here for student handout form. 9.)Practice Problems are included as attachments that can either be used as homework practice or as practice for students to work on in groups for cooperative learning.

 Attachments:**Some files will display in a new window. Others will prompt you to download. Exponential Growth and Decay Practice.rtf More Practice on Exponential Growth and Decay.rtf PRACTICE with exponential growth and decay.rtf
Assessment
 Assessment Strategies Students can go to this website to take a short online quiz to assess understanding of the concepts taught. You will click here for the quiz. Each student will solve the problems here and turn them in for a grade. Each student will need to print a copy of the KWL chart to fill out. This chart helps to assess what each student has learned. This chart can be found by clicking here.
 Acceleration: This lesson can be extended by having students research the population growth of the state in which they live over the past 100 years in order to figure the growth of the state. Intervention: Peer tutors can help students who may need more assistance.

Each area below is a direct link to general teaching strategies/classroom accommodations for students with identified learning and/or behavior problems such as: reading or math performance below grade level; test or classroom assignments/quizzes at a failing level; failure to complete assignments independently; difficulty with short-term memory, abstract concepts, staying on task, or following directions; poor peer interaction or temper tantrums, and other learning or behavior problems.

 Presentation of Material Environment Time Demands Materials Attention Using Groups and Peers Assisting the Reluctant Starter Dealing with Inappropriate Behavior
Be sure to check the student's IEP for specific accommodations.