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Station  11
Division 

Here’s an example of a division problem: Compute 276÷12276 \div 12.

And here’s a horrible way to solve it: Draw a picture of 276276 dots on a page and then circle groups of twelve dots. You will see, after about an hour, that there are 2323 groups of twelve in a picture of 276276.

Here’s a great way to solve it: Draw a picture of 276276 dots in a 1101 \leftarrow 10 machine and just see right away that there are 2323 groups of twelve in it!

Read and play on to see how we can do this!

Question 1

What is 3906÷33906\div 3?

That is, how many groups of 33 can we find in a picture of 39063906?

We can find 11 at the thousands level, 33 at the hundreds level, none at the tens level, and 22 at the ones level.

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Try it!

Drag the group of three dots in the card at the bottom right onto the machine to find groups of three dots in the picture of 39063906.

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Question 2

Calculate 402÷3402 \div 3 using the 1101 \leftarrow 10 machine!

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Question 3

Division by single-digit numbers is all well and good. What about division by multi-digit numbers? People usually call that long division.

Let’s consider the problem 276÷12276 \div 12.

Here is the representation of 276276 in the 1101 \leftarrow 10 machine.

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We are looking for groups of twelve in this representation of 276276. Here’s what twelve looks like.

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Actually, this is not right as there would be an explosion in our 1101\leftarrow 10 machine. We need to always keep in mind that this really is a picture with all twelve dots residing in the rightmost box.

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Okay. So we’re looking for groups of 1212 in our picture of 276276. Do we see any one-dot-next-to-two-dots in the diagram?

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Yes. Here’s one.

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Can you find more groups of one-dot-next-to-two-dots in the machine?

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Question 4

Try a few new cases to practice!

Use the dots-and-boxes approach to calculate 2783÷232783 \div 23!

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Question 5

A challenge:

Compute 3900÷123900 \div 12.

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Question 6

Let’s do another example. Let’s compute 31824÷10231824 \div 102.

Here’s the picture.

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Remember, all 102102 dots are physically sitting in the rightmost position of each set we identify.

Here’s my picture of the answer. I used different symbols for each group of 102102 that appears in 3182431824. Does my picture make sense?

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There are 33 groups | 11 group | 22 groups, that is, there are 312312 groups!

Try 31824÷10231824 \div 102 on the machine.

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Question 7

Compute 46632÷20146632 \div 201.

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Question 8

Here’s a tough challenge. It’s a problem that has a problem.
Can you make sense of the final answer you get?
(We’ll talk about this issue in the next station.)

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Great! Now we are ready for the next station.

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