problem solvers: problem
C y n t h i a Ta y l o r, K e l l e y B u c h h e i s t e r, a n d C h ri s t a Ja ck so n
YALCINSONAT/VEER
What is the best option?
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Some families pay a monthly Homeowners’ Association (HOA) fee. These funds are typically collected to pay for exterior maintenance, such as landscaping and snow removal. The association’s board of directors is normally responsible for setting the rates for the monthly HOA fees. During this investigation, students will justify which electric plan the HOA should choose to minimize the cost of heating the community pool.
Problem scenario The Clarksville Homeowners Association (HOA) maintains a 13 × 36 foot outdoor pool for the residents of the Clarksville community. The pool is kept at 78 degrees Fahrenheit from November through February. The HOA wants to keep down the cost of heating the pool. Three electric service providers, Green Planet Energy (GPE ), Peoples
Power and Energy Company (PPEC ), and Nixon Electric Partners (NEP ) have made proposals to supply electricity to heat the pool. The HOA wants to select the least expensive plan. Which provider should the HOA choose? Justify your answer with words, pictures, and/or graphs. See the student activity sheets (pp. 219–20) for the electric providers and rates.
Classroom setup To introduce students to the idea of energy usage and kilowatt hours (kWh), have them make a list of the items in their house that use electricity. Possible items include televisions, refrigerators, air conditioners, fans, or lights. Explain that we use kilowatt hours to measure the amount of energy (like electricity) that we consume. A kilowatt hour is a unit of energy expended for one hour of time and is equal to 1000 watt-hours. (For more information, see http://www.energylens
216Ë !ÜjMjÁËÔå¤ÏËVËteaching children mathematics | Vol. 20, No. 4 www.nctm.org C o p yright © 2013 The Nat i o n a l C o u n c i l o f Te a ch e r s o f M a t h e m a t i c s , I n c . w w w. n c t m . o r g . A l l r i g h t s r e s e r v e d . T his material may n o t b e co p ied or distr ibut ed el ec t r oni c al l y or i n any o t h e r f o rma t w i t h o u t w ri tt e n p e rmi s s i o n f ro m N C T M.
.com/articles/kw-and-kwh.) Each appliance that students list uses a different amount of energy. For example, a typical clothes dryer uses approximately 80 kWh each month; a CD player might use approximately 7 kWh each month. Discuss that electric providers charge consumers a certain amount for the number of kWh used each month. Ask students how they would calculate the monthly electric bill for the clothes dryer and the CD player if the cost per kWh were 7¢. You should anticipate that students will think about how many hours per day these devices are used. Students may then consider the days in a month and find the cost per month. Some students may want to work in dollars, and others will use cents. Such calculations challenge students’ multiplicative thinking and build number sense. Ask students to predict the cost for heating an outdoor pool for the month of November (see the student activity sheets on pp. 219–20). Have students discuss their prediction and justify their reasoning. For example, a student might say that to heat the outdoor pool would cost $200 because the size or volume of the pool is a lot larger than a clothes dryer. Therefore, the cost must be more. Other students may factor in the average temperature in their local area when making a prediction. For example, the typical outdoor temperature in Miami, Florida, in November is 80 degrees, and students may reason that there would be no need to heat an outdoor pool. (If this is the case, you may want to challenge students to predict the cost for heating an outdoor pool in another area of the country.) Encourage students to be specific; record their predictions so that the class can discuss the range. Share that the outdoor pool uses approximately 14,000 kWh during the month of November. Direct students to calculate the cost to heat the outdoor swimming pool for November if kWh cost 7¢ each. How did this compare with students’ predictions? How could we determine the cost to use this provider from November through February? Note the costs for each month on the board after students have had an opportunity to make their calculations.
Exploration Introduce students to the problem scenario and explain that they have already estimated the cost for one potential provider on the basis of www.nctm.org
Where’s the math? In the introduction, students think about how much electrical energy is used by common household appliances, and they calculate the cost of the kWh. Students engage in estimation when they use their previous calculations to predict the cost of heating the pool in November. Comparing predictions to actual costs provides students with opportunities to develop number sense. Students are expected to explain their reasoning through various mathematical representations, such as pictures, graphs, or words. Students also have the opportunity to reason about large quantities. The number of kWh used is quite large, and students will have to partition the quantity and make sense of units of dollars and cents to make the comparisons. Student letters explaining their selection to the HOA will integrate mathematics and language arts. Writing such a letter provides students with experience using mathematics to influence decision makers. Letters could include computations or visual representations of data to support students’ recommendations. Students can then identify and discuss characteristics of persuasive letters.
last year’s kWh usage. Explain that the other providers have different charges; note these plans on the board. Read through the two remaining plans and have students predict which plan (including the current one) they believe will cost the least. Hand out the student activity sheets. Remind students that they are expected to be able to explain and illustrate how they select the plan with the lowest total cost for the four months. Walk around as students work, answering questions and observing how students make sense of the problem and represent the data. For example, an appropriate representation may include a triple bar graph or side-by-side dot plots, either by month or provider. Let students be creative with how to represent the cost for each month. Be sure that students do not add the kWhs for all four months to calculate the total electric bill, because providers bill the HOA monthly. As students finish, remind them to be prepared to demonstrate their reasoning.
Summary Select different student-developed representations, and have them share their reasoning. Discuss the representations and justifications, focusing on clarity and allowing time for questioning. Questions might focus on how to label the work or the graphs they used or on computation strategies. Students could also explore Vol. 20, No. 4 | Íj?W~ËWaÁjË?Íj?ÍWÄËVËNovember 2013
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MONKEY BUSINESS IMAGES/VEER
problem solvers: problem
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Try this problem in your classroom. We are interested in how your students responded to the problem, what problem-solving strategies they used, and how they explained or justified their reasoning. Send your thoughts and reflections—including information about how you posed the problem, samples of students’ work, and photographs showing your problem solvers in action—by February 15, 2014, to either of the Problem Solvers department editors: Signe Kastberg, Purdue University, 100 North University St., West Lafayette, IN 47907-2098; or Erin Moss, Millersville University, P.O. Box 1002, Millersville, PA 17551-0302; or e-mail either of them. Selected submissions will be published in a subsequent issue of Teaching Children Mathematics and acknowledged by name, grade level, and school name unless you indicate otherwise. R E S OU R C E S
the efficiency of different approaches. As students present their solutions, ask the following questions: s How did you calculate the cost of heating the pool over the winter months? s Would it save money to switch providers? If so, which company should the HOA choose? Explain your reasoning. s Did the most cost-effective provider differ each month? Following the discussion, have students write a persuasive letter to the HOA identifying the plan they would choose and explaining and illustrating why they made the selection they did.
Extensions Go to http://www.glendalewaterandpower .com/rates/appliance_operating_costs.aspx to find information regarding how much it costs to run various appliances and items around the house that use electricity. Students could determine how much energy (in kWh) it costs to run their house for a month and compare their estimate with their parent’s electric usage on the monthly statement. What reasonable suggestions would students make to their parents if they were to suggest ways to cut energy usage in their households? Explain. 218Ë
Share your students’ work
!ÜjMjÁËÔå¤ÏËVËteaching children mathematics | Vol. 20, No. 4
BizEE Energy Lens. 2013. “kW and kWh Explained.” http://www.energylens.com/articles/kw-and-kwh City of Glendale Water and Power Site. “How Much Does It Cost to Operate My Appliances?” http://www.glendalewaterandpower.com/rates/ appliance_operating_costs.aspx Cynthia Taylor,
[email protected], is an assistant professor at Millersville University of Pennsylvania. Kelley Buchheister, buchheis@mailbox .sc.edu, is an assistant professor at the University of South Carolina. Christa Jackson, crista.jackson@ uky.edu, is an assistant professor in mathematics education at the University of Kentucky. All are former classroom teachers who are interested in creating a classroom culture that encourages students’ mathematical reasoning through worthwhile tasks. Edited by Signe E. Kastberg,
[email protected], a teacher of prospective elementary teachers at Purdue University in West Lafayette, Indiana; and Erin Moss,
[email protected], an assistant professor in the mathematics department at Millersville University of Pennsylvania. Each month, this section of the Problem Solvers department features a new challenge for students. Readers are encouraged to submit problems to be considered for future columns. Receipt of problems will not be acknowledged; however, those selected for publication will be credited to the author. Find detailed submission guidelines for all departments at www.nctm.org/tcmdepartments. www.nctm.org
➺ problem solvers activity sheet (page 1 of 2)
Name___________________________________
The Cost of Heating an Outdoor Pool The Clarksville Homeowners Association (HOA) maintains a 13 × 36 ft. outdoor pool for the residents of the Clarksville community. The pool is kept at 78 degrees Fahrenheit from November through February. The HOA wants to keep down the cost of heating the pool. Three electric service providers, Green Planet Energy (GPE), Peoples Power and Energy Company (PPEC), and Nixon Electric Partners (NEP) have made proposals to supply electricity to heat the pool. The HOA wants to select the least expensive plan. Which provider should the HOA choose? Justify your answer with words, pictures, and/or graphs. The HOA must choose among three electric service providers to heat its community pool. PPEC and NEP offer special rates for businesses and homeowners who use a large amount of energy each month, whereas GPE has a flat rate. See each supplier’s offer in the table on page 2 of this activity sheet. Part 1 From the following list, which item(s) do you think will have costs similar to the cost of heating the outdoor pool for the month of November? Explain. ______ The cost of 10 DVDs
_____
The cost of an iPod®
______ The cost of a new MAC laptop
_____
The cost of a new stainless steel refrigerator, stove, and dishwasher
______ The cost of a new car
_____
The cost of a new house
Explanation:
From the November 2013 issue of
➺ problem solvers activity sheet (page 2 of 2)
Name___________________________________
The Cost of Heating an Outdoor Pool Part 2 Complete the following table. Comparison costs for electricity providers Electric supplier Current Provider: Green Planet Energy (GPE)
Peoples Power and Energy Company (PPEC)
Nixon Electric Partners (NEP)
Provider plans 7¢ per kWh
$900 for the first 13,750 kWh used and 9¢ for every kWh used above 13,750
$810 for the first 11,500 kWh and 6¢ for every kWh used above 11,500
Cost for November (energy use ~14,000 kWh) Cost for December (energy use ~19,000 kWh) Cost for January (energy use ~16,500 kWh) Cost for February (energy use ~16,500 kWh) Total cost of plan per supplier
Which electric plan should the Clarksville Homeowners Association choose? Write a letter to the homeowners’ association, and justify your recommendation regarding which electric supplier the homeowners should choose.
From the November 2013 issue of
