Exploding Dots

5.2 Division

Lesson materials located below the video overview.

Let’s start slowly with a division problem whose answer might be able to see right away.


What is \(3906 \div 3\)?


The answer is \(1302\).


If you think of \(3906\) as \(3000+900+6\), then we can see that dividing by three then gives \(1000 + 300 + 2\).


And we can really see this if we draw a picture of \(3906\) in a \(1 \leftarrow 10\) machine. We see groups of three: 1 group at the thousands level, 3 groups at the hundreds level, and 2 groups at the ones level.

That’s it! We’re doing division and seeing the answers division answers just pop right out!


Try doing \(402 \div 3\) with just a dots-and-boxes picture. Do you see that unexplosions unlock this problem to reveal the answer \(134\)?


And if you want to think deeply about what is really going on in these pictures (is it really this easy?) skip to the section “Deeper Explanation” in this chapter.


But if you are feeling ready to keep going right now … then let’s keep going!


Long Division

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 a picture of \(276\) in a \(1 \leftarrow 10\) machine.

And we are looking for groups of twelve in this picture 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. Twelve will look like one dot next to two dots. (But 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.

Within each loop of \(12\) we find, the \(12\) dots actually reside in the right part of the loop. So we have found one group of \(12\) at the tens level.

And there are more groups of twelve.

We see a total of two groups of \(12\) at the tens level and three \(12\)s at the ones level. The answer to \(276 \div 12\) is thus \(23\).



Here’s are some practice questions you might, or might not, want to try. My answers to them appear in the final section of this chapter.


1. Compute \( 2783 \div 23\) by the dots-and-boxes approach by hand.

2. Compute \(3900 \div 12\).



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


Here’s the picture.

Now we are looking for groups of one dot–no dots–two dots in our picture of \(31824\). (And, remember, all \(102\) dots are physically sitting in the rightmost position of each set we identify.)


We can spot a number of these groups. (I now find drawing loops messy so I am drawing Xs and circles and boxes instead. Is that okay? Do you also see how I circled a double group in one hit at the very end?)

The answer \(312\) to \(31824 \div 102\) is now apparent.



Here are some more questions to try, if you wish.


3. Compute \(46632 \div 201\).

4. Show that \( 31533 \div 101\) equals \(312\) with a remainder of \(21\). 



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