Station M
Further Explanation

As one mulls on the long-division process you come to realize that there are subtle issues at play. (Maybe that is why you came to this station on the island?)

Let’s go!

Question 1

Let’s take some time here to think through division more slowly. And let’s start with the example whose answer we can write down right away.

What is $3906 \div 3$ ?

Answer: $1302$ .

What makes us able to see this answer so swiftly? It seems natural to think of $3906$ as $3000 + 900 + 6$ . It is easy to divide each of these components by three.

Dividing
$3906 = 3000 + 900 + 6$
by three gives
$3906 \div 3 = 1000 + 300 + 2 = 1302$ .

Great! And we see this natural decomposition too in a dots and boxes picture of $3906$ . We literally see $3$ thousands, $9$ hundreds, and $6$ ones.

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Dividing by three gives this picture.

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But let’s probe even deeper into the workings of this final division step. What really happened here?

We can think of division as a task of grouping. “What is $3906 \div 3 = ?$ ” is really asking

How many groups of three can you find among a collection of $3906$ objects?

We know that there one thousand groups of three among $3000$ dots, and three-hundred groups of three among $900$ dots, and two groups of three among 6 dots. And our picture of $3906$ actually shows this too.

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If we did all the unexplosions, the green loop of dots would unexplode to give one-thousand green loops in the ones place. Each pink loop of dots unexplodes to make one-hundred pink loops in the ones place, and as there are three pink loops we get a total of three-hundred pink loops in the ones place… We see that out picture is really one of one-thousand green loops, three-hundred pink loops, and two orange loops. We have $1302$ groups of three.

We can use tally marks to show that we have $1$ group of three at the thousands level, $3$ at the hundreds level, $0$ at the tens level, and $2$
at the ones level, again $1302$ groups of three.

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And these tally marks show what happens if we were to actually divide by three: each group of three dots becomes one dot. We’d get this picture.

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This final picture shows how many groups of three we had in the original $3906$ . But we don’t actually need to draw this final picture: the tally marks in the picture before it show this information too. So we can stop drawing once we’ve figured out all the tallies.

Try performing the division problem $3906 \div 3$ in the $1 \leftarrow 10$ machine.

Question 2

Draw a dots and boxes picture of the number $426$ and use it explain why $426 \div 2$ equals $213$ .

Try it on the app too.

Question 3

Let’s do another problem.

Let’s try $402 \div 3$ .

Question 4

We just showed that $402 \div 3 = 134$ .

What do you think is the answer, then, to $404 \div 3$ ?

Question 5

Compute $61230 \div 5$ by the dots-and-boxes approach. Try it on the app too.
(Does it get tiresome drawing dots? Do you have to actually draw them?)

Exactly the same thinking applies to multi-digit division too. In station 11 we looked at $276 \div 12$ .

Here’s picture of $276$ .

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Here’s what twelve dots look like.

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But in a $1 \leftarrow 10$ machine we’d really see them as one dot next to two dots after an explosion. (All twelve dots still really reside in the rightmost box.)

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