The use of a counting cloth is very much process based. The concept of times tables wasn't really strongly used until the days of pen reckoning with arabic numbers. The books introducing pen reckoning laid out pages and pages of times tables, though they did not call them that. On a counting board, you didn't have to be able to do much beyond counting and moving counters around to do some quite tricky multiplications, and you certainly didn't need to understand it, but you did have to be methodical and concentrate. It is very easy to go wrong if you are not. Watch your sleeves and don't lean on the table either!
Let's start with 11 x 4. Lay it out, put the bigger number on the left.
Now, we know that 11 x 4 = 22 x 2 = 44 x 1. Well, that's exactly the method we're going to use; it's called halving and doubling.
Double the left hand number and at the same time halve the right. To double, you just count the number of counters in each row and add the same again. To halve, you just take every other counter away on a line. To work, this requires there to be an even number of counters on every line. Don't forget to double every single line and halve every single line.
Here the left has been doubled and the right halved. Do it again, until the right hand side has only one counter in the 1s row.
One counter in the 1s row.
Tidy up, if you need to.
Let's do 12 x 6. Lay it out.
Oh dear, I've got single counters (or an odd number in at least one row). Let's try converting that 5 to 5 ones.
Oh good, an even number of counters on every row. I can halve that.
Uhoh, there are now three counters in the 1s row. What can I do?
This is where we start to use that third column on the board. Take a copy of the left hand side over to that third column, and take away one counter from the 1s row. We're now saying that 24 x 3 = ( 24 x 2 ) + 24.
We can now concentrate back on the first two columns; halve and double
We have 48 x 1 + 24. We can simplify by removing the "x 1"
Ah, a simple addition.
Better do some tidying, two stages needed here. Work from the bottom up.
And the answer.
If you have the bottom 1s row with even numbers, but higher up ones with an odd number in, you need to work those counters downwards and the problem is likely to disappear. You only need to do the copy over if you have an odd number of ones (the entire value you are representing is odd). If you already have a number in column 3, then you copy to column 4 and then add 3 and 4 together. I will do a worked example of this as soon as I have time.
Once you are good at keeping track, try Division
Go to:
Division
Introduction
Addition
Subtraction
