ࡱ>  "'`  bjbjLULU .?.?]?D,,,4`(((h(|l*`~jT+ ^.^.^.^.///v|x|x|x|x|x|x|$h|5,71/"/7171|^.^.%}333718^.,^.v|371v|33l,p^.H+ p{#NK(1n8z}0~n2ppp,~x,/L/63'0,S0///||;3^///~71717171```D$```$``` ALGEBRA 1 UNIT 3 MAP Exponential Functions 4 Weeks  ESSENTIAL QUESTIONSVOCABULARYDescribethe similarities and differences between exponential and linear functions. What similarities do all tables of exponential functions share? How doesthe value of a in theequationf(x) =a(bx) affect the graph and table of an exponential function? How do you recognizeexponential growth or decay and initial value from a graph, function rule, table,or real-world situation? How do you simplify expressions using properties of exponents? Asymptote Base Exponent Growth/Decay Multiplicative rate of change Power Starting/Initial Value COMMON CORE MATHEMATICAL PRACTICESMake sense of problems and persevere in solving them Reason abstractly and quantitatively Construct viable arguments & critique the reasoning of others Model with mathematicsUse appropriate tools strategically Attend to precision Look for and make use of structure Look for and express regularity in repeated reasoning OVERVIEW OF GRAPHIC ORGANIZERMany 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 asf(x) = abx. In this rule,arepresents the starting value whenx=0, andbrepresents 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 exampletn+1=tnxborNEXT=NOWxb,including initial value fort0or 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 STANDARDSUNPACKED STANDARDS AS RELATED TO EXPONETIAL FUNCTIONSSeeing Structure in Expression Interpret the structure of expressions A-SSE.1a. Interpret parts of an expression, such as terms, factors, and coefficients. Write expressions in equivalent forms to solve problems A-SSE.3c. Use the properties of exponents to transform expressions for exponential functions. Seeing Structure in Expression Interpret the structure of expressions A.SSE.1a. Students will be able to identify the initial value and growth/decay factor given an expression. Write expressions in equivalent forms to solve problems A.SSE.3c Students will be able to use properties of exponents to write an equivalent form of an exponential expression. Creating Equations Create equations that describe numbers or relationships A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Creating Equations Create equations that describe numbers or relationships A.CED.2 Students will be able to create and graph equations in two variables to represent relationships 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 A-REI.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. &Represent and solve equations and inequalities graphically A.REI.11 Students will be able to use a graph of an exponential equation to approximate the solution to a problem. Students will be able to use a graphing calculator to graph an exponential equations and create a table of values Interpreting Functions 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) assigns to each element of the domain exactly one element of the range. If f is a function and x is an 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. 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 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-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.& 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.& F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F-IF.8b. Use the properties of exponents to interpret expressions for exponential functions. F-IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions) Interpreting Functions Understand the concept of a function and use function notation F.IF.1 Students will be able to identify the domain and range of an exponential function using an equation, table, and graph. 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 Given an exponential function in context, students will be able to identify key features in graphs and tables including: initial value, rate of growth/decay (constant ratio), horizontal asymptotes. Given the initial value and rate of change of an exponential function, sketch the graph. 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 Students will be able to graph exponential functions expressed symbolically and show the initial value and rate of growth/decay (constant ratio). Students should be able to graph by hand and by using a graphing calculator. F.IF.8 Write a function in equivalent forms to show different properties of the function. Explain the different properties of a function that are revealed by writing a function in equivalent forms. 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 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.1a. Determine an explicit expression, a recursive process, or steps for calculation from a context. Building Functions Build a function that models 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 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 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. F-LE.1a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal 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.2. 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-LE.3. Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 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. &-Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbolLinear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems F.LE.1 Given a contextual situation, students will be able to describe whether the situation has a linear pattern of change or an exponential pattern of change. F.LE.1a Students will be able to show that exponential functions change by equal factors over equal intervals. F.LE.1c Students will be able to describe situations where a quantity grows or decays at a constant percent rate per unit interval as compared to another. F.LE.2 Students will be able to create exponential functions given the following situations: - a graph - a description of a relationship - two points, which can be read from a table 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 Based on the context of a situation, students will be able to explain the meaning of the coefficients in an exponential function. SUGGESTED PACINGA 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. 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. 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 RESOURCESMathematics Assessment Project Tasks: Linear and Exponential Models:A set of three short questions asking students to distinguish between linear and exponential using multiple representations.  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://map.mathshell.org/materials/tasks.php?taskid=296&subpage=novice " 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.  HYPERLINK "http://map.mathshell.org/materials/tasks.php?taskid=278&subpage=expert" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://map.mathshell.org/materials/tasks.php?taskid=278&subpage=expert OTHER RESOURCESNCTM Illuminations( HYPERLINK "http://illuminations.nctm.org/)" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://illuminations.nctm.org/) 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.  HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?ID=U142" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://illuminations.nctm.org/LessonDetail.aspx?ID=U142 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 CivilWar, WorldWarI, and WorldWarII.  HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?id=L670" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://illuminations.nctm.org/LessonDetail.aspx?id=L670 PredictingYour 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 investments worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit card payments and debt.  HYPERLINK "http://illuminations.nctm.org/LessonDetail.aspx?id=L761" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://illuminations.nctm.org/LessonDetail.aspx?id=L761 Computer Lab Exponential Functions: This activity explores the parameter changes of an exponential function of the form f(x) = abcx.  HYPERLINK "http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/ExponentialFunctions_ComputerLab.pdf" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/ExponentialFunctions_ComputerLab.pdf 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?  HYPERLINK "http://mathforum.org/dr.math/faq/faq.doubling.pennies.html" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://mathforum.org/dr.math/faq/faq.doubling.pennies.html Desert Bighorn Sheep:Use exponential functions tomodel the population of Desert Bighorn Sheep in North America.  HYPERLINK "http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/BigHornSheep.pdf" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/BigHornSheep.pdf Exponential Decay:Collect data modeling exponential decay through an experiment measuring the temperature change is a cup of hot water over time.  HYPERLINK "http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/AlgII_6-2-3_ExpDecay_Student.pdf" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/AlgII_6-2-3_ExpDecay_Student.pdf 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.  HYPERLINK "http://demonstrations.wolfram.com/BounceTimeForABouncingBall/" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://demonstrations.wolfram.com/BounceTimeForABouncingBall/ Interest Rates:These two activities model exponential growth using compound interest rates.  HYPERLINK "http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/SavingMoneyMakingMoney.pdf" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/SavingMoneyMakingMoney.pdf  HYPERLINK "http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/GraduationPresent.pdf" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.math.tamu.edu/~jwhitfld/TQG_AlgII/2006-2007/Fall06/GraduationPresent.pdf StudentExplorations 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.  HYPERLINK "http://www.nctm.org/publications/article.aspx?id=22317" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.nctm.org/publications/article.aspx?id=22317 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.  HYPERLINK "http://www.geogebra.org/en/upload/files/Line/Exponential.html" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://www.geogebra.org/en/upload/files/Line/Exponential.html Texas Instruments ( HYPERLINK "http://education.ti.com/calculators/timath/)" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/timath/) 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. 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  sikdn$$Ifl|)0644 layt` d;$Ifgd6 h$If^hgd6 $Ifgd6 ZZ ZyZzZ{Z|ZZZZg[h[i[j[[[\\DZ~eDZHDZe.3h6h65B*CJOJPJQJ\^JaJph9jAh6h6B*CJOJPJQJU\^JaJph0h6h6B*CJOJPJQJ\^JaJph*h6B*CJOJPJQJ\^JaJph9jh6h6B*CJOJPJQJU\^JaJph+h6h60JCJOJPJQJ\^JaJ4jh6h60JCJOJPJQJU\^JaJ9j[=h6h6B*CJOJPJQJU\^JaJph\^^^^^^z_{_|_}___`D`aaaAbBbCbbbѴ紙fLѴ紙3h6h65B*CJOJPJQJ\^JaJph9jKFh6h6B*CJOJPJQJU\^JaJph+h6h60JCJOJPJQJ\^JaJ4jh6h60JCJOJPJQJU\^JaJ9jh6h6B*CJOJPJQJU\^JaJph*h6B*CJOJPJQJ\^JaJph0h6h6B*CJOJPJQJ\^JaJphbbbcccddddddeeeeeeef gDZ{a{{DZDDZ{a{9j;Oh6h6B*CJOJPJQJU\^JaJph3h6h65B*CJOJPJQJ\^JaJph0h6h6B*CJOJPJQJ\^JaJph9jh6h6B*CJOJPJQJU\^JaJph+h6h60JCJOJPJQJ\^JaJ4jh6h60JCJOJPJQJU\^JaJ9jJh6h6B*CJOJPJQJU\^JaJph  HYPERLINK "http://education.ti.com/calculators/downloads/US/Activities/Detail?id=3988" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/downloads/US/Activities/Detail?id=3988 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.  HYPERLINK "http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5653" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/downloads/US/Activities/Detail?id=5653 BounceBack (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.  HYPERLINK "http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=4085&MICROSITE=ACTIVITYEXCHANGE" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/downloads/US/Activities/Detail?ID=4085&MICROSITE=ACTIVITYEXCHANGE NUMB3RSActivity: Chains and PyramidsEpisode:  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.  HYPERLINK "http://education.ti.com/calculators/downloads/US/Activities/Detail?id=6895" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/downloads/US/Activities/Detail?id=6895 Comparing Linear and Exponential Functions(TI-Nspire): Compare data from two different scenarios -- linear and exponential growth.  HYPERLINK "http://education.ti.com/calculators/downloads/US/Activities/Detail?id=11977" \t "_blank"  INCLUDEPICTURE "http://oaklandk12-public.rubiconatlas.org/common_images/page_world.png?v=Atlas7.2.6.89" \* MERGEFORMATINET http://education.ti.com/calculators/downloads/US/Activities/Detail?id=11977 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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