I recently wrote about how I like to teach the Racks and Tubes: with a story! But in these interesting times, many of us are trying to teach the Racks and Tubes without any Racks and Tubes. What a nightmare! I think I have found an alternative, and I plan to try it with my students this year. It’s called Exploding Dots, and it’s genius. James Tanton, the creator of Exploding Dots is a long-time math educator who has come up with a huge range of interesting ways to explore math. I highly recommend all his work.
I’m tempted to simply point you to his website and leave it at that, but I’d like to share how you could use a story with the Exploding Dots. I suggest you spend a little time exploring his website to understand how Exploding Dots works, and then come back here for the story.
This time around, we’ll tell the story with tulip bulbs. Let’s imagine we have 443,592 bulbs, and we’re going to distribute them to buyers from 36 stores. The nursery doesn’t really want to lug around and count 443,592 individual tulip bulbs, so instead, they package them up. Here’s how they do it:
Unit: an individual tulip bulb
Ten: a bag of ten bulbs to be sold as a set to the customer
Hundred: a display box of ten bags of bulbs
Thousand: a case of ten display boxes
Ten Thousand: a pallet of ten display boxes
Hundred Thousand: a truckload of ten pallets
We can represent this on a hundred chart that looks like this:
Packaging of tulip bulbs on a base-10 chart
Now, since we don’t have beads and containers to represent all the tulip bulbs, we’ll put dots into the sections of this chart. You’ll notice that I’m using Montessori hierarchical colors. If your children are familiar with these colors or you are teaching in a Montessori classroom context, you may find this very helpful. If not, you can just use a single color. I wouldn’t use lots of colors at random, as this can distract from the point.
Dots placed on a base board
(N.B. you could do this with the Stamp Game, small objects in little cups, or other manipulatives, but James Tanton recommends a whiteboard and marker, as we’re going to want to be able to erase dots pretty soon.)
So now we’ve represented all our tulips, let’s think about the people. These tulips are being delivered to 36 buyers, which is a lot to organize. So instead, the folks from the nursery divide the people into teams of ten and each team chooses a captain. When each individual is supposed to get a single bulb, the team captain gets a bag of ten bulbs and then the team opens it and shares out the bulbs between them.
No one really wants to deal with lots and lots of individual bulbs, so the sellers and the buyers agree to start by distributing whole truckloads of bulbs. They look at their trucks, realized they only have four full trucks (plus a fifth one, partially filled with all the spillover), and agree they are going to have to open up the trucks and deliver pallets to each buyer instead. So each team of ten crowds around one truck, to get their pallets, and the sellers open up the last full truck and distribute a pallet to each of the remaining six people.
Whoops! We don't have enough pallets to give one to each of the six individuals. Fortunately, we have an extra truck that hasn't been touched, so they can open up that truck and there are ten more pallets inside, more than enough for everybody. Notice again that I’m using hierarchical colors to group the dots. You can do this if it makes sense for your children, or just use a single color.
Now everyone gets one pallet go tulip bulbs
Now every buyer has gotten a pallet of bulbs, and there aren't enough pallets left for everyone to get another one, so they're going to start getting cases of bulbs. This means each team gets a pallet to distribute between them. Uh oh! We don't have enough cases for the individuals, so we'd better open up a pallet or two and pull out the cases. Then we'll have plenty.
Now everyone has one pallet and two cases of bulbs, or 12,000 tulip bulbs to take back to their own store and sell to customers. We still have more than 10,000 bulbs left, which is quite a lot. There is only one pallet left, so we're obviously going to have to open that up and take out the cases, and then probably open some of those cases, too, to get everyone an equal number of extra display boxes.
At this point, every buyer has 12,300 bulbs, or one pallet, two cases, and three boxes, or 123 display boxes, and there are 792 bulbs left. Apropos of my post on context and estimation the other day, our tulip merchants probably look at those remaining boxes and bags of bulbs in the bottom of the last truck and shrug. They might not even notice them. So about 12,300 bulbs per buyer is a pretty good answer.
But for the sake of argument, let's suppose it's January 1637, we're in Amsterdam, and it's the height of tulip mania. Some tulip bulbs are worth more than a large house, so no one is about to let a spare bulb go to waste. (I know, they didn't have semi-trucks or cardboard display boxes. They also would most likely have auctioned off the bulbs, not divided them equally between buyers. Humor me, please.) In this case, we could simply continue the process of opening boxes and bags to share out evenly. If you continue this, you'll find that every buyer gets exactly 12,322 bulbs. Try it for yourself!
A technical caveat and a request for help
The astute reader may have noticed that this story describes partitive division (that is, division by sharing), but the pictures illustrate quotitive--or quotative--division (division by making groups). I rather like the fact that this shows the way that we can move between these two ideas easily, but I also worry that it could be confusing. Please share your experience in the comments as well as your ideas about how this story could be improved. Thanks!
PARENT COACHING
Struggling to teach your child division? I can help! I now offer coaching for parents. During these Zoom sessions, we will explore your particular situation exclusively, and come up with a plan for your next steps. Sign up today!