Visual multiplication is delightful

It's been ages since I last checked in with math-doodling funnylady Vi Hart's YouTube feed -- too long. The latest installment's a great technique for visual multiplication (and binomial expansion!) that I wish I'd known when I was in school, and a little rumination on good sentence structure in mathematics.

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

I write books. My latest is a YA science fiction novel called Homeland (it's the sequel to Little Brother). More books: Rapture of the Nerds (a novel, with Charlie Stross); With a Little Help (short stories); and The Great Big Beautiful Tomorrow (novella and nonfic). I speak all over the place and I tweet and tumble, too.

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knappa

I could never remember 7×6, either. I usually decomposed it as (6×6)+6, though.
Anonymous

Easier – harder? For whom? If you are factoring in the range of learning styles and natural intelligences – providing multiple approaches to “seeing” math and “doing” math increases the probability of success for ALL students. I HATE math – can do it – have to do it but would rather do toilets first. Seeing this playful way of engaging with numbers enchants me – intrigues me to wonder again about mathematics. Too bad for me that I am way way past school years – maybe I could have had a more loving relationship with math over the years IF someone had provided other alternatives to learning it.
Anonymous

I wish more people would see math this way. It’s not just number crunching and getting the answer. It’s the consistent unambiguous expression, whenever it’s drawing lines or using numbers, that you can get a result that makes it interseting.

It’s not numbers that makes math. The numbers, for example 0,1,2, etc,. didn’t exist until Fibonacci introduced and made popular around the 13th century. Until then much of the Western world was stuck with Roman numerals.

Yup, sigh. But we don’t teach primary school kids that for some reason.
Anonymous

I’m an English teacher who avoids math at all costs. However, I am totally intrigued by all the different ways one math problem can be explained. If I had been shown that back in the day, I might not have such math anxiety. Thanks math people.
Anonymous

Warning! IAAM (I am a mathematician).

The notation she gripes about isn’t ambiguous. It’s the same notation I was taught and saw used by dozens of teachers from grade school to graduate school. The written rules match what I was taught.

Excel and programing seem stuck with a longer notation, but it’s no more explicit.

Oh, and for the ultimate shortcut math trick, our differential equations professor nailed it. When asked how he would solve a particularly thorny problem, he said he wouldn’t … he’d let Mathematica do it.
- SamSam
  
  The notation she gripes about isn’t ambiguous, but it is arbitrary. The order of Division, Multiplication, Addition, Subtraction could certainly be any other order.
  
  5 – 3 + 1
  
  I think most non-mathematicians would read that as 5 – 3 = 2, 2 + 1 = 3. Because that’s the way we read it in. The mathematician says “oh no, by convention we add first and then subtract, so the answer is 2.”
  
  Would anyone really be surprised if a Kenyan mathematician said “huh, over here we always do subtraction first.”
  
  Clear parentheses (and showing division as a fraction instead of using the division symbol) make it clear to anyone.
  
  5 – (3 + 1)
  
  is as clear as day to anyone with middle school math. (Sure, the idea of parentheses also has to be taught, but it’s a lot less arbitrary.)
  - CH
    
    “The mathematician says “oh no, by convention we add first and then subtract, so the answer is 2.”
    
    They do? You must have different mathematicians in your neck of the wood then. Addition and subraction have the same weight where I live. http://en.wikipedia.org/wiki/Order_of_operations
    
    5 – 3 + 1 is the same no matter what order you do it. You can do
    (5 – 3) + 1 = 2 + 1 = 3
    or
    5 + (-3 + 1) = 5 + (-2) = 3
    See, same thing.
    
    Now, if you do mean 5 – (3 + 1) then it’s not the same thing as 5 – 3 + 1. If you want to check, then just remove the parenthesis and you get 5 – 3 – 1 which is indeed 1.
    
    Remember: Multiplication and Division have the same weight. Addition and Subtraction have the same weight. Multiplication and Division goes before Addition and Subtraction.
    
    And… Math, it works!
    - CH
      
      Oh, and it’s easy to see why the have the same weight if you think that there really is no subtraction, what you are doing when you do
      5 – 3 + 1 is really 5 + (-3) + 1, and therefore you see easily that you can do them in any order, and the result will be the same.
      
      And, that 5 + ((-3) + 1) is not the same thing as 5 – (3 + 1).
    - SamSam
      
      You’re correct. I remembered the mnemonic BODMAS for the order of operations, and forgot that addition and subtraction have the same weight. And your point about subtraction just being the addition of negative numbers is a good one.
      
      Can we go back in time and pretend that my post above was about multiplication and addition together?
      
      My point (and the video’s point) still remains. Order of operations is still fairly arbitrary, and is something that has to be memorized if you’re going to make sense of 4 + 2 x 2. It’s better to simply make your equations clear by using parentheses.
  - SamSam
    
    And when I say the mathematician would say the answer is 2, I mean the mathematician in my head who can’t solve 5 – 4 correctly… Derp.
Anonymous

I disagree without the awesomeness of this method. In theory I like the idea, but I still sweat a little because the accelerated video make me feel like I’m way back in math class again, having the instructor blaze through the tricks and concepts before I’m given a chance to really absorb it. Sure, the naturally gifted math whizzes picked it right up, but it left everyone else not quite understanding it. Years later when I taught as a TA at the college level, I grew to really dislike this method of speed-teaching because no matter how cute the instructor thinks they are, the fact is that absorbing new material takes time to really internalize. Can you imagine getting an hour of this?
jphilby

97*86, no paper:

100*86 = 8600
- 3*86 = 258 (3*80=240 + 3*6=18)
————
8342
Anonymous

This is pointlessly complicated. The old way is far simpler. Just because something is new or pretty doesn’t make it better. This also applies to spouses. ;)
bkad

The problem I always had in school was the “I know you got the right answer doing it your way but we want you to do it our way” dilemma…that coupled with them always forcing me to show my work when I could do the problem in my head and I really didn’t know how to express it on paper…the school instilled doubt was always my biggest setback.

I certainly can’t defend all teachers in all cases, but when I was a Graduate TA there were at least a couple reasons I would insist my students do things a certain way:
1. Since I know the future (instructionally speaking), I can anticipate if a method that works for the student in certain simple cases is going to impede the student when things get more complicated. Doing things in a way that scales avoids re-learning later.
2. When I saw wrong answers, I would want to do forensics to identify where the student’s thinking went awry (both to help that student and to improve my own teaching.) Too hard to do that if work is missing or notation is nonstandard.

All that said, when I was taking introductory classes, I would skip all sorts of steps, ignore units until the end, etc., and I seem to have turned out ok.
Anonymous

Learn the multiplication tables and do it the traditional way, it’s much easier than this.
Anonymous

I am a math teacher and I can’t imagine teaching a two digit multiplication problem drawing a bunch of lines taking up an entire sheet of paper. Well, maybe I would consider it if it was a substantial difference from the “traditional way.” What does ease up the process (assuming the student understand the easy of multiplying and dividing by powers of 10) is to break that exercise down into the following:

97*86
=(90+7)(80+6)
= 90*80 9*8*10^2 = 7200 (“72 with two zeroes”)
+ 90*6 = 9*6*1 = 540 (“54 with one zero”)
+ 80*7 = 8*7*10 = 560 (“56 with one zero”)
+ 6*7 = 42
Then you have to add them all up just like you would with her method, or any multiplication methodology for that matter.

The video is VERY wrong about the order of operations though. As someone previously stated, parentheses are done first, then multiplication and division from left to right, followed by addition and subtraction from left to right. 48 / 2(9+3) = 48 / 2 (12) = 24 * 12 = 288
Anonymous

this is infinately harder, slower and more convoluted than the traditional way.
Anonymous

this is much more difficult than the traditional method.
Anonymous

this is MUCH more difficult than the traditonal method.

also, there must be something to be gained from memorizing multiplication tables.
- travtastic
  
  There definitely is. You just learn to break it down mentally, normally into digits. This makes every piece a listing on a multiplication table. Then you just add.
HenryS

@Anon (IAAM): I have never seen anybody in graduate school use the division symbol. Which area of mathematics are you in?
- Anonymous
  
  Odd, we use divisions symbols all the time. We use the / version, which means the same as the obelus symbol (Ã·). Ironically, the / symbol can introduce ambiguity since it is also used with fractions, but the obelus is pretty clear. 3Ã·2x is clearer than 3/2x.
  - Anonymous
    
    Actually 3Ã·2x is not clearer than what you probably mean by writing 3/2x (really should be 3/(2x) for clarity). Unless you place the 2x in parentheses, 3Ã·2x means 1.5x (3 divided by 2 times x).
SamSam

Well put! Although I’m happy to say that I worked out the connection between the line method and the regular method when the line method was still shown to me.

I do think it’s great to emphasize, as she does, that at the end of the day you’re just multiplying one number by another number, and that all these methods are just recognized techniques for breaking the problem down into easier chunks. If you want to do it some other way, that’s just as accurate! And if a teacher showed their students how the “regular way” is, in fact, just breaking down that huge grid, as she shows, I think students would understand it much faster.

As it is now, I think most students just see multiplication as a series of arbitrary rules, and so it’s super-easy to forget the order of them, or to skip a step or something. It doesn’t have any relation to what you’re actually trying to achieve.
bwcbwc

The “awesome” part is really how all these methods do the same thing under the covers. Your average student or non-mathematical adult never really _gets_ the generalization.
swishercutter

The problem I always had in school was the “I know you got the right answer doing it your way but we want you to do it our way” dilemma…that coupled with them always forcing me to show my work when I could do the problem in my head and I really didn’t know how to express it on paper…the school instilled doubt was always my biggest setback.
chgoliz

Excellent video. I wish all school-aged kids could learn how to be this creative in thinking about math. Learning by rote has its place, but it’s always not the right tool.
or420

I’m probably a unique case, but there were a lot of stuff I really didn’t like in school that I came to enjoy outside of school. I got good grades in most subjects, but didn’t enjoy any of them. Now that I’m almost out of college, I read far more than I did in school (about one book a week), I like math, especially calculus, I read stuff on physics in my spare time, and I generally I learn more than I did during my school years. Nowadays, I get a kick out of learning new things and I tend to remember almost everything I learn, contrary to the self-destruct sequence that happens right after final exams.
WombatBaloney

Wow. Slooooow down for us slackers in the back of the class.
SamSam

To all you saying that the lines method is not simpler: that’s the video’s whole point!

The lines method was going around the web. Young folks everywhere were saying “OMG that’s like so frikking cool”… or something.

This video shows them “hey look, this new method sucks when dealing with big numbers. But let’s take a look at it more closely. Hey look, this “new method” is actually exactly what we’re doing when we multiply the regular way!”

So the video is saying

1) What you thought was just a neat trick with lines is actually a graphical representation of what you’re doing when you use the multiplication system of multiplying units first, then tens, etc.

2) The regular way is so much easier once you get to bigger numbers.
Jesse M.

I’ve totally forgotten the normal way of doing multiplication by hand, so it is kind convenient to realize that 97*86 = (9*10 + 7*1)*(8*10 + 6*1) = 9*8*100 + 9*6*10 + 7*8*10 + 7*6 = 7200 + 540 + 560 + 42. Probably the normal way is a little faster but I’d probably just forget it again, I’ll use this if I’m ever stuck without a calculator…

Is there an analogous quasi-algebraic trick for remembering how to do long division? I guess it would have something to do with remainders, maybe you break 97/86 down into (90/86) + (0.01)*(700/86) so that in each case the numerator is larger than the denominator but by less than a factor of 10. Then that breaks down into [86/86 + 4/86] + (0.01)*[688/86 + 12/86] = 1.08 + 4/86 + (0.01)*12/86, then if you again make the numerators larger than the denominators you have 1.08 + (0.01)*(400/86) + (0.01)*(0.1)*(120/86) = 1.08 + (0.01)*[344/86 + 56/86] + (0.001)*[86/86 + 34/86] = 1.08 + 0.04 + 0.001 + (0.01)*(56/86) + (0.001)*(34/86), etc. etc. But perhaps there’s a way to think about it that makes it less laborious?
- Anonymous
  
  Division is inherently an iterative process because you have to test the trial divisor at each step to see if it’s off by one or not.
  
  dividend = divisor * quotient + remainder. At each iterative step you calculate:
  
  dividend – (divisor * partial_answer) = divisor * (quotient – partial_answer) + remainder
  
  In the traditional method you keep track of “dividend – (divisor * partial_answer)” under the problem while writing down the partial_answer above the problem. Any other method works so long as you can easily keep track of the changes in partial_answer and their effect on the dividend. Just stop when dividend – (divisor * partial_answer) < divisor and that value is your remainder.
- travtastic
  
  97*86 = (100*86)-(3*86)
  97*86 = (8600) – (258)
  
  Granted, that is as easy one for that shortcut.