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TMC16 Reflections, Part One

I’ve been trying to sort out my thoughts from the past week in Minneapolis and I have found that one of the best ways for me to do this is to sit and type them out. I am thinking that I may partition these reflections into three or four parts over the next day or two so that the ideas I am wrestling with will feel more bite-sized to me. Lil’ Dardy just had a terrible dental appt this morning so I am home with her all day. This will give me some writing time as she just naps away her pain and discomfort.

 

First, I want to concentrate on a small roundtable discussion section that I had proposed. I called it Building our own MTBoS at Home. A little background helps. I work in a small independent, day and boarding, PK – PG, co-ed school. I have five full-time colleagues in my department in our high school. Our other campus is three miles away and that is where my two children attend school. There are few other independent schools in my area and I have not found a way to connect logically with the public schools in my region. When I proposed this session I was hoping to crowd source some wisdom. I LOVe the online community I have tapped into and I suspect that if you are reading this that you do too. I also know that as valuable as you all are as an online resource, it is even better when I can sit down face-to-face to share ideas and energy. That is one of the beauties of the Twittermathcamp (TMC) experience. So, I was hoping to gather some ideas about how to build outreach so that we can find some of the same sustenance that comes from TMC more regularly in our home areas.

 

One of the GREAT problems posed by attending TMC is that every session slot has multiple promising events occurring. I was happy to have five energetic folks come to my session. I know that Sam Shah (@samjshah) and Tina Cardone (@crstn85) had a session with a  similar theme happening the next day. I look forward to picking their brains to see what came out of their session. A couple of the folks in my room where newbies to the TMC experience and it was great to hear what was on their mind. Our speaker at the Desmos pre-conference challenged us to think of evangelist as part of our job title, so that was on my mind all weekend. As we chatted in my session this idea kept coming up. Glenn Waddell (@gwaddellnvhs) spoke eloquently about his journey building community in Nevada. The phrase that came to my mind listening to him was ‘death by a thousand paper cuts.’ He spoke of sending out emails with links to administrators and other teachers every Monday. Simple, short links with a friendly message along the lines of ‘I saw this in my feed and thought it might be helpful.’ Every Monday – this is the part that really resonated with me. Be persistent, be consistent, be short and to the point. There is a local group that runs a math competition in the spring here – usually during our spring break, unfortunately – and I want to reach out to them. I want to find the email address of math teachers at my local schools. My goal this year is to build a couple of email group with addresses of these folks and reach out and share on a regular basis. Currently, I have been in the habit of emailing (or tweeting) links to colleagues – both those in my building and my online community – whenever I see something interesting. I think that I am going to adopt Glenn’s idea and make it sort of a weekly roundup. Perhaps I will use this space as the forum for my online team to share out ideas I have gathered or developed in addition to sharing out my classroom experiences. The other big idea I took away from Glenn was that he arranged a sort of happy hour meeting with some teachers in his area and, through the help of some grant money was able to provide some appetizers. He said that he also shared out some ideas regarding improving personal efficiency through some nice applications in addition to discussing class ideas. So he summarized by saying that he was able to provide a space that age each teacher three things to takeaway – (1) Some free food; (2) Something to improve their own personal life; and (3) Something to improve their own classroom.

 

I am pretty confident that I have a model to emulate and I hope to be able to start small with a meeting of local math teachers so that we can start building a support group for each other here in NE PA.

 

I want to thank all of those who came and I am pretty sure that I got all the names correct. I apologize if I missed someone in my scattered notes or if I got your name wrong. In the room was Kathryn Ramberg (@KathrynRamberg), Chris Robinson (@Isomorphic2CRob), Stephen Weimar (@sweimar), and Mary Langmyer(@mlangmyer)

Please reach out to any of these folks to improve your own community or to continue this conversation of how to enrich our local spaces the way we have enriched the online community that continues to grow. As always, also feel free to poke at me through the twitters where you can find me @mrdardy

 

A String of Good News

Our school year ends early, we graduate the day before Memorial Day here. So, I have had some time to unwind AND to look ahead to next year. I’ve been thinking about my new Discrete Math text, problem sets for my AP Calculus BC class (thanks to inspiration from Lisa Winer (@lisaqt314)), and my upcoming trip to TMC where I will be hosting a brief session to discuss how to develop communities similar to our MTBoS back at home.

Recently, I received not one, not two, but THREE pieces of good news that has happily distracted me a bit from thinking about the fall.

Last summer I led a session at the Pennsylvania Teachers of Mathematics summer conference. I gave it the dramatic title Escaping the Tyranny of the Textbook and it is essentially my love note to the MTBoS community. The goal of the presentation was to have any participants in the session leave the room feeling empowered to write their own curriculum or to learn better how to crowdsource curriculum that is tailored for their classrooms. I was pretty happy with it but I know it needs to be punched up. What better motivation to improve something than to put yourself in a public position where you need to be up in front of people all over again? So, I sent proposals to two upcoming conferences and I learned in the past few weeks that I was accepted to both of them! I will be presenting at the fall conference of the Pennsylvania Association of Independent Schools in October and I will be presenting at ECET2NJPA in September. I am flattered that my proposal was approved by each of these conferences and I am excited to meet some new folks to expand my circle of colleagues even more.

Our school started a STEM initiative shortly after I arrived here in 2010. The first director of the program has decided to step down in large part due to other responsibilities that she has since taken on. She has put the program on firm footing and when he school announced this opening they committed to having a director and two associate directors. I received the great news last week that I will be one of the associate directors of the program. All of our freshman take a STEM class that was designed by the program director and some of our students. They created some lovely iBooks that are still works in progress and that I feel a kinship to since I created our text for Geometry in a similar fashion. We have been hosting guest speakers, alumni, panels of regional experts to discuss items of interest. It’s an exciting program and I am looking forward to being part of the team for the next year. If any of you have advice regarding possible directions for STEM programming, please share here in the comments or over at twitter where I am found @mrdardy

 

How to Succeed

Feedback from my students at the end of the year touched, in part, on the idea that many of my students take some time to adjust to my expectations in our course. Years ago, I wrote a document called How to Succeed in Calculus. This was adapted from a document I found online by a teacher I never met named Dave Slomer. I have modified that document for my Geometry class and I want to share my first draft here. I shared it with my Geometry team and we have a nice conversation started about how to introduce and integrate this document. The first reaction from one of my colleagues is that the document might be a tad too long and students might easily put it aside. I agree that it is a bit wordy but I also feel that there is not much that I want to cut out. I would love any constructive feedback either here or through my Twitter account over @mrdardy

Here is my first draft –

How to succeed in Geometry
KEEP UP WITH THE ASSIGNMENTS.
Over the years, I have found that the best indicator of a student’s success is whether they keep up with their assignments. Students who keep up will likely do well – students who don’t likely won’t. We will be together for a good amount of time this year and we will routinely refer back to ideas and skills that we have discussed together. If you do not keep up with your assignments then it will become increasingly difficult for you to master new skills.

REMEMBER THAT THE GOAL OF AN ASSIGNMENT IS TO UNDERSTAND THE MATERIAL – NOT JUST GET THE PROBLEMS DONE.
You understand the material best when you can do the problems – and get them right – BY YOURSELF. There is absolutely nothing wrong with asking questions or seeking help from me, from other teachers, or from your fellow students. Everyone will need help sooner or later in this course. However, you must have the integrity to realize that the goal of the assignment is NOT just to get the assigned problems done. When we write our problem sets we are aiming to make sure that there is sufficient practice for all of our students. However, there will be times when you will need more practice than this, and you must have the courage and integrity to realize it. When you ask for extra practice, we can provide you with assignments that will help you to master new skills.

TREAT ASSIGNMENTS AS “PRACTICE TESTS”.
If you take your homework problem sets seriously, if you spend time thinking and working through the problems we present to you, you will feel more prepared for tests and quizzes than if you do not. Hard work spent on daily practice pays off on test days. Athletes who take practice seriously are better prepared for game days. Musicians and actors who take rehearsals seriously are better prepared for performances. Students who take daily practice seriously are better prepared for assessments. We know this to be true.

USE EXTRA PAPER
Your problem sets have narrow spaces available. Do not try to squeeze all of your work in these spaces. It is unlikely that you will be able to read your own work when you look back at your work and it is very unlikely that I’ll be able to clearly see your work and understand your reasoning. Do your work on notebook or blank paper and give yourself space to draw and to think.

NEVER ERASE.
If you hit a “dead end” and want to start over, cross out the work you don’t want with a big “X” – do NOT erase it. It might turn out later to be correct. Also, if you come to me for help, the first thing that I will say is “Let me see what you have done so far.” If you tell me that you erased it, it will be much harder for me to help you. Erasing can be a big time-waster on tests (where time is very valuable).

READ THE BOOK.
This is important in every class, but in this class the text serves as a valuable supplement to what happens in class. Often your homework will be to read the book in addition to any of the problem sets that we have written. Read the book carefully with a pencil and paper nearby. Pay particular attention to the illustrations and examples. Study the examples carefully. All of you have access to a PDF of the text and some of you will also have opted to have a physical copy of the text as well. Use your physical copy, if you have one, for margin notes. Use your PDF regularly to follow hyperlinks to explanations and activities that have been built in to your text. These are valuable resources and we expect that you will attend to them when you are asked to read.

LEARN THE VOCABULARY AND SYMBOLS.
It is vitally important that we can communicate in the language of mathematics. As you read or participate in class, pay particular attention to the meaning of each new term and symbol. This is a course that is heavy on vocabulary, you need to spend time and energy on this aspect of your study of Geometry.

REVIEW REGULARLY.
Luckily for you, tests are cumulative, and we will review in class; therefore review is somewhat automatic. Don’t hesitate to go back to review or seek help on algebra skills or on earlier ideas from this course that you may not have mastered as well as you wanted to.

TAKE GOOD NOTES DURING EACH CLASS.
Good notes are essential for success in any technical field. They are essential for review – not only for tests, but also for the problems you will work that evening. It is far too tempting to sit and listen and watch during class. You may feel comfortable at times following our conversations this way. However, when you sit down at night to do your homework, you will be without a valuable resource and you may not remember well what the conversation was hours ago when we were together in class. Every study of learning that has ever been done suggests that the act of writing something down helps in strengthening our memory. It is my expectation that each of you will come prepared each day to take notes on our class conversations.
CLASS TIME IS VALUABLE.
You need to use the time at the beginning of class to get ready for geometry. Get out your books, assignments, notebooks, pencils, etc. I will usually have a question on the board or the TV monitor when you arrive in class. Get to work on that and get your mind in its math mode. Socializing may be more pleasant than math, but the goal is to make math more pleasant, and socializing often gets in the way. At the end of the discussion period, begin (or continue) the current assignment right away – what better time to get help if you get stuck? We only spend valuable class time on important topics, so take good notes constantly during class.

ORGANIZE.
Your success depends on your ability to recall (or find, relearn, and then remember) concepts and techniques that were introduced earlier. If your notes and assignments are scattered about, folded inside the covers of your book, papering the bottom of your locker or the floor of your bedroom, you’re sunk.

BECOME AS SELF-SUFFICIENT AS POSSIBLE.
There are many students, and just one teacher, and time is too valuable for you to just wait – stuck in neutral – for help. Look in your text and your notes for sample problems that might shed some light on your difficulty. Learn tenacity – don’t just “fold” at the first sign of difficulty. Is there another way to approach the problem? You can do it.

SEEK HELP AGGRESSIVELY.
Everyone, no matter how smart or proficient in math, will get stuck sometime this year. Perhaps there is a new concept or technique that just won’t fit into place in your brain. Tenacity and self-sufficiency are great attributes, but sometimes there is going to be a quiz on this stuff tomorrow. Sometimes there just isn’t time to be tenacious. Attend conference bells, ask questions in class, just be sure to get the help you need to succeed.

COMMUNICATE.
If you have a worry, complaint, suggestion, or concern of any kind let me know. I can’t fix it if I don’t know about it. Remember that just because a problem – or a solution – seems obvious to you, it may not be obvious to everyone. Speak up.

NOW, A PERSONAL NOTE…

There are some things I do in class that you may find unorthodox. If we understand each other early in the year, we’ll avoid a lot of stress later in the year. There are mathematical facts that I expect you to know and I will remind you that you should know them. There are times when you will ask a question and I may reply with a question. Or, I may redirect the question to someone else in class. This is not done to avoid answering a question, it is done to encourage a thoughtful discussion and to help you to develop important problem solving skills. I believe strongly that we understand ideas more deeply if we can explain our own thoughts to others. For this reason, we sit in groups facing each other, rather than having everyone face me or face the board. I expect that you will explain ideas to each other and that you will ask each other questions. Questions in this class will ALWAYS be answered; you may just have to be patient before the answer arrives.

A quote by Galileo Galilei

“Philosophy [nature] is written in that great book which ever is before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. The book is written in mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.”

 

Problems / Exercises

I wrote about this earlier today and I want to spend a few minutes trying to organize my thoughts.

A conversation on twitter today with David Wees (@davidwees) reminded me of a conversation with a former colleague. It also reminded me of a class I took in my master’s program. The course had the vague title Problem Solving and my professor (who was my advisor) had a long background in studying problem-solving. I remember (not clearly enough) that we had a working definition for what qualified as a problem. The definition revolved around the idea that there are certain questions we encounter in math where we immediately know what we are supposed to do – what formula to use, what definition or theorem to call upon – while there are other questions where we do not immediately know what we need to do. The first group we classified as exercises while the second group were called problems. It is not necessarily that problems are harder. I have certainly dealt with many challenging math questions where I knew exactly what I needed to do, it was just really hard to do it. I have a real fondness for problems in mathematics and I have developed the habit of writing homework assignments for my classes that should probably be called problem sets. For years, I was writing these for Honors Calculus, AP Calculus AB, AP Calculus BC, Honors Precalculus, and AP Stats. A few summers ago I wrote a Geometry text for our school and I wrote all the HW assignments as well. These students are fundamentally different in many ways from the students I was working with in those other classes. I do not necessarily mean that they are inherently less talented or anything like that. What I do believe is that they are younger, less experienced, and less patient in their problem solving. So, they are more likely to simply shrug off a problem and figure that we’ll talk about it the next day. Over the course of the year most of them have become more patient and they are aware that we will discuss these questions and that they will not be graded on their HW. After my twitter exchange this afternoon, I am (once again) rethinking this strategy and I am nowhere near a conclusion. I did share the tweets with my class after we struggled through the question I wrote about earlier today. I asked them to honestly share their opinion about whether it is a valuable exercise to struggle with questions like this one. A few were upbeat and said that they liked thinking about these questions and that it is helpful to try challenging problems. One student said something really striking. She said that it is really frustrating to work through these problems alone and that she wishes she could get the opinions/insights of others when she is struggling with these questions. This is certainly in line with what David suggested in our exchange and with what my former colleague (who I wrote about earlier today) mentioned as well. I have some thinking to do here. I do believe that it is powerful for students to wrestle with challenging questions. I do believe that by not grading HW I am helping to create a safer environment to struggle. I know that a number of students work together on assignments either here in our dorms or libraries at night or during study halls during the day. I also believe that conversations in class are richer when they have some ideas already thought out to toss around. However, I also recognize that this is frustrating for some students and may simply push them further away. I recognize that if I am going to say that I value collaboration that I need to commit to making the time for that when we are together. I also recognize that what works to motivate seniors in college level math classes might not work as well with 9th and 10th grade students in a required math class.

 

Lots of thinking to do, luckily the summer will afford me some valuable time.

Thinking Out Loud

Super brief post here – class starts in 15 minutes. One of my Geometry colleagues asked me about a HW question I had written. I asked the students to find two cylinders that were not congruent but that had the same surface area AND the same volume. I thought it was a pretty interesting question, but I realized I did not have a coherent strategy for discussing it other than playing with numbers. I threw the question out to Twitter and engaged in a terrific conversation with Matt Enslow (@CmonMattTHINK), John Stevens (@Jstevens009), Dave Radcliffe (@daveinstpaul), and David Wees (@davidwees) Some good math was tossed around, but what really has my brain bubbling is an exchange with David Wees. He said he thought it was a great question but he would not have used it as a homework question. When I asked him why he said something that reminded me of a conversation with a former colleague. My former colleague once said that he sees a difference between exercises and problems and that he liked to keep problems for times together with the students where they could work together. I find that I feel (hope) that meaningful conversations can happen in class more readily about a problem like this one if the students have had time to think about it first. However, his words carry some weight with me as do David’s. I feel as if there is some conclusion I want to reach, but I also suspect that there is no right answer to this. I would love to hear some opinions about this in the comments or through Twitter where you find me at @mrdardy

 

 

What Do Numbers Mean?

This week I wrote about experimenting with number base systems in my AP Calculus BC class. A question came into my head yesterday about repeating decimals in base ten and whether/how we could decide if that number is also repeating in different number bases. It was really hard and the calculations got pretty ugly. So, today I started class with the following idea. I wrote a repeating decimal in a different number base and then converted it to base ten. The calculations are clearly more manageable and I had a clear idea that this could link back to our conversations about infinite series. What excited me today was that my vision of the infinite series was different than that suggested by my student Megan AND it was entirely different than a suggestion by my student Elijah.

I started with the base 3 number 0.122122122… I saw this as three different infinite geometric series’ each with a ratio of 1/27 and I worked the problem this way. Megan saw this as one series made of the first three terms with a ratio of 1/27. We, of course, arrived at the same answer and I really liked the way that her techniques was only one series to calculate instead of calculating three different series’ the way I saw it. I will put a picture below here with a different example showing Megan’s technique.

image

The example above started with the base 5 number 3.021021021021…

Elijah had a completely different approach, one based on how we teach converting base ten repeating decimals into fractions. The picture below shows his approach.

image

A couple of notes here. First, Elijah is a terrific math mind and this is a really creative approach. Second, this approach models the approach that my students have already seen. Third, this tactic encourages you to actually live more thoughtfully in this different number base.

I just came away SO impressed by the thoughtfulness, the persistence, and the creativity of my students this morning.

Some Post AP Fun

My Calc BC kiddos took their AP test last Thursday and we still have classes through next Wednesday. So, I have some time to play with. This year is the first time through for me teaching a Discrete Math elective and one of the topics I ran through with that class was the notion of different number bases along with a little history about some counting systems and the symbols used. I decided that my Calc BC students deserved the opportunity to think about this as well and for the past two days we have had fun saying things like 5 + 4 = 13 (guess the base!) and things like 5 X 2 = A. My students have appreciated me joking that they should make sure to go home and tell their parents that I said 5 + 4 = 13. What I have appreciated is seeing the combination of discomfort and curiosity which turns into a bit of joy as my students wrap their heads around this topic. It is especially in testing to me to see that the BC kids, who are really the top math scholars here, are not inherently more comfortable with this topic than my Discrete students were. There is a pretty big gap in the comfort level with mathematical ideas between these two groups of students, but this notion of fundamentally reconstructing meaning for numbers is a great equalizer. In BC today I even threw out this question – convert the base 8 number 41.37 into a decimal number. Contextualizing the ‘decimal’ portion of this number was not obvious right away, but they were easily convinced once one of their classmates offered a rationale for it. I know that this is far from an earth-shattering ideas, but I also know that this is an idea that too many students are not exposed to in their high school experience and I am kind of pleased that I get to blow their minds a bit. Tomorrow we talk about the Mayans and the Babylonians and we wrestle with their numeration systems. A fun way to wind down the year.

Platonic Triangles

Too long ago I started a Geometry post by suggesting that I might have a two post day in me. Needless to say, it did not unfold that way and some combination of malaise, exhaustion, and the irresistible momentum of the end of the year has kept me away from this place of peace and comfort for some time now.

I want to share something from our Geometry class this year that was largely motivated by the work of Sam Shah (@samjshah) and his colleague Brendan Kinnell (@bmk2k)

At TMC Sam and Brendan shared boatloads of ideas and docs that they had created for their Geometry class and I am still in the process of digesting them. One that jumped out to me immediately was a document that they called The Platonic Book of Triangles that they were kind enough to share and to allow me to share in this space. Sam wrote about their process here and here.

What I did this year was try to de-emphasize naming the trig functions and just concentrate on the inherent similarities tying together right triangles as a lead in to discussing the inherent similarities relating all regular polygons and circles. Part out of a whole has become a mantra in my class these days. So, what I did was I went to a local copy shop and had them print out a class set of bound copies of the above referenced book of triangles. My students are referring to it as the magic book of numbers. We reference it regularly to set up proportions to solve right triangles. I had the book laid out so that each page had complementary angles on either side. So the students recognized – with a little prompting – that the side lengths on the triangle with the 38 degree angle marked matched up with the side lengths on the triangle with the 52 degree angle marked. I have been SO happy with how they have taken to this reference. In a way it reminds me of the trig tables I used to look up in the back of my book but this has a couple of major advantages. First, it is far more visual and helps the students orient themselves. Second, it does not rely on memorization of a mnemonic about the definitions of the cosine, the sine, or the tangent of an acute angle in a right triangle. I have been careful when I do use those terms to say as clearly as I can that for now they do not want to talk about these functions for anything other than acute angles in a right triangle. There is a whole world of trig excitement waiting for them after their experience in our Geometry class is a dusty memory.

From this conversation about solving triangles and using this to lead into explorations of regular polygons I wanted to make sure to introduce the idea of radian measures to my young charges. I came up with what seemed like a clever idea. It was a chilly, drippy day here in NE PA so I called up the weather bug applet on my laptop. However, what I did before class was I changed its unit of measure to celsius rather than fahrenheit. A student mentioned that it was unpleasant outside – with a little prompting – so I called up my weather bug and expressed surprise that it was only 13 degrees outside. Students quickly pointed out that this was simply a different way to measure the same thing, that there is a way to jump from one representation to the other. Aha, the hook was baited! I then launched into a pretty unexciting, standard representation tying together radians and degrees, relying on my mantra of part out of a whole over and over again. I am not fully convinced that they are buying in and there is evidence that many of my students seem to think that attaching pi to a degree measure is simply some sort of stunt. I am also seeing evidence that simplifying fractions, especially those where the numerator is already a fraction, is a serious challenge to too many of my students. However, what I am convinced of at this point is that a seed has been planted that has a better chance of blooming in precalculus than for those students who did not see the concept of a radian presented to them before. We have our unit test on Monday and I hope not to be disappointed.

 

A Geometry Explanation Idea

Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.

In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.

Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy

 

More Calculus Fun with Series

Many many thanks to the wonderful Mike Thayer (@mthayer_nj) who sent a link to this lovely video https://www.youtube.com/watch?v=XFDM1ip5HdU in response to yesterday’s post.

We started each Calc BC class today by revisiting the rational function that caused me so many problems yesterday. Innocently enough, I decided that it would be interesting to examine 1/(1-x) first by long division where the divisor is 1 – x and we got the power series 1 + x + x^2 + x^3 + … as the quotient. Then, we used -x + 1 as the divisor and got the series -1/x – 1/x^2 – 1/x^3 – 1/x^4 … as our quotient. Class ended too soon and I was not able to answer the question of how we could consider these two very different looking series as being equal to each other since they were each results of considering the rational function 1/(1-x).

So, after sending out my call for wisdom in yesterday’s post I went to GeoGebra and discovered something lovely. Something that I was sure my students would be able to discover for themselves. I created a GGB file and I planned our day today. Can I tell you how proud I am of my students for how they handles today? Well, I won’t wait for your permission, I will just come out and tell you how thrilled I am.

I started the day by quickly revisiting yesterday’s two division results and then I called up the GGB file with only the rational function showing. They saw that I had each of the other two series expressions typed in already (out to x^5 for each) and I asked them which of the two they wanted to see first. My morning class wanted to see the series with x as the ratio first, my after lunch class wanted to see the one with the ratio of 1/x first. In each case, after unveiling the function of choice and noticing the relationship between the rational function and this new series expansion my students made the following observations:

  1. The graphs only seem to match over a certain set of x values
  2. If I were to add more terms, that match would improve
  3. If we look at the graphs of both series then we will have a nearly complete match to the graph of the rational function

What my students realized – what I realized last night – is that the two series we can generate have completely opposite intervals of convergence. It was absolutely lovely to see geogebra help this intuition along and it was fantastic that they made this realization before seeing the second graph to confirm it.

After this breakthrough we watched the video together and all of our brains were a bit achy by the end. Amazing that Mike found/knew this link so quickly when I blogged last night.

Some other notes – A student in my after lunch class made this observation about the complementary nature of the intervals of convergence before we even looked at geogebra. I gave one last example, f(x)= x^3/(x+5) and the student who was bothered by -1 = 1 + 2 + 4 + … quickly converted x + 5 into a form of 1 – r so we could interpret the rational function as an infinite geometric series. Another student converted it to x^2/(1 + 5/x) and we, once again, had two different series’ that had complementary intervals of convergence. I have taught this course five or six years before this and never had this ‘discovery’ pop up. Now, we cannot avoid it. It provides a wonderful context for our upcoming conversations about Taylor series and it gives us the opportunity to be more aware of convergence expectations. A pretty great day!