Domino’s and Linear Equations

I taught the Mathalicious Domino Effect lesson today. It was awesome.

First, the lesson is free. Check it out here! I’m in the middle of a unit on linear equations, but it could be used as an intro with same basic background knowledge just as well.

The lesson:
The lesson has three main parts. First, it gives costs of 2-topping and 4-topping medium pizzas, and students calculate the cost of a topping and the cost of a plain pizza, and confirm their findings through an awesome interactive Desmos graph.

Second, it gives costs of two small and two large pizzas, and asks students to write equations for the cost of each size of pizza in terms of the number of toppings, and graph it. They then confirm their findings through another awesome Desmos graph.

Finally, students consider whether these equations are a reasonable model for pizza costs, and look at a graph that shows the actual costs — the costs for 1-4 toppings are linear, and toppings after that are free, which creates an interesting conversation.

What I did:
Mathalicious writes awesome student handouts. Really awesome. I use tasks from lots of other sources, like MARS and Illustrative Mathematics, where I rewrite my own student handout every time to emphasize the parts I want emphasized and provide space and scaffolding for student work. Still, I rewrote the student handout for this one to build the lesson I needed.

We did the first part of the task as a class. The lesson provides costs for 2-toppings and 4-topping medium pizzas. We made a table, found the unit rate, found the cost of a plain pizza, and put the equation together. I pushed them a bit by asking whether an 8-topping pizza would cost twice as much as a 4-topping pizza, and got an interesting mix of answers. Note that I need to return to linear vs non-linear functions. Then, we looked at the Desmos graph, played with the slope and y-intercept, and confirmed that our equation worked.

Then I gave them the costs of 2 small and 2 large pizzas, and asked them to find the equations. This was tough.

A bunch of kids got a pretty strong start, but many more needed help with what pieces of information the needed to write the equation, and there were lots of mistakes in finding the unit rate and starting value. Still, with some time to check in with kids who were struggling, some great learning happened. I didn’t have them graph, which may have been a mistake, but as we’ve been working with linear equations for almost two weeks, I wanted to push students to work in the abstract. We then went through the two equations together, and looked at the graphs of the small, medium, and large pizzas, comparing them and why the costs were different.

Finally, I asked them whether they thought these equations were the actual costs of pizzas at Domino’s. Interestingly, two of my students have siblings who work at Domino’s, and they told me the prices were right. Then, I showed the final graph:

Which compares the predicted cost vs the actual cost (the three dots). It was a fun example of how math that models the real world isn’t perfect — and asking students what that flat line meant was a great application of the meaning of slope in context.

Missed Opportunities:
I chose not to have students graph, to focus on abstract representations of linear equations, as they have done a great job with graphing, but struggled with tables and writing equations from a context. I think this was a missed opportunity — the equations can come first, but graphing will only reinforce knowledge.
The piecewise actual graph of Domino’s costs is a great opportunity to contrast linear and non-linear functions, as well as a preview of piecewise functions and an illustration of zero slope. I don’t know if one or all of these topics would’ve been the best choice, but I think there was a good opportunity there.

Revealed Misconception:
When we calculated the equation for the medium pizza, we had 2- and 4-topping pizza prices. So we found the difference between the two prices, and subtracted it to find the price of a cheese pizza (0 toppings). When students tried the small pizza on their own, given 1- and 3-topping pizzas, many students subtracted the difference between the two prices again, in this case finding the cost of -1 toppings, not the cost of a plain pizza. Another reminder of how quick students can be to imitate rather than understand.
Students still really struggle to relate “cost per topping” to rate of change and “plain pizza” to y-intercept. Solid progress today, but work left to do.
Lots of students thought that an 8-topping pizza would cost twice as much as a 4-topping pizza. Their logic amounted to, approximately, because 8/4 = 2. I wonder whether this is an example of a misconception, or if it’s really just lazy thinking. I want to explore that idea more once students have more examples of non-linear functions to work with.

Mathalicious is Great:
This lesson was very deliberately structured to promote both revealing misconceptions and student understanding. I loved that the first pizza was given with 2 and 4 toppings, then 1 and 3 toppings, testing their understanding of the y-intercept, then 1 and 4 toppings, testing their understanding of slope. I loved the Desmos interactives to illustrate slope and y-intercept. I love that the scaffolding was minimal — the problem didn’t tell students how many toppings each pizza had; they needed to read it from the information given. Those subtle differences — scaffolding earlier problems, and increasing complexity, is what really drives mathematical learning, and I was excited to see my students engage with it today.

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