Total 99 Posts

## Dissent Of The Day: Mike Manganello

Mike Manganello offers a useful critique of Car Talk, pseudocontext, and WCYDWT:

I can certainly accept working definitions that require clarification, but the Car Talk problem confuses the issue [of pseudocontext] (at least for me). I’ve only done a little tweaking to the Car Talk problem:

“The fuel gauge of an 18-wheeler is broken, so the driver decides to check the gas level of his cylindrical gas tank with a dipstick. When the level of the gas measures 20 inches high, the tank is completely full. What will the dipstick measurement be when the gas tank is one-quarter full?”

Based on the working definition of pseudocontext, this problem fails on both counts. It completely ignores reality: Why wouldn’t you just fix the gas gauge? Then the problem asks for an irrelevant measurement: Why would we need to know that the tank is one-quarter full?

I assumed the trucker wanted to know one quarter of a tank (rather than four fifths) so he’d know when he had to refuel. An arbitrary number maybe (why not one fifth of a tank?) but not irrelevant. As for ignoring reality, I know more about mid-century Russian architecture than I do about long-haul trucking, but it seemed plausible to me that the trucker couldn’t waste time fixing the gauge in the middle of a run. In both of these cases, I deferred to the authority of the radio hosts. If either of Mike’s complaints were valid, why wouldn’t the hosts have echoed them?

Mike has also misquoted the definition of pseudocontext in small but crucial ways.

Mike’s: “It completely ignores reality.”
Mine: “Context that is flatly untrue.”

Mike’s: “Problems that ask for an irrelevant measurement.”
Mine: “Operations that have nothing to do with the given context.”

Mike:

Personally, I find the Car Talk problem kind of boring and not very mathematically rich.

Once again we find that a problem’s basis in either pseudocontext or context has nothing to do with how much anyone enjoys or profits from it. (Seems only to fair to mention, though, that Alex’s class had the opposite reaction.)

Mike:

Another word of caution: Mathematics is part utility and part artistry. By limiting mathematical study to problems related to genuine physical phenomena can only serve to retard the growth of mathematics.

It’s worth clarifying my total agreement here. My blog covers math applications pretty much exclusively not because I think those are the only problems worth studying but because those problems are the easiest to create and teach poorly.

## [WCYDWT] Pure Performance

Can I get forty comments on this video? The rest of this post (which I’ll update after we cross that threshold) depends entirely on the modal answer to the question, “what question perplexes you about this video?” Ask your own question before you look at the others.

BTW:

My advance prediction for the responses was a million-way tie for third place with these two responses taking the top two spots:

1. How fast was the car moving?
2. How many frames were on the wall?

I couldn’t guess which would win and, in terms of our mathematical objective, it doesn’t matter. Both questions are tightly interwoven. (Many commenters, in fact, asked both questions.) It mattered to the production company, though. One is its premise. The other is its conclusion.

The results, through Yaacov’s comment:

Sarah Cannon: “How fast is the car driving?” +25 related
mirjam: “how many pictures?” +16 related
Brent Logan: “How did the car jump? There wasn’t even a ramp.” +7 related
Other questions. +7

The behind-the-scenes video is less helpful than you’d imagine so I spammed basically every production company that had anything to do with the ad, asking them for information. Everyone was extremely helpful (why is this such a revelation?) and eventually my query was routed to Peter McAuley, the visual effects supervisor, who dropped a pile of knowledge on me, which I’ve uploaded here and which he cross-posted in the fourth comment here.

Important questions your students may have to reckon with.

1. How does this effect even work? More to the point, how does film work. The camera on the outside of the car is taking photos very quickly – at a rate of 24 frames per seconds. When they’re played back, our brains interpolate the rest of the motion and it looks smooth.
2. Did the production company decide on the size of the frames and then figure out the speed or did they know the speed they wanted and then determine the frames? The production company figured the faster the speed, the more exciting the effect. At a certain speed, though, even the best digital sensor produced a “rolling shutter” effect (excellent animated explanation here) so they went with film, which looked good up to 40 kph. Which, bingo. That figure is crucial to every question.
3. How long is the actual ad? This is the extended cut of a TV spot that ran for thirty seconds.

What’s your guess? How many frames do you think you saw fly by? How wide do you think each picture had to be? Give a number you know is too high / too low.

Ideally, either on their own or throughout the class discussion, your students will realize they need a) the length of the ad [30 seconds], b) the speed of the car [40 kph], and c) the shutter speed of the camera [24 frames per second].

From there, the dimensional analysis is up to you and your students.

Zipped archive [135.2 MB] containing:

• the extended commercial,
• the behind-the-scenes video,
• a PDF of the Peter McAuley’s e-mail.

## [WCYDWT] Car Talk

Sometimes this stuff is just sitting there for the taking, like a gold brick in the middle of the road, and all it takes is the right kind of eye to see it. David Petro e-mailed me a link to an episode of the long-running radio series, Car Talk, which does all the heavy lifting for us. I just shot a couple of supplementary photos.

The fuel gauge [on my 18-wheeler] is completely useless. It’s completely unreliable. So I have gone with a very low-tech method. I have gotten a wooden dowel which I use as a dipstick. Now the tanks on my 18-wheeler are cylinder-shaped and they’re on their sides. Now when I stick the dipstick in and I mark the level when the tank is completely full, it happens to be 20 inches. I mark ten inches at the half. My question is how I do I accurately mark one quarter?

I’d crop this clip pretty tightly. Rich says, “It obviously isn’t five inches,” but we’d rather put that to our students, asking them, “Just gimme a guess: how many inches do you think?” Some students – the impatient ones, maybe even a few advanced students – will snap to “five inches.” Have them sketch out their solution. Ask them if they still like it. Ask for a number they know is too large, too small.

Important questions your students may have to reckon with.

1. What information do we care about here? What information don’t we care about here?
2. Why does the trucker care about the level of a quarter tank anyway?
3. If you try out your guess, is there more than or less than or exactly a quarter of a tank left?
4. What kind of shape does the gas form in the bottom of the tank?
5. How is that shape formed from other shapes and how could we find its area?

These questions and their answers will vary significantly with the particular math course you’re teaching.

We finally came up with the answer, that in order for his dipstick to work he’s gotta be about 5.96 inches off the bottom. That’s a quarter of a tank. And it seemed like it should’ve been a lot more than that.

The Goods

Voila: a zipped archive. Thanks, David.

Extra Credit Assignment

Someone cook up a dynamic Geogebra applet for this scenario and pass me a link.

BTW: Woo! Check out Tim’s.

BTW: Updated to fix an error in my math. Thanks, RM.

## [WCYDWT] Grocery Shrink Ray

You run a Dollar Tree franchise. Everything costs a dollar.

You sell shampoo for a dollar – a popular brand called “White Rain.”

Problem: times are tough and the people who sell you the shampoo need to raise their prices higher than a dollar. What are your options at this point?

Teacher: write down student suggestions on the board. When someone suggests “sell less shampoo for the same price” resist the urge to declare, “Ahhh … that one’s eeeeenteresting,” thereby cluing your students to the fact that they weren’t really brainstorming, they were playing another game of “guess what the teacher wants to talk about.”

Spend a few seconds talking about, for instance, the student’s suggestion to rename the store. To what? Somewhere Around A Dollar Tree? You’re getting students all across the spectrum to invest in the problem and it’s costing you – what? – a few minutes.

Lower a little more structure onto the problem.

It turns out that “sell less shampoo for the same price” is exactly how that went down.

Teacher: “So what’s an interesting question we could ask here?”

Teacher: “How much extra are you actually paying here? What should the smaller bottle actually cost?”

Teacher: take guesses.

Your location in the scope and sequence of proportions will determine how much direct instruction your students will need here. That’s on you.

Issue extensions.

I dug through the entire Consumerist collection. The most valuable entries included both price and quantity.

Some of the others are great for discussion, though. Notice Dial’s effort to conceal the decrease with a taller, thinner container:

Tropicana’s opaque containers offer them enormous flexibility in the quantity of orange juice they give you:

And, if your class feels like venturing into some 3D geometry, ask your students how Yoplait has managed to shrink the amount of yogurt by 33% in a new container that looks nearly the same as the old.

The answer to that question (along with all the lesson materials) is located in this zipped archive.

[h/t MPG for the idea]

## Toaster Regression, Ctd.

Okay, so if you let the toaster cool down in between rounds, it is (more or less) linear. (Contra Dave’s experiment.)

Here, also, is an array of toast:

Try this:

1. Equalize the white balance on the toast photos,
2. Desaturate them,
3. Blur the heck out of the images,
4. Sample the center point of each slice, and then
5. Check the brightness value.

You get, well, rather uninteresting results. Had to scratch that itch, though.