Up a Little, Down a Little
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1 Up a Little, Down a Little A Solidif Understanding Task One of the most common applications of eponential growth is compound interest. For eample, Mama Bigbucks puts $20,000 in a bank savings account that pas 3% interest compounded annuall. Compounded annuall means that at the end of the first ear, the bank pas Mama 3% of $20,000, so the add $600 to the account. Mama leaves her original mone ($20000) and the interest ($600) in the account for a ear. At the end of the second ear the bank will pa interest on the entire amount, $ Since the bank is paing interest on a previous interest amount, this is called compound interest Model the amount of mone in Mama Bigbucks bank account after t ears. Use our model to find the amount of mone that Mama has in her account after 20 ears. A formula that is often used for calculating the amount of mone in an account that is compounded annuall is: A = P(1 + r) t Where: A = amount of mone in the account after t ears P = principal, the original amount of the investment r = the annual interest rate t = the time in ears Appl this formula to Mama s bank account and compare the result to the model that ou created. Based upon the work that ou did in creating our model, eplain the (1 + r) part of the formula.
2 53 Another common application of eponential functions is depreciation. When the value of something ou bu goes down a certain percent each ear, it is called depreciation. For eample, Mama Bigbucks bus a car for $20,000 and it depreciates at a rate of 3% per ear. At the end of the first ear, the car loses 3% of its original value, so it is now worth $19,400. Model the value of Mama s car after t ears. Use our model to find how man ears will it take for Mama s car to be worth less than $500? How is the situation of Mama s car similar to Mama s bank account? What differences do ou see in the two situations? Consider our model for the value of Mama s car and develop a general formula for depreciation.
3 Up a Little, Down a Little Teacher Notes A Solidif Understanding Task Purpose: The purpose of this task is for students to connect their understanding of eponential functions with standard formulas for compound interest and depreciation. Students will consider how a percent change is written in a formula in both an eponential growth and deca situation. Students will develop a formula for depreciation. Core Standards Focus: A.SSE.1 Interpret epressions that represent a quantit in terms of its contet. a. Interpret parts of an epression, such as terms, factors, and coefficients. b. Interpret complicated epressions b viewing one or more of their parts as a single entit. For eample, interpret P(1+r) n as the product of P and a factor not depending on P. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate aes with labels and scales. F.LE.5 Interpret the parameters in a linear or eponential function in terms of a contet. F.IF Analze functions using different representations F.IF.7. Graph functions epressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maima, and minima. e. Graph eponential and logarithmic functions, showing intercepts and end behavior F.IF.9 Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). For eample, given a graph of one quadratic function and an algebraic epression for another, sa which has the larger maimum. For F.IF.7a, 7e, and 9 focus on linear and eponential functions. Include comparisons of two functions presented algebraicall. For eample, compare the growth of two linear functions, or two eponential functions such as =3 n and =100 2 n. Launch (Whole Class): Start the lesson b eplaining the general concept of compound interest (without giving a formula) and ensuring that students understand the eample given at the beginning of the task. Ask students
4 to work in their groups to model the amount of mone in Mama s bank account and complete the questions about compound interest on the task. Encourage the use of technolog for the task. Eplore (Small Group): Monitor students as the work, supporting them in using technolog to create tables, graphs, and equations for the model. Be sure that ever group has written an equation on their own before using the formula. Discuss Part 1and Re-launch(Whole Class): Using a graphing calculator or other technolog, graph the equation that students have written. Use the graph to find approimatel how much mone will be in the account after 20 ears and then ask a student to show how the used the equation to find the amount. Ask students what the notice about how Mama s mone grows when the look at the graph. Be sure that the discuss how mone grows slowl at the beginning and then the rate of growth increases dramaticall as the mone remains in the account over time. Emphasize the importance of this principal in saving mone. Ask students to predict how the curve would change using different interest rates. Use the formula to write and graph equations that model 5% interest and 10% interest. Displa all three graphs together and compare the amount of mone in the account over time with the various rates. How much difference does the rate make if the mone is kept in the account for 5 ears, 10 ears, or 20 ears? Net, describe the meaning of the term depreciation. Demonstrate the idea b working through the eample given about Mama s car. Be sure that students understand that depreciation reduces the value of the item, so that it is tpicall modeled with a decreasing eponential function. Once the basic idea of depreciation is understood, let students work on the rest of the task. Discuss Part 2 (Whole Group): Start the discussion b using technolog to displa the graph of the equation that students wrote to model the depreciation of Mama s car. When does the car depreciate the fastest? According to the model, will the car ever be worth nothing? Discuss the similarit and differences that students notice between the compound interest and the depreciation scenarios. Turn the focus of the discussion to the formula that students have written for depreciation. Is it the same as the compound interest formula? Wh do the two formulas generate such different curves? What is the meaning of the (1+r) or (1-r) part of the formulas? Where does the one come from? Aligned Read, Set, Go: Linear and Eponential Functions 9
5 LinearandEponentialFunctions 9 54 Read,Set,Go! Topic:Evaluatingequations Read Fill out the table of values for the given equations. 2012www.flickr.com/photos/civisi 1. = = = ½ = 6 5. = MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution NonCommercial ShareAlike3.0Unportedlicense.
6 LinearandEponentialFunctions 9 55 Set Topic:Evaluateusingtheformulasforsimpleinterestorcompoundinterest. Giventheformulaforsimpleinterest:i=Prt,calculatethesimpleinterestpaid. i=interestp=theprincipalr=theinterestrateperearasadecimalt=timeinears 7.Findthesimpleinterestouwillpaona5earloanof$7,000at11%perear. 8.Howmuchinterestwilloupain2earsonaloanof$1500at4.5%perear? Usei=Prttocompletethetable.Allinterestratesareannual. i = P r t 9. $11,275 12% 3ears 10. $1428 $5100 4% 11. $93.75 $1250 6months 12. $54 8% 9months Giventheformulaforcompoundinterest:A=P(1+r) t,writeacompoundinterestfunctionto modeleachsituation.thencalculatethebalanceafterthegivennumberofears. A=thebalanceaftertears P=theprincipal t=thetimeinears r=theannualinterestrateepressedasadecimal 13. $22,000investedatarateof3.5%compoundedannuallfor6ears. 2012MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution NonCommercial ShareAlike3.0Unportedlicense.
7 LinearandEponentialFunctions $4300investedatarateof2.8%compoundedannuallfor15ears. 15.Supposethatwhenouare15earsold,amagicgeniegivesouthechoiceofinvesting $10,000atarateof7%or$5,000atarateof12%.Eitherchoicewillbecompoundedannuall.The monewillbeourswhenouare65earsold.whichinvestmentwouldbethebest?justifour answer. Topic:Usingorderofoperationswhenevaluatingequations Go Evaluatetheequationsforthegivenvaluesofthevariables. 16.pq 6+10;whenp=7andq= 3 17.m+n(m n);whenm=2,andn=6 18.(b 1) 2 +ba 2 ;whena=5,andb=3 19.( (9 4));when=4,and= ( ( 3 ));when=7,and=2 21.an 4 +a(n 7) 2 +2n;whena= 2,andn=4 NeedHelp?Checkouttheserelatedvideos: mathematics.com/simple vs compound interest.html economics/core finance/v/introduction to interest 2012MathematicsVisionProject MVP InpartnershipwiththeUtahStateOfficeofEducation LicensedundertheCreativeCommonsAttribution NonCommercial ShareAlike3.0Unportedlicense.
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