Felicia's Drive

Task

As Felicia gets on the freeway to drive to her cousin's house, she notices that she is a little low on gas. There is a gas station at the exit she normally takes, and she wonders if she will have to get gas before then. She normally sets her cruise control at the speed limit of 70mph and the freeway portion of the drive takes about an hour and 15 minutes. Her car gets about 30 miles per gallon on the freeway, and gas costs $3.50 per gallon.

1. Describe an estimate that Felicia might do in her head while driving to decide how many gallons of gas she needs to make it to the gas station at the other end.
2. Assuming she makes it, how much does Felicia spend per mile on the freeway?

IM Commentary

This task provides students the opportunity to make use of units to find the gas need (N-Q.1). It also requires them to make some sensible approximations (e.g., 2.92 gallons is not a good answer to part (a)) and to recognize that Felicia's situation requires her to round up. Various answers to (a) are possible, depending on how much students think is a safe amount for Felicia to have left in the tank when she arrives at the gas station. The key point is for them to explain their choices. 

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This task helps illustrate Mathematical Practice Standard 2 in which mathematically proficient students make sense of quantities and their relationships in problem situations, create coherent representations of the problem given and attend to the meaning of the quantities.  Students will need to translate the description of the situation into symbolic representations by decontextualizing.  In the task at hand, they may ask themselves questions such as: “What do the numbers used in the task represent?” “How can I use units to understand the task and guide my solution?” “What operations will I need to use to solve this task?”  Students are asked to approximate the number of gallons of gas needed to make it to the gas station. As they determine this amount, they will conduct a reality check to make sure their estimate seems sensible including considering whether this situation may warrant over- or under-estimating quantities based on the context. Students will demonstrate flexibility in their thinking about this problem by moving fluidly between numerical representations and the context of the problem. Lastly, students also have the opportunity to demonstrate MP3, Construct viable arguments and critique the reasoning of others since there may be varying opinions regarding a safe amount of gas for Felicia to have in her tank.

Solution

1. To estimate the amount of gas she needs, Felicia calculates the distance traveled at 70 mph for 1.25 hours. She might calculate $$70\cdot 1.25 = 70 + 0.25\cdot 70 = 70 + 17.5 = 87.5 \mbox{ miles}.$$ Since 1 gallon of gas will take her 30 miles, 3 gallons of gas will take her 90 miles, a little more than she needs. So she might figure that 3 gallons is enough.

   Or, since she is driving, she might not feel like distracting herself by calculating $0.25\cdot70$ mentally, so she might replace $70$ with $80$, figuring that that will give her a larger distance than she needs. She calculates $$80\cdot 1.25 = 80+ \frac14 \cdot 80 = 100.$$ So at 30 miles per gallon, $3 \frac13$ gallons will get her further than she needs to go, so should be enough to get her to the gas station.

2. Since Felicia pays $\$3.50$ for one gallon of gas, and one gallon of gas takes her $30$ miles, it costs her $\$3.50$ to travel $30$ miles. $\frac {\$3.50}{ 30 \text{ miles}} \approx \frac{\$0.12}{1 \text { mile}}$, meaning it costs Felicia $12$ cents to travel each mile on the freeway.