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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?)

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÷33906 \div 3?

Answer: 13021302.

What makes us able to see this answer so swiftly? It seems natural to think of 39063906 as 3000+900+63000 + 900 + 6. It is easy to divide each of these components by three.

3906=3000+900+63906 = 3000 + 900 + 6
by three gives
3906÷3=1000+300+2=13023906 \div 3 = 1000 + 300 + 2 = 1302.

Great! And we see this natural decomposition too in a dots and boxes picture of 39063906. We literally see 33 thousands, 99 hundreds, and 66 ones.


Dividing by three gives this picture.


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÷3=?3906 \div 3 = ?” is really asking

How many groups of three can you find among a collection of 39063906 objects?

We know that there one thousand groups of three among 30003000 dots, and three-hundred groups of three among 900900 dots, and two groups of three among 6 dots. And our picture of 39063906 actually shows this too.


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 13021302 groups of three.

We can use tally marks to show that we have 11 group of three at the thousands level, 33 at the hundreds level, 00 at the tens level, and 22
at the ones level, again 13021302 groups of three.


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.


This final picture shows how many groups of three we had in the original 39063906. 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÷33906 \div 3 in the 1101 \leftarrow 10 machine.

Try performing the division problem 3906÷33906 \div 3 in the 1101 \leftarrow 10 machine.


Question 2

Draw a dots and boxes picture of the number 426426 and use it explain why 426÷2426 \div 2 equals 213213.

Try it on the app too.


Question 3

Let’s do another problem.

Let’s try 402÷3402 \div 3.


Question 4

We just showed that 402÷3=134402 \div 3 = 134.

What do you think is the answer, then, to 404÷3404 \div 3?


Question 5

Compute 61230÷561230 \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÷12276 \div 12.

Here’s picture of 276276.


Here’s what twelve dots look like.


But in a 1101 \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.)


And when we hunt for groups of twelve in 276276, we get this picture.


We see two groups of 1212 at the tens level and three 1212s at the ones level. The answer to 276÷12276 \div 12 is indeed 2323.

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