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

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Re: Visual Multiplication and 48/2(9+3) : A quick response to some mathy things going around.

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While I'm working on some more ambitious projects, I wanted to quickly comment
on a couple 'mathy' things that have been floathing around the internet
just so you know I'm still alive.
So there's this video that's been floathing around about how to multiply visually like this:
Pick two numbers, let's say: 12 times 31...and then you draw these lines:
one, two...three, one. Then you start counting the intersections.
One, two, three on the left. One, two, three, four, five, six, seven in the middle.
One, two on the right.
Put them together: three-seven-two. 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, there'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 times eight lines and six lines.
Now all we have to do is count the intersections.
One, two, three, four, five, six, seven, eight, nine, ten... Okay 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, that's just six times seven 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 six times seven
and now I'm going to have to figure out five times seven
which is... [mumbling] ...so that's 35 and then add the sixth 7 to get 42.
Wow! I really should have known that one.
Okay, 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 to. Ones. Tens. Hundreds.
You add them all up and: voilá!
Which is exactly the same kind of breaking down into single digit multiplication
and adding that you do doing the old boring method.
The whole point is just to multiply every pair of digits,
make sure you've got the proper number of zeros on the end,
and add them all up. But of course seeing 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 combinations' 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 is 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 a 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 of what's right or 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 parentheses
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.