Debating Divisibility

In our Precalc Honors class we are discussing exponential and logarithmic functions now. I want to relate a fun observation/suggestion from a student a few days ago and a debate that fired up in class today.

Our text defines exponential functions as any function of the form y = a*b^x as long as b is positive and not equal to 1. One of my students, a girl named Shailee, suggested that it would feel more logical to simply say that b is greater than 1. This way, functions with a base between one and zero would instead have a negative exponent. This might make it more consistent to think about positive exponents representing growth and negative exponents representing decay. This also feels like a smoother definition for b instead of having two qualifiers, we’d only have one. Kind of a nice suggestion and one that I will be adopting for our class conversations this year.

Today I ran an activity that was suggested by Henri Picciotto when he came to do a workshop with my department in May of 2016. I had a couple of containers of 10 sided dice. They were numbered 0  through 9. I assigned a rule for each of my three groups. One group was to roll all the dice and count the number of evens. They then dispensed any that were not and rolled again. Lather, rinse, and repeat. The idea was that the number of dice remaining should model a half-life for them. The second group was looking for primes. Again, exponential decay with a base this time of 4/10. The last group was asked to look for multiples of 3. Someone asked if 0 counts as a multiple of 3. I reflexively said no but then paused and thought out loud about it. I threw the question out to twitter and we went on our merry way. We gathered data, plotted it on Desmos in a table and asked fro regression equations of the form y = a*b^x. Worked pretty well except one group went from 40 something dice down to something like 8 right away when they were supposed to have a 1/3 chance. We then checked in on twitter where interesting things were being shared. I’ll clip a few tweets below:

A side conversation also occurred when Christopher suggested that the 0 on the die was a 10 not actually a 0. This, of course, would have prevented this whole interesting conversation from happening in the first place. Anyways, this got a heated debate going in class where my students just felt uncomfortable about the idea of 3 being a factor of 0 since this implies, by a simple extension, that EVERY integer is a factor of 0. I guess we all accept without much debate that 1 is a factor of every integer, but this feels off somehow. I went off to lunch to bounce this idea off of some folks and I might have scared a couple of colleagues who are less comfortable with math. A lively debate/discussion at lunch led one colleague to casually say ‘So much just happened there’ When I returned to my classroom and my twitter feed the conversation had moved into a modular arithmetic mode. Here is a taste:

So, let me first say what an honor and a treat it is to share in a conversation like this with my students, my colleagues, and my virtual faculty lounge of folks spread around the globe. It is a mind-blowing thing to think about how much this world of education has changed for me since I took the plunge to going twitter. I am convinced (CONVINCED!!!) that life is better for my students since I did. I also want to say that the idea of modular arithmetic is one that I love to share with my students and I am determined to figure out how to find time to do so with my precalculus students since this debate brought up these ideas. I also have to admit that I am just a tad uncomfortable saying that every integer is a factor of 0. One of the side conversations at lunch went like this : Me – If 0 is a multiple of 3 then that means that 3 is a factor of 0. Rachel (science dept colleague) – If we say 3 is a factor  of 0 wouldn’t we say that 0 is a factor of 0? Me – Uhmmm, this would imply that zero divided by zero is a thing, right? This reminds me of debates I had with a friend from my old college town debating the physical meaning of 0 ^ 0

So, a delightful lunch time conversation, right? Fun to lift the curtain a bit and have my students see a debate unfolding. Fun to get my brain agitated thinking about all of the implications of saying something as simple to my kids as ‘Look for multiples of 3’. Probably a lesson to think a little more carefully about my directions to them!

 

Many many thanks to Henri, Sam, Christopher, David, and Bryan for engaging in this conversation and for giving me the idea of this experiment.

Another Great Student Observation

Yesterday in Calc BC we were discussing L’Hopital’s Rule. I have mixed feelings about this conversation every year because it feels like this powerful idea about comparing rates of change, but I know many (most?) see this as a simple mechanical process. Of course, there are always cases where it gets misapplied just because it is a fun new tool. What is that old saying about a hammer? If all you have is a hammer, then everything looks like a nail, right? Suddenly, every rational function limit should be subjected to L’Hopital. A couple of years ago, I finally settled on a way that feels comfortable to start with the old classic limit of sin x / x as x approaches 0. GeoGebra is a good friend here.   

We all agree pretty quickly that it is clear that there is a limit here, despite the fact that the idea of dividing 0 by 0 causes us discomfort. The breakthrough I had a couple of years ago (and I am guessing many of you have had it as well) is to next look at the graphs separately. 

So, the conversation now centers on the fact that we can only see one graph as our eyes near the origin. An interesting debate arose. Students were convinced that these graphs clearly intersected each other, no question there. What became a bit fo a debate was whether there might be multiple points of intersection. I had the idea of creating a new function. If we look at the graph of y = sin x – x and see that there is simply one root we can put the question to rest. Before I could suggest this I was trying to get someone to voice my idea. Instead, one of my students (Jake S) made a contribution to the conversation. He said he was thinking about the inverse sine graph. He pointed out that if we snip off a piece of the sine graph we were looking at from the minimum on the left of the y-axis to the first Max on the right of the axis we would have a function that was constantly increasing. Since it started off above the line (which is moving at a constant rate) and the Neds up below it, it seemed clear to him that there was only going to be one point of intersection. I was kind of floored by the connections he was making on the fly. I do not expect my students to think of inverse trig functions without being directly urged to and even then there is great reluctance. I did not expect an argument that had this kind of subtlety on the fly in the class conversation. I have not thought deeply enough about his argument to determine whether there might be some odd functions that defy his thinking, but this graph sure did not.

I say this on this space periodically and I say this in my life to anyone willing to listen on a regular basis – I am spoiled by the kids in this BC class. I don’t mean to imply that I do not enjoy my other classes. I do. I’m just generally spoiled (fortunate / blessed / lucky / fill in the blank) I get a kick out of all of my classes at different times for different reasons. What I reliably get from teaching the scholars in BC is a daily reminder that I still have an awful lot of math to learn. I know that I know things that they do not yet, but I also know that they are coming to class with questions and insights every day that blow my mind.

New Ways of Thinking About Calculus

I have been teaching a long time now, so when students get me to think about a new way of doing something I am always excited. A super brief post here highlighting two solution techniques suggested to me by my AP Calculus BC students in the past week or so.

We are studying inverse functions and the relationships between their derivatives. We had settled on the fact that the function y = e^x is the function that is its own derivative. We also knew how to differentiate y = ln x based on this fact about the exponential function. I asked about the derivative of the function y = 5^x. I intended to derive a pattern for this derivative using the fact that we had derived to deal with natural log functions. Instead, one of my students suggested that we should think about the e^x function and the chain rule rather than develop a new rule. He pointed out that we know that e^x will eventually be equal to 5 so 5^x is simply a new power of e. He suggested that I write 5 = e^u and then differentiate the function y = e^(ux). I was delighted by this. Rather than add new rules to remember, simply this and rely on derived facts. I will always encourage my student any time that they can whittle down the number of rules to remember. Super excited by this.

On Thursday, I had a quiz in that class and one of the questions involved a rational function. I asked my students to verify that this function had an inverse that was also a function. I expected them to take the derivative and show that it never changed sign. I expected that because that is the way that I have thought about it and because I have taught them to approach this question through this lens. One of my students instead said that if f(a) = f(b) for some values where a and b were unequal, then that function is not one to one. He solved this equation showing that it was only true if a = b and concluded therefore that the function is one to one. Delightful! I tweeted about this and one of the responses congratulated him for relying on a definition instead of a technique. I applaud this as well and I hope to remember this well enough to present it as an alternative approach to answering this question.

Super proud of my students and I love that I get to make a big deal about the fact that I am learning from them as well as them learning from me. The big message, of course, will be that we should all be learning from each other.

How Do We Help our Students Ask Questions?

I have told this story to a number of friends and colleagues over the years. One of my favorite former students – he graduated in 1994 – gave me what I value as the best compliment I think I ever received about my teaching. He was a brilliant kid, school seemed effortless to him. I taught him for four years in a row culminating in AP Calculus BC when he was a junior. He took a math class at UF his senior year. About ten years ago I was living in Jersey and he was living in NYC so I had the chance to see him a few times there. Once we were having lunch and he told me a story. He worked in a small business doing financial analysis and he was frustrated by a problem he had been working on. He told his boss that he was going to take a long lunch to clear his head. When he came back his boss had left some notes for him on the file he had been working on. He told me that his boss reminded him of me. “He asks questions that I would not think of asking myself.” I walked away so happy about this. He did not remember a trig graph or a derivative or the fundamental theorem of calculus (although he probably did remember these things!), instead he remembered that I asked him questions that he did not think of asking himself. I felt SO good after that conversation. I was thinking of it today after school during our afternoon conference time built in to our day. All of our teachers are expected to be in our rooms for about a half hour after the end of the school day and many students make a habit of dropping by to ask questions. I was talking to two of my  Calculus Honors kiddos. This is our non-AP class that takes a deep year-long dive into Differential Calculus. We were looking at some problems on one of the problem sets I wrote and these two young women were saying that they understood the problems when we talked in class but they did not know how to start them on their own. I pointed out that almost all of the ideas in class came from the students, I rarely flat out TELL them how to solve a problem, we work through the question together. What I work really hard on is to ask questions of the students that prompt them to see connections and realize what they know about the problem. I want them to feel the power of being the ones who generate the answer. One of the girls said that she does not know what questions to ask herself when she is home working on these problems. So, the challenge is to figure out how to help her, and others, across that bridge. Is it enough to simply model an inquisitive mindset? Is it enough to be a good role model in persistently asking questions? How can I explicitly help my students develop that instinct and ability to push themselves along a solution path by asking meaningful questions? I would love to hear any wisdom on this front. I am going to share a meaningful quote that I ran across in my days as a doctoral student studying curriculum and instruction:

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

Why Do We Know What We Know?

In my AP Calculus BC class today I presented an activity from Christopher Danielson that I found through the MTBoS Search Engine

You should check out Christopher’s post. I asked the questions he proposed, but I had a new first one. I asked my students to discuss in their small groups what assumptions they were making about this function pictured here. I heard some good stuff. They talked about continuity, they talked about it being a polynomial function, hopefully one with an even highest power. They talked about the fact that circle C could not have a root since it is entirely above the x-axis (although one student raised the question of complex roots and this prompted a conversation about use of the word root versus calling them x-intercepts), they talked about the minimum number of critical values. In general, just some great recall. At our school, BC is a second year Calculus class so we were talking about ideas from last October/November. This led me to raise a question that I was a bit worried about. Earlier, we had made mention of the fact that polynomial functions with an odd highest power have all real numbers as their range. Sure, we know this. But why? Do we really have an idea why this is true? I was worried that this was too vague a question. I was worried that they would waive it away. We know this is true, Mr. Dardy. Why talk about this? Instead, we got some great GREAT conversations. I was told to think about limits, to think about derivatives. I jokingly asked if we should think about area between curves or optimization or some other time honored Calculus ideas. I was told to consider the limit as x grows without bound both for a positive leading coefficient and for a negative one. We discussed how all terms in the polynomial eventually become insignificant compared to the highest powered term. We talked about the derivative being even powered and what we know about those graphs. Man, I was just so pleased that they were willing to travel down this sort of hazy questioning path with me and reinforce what they know and WHY they know it. I say this every year, but this class absolutely spoils me.

 

Reflections – While thoughts are Fresh

In my two morning classes today we tackled the problem that I just blogged about (link here)

Some interesting observations first and then some questions that came from my learners.

When data is not presented as a table, there is a distinct extra layer of processing that has to happen. One student went straight to desmos to graph the points his group had. I liked that he wanted a visualization. A number of them, when I asked how to find the AROC between two pictures were flummoxed. Let me give you an example of what I saw from a number of students. Look at the picture below:

A number of students divided 1381.5 by 34.18 (more on that in a moment) and arrived at 40.418 as an answer. This matched the given information but they did not seem to notice that this did not answer any part of my question about AROC from point A to point B, from point B to point C, or from point A to point C. [This conversation makes sense (I hope!) if you have read the first blog post that is linked above]

I had to poke to get them to recognize that any time we talk of an average rate of change, we are talking about more than one data point to consider. A number of them were happy to enter a time like 30:56 as 30.56. This disappointed me a bit. For smaller minute data points I can see the mistake a little more clearly. For something as close to a full hour as 56 minutes, the willingness to enter .56 seemed more clearly wrong.

Once we ironed out the fact that we need to see time and distance as coordinates of a data point, then the slope idea for AROC fell into place more comfortably. Before talking about my challenging final question, I want to share a few questions and observations from the classes.

A student asked about the clock on the dashboard. Does it still count when the car is sitting? When I am sleeping? Great question, the answer was no. It only moved when the car was moving.

Conversations came up about the geography of Florida where I was driving. They guessed that some snapshots were taken in city traffic, others after highway travel.

Discussions came up about why the average on the dash did not change even when intervals had different AROCs. I relied on a baseball analogy. Late in the year a 0.250 hitter might have a great day and go 3 for 4 while not changing his overall average at all. That seemed to make some sense to them.

The final question I asked was more difficult for them than I had anticipated. I asked each group to consider the following situation. How long would I have to travel at 60 MPH to raise the trip meter average velocity to 42? In my mind I simply wanted to use the last data point as the jumping off point and add an unknown time ti to the x value (the time input) while adding a distance of 60t to the y variable (the distance output.) This idea did not organically appear in class and I pushed a bit more than I wish I had as I saw our 45 minute time together elapsing. I am often comfortable with questions being unresolved at the end of a class period. However, with this question at this time of the year (this was only our fifth class together) I really wanted a conclusion to the mystery. I will definitely revisit this experiment as we work more deeply on our ideas of rates of change and I will remind them of this conversation on a number of occasions. An unexpected bonus idea that came through loud and clear was the MVT even though we don’t have a name for it or a formula describing it yet.

Pretty pleased, I must say.

 

Exploring Rates of Change

One of the courses I teach this year is a course called Honors Calculus. It is a non-AP course and we made a decision about 6 years ago to make this a course in Differential Calculus. While the AP AB course completes a college semester in a high school year, this course completes an AP AB semester in a high school year. This allows us to remedy some bad habits, fill in some gaps in understanding or mechanics and, most importantly, really slow down and think about what we are exploring. I have an activity that I do on the first day of class where we explore motion. This year we went in the hallway and rolled a whiffle ball, a softball, and a lacrosse ball down the hall a fixed distance. We tried to roll each with approximately the same force and had a good conversation about what the data told us. We made some physics based observations that I did not plan for and we talked about what we knew and what we did not know about the rate at which the balls were rolling. My goal was to arrive at the conclusion that we could talk about average rates of change but not so much about the rate at a particular instant. The conversations went reasonably well, but we got distracted a bit by conversations about bounciness of balls and air resistance. In any event, I think I planted some decent ideas to consider as we embark on a conversation about  average rates of change of a function on an interval (our text calls this the AROC) and we are about to wrestle with the limitations of being able to know much about the instantaneous rate of change (the IROC)

This summer I rented a car whose dashboard gave information that I knew would work well for this class. A picture below will prompt your teacher brain as well I think.

I took six such photos during the course of my trip. Tomorrow, I intend to give each of my groups of students (I have them in groups of three) three of the pictures. I’ll scramble them up a bit so different groups should have different subsets of the data. I intend to ask them some pretty simple questions that should generate some good conversations. I want to ask them the following questions:

  1. What was my average speed between any of the two pictures? (So each group should have three answers for this)
  2. Can you determine my maximum velocity in that time interval?
  3. I want to raise the average to 42 MPH. How far would I have to travel at 60 MPH for this to happen?

 

These are not terribly deep questions, but they feel rooted in an example of real world data (I was inspired by Denis Sheeran’s wonderful book Instant Relevance for this data driven experiment) I also think that this will continue to scratch at the itch that will make the breakthrough of being able to find the IROC feel more meaningful.

I have all the photos together in a WORD doc on my dropbox. You can find that file here. I would love to hear any clever ideas about how to play with these images/this data.

 

Beginning of the Year Thoughts – Inspired by Christopher Robin

About three years ago I made a commitment to myself. I was still living with my family in one of the boys’ dorms on our campus and was living within a pretty strict clock regimen of dinners at the dining hall, study hall hours, lights out (ha!), and rotating dorm duties. I made a decision that I would close my computer when I left my classroom at 3:30 and not open it again until study hours started in the dorm at 8 PM. I managed to maintain that commitment (mostly) for the year. I was regularly up later than I really wanted to be due to dorm duties and noise. I was up early every morning catching up on work, but I had every afternoon free of staring at my computer. I hung out with my kiddos, we played on the dorm lawn, we ran around with friends on campus, I sat at the dining hall and caught up with folks. Part of what made it possible to carry out this goal was that I was teaching four classes that year, most years I have taught five.

This year I am teaching four classes again and, once again I made this commitment to myself. This past week was our last week before classes. I took the time to take each of my children to see movies. On Thursday I took my son to see the new Spike Lee film. It was terrific. On Sunday I took my daughter to see Christopher Robin. It was sweet and tender. A bit predictable, but filled with good nuggets. I got emotional on a number of occasions because I have become that dad. But I walked away renewed in my sense that I need to create space. I need to put that laptop away for awhile. Email will be answered eventually. My job as a teacher and a department chair is not THAT important that a few hours will get in the way of carrying out my responsibilities in a meaningful, humane, and productive way. I can just be dad and husband for a few hours. I can hang out on the hammock. I can sit and chat about music with my boy. I can watch my girl play with LoL dolls and her Baby Alive crew. I can sit on the porch and read. I can listen to some music and just be. If I do this the right way, then when I open my email and my files I will probably be a better colleague and a better teacher.

Sometimes ‘Doing nothing leads to the very best something’ as a silly old bear once said.

 

Some writing about teaching again soon. But it won’t be done between the hours of 3:30 and 7:30 PM!

Opening Day Activity – Evolving

I am not 100% sold on how I have divided the data, but I want to think out loud about how my opening day activity for Precalculus Honors is developing.

Last year, I took the advice of a number of online colleagues and divided my classroom into table sets of 3 for student groups. I anticipate that I will have five of these student groups in Precalculus Honors (PCH from now on in the post) so I set up five subdivisions of the data that I previously shared (in this post) and I will start off with the cleanest of the data sets. At the bottom of the post I will attach GeoGebra file links as well as the current status of my handout. I will present screen shots of the data subsets and discuss my hopes and dreams for how this activity will unfold.

Group #1 will have an image similar to this when they graph their data. I will share this exact one with the whole class. I set up five GeoGebra files with the same screen dimensions (at least I think the dimensions are exactly the same, hope so!) My hope is that this will feel largely quadratic to them. I do not think that they have previous experience with regression equations on their TI or with any online tool. I am suggesting that they use Desmos in class. I will present GeoGebra with the goals of exposing them right away to two of my favorite tools. We’ll discuss which one might feel better for different situations. In the past few years I have had a number of students adopt both programs as a natural part of their problem solving. On more than one occasion a student has written a note on a problem set along the lines of ‘Desmos agrees with my conclusion!’ A natural direction for them to go is to use point H as a vertex and develop a quadratic that fits this reasonably well. Hopefully, we can incorporate a discussion throughout the year of the messiness of real world data compared to mathematical models. I love quoting George Box here – ‘All models are wrong, but some are useful’

Group two will generate something similar to this image. Again, I am anticipating a quadratic guess. This time point E should be seen as the vertex. Playing a bit with my TI and anchoring my guess at point E yields the following promising picture. 

Now, we are getting somewhere!

Group three sees this data and has some decisions to make. This is where I am really questioning myself. I worry that there is too much here for a first day activity. I think I may tweak this data set so that it is also more deceptively quadratic in nature.

Group 4 picture above.

Group five data picture.

And, finally, the whole set together here –

As I write this and think about my goals for that first day, I am sure that I want to modify the data that group 3 gets. I don’t want this play to become frustrating on day one. Let’s save frustration for later on!

I am hoping to plant seeds for a number of interesting conversations to have over the course of the year. I want the students to think about decisions based on small sets of data versus larger sets. I want them to think about periodicity and where/when/why to expect that behavior. I want to present them with an intentionally open question on our first day together to set a tone for open questions together throughout the year. I want them to remember something fundamental about quadratics and I expect to present two forms (standard and vertex) on the board after a little nudging from the groups.

I worry, as I often do, that I am being too ambitious. We have about 30 minutes together on day one due to a whole school convocation that day. I am really debating whether to make this the activity and conversation for our first full class day together. In that case, I would pull out an old favorite problem to have as the conversation seed for day one and mix in a bit of the boring old syllabus, etc on day one. Any advice there?

Here is a link to a dropbox folder with the GeoGebra files as they currently are arranged as well as my Word handout.

As always, advice/comments/questions are welcome here or over on twitter where I am @mrdardy

Preparing for Precalculus

For the first time since the 2010 – 2011 school year, I will be teaching our Precalculus Honors course. Since I left the course we changed texts and five teachers in my department have arrived since then, including the colleague who I will be working with on this course. Now that it’s August, I sat down this morning with our text (we are using the Demana, Waits, Kennedy, etc) book and I have some thoughts/questions that I want to air out. I will understand my own thoughts better after writing and I anticipate some helpful wisdom coming my way through this site or over on twitter. We start off in Chapter Four of this text, doing right into trigonometry.

Things I know I don’t like

  • Any formula to convert angles to arc length. Just emphasize part/whole relationships!
  • Language of vertical or horizontal shrinks or stretches. I just want to talk about amplitude and period. It feels like this extra language just clutters things up.
  • Inverse trig function using the odd negative 1 power. I want to write arccos x or arcsin x. Pointing out where it is and what it looks like on their calculator is a necessary nuisance.

 

Things I think I don’t like

  • The text has an odd emphasis on the word sinusoid. I don’t know why I would want to use that word, not clear on any benefit.
  • DMS notation. Why? Not sure, other than in surveying, when they will encounter this.
  • Introduction of cosine as x / r and sine as y / r – I kind of want to talk about the fact that all triangles are similar and simply scale down to ‘unit right triangles’ with a hypotenuse of length 1. This feels like a natural lead into the unit circle.

 

Things I know I like

  • An activity my colleague shared with paper plates, strings, and discovery that an arc that is equal to the radius will be subtended by the same central angle no matter the size of the plate.
  • The opening day activity I am tweaking that involves the length of daylight hours as a function of days after January 1.
  • Conversations I am planning on having about why it is usually sufficient to solve a triangle by knowing three facts out of six (three side lengths and three angle measures) and when it would not be sufficient.

 

I have taught Precalc at each of the four schools where i have worked and I always enjoy the course despite its weird, buffet style curriculum. The kids are fun to work with, sophisticated enough to have serious math conversations. We do not have the AP calendar breathing down our necks and our new schedule that includes a 90 minute class once every seven school days really lends itself to some meaningful play time in this course. I’m excited.

As always, please share any opinions/advice/questions here in the comments or over on twitter where I remain @mrdardy