Gas-Bill Mathematics

Gas-Bill MathematicsAuthor(s): Eric WoodSource: The Mathematics Teacher, Vol. 88, No. 3 (MARCH 1995), pp. 214-218, 224-227Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27969278 .

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Eric Wood

ctivities

Gas-Bill Mathematics

Mathematics

is where you find it

TEACHER'S GUIDE Introduction: A pervasive theme in the current mathematics reform literature is that of connections. Research suggests, however, that many mathemat ics graduates have very little experience in making connections, both within and outside mathematics (Wood 1993); this background then shapes their

teaching. The purpose of this article is to illustrate the maxim that "mathematics is where you find it* and to furnish a series of activities that allow teach ers to connect a number of mathematical topics with a real-world situation. In addition, the development of the ideas in the various activities illustrates

ways in which students can make conjectures, investigate mathematics, and construct their own

understanding of mathematical ideas. Seeing how mathematics can come from seemingly mundane

objects and events will, it is hoped, encourage read ers to look for mathematics in their world as well.

In North America, many homes are heated by gas furnaces. A typical gas bill from Ontario is

reproduced on sheet 1. It is not immediately appar ent that much could be gleaned from a simple gas bill. However, a deeper examination reveals a num ber of mathematical topics, each of which can be the focus of an interesting activity for students.

Grade levels: 8-11

Objectives: To allow students to see mathematics in the world around them; to establish the mathe

matical meaning for the term degree day; to read and interpret bar graphs; to analyze real data; to examine relationships between variables; and to link the abstract ideas of slope andy-intercept to

practical examples

Sheet 1: Initially, students examine the bill and familiarize themselves with it. The bill gives differ ent kinds of information, such as the price of gas, the amount of gas consumed, and the energy profile for the past year. The bar graph furnished as part of the bill's energy profile can be useful in establish ing any obvious patterns or trends. The third ques tion leads to a discussion about the assumptions that are made about gas use and home heating and whether all houses have gas hot-water heaters or other gas appliances as well as furnaces. The fifth

question introduces the concept of degree days. Intuitively, degree days might appear to be some

measure of the coldness of the month. The exact definition can be found on the back of the gas bill: "A degree day is defined as the difference between 18 degrees [C]elsius and the average temperature of the day. For example, if the average temperature on a given day is 5 degrees [C]elsius, then the num ber of degree days would be 18 minus 5, or 13. If the average temperature for the day is 18 degrees [C]elsius or higher, then the number of degree days for the day would be zero.* It is reasonable, there fore, that a high number of degree days for a month means that the temperature was low for that month. If a utility company does not supply this kind of information directly on the bill, assign stu dents to find out what this term means and to bring the explanation to class the next day. Any local

heating company could furnish this information.

Eric Wood is an associate professor on the Faculty of Edu cation at the University of Western Ontario, London, ON

N6G 1G7. He is a teacher educator interested in develop ing in prospective teachers an appreciation for the connec tions between mathematics and the real world.

Edited by Timothy V. Craine, Central Connecticut State University, New Britain, CT 06050 Kim Girard, Glasgow High School, Glasgow, MT 59230 Guy R. Mauldin, Science Hill High School, Johnson City, TN 37604

This section is designed to provide in reproducible formats mathematics activities appropriate for students in grades 7-12. This material may be reproduced by classroom teachers for use in their own classes. Readers who have developed successful classroom activities are encouraged to submit manuscr?pts, in a format similar to the "Activi ties" already published, to the editorial coordinator for review. Of particular interest are activities focusing on the Council's curriculum standards, its expanded concept of basic skills, problem solving and applications, and the uses of calculators and computers.

Write to NCTM, Department P, for the catalog of educational materials, which lists compilations of "Activities" in bound form?Ed.

214 THE MATHEMATICS TEACHER



In Canada, Celsius temperatures are used; point out this fact to students as they examine the bill. If

samples of gas bills are used that record tempera tures in degrees Fahrenheit, the definition of degree days will use the pivot temperature of 65?F, which is equivalent to 18?C. Use this opportunity to dis cuss conversions between the two systems of mea surement. Note that because the definition of

degree days involves differences between tempera tures, converting from Celsius degree days to Fahrenheit degree days does not require a consider ation of different starting points?0 and 32?but rather is a simple 5:9 ratio. Degree days can also be defined for cooling purposes; electric utilities use this concept to estimate electrical power demands for air conditioning.

Sheet 2: In this activity, students graph values that they have calculated from the definition of degree day (fig. 1).

Average Daily Temperature (?C)

Rg< 1 Graph of degree days versus temperature

The graph that results is interesting because it is not the typical smooth curve or simple straight-line relationship with which most students are familiar. In fact, it consists of two different functions, each defined over a particular domain. This graph can then help students develop a formula for the rela

tionship between the number of degree days and the average daily temperature. The formula that results is also different from many of the equations that students typically see because it is defined in a

piecewise fashion. Although students do see this kind of definition of functions in other contexts?in absolute value, for example?they often find it diffi cult and confusing because it seems so unusual and theoretical. In this example, the functional formula

if*>18,thend = 0

iff <18,thend = 18-f,

where t is the average temperature for the day and d is the degree days for that day, is easy to under stand and makes intuitive sense because people do not heat their homes once the temperature is above 18 degrees Celsius. Consequently, it is to be expect ed that the number of degree days should be zero when the average daily temperature exceeds 18

degrees Celsius. Furthermore, it seems reasonable that the colder the temperature, the more degree days will occur, if this quantity is supposed to give a rough measure of how cold it gets. If graphing calculators are available, students may also exam ine how these tools handle piecewise continuous functions.

Sheet 3: Most students should now realize that if the number of degree days is related to the coldness of the day and the coldness of the day is related to how much gas is consumed, then the amount of gas consumed ought to be related to the number of

degree days. Students can investigate this intuitive

relationship using real data simply by examining a number of gas bills and drawing a graph. The data in figure 2 come from one homeowner's gas bills over several years, excluding the summer months. This user did not have a gas hot-water heater at the time, so the gas consumption comes exclusively from heating demand. Alternatively, students may get data for their own house either from their par ents' bills or from the utility company that supplies the gas.

BOO 800 ?OOO OaysyMonth

Students can calculate the slope and the

y-intercept from their graphs to establish a formula that relates the gas consumption, g, to the number of degree days, d. In fact, doing this procedure by hand using graph paper with centimeter squares subdivided into millimeters produced the formula

g = 0.46 ? d - 50. Alternatively, students can use

the regression program built into many calculators to produce the formula g = 0.45

? d - 49. Finally, a

The value

of mathematics

becomes

obvious

Vol.88,No.3 March 1995 215



spreadsheet or graphing program can be used to

produce graphs and do a curve fit. Either way, this

process makes a nice student investigation of real

mathematics. Students can then use this formula to

make predictions and check them against actual values from a variety of gas bills to establish the

accuracy of the mathematical heating model sug

gested by the relationship. The fact that the model is not perfect leads naturally to a discussion of what

factors could lead to these inaccuracies. Clearly, some people might heat their homes to a higher level than others, and some have thermostats that

set the temperature lower at night. Both actions would tend to make the predictions of the model, which is based on average heating patterns, differ

ent from the actual values. In fact, the data in fig ure 2 come from a homeowner who has a setback thermostat that lowers the temperature during the

night. Access to the heating bills from several homes

leads to an interesting extension for this investiga tion. The new data can be used to predict a formula

similar to the previous one. The set of data in figure 3 is for the same house used for figure 2 after a

large addition was constructed. The overall result

is that the "new house" is much larger than the "old

house*; this difference is manifested in increased

gas consumption. The graph that results in this case is given in figure 3, with formulas of g = 0.69?

d-85 andg = 0.67 ? d - 83 done by hand and by calculator, respectively. An interesting discussion can then ensue about those factors that affect heat

ing a house, such as size, amount of insulation, thermostat setback, and so on. This discussion can

also be an opportunity to integrate some considera

tion of energy-conservation issues into the mathe matics curriculum.

Realistically, gas companies that supply gas by

pipeline are not particularly interested in this kind

of relationship because gas is always available to

the consumer. However, oil companies and gas com

panies that deliver gas to rural customers with

storage tanks do use these concepts to establish when to deliver oil or gas to their customers. Stu dents may be asked if they have ever run out of

heating oil in the winter and how they think the oil company knows when a customer needs a delivery.

Suppliers keep careful track of the number of

degree days, and this information guides their

delivery schedules. Once again, the value of mathe matics becomes obvious.

Sheet 4: As a final activity, students can investi

gate how the cost is related to the consumption of

gas. The data from one homeowner's heating bills

produce the graph in figure 4?

A formula can be developed from this graph, which in this example is y = 0.1826* + 7.25 when

done manually and y = 0.181862* + 7.24757 with a

calculator's regression program. The resulting

graph clearly illustrates the linear relationship between the variables but also shows that even a

consumption of zero has a cost. In fact, the gas bill

does give the monthly minimum charge and the cost per cubic meter; these values produce the for

mula y = 0.1818490* + 7.25. The slope of the line

represents the unit cost of gas, and they-intercept represents the fixed minimum charge per month. The investigation that students have done on their own matches the exact values very closely. This

fact gives further credence to the students' own

work. This last graph also affords students the

opportunity to compare and contrast a graph pro duced by a formula and a graph produced by empir ical data. In the last graph, the data points are gen erated by the cost function, whereas in the second

graph, the function is developed from the data

points. It is useful to have students realize that

many mathematical models, although useful, are

only models and may not be exact representations of data in the real world.

THE MATHEMATICS TEACHER



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Assessment: Given the objectives of these activi

ties, giving students a test and asking a series of

questions about degree days and home heating costs would make little sense. A more meaningful way to gauge students' appreciation for how mathe matics can be used to model real-world phenomena would be to assign a group project that requires them to collect and analyze some data to establish what relationships are in evidence. One project, related to what has been discussed in these activi

ties, would be to have them establish a relationship between the number of degree days in a heating period and the heating cost for that period. Assum

ing that the rate does not change, the relationship should be reasonably linear, given that degree days are linearly related to consumption and consump tion is linearly related to cost. Students could also be asked to prepare a series of questions based on

this analysis, along with possible answers. These

questions can then become part of the class discus sion during the presentation of results. Many sta tistics books have examples of variables that are

correlated, and these examples can be used to give students some ideas.

Activities that include connecting, exploring, and

investigating are often difficult to realize in a typi cal classroom setting. This article has suggested some avenues of approach that show how even a

gas bill can be the source of some interesting and relevant mathematics.

Selected answers: Sheet 2:

3.

Day MTWTF SSMTWTF S S Average Temper

ature 20 19 17 16 16 15 18 19 20 25 23 22 15 9

Degree Days for the Day 00122300000039

Sheet 3:5. By hand: gas = 0.46 ? d - 50; by calcu

lator or computer: gas = 0.45 ? d - 49.7. Using the

formula from the regression equation gives a gas

consumption of 145.5 cubic meters, which is 25 per cent higher than the actual value of 116 cubic meters. 12. By hand: gas = 0.69

? d - 85; by calcula tor or computer: gas = 0.67

? d - 83.

Sheet 4:3. By hand: cost = 0.1826 ? gas + 7.25; by

calculator or computer: cost = 0.1818 ? gas + 7.25.

4. It is the fixed cost, $7.25.

REFERENCE Wood, Eric F. "Making the Connections: The Mathe

matical Understandings of Prospective Secondary Mathematics Teachers." Ph.D. diss., Michigan State

University, 1993. @ (Worksheets begin on page 218)

HIGH SCHOOL MATH EDUCATORS D&S Marketing Systems, a well-established publisher of AP Calculus study material, is now accepting written manuscripts and math-based application software for evaluation and publication by our new Math Publications Division. Subject matter should focus on a specific high school math discipline and must include graphing calculator-active material. Workbooks, study material, and software are our

primary interest. Please send your resume and a cover letter pertaining to your material, in confidence, to:

Leonard Zack, Recruitment Manager D&S MARKETING SYSTEMS, INC. 1325 East 17th Street, Suite 619 Brooklyn, NY 11230

Vol.88,No.3 ?March 1995

Wood, Eric F. "Making the Connections: The Mathe matical Understandings of Prospective Secondary Mathematics Teachers." Ph.D. diss., Michigan State Universitv,1993.

(Worksheets begin on page 218) |

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Vol. 88, No. 3 e March 1995 217



RELATIONSHIPS SHEET 1

A Typical Ontario Gas Bill

The bill reproduced here is from the gas company in London, Ontario, for a month in the spring. As you can see, a lot of mathematical information is included. Examine the bill and answer the

following questions:

1. What three different kinds of mathematical information can you find on this bill?

2. Look at the bar graph at the bottom of the bill. Which months seem to require the most gas? The least gas? Why do you think this situation occurs?

3. Why does the graph not show gas consumption in the summer months? In what situation

might a graph show consumption in the summer months?

4. This bill is for a home close to London, Ontario, with latitude 43? north. Suppose the family lived in Christchurch, New Zealand, with approximate latitude 43? south. What would the

graph look like?

5. It appears from the chart beside the bar graph that the number of degree days for a month is smaller when the average daily temperature is higher. What is a degree day?

UNION GAS LIMITED RETAIN THIS PORTION

From the Mathematics Teacher, March 1995

(Continued on page 224)



(Continued from page 218)

PIECEWISE DEFINED FUNCTIONS SHEET 2

A degree day is defined as the difference between 18?C and the average temperature of the day. If the average temperature for the day is 18?C or higher, then the number of degree days for the day is zero. For example, if the average temperature for the day is 12?C, then the number of degree days for that day is six. If the average temperature for the day is 20?C, then the number of degree days is zero.

1. Does it matter whether the mean daily temperature is 20?C or 30?C as far as the number of degree days is concerned? Why or why not?

2. Why is the number of degree days zero whenever the mean daily temperature is 18?C or above?

3. The daily average temperature for a two-week period is given in the chart below. Fill in the number of degree days for each day.

Day_MTWT FSSMTWT FSS Average

Temperature 20 19 17 16 16 15 18 19 20 25 23 22 15 9

Degree Days for the Day_

4. Some relationships are easier to understand if they are shown pictorially. Use the foregoing chart to graph average daily temperature on the x-axis and the number of degree days on the y-axis.

5. What is unusual about this graph compared with others that you have seen?

6. Try to establish a formula for this graph.




DEGREE DAYS AND HEATING SHEET 3A

1. If the number of degree days for a month is high, what impact will this situation have on the gas consumption for that month?

2. To get a more precise idea of the relationship between the number of degree days and gas consumption, graph the data given in the following chart.

d 613 760 716 687 318 161 119 194 425 599 614 Gas

(m3) 265 259 273 264 113 28 2.8 22.5 116 192 254

3. What kind of relationship appears to exist between the number of degree days per month and the gas consumption for that month? Draw the best-fitting line.

4. Calculate the slope and the /-intercept for your line.

5. Give an equation for the relationship represented by this line.

6. Although we can mathematically get a negative value for the /-intercept, we cannot consume a negative amount of gas. What might cause this apparent contradiction?

7. If the month of February had 425 degree days, how much gas would you expect to use? How close is this value to the actual value in the chart? What factors might be responsible for this discrepancy?

From the Mathematics Teadier, March 1995



DEGREE DAYS AND REATWG-Continued SHEET 3B

8. Could this same formula be used to make predictions for your house? Why or why not?

9. Graph the data from a different house given in the following chart.

d 678 402 233 373 734 596 509 467 177

Gas

(m3) 366 189 68 141 431 293 268 254 51 y

10. What kind of relationship appears to exist between the number of degree days per month and the gas consumption for that month? Draw the best-fitting line.

11. Calculate the slope and the y-intercept for your line.

12. Give an equation for the relationship represented by this line.

13. How does the formula for this house compare with the one for the first house? What factors

might account for these differences?




GAS CONSUMPTION AND COST SHEET 4

1. What will happen to a gas bill when more gas is used?

2. To get a more precise idea of the relationship between gas consumption and cost, graph the data given in the following chart. Put gas consumption on the x-axis and cost on the y-axis.

Gas Used (m3) 140.9 431.1 276.1 293 267.7 253.6 50.7

Monthly Bill ($) 32.87 85.65 57.46 60.53 55.93 53.37 16.47

3. Calculate the slope and the y-intercept for the line. Use these values to develop an equation that relates the cost of gas to the consumption.

4. What is the cost when the gas consumption is 0? Is this amount reasonable? Why? What mathematical quantity is this amount?

5. Look back at the original gas bill and try to establish the meaning of the slope of this line in terms of the cost of gas.

6. Why do the points that were plotted seem to be exactly on a line in this example, although they were approximate in the other graphs?




Gas-Bill Mathematics

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