Introduction / Background
Students often have difficulty making unit-based calculations, plotting data, and analyzing graphs. These basic skills are essential to all disciplines of science. This case study is designed to help students improve these skills in the context of a relevant and timely issue—saving money on gasoline and protecting the environment.
The case begins with a family’s dilemma about which car to replace to save money on gas. One way to reduce fuel consumption is to buy a more fuel efficient vehicle. In the U.S., fuel efficiency is measured with a “miles per gallon” or MPG rating. The problem with this rating is that people falsely assume that every increase in MPG gives an equal decrease in the amount of gas consumed in a certain distance. In other words, there is a common misconception that gas consumption decreases as a linear function of MPG rating (Larrick & Soll 2008). This case study is based on this misunderstanding.
There is actually a non-linear relationship between gas consumption and fuel efficiency. However, this concept is difficult to visualize without graphing the data and making a few mathematical calculations. A graph of gas consumption or cost versus MPG ratings shows a downward sloping curve. Looking at the graph helps students see that a small gain in MPG at the lower end of the fuel efficiency scale results in much larger fuel savings than a large gain in MPG at the higher end.
Another way to look at the data is to invert the MPG ratio to “gallons per miles” (GPM) driven (Larrick & Soll 2008). When cost to drive a certain distance is plotted as a function of GPM, the graph is a straight line. This graph clearly shows that small increases in mileage for the least fuel efficient vehicles result in large fuel and cost savings. Understanding this will help people see the value of replacing their least efficient vehicles to save on fuel costs.
In completing this case, students learn how to analyze fuel efficiency in terms of “gallons per miles” driven. The activities associated with the case correct common misconceptions about MPG ratings and help students learn graphing skills using graphing software. Finally, students see how a non-linear graph can be converted into a straight line. A homework assignment about how speed relates to travel time is also provided that reinforces for students what they have learned about non-linear graphs.
Objectives
- Practice dimensional analysis and mathematical calculations.
- Analyze data.
- Create graphs using graphing software programs.
- Practice interpreting inverse or in-direct graphs and direct or linear graphs.
The case was developed for use in a high school general science course. It could be adapted for use in introductory physics, chemistry, algebra, or environmental science courses at the high school or college level. It would be especially appropriate for a general science course for non-majors and also for science, technology, engineering, and math (STEM) workshops for high school students.
Classroom Management
The case is designed to be taught early in the first semester of a general science course to help explain different types of graphs and to teach students how to use graphing software. The case consists of four parts followed by a take-home assignment. The case itself could be completed in 1.5 to 2 class periods.
Part I – A Drop in the Bucket introduces the characters and story line, and takes about 10 minutes to complete in class. After answering the questions, poll the class for their opinions about Question 1 and record the responses on the board. Students will most likely choose to trade in the Corolla for the hybrid sedan. Question 2 brings safety and vehicle cost issues into the discussion.
Part II – Gas Guzzler involves making calculations about the cost and quantity of gas consumed by four different vehicles. This would make a good homework assignment or, if time permits, a good in-class, small-group activity. Discuss the data with the class before advancing to Part III of the case. Have students explain their reasoning for which combination of vehicles would save the most money on gasoline.
Part III – The Graph Explains All is a step-by-step graphing exercise that shows students how to use a graphing program such as Logger Pro or Excel. This part of the case is designed to be done in class, with students working independently or in pairs at a computer. Students not only learn how to graph data, they also learn how to transform data, i.e., “crunch numbers” with the computer. The activity helps students learn about inverse (non-linear) graphs and interpret the connection between fuel efficiency and fuel cost.
Part IV – Straighten Up! demonstrates how data can be transformed to convert an inverse (non-linear) graph into a straight line. When a graph is linearized, there is no numerical change to the data. Instead, the “miles per gallon” data is reciprocated. This makes the graph look different. But, the resulting function/equation is equivalent to the nonlinear equations discussed in Part III. Students should begin to realize that thinking in terms of gas consumed over a specified distance is a less confusing measure of fuel efficiency than the MPG rating.
Homework Assignment: The homework assignment (available in MS Word format as gas_mileage_hw.doc) should be handed out after the case is completed to help reinforce the graphing skills learned above. It will also help students think about how speed effects travel time, fuel consumption, and safety. This activity demonstrates for students the relationship between fuel efficiency (mpg) and speed (mph). In other words, cars are generally the most efficient between 40 and 60 mph. Speeds greater than 60 mph result in decreased fuel economy. Because of issues such as fuel economy and increased safety, students will begin to understand the reasoning behind government-imposed reductions in highway speed limits. Answers to the homework questions are contained at the end of the Answer Key.
Assessment
In addition to assessing student participation in the case study in class, the homework assignment serves as an additional assessment tool for this case. In this assignment, students use what they learned in the case to make similar mathematical calculations and non-linear graphs.
Blocks of Analysis
Fuel Efficiency
Different vehicles use different amounts of fuel (gasoline, diesel, etc.) depending on engine properties, the weight/shape of the vehicle, how the vehicle is operated, etc. (How Stuff Works 2000). Some cars are more efficient than others, which mean that they use less fuel to go a certain distance. The term “fuel efficiency” is usually represented as a ratio. In the U.S., fuel efficiency is measured as “miles per gallon” or MPG. In Canada and Europe, fuel efficiency is measured as “liters per 100 kilometers” or “liters per mile” (Wikipedia: Fuel Efficiency n.d.).
Petroleum
Petroleum, which comes from crude oil, is used to make a variety of products. Most petroleum is used to make gasoline for cars, jet fuel, diesel fuel, and home heating fuel. It is also used to make plastics, synthetic fibers, detergents, cosmetics, and pharmaceuticals. According to the Energy Information Administration, the U.S. consumed a total of 20.7 million barrels of petroleum per day in 2007 to make petroleum-derived products (Energy Information Administration 2008). That same year, nearly half of this volume, or 9.29 million barrels, was consumed each day as motor gasoline. Worldwide, about 85 million barrels of petroleum were consumed per day in 2006, with the U.S. and China consuming the most oil (Energy Information Administration 2009).
Renewable Fuels
Currently, the world relies heavily on non-renewable, fossil fuel resources for powering vehicles. Rising gas prices and concerns about supply have stimulated the search for alternative energy sources. One such renewable resource is biomass, which refers to living or recently living materials, such as grassy plants, algae, and wood chips. Biomass is used to make biofuels, such as ethanol and biodiesel.
In recent years, more renewable fuels have become commercially available and are often touted as being better for the environment than fossil fuels. This is because biofuels recycle carbon dioxide. In other words, plants remove carbon dioxide from the air as they grow. Then, cars release carbon dioxide back into the air when the fuel burns. The next season’s crop removes that carbon dioxide from the air. In contrast, burning petroleum-derived fuels just adds carbon dioxide to the atmosphere.
Though biofuels have the potential to be more environmentally friendly than petroleum, there are still many other concerns about this resource. For example, various research studies have indicated that it takes more energy to make biofuels than we get from burning them. Land use issues have raised concerns about how biofuels will affect food prices. In addition, people are concerned about the environmental impact of using fertilizer and water to grow the plants for biofuels.
Greenhouse Gases and Climate Change
When petroleum products and biofuels are burned, they release carbon dioxide and other pollutants, such as carbon monoxide, nitrogen oxides, and particulate matter, into the air. These emissions contribute to air pollution. Many environmental laws have been passed to limit the emissions from cars and trucks because of the damaging effects of these pollutants on the environment and human health.
Combustion of fossil fuels produces the majority of greenhouse gas emissions. Greenhouse gases are chemicals in the Earth’s atmosphere that absorb the infrared radiation (heat) that is reflected from the Earth’s surface, trapping this heat in the atmosphere. Common greenhouse gases include carbon dioxide, water vapor, methane, and nitrous oxide. These gases play an important role in maintaining the temperature of the Earth, keeping it warm enough to sustain life. However, rising levels of greenhouse gases are contributing to global warming. The Earth has warmed by 1 degree Fahrenheit in the last century (PBS Online 2009). Scientists believe that this warming has been caused by human activity (U.N. Intergovernmental Panel on Climate Change n.d.), especially because of the increased emission of carbon dioxide from car exhaust and nitrous oxide and methane release from industrial plants. Global warming affects climate change around the world. If greenhouse gases are not reduced, we could see flooding in coastal areas and less snow pack and more intense heat waves in North America (PBS Online 2009).
Ideal Gas Law
One way to reduce greenhouse gases is to drive less. Another option is to drive more efficient vehicles that burn less fuel and release fewer emissions. Though carbon dioxide is naturally found in the air, it is also released from burning fuels, and atmospheric levels have increased by 36 percent in the last century, as estimated by the U.S. Environmental Protection Agency (2008). Despite this fact, most people are unaware of the quantity of carbon dioxide that their cars emit because it is impossible to visualize the volume of this colorless, odorless gas. This case study could be related to a lesson on the ideal gas law by using this equation to estimate the volume of carbon dioxide emissions from various vehicles. This volume could then be converted into something tangible, such as the volume of semi truck trailers or train boxcars. This image by itself for a single car may not seem that significant until students translate this information to the amount of carbon dioxide emitted each year from the total number of passenger vehicles in the U.S., which was 250,851,833 in 2006 (Wikipedia: Passenger Vehicles in the United States n.d.). This visual representation of carbon dioxide emissions would help students understand the magnitude of this growing problem.
Answer Key
Answers to the questions posed in the case study are provided in a separate answer key to the case. Those answers are password-protected. To access the answers for this case, go to the key. You will be prompted for a username and password. If you have not yet registered with us, you can see whether you are eligible for an account by reviewing our password policy and then apply online or write to answerkey@sciencecases.org.
References
- Energy Information Administration. 2008. Petroleum Products Consumption. Last updated Sept. 2008. Last accessed July 2009.
- http://www.eia.doe.gov/neic/infosheets/petroleumproductsconsumption.html
- Energy Information Administration. 2009. Petroleum Basic Statistics. Last updated July 2009. Last accessed July 2009.
- http://www.eia.doe.gov/basics/quickoil.html
- How Stuff Works. Sept. 27, 2000. “What speed should I drive to get maximum fuel efficiency?” Last accessed July 2009.
- http://auto.howstuffworks.com/question4771.htm
- Larrick, R.P., and J.B. Soll. 2008. “The MPG illusion.” Science 320(5883) 1593–1594.
- PBS Online NewsHour Science Report: The Global Warming Debate. March 17, 2009. Last accessed July 2009.
- http://www.pbs.org/newshour/indepth_coverage/science/globalwarming/explainer_update.html
- U.N. Intergovernmental Panel on Climate Change. IPCC Assessment Reports. Last accessed July 2009.
- http://www.ipcc.ch/
- U.S. Environmental Protection Agency, Climate Change Science: “Atmosphere Changes.” Last updated Dec. 17, 2008. Last accessed July 2009.
- http://www.epa.gov/climatechange/science/recentac.html
- Wikipedia: Fuel Efficiency. n.d. Last accessed July 2009.
- http://en.wikipedia.org/wiki/Fuel_efficiency
- Wikipedia: Passenger Vehicles in the United States, Last accessed July 2009.
- http://en.wikipedia.org/wiki/Passenger_vehicles_in_the_United_States
Additional Web Resources
- Fuel Economy Guides (MPG data from EPA estimates). Last accessed July 2009.
- http://www.fueleconomy.gov/feg/FEG2000.htm
- “Warner: Reduce speed limit to save gas?” Associated Press, July 4, 2008. Last accessed July 2009.
- http://www.politico.com/news/stories/0708/11525.html
- http://www.bloggingstocks.com/2008/07/07/u-s-sen-john-warner-talks-up-55-mph-national-speed-limit/
- Wikipedia: National Maximum Speed Law. Last accessed July 2009.
- http://en.wikipedia.org/wiki/55_mph
- Treehugger: “55 MPH: It's time to bring it back,” by Lloyd Alter, Feb. 6, 2008. Last accessed July 2009.
- http://www.treehugger.com/files/2008/02/55_mph_its_time.php
- MPG for Speed: “Speed Kills MPG.” Last accessed July 2009.
- http://www.mpgforspeed.com/
- Fuel Economy: “Driving More Efficiently.” Last accessed July 2009.
- http://fueleconomy.gov/feg/driveHabits.shtml
- “Study: As Gas Prices Go up, Auto Deaths Drop,” by Joan Lowy, Associated Press, July 11, 2008. Last accessed July 2009.
- http://www.usatoday.com/money/autos/2008-07-11-auto-deaths-gas-prices-study_N.htm
- Fatality Analysis Reporting System: Fatalities and Fatality Rates by State. Last accessed July 2009.
- http://www-fars.nhtsa.dot.gov/States/StatesFatalitiesFatalityRates.aspx
Acknowledgements: This case was published with support from the National Science Foundation under CCLI Award #0341279. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Date Posted: August 26, 2009.
Originally published at http://www.sciencecases.org/gas_mileage/notes.asp
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