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Arithmos
Arithmos
Arithmos

Station  H
Long Multiplication 

Is it possible to do, say, 37×2337\times 23, with dots and boxes?

Question 1

Here we are being asked to multiply three tens by 2323 and seven ones by 2323.

If we are good with our multiples of 2323, we will realize that this must give must give 6969 tens (since 3×23=693\times 23 = 69) and 161161 ones (since 7×23=1617\times 23 = 161).

The answer is thus 6916169|161.

A machine like this one might help you.
With explosions this becomes 851851.

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Question 2

When Suzzy thought about 37×2337\times 23 for a little while, she eventually drew the following diagram:

I3SH-image94

She then said that 37×23=62321=8321=85137\times 23 = 6 | 23 | 21 = 8 | 3 | 21 = 851.

Can you work out what Suzzy was thinking?

What diagram do you think Suzzy might draw for 236×34236\times 34 (and what answer will she get from it)?

Question 3

Using Suzzy’s approach do 37×2337\times 23 and 23×3723\times 37 give the same answer? Is it obvious as you go through the process that they will?

Do 236×34236\times 34 and 34×23634\times 236 give the same answer in Suzzy’s approach?

Question 4

Here’s another fun way to think about multiplication. Let’s do it in a 121\leftarrow 2 machine this time.

Let’s work out 13×313\times 3.

Start by representing thirteen in a 121\leftarrow 2 machine.

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Question 5

We’re being asked to triple everything. So each dot we see is to be replaced with three dots.

I3SH-image106

And now we can do some explosions to see the answer 3939 appear (which is 100111100111 in the 121\leftarrow 2 machine).

Try this on the machine above and show that the answer 100111100111 would appear if we had the sixth box in the machine.

Alternatively, we can notice that three dots in a 121\leftarrow 2 machine actually look like this.

I3SH-image107

So we can replace each dot in our picture of 1313 instead by one dot and one dot one place to the left. (I’ve added some colour to the picture to help.)

I3SH-image108

Now with less explosions to do, we see the answer 100111100111 appear.

Try this too on the machine above.

You can either play with some of the optional stations below or go to the next island!

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