My 2019 Mathematics A To Z: Unitizing


Goldenoj suggested my topic for today’s essay. It delighted me because I had no idea what it was. It wasn’t even listed on Mathworld, where I start all my research for these essays. It turned out to be something that I use all the time, but that I learned so long ago that it’s faded to invisibility. I didn’t even know that the concept had a name. So that makes it a great topic for an essay like this. I hope.

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Unitizing.

I once interviewed for a job I didn’t expect to get (or take). I would have taught for a university that provided courses for United States armed forces dependents. One bit of small talk that I thought went well had my potential department head mention a weird little quirk. United States-raised children were unusually good in multiplying stuff by 25. I had a ready hypothesis: the United States (and Canada) have a quarter-dollar coin. Many other countries just don’t, making do with 20-cent and 50-cent pieces instead. The potential department head said that was a good observation. United States-raised kids got practice turning four 25’s into a block of 100.

And this is the thing labelled as unitizing. A unit is, in this context, the thing we think of as “one thing”. This can be dollars, or feet of distance, or loaves of bread, or weeks of paid vacation. Whatever we need to measure. A unit often is made up of tinier pieces, cents or inches or slices or days. It can often be bundled up into bigger ones. Unitizing is about finding the bundle of things that makes the work one wants to do easy to understand.

This is a difficult topic for me to write about. I find it hard to notice myself doing it. But, for example, consider counting. Most people have a fair time counting up to five or six things at a glance. Eighteen things? There’s no telling that at a glance. What you can do, though, is notice that they group together, a block of six things here, another six here, another six there. Then the mass of things has turned into a manageable several collections of manageable counts of things. And, if we need to reverse the process, we can do that. Recognize that the 36 little triangular-wedge game tokens can be given out nine each to the four players. They can in turn arrange six of the tokens into an attractive complete wheel, and make do with the three remainder.

Slices of things turn up a good bit in thought about unitizing. One of particular delight that I found is this paper, by Susan J Lamon. It’s The Development of Unitizing: Its Role in Children’s Partitioning Strategies. Lamon investigated how children understand quantity, and the paper describes several experiments. A typical example is asking children how to evenly divide four pizzas among six people. And how their strategies change if all the pizzas are cut beforehand, versus whether they have to make the cuts themselves. Or how the question changes if things that are not pizza are considered. One child had different cutting strategies for four pizzas versus four cookies. The good reason: cookies are harder to slice than pizzas. You need to be more economical with your cuts so you don’t ruin the food.

And what kids found to be units depended on what was being divided. Four pizzas with different toppings would be divided differently from four identical pizzas. Four Chinese dinners were split by different strategies too. One child explained it just didn’t seem right to call what each person got four-sixths of each dinners. Lamon speculates this reflects cultural conventions about meals that are often eaten in common, and that feels right to me.

There’s obvious uses to this unitizing, in figuring how to divide pizzas and cases of 24 pop cans. There are subtler uses. Positional notation depends on unitizing. We group ten individual things into a new block, and denote it as something in a tens column. Or ten individual blocks-of-ten, which we denote as something in a hundreds column. And we go the other way as we need, when subtracting or dividing.

When I was learning base-ten (and other) arithmetic, they taught me to think of exchanging ten pennies for a dime, or ten dimes for a dollar, or back the other way. To someone hoarding pennies so as to afford things from the bookmobile the practice working out units worked well.

With that context you see why it’s hard to point out what’s happening. You aren’t reading a pop mathematics blog unless you’re quite at ease with calculation. That there is a particular skill done becomes invisible due to its ubiquity. It takes special circumstances to see it again.


Thanks for reading. This and the other essays for the Fall 2019 A to Z should appear at this link. I hope to publish the letter V on Thursday. And all past A to Z essays ought to be at this link.

Making Lots Of Change


John D Cook’s Algebra Fact of the Day points to a pair of algorithms about making change. Specifically these are about how many ways there are to provide a certain amount of change using United States coins. By that he, and the algorithms, mean 1, 5, 10, 25, and 50 cent pieces. I’m not sure if 50 cent coins really count, since they don’t circulate any more than dollar coins do. Anyway, if you want to include or rule out particular coins it’s clear enough how to adapt things.

What surprised me was a simple algorithm, taken from Ronald L Graham, Donald E Knuth, and Oren Patashnik’s Concrete Mathematics: A Foundation For Computer Science to count the number of ways to make a certain amount of change. You start with the power series that’s equivalent to this fraction:

\frac{1}{\left(1 - z\right)\cdot\left(1 - z^{5}\right)\cdot\left(1 - z^{10}\right)\cdot\left(1 - z^{25}\right)\cdot\left(1 - z^{50}\right)}

A power series is a polynomial. The power series for \frac{1}{1 - z} , for example, is 1 + z + z^2 + z^3 + z^4 + \cdots ... and carries on forever like that. But if you choose a number between minus one and positive one, and put that in for z in either \frac{1}{1 - z} or in that series 1 + z + z^2 + z^3 + z^4 + \cdots ... you’ll get the same number. (If z is not between minus one and positive one, it doesn’t. Don’t worry about it. For what we’re doing we will never need any z.)

The power series for that big fraction with all the kinds of change in it is more tedious to work out. You’d need the power series for \frac{1}{1 - z} and \frac{1}{1 - z^5} and \frac{1}{1 - z^{10}} and so on, and to multiply all those series together. And yes, that’s multiplying infinitely long polynomials together, which you might reasonably expect will take some time.

You don’t need to, though. All you really want is a single term in this series. To tell how many ways there are to make n cents of change, look at the coefficient, the number, in front of the zn term. That’s the number of ways. So while this may be a lot of work, it’s not going to be hard work, and it’s going to be finite. You only have to work out the products that give you a zn power. That will take planning and preparation to do correctly, but that’s all.