Day 2 – The One Cut Problem Leading to Great Mathematical Dialogue


Today was day 2 for school, but day 1 for my Honors Precalculus class. I wanted to start with a problem that felt playful at first but then might have some really cool deep math attached to it. So we did the “one cut” problem where you try to fold up a paper so that you can make ONE SINGLE SNIP and cut out a certain shape.

They just started working on some shapes and 10 minutes in I asked them… what’s going on here? What mathematical principles are happening? This was most of the conversation in the next 20 minutes or so. It was fairly quiet, but students progressed further and further.

  • One student said “symmetry”. Great. What about symmetry?
  • Another student “Well, this has to do with angle bisectors.”  We talked about why that makes sense and then that led many students in the class to nail the scalene triangle.
  • “Well if those are angle bisectors, that means that the point here that matters in the incenter.” Yup!
  • The next piece was an irregular quadrilateral. “Well, does it make sense for a quadrilateral to have a single incenter?” The class decided, no, not really.
  • Is this shape possible? I shared that there’s a theorem here, that EVERY shape is possible. This might have jumped the gun a bit and stolen some thought from them, but I wanted them to keep persisting with the shape.
  • They worked for a while and a few students got an acceptable solution for the quadrilateral. One student’s idea was to draw the angle bisectors, find the pairs that connected, call these two “incenters” in a way, connect those and fold along all the lines (see the diagram above).
  • One student: “Okay, so if an incenter is the center of a circle, do these have to do with that? Can you inscribe a circle in a quadrilateral so that it touches all four sides?” We decided no. A simple rectangle helped us with that.
  • “Well if there are two incenters in a quadrilateral, I wonder if they are the foci of an inscribed ellipse.” Whoa. Probably not, but cool idea!
  • Then, right at the end, a student noticed that the incenters themselves form a smaller quadrilateral inside the shape. The shape looks similar to the outside one. Is it!?? Could that help with the solution?

This had nothing to do with what we are going to do soon (i.e. tomorrow), but man – what an amazing progression of mathematical ideas that I didn’t see coming. It shows what happens when you start with a beautiful, rich problem that you might not even know how to solve.

Day 1 – Four 4’s Leading to a School-wide Competition

Welp, here it goes. Gonna try to do the 180 blog this year.


Today, in my Problem Solving 2 class (it’s an Exeter-aligned integrated math course for mostly sophomores) we started with the Four 4’s problem. It’s very easy to state – make all of the numbers 1 through 20 with exactly four 4’s and any operations you can think of (ex: 1 = (4+4)/(4+4)). I would say about 80% of them got pretty into it.

I really wanted to do this though so that we could think about “problem solving” and what makes good problem solvers. Here were their reflections based on this activity:

  • You need to be patience and tenacious
  • You don’t need to work in order – do what you can do, solve easier problems and they help you solve the harder ones
  • Sometimes you just have to try stuff even if you don’t know exactly where you are going
  • You can use some answers to get others and some patterns to find other answers
  • Sometimes you need to take a break and come back later

They got all but 11, 13 and 19.

I had them write solutions on these strips of paper because my secret goal is to put this up in the math hallway (with empty strips for all the numbers up through 100) and start a school wide competition. So basically, when you get an answer, you take down the strip, write the solution and put it back with your name. We’ll see how it works. I could see it tanking, but it might be cool!

Day 135 – Calc Project Spotlight: Middle Earth Math

One of my students is REALLY into Lord of the Rings. What she is doing for her final project is tracking Frodo's journey from the Shire to Mordor. She is trying to calculate how fast he is going at each point in the journey using km per day as her unit, and then comparing to when she's using km per page as her unit. She has all these other ideas about how to analyze the data, and I'm really curious to see them!

I'm having a dilemma about these projects, which is how much calculus should they really include? This one is a good example… I don't know how complex the Calculus will be in this, but the idea is so awesome I want to encourage it and see if some Calculus can pop up!


Day 134 – Calc Project Spotlight: Dawn of the Living Calculus

We are working on final projects all week in Calculus. Kids have split up into groups and are working hard. Even when I give them a million options to do, I love that they want to do something that is their own. I have really creative students this year who have some great ideas. So every day this week I'm going to profile a cool final project idea.

The first: Dawn of the Living Calculus! Two students sent out a link to WordPress page where it tells the visitor that they've been infected. Then there are instructions to send that to 3 more people. They are going to track the infection spreading by looking at the WordPress stats. Tomorrow, I'm going to see if they can try to make a predictive model based on a day of growth. I'm curious to see where this one goes!


Day 133 – Post AP Debrief

Today, I had all my AP students tell me which topics they felt great about and all the topics that they felt like they needed more work on. This will help me teach the class a bit better next time around. I am excited to see the free response questions tomorrow!


Day 132 – AP Calc Exam! And Faculty Stand Up Comedy Show…

It's here! The AP Calc exam! Woohoo! I feel really confident about how prepared most of my students are. Nothing more that I can do!

Today, we also had a Faculty Stand Up Comedy Show. It was overall hilarious, and my first stab at stand up comedy was fairly successful. I toed the inappropriate line, but my headmaster complimented me on the performance today, so I still have my job. I found this picture on the Instagram account “@makebowmanfamous” which is a Calc project from a pair of students. They are publicizing me and seeing if they can track my followers and likes to see how “famous” I can get. Life as a high school treachery is hilarious.


Day 131 – Final Project Intro and continuing the Drawing Project

Today, I introduced our final project in Calculus. Students can do whatever they want as long as they are investigating and interesting question with mathematics! This year, my students are uber creative and really enthusiastic, so they proposed some great, great ideas. One student is going to investigate the Calculus of Frodo's expedition to Mordor. One group is going to stage a fake CSI murder. One group is going to track data about the nurses office at school to make a model for how often people visit. Great stuff!
The integration drawing projects are starting to come in too… Here is my current favorite.

(Whenever I forget that the creatures in front of me are not yet adults, they do stuff like choose Disney characters for projects. The best.)