Real World Performance Tasks
Real World
Real Life, Real Data, Real-‐Time -‐ These activities put students into real life scenarios where they use real-‐time, real data to solve problems. In the Car Crazy series, we use data from automobile manufacturers and update our data regularly.
Note -‐ some data has been rounded or simplified in order to adjust the math to the appropriate level.
Engaging
Relevant – Students today are very familiar with automobiles, making these activities very relevant to their everyday lives. To pique their interest further, try asking the Your Challenge question to the class first.
Authentic Tasks -‐ Through these activity sheets students learn how compound interest works and are prompted to form opinions and ideas about how they would solve real life problems. A glossary is included to help them with the unfamiliar terms used.
Student Choice -‐ Each set of activity sheets is available in multiple versions where students will do the same activities using data for different automobile manufacturers (e.g. Honda, Ford, Toyota). You or your students can pick the brand that most interests them.
Modular
Principal Activity -‐ The activity sheets always start with repeated practice of a core skill matched to a common core standard, as set out in the Teacher Guide. This principal activity (or Level 1 as it is labeled to students) can be used in isolation. This should generally take around 10-‐15 minutes.
Step Up Activity -‐ For the Level 2 questions, students are required to integrate a different skill or set of skills with increasing complexity. The additional skills used to answer these questions are set out in the Teacher Guide. This should generally take around 20-‐30 minutes.
Challenge -‐ This is designed to require critical thinking skills and stretch students to reason with math and data to come to conclusions. They are matched up with one of the Common Core Standards for Mathematical Practice. These activities work well with students in pairs or small groups where they can discuss the math.
Cross-‐Curricular Activity -‐ Every activity sheet also includes a finale that you can use to extend the math lesson into another subject (usually ELA). These could be assigned in a second lesson or for homework.
Customizable
All of the activity sheets are provided in Word so that they can be differentiated to add, remove, or edit questions or even add space for students to show their work. Suggested customizations for each activity sheet are given in the Teacher Guide.
Community
We would love you and your students to tell us about your experience. Join the conversation on Twitter starting your tweet with @nextlesson and using #CarCrazy.
BMW
Teacher Guide
Sound Bite for Students: “In the real world, we use exponential functions to help make financial decisions.”
Skills Practiced:
Principal Activity (Level 1):
-‐ Evaluating exponential expressions Step Up Activity (Level 2):
-‐ Evaluating exponential expressions -‐ Solving exponential expressions
Common Core Math Standards Addressed:
Principal
Activity: HSF-‐IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
Step-‐Up Activity:
HSA-‐CED.A.1 Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
HSF-‐IF.C.8b Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
HSF-‐LE.A.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).
HSF-‐LE.A.4 For exponential models, express as a logarithm the solution to ab ct = d where a , c , and d are numbers and the base b is 2, 10, or e ; evaluate the logarithm using technology.
Differentiation Tips:
You can edit any of the activity sheets to:
-‐ Alter the tasks (e.g. remove car types in order to reduce the amount of work; remove number of years of depreciation in Level 2 to further reduce the work load)
-‐ Cue for differentiation purposes (e.g. show examples of expressions correctly substituted with variables; complete the calculations for one type of car)
-‐ Utilize TI graphing technology
Due to school paper restriction, the spacing provided is only for answers. However, you could modify the spacing to add room for work if desired. Students should be encouraged to show their work where possible.
Updates:
At NextLesson we strive to engage students with data that is real and real-‐time. This lesson uses data as of June 2015. Please come back for the most recent updates.
Exponential Functions
Name: ________________________________
You are a Journalist for Popular Mechanics magazine, and you are doing a feature on vehicle financing of BMW automobiles.
Your Challenge: How much does a car really cost?
LEVEL 1
To begin your research, you are going to consider the costs and promotions BMW offers to determine their monthly payments and overall costs.
In order to calculate monthly payment, you must be able to evaluate the following exponential equation:
𝑚𝑜𝑛𝑡ℎ𝑙𝑦 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 𝐴 × 𝑟
12 × 1
1 − 1 + 𝑟
12
!!
where A is the loan amount, r is the interest rate, and n is the number of months to repay the loan Answer the following questions in the table below.
1. Evaluate the equation to determine the monthly payment cost.
2. Multiply the monthly payment cost by the number of months to determine the overall cost of the loan.
3. Calculate the total amount of interest you will pay for the loan.
Vehicle Cost Interest Rate and Term
Monthly Payment
Overall Cost of Loan
Amount of Interest Paid
3 Series $30,980 3.08% for 36
months
4 Series $44,726 3.23% for 48
months
2 Series $47,342 3.23% for 48
months
X4 $44,151 3.08% for 36
months
X5 $54,186 3.38% for 60
months
X6 $59,769 3.38% for 60
months
7 Series $81,165 3.53% for 72
months
6 Series $81,397 3.53% for 72
months
BMW
2
4. Based on the calculations, which car will have the highest monthly payment?
5. Calculate the difference between the highest and the lowest monthly payments.
6. Determine the difference in the interest paid for the X5 and for the 2 Series.
7. Based on data from the table, which variable most affects the total interest paid? Use evidence to support your reasoning.
BMW
LEVEL 2
The tricky thing with automobiles is how quickly they depreciate. In your article, you want to explore how the value of your new automobile would fall. To do this you will use the exponential equation:
𝑉 = 𝐶 1 − 𝑟 !
where V is the current value of the car, C is the original cost of the car, r is the rate of depreciation, and t is the time in years.
8. In order to estimate the value of the cars, you will use the national average depreciation rate of 15%. Write an expression and determine the value after each time period.
Vehicle Cost Equation Value After
1 Year Value After
3 Years Value After
5 Years Value After 10 Years
3 Series $30,980
4 Series $44,726
2 Series $47,342
X4 $44,151
X5 $54,186
X6 $59,769
7 Series $81,165
6 Series $81,397
9. Calculate when the 4 Series will have a value of $10,000. Round to the nearest hundredth of a year.
10. Calculate when the 3 Series will have a value of $6,000. Round to the nearest hundredth of a year.
11. Calculate when the X6 will have a value of $22,000. Round to the nearest hundredth of a year.
Trucks, SUVs, and vans hold their value a little longer and depreciate a little more slowly.
Suppose the X4 only depreciates at 10% per year.
12. Determine the X4's value in 5 years and compare it to the cost in the table. Is it a significant difference?
BMW
4
Challenge
In order to write a powerful conclusion to your article, you want to calculate the point at which you are paying more than your car is worth.
1. Utilize the monthly payment that you’ve calculated to develop a linear equation that models the annual cost of each automobile over x years.
2. Using either graphing or algebra, determine the point for each automobile where the linear model and the exponential model are the same.
Vehicle
Linear Equation for
Annual Cost
Exponential Model for Depreciation
When will the Cost equal the
Value?
What will the Value be when
they are the same?
3 Series 4 Series 2 Series
X4 X5 X6 7 Series 6 Series
3. Consider how much longer it will take a person to pay off the car once the cost and value are the same. Compare the different automobiles. Do they have similar results or are they significantly different?
4. Based on the calculations, which car do you believe is the best buy for a new car? Use evidence to support your opinion.
5. Consider how different the results would be if a car buyer was able to pay in cash instead of financing a loan. Choose one car from the list and complete a hypothetical analysis.
BMW
Finale
You could give students one of the following finale options or have them choose themselves.
1. Investigate how financing and depreciation of used cars differs from new cars. Use online car dealers to determine figures and estimate monthly payments and depreciation. Write a blog article that compares the difference.
2. A hybrid car has significantly higher gas mileage than its competitors. If you consider the true cost of ownership, determine when the cost of an ordinary model exceeds the value of a hybrid model. Create a Prezi that shares your findings with your peers.
3. In order to make calculations for many automobiles at the same time, develop a spreadsheet that will automatically calculate monthly payment when variables are inputted. Prove that your spreadsheet will work with five different examples.
BMW
Glossary
Depreciate – the amount a product’s value decreases over time.
Hybrid – a car with an engine that uses electric power part of the time.
Journalist – someone who writes articles professionally for a newspaper or magazine (see also Job Background section).
Interest rate – the percentage you must pay a loan originator in addition to the loan amount in order to secure a loan.
True cost of ownership – the total cost of a vehicle factoring in the average costs of gas and routine maintenance.
BMW
Job Background
Journalists fall into the job category of “Writers and Authors” according to the United States Bureau of Labor Statistics.
Quick Facts: Writers and Authors
2012 Median Pay $55,940 per year
$26.89 per hour
Entry-‐Level Education Bachelor’s degree
Work Experience in a Related Occupation None
On-‐the-‐job Training Moderate-‐term on-‐the-‐job training
Number of Jobs, 2012 129,100
Job Outlook, 2012-‐22 3% (Slower than average)
Employment Change, 2012-‐22 3,800
What Writers and Authors Do
Writers and authors develop written content for advertisements, books, magazines, movie and television scripts, songs, and online publications.
Work Environment
Writers and authors work in an office, at home, or wherever else they have access to a computer. Most work full time. However, self-‐employed and freelance writers usually work part time or have variable schedules.
About two-‐thirds were self-‐employed in 2012.
How to Become a Writer or Author
A college degree is generally required for a full-‐time position as a writer or author. Proficiency with computers is necessary for staying in touch with sources, editors, and other writers while working on assignments.
Excellent writing skills are essential.
Pay
The median annual wage for writers and authors was $55,940 in May 2012.
Job Outlook
Employment of writers and authors is projected to grow 3 percent from 2012 to 2022, slower than the average for all occupations. Strong competition is expected for full-‐time jobs because many people are attracted to this occupation.
BMW
ANSWER KEY
LEVEL 1
1. See Column 4 below.
2. See Column 5 below.
3. See Column 6 below.
Vehicle Cost Interest Rate
and Term Monthly
Payment Overall Cost
of Loan Amount of Interest Paid
3 Series $30,980 3.08% for 36
months $902.03 $32,473.03 $1,493.03
4 Series $44,726 3.23% for 48
months $994.53 $47,737.58 $3,011.58
2 Series $47,342 3.23% for 48
months $1,052.70 $50,529.73 $3,187.73
X4 $44,151 3.08% for 36
months $1,285.52 $46,278.78 $2,127.78
X5 $54,186 3.38% for 60
months $982.83 $58,969.72 $4,783.72
X6 $59,769 3.38% for 60
months $1,084.09 $65,045.60 $5,276.60
7 Series $81,165 3.53% for 72
months $1,252.53 $90,182.44 $9,017.44
6 Series $81,397 3.53% for 72
months $1,256.11 $90,440.22 $9,043.22
4. X4
5. $383.49 6. $1,595.99
7. Answers may vary. Accept responses that…
• Identify a pattern from the data to make the conclusion.
• Provide at least two examples, supported with data, to support the response.
Example: “Looking at the data, the original cost of the car has the highest impact on the interest paid. To arrive at this conclusion, I compared the difference in interest between the 3 Series and 4 Series with the difference in interest between the 2 Series and X4. The 3 Series and 4 Series both have the same interest rate and loan term, so the difference in interest (which comes to $1,518.55) must stem from the difference in cost of these cars, which is $13,746.00. The 2 Series and X4 likewise share identical interest rates and loan terms, so their difference in interest ($1,059.95) must also be attributed to their difference in starting cost, $3,191.00. From this data, we can see that when all other variables (interest rate and loan term) are kept constant, a higher starting cost of the car means more interest paid.”
BMW
ANSWER KEY
LEVEL 2
8.
9. In about 9.22 years.
10. In about 10.10 years.
11. In about 6.15 years.
12. At the new depreciation rate, the value will be $26,070.72 in 5 years. Yes, the difference of $6,480.69 is significant.
BMW
Vehicle Cost Equation Value After
1 Year Value After
3 Years Value After
5 Years Value After 10 Years
3 Series $30,980 V = 30,980(1-0.15)t $26,333.00 $19,025.59 $13,745.99 $6,099.17
4 Series $44,726 V = 44,726(1-0.15)t $38,017.10 $27,467.35 $19,845.16 $8,805.40
2 Series $47,342 V = 47,342(1-0.15)t $40,240.70 $29,073.91 $21,005.90 $9,320.43
X4 $44,151 V = 44,151(1-0.15)t $37,528.35 $27,114.23 $19,590.03 $8,692.20
X5 $54,186 V = 54,186(1-0.15)t $46,058.10 $33,276.98 $24,042.62 $10,667.84
X6 $59,769 V = 59,769(1-0.15)t $50,803.65 $36,705.64 $26,519.82 $11,766.99
7 Series $81,165 V = 81,165(1-0.15)t $68,990.25 $49,845.46 $36,013.34 $15,979.31
6 Series $81,397 V = 81,397(1-0.15)t $69,187.45 $49,987.93 $36,116.28 $16,024.99
ANSWER KEY Challenge
1. See Column 2 below.
2. See Columns 4 and 5 below.
Vehicle Linear Equation for Annual Cost
Exponential Model for Depreciation
When will the Cost equal the Value?
What will the Value be when
they are the same?
3 Series y = 10,824.34x V = 30,980(1-0.15)t 2.05083 $22,198.89
4 Series y = 11,934.40x V = 44,726(1-0.15)t 2.49741 $29,805.08
2 Series y = 12,632.43x V = 47,342(1-0.15)t 2.49741 $31,548.36
X4 y = 15,426.26x V = 44,151(1-0.15)t 2.05083 $31,636.64 X5 y = 11,793.94x V = 54,186(1-0.15)t 2.87802 $33,943.21 X6 y = 13,009.12x V = 59,769(1-0.15)t 2.87802 $37,440.51
7 Series y = 15,030.41x V = 81,165(1-0.15)t 3.20674 $48,198.61
6 Series y = 15,073.37x V = 81,397(1-0.15)t 3.20674 $48,336.38
3.
Vehicle Months Needed to Pay Off From Equal
Cost and Value Point
3 Series 12
4 Series 19
2 Series 19
X4 12
X5 26
X6 26
7 Series 34
6 Series 34
BMW
ANSWER KEY
Answers may vary for the explanation. Accept responses that…
• Identify major differences in pay-off times that depend on the number of months the car is being financed.
• Include calculations regarding the pay-off times to support the answer.
Example: “The car models do have significantly different pay-off times from when cost and value are equal. The 3 Series requires 12 more months of payments, for example, while the 6 Series requires 34 more months. The difference stems from the fact that the 3 Series has a 36 month loan term while the 6 Series has a 72 month loan term. Any models with the same loan term have the same pay-off time from the point of equal cost and value.”
4. Answers may vary. Accept responses that…
• Draw reasonable conclusions from the data to select the model or models that are a best buy.
• Provide multiple pieces of data and/or calculations to support the decision.
Example: “In my opinion, the 3 Series and 4 Series are the best buys. Both of these cars are paid off only 12 months after the cost and value are equal. The model with the next quickest pay-off time is 19 months, a whole 7 months later. Paying off the car soon after its value drops below the cost is smart because the buyer ends up losing less money on their investment in the vehicle.”
5. Answers may vary. Accept responses that…
• Explain that because the full cost of the car is paid up front, the time when the cost and value are equal is at the time of purchase.
• Provide data and/or calculations from both the financed vehicle and the one purchased with cash.
Example: “Whenever a buyer pays cash, he or she is paying the entire cost up front. For this reason, the cost of the car and its value will be equal at the moment of purchase. In the case of the 3 Series, for instance, the buyer pays the full cost of $30,980 at the time of purchase. This moment is the only time when the value of the car will be equal to the purchase price. The moment the car leaves the lot, it starts depreciating. Therefore, when a buyers pays cash for the 3 Series the cost and value are equal at t=0, and the value at the point is the purchase price, $30,980.
However, when financing the 3 Series instead of paying cash, it takes 2.05083 years to reach an equal value and cost of $22,198.89.”
