# Blog Archives

## Integration Drawing Projects ’12

I wrote about this project back on Sam’s blog this summer when Sam gave me reign of his kingdom for a month or so, but I wanted to share the student work that I got this year from it, because it was much better than last year, and some of the work is actually really beautiful/cool/interesting (Math Art, MArTH anyone?).

The basic premise of the project is to RECREATE A PICTURE USING INTEGRALS by doing the following:

1. Upload a picture into GeoGebra.
2. Place points around all the outlines making sure to hit critical points
3. Fit functions to the outlines.
4. Use integrals to shade in the areas between the outlines.

I initially waffled about whether this was a worthwhile problem or just an exercise in integrals, but having taught AP Calculus this year, I realize how these problems of just finding the area of a weird shape are interesting and important for deep understanding of the connection between a Riemann sum and how the integral actually calculates area. So basically, if you think that this is a worthwhile problem…

Find the Area of R and S given that f(x) is blah blah and g(x) is blah blah blah squared.

…then this project is just a glorified, more interesting, more complex version of that problem. If you don’t think that problem is worthwhile, well, then you probably wont like this either. Regardless, it was a great thing to do to hammer in ideas about finding the area between curves, and a great learning mode while AP’s were occurring because attendance did not really matter all that much. It took most students 3 and a half 45-minute class periods (so about 2.5 hours), though I think that more efficient students not freaking out about standardized tests, and consistently present in the classroom, might be able to do it a little quicker.

## ALL OF THE STUDENT WORK:

(the good, the bad, the ugly!)

## Drawing in Math Class

One of my favorite ways to start class is by putting out whiteboards with a problem paper-clipped at the top, and names of random groups.  I love it most  because every single person is engaged in mathematics within 30 seconds of class starting. In fact, students always ask me a minute or two before class starts “can we begin?” They can’t seem to resist the markers and the problem in front of them. Also, I found when I wanted to use whiteboards in the middle of class and put students in random groups that it just ate up a few minutes in each class, so this just feels more efficient (I’m kind of neurotic in terms of efficient use of class time).

Continuing my experiments with different modes of math whiteboarding, a great whiteboard warm up I tried was having them illustrate related rates type situations for objects that are changing in different ways. For example:

A pumpkin grows in a garden…
1. With a constant increase in the radius of the pumpkin
2. With a constant increase in the volume of the pumpkin

Then I had them describe what is happening to the rate of change of the important variables (so if dV/dt is constant, what is happening to dr/dt?). We then had a really good full class discussion where students explained their situation. I think this helped clarify for a lot of students the difference between “V” increasing and “dV/dt” increasing, or how just because “dV/dt” is decreasing it doesn’t mean the volume is decreasing.

This was part of a larger goal of mine to focus on big ideas and deep understanding this year – I’ve always asked students interpretation questions on tests (my final this past term had a crap-ton of writing) but I never felt like I actually directly taught them these sorts of things. For Related Rates, we solve all these problems and come up with all these numbers, but never actually talk about why they are interesting problems – the fact that as one aspect of a situation changes, another may change at a totally different rate, and that there is a relationship between all these rates that explain how things change the way they do. And honestly, I think this little activity made a huge difference – on the interpretation question on the Related Rates quiz, tons of students drew pictures to aid their explanations. 15 minutes well spent!