After another round of conversation on twitter that included tips from a student who just graduated from our school, I have been playing with three different data sets. I want to play with average daily temperature, high tide level, and with daylight hours. The data for the first two came from wunderground and the third data set came from dateandtime. Below are pictures of the data and Desmos links to the tables.

This picture (above) shows the average daily temperature in my town during 2017 at 10 day increments. You can see the Desmos table here. There are some things I like about this picture. I like the fact that the general shape can be inferred. We can talk about why it fluctuates on a number of different levels. However, I don’t think that this is a great data set for beginning to develop an idea of periodic functions. It feels too noisy to me.

Here is the picture for high tides.

This table was built on data at 6 day intervals from the beginning of the year through June/July. You can see the Desmos link here. I would definitely ask the students to play with window sizing and I think that some powerful ideas about amplitude and vertical shift can quickly come out of such a conversation. I picked 6 day increments thinking that I would be slightly off phase with what I thought would be a period related to the full moon. A quick survey on google just talks about a 12 hour plus period so I may have been making this up in my mind. This picture feels more friendly with just a little noise involved. I might use this one early – maybe even on day one. The next one is much cleaner looking. Here is the data on length of daylight hours during 2017.

This is where my mind was when I created the demo lesson. However, this data is for our hometown here. You can see the Desmos link here. My thoughts about this data set go down two different paths. One thought is that this is clean and clear and easily explainable. One tweak I might make based on my conversation with Bonnie is that I might extend past one full year (say about 400 days or so) to make the periodicity visible not only intuitively meaningful. My second thought is that this might be too clean that it might lead my students into expecting such clean, clear periodicity in a messy world. I am probably overthinking this on the second train of thought.

I expect to have five table groups of students (groups of three in my classroom after long debating it, I accepted the wisdom shared by a number of MTBoS folks – especially Alex Overwijk (@AlexOverwijk) and my classes were better this year because of that change!) and I am thinking that each table group should have different data. I am playing with the idea of mixing up sunlight or tidal subsets of data versus simply subdividing one larger set. For example, if I go with the cleaner daylight data I can extend it to about 450 days or so and give different table groups subsets of about 35 data points each. I feel that they would benefit from seeing how the data ‘fits together’ and that individual table group decisions about amplitude, vertical shift, and period all match each other pretty well.

I would love some feedback/suggestions/questions and I thank Bonnie again for her valuable thoughts. You can drop comments here or over on twitter where I am @mrdardy

## 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.

## MTBoS New Year’s Resolution

If you have not seen my Geometry book yet and want to take a look, you can download it from my Dropbox at this link. If you want to look at my Chapter Six pacing calendar, entrance slips, and HW assignments, you can find them all in this Dropbox folder

I am hoping that January will be a productive month for this blog space as I reflect and report on how this experiment unfolds.

## TMC 15 Reflections, Part Two

So, in my last post I gathered some of my thoughts about the days at TMC up through the post-lunch keynotes. I did not touch on the My Favorites sessions, the afternoon sessions, or the evenings. I’ll tackle those in this post.

Each morning and afternoon we were treated to quick presentations by volunteers (well, we were all volunteers there, really) who wanted to share something brief about their classroom practice, about some project they were working on, or to share some other insights. These were all fun nuggets and I don’t want to ignore any of the treats that were shared, but I do want to single one of them out. Glenn Waddell talked about his commitment to high fives with his students. When I was a young teacher (a LONG time ago) I was encouraged to stand outside my door between classes. I try to remember this and use this as an opportunity to greet my students and other students passing by in the hallways. However, I do not have the kind of commitment to this and energy that Glenn shared. He greeted each student with a high five and discussed how he broke down the resistance of a particular student who finally joined in the fun. He talked about how on days when he had to miss class, the students would ask for two high fives the next day. I had the pleasure of picking Glenn’s brain about all of this one evening and he told me that would tell his students that he was congratulating them because they were about to do something awesome. He also said that he greeted each class with a rousing ‘Good Morning!’ no matter what time of day he met them. His enthusiasm was infectious and I could not resist giving him a high five every time I saw him the rest of the week.

I went to two afternoon sessions each day and I want to highlight two particularly fantastic ones. Dylan Kane (@math8_teacher) conducted a session called Arithmetic to Algebra: Key Building Blocks in Abstract Thinking. Dylan shared a series of fantastic questions that he had developed to extend student thinking and give him insights into their thought process. Dylan is bubbling with energy and enthusiasm and he asks great questions. He has been on fire on his blog lately – if you have not read his stuff you should change that and visit him regularly. A great conversation ensued in the room and I am looking forward to using some of his questions as class warm ups this year. Meg Craig (@mathymeg07) conducted a session called Function Transformations Without Tears. I have raved about Meg and her blog recently where she has been on a tear recently sharing tons of great files. She conducted a thought provoking session on how to approach function transformations together. What was as impressive as the thought that went into her presentation was the energy of the conversations in the room. A number of us were sharing ideas about what language to use and how to incorporate this into our classroom. Meg gave us room to debate/discuss even though it may have taken time away from what she was thinking of doing in her session. Both Dylan and Meg stepped back and let us, their ‘students’, take the conversation where we needed/wanted it to go. Nice modeling of positive teaching behavior on their part!

This just leaves dinner time and evening fun to recount. On Thursday I had the pleasure of meeting my brother for dinner. It has been well more than a year since we had seen each other. He has lived in LA for nearly twenty years and just recently bought a house outside the city. We met in a nearby town that was roughly equidistant from his home and my hotel. We were able to spend nearly four hours catching up before I started winding down and headed back ‘home’. When I arrived at the hotel a little after 10 PM I walked into the hotel courtyard to find about 40 or so conference members hanging out and catching up with each other. I was greeted warmly by three of four folks as soon as I walked through the doors and my energy was lifted. I spent some time chatting – especially with Brian Miller (@TheMillerMath) who I met last summer. Brian became my car buddy for a number of events over the time there and I tried to help him with an unfortunate battery problem his car developed. I was struck once again by how meaningful friendships can feel even though we had seen each other in person for only three days in Oklahoma. Meaningful conversations with Alex Overwijk (@AlexOverwijk), Jasmine Walker (@JazMath), and others over the time there made the hotel courtyard a warm and wonderful place in the evenings. One evening featured a barbecue at a nearby park that was funded by the wonderful folks over at Mathalicious and prepared by local hosts. It was a beautiful night to be outside, great food, good conversations, and wonderful views of the mountains nearby. Having spent most of my adult life in Florida, I am taken by mountains.

On my final night there I ended up having dinner and a cupcake run with a group of 8 terrific women (lucky me) before retiring to the hotel courtyard again. I had a very early flight back across the country but I kept myself up until almost midnight talking with fascinating folks before finally retiring for the night.

I realize I am just barely scratching the surface of the energy and spirit of the camaraderie of the folks there. I will try to capture it this way. I left for California on a Wednesday and the rest of my immediate family left the Friday before for an adventure. I was away from my wife and kids for nine days and missed them terribly. While I was at TMC15, instead of primarily dwelling on missing my family (which I still did, for sure) I felt so immersed in a different kind of family. Sure, we are not bonded by blood, but we are bonded by a love of idea, by a love of our work, and by a generosity and, dare I say it, love for each other. Although I have spent six days or less in the physical presence of this community, I feel a deep bond of trust and friendship. I have seen over and over again how a blog post about challenges and questions in my job results in thoughtful replies and sharing of resources. I have seen how a tweet sent out about a question results in quick, thoughtful replies. I have seen how the simple accident of sitting in the same place in an auditorium for three days in Oklahoma leads to a sense of camaraderie that spans the width or height of our continent. I have also seen how my wife is already making plans for me to attend TMC16 in Minnesota and how to turn it into an extended family adventure. She sees what I clearly feel. This professional community helps me grow, helps me feel supported, and tickles my brain.

Next challenge – a new school year starting in seventeen days!

## Highlights of a Stressful Week

So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.

1. My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
2. I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
3. In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.

## Borrowing from the MTBoS

I’m guessing that most of you reading this are familiar with the awkward acronym for the Math Twitter Blog-oSphere – one of the joys of tapping into this community is that they are remarkably generous about sharing ideas and resources. Today in our Geometry classes (I teach only one of the five sections we have at our school) we used an activity written by Kate Nowak (@k8nowak on twitter). It is an activity based in GeoGebra and allows the student to explore the ratio between lengths of legs in a right triangle. You can find the document we used here . I modified (very slightly) the document that Kate originally posted here. Next time I use it I will tweak it a bit. I have only twelve students in my class and chose not to explicitly team them up. They talked with their neighbors as they are usually encouraged to. However, the directions either need to be tweaked so that team references are excluded or I need to clearly team them up. I am also debating question 7. A number of students did not make the explicit leap from using the ratio they found on page 1 and using it here. I don’t necessarily want to give away too much but I may add a little prompt that they should consider the work that they have already completed. We set up a google spreadsheet and in the next couple of days I will refer to this repeatedly to show that different students working on different triangles were arriving at the same ratio. We make a big explicit deal about scale factors between similar figures. I do not think we spent enough time pointing out that scale factors within figures will also match up for similar figures. I will definitely make this more of a point of emphasis next time through my text.

I cannot thank Kate enough for sharing this activity. My students worked well and I am convinced that they will have a more solid grasp of trig ratios moving forward. As I plan out the rest of the unit I am also going to be borrowing from Sam Shah’s latest post about trig. You can find that over here.

Man – the benefits my students are reaping from people that they will never meet – such as Kate Nowak, Jennifer Silverman,  John Golden, Jed Butler, Sam Shah, Pamela Wilson, Meg Craig, and so many more – is just remarkable.

## Why Do I Blog?

Once again, Dan Meyer has me thinking. This time the blame can be passed along to Michael Fenton. Michael raised a question on twitter.  His question to Dan was

Could you have written a list of 5 reasons you blog 5 years ago? And a list of 5 reasons you blog now? Lists match? What’s changed?

As usual, Dan’s page is generating some great responses. You can jump to that page here.

As I write, I am on my first day of spring break. It is a quiet, beautiful morning on my mom’s back porch with everyone else sleeping. Let’s see if I can make sense of this question.

So, I realize that most of what i have just said explains why I read blogs and prowl twitter. Why do I write my blog? A much simpler answer. I think that there are two reasons.

1. I want to give back – at least a little – to this rich world of ideas.
2. I want feedback on my ideas as they develop.

Not profound, but it feels good to chime in. Now, I’m off to breakfast.

## Summertime

That’s right. It’s wintery and snowy here in NE PA and I am thinking about the summer. It’s not because I want this school year to end in a hurry. I am enjoying my classes and students too much for that. No, I’m thinking about the summer because an amazing opportunity has presented itself. Tina Cardone – she of the terrific Drawing on Math blog and the mastermind / editor of Nix The Tricks recently reached out to me and asked me to attend Twitter Math Camp ’14 and to help her in running the Precalculus offering in the mornings that week. I’m flattered beyond my ability to put it in words. Looking at the list of people who’ll be there makes me think I need to go around with a sharpie to get autographs on my lovely MTBoS shirt that Jen Silverman designed. We worked up a description of what we hope to accomplish through a lively google doc discussion. We ended up with the following

Not sure if your course falls under the title of PreCalculus? Tweet us and ask!

Tina @crstn85 and Jim @mrdardy

The camp meets in Jenks, OK (near Tulsa). I have felt so energized by the connections I’ve been making with many of the people I’ll finally meet there. Can’t wait and I am doing a quiet countdown in the back of my head already.

## The Final Mission – MTBoS #8

The challenge this week is a sort of meta-sharing challenge. Sam has asked us to share about what/how we share.

One of my roles here in my school is that I am the Math Dept Chair. I also chair a technology committee here. One of the really joys of diving into the MTBoS community is that I am finding resources to share with my colleagues. I send out multiple emails per week sharing ideas that I have run across.

For example, in the last few weeks, here are some of the links I have sent out

To a tech committee colleague – http://blogs.kqed.org/mindshift/2013/11/inquiry-learning-ideas-for-math-and-science-with-ipads

To the whole tech committee – http://www.thethinkingstick.com/why-i-still-want-ms-and-hs-to-have-a-laptop/

To our STEM director – http://emergentmath.com/2013/10/30/a-problem-based-learning-starter-kit

These are just a few examples of how I’ve been inspired to share some of the wealth of riches I’ve found on the web through the MTBoS community.