PD with Dan, modeling with mathematics

With the help of the NYCDOE, this past Wednesday was the third of three workshops that I’ve attended with Dan Meyer this school year. I’ve written about the first two, so why not cap off the trilogy? Read more here and here. In an effort to solidify my experience and have something to reflect upon at a later date, here’s a recap of the session.

The focus was on mathematical modeling. This is one of my weaknesses, so when I walked in and looked at the agenda, I was pretty happy. Teach me Dan! Things opened up with us sharing what we thought it meant for students to model with mathematics. The answers varied, but after feeling us out, Dan emphasized the difference between model the noun and model the verb. I had to catch myself here too as I hadn’t played close enough attention to this subtly. He used Desmos to collect our responses and our comfort level with mathematical modeling.

We then dove into the Penny Pyramid 3-Act. Dan worked his magic, as usual. During the debrief, he pulled from the CCSS and presented five actions that are necessary for students to be part of modeling experiences:

  • Identify variables
  • Formulate models
  • Perform operations
  • Interpret results
  • Validate conclusions

We talked about how the Penny Pyramid, while probably not ideal, works to get students to do all of these things. Our questions helped us to identify variables. Interestingly, Dan highlighted the vocabulary that we chose to represent the variables. This is important because, indirectly, he demanded precision and consistency from all of us when it came to describing key parts of the experience. Our tabular, algebraic, and physical representations of the pennies were the models, which we used to calculate the number of pennies in the pyramid. We spent time as a group discussing the meaning of our calculations and parts of our models – even those that weren’t correct. For example, one group landed on 22,140 pennies, which didn’t take into account that each stack was 13 pennies. Also: what does the 612 mean? The weakest part of our models was the validation. It’s not like we could actually count all the pennies. That said, we would compare outcomes of different models. There was also a newspaper article on the pyramid which helped with the actual count of pennies. Validation runs on a spectrum.

The afternoon was spent creating and improving some modeling tasks. We started with 25 billion apps and then went into a few graphing stories. We spent time thinking about how to structure modeling activities using these tasks. A great conversation occurred while exploring Distance from Camera. Someone brought up whether the graph should be piecewise linear or rounded periodic (like the solution). It was a great discussion, worthy of its own post. Dan remained vulnerable and eloquently showed us this ferris wheel Geogebra applet from the awesome John Golden. (Check out his full collection of Geogebra resources.)

We brainstormed how technology can help make our modeling dreams come true. The list was plentiful. Staying away from “textbook traps” topped the list. These include giving up numbers, tools, and other key information far too early in the modeling process. It’s also ironic that, because of technology, Dan was behind a desk for extended periods of time while we worked (he was typing our insights and using Desmos). Without technology, this teacher move is frowned upon.

The culminating task had us dig into several different versions of Barbie Bungee. Having never actually done this modeling activity, I learned not only about how to do, but also what not to do. Truth be told, if I would have looked at the activities before the workshop that day, I wouldn’t have suggested any changes. I would have been excited to implement them as is. But after purposefully rethinking modeling with mathematics with Dan for five hours, I felt very different.


  • The Penny Pyramid task would serve as a great introduction to summation notation.
  • Dan: “When there’s a great classroom experience, I ask myself: how could I have ruined this?
  • Not every aspect of modeling needs to every lesson. The goal is to feed students a healthy diet of modeling verbs.
  • Focus on broad questions. These will lead to more specific, granular questions. Avoid the reverse.
  • Seeing every teacher move I make as an investment into the lesson. Which moves are worth their investment? Which aren’t? This reflects the gravity that every decision we make. I need to be more deliberate.
  • Depending on what info we give kids, the level of modeling that they do could be very different. Don’t do the modeling for them! Give them procedural stuff.
  • It’s not if to give a handout, but when. This outcome was directly tied to Dan’s first session back in December, which helped things come full circle.
  • Once again, I was impressed with how Dan managed the audience. There were so many slick teaching moves (see notes).
  • I loved catching up with Sahar during lunch. We talked about her experiences visiting students homes with her school. She also gave me a tip about taking photos while students are working during the first couple of days of school to showcase how mathematical discussions should happen amongst students.
  • This wonderful periodic modeling Desmos activity.
  • If I ever do Barbie Bungee, I need to use Dan’s intro video.
  • Be mindful when moving between the real and math worlds. Don’t get lost on your travels.

Dan’s Google Doc for the session.


Exploring the roots of quadratic functions

In my algebra 2/trigonometry class, I wanted to spend a day with them exploring quadratic functions, their roots, factors, and how everything is related. This is taught in algebra 1, but the students always seem forget all this after geometry. It’s a perfect intro to our quadratics unit.

Part 1. Building off Dan Meyer’s approach to factoring trinomials, I first had the students find all values that make quadratic expressions equal to 0.

find values

Most found this straightforward and doable, which was great because that was the point: accessibility. The third expression had one non-integer value as an answer, which I included on purpose to throw them off. A couple kids actually factored and used the zero-product property, which was ahead of the game…they actually remembered something from algebra 1!

Part 2. I had them use Desmos to examine the graphs of the three functions and find a relationship between the graphs and the values they found that made each equal 0. It took several minutes, but there were definitely some ah-ha! moments when they saw the connection, which was cool.

Screen Shot 2015-12-02 at 4.35.50 PM

I then re-introduced them to be the term “roots” as a way of describing these magic numbers.

Part 3. I also wanted them explore the relationship between the factors and the roots. Because of time, we more or less did this together (instead of them working it out in groups). We first factored all the expressions. I then asked how the factors relate to the roots of each function. Most of the class realized that when each factor is set to 0, the roots result from these “mini” equations.

Screen Shot 2015-12-02 at 4.36.45 PM

Overall, the lesson was solid. I really liked that, other than imparting the term “root,” there was no need for me to lead any part of the lesson. I simply provided resources and asked the right questions that spurred deep thinking.

Exit slips showed their understanding of the connection between the roots and factors wasn’t strong. This was probably due to the fact that the lesson was a bit rushed at that point. The next lesson focuses strictly on finding roots by means of factoring quadratic equations, so hopefully that helps. I also felt the lesson flip-flopped around the term expression and function too much. Leaving the lesson, the difference between the two could have been unclear and may cause some confusion amongst the kiddos. Another thing I would have changed is not having all trinomials…the kids could possibly generalize that all quadratic functions are trinomials, which is obviously not true. Even if they don’t go that far, a variety quadratic functions still would have been better for them to explore.

Here is the document.



Art & Desmos

Desmos Screenshot cat

Recently I had my precalculus students complete an art project using Desmos. We were finishing up our unit on conic sections. I paired them up and gave them two class days and the weekend to conjure something good. They didn’t disappoint. Props to Bob Loch who helped provide the structure.

Desmos In Class 1Desmos In Class 2Desmos In Class 3

The guidelines were pretty simple:

  • Include at least one of each conic section in your art work
  • Place restrictions on the domain and/or range of at least two of your graphs
  • Solve a system of equations resulting from your graph

The grade was based on the above criteria and how complex their artwork was. I loved this activity because it was so open ended. I usually don’t do a great job allowing my kids to showcase their creative side during class activities. I was impressed with some of the art they managed to create.

Desmos Screenshot Tux Desmos Screenshot mushroom Desmos Screenshot bullseye

I’ve been pleasantly surprised with my increased usage of Desmos this school year. It’s an excellent tool. I literally can’t imagine teaching without it.