Real to me. My wife and I were on a beach recently and found ourselves in this math problem. This happens to every math teacher, I’m sure. We use our own product. We employ mathematical reasoning in our own lives in obvious and subtle ways. I’ve tried to discipline myself not to miss those moments, to instead write them down, photograph them, and turn them into a task where students experience the same dilemma my wife and I did.
Google Maps. The game here is to screenshot a bunch of tiles from Google Maps, align and stitch them together in Photoshop, and then fly around that large image in AfterEffects.
Use appropriate tools strategically. The sequels aren’t optional here. One sequel suggests that the cart will start moving towards you and asks “at what location will both paths take the same time?” The other asks for an even faster path than either of the two originally posed.
In both cases, I enjoyed setting up and solving the algebraic models.
But as I contemplated solving one equation and finding the minimum of another, symbolic manipulation never occurred to me. Without any teacherly presence hovering over me, nagging me to rationalize my roots, the most obvious, practical solution was Wolfram Alpha — no contest.
A teacher at a workshop pulled off a similar move this week and felt embarrassed. He said he had “cheated.” Tools like WolframAlpha require us to come up with a more modern definition of “cheating.” (And of “math” for that matter.)
The ladder of abstraction.
- the dog and the ball are represented by points; their dogness and ballness have been abstracted away,
- very little of the illustration looks like the scene it describes, for that matter; the water and sand are the same color; the image of a dog swimming after a ball has been turned into the remark “1 m/s in water,”
- points have already been named and labeled,
- important information has already been identified and given,
- auxiliary line segments have already been drawn; the segments AB and BC and DC don’t actually exist when the dog is running to fetch the ball; they have been abstracted from the context later.
My version of the task starts lower on the ladder. You see the sand and the sidewalk. You see what it looks like to walk in each. They aren’t abstracted into numerical speeds until the second act of the problem, after your class has discussed the matter. I do draw a triangle on the video, which is a kind of abstraction. I didn’t see any way around it, though.
BTW. Andrew Stadel also has a nice task involving the Pythagorean Theorem and rates.