Exploding Dots

3.5 (Optional) Multiplication by 10

Let’s answer one of the previous practice questions here.

Why must the answer to \(26417 \times 10\) look like the original number with a zero tacked on to its end?

 

I remember being taught this rule in school: to multiply by ten tack on a zero. For example,

\(37 \times 10 = 370\)

\(98989 \times 10 = 989890\)

\(100000 \times 10 = 1000000\)

and so on.

 

This observation makes perfect sense in the dots-and-boxes thinking.

 

Here’s the number \(26417\) again in a \(1 \leftarrow 10\) machine.

Here’s \(26417 \times 10\).

 

Now let’s perform the explosions, one at a time. (We’ll need an extra box to the left.)

 

We have that \(2\) groups of ten explode to give \(2\)  dots one place to the left, and \(6\) groups of ten explode to give \(6\) dots one place to the left, and \(4\) groups of ten explode to give \(4\) dots one place to the left, and so on. The digits we work with stay the same. In fact, the net effect of what we see is all digits shifting one place to the left to leave zero dots in the ones place.

Indeed it looks like we just tacked on a zero to the right end of \(26417\). (But this is really because of a whole lot of explosions.)

 

3. a) What must be the answer to \(476 \times 10\)? To \(476 \times 100\)?

     b) What must be the answer to \(9190 \div 10\)? To \(3310000 \div 100\)?

 

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