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Station  11Division

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

And here’s a horrible way to solve it: Draw a picture of $276$ dots on a page and then circle groups of twelve dots. You will see, after about an hour, that there are $23$ groups of twelve in a picture of $276$.

Here’s a great way to solve it: Draw a picture of $276$ dots in a $1 \leftarrow 10$ machine and just see right away that there are $23$ groups of twelve in it!

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

Question 1

What is $3906\div 3$?

That is, how many groups of $3$ can we find in a picture of $3906$?

We can find $1$ at the thousands level, $3$ at the hundreds level, none at the tens level, and $2$ 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 $3906$.

Question 2

Calculate $402 \div 3$ using the $1 \leftarrow 10$ 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 $276 \div 12$.

Here is the representation of $276$ in the $1 \leftarrow 10$ machine.

We are looking for groups of twelve in this representation of $276$. Here’s what twelve looks like.

Actually, this is not right as there would be an explosion in our $1\leftarrow 10$ 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 $12$ in our picture of $276$. Do we see any one-dot-next-to-two-dots in the diagram?

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 $2783 \div 23$!

Question 5

A challenge:

Compute $3900 \div 12$.

Question 6

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

Here’s the picture.

Remember, all $102$ 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 $102$ that appears in $31824$. Does my picture make sense?

There are $3$ groups | $1$ group | $2$ groups, that is, there are $312$ groups!

Try $31824 \div 102$ on the machine.

Question 7

Compute $46632 \div 201$.

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?