Trying to find math inside everything else

Archive for the ‘algebra’ Category

What does “mean” mean?

In both geometry and calculus this year the opportunity has come up to ask, “What is a mean average?” Students can usually pull together an answer that gives me a procedure (“add them up and then divide by how many there are”) but not one that really explains what that procedure shows.

With my tutoring students, I usually get to show them the method that my mother taught me when I was young, that I’ve never seen elsewhere. The procedure works by answering the question from the previous paragraph: a mean average is the value you’d have if the quantity were distributed evenly. (If we have 20 total cookies, how many to give each person so they are the same. If we have a certain budget for salaries, how to adjust them so everyone gets paid the same.)

So the method my mother taught me works that way – not adding and dividing, but redistribution. Let’s see an example.

Let’s take these five numbers I got from rolling a d100 five times. I want to average them. First, I’m gonna take a guess of around what the average is. The 10 is gonna pull it down a lot, so let’s guess the average is 70. So first I’ll redistribute the numbers from the ones larger than 70 to the ones lower than 70, like so.

Okay, now 4 of the numbers are the same, but the last one is too low. At least now when I redistribute I can take the same amount from all of them. Let’s do 5.

Pretty close! We have that 1 spare, so we’ll break that into a fraction, giving us a mean average of 65.2.

This idea of redistribution helps clarify the average value of a function or the average rate of change in calculus. Average value of a function is the y-value we’d have if we had the same total value (area) but redistributed so that all the y-values are the same (a constant).

The function in black, and its average value in purple.

The average rate of change of a function is the rate we would be going if we were going at a constant rate (aka draw a straight line, the secant).

You may be thinking that’s great, but often adding and dividing will be a cleaner and faster algorithm, and you’re not wrong. However, this method really shines when you have a question like one of these.

So let’s think. We have four tests, but we don’t know the fourth one.

We do know that we want an 89 average, which means we want all of the tests to be equal to 89. So let’s do that.

So we need a total of 10 extra points to get to that 89 average. Those points can’t come from nowhere – they have to come from the 4th test.

Therefore, that last test must be 99. Perfectly balanced, as all things should be.

Slopes and Lattices Game

Okay, here’s a game I came up with off the cuff today. It kinda worked, but I guess if other people tried it and gave feedback, that’d be swell.

Players: 2 (or 2 teams), each with two colors

Board: A 10×10 grid.

The game is played in two phases. In the first phase, each team takes turns placing points on the grid, until each team has placed 5 points. The origin always is claimed as a neutral point. Every point has to be on a lattice point. (In the example below, I was blue and my student was yellow.)

In the second phase, on their turn, each player may place a new lattice point and form a line with one of their original 5 points. If that line then passes through one (or more!) of the opponent’s original 5 points, those points are stricken. If one player can strike out all of the other player’s points first, they win. (If not, then whoever strikes out the most.)

There is one caveats to round 2 – when a line is drawn, determine the slope of that line and write it below. That slope can’t be used again.

Image
A game in progress. I was blue&red, and have struck out 4 of my student’s points. They were yellow&black and have struck out two of mine. It’s their turn.

After playing the first time, it became clear that much of the game came down to placing the points. If you could place one of your points so it was collinear with two of your opponents, you can strike them both with a single line. (But this only works if there is space for a 4th, alternate color point in phase 2 to form the line.) You also want to place your points defensively, with weird slopes that don’t pass through a lot of lattice points, to keep them safe. The second player definitely has an advantage when placing points, but the first player has an advantage when drawing lines, so I’m hoping those balance out.

Thoughts?

Vimes’ Theory of Socioeconomic Injustice

As I was planning my linear functions unit, I noticed a problem about someone choosing between two electric companies, with the standard idea of one having a larger start-up cost but a lower monthly rate. I realized these problems are very common, and they reminded me of of a quotation from my favorite author, Terry Pratchett:

The reason that the rich were so rich, Vimes reasoned, was because they managed to spend less money.

Take boots, for example. He earned thirty-eight dollars a month plus allowances. A really good pair of leather boots cost fifty dollars. But an affordable pair of boots, which were sort of OK for a season or two and then leaked like hell when the cardboard gave out, cost about ten dollars. Those were the kind of boots Vimes always bought, and wore until the soles were so thin that he could tell where he was in Ankh-Morpork on a foggy night by the feel of the cobbles.

But the thing was that good boots lasted for years and years. A man who could afford fifty dollars had a pair of boots that’d still be keeping his feet dry in ten years’ time, while the poor man who could only afford cheap boots would have spent a hundred dollars on boots in the same time and would still have wet feet.

This was the Captain Samuel Vimes ‘Boots’ theory of socioeconomic unfairness.

– Men at Arms

I wanted to share this concept with my class, so I looked for problems. One thing I realized, though, was that many of these problems involved one person choosing between two things. This makes it so there is one clearly correct answer.

But oftentimes, in the real world, people don’t have a choice. One person can afford the upfront cost to pay less in the long run, but another person can’t, and winds up paying more overall, as in the Pratchett quote above. So I decided to reframe the problems as comparisons between two people, to highlight that injustice.

The lesson started with a model problem, then I gave each group a different problem from the set below.

(I went through this page of unisex names to make all of the problems gender-neutral.)

Each table worked on a different problem (with some differentiation on which group worked on which), then they jigsawed and, in their new groups, they shared out their problems and how they solved them. Then, most importantly, they looked for similarities and differences between the problems.

We then read the Pratchett quote and discussed its meaning, and students had to agree or disagree (making a claim and warrant for each). We had a good discussion on whether it really applied to today’s world or not.

Potluck Math

I was talking to one of my co-workers about a “Friendsgiving” she is holding, and how the food bill is getting up there as more people are invited. But some of those people are also bringing food – and everyone is worried about having enough.

I realized this is a very common problem with potluck meals. Everyone wants to make sure they have enough food, so the more guests, the more they make. But think about this –

At a 4-person meal, each person makes a dish that feeds 4. (4 servings). So each person then eats 4 servings of food. (Which seems like a normal amount.)

Now it’s a 20 person meal, and each person makes a dish that feeds 20. So now each person eats 20 servings? That seems unlikely – it’s much more likely that people eat 3-6 servings, for 60-120 servings eaten, leaving 80 servings of food left over.

The problem here is that each attendee is treating the problem linearly, when it would better be modeled quadratically. Of course, this is complicated by the one hit dish that every eats a full serving of, and that other dish that no one eats, and everyone wanting to try a little of everything, so figuring out how much to cook can get complicated pretty quickly.