I will start today’s class by giving students the answers to yesterday’s exam review and asking if they have any questions for tomorrow’s exam. I plan to pay particular attention to their conceptual understanding of the trigonometric topics. It can be easy to rely on strictly procedural understanding when talking about trig, but you will have to build these discussions. For example, if a student asks about how to write an algebraic expression for sin(arctan x), I stress that arctan x is an angle measure, so we merely want to find the sine of that angle.
I got the idea for this Car Talk problem from Dan Meyer’s blog a few years ago. The first time I tried this with my class still stands as my favorite lesson I have ever taught. Students were engaged, there were great conversations, and it seemed like every single student could contribute to the problem solving. One aspect of a great problem is that every single student has an entry point and they all can be engaged- that is certainly the case for this problem. I had a handful of students who stayed after school for about 45 minutes because they wanted to know the answer to this question so badly. This is definitely a problem you want to solve before students work on it. It is very complex so you want to be prepared for what problems will arise.
The premise of this problem is taken from the NPR show Car Talk. Here is the actual audio from the radio show. Play this for your students to get them thinking about the problem. WARNING: the audio clip gives the answer, so make sure you listen to it beforehand and note where you should stop it so that students don’t know the answer before beginning.
There are a lot of different math topics at work in this problem! Circles, sectors, triangles, inverse trig, and graphing are all part of the solution path. I explain more in the video below.
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This PowerPoint has some information about the problem that you can share with your class. One the second slide is a written description of the problem. Give students a minute to discuss with their groups what height they think will correspond to ¼ of the fuel remaining in the tank. On the third slide, gather some guesses from the students in the class. One thing that will probably come up is that it has to be less than 10 inches.The purpose of this is just to generate some reasonable answers. If a students derives an equation and get 32.9 as the answer, they will have already filtered some unrealistic answers and will hopefully understand that 32.9 can't be the answer.
After I have some estimates from the class, I put students in groups of three to work on this problem. I explain to them that it is a very challenging problem and that I expect them to work through it and keep a record of their work. I also explain that I want as much precision as possible. Merely saying around 7 inches is not sufficient, I want an answer that is accurate to three decimal places. After assembling groups, I give students at least 10 to 15 minutes to work undisturbed. I do not want to lead them down a particular path until they absolutely need it.
There are usually a few hints I will give students. (You need to make the decision if these are necessary or not.) Ideally, I want students to do the most independent work that they are capable of. Sometimes it will be necessary to intervene with a few groups - other times the entire class may be at a roadblock and you will need to get everyone on track. I don’t want to give away too much, but here are some hints that I will usually give students either in groups or with the class.
Ask students why we are only focusing on a circle when the tank is actually a cylinder. Then ask them to think about what in the diagram are we actually trying to find. They will say the height of the fuel, so you can update your diagram to look like what is shown below.
Next, ask them about the area of the green portion of the circle. They will likely say that it has to be 25% of the total area. Since we know the diameter of the circle is 20 inches, we know that the area of the green portion is 25pi square inches.
At the end of class, explain to students that we will be working on this problem the day after the exam. Students will have some more time to work on this with their group.