In this lesson, students apply the familiar formula y=mx+b to a relevant real-world scenario: buying a car.
Students in U.S. high schools can get free digital access to The New York Times until Sept. 1, 2021.
Lesson Overview
Featured article: “Electric Cars Are Better for the Planet — and Often Your Budget, Too” by Veronica Penney.
Last month, General Motors announced that it will sell only zero-emissions vehicles by 2035. The announcement represents a shift toward not only a more sustainable energy model, but also a more sustainable business model. As electric car technology becomes increasingly efficient and affordable, more Americans are choosing to buy electric vehicles.
This trend raises a question familiar to many drivers: Are we there yet? Specifically, have electric cars become both more environmentally-friendly and more affordable than standard gas cars? In this lesson, you will explore this question algebraically using systems of linear equations. In addition to analyzing broader trends, you’ll analyze your own family’s (or a friend’s) car to see if “going green” with an electric vehicle would mean saving some “wallet green” too.
Note to Teachers: This lesson plan is available as a PDF for students.
Warm Up
Often the decision to buy a gas or electric vehicle is framed as a trade-off between the cost of the car and its pollution potential. First, we’re going to consider this trade-off with respect to two cars: the Tesla Model 3 (electric) and Toyota Camry XLE (gas).
Let’s consider economic costs. Respond to the following questions in writing or in class discussion:
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Which car do you believe costs more to buy? Why?
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Which car do you believe costs more to drive (after initial purchase)? Why?
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Which car do you believe costs more over all (to buy and to drive) over its full lifetime? Why?
Next, let’s consider environmental impact. One central way to measure environmental impact is by measuring greenhouse gas emissions. Greenhouse gases, such as carbon dioxide (CO2), contribute to climate change.
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How do cars emit greenhouse gases? What is the most common greenhouse gas that gas-powered cars emit? (Hint: It’s mentioned above.)
Greenhouse gases are not just created while driving. For example, most manufacturers have to emit greenhouse gases when creating the materials for, assembling and shipping their newly-sold vehicles. These are considered emissions made “during production.” In addition, in many areas, greenhouse gases also power the city’s electrical grids that charge electric car batteries.
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Which car (Tesla Model 3 or Toyota Camry XLE) do you believe creates more greenhouse gases in its production? Why?
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Which car do you believe requires more greenhouse gases to create its power source (fuel/electricity)? Why?
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Which car do you believe emits more greenhouse gases over all (during production and while driving) during its full lifetime? Why?
Activity: Modeling the Tradeoffs
Model the Costs
Read the article from the beginning … to this paragraph:
They found electric cars were easily more climate friendly than gas-burning ones. Over a lifetime, they were often cheaper, too.
Answer the following questions:
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What factors might make electric cars more expensive than gas cars initially? What factors might make electric cars more affordable over time?
Now we’re going to model the costs of our two chosen cars (the Tesla and the Camry) with data from the carbon counter tool used in the article’s analysis. First, let’s consider the upfront costs: The Tesla Model 3 retails for $37,990. The Toyota Camry XLE retails for $29,870. We can start modeling the total costs of each car as follows:
Tesla: Total Cost=37990
Camry: Total Cost=29870
The Toyota Camry is about eight thousand dollars cheaper. So, it looks as if we should buy the Camry.
But … not so fast.
We can use the carbon counter tool (with the reasonable settings used in the “customize” tab here) to estimate the yearly cost of driving these vehicles. For the Tesla, the estimated price for electric charging and maintenance is $1,016 per year. For the Camry, the estimated price for fuel and maintenance is $2,038 per year. How would you include these estimates in the total cost model that we built above?
Answer: Where x represents the number of years you’ve owned the car:
Tesla: Total Cost= 1016x+37990
Camry: Total Cost= 2038x+29870
Let’s briefly break down the mathematical model of total cost ($) for the Tesla:
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If you bought the Tesla today, what would the value of x be? What would the total cost be? Hint: No years have passed.
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If you’ve had the car for one year (x=1), what is the total cost? How about after two years (x=2)? After three years (x=3)? By how much is the total cost increasing each year?
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If you’ve taken some algebra classes, the above models might remind you of a familiar formula: y = mx+b. In this formula, m represents the slope or rate of change; b represents the y-intercept or starting value. For the Tesla model, what is the rate of change (in $ per year)? What is the starting value (in $)?
Let’s imagine that we’ll own these cars for their entire lifetimes, which we’ll assume to be about 12 years. Which car will have a lower total cost over 12 years of ownership? Exactly how much cheaper will it be? Use the mathematical models to find your answers. Show your work.
Answer:
Tesla: Total Cost=1016(12)+37990
=12192 + 37990
= $50,182
Camry: Total Cost=2038(12)+29870
=24456 + 29870
= $54,326
The Tesla is less expensive by $54,326 - $50,182 = $4,144
Answer the following in writing:
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In two sentences or less, explain to someone what made the Tesla cheaper than the Camry over time.
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What does the above example show about the costs of electric cars?
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How does the example illuminate why General Motors may have decided to sell only zero-emission vehicles by 2035?
Using Tables and Graphs
Sometimes we might consider owning cars for a shorter period. This raises the question: How long would I need to own a Tesla for it to “pay off” and become less expensive than the Camry? This question can be answered directly through algebraic solutions (e.g., substitution and elimination). Another method is to explore a table of values and graph:
Provide a written response: At approximately what year does the Camry start to become more expensive to own than the Tesla? Use evidence from both the table and the graph to support your answer.
Answer: At about eight years of ownership, the Camry surpasses the expenses of the Tesla in the table of values. In addition, the lines in the graph representing our models “cross” at Year 8 — at this point, the Camry’s higher rate of cost per year finally “catches up” with the Tesla (which had a higher starting cost).
Model the Emissions (Student Practice)
In addition to costs, we can also model the total greenhouse gas emissions associated with each vehicle over time. Although Tesla is better for the climate over time, the Camry puts out less emissions during its production. Answer the next questions to explore the emissions trade-offs for these vehicles:
1. The upfront vehicle and battery production for the Tesla creates emissions in the amount of 11,074 kg CO2-eq (this unit of measurement means “emissions equivalent to the global warming potential of 11,074 kilograms of CO2”). The power sources that create electricity to charge the battery expend about 1,368 kg CO2-eq per year. Create a mathematical model to determine total emissions (in kg CO2-eq) for the Tesla:
Answer: Where x represents the number of years you’ve owned the car:
Tesla: Total Emissions =1368x+11074
2. The upfront vehicle production for the Camry creates 7,402 kg CO2-eq of emissions. The production of fuel and burning of fuel needed to drive the car produces 4,525 kg CO2-eq per year. Create a mathematical model for the Camry:
Answer: Where x represents the number of years you’ve owned the car:
Camry: Total Emissions=4525x+7402
3. Approximately how many years will it take for the Camry to produce more total emissions than the Tesla? Show your work using a table of values and a graphical sketch.
Answer: The Tesla becomes more emissions-efficient shortly after the first year of ownership. By Year 2, according to the table, the Tesla has lower overall emissions. In the graph, we see the lines “cross” shortly after Year 1, as the Tesla begins to have lower total emissions because of its lower rate of emissions per year. Hands down, the Tesla is the more environmentally-friendly option, and it grows more environmentally friendly with each passing year.
Going Further
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Analyze your own family’s car. Use the carbon counter tool to compare your own family’s car with the emissions and costs of electric, hybrid or gas alternatives. Find the car model you’d like to compare by using the search bar. Then, use the “bar chart” feature to directly compare the selected cars. Change different model assumptions (miles driven per year, tax benefits for electric vehicles and so forth) to see how your estimates change. Note: The tool displays data for only new cars.
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Broaden your analysis by looking at city-level data. Explore this interactive map of auto emissions in different cities in the United States. Find your local area on the map to analyze auto emissions in your community. Use the line charts to examine the relationship between total emissions and emission rates (per person).
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Look for trends among many electric and gas cars. See the recently published “What’s Going On in This Graph?” to learn more about comparing the cost and efficiency of many cars. What trends do you notice among the relative cost and fuel efficiency of gas, hybrid and electric cars?
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Write a proposal to your family about buying an electric car. Explain the trade-offs between cost and efficiency compared with a gas-burning vehicle. Then, make a case for why buying an electric vehicle would be better or worse in your family’s specific situation.
This lesson was written by Dashiell Young-Saver, who is a high school statistics teacher and the founder of Skew The Script. Important contributions were made by Matthew DiSorbo and by Sharon Hessney, who writes the Learning Network’s weekly feature “What’s Going On in This Graph?”
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February 16, 2021 at 04:00PM
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Gas or Electric? Thinking Algebraically About Car Costs, Emissions and Trade-offs - The New York Times
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