Trig Identities

This year I am teaching Precalculus Honors at my school (in addition to two different levels of Calculus) and I have not taught this course since the 2010 – 2011 school year. Last weekend, as I was planning ahead a bit, I realized that trig angle addition identities were coming our way. I have to admit that I have been entirely unsatisfied with how I dealt with this in the past. Most texts have some sort of distance formula based derivation of the formulas and I have read through them over and over never really satisfied that I could add much to the presentation. I generally presented these as facts and tested out a handful of examples to see that the formulas verified what we already knew to be true from the unit circle. A pretty unsatisfying situation. So, I did what I do. I sent out a call to twitter for help and got the typical handful of helpful responses. One really stood out and I tried it out in three of my classes. Tim Brzezinski (@dynamic_math) sent me a link to one of his lovely GeoGebra explorations. You can find that link here I am including a screenshot below to help you understand what we were able to accomplish due to Tim’s clever design (and his endless willingness to share!!!)

The front page of Tim’s GeoGebratube link

The students are presented with the above image and the very simple facts that this is a rectangle and that the two yellow triangles are similar. The point on the right side of the triangle is movable. A few things right off the bat struck me as wonderful here. We talked about WHY we could know that the yellow triangles were similar. So, we had the opportunity to remember the AA postulate. A student in one of my classes knew that the upper angle is alpha + beta because it is the alternate interior angle of the lower left corner angle. Super sweet! I was going to present a boring conversation about 90 – alpha and 90 – beta on the top. So, I liked that aspect right off. I also LOVED the aspect of how open this construction is AND the fact that it was not at all obvious to my students what we were about to discover. Pretty cool.

I ran this first for my Precalculus Honors kiddos and had each small group discuss where the wages need to go then we put our thoughts together. After a (very) gentle reminder of the structural properties of rectangles, we realized that we had discovered the angle addition formulas for cosine and for sine. An interesting response followed. One of my more curious and driven students asked ‘Don’t we have to prove this?’ I think that this speaks volumes about the natural response to the idea of ‘proof’ in our students. This exercise seemed clear and concise. Couldn’t qualify as a proof, right? Now, I am not fooling myself here. There is still a great deal of simply committing these formulas to memory at the end of the day. But I am convinced (CONVINCED!) that this feels more meaningful now. My kids were able to see and derive for themselves these relationships. They stopped and thought about similarity, about ratio definitions for the cosine and sine functions, and about the structural requirements of calling something a rectangle. I went on to tell them that they do not need to commit to memory double angle formulas because they come straight from here. Most students don’t take my advice on things like this, they feel safer simply consuming memory space with formula after formula, but that is another issue entirely.

After this went SO well with my precalc honors kiddos I unveiled it in my Calculus Honors class. We were just getting to the point where we were dealing with derivatives of trig functions and I knew that the chain rule was about to be laid on top of this. I guessed that this would be a great exercise to jog their dormant trig memories from last year. Again, in each section of Calc Honors, small group conversations led directly to sharing of ideas and a quick dissection of the diagram. I am pretty sure that these conversations woke up some sleeping facts in their brains and I hope it pays off in the form of quicker recall and comfort when we lay the chain rule on top of the standard trig derivatives soon.

Many thanks to Tim and to all the others who shared out ideas when I sent out my call for help. My students don’t really understand how much better their education in my room is due to the network of supportive, smart, creative folks out there. I do make an explicit point of telling them when I am using ideas/activities from others to help make all of this clear. The subtext I hope sinks in is this – If you have an interesting questions, send it out to the world. You’ll get some interesting feedback.

I’m Spoiled

This post is inspired by my AP Calculus BC class. At the beginning of each of our seven day rotations, I give them a problem set. These problem sets are pretty wide ranging, some Calculus questions, but mostly wide open fun problems to explore. Here is a problem from the set that was due today (actually due yesterday, but we were snowed out):

Starting at the origin, a bug jumps one unit either left, right, up, or down. He jumps once each second. List the possible locations (and probabilities associated with those locations) that the bug could be after 6 seconds.

I have given them variations of this where the intrepid bug could only move left or right. I thought that this would be an intriguing extension. During my library duty nine nights ago one of my students spent about 15 minutes with me bouncing ideas around. We came up with the GeoGebra sketch below:  

This seemed pretty daunting and I admitted to him that I may have overreached with this problem. We shared our conversation the next day with the whole class, including showing this intimidating graph. Trying to figure out how many of the 4096 pathways could lead to each point felt overwhelming. A couple of students started tossing out ideas but I kind of encouraged them to let this problem slide and blame me for poor planning.

When the problem sets came in today three of my students discussed some coding attacks that they took. Two of them were collaborating late into the night working on an approach to the problem and the output from their attack is below:

They took over the class conversation today pointing out that they recognized a pattern based on a multiplication table where the column header AND the row header were each the 6th row of Pascal’s Triangle. The numbers in their printout above are the products of this table. A long explanation invoking a three dimensional Pascal’s Triangle (I thought of layers like oranges in a conical pile) was presented with great enthusiasm. Some of the kids in class kind of glazed out, but a number were fully engaged. I was engaged, awed, and slightly confused. I have asked the two who collaborated on this if they would be willing to be guest bloggers here and they seem amenable to that idea. I hope to have a follow up, in their words, in the next day or two.

I am SO spoiled to be able to sit back and learn from students who could have easily left this problem alone, but were unsatisfied with that notion.

 

TMC17 Reflections

Beginning my reflections on the latest TMC experience (I am fortunate enough to have been for the past four years) I find myself focusing more on the personal experiences in ATL than the mathematical ones. That being said, I LOVED the presentation on base-8 math by Kent Haines (@kenthaines) and I am beginning to shift away from my strict aversion to multiple-choice questions based on Nik Doran’s (@nik_d_maths) advice in his morning session.

 

Last year in Minneapolis I allowed myself to dwell on the fact that there were social happenings that I was not part of. I KNOW that this is an inevitable fact when any large group of people gather together. It was especially true since we were housed in different places AND I was not equipped with technology that allowed me to tune in to everything going on around me. It was not until November of this past school year that I had a smart phone. Looking back, I KNOW how foolish this was. I had lovely dinners and chats with folks. I went out within hours of arrival to a lovely pub with Brian Miller, his school colleague Wilson, and Henri Picciotto. I had an amazing talk at dinner one night with Dave Sabol, who is kind enough (or crazy enough) to be one of the hosts for TMC18. I had fantastic math conversations and life conversations and came home a richer person than I arrived. However, I have allowed myself to dwell on what did not happen.

 

This year, armed with a smart phone (that I did not end up using much at all, really), going to a hotel where (almost) everyone was staying, and being in a city I knew, I went in with an agenda for myself. I knew I would be away on Friday night visiting an old high school buddy who was also my first college roommate. I made a commitment to myself. I was not going to hang around and see what happened about lunches or dinners. I sent out a tweet on Wednesday night inviting folks to join me at a restaurant I found called Smoke and Duck Sauce. Wednesday night ended up with a large gathering at Rose and Crown that was a great deal of fun. I sent out a call on the #tmcplans for Thursday night and had a great dinner with a fun group. On Friday night I had a lovely meal with my old friend and his family and returned to the hotel to stumble in on a deeply meaningful conversation with a fantastic group of friends. I was drawn over by seeing Brian Miller and Jasmine Walker (a couple of my favorite TMC pals) and ended up awake far later than I intended to be as a sprawling group of folks in a corner of the lobby bar really dug down deep on some personal and professional issues in a sensitive and vulnerable way. My had was spinning as I went to be. On Saturday night, I sent out another call on #tmcplans and ended up at Cowfish with a dozen folks. A LOVELY meal, great conversation, laughs as we celebrated a fake birthday, and a great sense of belonging and satisfaction as people piled into my rental car there and back on each evening. I went along to a breakfast at Waffle House based on an open invite. I had lunch with different folks every day at the campus of Holy Innocents. I had a quiet breakfast by myself the first morning of the conference enjoying southern grits and getting my head focused for the upcoming adventure.

 

I am not going to dip my feet into the mini controversies that came up during the week about hashtags and inclusion. I just want to say that I know that when I took it upon myself to be responsible and engaged in the community I enjoyed myself far more than when I was passive about it. Even though I also enjoyed myself then!

 

Of course, the social aspect and the connections are only part of the reason to come to TMC. There is also some sweet math to be experienced. My morning session with Nik thinking about hinge questions has me seriously re-thinking my bias against multiple-choice questions and recognizing their value if they are thoughtfully constructed and are treated as important data points in understanding what my students understand. His energy, intelligence, and good cheer made the morning sessions well worthwhile. I had two moments of mathematical epiphany during the week. On one of David Butler’s afternoon sessions he introduced us to some of his puzzles from 100 Factorial. I worked in a group with Jasmine, Joe Schwartz, and a new pal Mo Ferger on a fantastic problem called skyscrapers (you can find a link here!) We worked doggedly, and successfully, on this problem. On an afternoon session with Kent Haines I worked on some problems and pattern finding in base eight arithmetic. Again, working with some folks in the room (I wish I could remember who!) we poked around and noticed and wondered and fought the frustration that many of our students must routinely feel as we tried to find a comfort level in this realm of mathematics.

 

After a busy, happy, and rewarding three days with my #mtbos family in Atlanta, I am now relaxing with my (much smaller) family on vacation counting down the days to the new school year. I know I will still have some of this energy fresh in my mind in a few weeks. The challenge is to keep it fresh in my mind all year.

Long Overdue Thanks

I will be cross-posting this over at the One Good Thing site as well.

 

I graduated high school in 1982 and I just started my 30th year as a high school math teacher. During my time in school there were three particular teachers that had a huge positive impact on me. My junior and senior English teacher, Mrs. Myra Schwerdt; my junior Honors Introduction to Analysis teacher, Mrs. Sally Giles; my senior AP Calculus BC teacher, Mr. Barry Felps.

I was fortunate enough to have run into Mr. Felps about 8 or 10 years into my career at a workshop. I graduated high school in Jacksonville, FL and I went to college and started my teaching career in nearby Gainesville, FL. My graduate advisor at the University of Florida was also Mr. Felps’ advisor when he had been in school. This made me feel that there was some sort of deep connection there and, in an odd way, reaffirmed my decision to think about teaching. Anyway, I was able to see Mr. Felps in person and thank him for his influence. I hope that this meant something to him.

I never saw either Mrs. Schwerdt or Mrs. Giles again after graduating. I fear that Mrs. Schwerdt may no longer be around (this was a long time ago that I graduated!) but recently a friend and classmate sent me a link to a profile of Mrs. Giles. She had changed careers at some point and was working for a local agency in Jacksonville called Cathedral Arts Project. A local paper wrote a feature on her career there as she was preparing to retire. I reached out to the communications director at that program and she shared Mrs. Giles’ contact information with me. I just finished writing an email to Mrs. Giles thanking her for what she did for me and letting her know that she is a major reason that I chose to do what I do.

I know that it means a great deal to me when former students reach out to thank me or simply to share some story as a way to keep in touch. I have no real way of knowing whether Mrs. Giles will remember much about me as it has been over 30 years now since I was in her class. I also know that she no longer teaches, so this note will not serve as a pick me up on a tough teaching day. But I also know that I fell MUCH better having written this note and I hope that in some way it brightens her day.