Friday, January 25, 2013

Gridding and Chunking the new maths multiplication and division

Either it's a slow news day or a producer at the BBC has finally come across the new method of multiplication and division now being taught in schools. Either way in a feat of sheer brilliance they decided to explain it to parents by cramming in a short session before the local news at 8:30; you know when parents are out dropping their kids off at school. Due to the short session they also did a poor job of explaining it; or to be precise why these new methods are being taught and why they might be better in the long run.


I'll guess that the majority reading this are familiar with the traditional method of multiplication

 13
*12
130
+26
156

The new method is based on turning the problem into a "grid" and separating it out into simpler numbers

XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX 
XXXXXXXXXX|XXX
XXXXXXXXXX|XXX


So we have a 10*10 grid, a 3*10 grid, a 10*2 grid and a 3*2 grid Multiply those simple numbers together and add. That's the first stage, but drawing a whole set of points is tedious and if done a person might as well count them as they went along; thus the next step is to simplify that as follows:

    10| 3
10|100 30
 2| 20  6

Note this is exactly the same as the previous grid, but in a far more compact form. The answer is found by adding all the values within the table. Now this may seem a long winded method of multiplication, but it has two major benefits. The first is that it becomes obvious how numbers multiply together. Consider the tradition method again step by step

 13
*12
  0

Why is there a zero there? One could answer that it's to allow us to reach the first "1" of "12" or that because the "1" is really a "10" it has to be added, but neither is an intuitive explanation. It becomes "what you need to do to multiply" a black box formula that you don't need to understand simply the ability to process. The grid method shows exactly why every number is where it is. The second reason is that this method can be used in exactly the same way for algebra. Try multiplying x+2 by x+4 using the traditional method

 (x+2)
*(x+4)

Where would you even start? Now try it via "gridding"

   x| 2
x|x² 2x
4|4x  8

and the answer of x²+6x+8 drops out instantly. This method is known as the distributive property (amusingly causing right-wing panic in the USA). Now while gridding is fine, it's still a little clumsy and thus the final step needs to be added:

13*12 = (10+3)(10+2)=(10*10)+(3*10)+(3*10)+(3*2)

That is remove the grid and, ironically, return to the same traditional method taught in algebra

(x+2)*(x+4) = (x*x)+(x*4)+(2*x)+(2*4)

So that's gridding what about "chunking"? The sum used as an example at the BBC was 987/7 in traditional full long diversion that would appear like so:

  141
7|987
  7
  28
  28
   07
    7
    0

This seems obvious, but as with the initial "0" of multiplication there's a question of "why do you drop the numbers down?". The initial stage of "chunking" is to remove "chucks" from the number i.e. multiples of 7 like so

 987
-700 (100)
-280  (40)
-  7   (1)

Then add the multiples together. This can also be shown via traditional long diversion as follows:

  141
7|987
  700
  287
  280
    7
    7
    0

While this shows where every number comes from, unlike gridding this results in a set of far more difficult maths (287/7 rather than 28/7) It also causes problems when numbers don't even divide take 988/7 in traditional form

  141.14 
7|988.00
  7
  28
  28
   08
    7
    1 0
      7
      30
      28
       2

And so on. Try that using chunking

 988
-700 (100)
-280  (40)
-  8   (1)

141 and 1 remaining. While simply stating and 1/7 is fine at the basic mathematics level how would that be expressed as a decimal if the only knowledge of division a person has is via chunking? Try chunking with algebra using our multiplication example; how many (x+2) "chunks" fit into x²+6x+8?

x²+6x+8
x+2 (?)

Try it using traditional long diversion:

    x + 4  
x+2|x²+6x+8 
    x²+2x

     0+4x+8
       4x+8
          0

The same method can be used with minor modifications (how many x in x²) Okay it can be done via chunking with some additional lines :


x²+6x+8
x²+2x   (x)
   4x+8 (4)

Which again results in x+4, but is far more difficult to see.

As I see it "gridding" leads naturally to more advanced forms of alegbra; whilst "chunking", while being a better demonstrative method, may lead to complications further along.

3 comments:

Warner Carter said...

How about a genetic analogy? Let’s say that being even is a dominant allele, and being odd is recessive. Whenever a multiplication has at least one dominant allele, the result is even. Only when both multiplicands are odd, is the result odd. math

A.K. Fortis-Evan said...

I don't get it. If half the number of students surveyed said that the findings didn't add up, what did the other 54% say?

FlipC said...

I'm unsure how these comments relate to the post.

For the genetic analogy that works when we treat odd and even numbers as the only two alternative phenotypes, for basic genetics it's a fine analogy.

As for half and 54% to what does that relate?