## Taxicab Geometry – A Brief Exploration

We are officially on spring break here at my school and we end the term with a week of what are called test priority days. The idea from the school’s end is that we want to protect students from having says with three (or more) major assessments as the winter term comes to a close. With a two – week break most teachers try to put a little bow on their material before taking off so as not to simply start off again on March 14 repeating a week’s worth of material. However, this leads to some awkward scheduling. My last test priority day was Monday and I met classes on Tuesday, Wednesday, and Thursday. I sent out a call for ideas on twitter (like you do, right?) and I received a handful of great suggestions. From a conversation with Henri Picciotto (@hpicciotto) and Becca Phillips (@RPhillipsMath) I decided to spend a few days with Geometry AND with Discrete Math on an intro to taxicab geometry. Henri shared a great link to one of his pages (I encourage you to download that file from Henri – the relevant ones for this discussion are labs 8.4 – 9.1) and I modified some of those ideas and created two handouts of my own (here is #1 and here is #2)

I want to take a moment here to reflect on how our two and a half days with this unit went. We worked Tuesday and Wednesday in each class and wrapped up our conversations before tackling the cool problem I wrote about here to finish our time together on Thursday. First, I want to comment on my documents and how I intend to tweak them before using them again. Then I will comment on the class action these days.

Handout #1 – First change I would make is that point B would be the point (5,4) instead of (4,5). I do not know if any student caught this, but when I imposed the map of Gainesville, FL on the situation I described, Anne is not at the point (4,5). This tweak would solely be for my comfort. I do know that students in all three periods had trouble deciding whether street location should be an x coordinate or a y coordinate. It should have been an easy decision, I think. I like the introduction of the Manhattan map as a way to discuss what a city block might mean, but I did too much talking this first day. I need to introduce the idea then get out of the way and let the students ask these questions. I also should change some of the coordinates I suggested. My students really wanted these points on the section of the grid I provided. I should probably adjust for that. Finally, I have to admit that I am pleased with the questions I asked here. I think that there is a pretty nice balance of practice, of comparing taxicab and Cartesian distances, and of asking some nice guiding questions. By the end of the day Tuesday I felt that my students were in a pretty good place.

Handout #2 – I like that I start off with the same image and the same text to reframe the conversation. I think I will take away the text here that defines a circle and make sure that this definition arises from conversation – either whole class or in small groups. I love the sense of discovery that emerges as the students begin to realize what a taxicab circle will look like. I had GeoGebra fired up on the projector and started taking ordered pair suggestions so we saw the shape emerge together. I am happy again with my questions here even though I am unsure of whether there is actually a clear formula for the number of lattice points inside the border of a Cartesian circle. We did stumble upon a formula for the lattice points inside a taxicab circle and it was pretty darned exciting to see this unfold. Since this was the last night of class work I had very little evidence that any of my students had entertained this question on their own. We had a nice enough conversation about it in class, but it would have clearly been more energetic if there had been some reflection on their own by any of my scholars.

My Geometry students seemed more engaged and interested in how the ideas unfolded in this exploration. Perhaps this is due to its clearer relationship to our ‘normal’ material for the course. My Discrete kiddos were willing to have these discussions, but they were clearly less excited/annoyed/engaged/frustrated/surprised by the discovery of the fact that circles are now squares. I felt pretty committed to the idea that we should agree on whether we wanted to limit ourselves to only considering lattice points when deciding about the nature of the taxicab circle. I had been rooting for a loosening of the idea of points here so that we would have a continuous boundary in the taxicab world as we do in our Cartesian world. Since I had so clearly framed the conversation the day before in terms of city streets and avenues almost all of my students wanted to stay with that restriction and they voted clearly to restrict to lattice points. There have been a few other places where the Geometry students were asked to agree on definitions. We agreed that a trapezoid should have only one pair of parallel sides and we agreed that kites should not have four congruent sides, they should have two pairs of congruent sides that were not congruent to the other pair. There is a clear pattern of wanting to agree to more restrictive definitions here. I have discussed this with one of my Geometry teammates and he seems a bit bothered by my willingness to allow these restrictive definitions. I understand his point about definitions later on in math, but I feel pretty committed to letting the students come to these agreements together at this level. I hope I am not undermining their future as mathematicians here. I like the placement of this material in a short, unconnected time span on our calendar. We could have this conversation at a number of times in the year and I want to keep this in my back pocket to uncover when time allows/demands a unit such as this one. I think that the fact that the students knew that this would not be part of an immediate assessment allowed them to relax a bit and just play with some of these ideas. I also think that this fed into the near complete lack of work done on finishing the questions I presented after class discussion time. I think I am willing to accept this limitation as long as the benefit of relaxation comes along with it.

I want to thank Henri and Becca for helping push me into this and I want to thank my teammate Mary who was willing to dive in and try this unit as well. My other two teammates tried some different ideas and I want to pick their brains to see how life went in their classes for these three odd days. I also want to say that I am fairly happy to have a bit of a break now and I hope to return to school on March 14 with at least a couple of weeks planned out carefully for both the Geometry and Discrete Math classes.

So, our school works on a trimester system with Thanksgiving Break (a full week) marking the end of the fall term. We also have fall term finals, so my last full day of classes was November 12. I set myself some lofty goals for the break and met about 80% of those goals. My number one goal, by far, was to do what I could to plan out our next fourteen days for all three of my preps. We have fourteen days of class until the long winter break begins.

I found out late in the summer that I was teaching a new course (around August 10) and I also have two brand new colleagues in my  department. I have not been able to spend as much time mentoring them as I had planned to. The combination of this disappointment, along with perpetually being only a few days ahead of my Discrete class made the fall term a pretty stressful one. I have three preps, five sections, and my chair responsibilities. Luckily, I have a pretty light student load this year.

So, I have my calendar mapped out for Geometry and AP Calculus BC and I have about ten of the fourteen days of Discrete taken care of. Overall I am pretty pleased. Add in the naps and the time with my wife and kiddos and it has been a good break with just enough productivity thrown in.

## Trolling for Ideas

We have started a STEM initiative at our school. I am hoping to gain some traction for a conversation centered on a joint Physics/Calculus curriculum. We have two courses in place at our school that are joint teacher operations.  We have a course called Seminar that is co-taught by our history and english department chairs. We also have a course called Creative Spirit co taught by a studio art teacher and our music director. So, we have the vision in our school to create courses that break the mold a bit. I would love to try and launch a course combining physics and calculus ideas. I am certain that there are schools where such a program exists and I would love to have some curricular conversations along these lines. Anyone out there with ideas they’d love to share?

## New Challenge – Curriculum Development

Wish us luck and lend a hand where you can!

## Stepping Outside my Little Corner of the World

Our school had a day off on Friday (and a day off today as well) for a long fall weekend. We were asked by the powers that be to use Friday for professional development. I chose to drive a couple of hours in the morning to go visit another school. When I did my last job search in the early months of 2010 there were a number of schools that caught my eye and the school I visited on Friday was one of them. The chair there was remarkably kind and helpful in setting up a too short visit that morning. Since I had kid pick up duty that day AND we had agreed to house sit for some friends to look after their dog AND my boy had an ice skating birthday party to go to (there is a theme here about how life unfolds in the dardy household) I did not have quite the leisure I had hoped for. I arrived at 8:30 ish for a warm, quick chat with my host, I saw a Geometry class, then I saw a Precalculus class, then I saw an AP Stats class. A nice follow up chat and lunch with the chair, then I was off to home.

I always enjoy seeing classes – it is a part of my job as a chair that unfortunately gets buried under other tasks. It is fun to  pick up tricks from other teachers. In this case, the geometry teacher had a lovely way to highlight parts of the parallel line with transversals problems that they were working with that morning. She had spools of different color tape that looked like athletic trainer tape. She pulled off two of one color to highlight which lines in the diagram were parallel to each other and a different color for the transversal. It was SO COOL to see this way of making the relevant information in the diagram just pop out to the kids. It was also fun to see her improvise. The kids were checking their work from the night before and were having disagreements about measures they had taken. Out the window went the lesson plan for the day and out came a class set of protractors so that they could practice with their measuring skills. The teacher confided in me that some of her attitude about this was strongly influenced by her husband who is a woodworker. In the AP Stats class I was privileged to watch someone who was a real, honest to goodness statistician before entering the classroom. As a stats novice myself, it was great to chat with her beforehand and to watch her in action. I think that she convinced me to try an activity that has been previously pretty intimidating to me. The precalc class was fun to watch as well as the kids were hanging in there working through some complex polynomial graphing ideas.

I know that I have a tendency to look at my world and see the potential for excellence in the people around me. I know that I focus at times on what is not quite right instead of celebrating what is right. A visit like this worked wonders for me on a number of fronts.

1. It’s always great to reach out to more people to bounce ideas off of

2.  It’s fun to watch kids at work – especially when I have no preconceived notions of who they are or what they SHOULD be doing

3. It’s rewarding to talk to others who are working through some of the very same struggles. How do we accurately place test kids who are new to a school? How do we balance ambitions for kids with their abilities and previous track record of achievement? How do we find TIME in the school day/week/year for meaningful problem-solving while still serving an ever expanding curriculum? The chair I met with is thoughtful, experienced, and intelligent. The fact that she is struggling with these questions as well makes me feel better.

I’m proud of my school, our students, and my colleagues. I believe that we can all be better than we are but I want to try and focus on what we’re doing right and I think that this experience on Friday can help me with that.