In the previous lesson, we were looking at Egyptian fractions; now we're going to jump ahead four thousand years to see how rational functions are used in a completely different setting.
What's a "gas guzzler"? Why do we care? Do cars get the same "mileage" out of a gallon of gas under all conditions?
These simple questions should be enough to remind everyone of the notion of fuel efficiency. They're probably familiar with the term "mileage", but they might not know how to calculate it and might not be familiar with the terms "fuel efficiency" or "fuel economy".
Once I'm satisfied that the whole class understands that fuel economy refers to the miles a car can drive on a gallon of gas and that different driving conditions result in different fuel economy values, it's time to get the lesson started.
Hand out Combined Fuel Economy and let the students begin working individually or in small groups. Once the situation is fully understood, students should find the first few questions relatively easy. The "Quick Question" in the middle of the problem description simply asks students to calculate the mileage for a single trip. This quick check for understanding is important before students move on to the more complicated formula.
I expect students to begin to have some trouble when they reach the third problem, and I may find myself explaining the intention behind the problem with a similar example. Since I want to avoid doing too much of the think for the students, I'll use an example like 23/4 = 5 + 3/4. Once we explicitly identify the quotient and remainder in this simple case, students should have enough information to tackle the problem at hand. This is a good time to encourage them to make sense of the problem and persevere in solving it! (MP 1)
This problem set is adapted from Illustrative_Mathematics; please refer to this for the solutions and some excellent commentary and solutions.
I don't expect students to have finished the problem set yet, but I'll take the final ten minutes to bring everyone together for a summary discussion. This serves as a nice formative assessment for me, and helps the students to make sense of what they've accomplished.
First, we'll review the answers to the first 2 questions and make sure that everyone has a chance to ask any questions they might have. I imagine some will ask where the formula comes from, but besides saying that it is the harmonic mean of the two fuel economy values, I think we'll just have to take it for granted. At the same time, I'd encourage any interested students to research the harmonic mean and try to figure out why it might be used in this case. (For some insight, you might check out this article at Cut-the-Knot.)
Next, we'll discuss the initial set-up for the equation in question 3. I want to make sure that everyone understands the substitution of (x + 10) for y. (See this video for some more thoughts on this.)
Finally, I simply want to make sure at this point that everyone understands what they are being asked for in question 4. Making sense of the problem is challenging enough, that I'm happy at the end of this lesson if we've accomplished that.