At the opening of trig, most Pre-Cal texts will start with radian measure. You do some converting of angles between the two measuring systems and boom, a random digression about linear and angular speed. Most of the problems are like this:
This unit sticks out a bit, and the problems are quite random in the midst of all the triangle talk. For years it's been a throwaway for me and I didn't have a great way to look at these concepts. About a year ago, I handed out a really lame assignment for this. The issues I had with that assignment are the same as this saw blade problem: information given up front, explicit instruction, no opportunity for curiosity.
In our math department office we have some measuring tapes. They're 100' long and can be rolled up with a handle. They come in handy for lots of things and are especially useful here.
Instructions were simple. One person got our their phone and opened the stopwatch. The other person manned the tape. The timer stops when all 100' are rolled up. The person doing the rolling had to keep track of their rotation count.
Goal: Can you spin the handle at 1 mph?
Most students experience miles per hour in a car. The challenge would imply "I should spin pretty slow." Yet, the handle isn't very long. And there's the curiosity. What DOES 1 mph look like at this small scale?
So they spun.
Real data in hand, we walked through turning their rotation counts and times into a rotations per minute number. Then we discussed distance. What distance did the handle cover? Most were able to see that as circumference. A few unit conversions and we were able to see how everyone did. Some were pretty close. Many overshot.
Now with some reference about what 1 mph feels like, I gave a second task and had them do all the work to determine their speed. So, you think you can spin the handle at 3 mph?
I collected the data publicly. For the initial test, I had them calculate RPM on their own with a formula reveal to check work. Same for the final speed. They told me, then ran a formula to verify.
Fun task. Lots of interest from the kids from start to finish. Cut all the stuff that lacks real context (looking at you, radians per minute and πft/s). This unit is still a bit of an oddball, but it was more interesting this time.