### Modeling Prompt: A New Heating System

#### In Class Launch

Use after Unit 2, Lesson 23.

Solicit information that students already know about heating buildings, which they will use in this task. Here are some possible questions for discussion:

• “How are buildings heated?” (fireplaces, boilers, gas furnaces)
• “What kinds of energy can be used to heat buildings?” (electricity, natural gas, solar power)
• “When in the year do you think our school uses the most power for heating?” (probably right before or after winter break, when it’s pretty cold and there are still people in the building for most of the day)

Tell students that in this task they will investigate energy costs for some different heating systems. They will begin by considering a homeowner’s current heating system, and compare some other systems the homeowner could use instead. Tell students that energy is measured in kilowatt-hours, which is abbreviated kWh.

Then give students a preview of some of the calculations they’ll need to do by showing why the homeowner currently pays \975/year to heat the house. Tell students that the current system is 60% efficient, which means that for every 100 kWh it uses, it only produces 60 kWh of heat. Display this statement for all to see: “For every 100 kWh of energy a certain heating system uses, it produces 60 kWh of heat. If the system has to produce 11,700 kWh of heat to heat a house for the winter, how many kWh of energy will it use?” Ask students to think about how they would find an answer to this question. They do not need to calculate an answer, only think of a strategy. After some quiet think time, ask students to share their thoughts with a partner. Then invite students to share strategies with the class. Use one of the suggested strategies to calculate the answer, and write the steps for all to see. Here is one possible way: \displaystyle \begin{align*}{} \dfrac{100 \text{ kWh input}}{60 \text{ kWh output}} &= \dfrac{x \text{ kWh input}}{11,\!700 \text{ kWh output}} \\ \dfrac{100 \text{ kWh input}}{60 \text{ kWh output}} \boldcdot 11,\!700 \text{ kWh output} &= x \text{ kWh input} \\ 19,\!500 \text{ kWh} &= x \end{align*} To find out how much this will cost the homeowner, we need to know how expensive the fuel is. If we assume natural gas costs \0.05 per kWh, then multiplying 19,500 by 0.05 will give us the total cost of \\$975.