Order All The Pizzas!

In Dan Meyer’s recent talk at NCTM, he shared some contrived examples of “real world” math, including this one about congruent triangles found on the tail of an orca:

Pretty ridiculous, right?

But then some days you really do find some math out in the real world, and you can’t help but snap a picture:

I mean, holy cow! So many boxes – and one would presume – so many pizzas! I couldn’t help but take a picture and share on Twitter. The photo grabbed the attention of a few folks:

What makes this image so much more compelling than the whale tail? Both are photographs and therefore “real world.” Both have connections to math concepts. And yet one is ridiculous (not in a good way) while the other prompts thoughtful notice and wondering.

To me the difference has to do with two things – novelty and narrative. While there is a tourism industry around whale watching in person, there is nothing particularly novel about seeing a photo of a whale’s tail sticking out of the water. In addition, the textbook photo doesn’t even hint at a story. It’s a tail. It’s sticking out of the water. It’s likely going to go back in the water. Even worse, that flimsy narrative has nothing at all to do with congruent triangles.

The pizza picture, on the other hand, is extremely novel, assuming you don’t work at a pizza parlor. So much so that I felt compelled to not only stop and take a picture but also post it on Twitter for others to see. The picture taunts you with a narrative. What’s going on here? Why are there so many pizza boxes stacked on this table?

I couldn’t help but get to the bottom of it.

As I ate lunch, I watched as the guy put together even more pizza boxes. He eventually spread over two tables, and he kept consulting these long receipts.

I couldn’t help myself. I finally went over and asked who the order was for. It turns out a hospital had ordered 78 pizzas. 78!! Not only that, they had an order for 88 pizzas that afternoon followed by another order of 78 pizzas. And(!) they had an order for 88 pizzas the night before.

I asked how long it would take to make all 78 pizzas. I couldn’t believe my ears when she told me an hour to make them all and 40 minutes to bake them. Holy cow! 78 pizzas in less than two hours?! It just boggles the mind.

And why is a hospital ordering so many pizzas? Here’s a wonderful idea shared on Twitter. I hope it’s true.

Novelty and narrative, two factors that make the real world real and interesting to talk about in math class.

If you happen to want to share this with your students to see what they notice and wonder, here’s the final photo I took of all the boxes stacked up:

And here’s a photo with some additional information about the sizes of pizza and the number of slices. By the way, all of the pizzas in this order were large.

My First Three-Act Lesson

My co-worker Regina and I took a stab at our first three-act math lesson. Well, we took a stab at writing a lesson to provide some motivation for learning about measuring liquid volume, and it sort of morphed into a three-act lesson along the way. However we got there, it was fun to write, and the teachers we shared it with at a PD session in March really enjoyed it. Here’s hoping I get the chance to develop another three-act lesson sooner rather than later!

Writing this lesson came hot on the heels of spending a day with Dan Meyer at the recent Texas Association of Supervisors of Mathematics meeting. He offered some advice for designing engaging learning experiences that I couldn’t wait to try out:

• Start a fight
• Turn the math dial down

If you’re intrigued by his advice – and I hope you are – I recommend checking out his recent talk at NCTM. You’re only going to get about 45 minutes with his ideas about engagement instead of the 6 or so hours I got, but I guarantee it is still time well spent.

A Gallon of Ice

Standards

• Texas: 3.7D and 3.7E
• CCSS: 3.MD.2

Act 1

Watch the video.

1. What do you notice? What do you wonder?
2. How long do you think it will take for all of the ice to melt? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.
3. How much water will be in the jug after all the ice melts?

I recommend bringing in an empty milk jug so students can draw small mark and their initials on the side of the jug to show their estimate. Start a fight! The students will want to know if their answer is correct. I did this with teachers during a PD session, and they had quite a range of answers. At this point, the math dial is turned down low, so we did not talk about units of measurement, just an estimate of how high the water will fill the jug once the ice is melted.

Act 2

Watch the video.

1. How long did it take the ice to melt? (Sadly, it finished melting while I was sleeping, so the most precise answer we can give is longer than 11 hours but less than 20 hours, since I checked the jug again at 7:00am.)
2. Whose estimate was closest to the actual height of the water in the jug? (Resolve the controversy!)
3. How much water is in the jug? Estimate – Write an estimate that is too low, an estimate that is too high, and your just right estimate.

This is where you start to slowly turn up the math dial. Question 3 is a great question to find out what your students already know about units of volume. They might very well be stumped depending on their prior experiences. You might have them imagine other packages and containers that have liquids in them and think if there are any words they know that describe how much liquid is inside. It’s totally fine for the estimates to be sort of weak here.

The whole purpose of this question is to create a bit of a headache – get the class to a point where you (or your students!) can say, “I think we need to know a bit more about measuring liquids so we can come up with estimates we’ll feel confident about,” and then take a break from this three-act lesson to do some explorations of measuring liquid volume. After doing that, which might take a day or two, show the Act 2 video again and then give the students a chance to add on or revise their estimates.

Here are some estimates made by 3rd grade teachers at our PD session:

I can tell the teachers were hooked when they reacted in shock when they found out I wasn’t going to reveal the answer right away. Just like with students, we took a detour away from this lesson. We wanted to spend a bit of time sharing ideas for how students can explore measurements of liquid volume. But they wanted to know the answer! One of them was really worried and wanted to make sure we would tell them before they left the PD session.

I couldn’t have been happier.

Act 3

All is revealed! Now that your students have some personal experiences with measuring liquids using various units and you’ve given them a chance to add on or revise their estimates, it’s time to find out the actual volume!

And of course I spilled some water! When I was first filling the jug, I had to cut a flap in the top to make the opening wider for ice cubes to fit. Unfortunately, I forgot about it when I was doing my first pour and water did not come out like I was expecting. Thankfully it was only a small amount.

There’s so much going on in this video! You’ve got quarts, and half gallons, and cups, and fractions of cups. All great stuff to talk about! But I purposefully tallied the number of cups throughout the video so that students could at least come up with 8 2/3 cups. However, this is a great opportunity to talk about how we can read measurements differently depending on our units. For example:

• 8 2/3 cups
• 1/2 gallon and 2/3 cup
• 69 1/2 ounces
• 2 quarts and 2/3 cup

This is different from making conversions; it’s more about the choices available when reading a measurement off a tool. You don’t have to go here, but I think it is important for students to know that they do have choices in how they read a measurement given the options provided by the tool. Learning that flexibility here is only going to help them when they start encountering questions related to measurement conversions down the road.

And that’s a wrap! If you try out this lesson in your own classroom, I’d love to hear about it in the comments.