Hello! Though Sam may refer to me as Kiki, don’t be fooled. My name is Bowman and I’m an American dude teaching MATH at a 9-12 co-ed boarding school in Amman, Jordan. I teach mostly Jordanian kids, though we teach an American-style curriculum in English, with sort of international school type outlook. For the past two years I have taught Physics, then last year I picked up Calculus, and next year I’m dropping the Physics to pick up AP Calculus AB. All of my friends can’t really understand why I’m so pumped about this because they think I’m the only person in the world that gets giddy about Calculus. False.
I love the math blogging community and am excited to be delving into it. Though I already have an “I-don’t-live-in-America” type blog about my time in Jordan, I have relied heavily on edublogs to develop as a teacher and I’m looking forward to repaying my debt. And probably like you, Sam’s blog is my fave, so I’m honored that he would give me some airtime. If you ever get a chance to meet him in person, consider yourself lucky. Since the thing that first drew me here was the wealth of practical lesson planning ideas in his Virtual Filing Cabinet (which I check pretty consistently before I plan a unit), I thought I’d begin repaying my debt by sharing some of the creative ideas I have used to present specific material this past year. Acknowledging my youth and paucity of teaching experience, I’m going to title these posts “Make It Better” to indicate that while I think these ideas have a lot of potential, I’m looking for ways to improve them. Enough introduction… on to the math!
Drawing with GeoGebra
For anyone who has not discovered the magic of GeoGebra yet, download it right now and then spend some time this summer playing around with it. It has a nice mix of geometric and algebraic capabilities, with fancy looking sliders and animations to help students visualize or experiment with mathematical concepts. These can be used in front of the class or on individual students’ computers. I ended up using it so much throughout the year in student directed learning that when we did end-of-the-year individualized projects, a majority of my students pulled out GeoGebra on their own to graph something or fit a model to some data. Sweet.
One of my favorite GeoGebra exploits this year was a “drawing” project, where students converted an actual picture into a mathematical picture by fitting functions around the outlines and then using integrals to shade in the area between. For example, they could a picture of a guitar and turn it into a sexy mathematical image, like this:
>>> 
Basic steps (more detailed procedure below):
- Upload a picture into GeoGebra and scale the axes to the right scale.
- Place points around all the outlines making sure to hit critical points
- Fit functions to the outlines.
- Use integrals to shade in the areas between the outlines. The basic syntax looks like this…
which means the integral between the [top function f, and the bottom function g, from x=1, to x=4].
- The hardest part about integrals is setting them up and that’s all students practiced in this project. They did absolutely no calculation. (In the age of Wolfram Alpha and TI’s could that be sooomewhat a thing of the past?)
- The visual nature of the project gave immediate feedback to wrong inputs. If a student chose the wrong endpoints or the wrong functions, the wrong integral that they typed in would show up. They could see what was wrong about it to hopefully figure out how to correct their input, and correct their misconception. The tinkering aspect was maybe my favorite part because they often don’t understand that just by trying something to solve a math problem, it can point you in the right direction to solve the problem even if it’s “wrong.”
- It unearthed deep misconceptions about integration. Some students were conceptualizing integration diagonally, some would choose endpoints at completely wrong spots and some couldn’t conceptualize what areas they were trying to “color in” in the first place. I had lots of great conversations to address misconceptions that were at the same time a bit scary because we had already been integrating for a few weeks by that point.
- Everybody’s problem was different. Each student was forced to visualize what he or she needed to do and had to attack a rather large problem by breaking it up into much smaller pieces.
- The whole thing was kinda fun. Sometimes I pretend I’m above this, but each student chose a picture they were interested in and then we hung them up in the classroom at the end. And you can color the integrals whatever color you want. Pretty!
- Some students totally copped out and chose really easy picture. I wasn’t very clear with my expectations (well, I actually didn’t know what to expect) and as a result a few students chose dumb things. One student did a watermelon… uncut…. like, a whole green watermelon. Or other students didn’t know what would be a “good” picture and chose something that ended up being way too hard, or uninteresting.
- The function fitting part was a bit ridiculous. You can go really high with the degrees of the polynomials for the function fitting so people would just put points along a really curvy surface and then pick a 73rd degree polynomial that nicely fit the whole thing. I don’t know if this is actually bad, but it felt weird to me.
- The problem was slightly meaningless. I had them add up the integrals at the end to find the total area, but this was a bit meaningless unless they had chose a flat object.
I can picture something like this being done for lower levels too. The concepts that come to mind immediately to me are piecewise functions and transformations. Instead of having them fit functions to points, you could give them a basic set of functions and force them to manipulate them with various transformations to fit outlines (and then ignore the coloring in part). There are easy ways to limit the domains of functions in GeoGebra. Below are GeoGebra instructions for the various steps and waaaay below are some more examples of student work.
But first, the whole point of this post…
Make it Better.
What do you think? What would you do differently? Do you think this holds educational value even though the problem is contrived? How could I make this more meaningful (i.e. make the result, the integral, not just the picture, actually hold value itself)? Should I give them a set of predetermined images for them to choose from to avoid “bad” choices? How else could I avoid “bad” choices? With what other material could something like this work? Would you use this in your classroom?
from @bowmanimal
The procedural instructions for the various tasks:
Examples of student work (I had to include the watermelon because I mentioned it):
Continuous Everywhere but Differentiable Nowhere 
What a fun idea! I love that is is self-correcting as well. I would love to improvise this for 6th and 7th. Maybe I could have them use geometric shapes and find the equations of the lines around their shape? I need to learn Geogebra! Thanks for the inspiration!
Actually I really like the choice of a watermelon. The melon seems like such a pleasing, regular shape: is it close to some natural mathematical curve? How does it vary with the size of the melon, within a species? It’s rotationally symmetric, so if you know it’s area, can you determine its volume? How? (Or why not, if it’s impossible?)
Of course I would have been less impressed had they chosen a round melon.
wow, thanks for this. i was thinking the watermelon was bad because you fit two functions and compute one integral (so dumb in the context of this project), but i guess what i take from you comment is that there are ways to make the function fitting part more meaningful (rather than just making outlines). it would be fin to find the volume of the melon by fitting a function and rotating and then find the actual volume of the melon and compare.
I love the student work. Your students have a refined sense of color matching.
As a critique of this assignment, this is a TASK and not a PROBLEM. Sure you can find the function that makes a shape like the picture and use that to find the area, but why would you do so? You pointed this out yourself. I’ve always been suspicious of using functions to draw pictures.
But maybe your students aren’t so suspicious and they had fun with this one. You be the judge on that. The ideas of modeling with functions and finding area are definitely important.
To “make it better,” I would see if you can use the technical skills your kids developed to accomplish the picture task with a new problem where there is a clear need and meaning to finding area. Good luck!
great point about it possibly being useful just to teach the technical skills of modeling. we did do modeling later in the course, but the trouble i faced was that some students didn’t really think about the actual function when fitting it. for example, one group was modeling a rocket’s flight and instead of splitting up the function into a few different ones, they fit like a 63rd degree polynomial and said “wow, look how close it fits!” without really thinking of “well why would that be a 63rd degree polynomial?” i guess i need to be more conscious with designing this if the goal is to teach those skills… thanks!
Excellent activity Bowman. Been using GeoGebra for a long time, but I haven’t thought of this one. By the way, if you have not visited my site yet, I have written more than 50 GeoGebra tutorials there:
http://mathandmultimedia.com/geogebra/
Maybe you can refer some of your students there to learn GeoGebra.
your tutorials are great! my goal is to figure out the html form where students can input things in a web environment before I get back to school. Thanks Guillermo
You just have to learn some basic html input (input, text boxes, drop down list, etc) and a little bit of javascript. I’m reviewing a little bit myself. By the way, GeoGebra 4.0 has command buttons and text boxes, so I think that will save us some trouble coding.