# Mathematics of Energy Efficiency: Use Less, Save More

byKenya L. Lawrence

# Introduction

In the media and on the political stage, the main topic of discussion is renewable energy and its environmental and economic benefits. There is such a great urgency to switch, and to switch now! So, why doesn't our society make the leap and switch from non-renewable to renewable energy sources? Well, it's not that easy. To fully understand requires a deeper knowledge of and exposure to what are renewables and what is their overall impact to our communities. Thus, it is our job as educators to be informed and to provide awareness of non-renewable and renewable energy options early and often to our students. Gaining awareness of the options gives our students the ability to make conscious decisions about their role in the local/global effort to be environmentally friendly. In this unit, students will make real-world mathematical connections to the origin of our electricity, the impact of our use on the environment, the potential of wind energy and biodiesel, and the evaluation of our energy practices. Through demonstrations and inquiry-based learning opportunities, students will acquire and retain mathematical concepts which include writing, solving and graphing linear equations and inequalities, systems of linear equations and inequalities, and direct and inverse relationships. Students will also analyze data statistically using box-and-whiskers plots, mean absolute and standard deviation, z-scores, and curve of best-fit.

Non-renewables are energy sources that can be used up like gasoline in our cars. Gasoline is made from petroleum, remains of creatures that lived many years ago and are now buried deep in the earth, subjected to high temperatures and pressures which caused physical and chemical changes, and for that reason it cannot be replenished. Examples of non-renewables include petroleum, coal, nuclear and natural gas. Renewables are energy sources that can replenish itself like wood for a fire. Wood comes from trees; new trees grow to replace those harvested, hence, it is renewable. Examples of renewables include biomass, wind, solar, geothermal, and hydropower. Each of these proposed renewable energy sources is expensive to implement and has unique environmental challenges. Combine this fact with no widespread infrastructure to use supposed sources and you arrive at the core reason why we have not made the leap and switch from non-renewables to renewables. Consequently, my students will come to the inescapable conclusion that they are limited to our current source of energy. However, by the end of the unit, perhaps my students will realize that with a few personal adjustments in habit we can use energy wisely and carefully through energy conservation, also known as the 'fifth fuel.' 1 Energy conservation is decreasing energy use, not depriving yourself of life's simple pleasures. However, according to Daniel Yergin, author of The Quest: Energy, Security, and the Remaking of the Modern World, it is "applying greater intelligence to consumption, being more clever about how energy is used—using less for the same or greater effect." 2

So, who has time for energy conservation? WE ALL DO!! My Algebra I students meet in 90 minute block classes and are mostly comprised of 9 th graders with a few 8 th and 10 th graders who will discover we can conserve with just a few minor adjustments. This unit will be implemented throughout the academic year following the Algebra I pacing chart established by Richmond Public Schools. At the culmination of this unit, my students, newly named Energy Inspectors, will conduct an audit of their home by making a list of items used in their home or on the go that use electricity and evaluate the cost of usage at a particular point in time and over a year's time, taking into account the time of day, year or season in which they are most likely to use these items. Then, they will write a proposal to their caregivers including a top ten list of ways to conserve energy.

# Rationale

Currently, I am in my seventh year of teaching secondary mathematics, specifically Algebra I and II, to inner city youth at Franklin Military Academy (FMA), the nation's first public military school founded in 1980. FMA is a 6 th -12 th grade choice-school located in Richmond, Virginia. Students apply to FMA in the 6 th grade and again in the 9 th grade. Middle school students are not guaranteed a slot in the high school, but, are strongly encouraged to apply. Through an extensive application process, we accept children all over the city with various socioeconomic backgrounds and family dynamics. So, our students are neither the cream of the crop nor your child in need of strong discipline. A high percentage of our students are economically disadvantaged and/or are in circumstances in which their caregivers are incapable of helping with homework, researching, preparing for tests, college or the work force. In addition, FMA just concluded our 2 nd year of school improvement with an emphasis on mathematics. We are awaiting the final results of testing this year; however, the preliminary results suggest we continue to struggle in mathematics.

After reflection on my teaching practices, I plan to employ inquiry based learning activities more vigorously to increase engagement, foster critical thinking and, therefore, increase retention of mathematical concepts and skills. I hope to achieve this through interdisciplinary planning under the theme of energy. Some activities in this unit are repeated in a chemistry or physical science class; so, do not feel like you are stealing the science teacher's thunder. Most science teachers will appreciate the math teacher giving students a foundation of scientific concepts and principles in context. Also, to increase achievement across grade levels, my colleagues and I are having meaningful discussions about vertical articulation, early intervention, and professional development opportunities that encourage collaboration amongst mathematics and science teachers.

President Obama intends for energy efficiency to play a major role in cutting carbon pollution while keeping our economy strong. 3 Energy conservation is not free. It requires investment of time and money. The Obama administration's energy platform is focused on energy efficiency investments as a tool to strengthen our economy and environment. 4 As a result, we need individuals who are trained with the necessary knowledge and skills for a career in residential and commercial energy efficiency related occupations. This unit will expose my students to this sector of our economy. This year, I want the industry of energy efficiency to be considered as a possible career pathway.

# Background

When I come home, the first thing I do is flip the light switch and my lights are on, like magic. Assuming I pay the electricity bill on time, what process allows the lights to come on upon the flip of a switch? Likewise, I am able to plug in one of my various appliances and I can bake a peach cobbler, wash my clothes, or watch television. I perform these activities without regard for where the electricity comes from or the impact of my use on the environment. For many adults and students, this is their reality. The background information will answer the question "Where does our electricity come from?" The focus is in on Virginia; however, you may use some of the same resources to find information specific to your state. Then, I will provide information about the environmental effects of coal mining and some renewable energy options namely biodiesel and wind. Finally, my research will look at reasons and ways to increase energy conservation practices related to electricity and fuel usage. While I go through the background, perhaps you may get some great ideas for lessons/activities you can do beyond what is presented in the lesson plans.

My seminar, Energy Sciences, helped me to understand the scientific and social factors associated with non-renewables and renewables. Yergin describes the evolution of renewable best when he says "it is one of innovation, entrepreneurial daring, political battles, controversy, disappointment and despair, recovery and luck." 5 While the campaign for renewables has restarted after somewhat of a hiatus, it is important to recognize all primary energy sources have positive and negative impacts on the environment. We can use algebraic reasoning to model, project and verify current concerns with both types of energy sources. From a social aspect, we can consider Dr. David MacKay's comment which cleverly outlines the realistic views of the British on renewables; I imagine the sentiments are similar in the U.S.:

Wind: Not in my backyard! Wind farms? "No, they're ugly noisy things." Shallow offshore wind: Not near my birds! Offshore wind? "No, I'm more worried about the ugly power lines coming ashore than I was about a Nazi invasion." Biomass: food, biofuel, wood, landfill gas: Not in my countryside! 6

How do Virginian's obtain their electricity?

First, let's define the terms energy, power and electricity. Energy is the ability to work. There are two types of energy: kinetic, also known as working energy, and potential, also known as stored energy. Energy is in everything and comes in different forms: heat (thermal), light (radiant), motion (kinetic), electrical, chemical, nuclear and gravitational. Power is the measure of energy used over time; power, P, varies directly with energy, E, and inversely with time, t, P = E/t. Electricity is the flow of an electrical power or charge through a conductor. It is actually a secondary source of energy or energy carrier because it is the result of the conversion of primary sources of energy such as coal, wind, solar energy, etc.

Figure 1. Breakdown of Virginia's energy sources 7

Coal and natural gas make up 57% of the primary energy sources of Virginia's total electricity production. The rest is nuclear power and renewables. Virginian's pay an average retail price of 8.87 cents per kilowatt-hour and are ranked 23 rd lowest cost in the U.S. The unit will specifically look at the production and consumption of coal.

Dominion Virginia Power runs Chesterfield Power Station in Chesterfield, VA, about 15 miles south of Richmond on the James River. It is the largest fossil-fueled power station in the state and generates 1,600 megawatts (MW). The average daily consumption of coal is 8,400 tons. Nevertheless, Virginia imports more electricity than any other state except California and the demand keeps growing. 8

Coal is plant material buried deep in the earth many years ago, subjected to high temperatures and pressures which caused physical and chemical changes, altering the plant material into peat and then into coal. Coal reserves are discovered through geological explorations which include creating a geological map of the area, conducting geochemical and geophysical surveys, and then, drilling. The area will become a mine if it is large enough and of sufficient quality so the coal can be retrieved at a reasonable cost. Coal is mined in two ways: surface (which includes mountain top mining) and underground mining (which includes strip mining). Once mined, coal is washed of its impurities, treated depending on its intended use and arrives at the power station by truck, train or ship.

So, how is coal, a primary energy source, processed to get electricity, a secondary energy source? It is placed into a hopper where it is crushed into small chunks and the conveyor belt moves the coal to the coal pile which holds a 30-day supply. From the coal pile, the conveyor belt moves the coal to the station and feeds it into a pulverizer that grinds the coal into fine pieces, like baby powder. Coal contains potential chemical energy and is combustible, which means it can catch on fire and burn easily. The powdered coal is blown into the combustion chamber of a boiler where it instantly combusts and releases kinetic thermal energy. The walls of the boiler are lined with tubes which contain purified water. The heat turns the water into high pressure steam which is used to turn the turbine, a massive drum with propeller like blades. This transforms kinetic thermal energy into kinetic motion energy. The turbine turns a magnet wrapped in copper coil to produce electricity in the generator, turning kinetic motion energy into kinetic electrical energy. Electricity is sent to a transformer that increases the voltage to 30,000-400,000 volts (like you increase water pressure). Before the power can be delivered to your homes, the electricity is lowered to safer voltage at an electrical substation where a transformer reduces the voltage to 100-250 volts (like you decrease water pressure). Finally, the electricity moves over distribution lines to your home. 9 Flip the light switch and the lights come on. Surprisingly, while living things are able to store energy from the sun, mankind has yet to find an efficient way to store electricity, so the national grid must balance supply and demand on a minute-by-minute basis.

## Coal Mining Experiment

Coal mining companies are often the largest employers in the area which makes their existence essential to the families in the area. On the other hand, their existence means large areas of land are disturbed. This creates environmental challenges including soil erosion, dust, noise and water pollution, removal of all vegetation, and release of greenhouse gases such as methane and carbon dioxide. As a result, the coal industry is charged with rehabilitation, also called reclamation, of coal mine lands during and after use. Power companies are also charged with reducing release of pollutants and greenhouse gases such as such as mercury and sulfur when burning coal. New technologies are constantly developed to reduce the environmental impact of both coal mining and burning. 10

Students will simulate mountain top coal mining and its effect on the surrounding land. They will use algebraic models to describe the effects of production and consumption of coal on our environment using greenhouse gas emissions data from the Energy Information Administration (EIA). Problems may contain conversions depending on how the data are presented.

## Wind Energy: Building windmills

Wind is air in motion created by the unequal heating of the earth's surface by the sun's radiant energy. Think of the power exhibited by the wind speeds of a hurricane or a tornado. Imagine we are able to harness this energy, a renewable source, to power communities. In fact, this idea is not so novel. In the mid-west of the U.S., windmills are used to grind grain, pump water and provide electricity. The wind's velocity, which can be measured with an anemometer, is a factor in calculating the amount of electricity wind turbines can generate. Wind power, P, varies jointly with half of air density,Ρ, area swept by the turbine blades, A, and the cube of the velocity, V, P = (1/2)ΡV 3. Air density is the mass per unit volume of earth's atmospheric gases and measures 1.25 kg/m 3 at sea level and 68 °F. Windmills are tall, and the air density decreases at higher altitudes and higher temperatures. The air density decreases by 3% for every 1000 additional feet in height and 1% for each additional degree in temperature. 11

On June 25 th, President Obama introduced his Climate Action Plan. The document lays out his vision to slow the effects of climate change. Wind energy plays a significant role in this endeavor. 12 The Department of Energy is planning to build a research facility off the Virginia coast near Virginia Beach, Va. The intent of the research facility is to test technologies such as remote sensing designed to determine the potential power of offshore winds. Researchers hope the results of their findings will lead to increased capital from investors by providing valid data, thus, increasing production of wind plants, also called wind farms. 13 Virginia has a strong wind resource off the coast. Preliminary research completed by Virginia Coastal Energy Research Consortium showed almost 10% of our electrical power can come from wind energy. Onshore, Dominion Virginia Power is currently developing the Bluestone River Wind Project in Tazewell County, southwest Virginia, which could produce nearly 80 MW. 14 If Dominion is successful, Bluestone River Wind Project will provide electricity to approximately 80,000 people, that is one megawatt per 1,000 people. 15

Significant environmental benefits to wind energy include no release of harmful greenhouse gases and it does not interfere with agriculture. Another significant benefit is the ability to produce power in remote locations all over the world. There are generators transforming the wind's kinetic energy into electrical energy for people who otherwise would not have access to electricity. However, like coal production and consumption, there are negative environmental impacts of wind farms. For example, other than the coast, most of the windy areas in Virginia are in national parks, like Blue Ridge Parkway. 16 This is an obvious conflict of interest; after all, national parks were established "to conserve the scenery and the natural and historic objects and the wild life therein and to provide for the enjoyment of the same in such manner and by such means as will leave them unimpaired for the enjoyment of future generations." 17 Other impacts include the consistency of wind power, as it is not always windy which makes it difficult to implement into existing power grids (recall energy supply and demand must be balanced by the minute), the aesthetics to the landscape or ocean view, the harm to birds and bats, and, in some cases, the noise. In addition, there is a large transportation cost because the majority of wind turbines are mass-produced in Europe or in the Midwest. Hence, there is a large need for skilled workers to support the development of a wind energy supply in our region. 18

Students will build a device that can measure wind speed and a windmill that can do work by picking up an item. Students will then calculate the amount of work and power needed to pick up the weight. Work is the measure of the weight lifted through height; Work, W, varies jointly with mass, m, acceleration of gravity, g, and height, h, W = mgh. Students will also calculate the wind energy created by a fan set on different speeds: low, medium and high.

## Biodiesel: Alternative Fuel

Have you heard the story about the guy who ran his car on Canola oil? Too good to be true, right? Correct, that guy did a little more to the composition of the Canola oil than just putting it into his fuel tank. The simple chemistry involved using oil, methanol and sodium hydroxide. The product is biodiesel and glycerol which separates well with biodiesel on top and glycerol, a heavier substance, at the bottom of the container. Methanol and sodium hydroxide are dangerous substances. Glycerol is then removed because it will damage engines. Water is used to wash it out of the fuel, leaving biodiesel only. With no engine modifications, biodiesel can be used to power any diesel engine.

Biodiesel is a renewable fuel produced from vegetable oils, animal fats or recycled restaurant grease. In the early 1900's, Dr. Rudolf Diesel demonstrated the diesel engine and ran it with peanut oil. Diesel engines were marketed to farmers with the notion they could grow their own fuel. Indeed, farmers ran their large trucks, tractors and machinery on vegetable oil until the arrival of low cost and convenient petroleum diesel. Since 2004, West Point, VA has housed a biodiesel plant built by Pacific Biodiesel that uses soybeans made in Virginia. 19 Governments back the manufacturing of biofuel due to concerns about the environment and security of oil. Using biofuels instead of gasoline reduces carbon emissions, creates jobs and keeps money within the community. 20 While biodiesel is good for your vehicle because it provides increased fuel lubricity and extends the life of the engine, there are some concerns. Biodiesel gels like petroleum diesel does in cold temperatures. This can be avoided by purchasing biodiesel treated for winter use. There is also a conflict with the food industry because of increasing demand on the industry to supply both food and fuel to the world. The U.S. Department of Agriculture expected world grain use to reach 20 million tons in 2006, 14 million tons for the fuel industry in the U.S. and 6 million tons to supply the world's food necessities. 21 As a result, the price of food is up and it is the highest it has been in 20 years, since almost everything we eat can be used to create biodiesel. 22 The World Bank is concerned about high prices putting an additional 44 million people into poverty. 23 Moreover, most governments do not have measures in place to limit the amount of land used for biofuels as not to compete with the food industry, thus, strengthening hunger in impoverished countries.

Students will make biodiesel from old and new oil then perform titrations to determine if the biodiesel has the correct pH. Students will use compound inequalities to describe the acceptable range of pH, fuel efficiency of biodiesel, crop production and sales, and oil content of crop. Students will evaluate the sustainability of a crop by calculating the energy return on energy investments to ensure that the energy input is significantly less than the energy output. Students will also use systems of equations to model and solve problems related to the production and production cost of biodiesel.

## Energy Efficiency: Use Less, Save More

Figure 2. Breakdown of how energy is used in Virginia 24

Dominik Saner, researcher with the Institute of Environmental Engineering in Switzerland, says residential and private transportation energy use accounts for more than 70% of carbon dioxide emissions. 25 The energy Virginians use in the residential and transportation sector make up 57% of the total energy generated. The future of energy in Virginia requires a comprehensive approach that combines efficiency programs with renewable energy sources. Many of us are not in situations where we can construct a windmill for our individual use. So, we are left to think of ways to make our current non-renewable primary energy sources last longer by decreasing the demand for energy. If we are doing our part at similar commitment levels, the ratio of consumption described could stay the same, but the total amount of consumption is reduced.

There are a variety of ways to conserve on the road that does not include converting your car to a diesel engine and making your own fuel. If you need incentive, gas in New Haven, Connecticut is currently \$3.93/gallon. Gas in Virginia is about sixty cents cheaper, temporarily. So, until there are significant changes in the manufacturing of automobiles, we are faced with the high gasoline prices. One way to beat the cost is to drive at slower constant speeds and accelerate/decelerate at a slower rate when merging into traffic on a highway or local road. We waste gasoline driving at higher speeds due to rolling and internal vehicle friction and drag force. Also, turn off your car instead of idling; restarting your car takes the same amount of gas used in 0.2 seconds of idling. Finally, keeping your tires properly inflated will save money and make your gasoline last longer. 26

Students will perform an energy audit of their home, write a proposal to their caregiver making suggestions and create a flyer describing 10 things we can do to conserve energy. We will use the formula: C = WNHDP/1000, where W is the power in Watts consumed by a device, N is the number of devices in use, H is the number of hours per day the device is in use, D is the number of days in a year when the device is in use, P is the price of electrical service in dollars per kilowatt-hour (\$0.0887), and C is the total cost in dollars per year for using the device. We divide by one thousand to convert Watts to kilowatts. This formula accounts for other factors that simpler energy formulas do not take into account.

# Teaching Strategies

I need something to connect our students on day one to our class and the content in a positive way. Therefore, I will start the year and unit with a survey. Dr. Judy Willis, author of Learning to Love Math: Teaching strategies that change student attitudes and gets results, believes collecting information about your students is an effective way to get what we need to link them to topics through interest or integration of information about them into math problems. 29 My survey will serve a dual purpose of collecting biographical information about my students' prior experience in math class and an assessing their knowledge of energy and energy efficiency habits. The survey may pique students' interest in how the information will be used. Teachers can encourage real-world connections through inductive reasoning by having students make predictions and, as we go through implementation of the unit, provide prizes to those who were accurate or close. Making real world connections integrates creative problem solving, communication, collaboration, and critical analysis, all of which are 21 st century skills we want our students to have. 30

According to Willis, 50% percent of high school students living in large U.S. cities drop out of school. This creates a phenomenon in which there is a likelihood the parents of our students will have graduated from high school rather than our students graduate. 31 One reason for high dropout rates is boredom and relevance of content to their life. Speakers can help students recognize how the math concepts and skills learned in class relate to careers they might find appealing. 32 Dominion Virginia Power's Speakers Bureau conducts free educational presentations to schools. I will invite them to my class for two visits. During the first visit, the speaker will focus on the basics of electricity (how it is generated, distributed and used) and renewables (how they play a part in the diverse mix of energy sources). The second visit will focus on energy conservation and careers with one of the nation's leading energy companies.

Additional teaching strategies will include demonstrations, hands-on-activities to include inquiry-based learning opportunities, and discrepant events that illustrate scientific principles related to energy. Grabbing the attention of our students at the beginning of the lesson and throughout the lesson through one of these media creates cognitive dissonance. By creating disequilibrium, our students innately feel this vehement need to find a greater understanding of why the outcome did not match their prediction. Students are totally engaged at this point and it is our job to pounce on the opportunity to educate directly or indirectly. Some of the activities I can facilitate/perform and explain while others will be explained by our science teachers.

# Lesson 1: Coal Mining Experiment

For the Warm-Up, find out what students know about the origin of their electricity, environmental concerns related to energy, and energy efficiency practices by giving an energy survey. You can use the information to determine groupings. Students will then view a video that connects coal to electricity. 33

## Activity 1: Coal Mining Experiment

For this activity, groups will consist of one person who is fairly knowledgeable about energy and another person who may not be as proficient. Students will simulate mountain top coal mining using a chocolate chip cookie and a toothpick or paperclip. The cookie is the mountain, the chocolate chips are the coal and the toothpick or paperclip is the retrieval tool. Students will estimate the number of chips they can retrieve without picking up the cookie; remind them they cannot pick up a mountain. In the real-world, the area will only become a mine if it is large enough and of sufficient quality so the coal can be retrieved at a reasonable cost. Have students write down the factors they considered in order to arrive at their estimate. Then, students will use the toothpick or paperclip to retrieve the chocolate chips. After they retrieve all the chips they can, have students compare their actual number of chips to their estimates and think about was it difficult to recover the coal from the ground? What impact does mining have on the surrounding land? Is the land suitable for future use? Is there another way to make mining have less of an impact on the land? 34

## Activity 2: Coal Mining Math

According to EIA, the average emission rates in the U.S. from coal-fired generation are: 2,249 lbs/MWh of carbon dioxide, 13 lbs/MWh of sulfur dioxide, and 6 lbs/MWh of nitrogen oxides. 35 For each 0.00053 short tons or 1.07 lbs of coal, one kilowatt-hour (kWh) is generated. 36 Students will use this information to write an algebraic expression to model the relationship between the production/consumption of coal and the average emission rates of a greenhouse gas. Teachers can tier the information by providing the data with no conversions or scaffold it by having students do the conversions one step at a time using dimensional analysis. Important conversion factors are 0.001 MWh = 1 kWh and 2000 lbs = 1 short ton. It is important for teachers to be intentional about students seeing the units cancel. For example, the algebraic expression for the amount of nitrogen oxides emissions in pounds for any amount of short ton of coal produced/consumed is

where C is the amount of coal produced/consumed in short tons. Students will then evaluate the expression for given replacement values and look at long term data to compare emissions in various years.

For homework and in preparation for the Dominion Virginia Power speaker, 37 students will view the videos below, which contain different perspectives and give unique information, to construct a verbal or pictorial sequence of where energy comes from. I will split the class into two large groups who will transfer their individual information to a large common paper.

Dominion Power:

National Grid: How we get electricity and gas using types of energy and an

RCC Power:

A possible extension can include students calculating the curve of best fit using the coal production/consumption data to calculate and extrapolate the amount of greenhouse emissions, coal production or coal consumption in the future. Students can also use linear equations to model supply and demand and solve them using systems of equations. In addition, since coal contains potential chemical energy and is combustible, teachers can do a combustion demonstration. For safety reasons (and a chance for the science teacher to remind students about safety), collaborate with the science teacher on this demo. Finally, you can arrange a field trip to the local power plant. I plan to take my class to visit Chesterfield Power Station, the largest fossil-fueled power station in the state. It generates over 1,600 MW and has an average daily consumption of 8,400 tons of coal.

# Lesson 2: Wind Energy: Building Windmills

For the Warm-Up, students will construct a device that measures wind speed and then measure the wind speed at different times during the day for a week and record their answers. Students will display measurements in a box-and-whiskers plot and determine if wind is a good source of energy in Churchill, Richmond, Va. 38 In order to use the wind as a source of energy, we must have a steady source of wind.

## Activity 1: Wind Energy: Building Windmills

In small groups, students will construct a windmill that can do work by picking up a certain amount of weight. Students will then calculate the amount of work and power needed to pick up the weight. Work is the measure of the weight lifted through height; work, W, varies jointly with mass, m, acceleration of gravity, g, and height, h, W=mgh. Power is the measure of energy used over time; power, P, varies directly with energy, E, and inversely with time, t, P = E/t. The two equations are related since energy is the ability to do work, so E=W. Students will then work in groups to complete a virtual lab 39 on wind energy; there will be a competition of who creates the most efficient and economical wind farm.

## Activity 2: How much wind power does a fan generate?

Students will calculate the wind power created by a fan set on different speeds: low, medium and high. Wind power, P, varies jointly with half of air density, Ρ, area swept by the turbine blades, A, and the cube of the velocity, V, P = (1/2)ΡV 3. Air density is the mass per unit volume of earth's atmospheric gases and measures 1.25 kg/m 3 at sea level and 68°F.

For homework, students can write, graph and evaluate linear models for the following individual facts: wind integration costs are approximately \$10/MWh, the decrease in air density is 3% for every 1000 additional feet in height and 1% for each additional degree in temperature (air density is 1.25 kg/m 3 at sea level and 68 °F), and the Bluestone River Wind Project in Tazewell County, southwest Virginia, could produce nearly 80 MW (1MW of wind energy will provide electricity for approximately 1,000 people).

A possible extension can include looking at a map of wind power potential along the Virginian coast and having students write a system of linear inequalities to describe the area. Students can also use a system of equations to find the intersections of the boundary lines. Students can write energy equations using the language of direct and inverse relationships. They can also make tables and graph direct and inverse relationships. Have students pay attention to independent and dependent values. In some cases, negative numbers do not exist in the domain or range. Finally, students can calculate a curve of best fit for wind energy related data and extrapolate the wattage of future wind power.

# Lesson 3: Biodiesel: Alternative Fuel

For the Warm-Up, students will be presented with a table that has no headings but describes pipeline incidents and related injuries and fatalities. 40 They will predict what the data could represent. Teachers can only answer yes or no questions. This activity encourages students to look at possible trends in data and other details that may be of use.

## Activity 1: Biodiesel: Alternative Fuel

Students will prepare for lab by gathering information on biodiesel (history, composition, prevalence, advantages and disadvantages, etc.). Students will make biodiesel from two types of oil (used and new). Encourage students to retrieve a variety of waste oils and label with the location. New oil requires 4 grams of lye per liter of oil, while used oil requires more depending on the quality. Students will determine by chemical analysis how much lye is needed for the old oil.

## Activity 2: Biodiesel Math

Biodiesel has to be a specific pH, 8.5, but has a tolerance pH of 0.5. Students will use a compound inequality to describe the allowable range of the pH (between 8 and 9). Compound inequalities will also be used to describe fuel efficiency of cars which use biodiesel, crop production and sales of soybeans, and expected yield of oil content from soybeans. Students will use a quadratic model to maximize fuel efficiency of biodiesel and systems of equations to model and solve problems related to the production and production cost of biodiesel. Finally, students will evaluate the sustainability of a crop by calculating the energy return on energy investments.

A possible extension can include students analyzing yearly oil and biodiesel production statistically using mean absolute deviation, standard deviation and calculating the z-score for a certain amount of gallons. Teachers can use this opportunity to explain the difference between mean absolute and standard deviation.

# Lesson 4: Energy Efficiency: Use Less, Save More

For the Warm-Up, each student in the class will view different videos from Dominion Virginia Power in their energy efficiency series. 41,42 Students will write down notes and report out to the class.

## Activity 1: Energy Efficiency: Use Less, Save More @ Home

Students will act as Energy Inspectors and conduct an energy audit of their home. First they will make a list of items used in their home or on the go that use electricity. They will gather information about usage such as what time during the day they use this item and how many times during the year is it used (seasonal, monthly, weekly, etc.). They will then use this information to evaluate the cost of usage. We will use the formula: C = WNHDP/1000, where W is the power in Watts consumed by a device, N is the number of devices in use, H is the number of hours per day the device is in use, D is the number of days in a year when the device is in use, P is the price of electrical service in dollars per kilowatt-hour (\$0.0887), and C is the total cost in dollars per year for using the device. We divide by one thousand to convert Watts to kilowatts. For example, a washing machine has an average rating of 1150 W. Let's assume you only have one washer and you wash clothes every 2 weeks. A wash cycle lasts about 33 minutes and you usually have 3 loads: white, light and dark colors. This means the washer runs for approximately 1.65 hours. How much money does the washer cost in 1 year? The calculations show

Once students calculate this they can investigate if there are more cost efficient options and calculate how long it would take to get the money back if an investment is made in a more cost efficient option.

## Activity 2: Energy Efficiency: Use Less, Save More on the Road

Given a table, students will model the following for 3 types of cars: rolling friction force as a function of velocity, gas used as a function of rolling friction force, constant speed drag force as a function of velocity, gas used as a function of constant speed drag force, and total gas used to overcome forces as a function of velocity. Then, they will calculate how much money can be saved by the reducing rolling friction force and constant speed drag force.

For homework, students will make a top ten list of ways they are able to conserve energy in their household and write a proposal to their caregivers. They will use research from the internet to support their proposal.

A possible extension could include student modeling acceleration/deceleration and use of gasoline; then, calculate how much money can be saved by accelerating/decelerating at a slower rate when merging into traffic on a highway or local road. Students can also calculate the amount of money a taxi driver can save by turning off their car instead of idling. Restarting your car takes the same amount of gas used in 0.2 seconds of idling. 43 Students can research occupations in the field of residential and commercial energy efficiency on the Energy Efficiency and Renewable Energy (EERE) website. 44

# Appendix A: Content Objectives

Virginia DOE did not adopt the Common Core, so teachers are held to the 2009 Virginia Standards of Learning for Algebra I, which are comparable to the Common Core.

A.1: The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values using data related to emissions of greenhouse gases during the consumption and production of coal. The relationship between consumption/production of coal and emissions of greenhouse gases will become apparent through concrete, pictorial, symbolic and verbal representations.

A.4cf/A.5cd: The student will solve real-world quadratic and multistep linear equations and inequalities in one/two variables, and systems of equations and inequalities to determine the allowable range of the pH, fuel efficiency of cars which use biodiesel, crop production and sales of soybeans, and expected yield of oil content from soybeans and production and cost of production for biodiesel. Finally, students will evaluate the sustainability of a crop by calculating the energy return on energy investments. Students will evaluate the reasonableness of a mathematical model and use graphing calculators to solve problems and to verify algebraic solutions.

A.8: The student will analyze real-world relations to determine whether a direct or inverse variation exists, and represent a direct or inverse variation algebraically given a table of data related to gas consumption and velocity or energy output over time.

A.9: The student will interpret variation and calculate and interpret mean absolute deviation, standard deviation, and z-scores for data related to oil and biodiesel production and determine the implications from which the data derive. The student will compare and contrast mean absolute deviation and standard deviation.

A.10: The student will compare and contrast multiple univariate data sets, using box-and-whisker plots created from data collected by measuring wind speeds with different tools.

A.11: The student will collect and analyze data, determine the equation of the curve of best fit in order to make predictions, and solve real-world problems related to coal production/consumption, greenhouse emissions and wind energy, using mathematical models.

# Appendix B: Lesson Handouts

Figure 3. Lesson 1 - Energy Survey and Coal Mining Experiment

Figure 4. Lesson 1 - Coal Mining Math

Figure 5. Lesson 2 - Wind Power in Churchill

Figure 6. Lesson 2 – Building a Windmill and Wind Math

Figure 7. Lesson 2 – Power of a Fan

Figure 8. Lesson 3 - Unnamed table

Figure 9. Lesson 3 – Making Biodiesel

Figure 10. Lesson 3 – Biodiesel Math

Figure 11. Lesson 4 – Home Audit: Save Home

Figure 12. Lesson 4 - Save on the road

# Bibliography

DOE/Pacific Northwest National Laboratory. "A new point of reference for offshore energy development." Science Daily. http://www.sciencedaily.com/releases/2013/01/130108123038.htm. (accessed July 13, 2013).

"Alliance Applauds Efficiency Focus in President's Climate Plan , Alliance to Save Energy." Alliance to Save Energy , Creating an Energy-Efficient World. http://www.ase.org/news/alliance-applauds-efficiency-focus-presidents-climate-plan (accessed July 10, 2013).

American Chemical Society. "Big Environmental Footprints: 21 Percent of Homes Account for 50 Percent of Greenhouse Gas Emissions." ScienceDaily. www.sciencedaily.com/releases/2013/06/130626142944.htm (accessed July 15, 2013).

Brown, Lester Russell. Plan B 2.0: rescuing a planet under stress and a civilization in trouble. New York: W. W. Norton & Co., 2006.

"Chesterfield Power Station." Chesterfield Power Station - Fossil-fueled Power Stations , Dominion Power. https://www.dom.com/about/stations/fossil/chesterfield-power-station.jsp (accessed July 13, 2013).

"Energy Efficient Mortgages: ENERGY STAR." Home: ENERGY STAR. http://www.energystar.gov/index.cfm?c=mortgages.energy_efficient_mortgages (accessed July 1, 2013).

"Facts & Figures - according to EIA data , America's Power." America's Power ,. http://www.americaspower.org/according-to-eia-data (accessed July 13, 2013).

"Pacific Biodiesel - Pacific Biodiesel." Pacific Biodiesel - Pacific Biodiesel. http://www.biodiesel.com/ (accessed July 14, 2013).

MacKay, David J. C. Sustainable energy—without the hot air. Cambridge, England: UIT, 2009.

Rosenthal, Elizabeth. "Rush to Use Crops as Fuel Raises Food Prices and Hunger Fears." New York Times (New York City), April 7, 2011, New York edition, sec. A.

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Virginia Center for Wind Energy. "Where are the wind farms in Virginia?" Virginia Center for Wind Energy. http://windpowerVA.org (accessed July 14, 2013).

Willis, Judy. Learning to love math teaching strategies that change student attitudes and get results. Alexandria, Va.: ASCD, 2010.

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Veigele, William J. How to save energy and money at home and on the highway: the mathematics and physics of energy conservation and reduction of consumer energy costs. Boca Raton, Fla.: Universal-Publishers, 2009.

Yergin, Daniel. The quest: energy, security and the remaking of the modern world. New York: Penguin Press, 2011.

# Teacher Resources

"3M Science of Everyday Life - Discovery Education." 3M Science of Everyday Life - Discovery Education. http://scienceofeverydaylife.com/students/ (accessed July 25, 2013).

Crowley, Grace, Nick Ford, Owen Harris, Nicole Kendle, and MacKailah McKinley. "Learning from Coal." Cargo - Gallery. http://cargocollective.com/learningfromcoal#Pictures-in-Motion-Part-1 (accessed July 24, 2013).

"EERE: Energy Education and Workforce Development Home Page." U.S. DOE Energy Efficiency and Renewable Energy (EERE) Home Page. http://www1.eere.energy.gov/education/ (accessed July 14, 2013).

Glencoe algebra integration, applications, connections. New York, NY: Glencoe/McGraw-Hill, 2001.

"How much coal, natural gas, or petroleum is used to generate a kilowatt-hour of electricity? - FAQ - U.S. Energy Information Administration (EIA)." U.S. Energy Information Administration (EIA). http://www.eia.gov/tools/faqs/faq.cfm?id=667&t=2 (accessed July 24, 2013).

"Issue Brief 23 , Pipelines Are Safest For Transportation Of Oil And Gas." Manhattan Institute for Policy Research. http://www.manhattan-institute.org/html/ib_23.htm (accessed July 25, 2013).

Pfaff, Thomas J. "TJP Sustainability Page." Ithaca College, Ithaca, NY. http://www.ithaca.edu/tpfaff/sustainability.htm (accessed July 25, 2013).

Reilly, Kathleen M. Energy: investigate why we need power & how we get it: 25 projects. White River Junction, VT: Nomad Press, 2009.

Dominion Power. "Speakers Bureau Program." Dominion Power. https://www.dom.com/about/education/speakers-bureau-program.jsp (accessed July 24, 2013).

Willis, Judy. Learning to love math teaching strategies that change student attitudes and get results. Alexandria, Va.: ASCD, 2010.

# Notes

1. Daniel Yergin, The quest: energy, security and the remaking of the modern world, 620.

2. Ibid, 620.

3. Alliance to Save Energy, "Alliance Applauds Efficiency Focus in President's Climate Plan," http://www.ase.org/news/alliance-applauds-efficiency-focus-presidents-climate-plan.

4. Daniel Yergin, The quest: energy, security and the remaking of the modern world, 621.

5. Ibid, 7.

6. David J C MacKay, Sustainable energy—without the hot air. 122-123.

7. Data used in pie chart came from America's Power, "Facts & Figures - according to EIA data,." http://www.americaspower.org/according-to-eia-data.

8. Dominion Virginia Power, "Chesterfield Power Station," https://www.dom.com/about/stations/fossil/chesterfield-power-station.jsp.

9. World Coal Institute, The coal resource: a comprehensive overview of coal, 2-21.

10. Ibid, 27-29.

11. William Veigele, How to save energy and money at home and on the highway: the mathematics and physics of energy conservation and reduction of consumer energy costs, 60.

12. Wind Program: Wind Powering America, "Wind Powering America: President Obama Unveils Climate Action Plan," http://www.windpoweringamerica.gov/filter_detail.asp?itemid=3900.

13. Science Daily, "A new point of reference for offshore energy development," www.sciencedaily.com /releases/2013/01/130108123038.htm.

14. Virginia Center for Wind Energy, "Where are the wind farms in Virginia?" http://windpowerVA.org.

15. Lester R. Brown, Earth Policy Institute – Building a Sustainable Future, "Eco-Economy Indicators - Wind Power- Wind Electric Generation Soaring, EPI," http://www.earth-policy.org/indicators/C49/wind_power_2002.

16. Ibid.

17. National Park Service, "America's National Park System: The Critical Documents - Edited by Lary M. Dilsaver," http://www.cr.nps.gov/history/online_books/anps.

18. Virginia Center for Wind Energy, "Where are the wind farms in Virginia?" http://windpowerVA.org.

19. Pacific Biodiesel, http://www.biodiesel.com/.

20. Lester Brown, Plan B 2.0: rescuing a planet under stress and a civilization in trouble.

21. Lester Brown, "Plan B Updates - 55: Supermarkets and Service Stations Now Competing for Grain."

22. Elizabeth Rosenthal, "Rush to Use Crops as Fuel Raises Food Prices and Hunger Fears," A21.

23. Ibid.

24. Information in the pie chart came from U.S. Energy Information Administration (EIA), "Virginia Profile," http://www.eia.gov/state/print.cfm?sid=VA.

25. American Chemical Society, "Big Environmental Footprints: 21 Percent of Homes Account for 50 Percent of Greenhouse Gas Emissions," www.sciencedaily.com/releases/2013/06/130626142944.htm.

26. William Veigele, How to save energy and money at home and on the highway: the mathematics and physics of energy conservation and reduction of consumer energy costs, 46, 54, 67-8.

28. ENERGY STAR, "Energy Efficient Mortgages: ENERGY STAR," http://www.energystar.gov/index.cfm?c=mortgages.energy_efficient_mortgages.

29. Judy Willis, Learning to love math teaching strategies that change student attitudes and get results, 117.

30. Ibid, 125.

31. Ibid, 94.

32. Ibid, 126.

33. Video created by Grace Cowley, Nick Ford, Owen Harris, Nicole Kendle, and MacKailah McKinley, "Learning from Coal," http://cargocollective.com/learningfromcoal#Pictures-in-Motion-Part-1.

34. Kathleen Reilly, Energy: investigate why we need power & how we get it : 25 projects, 56.

35. U.S. Energy Information Administration (EIA), U.S. Coal Consumption, http://www.eia.gov/emeu/aer/coal.html Table 7.2, 7.3 and U.S. Coal Production.

36. U.S. Energy Information Administration (EIA), "How much coal, natural gas, or petroleum is used to generate a kilowatt-hour of electricity? - FAQ - U.S. Energy Information Administration (EIA)," http://www.eia.gov/tools/faqs/faq.cfm?id=667&t=2.

37. Dominion Power, "Speakers Bureau Program," https://www.dom.com/about/education/speakers-bureau-program.jsp.

38. Glencoe algebra integration, applications, connections, 29-31.

39. Discovery Education, "3M Science of Everyday Life - Discovery Education," http://scienceofeverydaylife.com/students/.

40. Manhattan Institute for Policy Research, "Issue Brief 23, Pipelines Are Safest For Transportation Of Oil And Gas," http://www.manhattan-institute.org/html/ib_23.htm.