## Exploding Dots

### 3.6 (Optional) Long Multiplication

Check out Goldfish & Robin’s “Where Young Minds Collide” videos taking the ideas of this lesson further: Multiplicaton in a 1 <–2 Machine ;  Multiplication in 1 <– 3 Machine ;  More than just Multipilcation in a 1 <– 2 Machine

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 you are good with your multiples of $$23$$, this must give $$3 \times 23 = 69$$ tens and $$7 \times 23 = 161$$ ones.  The answer is thus $$69|161$$. With explosions this becomes $$851$$.

But this approach seems hard! It requires you to know multiples of $$23$$.

Thinking Exercise:

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

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

a) Can you work out what Suzzy was thinking?

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

c) 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 work with a $$1 \leftarrow 2$$ machine this time.

Let’s compute $$13 \times 3$$.

Here’s what $$13$$ looks like 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).

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 a second dot one place to the left. (I’ve added some color to the picture to help.)

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

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