Continuous Everywhere but Differentiable Nowhere

I’m putting a call out to calculus teachers and calculus aficionados out there. I want to know how you transition to teaching integration, and why you cho0se to do it that way. And if you have any activities, investigations, etc., that you can send me, I’d love to have them (and post them here for other calculus teachers).

I’m not super pleased with, but I don’t hate, what I’m going to be doing tomorrow.

Here’s the deal. I just gave my last test on differentiation today, and tomorrow I’m transitioning to teach integration. I teach a regular (non AP) calculus class, so we can take our time. At the moment, I’m grappling with two things: (1) whether to teach anti-differentiation first and the notion of “area under the curve” second, or vice versa, and (2) how to make integration intuitive.

Last year, I transitioned by giving students a graph of and told them to find the shaded area. Those were my only instructions.

Some students made triangles, some students guestimated, some students made rectangles. I don’t remember all the different approaches. But then we had a discussion about how they estimated their areas, which then led to me transitioning to Riemann sums and a general introduction to the whole new unit. The thing I emphasized: “In all your previous math classes, you only learned how to find areas and volumes of silly little figures, like squares and cubes and maybe you remember a nonagon or cone. But what about crazy, strange, weird areas? Volumes of crazy, strange, weird figures? Did you ever wonder where the formula for the volume of a sphere come from? Calculus not only can answer questions about position, velocity, and acceleration, and how to maximize and minimize quantities, but it can do all this other stuff too.“

This year I’m not going to talk about areas under curves (yet). I’m going to start with two days of practicing antidifferentiation. I’m not going to say much to transition to this new material except to say that derivatives were the first part of the course and antiderivatives will be the second. And that we’ll soon be able to do a lot with them, like we found out we could with differentiation… Then I’m going to introduce the idea of the “opposite of differentiation” and spend the entire period having students build their intuition.

First, they’re going to do a matching game in pairs (PDF). We’ll then quickly debrief, but not really go into depth about any question.

Second, they’re going to work in a different set of pairs on just playing around with finding the antiderivative, by intuition and guess and check. I want them to learn to think through a problem. So I typed up what goes through my head when I try to do an antiderivative.

And then I’m letting them loose on a set of problems which should hopefully introduce them to some basic integration rules (PDF). I think it’ll take the whole period. And we’ll spend the next day debriefing. I want them to struggle through integration now. I want them to see why instead of memorize the power rule. I anticipate it to be kind of hellish for them; they — like most students — want formulaic ways to do calculus.

But just as I struggled to hone my students intuition (see my previous blog post) for differention, I wanted to make something similar for integration.

We’ll see what happens tomorrow.

This post refers to the pendulum lab I recently posted about. I had my students collect data, and this is what they got:

The data was pretty consistent among the various groups. Remember we did this lab in the context of parabolas. However, when the groups plotted their results, they were getting:

What?! The data doesn’t look quadratic. If anything, it looked to us like a square root or a quadratic with a negative coefficient. We used our calculators to do a quadratic regression, and got (for one set of data):

I tricked my class into believing this was a good model. I mean, look at it! The parabola fits the data so well! 

But then we looked at the coefficient and saw it was negative and nearly zero. And then when we expanded our domain, we got:

It was at that point that students saw how our model sucked. Because they said that if we increase the length of the pendulum, the period should increase too.

So we went back to the drawing board. I suggested  that we plot period versus length, instead of length versus period. (Next year I’m going to have us discuss this idea more — the swapping of x and y coordinates, and how something that looks like a square root might look like a parabola if we do that. Because of timing issues in this class, I just told them that was what we were going to try. Sigh.)

And we did, and found the quadratic that modeled it, and saw: 

And then we extended it to see that if we increased our period, if the length would increase too… 

It does! It makes conceptual sense too! (We also talked about whether it should hit the origin and why our model does or does not hit the origin.)

With our newfound analysis, I had students answer the following question based on their “good” quadratic model (in our case above: ):

(a) If your pendulum has a period of 1.5 seconds, estimate the length of the pendulum.
(b) If your pendulum  has a period of 20 seconds, estimate the length of the pendulum.
(c) If your pendulum has a length of 10 inches, estimate the period.
(d) If your pendulum has a length of 1,200 inches, estimate the period.

What is nice is that (a) and (b) just involve students plugging in and into their model. And all the groups got very similar answers for the first length, and really different answers for the second length. So we got to have a short (I wish it could have been longer) discussion of why that is so. (We talked about interpolation versus extrapolation.)

And then (c) and (d) involved students solving a messy, real world quadratic because they’re setting and . The same thing that happened in part (a) and part (b) happened in part (c) and part (d); all the groups got very similar answers for the first period, and really different answers for the second period.

What we didn’t get to talk about, unfortunately, is the theoretical answers, based on physics. The formula for the period of a pendulum is where , , and are in standard metric units. So I was hoping we’d get a chance to do some unit conversions to see how our experimental data relates to to theoretical data.

I did get to show my students how their values compared with the theoretical data:

I think they were impressed, though I didn’t get the ooohs and aaahs I was hoping for. I’ve plotted the theoretical (purple) with the actual data (yellow) so you can see how good the experiment was. I am not plotting it on a period versus length graph, though if I were to show my students, I would do that because that’s the way we analyzed the data (we got a parabola).

And with that, we finished our lab.

The Technology Review magazine has an arresting photo essay on Moore’s Law — as told through a bunch of stunning pictures of computer chips. Click on the link above to see all the other circuits. For those who don’t know, Moore’s Law says that about every two years (some say 18 months), the number of transistors that can fit on a circuit doubles (for Wikipedia article, click here).

The only thing I wish about the photo essay is that there was some sense of scale for each picture. Regardless, the captions tell the year each circuit was created, and the number of transistors on each circuit. The data are:

So of course, even though this data isn’t perfect nor complete, I thought I’d see how it’d look graphed.

Ohhh, it looks like it could be exponential… Let’s plot it on a log-scale. If it’s exponential, we should get a straight line:

Ohhh, this looks pretty linear! I wasn’t sure that it was going to work out. 

The exponential line of best fit is: . When I plot the data (pink) and the exponential line of best fit (blue) on the log-scale graph, you’ll see that Moore’s Law looks like it has some serious bite to it.

Doing a little algebra with the exponential model we came up with, it appears that the number of transitors doubles about every 1.75 years.

And if you cared, Wikipedia gives their own following graphical illustration of Moore’s Law:

I’m going to be teaching exponential functions in a bit. I hope we’ll have time to do regressions. If so, I’ll probably make a 2-day investigation out of Moore’s Law.

Other posts I’ve made about logarithmic and exponential functions:

Logarithmic Graph in the News
Earthquakes, Richter Scale, and Logarithms
The Supreme Court, Linear  and Exponential Growth, and Racial Segregation
The Origin of Life on Earth and Logarithms 
Paper Folding and Exponential Functions 

This week I’ve had one and a half Algebra II classes to “kill” because I’m ahead of the other teacher and we need to sync up again. Since we’re working on parabolas, I thought we could do something fun.

A while ago, I watched this video:

And I decided, perfect! I’m going do the pendulum thing in class. I got some string and washers, and put masking tape on the string every 6 inches. And I had student calculate the period of the pendulum when the length of the string was 6 inches, 12 inches, 18 inches, … , 60 inches.

To minimize the error generated by a student not exactly stopping the stopwatch when pendulum swung back and forth, I had students have the pendulum swing three times. That way any reaction time error of the person operating the stopwatch gets reduced by a third! And I had students do 3 trials for each length of string, to further minimize error.

It took all 50 minutes for students to collect all their data, plot it on a graph, and enter the data in their calculators.

Tomorrow we get to have fun. To warm up, we’re going to talk about sources of error. Then each group will get to share their graphs and talk about their findings. Then we’re going to perform a quadratic regression on our calculators, talk about if we have a good or bad model and ways to decide, and then use our model to make some predictions. (If we know the period, can we find the length of the pendulum, and vice versa.) Then, I’m going to conclude by showing students the theoretical formula for the period of a pendulum () and we’re going to see if their collected data matches up with the theoretical predictions.

The best part about this is that all the groups data seems in line with each other, and in fact, they are all really close to the theoretical predictions. I can’t wait to see if there are oohs and aahs about how accurate these data points they got are to the theoretical predictions.

I’m excited for tomorrow!

UPDATE: my data collection sheet (PDF), my lab debrief sheet (PDF).

Dan Meyer has recently been asking teachers to consider (a) the problems with textbook application problems and (b) finding or creating compelling content for the classroom.

I’ve been thinking about this since he’s started posting, and I’m trying to think of where I can find — but not create — the interesting digital media that would satisfy his criteria.

In the meanwhile, I decided to look on youtube for some videos that would naturally lead to a particular discussion/topic for some high school math class. Some are better than others. Without further ado…

In Algebra II,, we’ve recently been delving into quadratics. I recently blogged about how I taught completing the square and the quadratic formula, and put up a bunch of resources. Since then, we’ve moved on to graphing quadratics, followed by inequalities.

The complete topic list for inequalities is:

I’ve been trying something new, which is creating packets for students to work on. In essence, I’m creating my own textbook for these sorts of questions. I thought I’d share them with you in case they prove useful [1]. I’m pretty proud of them — and they way they fit together and build up understanding, not just providing a method to solving problems.

1. PACKET 1: Linear and Quadratic Inequalities on the Number Line (PDF version)
2. Additional Homework on Quadratic Inequalities (PDF version)
3. PACKET II: Linear and Quadratic Inequalities on the Coordinate Plane (PDF version)

4. PACKET III: Systems of Inequalities (Linear and Linear-Quadratic) (PDF version)
5. Additional Homework on Systems of Inequalities (PDF version)
6. Pop Quiz on Inequalities and Quadratics (PDF version)

Hopefully they’ll be useful to someone else out there!

[1] The formatting might be a bit off for you… It looks slightly off (meaning the pages don’t end where I intended them to end) on my mac but fine on my PC. I think you need to make sure that on a Mac you select all and convert the font to “Gill Sans” (on a PC, I think it’s called “Gill Sans MT”, which is creating the problem).

UPDATE: PDFs posted, without typographic weirdnesses.

So my school offered the AMC 10 and the AMC 12 (math competitions) last week. And although we have a number of pretty strong math students, none of them broke the 100 point mark. No student in recent history — apparently — at my school has done that. The fact is that to do well on these competitions, you have to be familiar with the types of questions and methods to solving them. There are techniques to doing well, tricks that any student who has seen enough of these can put away in their mathematical arsenal. These contests require a different way of thinking, a different way of approaching problems. 

I am the faculty adviser for my school’s math club, and I just go with the flow. I listen to what the students want and we do it. Sometimes students bring an interesting problem or an extension of a problem. Sometimes I bring a problem. Sometimes we watch a video. Sometimes we work on contest problems.

Last year and this year, the students haven’t wanted to dedicate time outside of math club to doing math. We have only 25 minutes a week to meet. Well, frankly my dear, you can’t get through much in that time — especially if students don’t want to concertedly work on problems outside of math club, and then use math club to present solutions or failed methods of attack. When it comes down to it, to do well on these contests, you need to practice. 

I’m okay with students not wanting to spend time outside of mathclub working on math problems. They are all busy and well-rounded and are juggling a ton of different things.

Still… I am waiting for the day when someone in mathclub says: “Everyone, I am going to solve every one of these 25 competitions problems by next week” and goes at it. Whether or not they succeed, it’s irrelevant to me. That’s the kid I want to take under my wing. 

PS. I finally got around to taking the AMC12 under testing conditions (75 minutes, no calculator). I scored a 108. Which is around the same score I got last year and when I was in high school.

I’m teaching double integrals, and there was this really great problem:

over the region .

The problem is supposed to show that changing the order of integration can make the problem really easy or really hard.

is easy to solve.

is hard to solve.

However, I don’t think the second integral should be impossible to solve. It’s been so long since I’ve really dug into integration, and we tried in class and couldn’t find a quick way to solve the second integral. Any ideas?

For those who know calculus but not Multivariable Calculus, the hard part of the second integral is simply being asked to solve: , where is simply a constant.

I’m a big fan of anonymous student feedback, because I know I always have things to learn. I struggled this year when decided what format I would use to solicit this feedback. It was a decision between:

1. a set of short answer questions and numerical responses (e.g. “On average, how long do you spend on homework each night?” or “If you could change any 3 things about this course, what would they be, and why?”)
2. an open-ended narrative comment

This year (as I did last year), I decided to have students write the open-ended narrative. The reasons I made this decision were the same as last year. First of all, if I pose the questions, I’m pretty much telling students what to focus their criticism on. I’m telling them what I think is important. Instead, I want them to tell me what’s important. Second, I write comments on my students at least twice a year. Students should have the opportunity to write comments on me too. At the beginning of the year, I told my students that we are making a contract with each other. I want them to know that I care if I’m keeping up my end of the bargain.

My charge to them:

Hi all,

Your homework is to write a comment about the class and about my teaching. I am always looking to improve and to keep on with what I’m doing well, and so I really appreciate you taking the time to write these comments. You don’t have to make it anonymous. It’s totally up to you, but you should feel more than free to do so. Please type it into the form attached, and print it out before class to hand it!

Always my best,
Mr. Shah

Write an comment about your experience in this course. You can write about anything — things that are going well, things you’d like to see changed, aspects about my teaching that work and don’t work, your opinion of smartboard, anything.

And that’s it. I’m not leading them to talk about anything specifically. And I think that the feedback I got has been really valuable. More than observations from administrators, student feedback is the most valuable.

Honestly, there’s a terrifying aspect to these evaluations. I’ll put it this way… I asked for these recommendations in January. I didn’t look at them until early February. I couldn’t look at them until early February. I carried them with me to and from school every day, waiting until a moment when I felt like I could face them. What if they were all terrible? Or they focused on areas of my teaching that I’m sensitive about? What if I am a terrible teacher?

But when I finally countenanced them, it was after I realized “even if I’m bad at what I do, I need to know that in order to do better.” I mean, it’s what I tell my students — you have to know where you are weak in order to know how to become strong. You can’t just bury your head in the sand and hope things will change. You have to change them.

Here’s the rundown of my Algebra II student evaluations. Which made me realize the anxiety was uncalled for.

I. Am. A. Good. Teacher. (Not that I don’t have places to improve.)

MY ALGEBRA II EVALUATIONS

strengths

• “[M]y experience has been… surprisingly positive for a math class”
• “[Y]our fast-paced teaching style… suits me”
• “[A]bility to interpose math applications to real life” *
• “Smartboard [will be] a valuable reference for studying for midterms”
• “I appreciate your ability to interpose math applications to real life, like the fractals activity” *
• “I can always count on… your organization. You always have your [Smartboard] ready and you are always punctual”
• “You make math fun”
• “[Y]ou have a lot of energy, and that makes the class really interesting. Unlike some subjects, every day class is different, this makes learning fun and enjoyable”
• “[I]t’s helpful that you are so willing to meet for extra help”
• “Rather than just writing ont he board what we can earily just take notes from in the textbook I feel you are really teaching us algebra”
• “I can also tell that you really care that your students learn what you are teaching”
• “Our class gets along very well which makes this class interesting as well”
• “[Mr. Shah] is probably the first teacher that compels me to work in class rather than fall asleep. This is for a variety of reasons; it could be his dress, which is always vibrant, his enthusiasm, or his witty power points”
• “Generally I have found the class to be an extremely gratifying experience. It is well-paced, well-taught, and well-structured”
• “I appreciate how you tell us about assessments in advance, and the use of smartboard shows me that you do care about the students and teaching”
• “I feel accomplished when I do well on one of your tests”
• “[Y]ou do a really good job of keeping the class focused and on top of what’s going on. You always make sure that everyone knows what is going on”
• “[M]eeting with you is SO helpful… I know for sure that meeting with you has boosted my test scores up”
• “Math class is one of the very few classes I don’t dread going to”
• “I like that with the [Smartboard] presentations the materail is broken down into manageable pieces at a time and then throughout the period we work up to the whole concept”

areas for improvement

• “I can imagine… if I were ever to miss a day of class… it could be a problem”
• “[Change] the amount of time we spend going over the homework”
• “[C]hange the amount of practice problems we do. Sometimes, I feel that we do the same type of question a lot of different times”
• “[I]t might be helpful if we could slow down a bit”
• “As a teacher I’d give [Mr. Shah] a B+/A- (more A-)… because sometimes our questions are unanswered”
• “At some points I feel like the class is moving too fast for me”
• “[T]he class seems boring at times… I know math is not boring and can be made fun”
• “I would like to suggest that it’s ok to go off on tangents and talk about topics that are slightly unrelated”
• “I’d recommend that [it] would benefit me… to slow down a little”

*I actually do not do this much, but a few students did mention it! So I guess I should look for applications, and try to integrate them into what we’re doing more often.