02 Apr Dar’s Trick
I met Dar a couple years ago. Vicky had scheduled me to spend an hour with a new student, just playing around with math. I didn’t know anything about him except that he was six. I brought along my number cards, which are pretty great (if I do say so myself) to have handy, when I don’t really know what to expect.
That first day, I started Dar off with some two-card single-digit addition. He had no problem with that, so we switched over to subtraction – still no trouble. I put the low-number deck aside and grabbed the deck with numbers 1-60. This gave me an opportunity to learn a little about how his mind works. He still had very little trouble, but it took him a just a bit longer. The important thing was that, it gave me an opportunity to ask him how he was getting to his answers. He was really great at explaining his steps, all of which were nicely and analytically reasoned out. I was really impressed. After some time playing with addition and subtraction with the higher numbers, we went back to the 1-12 deck and I had him do some multicard addition – questions like, 12 + 4 + 7 = ?. He really enjoyed it and kept asking for more cards. From three cards, we went to four, then five, then six and finally seven. Dar was great at finding ways to combine the cards to make tens to quickly arrive at a solution. For example, if the problem was 4 + 8 + 12 + 7 + 6 = ? he was lightning fast at seeing that 12 + 8 was 20 plus 4 + 6 = 10, for 30 plus the leftover 7 to equal 37. And not only was he great at doing it, but he was great explaining how he did it, which I found pretty impressive.
After that day, I began meeting with him once a week. We quickly advanced from that multicard addition to multiplication and then division and division with fractional remainders and then further exploration of fractions and more. I moved him forward fairly quickly, because he was great at mastering everything I threw at him. Eventually, I found a level that was challenging for him, where we could slow the pace and dive a little deeper, but it was well beyond where I would ordinarily have expected it to be. The important thing – to me anyway – but I think to him too – was that we emphasized fun – To Dar though, fun includes being challenged.
So, fast-forward a year or so and Dar and I were still meeting once a week. We had been memorizing the perfect squares up to 25 x 25. A couple weeks earlier, I had mentioned to Dar, who had already memorized 12 x 12 = 144, that 132 = 169, which was kind of easy to remember because 142 = 196, same digits, just rearranged – so you get two for the effort of memorizing one – a pretty good deal, I thought – and he agreed. I think I also pointed out that 152 = 225, and from there he just wanted more. He and I had been working on two-digit multiplication, so it wasn’t too difficult for him to figure out for himself 16 x 16, by doing (16 x 10) + (6 x 10) + (6 x 6) = 256, but the idea of memorizing the answers appealed to him, so that first day, I think he memorized up to 17 x 17, and by the next time I saw him, he had laid out a schedule for himself that would have him memorize three per week until he was up to 252.
That’s about where he was, one day when he came in and noticed something interesting up on the board. His session that semester was immediately after my algebra class, and during that class, I had mentioned to the students a neat little trick for finding the squares of numbers that end in 5. It turns out that to find the square of a number that ends in five, you simply need to multiply the part of that number that is not 5, times that part plus one, and then stick a 25 at the end. Let me show you what I mean:
Let’s find 652. We take the part of 65 that is not 5, so that’s 6. Then we multiply that times that part plus one, that’s 7. Then we stick 25 on the end. So the whole math problem is 6 x 7 = 42, so 652 is 42 with a 25 stuck on the end 652 = 4225!
Likewise, 752 = 5625 and 852 = 7225
Now, that’s a pretty neat trick, but it’s not Dar’s trick. The interesting thing about tricks – well, one of the interesting things – is why they work. I mentioned to Dar that this trick for finding the squares of numbers that end in five, is related to the fact that while 7 x 7 = 49, 6 x 8 = 48 or that while 5 x 5 = 25, 4 x 6 = 24. I pointed out to him that when multiplying numbers that have a difference of two, they will always be one less than the square of the number in between. I then went ahead to write out other problems on the board – what about numbers that are four away from each other? what about 6 away? And so on. Dar quickly saw that the solutions were always the square of the number in the middle minus the square of the difference between that number and the ones you’re multiplying. We better look at some examples.
7 x 7 = 49, so 6 x 8 (7 + 1 times 7 – 1) = 48 or (72 – 12) and for the same reason, 5 x 9 = 45 (7<2 – 22 and 4 x 10 = 40 (72 x 32) and so on.
Dar didn’t know any algebra in those days, and so he definitely didn’t know about factoring special quadratics, which meant that I couldn’t just explain this as the difference of squares, so I drew grids of various dimensions and diagrammed relationships and got into a ridiculously involved explanation about why this works, and he ate it up. The thing was, that while this just seemed like a neat, but sort of useless (I mean how often do you need to multiply numbers that are equidistant from a given number that you just happen to know the square of? and when you do, is that really the easiest path to the solution?) property of numbers to me – Dar, could see that this was something he could apply his newfound fondness for squares to. He grabbed hold of this and ran with it. In those days, we were spending a fair amount of our weekly hour together practicing two digit times two digit multiplication, and while I would always sort of rely on my old methods of just distributing things out and adding the results, Dar immediately adopted this new method of finding the difference of squares. So for example.
If the question was 17 x 23, I might do 17 x 25 = 425 and then subtract 17 x 2 = 34, which I’d have to convert in my head to 425 – 25 – 9 = 391. Or maybe I’d do it as 17 x 20 = 340 + 17 x 3 = 51 and then get 391, but Dar would do 202 – 32 (400 – 9) and be at 391 almost as quickly as I was. His speed increased dramatically! Which made me reconsider my methods – because I wasn’t ready for him to start getting the answer first – I can be competitive too!
Let’s look at a few more examples of Dar’s trick in action;
27 x 21 = 242 – 32 = 576 – 9 = 567.
28 x 14 = 212 – 72 = 441 – 49 = 392.
32 x 18 = 252 – 72 = 625 – 49 = 576. Wait, what?! That’s 242! 32 x 18 = 242! How does that work? Okay, we better not get into that now, but here’s where I’d start if I was you and I wanted to get to the bottom of it – look at the prime factors of 32, 18, and 24 – is there a trick in there waiting to be discovered?
16 x 22 – 192 – 32 = 361 – 9 = 352, but of course you knew that because you can double 16 and cut 22 in half and then use Xinwan’s trick.
So, just as Xinwan’s use of her trick made me reconsider the way I multiply by eleven, Dar really sold me on his way of solving these problems – and it’s surprising how often they come up, once you start looking for them. Of course it only really works if you’ve got the squares memorized at least up to 25 x 25, but it’s a pretty good motivation to do so.
Sometime soon, I’ll try to write a post going into a little more of an explanation about the math of this trick, so we can get to the bottom of why it works.