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.




A wonderful Problem

Today was our last day of school before a loooong spring break – we do not return until March 14. We were asked not to have any assessments today as some students have term finals tomorrow. So, I wanted to find a flexible problem that all 3 of my courses could wrestle with today. I teach AP Calculus BC, Discrete Math, and Geometry so this was a bit of a challenge. I found a lovely problem here : Screen Shot 2016-02-25 at 2.55.39 PM

I was so delighted by how my students engaged with this problem today. A little background first. My BC kids are on the verge of learning about power series so a series/sequence question is right up their alley right now. We have been talking convergence and divergence tests. I also had some competition problems in my back pocket because I knew this would not take them very long. My Geometry class just finished a chapter on similarity and we have spent the past two days playing with Taxicab Geometry. A blog post on that adventure is coming tonight or tomorrow night. My Discrete kids just finished their winter term where we studied patterns (numeric and visual) as well as some theory about voting and ballot strategies and they, too, have played with Taxicab Geometry this week.

I want to share a few of the insightful comments that some of the students made about this list of sequences. I prompted each class with one question first: Why does they say that these are related sequences?


In all of my classes students first focused on the rules for each arithmetic sequence and made observations about the pattern of differences moving from 2 to 4 to 8. In one of my BC classes a student instantly said ‘Each first term is 2^n and then you add 2^(n+1)’ Amazingly fast pattern recognition, but more than I hoped for right out of the gate. Most of his peers were taken aback and seemed happy to focus on smaller pieces. In each Discrete class and in my Geometry class I had students noticing the doubling pattern from one sequence to the next. Only my Calc students used recursive language technically, but all classes had students recognizing that pattern. It is interesting on reflection to see how formula driven (or is that formula comfortable) my Calculus students are compared to the other classes.

I tried to get a series of ‘what do you notice?’ comments going and the following popped up in every class;

  • The first sequence is the only one with odds
  • They are all arithmetic series (either by description or by use of the formal language)
  • The difference in each sequence is increasing by an twice as much as the difference from the previous sequence
  • The first term is a power of 2 (my Geometry kids needed prompting to remember about the 0 power)
  • All the sequences other than the first have only even numbers


After gathering a series of observations about the sequences, we directed our attention to the charge of finding where the number 1000 might be hiding. Luckily no one wanted to list all the terms of a sequence until 1000 arrived or was passed by. So the following suggestions came my way;

  • Subtract the first term from 1000 and divide by the common difference to see if 1000 is on the list
  • Divide 1000 by 2 repeatedly until we arrive at a term that is more manageable and more clearly on one of the lists
  • 1000 is 10^3 so we need to find 5^3 since 10 = 5 * 2 and we know that 2s are built up row by row


I was really pleased by the focus on 1000 being built up by factors of 5 and 2. One of the discrete classes built up to 1000 while the other kept dividing by 2 to bring it down to the 125 necessary. Once we were focused on 125 it was clear in all classes that the first sequence was the only place that 125 could live. My first class of the day is one of my AP Calculus BC classes and after realizing where the 1000 is there was no discussion of whether that 1000 could appear anywhere else. In my second class, one of my Discrete Math classes, they focused on the plural in the question and wondered whether there might be multiple landing spots for the 1000. We counted out 1 – 20 together on the lists and noticed that no number was repeated. We were pretty confident that this pattern would hold. In my second AP Calculus class – the one where a student generated a formula right away – he stepped up and showed a terrific proof that this had to be a unique solution. Writing each term as 2^n + (2^(n+1))*k where k represents some multiple of the number of differences in the sequence. By setting this equal to 1000 and factoring out a 2^n he made the argument that 1000 needed to be written as a product of a power of 2 and an odd number of the form 1 + 2k. Listing factors of 1000 it was pretty clear that only 8 * 125 satisfied the conditions of the problem.

Finally, my Geometry kiddos had the opportunity to dig into the problem and I was pretty darned pleased, I must say. It was the last period of the last day of school before a two week spring break. They are the youngest of all my students and they are the least experienced mathematically. What I saw today was real evidence that these students have been growing as problem solvers. They are more patient and persistent than they were in the fall and they are more willing to make guesses out loud than they were when we started the year together. I am so happy about the conversation we had. Other than the concern about whether 1000 exists in any of the other lists, they were able to nail all of the important pieces of this problem.


I discovered the problem at about 5:45 this morning and I could not be more pleased about the conversations I had with my students today.



More Similarity Adventures

Yesterday we had a two hour delay and I was looking around for an idea to engage my Geometry students at the end of the day. As I have been writing about for awhile now, we are engaged in conversations about similarity. We had some Kuta skills practice, we had some problem sets I wrote for the students for HW practice and today I wanted to have a little activity where I could introduce a question and get out of the way and listen to them debate/discuss/discover some important ideas. I looked at my own Virtual Filing Cabinet and rediscovered a great question posed by Nat Banting (@NatBanting) over on his blog called Musing Mathematically. The question I pulled was from a post last year looking at coffee cups. That particular post can be found here. Below are the two key photos that prompted the conversation yesterday. Screen Shot 2016-02-17 at 8.43.34 AM

Screen Shot 2016-02-17 at 8.52.31 AM

I posed the following questions on my class handout :

  1. Show that these cups are not similar.
  2. If the small cup of coffee costs $0.99, how much would you expect the large cup of coffee to cost?
  3. Since they are not similar, change the height of each cup – maintaining the diameter of the top – so that the cups are similar to the small size.
  4. Now, instead change the height of each cup – maintaining the diameter of the top – so that the cups are all similar to the extra large size.


Before presenting these questions/challenges I prefaced the conversation by talking about the habit of upsetting, like at a movie theater concession stand, and pointed out that larger sizes are (almost) always the better value but usually not necessary. Another important note is that I allowed them to ‘cheat’ a bit by presuming that the coffee cups are cylindrical. We have not officially touched on much in the way of volume conversations so we needed to come to an agreement, which we did quickly, on what the volume of a cylinder ought to be.

I am so thrilled with how our conversation unfolded and with the ideas that popped up during our chat. The students were quick to notice that the large and extra large cups each have the same diameter, so similarity there is thrown out the window. A student quickly nominated the ratio of height to diameter as the scale factor that was important. They were shocked by the theoretical cost for the large cup of coffee. I suggested that we ignore the pi in the calculations of volume and at first they were happy with my reason why and then they balked at the idea of just throwing it out of the calculations. This seemed to be a nearly perfect length for an exploration on a silly thirty minute class day schedule. I only hope that they remember nearly as much of the conversation as I do.



Proportions Follow-Up

Earlier this week I was excited to see the problem posted on twitter by Megan Schmidt (@veganmathbeagle) and I wrote about revising my Wednesday plans due to snow and this intriguing problem. I was not the only one thrilled by the problem. Later on Wednesday I saw another post by Joseph Nebus (@nebusj) featuring the same problem! You should definitely check out his post that I linked to and his other writing as well. So, my plan was to play with this problem with my two Discrete Math classes. I borrowed a bag of dice from our new AP Stats teacher and had enough so each student would have four different colored dice. I had each student run twelve trials under two separate sets of rules. One group could manipulate their four dice in any way to try to set up a ratio equivalence. The other group had to decide up front where each color would go. A nice conversation about the problem before we ran trials uncovered some uncertainty about our goals and some debate about whether the additional condition on one group made a difference. It did not take long for us to agree that the group with more freedom should see more success. What happened in each class is what usually happened when I taught AP Stats for five years. Small sets of data do not conform terribly well to theoretical long-term probabilities. I like this in theory but it is frustrating to see the data defy us so often, especially when it feels that the students are really fragile in their understanding of the problem at hand. I had shared this problem with the math team at my school and our AP Computer Science teacher wrote a quick applet and ran thousands of trials. He reported to me that the data he was gathering showed about 6 – 7% success rate with the restriction and about 12% success without. Our data did not match this at all.

Later in the day I saw one of my former students in the hallway. He is a math problem fiend and I figured I would pique his interest by asking him to ponder the problem. He dug in and spent about forty minutes after school with me attacking the problem. Interestingly, he did not grasp my language right away when I spoke about order mattering in one case. When I showed it to him with the colored dice he lit up and immediately attacked my board with renewed vigor. He set up a simple grid and color coded possibilities and we ran through the outcomes that satisfied our requirements. The results we came up with – unfortunately, I did not jot down the numbers – were in line with the results my Comp Sci colleague arrived at in his data. So, I feel like our work was pretty good but it was not elegant at all. Another one of my wizard students suggested using an approach based more in the language of permutations and combinations. I hope to follow up by picking his brain a bit about this. It is almost certainly more elegant than the scribbling we did on the board Weds afternoon.

It was fun to have my brain tickled by a problem that is so easy to approach but so stubborn in giving up an answer. It was also fun to see others on the web tickled by the same problem and now I have another blogger and tweep to keep track of!

Snow – and proportions – are in the air

We had our first snow of the winter yesterday and we have our first snow delay today. We have 30 minute classes so I am scrambling a bit to modify my plans for the day. Luckily, my pal Megan (@veganmathbeagle) sent out a great question yesterday on twitter. Screen Shot 2016-02-09 at 9.49.25 AM

We are working on proportional logic in Geometry as we introduce similarity for figures. In my Discrete Math class we had a test yesterday. I have options now for three of my five classes today! I borrowed dice from my AP Stats colleague and we will play and do some data gathering today. Can’t wait to report on how it goes.


PS – I asked Megan if order mattered. She replied by asking me if orders matters. I will extend the favor to my students and ask them to decide if order matters and if that changes the answer to the question.

Modeling Good Behavior

I think about this all the time as a dad – my lil ones are a 12 year old boy and a 6 year old girl – and I often criticize myself for falling short. I think about this all the time as a teacher and as a colleague. Again, I often criticize myself for falling short. Don’t get me wrong, I think I am doing the right thing much of the time, I just wish it were easier – or more manageable – to do the right thing all the time. Our school Reverend delivered a chapel today that made me really dwell on this and I remember an important quote that I keep on my bulletin board. It is a quote that the wonderful Meg Craig (@mathymeg07) shortened for a poster in my room. I want to share the quote to help me stay focused and, hopefully, to help anyone else reading this stay focused as well.

Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.


John Mason

Source on the web here

Eyes on the Prize

In Geometry today we were reviewing for tomorrow’s test. A student asked to go over a proof about kites that we did together last Friday. It got me thinking about which proofs are really essentially interesting in Geometry AND it got me thinking about some former colleagues in Florida. I worked with a history teacher who had an interesting habit. The day after any test he handed out the essay question that was going to be on the next test in a few weeks. He checked back in on the question during the unit and used this question as a guide to their discussion along the way. At the same school I worked with a photography teacher. At the beginning of any project assignment she would hang up the best photos from previous years and referred back to these as a guide for her students.

So, today it occurred to me that there might be ways for me to model for my students what the goal posts are as we move along through the course together. I think I wrote about this already, but one of the changes I have made this year is that I am handing out previous tests that I have written a few days before our test day this year. My students take this HW assignment more seriously than any other assignments. I wrote the Geometry book we use and I have written all of the problem sets we use. I hope that my students feel that this problem sets are meaningful and worth their time. However, I understand that there are calculations to be made about how to spend time and my students feel that available time is at a minimum. I think that I will not wait until the week of my next test, I think that tomorrow I am going to hand out last year’s test even though our next test does not occur for a few more weeks. I will check back in on this test every couple of days to give the students a sense that we are making progress. I also want to spend this summer compiling about twenty or so proofs that I think are particularly interesting. I will not include the proof itself, I will simply put together a set of diagrams and given information along with the conclusions to be drawn. I think that I want to hand this packet out at the beginning of the year next year and use this as a regular reference during the year.

I would love to hear any opinions about the benefits or drawbacks of these ideas.