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.
- 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.