Question

Multiply using duplication and mediation

8 x 169

Answer 1

I suspect you want something like this:

8 x 169
= 4 x 338
= 2 x 676
= 1 x 1352

= 1352

I halved the first factor and doubled the second factor until the first factor becomes 1
Most people can double a number and half a number in their heads.

You would obviously start halving the smaller number.
The problem arises when the "halved" number is odd.
I used to teach this method as a novelty and called it "Newfie" multiplication, (a non-politically correct reference to my fellow countrymen from Newfoundland)

Here is how you would handle a problem like
34 x 56
= 17 x 112
= 8 x 224 , should have been 8.5, but oh well , just drop the .5
= 4 x 448
= 2 x 896
= 1 x 1792

Now drop all the rows where the first factor ( the one you halved) is even, leaving
17 --- 112
1 --- 1792

add the numbers in the right column
112+1792 = 1904

so 34 x 56 = 1904

Using this method, one does not have to know the multiplication table, only being able to double a number and to take half of a number.

My challenge to the students was to explain why this worked.