2 digit multiplication- how?

[ampere]

DS2 is in Y5 and can allegedly do 2 digit multiplication. However, he is trying to calculate areas and appears to have no idea how to do it! DH is trying to show him- but only via our oldie way, which is

12
13x

where you do 2x3 = 6
1x3 = 3

put down '0'
1x2 = 2
1x1 = 1

then add up the columns.

This obviously works BUT I feel DS2 needs to understand the method before he uses it! I mean, why the zero? OK, I know but I doubt DS2 would get it.

I want the method where you break it down into multiplying the tens, then multiplying the units, then adding the answers together.

Something like 10 + 3 = 13

So, 12 * 10 =120

3 * 12 = 36

120 + 36 = 156

I love the grid method which my DSDs taught me - I use it myself now! Hard to describe but you draw a little square divided up into 4 (ie 2 by 2)

You put 10 above the first column and 2 above the second column.

Then put 10 to the left of the top row, and 3 to the left of the bottoms row. (basically dividing into tens and units)

Then you fill in each square with the corresponding product so 10x10=100 etc, you end up with this:
100 . 20
30. . 6
In the square.
Then just add the 4 numbers.

Ooh in fact as you mentioned area, you could give him a literal demonstration of how it works - if you have a 12 by 13 rectangle you could divide it up into a 10x10 square, 2x10 rectangle etc.

It's a good method as you can do bigger numbers too, you just have more numbers to add at the end.

Thanks.

DS2 suddenly recalled he remembers to grid method when I drew a grid out for him!

The grid method is brilliant - it makes so much sense.

Is there another method where you multiply the 100's (in the case of 3 digit numbers), the 10's, then the units, then add the whole shebang together?

Well, you can do that with the grid method, just add another column - the way you describe is exactly how the grid method works.

Yep you would have a 3 by 3 grid if multiplying Eg 153x326

You can also do, say, a 2digit by a 3digit, it just means you have a rectangular grid instead of a square one. I'm doing a maths degree and I used the method in my recent non-calculator exam!

Once they understand the grid method, you can show them that the traditional method does do the same thing.

So when you are doing 12x13 the 'old way', the first row is in fact 12x3, and the second row is 12x10. You put the extra '0' down, because then you are going to actually do 12x1 and multiply it by 10. (This is not a great example really because children won't see the point of multiplying by 1 and then adding a zero in order to multiply by 10. Better to use an example where the bottom row is multiply by 20, say, and then you can show them how you only need to multiply by 2, and then put the extra zero down so that you've really multiplied not just by 2, but by 2x10 (i.e., 20). This sounds confusing to say in words here, but it's much easier when you are showing it!).

If you do several examples both ways, you can show how the first row of the traditional way is in fact the same answer as two of the grid values added up, and the second row of the traditional way is the same as the other two of the grid values added up.

The grid method is fine, but it can get cumbersome when it's quite long, and some children then need to write out the addition at the end, which takes longer, and they can struggle with getting all the place values lined up.

Another method that can be useful is sometimes called 'Napiers bones' (or 'lattice method'). You also draw a grid that is two boxes by two boxes, if you are doing a 2-digit by 2-digit sum. Then you divide each box diagonally into two, from top right to bottom left. Above to the boxes, you split the first number of the question into tens and units, and write the digit of the tens abvoe one column and the digit of the units above the other. Do the same for the number you are multiplying by, but put it at the right. Then multiply the numbers in the grid, and put the tens value on one side of the diagonal line, and the units on the other. You don't have to worry about adding extra zeros and so on, just doing it according to the single digits. When you are done, you add up along the diagonals of all the boxes, carrying into the next one if needed. It is a little hard to explain in words, but it is quite easy to show.

So you'd set up 36 x 43 like this:
(None of the underscores or commas mean anything, but I just had to put some kind of symbol in there, as spaces seem to be ignored and then the formatting doesn't work).

 3 <span class="underline">_</span> 6

---
|1/|2/| 4
|/2|/4|
---
|0/|1/|
|/9|/8| 3
---

Then add the diagonals, starting from the right: (carrying the '1' from the '14' that the second diagonal adds up to: 9+1+4 into the third diagnoal, so it becomes 2+2 + the carried 1 = 5)

,, 3 ,, 6
,, ---
,, |1/|2/| 4
1 |/2|/4|
,, --
, |0/|1/|
5 |/9|/8| 3
,, ---
, 4 , 8

= 1548

An advantage of that method, is that it is quite easy to add decimal points in. You can with the grid method too, but sometimes the boxes get very small by the time you've added all the zeroes that are needed in one direction or the other, and it can take a little thinking to work out exactly where to put the decimal point in each one. So this one is much less cumbersome in terms of the actual writing out and adding up side of things.

However the grid method, particularly with decimals, has the advantage of making sure the child really understand the whole concept of place value, which this method somewhat obscures.. So it is useful to teach the grid method first, and this lattice method as a short cut once place value is thoroughly understood.