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# Re: Visual multiplication and 48/2(9+3)

## Video transcript

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.