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Station  11

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.


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.


Question 2

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


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.


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


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.


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?

I5S11 - Image003

Yes. Here’s one.


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


Question 4

Try a few new cases to practice!

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


Question 5

A challenge:

Compute 3900÷123900 \div 12.


Question 6

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

Here’s the picture.


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?


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

Try 31824÷10231824 \div 102 on the machine.


Question 7

Compute 46632÷20146632 \div 201.


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.)


Great! Now we are ready for the next station.

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