# Let the Random Problem Ideas begin…

One of the ways that I want to use this blog is to throw problem ideas out there and get feedback. I don’t want to make a big deal out of any of them, like putting them in immediately usable form, because if I require myself to spend inordinate amounts of time to publish them, this blog will die a very quick death. So I will just toss ideas out there and see what happens. A lot of them will suck, by the way. Hey, I sound like EmergentMath right now. I hope he won’t mind.

So here goes my first one. I was at a baseball game yesterday and in the parking lot saw this vehicle with a dirty rear window:

So, simple enough, here’s the problem:

- How well-designed is the wiper system? (or what is the ratio of clean:dirty)?
- Design a better wiper system.

I could also take many pictures of different vehicles to give variety (I like problems to be messy and to not personally know any “answers” ahead of the students).

One other note: when I worked for Motorola developing automotive software, I drove a Mercedes E320 for the weekend, and it had a single-wiper system for the front windshield. There could be something there too.

My own critique would be that it seems too easy for a full-on problem, but too hard for a bellringer. It might not be engaging enough either. But I’ll leave the majority of bashing to others…please.

[Also, I know that I have absolutely no following on this blog, and what I do have is probably more interested in SBG than math at this point…so feedback will be close to nil, but I have to start somewhere, right?]

I certainly don’t have a monopoly on potentially sucky, half-baked ideas. That said, this isn’t one of them. I see concepts of area, ratio, and annuluses (annuli?) popping up here. Give each group a different car model to design the “idea” wiper length?

Thanks, I definitely think it’s a good idea to give multiple pictures.

And nice use of the “annulus”, math’s answer to the boring old “donut”.

When I taught high school math, I found myself wondering, “why don’t my students see the math that is all around them?” Then it hit me: I failed to model the ways in which I see the math that is all around me. Your example here is one that will likely resonate with students, because they’ve seen it before. Heck, you might ask them to take a picture of a car from their residence and bring it in the next week to spice it up a bit, too.

Another possible prompt:

Why do you think Jeep (and so many other manufacturers) makes windshield wiping systems this way? (might be a good cross-discipline connection)

Nice idea, both on having them bring pictures and the second (more open) prompt.

When it comes to modeling, you make a great point. I find that a good number of the mistakes I make in class come from failure to model for the students. Something to always reflect on.

I think I might use this in my pre-calc course (community college). I don’t think it’s easy at all (for students). Thanks!

I meant not too easy. I just searched google images for windshield wipers, and then dirty windshield. Did you know there’s a whole … genre … of windshield art?!

Hi Sue,

Oddly enough, I stumbled on the windshield art the same way you did…trying to find more pictures of dirty windshields! Some of it is pretty amazing.

The only way I would consider it to be “too easy” is if the students already had a grasp of all of the mathematical concepts, rather than discovering them through the problem. But then again, it’s a bit too hard for a bell ringer too. Thanks for your input.

This looks like a great idea for a problem! In addition to considering the ratio of clean-window-to-dirty-window, I’m also wondering if the wiper is in the optimal position for the driver. For the purpose of being able to see out of their rear view mirror, is it better for the single wiper to be positioned in the center or somewhere slightly to the left or right of center?

Keep posting those half-baked ideas — I will be, too!

What a great idea – low threshold, high ceiling (students could devise a formula of the length of the wiper against the area cleaned – does it have a max value?) Thank you!

So Jeff, Did you ever implement this idea? Curious how the students embraced it.

Hi Monique,

I did actually implement it a few weeks ago and plan to write about it soon. In some ways it went better than expected, and in other ways worse. More to come. Thanks for your question.

I think starting with the question of what makes the best windshield wiper could be interesting. Have the students come to the conclusion that it is the area of the clean over the total area, or something like that. Then maybe have them compare different cars – whose got the best windshield wipers? Maybe the students bring in pictures of their parents cars…. The extension would be to then create the best possible windshield wiper using one or more wipers.