The year is coming to a close and I’ve found something to entertain my seniors. They’re taking regular calculus. More than likely, most of them will never take a math class again. If they are going to take math in college, chances are they’re going to be taking calculus over again (I don’t teach the AP calculus classes at my school).
My school treats seniors with the deference that seniors think they deserve. They don’t have to take final exams, they don’t go to classes after May 22nd (don’t ask), and they miss a lot of May to AP exams. All in all, because of these restrictions, May is pretty hard to plan, if you teach a senior class.
I gave my last quiz recently, and I’m having students use their class time to work on a calculus project.
I only have 7 students in this class, so I decided to do something pretty radical. I pretty much gave them free reign on their project. I told them they could do anything they wanted — just as long as they’re passionate about it. They have to do something they’re going to enjoy doing. They could also choose the point value of the project (a large quiz grade or test grade).
At this point, the only way I’m going to get them to do anything is by tapping into things they like.
So I had them brainstorm, we met individually so I could guide them, and they’re off to the races, with some great projects:
- One student is doing a study of Newton’s method (we didn’t cover it in class) to find the zeros of a polynomial. She’s going to compare whether Newton’s method to finding zeros is “better” than a more simplistic method of finding zeros. That method, in case you were wondering, has you find an interval where you know there is a zero (e.g. for example, say you know there’s a zero on [-1,1] because the function is negative when x=-1 and positive when x=1). Then you divide the interval in half (into [-1,0] and [0,1]) and you find which of those two intervals has the zero. Then you divide that interval in half, and find which of those two intervals has the zero. On and on and on…
- Another student is doing a study of rainbows, which involves calculus. (Awesome resources here and here.)
- Another student really liked learning the intuitive version of the chain rule that I taught (post one and two), and wanted to make a lesson for my students next year on that! So she’s making a video tutorial and worksheet to accompany it.
- In the same spirit of teaching, one of my students wanted to do something similar by making a video tutorial on the formal definition of the derivative.
- One student is taking AP Physics B, but throughout the course, has noted connections between what he’s learned in his non-calculus-based physics class and what we’re doing in calculus class. One connection he made was between Pressure, Volume, and Work. He (rightfully) noted that
. So he’s going to be making a presentation on this relationship by doing a bit of research and bringing application to the class.
- Another one wanted to learn something “new” so I suggested he do some research on a hanging string. More notably, if you hold up a string (like a necklace), it will hang down due to gravity. Surprisingly (or not?) the shape is not a parabola. It turns out that it’s this funky shape called a catenary. He’s investigating why that’s the case, and how to derive the formula.
- Last but not least, one of my students had difficulty with the sections on surface area and volume, because she couldn’t visualize the regions/spaces being formed. So she’s making two mechanical thingamajiggers out of wire. You bend the wire to be whatever function you’re going to be rotating, and then there’s a handle that rotates the wire. I am so excited about this one — I hope it works out so I can use the model next year in class!
Continuous Everywhere but Differentiable Nowhere 
How do you make students passionate like this? How had you ensured that all students were passionate about something?
Was it luck of the draw, or of the class size?
I’m a senior in AP Calculus and my teacher is basically doing the same thing. We all love it (and most of us are using as an excuse to bring in some food). It’s also a chance for us to explore the real applications of calculus and relate it to other areas of our lives.
I have a special place in my heart for these seniors — it’s a class of 7 seniors, in a school with about 80 seniors. So everyone knows each other really well, after being “stuck” with each other for years (small class). And most of them are friends outside of class (luck of the draw). So we have a merry little bunch. They also are pretty driven to do well, even though most of them don’t consider themselves “math people.”
That helps. A lot.
Also, I try to go into class being enthusiastic about almost everything we teach. By the end of the year, my kids said to me (when I was teaching them partial fractions and integration), “Mr. Shah, you think EVERYTHING is the most amazing thing ever.” But, in fact, if you take a minute and step back and look at what you’re doing, you can’t end up seeing how cool some of the stuff is that you’re doing.
I tell them, for example, when we’re doing the length of a curve in calculus, that previously, there were only TWO types of curves they knew the length of: a straight line segment and a circle. And what’s amazing is that in class today, they’re going to learn how to find the line of ANY curve, no matter how funky looking. They dig finally seeing it all come together, I think.
I think that one thing I’ve learned in my student teaching and this year is that enthusiasm is infectious. If you have it (or feign it, as I sometimes have to do with the more mundane topics), you’re students will pick it up.
But other thing, for this particular assignment, is that I literally let them do whatever they wanted. The only requirement was that it had to be something they were interested in. This is how it specifically went down…
For homework one night, I asked them to write 1-2 paragraphs describing ANY project at all they would want to do, and I gave some basic examples, but I purposefully spoke vaguely so they wouldn’t be constrained in their thinking. Then I met with them individually to go over what they chose…
One of the students who is teaching the chain rule, for example, wants to become a teacher, and really loved learning the chain rule. It makes sense that she chose her project on teaching the chain rule. The one doing the mechanical model of surface area/volume is super creative and artistic, so we took her embryonic idea [“I want to do something to help other students visualize this”] and made it into something concrete. The one who is doing the rainbows didn’t know what she wanted to do, but she wanted “relevance.” She is a budding poet, so I thought why foist some random applied physics or economics thing on her? So I found this project and presented it to her, thinking that it would appeal to her sensibilities. It did. (I think.) The physics student came up with the idea totally on his own. The Newton’s method project was chosen by my other student from a book of calculus projects that I have, because she loves the “puzzle” of math and the involved proofs and drawing connections. She pegged this project out of the book because it looked the most interesting.
So basically, because I have so much faith in them at this point, have let them go. We’ll see when I get their final projects if they’ve flown or not.
But yeah, I couldn’t do this type of thing with my Algebra II kids, because I don’t have that type of time to guide them, the class size is larger, and also because honestly, many still haven’t come around to becoming converts to the material like my calculus class has.
Hi Sam
The ‘other method’ your student is comparing Newton’s method to is called the Bisection method. You probably already know this but I thought I would mention it since you didn’t refer to it by name :)
If she has any leanings towards computer programming then it might be fun to gently push her towards ‘discovering’ some fractals that arise from Newton’s method.
http://facstaff.unca.edu/mcmcclur/mathematicaGraphics/Newton/index.html
Cheers,
Mike
Thanks Mike. I will definitely send her some fractal info! She’ll love it.
And I don’t think I ever learned that name, so I’m glad you told me. Huzzah!
dear sir/madam
can you please esnd more information & examples on how to understand bisection method.
i want to do a project on basic calculus-differential fr my scul fest. m jst n 9th grade bt i seriously want to make my project special . but i dont no howto. any ideas?????
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I might have to do some similar things for my math challenge problems.
Any suggestions on additional project topics? I have 19 students with major senioritis. Trying to find something engaging for 2 weeks before we prepare for finals.
Sorry! I did this in my first year of teaching, and with only a handful of students. But I’m not teaching calculus right now so I don’t really remember if there were any great resources. I would bet googling would be useful, but you’d have to sift through garbage.
I used to do this at the end of the year: https://samjshah.com/2012/06/07/wealth-inequality-a-calculus-investigation/
I did this during our optimization unit that was fun: https://samjshah.com/2012/03/15/optimization-an-introductory-activity-project/
Good luck!