# Station H**Long Multiplication**

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

Station H

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

Here we are being asked to multiply three tens by $23$ and seven ones by $23$.

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

The answer is thus $69|161$.

A machine like this one might help you.

With explosions this becomes $851$.

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

She then said that $37\times 23 = 6 | 23 | 21 = 8 | 3 | 21 = 851$.

Can you work out what Suzzy was thinking?

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

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

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

Here’s another fun way to think about multiplication. Let’s do it in a $1\leftarrow 2$ machine this time.

Let’s work out $13\times 3$.

Start by representing thirteen in a $1\leftarrow 2$ machine.

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

And now we can do some explosions to see the answer $39$ appear (which is $100111$ in the $1\leftarrow 2$ machine).

Try this on the machine above and show that the answer $100111$ would appear if we had the sixth box in the machine.

Alternatively, we can notice that three dots in a $1\leftarrow 2$ machine actually look like this.

So we can replace each dot in our picture of $13$ instead by one dot and one dot one place to the left. (I’ve added some colour to the picture to help.)

Now with less explosions to do, we see the answer $100111$ 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!