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Author: Hannah Knisely, Kent Island High School, Queen Anne’s County Public SchoolsBackground InformationSubject:Identify the course the unit will be implemented in.Algebra IIGrade Band:Identify the appropriate grade band for the lesson.10-12Duration:Identify the time frame for the unit.One 90 minute class periodOverview:Provide a concise summary of what students will learn in the lesson. It explains the unit’s focus, connection to content, and real world connection. In this lesson, students learn the real-world application of exponential models. In subsequent lessons they will learn how to solve exponential and logarithmic equations as well as base e, natural logarithms, and exponential decay and growth problems.Background Information:Identify information or resources that will help teachers understand and facilitate the lesson.Teachers should understand how exponential functions model both growth and decay in our everyday world. Applications of exponential decay can be found with the following examples: medications/caffeine leaving the body, radioactive decay, half-life, carbon dating, depreciation of material objects, etc. Applications of exponential growth can be found with the following examples: Population growth, bacteria growth, appreciation of material objects, compound interest, etc.STEM Specialist Connection:Describe how a STEM Specialist may be used to enhance the learning experience. STEM Specialist may be found at STEM Specialist can:Help students interpret the results of their dice activity. Engage students in a discussion of how exponential growth/decay can be found in our everyday lives. Encourage students to look for patterns in the real-world application problems.Discuss careers in which application of exponential models is prevalent (Biology, Anthropology, Banking, Investments, Financial Advisors, Real-Estate Agents, Census workers, Food Production, etc.)Enduring Understanding:Identify discrete facts or skills to focus on larger concepts, principles, or processes. They are transferable - applicable to new situations within or beyond the subject.Mathematical models are used to develop solutions to real-world problems.Exponential models carefully define the percent rate of change in real-world applications.Essential Questions:Identify several open-ended questions to provoke inquiry about the core ideas for the lesson. They are grade-level appropriate questions that prompt intellectual exploration of a topic.How can exponential functions be used to model real-world problems and solutions?How does a STEM professional use exponential functions?Student Outcomes:Identify the transferable knowledge and skills that students should understand and be able to do when the lesson is completed. Outcomes must align with but not limited to Maryland State Curriculum and/or national standards.Students will be able to:Graph exponential functions expressed symbolically and show key features of the graph by hand and using technology. Determine if a graph/equation is representing exponential growth or exponential decay.Explain the application of exponential models in real-world situations.Product, Process, Action, Performance, etc.:Identify what students will produce to demonstrate that they have met the challenge, learned content, and employed 21st century skills. Additionally, identify the audience they will present what they have produced to.Students will accurately graph and explain the results of the dice activity.Students will think through a problem and persevere in solving an exponential equation given bank account information over a period of time.Students will analyze a situation (exponential model) and make sense of the situation by solving and explaining the solution.Audience:?Peers ?Experts / Practitioners ?Teacher(s)?School Community?Online Community?Other______Standards Addressed in the Unit:Identify the Maryland State Curriculum Standards addressed in the mon Core Algebra II Standards:Domain: Modeling with FunctionsCluster Statement: Interpret expressions for functions in terms of the situation they model.Standard F.LE.5 Interpret the parameters in a linear or exponential function in terms of context.Cluster Statement: Construct and compare linear, quadratic, and exponential models and solve problems.Standard 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 (including reading these from a table).Cluster Statement: Analyze functions using different representationsStandard 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.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.Cluster Statement: Analyze functions using different representations.Standard F.IF.9: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.Suggested Materials and Resources:Identify materials needed to complete the unit. This includes but is not limited to websites, equipment, PowerPoints, rubrics, worksheets, and answer keys.Equipment:computer with internet accessprojectorWebsites* Magnitude of an Earthquake ()The Science of Overpopulation ()Futurama video clips: ()This particular website has been edited into video clips for convenience in the classroom. Those clips can be found as media clips in the PowerPoint and/or file folder associated with this lesson.* The sites have been chosen for their content and grade-level appropriateness. Teachers should preview all websites before introducing the activities to students and adhere to their school system’s policy for internet use.Materials:Dice (36 per group of 2-4)Dice ActivityExponential Functions PowerPointFry's Bank Account Video Questions HYPERLINK \l "Fry_Bank_Account_Answer_Key" Fry’s Bank Account Video Questions Answer KeyFry's Bank Account (First Clip)Fry's Bank Account (Second Clip)Exponential Functions HomeworkExponential Functions Homework Answer KeyLesson Overview: Students will discover the use of exponential function in the real-world. They will work through several examples independently as well as a class. Additionally they will view video clips explaining the real-world use of exponential modeling.Duration: 90 MinutesLearning Experience5E ComponentIdentify the 5E component addressed for the learning experience. The 5E model is not linear.DetailsStandards for Mathematical Practice?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Dice (36 per group of 2-4)Dice Activity worksheet (1 per group)ComputerProjectorExponential Functions PowerPointPreparation: Gather enough dice needed to complete the activity for your class.Copy the Dice Activity worksheet for each of your groups. The groups could be pre-determined or you may allow students to choose; the preference is yours based on your classroom dynamics.Facilitation of Learning Experience:Allow students time for experimentation/exploration. Students will start with 36 dice. Each time a 6 is rolled that die should be removed from the bunch and the remaining dice should be recorded in the table. After 15 trials (or when the dice run out), the students will be able to show you an exponential decay regression graph. They will then answer the following questions:What do you notice about the graph?What type of function do you think this is?Transition:The questions from the activity will be discussed as a class once each group is finished with their experiment.This discussion should naturally transition into the remaining aspects of the PowerPoint.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.?Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Exponential Functions PowerPointComputerProjectorPreparation: Go through the PowerPoint as the instructor before presenting the information with your students to ensure your own understanding of the material. Facilitation of Learning Experience:Discuss, in detail the difference between linear and exponential functions. The graphs on the PowerPoint will provide clarification. Emphasize that linear graphs have a constant rate of change and exponential graphs have a percent rate of change.Work through the job offers example together as a class. Discuss the choices present: Which is better immediately? Which is better in the long-term?Allow students time to complete the table from the savings account example on their own; then discuss the results as a class. Help them derive the equation for the model of the bank account. Once finished with this, allow the students to be the ones to determine how much money is in the account after 20 years. Compare answers as a class.Transition:Discuss how exponential models are used in the real-world. Allow students time to think about how these models are used in the real-world before giving them the answers. OrA STEM Specialist can be used to help students understand how exponential functions are used in the STEM workforce. The Specialist will engage students in a hands-on learning experiences that demonstrates how he/she employs exponential functions in his/her field. To find a STEM Specialist visit .?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Exponential Functions PowerPointComputerProjectorPreparation:Familiarize yourself with exponential equations: fx=abx.Know that a is the constant in each exponential equation. This relates to the y-intercept of the graph (initial value).b is the base of the equation and tells us the rate in which the function increases/decreases.The variable is in the exponent allowing us to calculate values per interval of time (usually time).Facilitation of Learning Experience:Go through each slide of the PowerPoint explaining the symbolic meaning of an exponential function as well as the graphical meaning.Show a graphical representation of an exponential function and explain how you can tell if the function is an increasing/decreasing exponential function based on the graph alone.Break down the individual pieces of the equation of an exponential function and stress what each means.Discuss with the class the meaning behind the mathematical language of fx=a?bx for b>1 (growth) and fx=a?bx for 0<b<1 (decay).Challenge student understanding of the meaning behind the symbolic representation of an exponential function by asking, “Can you automatically conclude that an exponential function models decay if the base of the power is a fraction or decimal?”Transition:Discuss the answer to the question posed above. You cannot conclude that a fraction/decimal implies exponential decay. This is true because not all fractions/decimals are less than 1. Some fractions/decimals represent numbers larger than 1 which would imply exponential growth.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:ComputerProjectorExponential Functions PowerPointFry's Bank Account Video QuestionsFry's Bank Account (First Clip)Fry's Bank Account (Second Clip)Fry’s Bank Account Video Question Answer KeyPreparation:Preview each of the following video clips to ensure they are appropriate for your classroom plete the Fry’s Bank Account Video Questions in advance to prepare for any questions that may arise from your students.Facilitation of Learning Experience:Pass out Fry’s Bank Account Video Questions to each student (the supplement provided is cut into two—half sheet per student should suffice).Allow the students to watch the video clip from the TV show Futurama. If students have not seen the show explain that the character Fry is from the past. The show takes place in the year 3000. Fry is from the year 2000. This will help with the understanding of the clip.After Clip 1 has been shown have students guess how much is in Fry’s bank account. They should write down a number they know is too small and a number they think is too large. The actual number will be bleeped out in the video clip to allow for students to think about it.Once all the students have answered take a quick poll of the class to see what kind of numbers they came up with.View video clip 2. In this clip they say that Fry’s bank account with a balance of $0.93 with 2.25% interest over 1000 years has accumulated to $4.3 Billion. This amount can be confirmed using the equation A=P1+rnn?t. Assume the account compounds interest once, annually. Show the students the mathematics behind this amount manually and using a calculator (reinforce the importance of the order of operations).Students should then answer the remaining questions: It took Fry 1,000 years to get that much money. How long will it take him to double it?How long will it take him to get a trillion dollars?Be sure to let the students figure these questions out independently or with a partner before discussing the answers as a class.After the Futurama video clips there is one more real-world example in the PowerPoint. This helps students see how slow (sometimes fast) dilution of medication happens in the blood system.This would be an interesting place to discuss how all medications/supplements affect the human body (a great example is caffeine). Work this example through with the students as a class and discuss how to find a growth factor.If time allows, show students the video clip discussing the magnitude of earthquakes and how exponential models help us mathematically define the epicenter.If additional time allows, show students the Science of Overpopulation clip. It is lengthy put does a nice job outlining how human population will one day outgrow the world’s ability to produce food.Transition:Re-emphasize to the students how examples of exponential decay/growth can be found everywhere in the real-world.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.?Engagement?Exploration?Explanation?Extension?EvaluationMaterials:Exponential Functions HomeworkExponential Functions Homework Answer KeyPreparation:Take time to complete the exponential functions homework assignment before handing it out to students. Be prepared to ask any questions the students may have about the assignment.Feel free to alter the questions to the ability level of your students.Facilitation of Learning Experience:Allow the students time to discover information/work through problems on their own.?Make sense of problems and persevere in solving them.?Reason abstractly and quantitatively.?Construct viable arguments and critique the reasoning of others.?Model with mathematics.?Use appropriate tools strategically.?Attend to precision.?Look for and make use of structure.? Look for and express regularity in repeated reasoning.Supporting InformationInterventions/EnrichmentsIdentify interventions and enrichments for diverse learners.Struggling LearnersGroup students based upon ability, learning style, or other appropriate criteria, so all students can equally contribute to group work.If the questions asked in class are too vague, try guiding your questioning strategies to allow students to look for repeated reasoning.Class time should be effectively used—let students know the time frame allotted for each activity. Provide resources to define and/or pronounce difficult vocabulary.Provide additional time for work completion.English Language LearnersStrategies to help English Language Learners are similar to those listed above.Provide resources to define and/or pronounce difficult vocabulary. A native language dictionary may also be beneficial.Use visuals.Read directions and documents aloud to students, when appropriate.Gifted and TalentedAsk students to research further a particular situation that appears to have an exponential relationship.The instructor should foster independent thinking.Higher level thinking questions should be asked throughout the lesson with the expectation of responses that are thoughtful and elaborate.Encourage students to develop discussion questions for the STEM Specialist.Think about careers that use exponential modeling and explain how that modeling is used.Names: ______________________________________________________DiceEquipment Needed: 36 number cubesIn the table below, the original number of dice (36) has been recorded.Roll these dice.Remove any die that show a 6 on the top face.Count the remaining dice and record that number in the table below.Using only the dice remaining, repeat steps 2-4 until there are no die remaining or until you have rolled 15 times, whichever occurs first.Number of RollsNumber of Dice Left036123456789101112131415Using the results from your table sketch a graph of your results here:What do you notice about the graph?What type of function do you think this is?Fry’s Bank Account (From “Futurama”)Name: ____________________Clip 1:How much money does Fry have in his bank account?Write down an answer you know is too high.Write down an answer you know is too low.-37211027940000Clip 2:It took Fry 1,000 years to get that much money. How long will it take him to double it?How long will it take him to get a trillion dollars?2095538481000Fry’s Bank Account (From “Futurama”)Name: ____________________Clip 1:How much money does Fry have in his bank account?Write down an answer you know is too high.Write down an answer you know is too low.-37211027940000Clip 2:It took Fry 1,000 years to get that much money. How long will it take him to double it?How long will it take him to get a trillion dollars?Fry’s Bank Account (From “Futurama”)Answer KeyClip 1:How much money does Fry have in his bank account?Write down an answer you know is too high.Students guessWrite down an answer you know is too low.Students guess-37211027940000Clip 2:It took Fry 1,000 years to get that much money. How long will it take him to double it?A=P1+rnntRemember: A= amount in the accountP=Principle (amount deposited)r= interest rate (as a decimal, not a percent)n= number of times compounded yearlyt= number of years4,300,000,000≈0.931+0.022511?1000So… to double it we are looking at 8,600,000,0008,600,000,000≈0.931+0.02251t9247310000=1.0225tlog9247310000=t?log?(1.0225)t=log9247310000log?(1.0225)t=1,031.32 yearsHow long will it take him to get a trillion dollars?1,000,000,000,000≈0.931+0.02251t1075270000000=1.0225tlog1075270000000=t?log?(1.0225)t=log1075270000000log?(1.0225)t=1,245.07 yearsName: _____________________Exponential Functions HomeworkDirections: Read the following situations and answer the questions that follow using the information learned in class today.A golf ball manufacturer packs 3 golf balls into a single package. Three of these packages make a gift box. Three gift boxes make a value pack. The display shelf is high enough to stack 3 value packs on top of the other. Three such columns of value packs make up a display front. Three display fronts can be packed in a single shipping box and shipped to various retail stores. How many golf balls are in a single shipping box?The cost of a pair of sneakers increases about 4.9% every year. About how much would a $60 pair of sneakers cost 30 years from now?The initial number of bacteria in a culture is 10,000. The number after 5 days is 320,000.Write an exponential function to model the population y of bacteria after x days.How many bacteria are there after 10 days?Tom opened a savings account that accrues compound interest at a rate of 2.10% quarterly. Let P be the initial amount Tom deposited and let t be the number of years the account has been open.Write an equation to find A, the amount of money in the account after t years. (Hint: we discussed this in class today) If Tom opened the account with $450 and made no deposits or withdrawals, how much is in the account 10 years from now?A university with a graduating class of 4,000 students in 2013 predicts it will have a graduating class of 4,862 in 4 years. Write an exponential function to model the number of students y in the graduating class t years after 2013.42481556451500245618059626500Sketch the graph of each function. Then state the function’s domain and range.y=32xb. y=212xc. y=1.50.4x4486910000Domain: Domain:Domain:Range: Range:Range:Determine whether each function represents exponential growth or decay:y=38xb. y=22.5xc. y=0.13xd. y=10-xe. y=4910xf. y=5?4-xAnalyze #6b and #7b. How do each of them compare and contrast?Mark is one of 1,200 workers at the UPS warehouse in Baltimore, which operates seven days a week. He arrives at the factory on Monday with a slight fever. While working with three others on a project, he becomes so ill that he goes to the nurse and is sent home. Later that day, the nurse determines that Mark has a contagious virus that will appear one day after exposure in those with whom he has been in close contact. An epidemic is about to happen. The warehouse must close if more than 40 percent of its workers are ill. If the three workers with whom Mark worked on Monday come down with the virus on Tuesday and each day each worker infects three others, when will the factory need to close?Complete the following table and describe the pattern of the virus.DAYMondayTuesdayWednesdayThursdayFriday# of workers infected?????Describe the pattern:BONUS: If each sick worker must stay home for three days, when will the factory reopen?In the science-fiction novel The Pride of Chanur, author C. J. Cherryh imagined an alien race, the stsho, which has three sexes instead of two as we humans do. Thus, each stsho has three parents, which will be referred to as X, Y, and Z instead of "mother" and "father." Create a family tree for a certain stsho, named Tle-nle, showing three parents (XYZ), three grandparents (GX,GY,GZ), and three great-grandparents (GGX,GGY,GGZ). How many ancestors does Tle-nle have, going back through the sixth generation of the family tree?Name: _____________________Exponential Functions Homework Answer KeyDirections: Read the following situations and answer the questions that follow using the information learned in class today.A golf ball manufacturer packs 3 golf balls into a single package. Three of these packages make a gift box. Three gift boxes make a value pack. The display shelf is high enough to stack 3 value packs on top of the other. Three such columns of value packs make up a display front. Three display fronts can be packed in a single shipping box and shipped to various retail stores. How many golf balls are in a single shipping box? golf ballsThe cost of a pair of sneakers increases about 4.9% every year. About how much would a $60 pair of sneakers cost 30 years from now?≈$252.00The initial number of bacteria in a culture is 10,000. The number after 5 days is 320,000.Write an exponential function to model the population y of bacteria after x days.y=10,000e0.693147xHow many bacteria are there after 10 days?10,240,000Tom opened a savings account that accrues compound interest at a rate of 2.10% quarterly. Let P be the initial amount Tom deposited and let t be the number of years the account has been open.Write an equation to find A, the amount of money in the account after t years. (Hint: we discussed this in class today) A=P1+0.021044tIf Tom opened the account with $450 and made no deposits or withdrawals, how much is in the account 10 years from now?$554.85A university with a graduating class of 4,000 students in 2013 predicts it will have a graduating class of 4,862 in 4 years. Write an exponential function to model the number of students y in the graduating class t years after 2013.y=40001.05tSketch the graph of each function. Then state the function’s domain and range.y=32xb. y=212xc. y=1.50.4x3333751143000 44005501905Domain: All Reals Domain: All RealsDomain: All RealsRange: y > 0 Range: y > 0Range: y > 0Determine whether each function represents exponential growth or decay:y=38xb. y=22.5xc. y=0.13xgrowth growth growthd. y=10-xe. y=4910xf. y=5?4-xdecay decay decayAnalyze #6b and #7b. How do each of them compare and contrast?#6b represents an exponential decay equation. #7b represents an exponential growth equation. Both have a b-value (y=abx) that is a fraction/decimal but the difference is that in #6b the fractional value is between 0 and 1. #7b the decimal is a number larger than 1, thus it is an exponential growth equation.Mark is one of 1,200 workers at the UPS warehouse in Baltimore, which operates seven days a week. He arrives at the factory on Monday with a slight fever. While working with three others on a project, he becomes so ill that he goes to the nurse and is sent home. Later that day, the nurse determines that Mark has a contagious virus that will appear one day after exposure in those with whom he has been in close contact. An epidemic is about to happen. The warehouse must close if more than 40 percent of its workers are ill. If the three workers with whom Mark worked on Monday come down with the virus on Tuesday and each day each worker infects three others, when will the factory need to close?Complete the following table and describe the pattern of the virus.DAYMondayTuesdayWednesdayThursdayFriday# of workers infected1??3?927?81?Each day that passes, 1 infected person infects 3 new people. This is an exponential model. So, an exponential function to model this situation is f(x)=3x where fx is the number of persons with the illness and x is the number of days since the illness first came to work. It comes out to be that 243 have the illness on day 5 and 729 have the illness on day 6.? So, the factory would have to close on the Sunday following the Monday after Mark came to work since 40% of 1,200 workers is 480. That number of people are infected somewhere between Saturday and Sunday.But if you look closer at the situation it breaks down as follows:on day zero, no one is at homeon day one, one person is at homeon day two, 1 + 3 = 4 people are at homeon day three, 1 + 3 + 9 = 13 people are at homeon day four, 1 + 3 + 9 + 27 - 1 = 39 people are at homeon day five, 1 + 3 + 9 + 27 + 81 - 1 - 3 = 117 people are at homeon day six, 1 + 3 + 9 + 27 + 81 + 243 - 1 - 3 - 9 = 351 people are at homeThere are actually less than 40% of the factory workers who have the illness on day six and the factory won't close on day six but on day seven instead.on day seven, 1 + 3 + 9 + 27 + 81 + 243 + 729 - 1 - 3 - 9 - 27 = 1053 people are at home and the factory closes.BONUS: If each sick worker must stay home for three days, when will the factory reopen?Since the workers must wait 3 days to return to work after the illness breaks out the factory reopens on day 10.? The factory can reopen the second Saturday after the Monday that Mark came to work with the illness.In the science-fiction novel The Pride of Chanur, author C. J. Cherryh imagined an alien race, the stsho, which has three sexes instead of two as we humans do. Thus, each stsho has three parents, which will be referred to as X, Y, and Z instead of "mother" and "father." Create a family tree for a certain stsho, named Tle-nle, showing three parents (XYZ), three grandparents (GX,GY,GZ), and three great-grandparents (GGX,GGY,GGZ). How many ancestors does Tle-nle have, going back through the sixth generation of the family tree?(See Answer on Next page)348615023812533909002286001466850219075Tle-nle55626001333505524500171450547687519050034194751524003390900200025325755014287513811251524001171575190500962025123825X Y Z590550019939058959751803405886450170815568642516129055816501612905372100151765511492518034048863251803404648200161290362902518034036385501612903629025161290341947517081534194751803403248025161290299085017081527717751612902505075170815147637512319014287501231901400175189865120967519939011620501708151095375170815876300218440638175227965381000161290GX GY GZGX GY GZGX GY GZGGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZ GGX GGY GGZHow many ancestors does Tle-nle have, going back through the sixth generation of the family tree?fx=3x=36=729 ancestors ................
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