## Reading the Comics, April 18, 2017: Give Me Some Word Problems Edition

I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.

Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.

Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.

(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)

J C Duffy’s Lug Nuts for the 16th is a Pi Day joke that ran late.

Jef Mallett’s Frazz for the 17th starts a little thread about obsolete references in story problems. It’s continued on the 18th. I’m sympathetic in principle to both sides of the story problem debate.

Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, \$2.25 per pound had to be divided by three.

But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.

And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.

So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?

Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.

Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.

John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.

## The End 2016 Mathematics A To Z: Image

It’s another free-choice entry. I’ve got something that I can use to make my Friday easier.

## Image.

So remember a while back I talked about what functions are? I described them the way modern mathematicians like. A function’s got three components to it. One is a set of things called the domain. Another is a set of things called the range. And there’s some rule linking things in the domain to things in the range. In shorthand we’ll write something like “f(x) = y”, where we know that x is in the domain and y is in the range. In a slightly more advanced mathematics class we’ll write $f: x \rightarrow y$. That maybe looks a little more computer-y. But I bet you can read that already: “f matches x to y”. Or maybe “f maps x to y”.

We have a couple ways to think about what ‘y’ is here. One is to say that ‘y’ is the image of ‘x’, under ‘f’. The language evokes camera trickery, or at least the way a trick lens might make us see something different. Pretend that the domain is something you could gaze at. If the domain is, say, some part of the real line, or a two-dimensional plane, or the like that’s not too hard to do. Then we can think of the rule part of ‘f’ as some distorting filter. When we look to where ‘x’ would be, we see the thing in the range we know as ‘y’.

At this point you probably imagine this is a pointless word to have. And that it’s backed up by a useless analogy. So it is. As far as I’ve gone this addresses a problem we don’t need to solve. If we want “the thing f matches x to” we can just say “f(x)”. Well, we write “f(x)”. We say “f of x”. Maybe “f at x”, or “f evaluated at x” if we want to emphasize ‘f’ more than ‘x’ or ‘f(x)’.

Where it gets useful is that we start looking at subsets. Bunches of points, not just one. Call ‘D’ some interesting-looking subset of the domain. What would it mean if we wrote the expression ‘f(D)’? Could we make that meaningful?

We do mean something by it. We mean what you might imagine by it. If you haven’t thought about what ‘f(D)’ might mean, take a moment — a short moment — and guess what it might. Don’t overthink it and you’ll have it right. I’ll put the answer just after this little bit so you can ponder.

So. ‘f(D)’ is a set. We make that set by taking, in turn, every single thing that’s in ‘D’. And find everything in the range that’s matched by ‘f’ to those things in ‘D’. Collect them all together. This set, ‘f(D)’, is “the image of D under f”.

We use images a lot when we’re studying how functions work. A function that maps a simple lump into a simple lump of about the same size is one thing. A function that maps a simple lump into a cloud of disparate particles is a very different thing. A function that describes how physical systems evolve will preserve the volume and some other properties of these lumps of space. But it might stretch out and twist around that space, which is how we discovered chaos.

Properly speaking, the range of a function ‘f’ is just the image of the whole domain under that ‘f’. But we’re not usually that careful about defining ranges. We’ll say something like ‘the domain and range are the sets of real numbers’ even though we only need the positive real numbers in the range. Well, it’s not like we’re paying for unnecessary range. Let me call the whole domain ‘X’, because I went and used ‘D’ earlier. Then the range, let me call that ‘Y’, would be ‘Y = f(X)’.

Images will turn up again. They’re a handy way to let us get at some useful ideas.

## Making The End Of The World Quantitative

I haven’t forgot my little problem about working out where the apparent edge of the world was, from my visit to the Sleeping Bear Dunes in northern (lower) Michigan. What I have been is stuck on a way to do all the calculations in a way that’s clear and that avoids confusion. I realized the calculations were reasonably clear to me but were hard to describe because I could put into similar-looking symbols a bunch of things I wanted to describe.

So I’ve resolved that the best thing I can do is take some time to describe the things I mean, and why they’ll get the symbols that they do. The first part of this is drawing a slightly more mathematical representation of the situation of standing on top of the dune and looking out at the water, and seeing the apparent edge of the dune as something very much closer than the water is. This is what’s behind my new picture, a cross-section of the dune and a person looking out at its edge.

## The End Of The World, Qualitatively Explained

So I want to understand the illusion of being at the edge of the world at the Sleeping Bear Dunes in northern Michigan. Since I like doing mathematics I think of this as a mathematics problem; so, I figure, I need to put together some equations. Before I do that, I need to think of what I want the equations to represent, which is the part of the problem where I build a model of the dunes. In the process I should get at least a qualitative idea of the effect; later, I should be able to quantify that.

What’s a dune? Well, it’s a great whomping big pile of sand right next to the water. There’s more to a dune than that, but since all I’m interested in is how the dune looks, I don’t need to think about much more than what the dune’s shape is, and how it compares to the water beside it. If I wanted to understand the ecology of a dune, or fascinating things like how it moves, then I’d have to model it in greater detail, but for now I’m going to try out this incredibly simple model and see what it gets me.

## Just How Far Is The End Of The World?

I got to visit the Sleeping Bear Dunes National Lakeshore earlier this month. I thought I knew what dunes were, from the little piles of sand that accumulated on the Jersey Shore sometimes, but, no. These dunes, at the northern end of Michigan’s lower peninsula, look out on Lake Michigan from, at one spot on the Pierce Stocking Scenic Drive, about 450 feet above sea level. That’s a staggering height, that awed me, particularly as we approached it from the drive, and so had a previously quite nice enough drive through lovely forests come to a sudden, almost explosive, panorama of sand towering far above the ocean.

Maybe a quick working definition of a “scenic overlook” is a spot of land from which you can look noticeably down and see birds flying. This is vey scenic: had there been a Saturn V moon rocket, on the launchpad, at sea level, we would have been looking noticeably down at its escape rocket. For that matter, if there had been a Saturn V moon rocket, on top of which was somehow perched a Gemini-Titan rocket, we’d still be … well, we’d have to look up to see the crew in the Gemini capsule, but we would be about eye level with the top of the Titan booster, anyway.

Something that I imagine no picture except a three-dimensional one is going to capture, though, is the sense that one is standing at the edge of the world. From the top of the dune, the end of the sand seems to be very nearby, maybe a couple dozen feet off, and the water below is so clearly distant that it feels impossibly far away. Walking toward that edge makes the edge of the world recede, of course, but it never quite loses that sense of being on the precipice until quite far along.

Some mad souls do follow a trail all the way down to the beachfront of Lake Michigan; I wasn’t among them. The difficulty in walking back up — all on sand, on a pretty significant slope — from just walking a little near the edge and maybe dropping twenty feet or so in altitude convinced me not to carry on. I didn’t know it was a full 450 feet up, but it was obviously far enough.

The geometry of all this, though, has captivated me, and I hope to spend a couple essays here working out such questions as just how the optical illusion of this edge of the world worked, and how its recession works.