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Re: Visual multiplication and 48/2(9+3)
While I'm working on some more ambitious projects, I wanted to quickly comment on a couple of mathy things that have been floating around the internet, just so you know I'm still alive. So there's this video that's been floating around about how to multiply visually like this. Pick two numbers, let's say, 12 times 3. And then you draw these lines. 12, 31. Then you start counting the intersections-- 1, 2, 3 on the left; 1, 2, 3, 4, 5, 6, 7 in the middle; 1, 2 on the right, put them together, 3, 7, 2. There's your answer. Magic, right? But one of the delightful things about mathematics is that there's often more than one way to solve a problem. And sometimes these methods look entirely different, but because they do the same thing, they must be connected somehow. And in this case, they're not so different at all. Let me demonstrate this visual method again. This time, let's do 97 times 86. So we draw our nine lines and seven lines time eight lines and six lines. Now, all we have to do is count the intersections-- 1,2, 3, 4, 5, 6, 7, 8, 9, 10. OK, wait. This is boring. How about instead of counting all the dots, we just figure out how many intersections there are. Let's see, there's seven going one way and six going the other. Hey, let's do 6 times 7, which is-- huh. Forget everything I ever said about learning a certain amount of memorization in mathematics being useful, at least at an elementary school level. Because apparently, I've been faking my way through being a mathematician without having memorized 6 times 7. And now I'm going to have to figure out 5 times 7, which is half of 10 times 7, which is 70, so that's 35, and then add the sixth 7 to get 42. Wow, I really should have known that one. OK, but the point is that this method breaks down the two-digit multiplication problem into four one-digit multiplication problems. And if you do have your multiplication table memorized, you can easily figure out the answers. And just like these three numbers became the ones, tens, and hundreds place of the answer, these do, too-- ones, tens, hundreds-- and you add them up and voila! Which is exactly the same kind of breaking down into single-digit multiplication and adding that you do during the old boring method. The whole point is just to multiply every pair of digits, make sure you've got the proper number of zeroes on the end, and add them all up. But of course, seeing that what you're actually doing is multiplying every possible pair is not something your teachers want you to realize, or else you might remember the every combination concept when you get to multiplying binomials, and it might make it too easy. In the end, all of these methods of multiplication distract from what multiplication really is, which for 12 times 31 is this. All the rest is just breaking it down into well-organized chunks, saying, well, 10 times 30 is this, 10 times 1 this, 30 times 2 is that, and 2 times 1 is that. Add them all up, and you get the total area. Don't let notation get in the way of your understanding. Speaking of notation, this infuriating bit of nonsense has been circulating around recently. And that there has been so much discussion of it is sign that we've been trained to care about notation way too much. Do you multiply here first or divide here first? The answer is that this is a badly formed sentence. It's like saying, I would like some juice or water with ice. Do you mean you'd like either juice with no ice or water with ice? Or do you mean that you'd like either juice with ice or water with ice? You can make claims about conventions and what's right and wrong, but really the burden is on the author of the sentence to put in some commas and make things clear. Mathematicians do this by adding parenthesis and avoiding this divided by sign. Math is not marks on a page. The mathematics is in what those marks represent. You can make up any rules you want about stuff as long as you're consistent with them. The end.