Cucumber Day in Calculus, 2014 Edition

If I’ve used an activity for three years in a row, I guess it’s a tradition! My calculus students have done this one for several years now, and it’s always been a hit.

To introduce the concept of volumes of rotated solids, I have my students slice cucumbers into disks and measure the volume of each slice. Then they add them up to approximate the total volume. It’s a nice way to help them visualize objects in 3D. You can read about what we do in a couple of earlier posts, here and here.

Here are the basics of the lesson:

1. Review the formula for the volume of a cylinder:

2. Pair up the students, and distribute the cucumbers.

I used small seedless cucumbers that came six to a pack, and were relatively similar in size:

Have a discussion on whether we can use the cylinder volume formula to calculate the volume of a cucumber. This should (hopefully) lead to a conversation on how we could make the cucumbers approximate cylinders if we cut off the ends. They soon realize that the radius is not constant for the entire length of the cucumber, which usually inspires someone to suggest slicing them into many small disks. (They are already familiar with approximating the area under a curve by adding many small rectangles or trapezoids.)

3. Have everyone agree on what data needs to be collected (radius and height of each slice), and what units of measurement we will use (inches or cm).

4. Distribute plates, paper towels, rulers, and knives. Have them get to work!

5. Post the results for everyone to see:

As you can see, they are fairly consistent, falling in a range from 78 to 105 cubic cm.

To verify the accuracy of an approximation, you can drop the slices of a cucumber into a beaker that is half-full of water, and see how much additional volume the slices add.

6. I always provide a bottle of ranch salad dressing, so my students can enjoy a snack after they have finished measuring and calculating.

I usually transition from this activity to determining the volume of a simple function rotated about the x-axis, such as this problem:

Since I began doing this activity, I have noticed a marked improvement in my students’ understanding of the basic process of using integration to calculate volumes of rotated solids. Having actual cucumber disks to measure and add together really drives home the concept of adding an infinite number of disks together using an integral. Before we finished working through the above example, one student was already wondering what would happen if the axis of rotation was vertical, and another was figuring out how to find the volume of a solid with a hole through it!

A possible tweak I want to try next year: cut a cucumber in half, lengthwise, and trace its outline on graph paper. Then try to derive a function that approximates the outline, and compare the volume calculated from the function with the volume found by adding disks.

Here are pictures from yesterday’s class:

I’ve been on Spring Break this week, and we’ve stayed home this year, which has been nice. I’ve had a chance to catch up on some reading, and I really enjoyed two books in particular – one nonfiction, and one fiction. Both will appeal to teachers and lovers of mathematics.

First, the nonfiction book: Amanda Ripley’s The Smartest Kids in The World and How They Got That Way. I’ve read many books about education, and how American schools are failing our students, but this one is the most eye-opening and refreshing take on that subject I’ve seen. Ms. Ripley shares her data and conclusions through the personal stories of three American high school students who participate in foreign exchange programs. Kim leaves Oklahoma to spend a year in Finland, which has been in the news lately because of its students’ extraordinary performance on the PISA (Program for International Student Assessment). Eric travels from Minnesota to Korea, and Tom goes from Gettysburg, PA to Poland. Among the trials and tribulations Kim, Eric, and Tom undergo are lots of hard data on their host countries educational systems.

Finland’s secret boils down to recruiting the best and brightest college students to be teachers, and making an education degree as demanding as a medical one. The Finns drastically cut the supply of teachers, which increased their value. My favorite quote is from a sixth-grade teacher in the course of a conversation with Ms. Ripley:

Vourinen was visibly uncomfortable labeling his students. “I don’t want to have too much empathy for them,” he explained, “because I have to teach. If I thought about all of this too much, I would give better marks to them for worse work. I’d think, ‘Oh you poor kid. Oh well, what can I do?’ That would make my job too easy.”

Poland is interesting, because they put into place a radical change to the way they tracked their students; they waited an extra year until they were 16 before deciding if they went to vocational school or a university. They had almost immediate improvement in their PISA scores, but it looks like they may slide backwards. The party of reform was voted out in recent elections.

Korea is simply a nightmare. Students attend public school during the day, then attend a private “tutoring” hagwon every evening to prep for the big exam that determines if they will get into one of three elite universities. Basically, every student attends two days’ worth of school every day! The government has enacted a ban on hagwons staying open past ten p.m., but it hasn’t been effective.

Over and over again throughout the book the message is, students will rise to meet high expectations. The fact that almost all of our students in America are bored and not challenged is an indictment of our educational system. Also, technology appears to have no effect on students’ academic achievement; it may even be counterproductive. And speaking of counterproductive, our culture’s obsession with sports comes in for a lot of criticism. Ms. Ripley also has some concrete things parents can do to help their children succeed in today’s competitive environment. They are surprisingly simple and don’t cost a thing.

The fiction work is Yoko Ogawa’s The Housekeeper and the Professor. I’ve enjoyed this twice now; it is one of the most moving and beautiful stories I’ve ever read. It is set in contemporary Japan, and I’m not giving anything away when I explain the premise: a young, single mother has a job with a housekeeping agency, and she is hired to care for a professor of mathematics. He has suffered head trauma, and his memory only lasts for eighty minutes at a time. Her remembers everything he did before the accident, but nothing since. Every eighty minutes he forgets what he has just experienced.

The  housekeeper, the professor, and her son struggle to develop some sort of relationship, and Ogawa makes the reader question what constitutes a family; is it possible to deeply care for someone when there are no shared memories? Along the way, Ogawa uses the professor’s love of mathematics to share her wonder at the beauty of numbers. Here are some of my favorite quotes:

He had a special feeling for what he called the “correct miscalculation,” for he believed that mistakes were often as revealing as the right answers.

“The truly correct proof is one that strikes a harmonious balance between strength and flexibility. There are plenty of proofs that are technically correct but are messy and inelegant or counterintuitive. But it’s not something you can put into words – explaining why a formula is beautiful is like trying to explain why the stars are beautiful.”

“The square root sign is a generous symbol, it gives shelter to all the numbers.”

When he is helping the housekeeper’s son with a word problem, he says this:

“You’re right. This is the trickiest one in your homework today, but you read it well. The problem consists of three sentences. The handkerchiefs and socks appear three times each, and you had the rhythm just right: so many handkerchiefs…so many socks…so many yen; handkerchiefs…socks…yen. You made a boring problem sound just like a poem.”

And finally, as the Housekeeper goes to the library to try to understand a simple formula the Professor has written down:

In my imagination, I saw the creator of the universe sitting in some distant corner of the sky, weaving a pattern of delicate lace so fine that even the faintest light would shine through it. The lace stretches out infinitely in every direction, billowing gently in the cosmic breeze. You want desperately to touch it, hold it up to the light, rub it against your cheek. And all we ask is to be able to recreate the pattern, weave it again with numbers, somehow, in our own language, to make even the tiniest fragment our own, to bring it back to earth.

Ogawa’s prose is delicate and understated, which makes the story even more powerful as it unfolds.

Every summer, my school has an “All School Read”, where each upper school student and faculty member reads the same book, and we discuss it in small groups. Last year, the history department sponsored Ruta Sepetys’ Between the Shades of Gray, the story of the Soviet genocide of the Lithuanian people. This year, it’s the math department’s turn, and we will be reading The Housekeeper and the Professor.

Math Is So Romantic

I’ve been using a new review technique with the students in my Calculus class – speed dating! I wish I could take credit for it, but one of the incredibly creative teachers in the Harpeth Hall math department, Maddie Waud, introduced me to it. The first time I tried it, I was very pleased with how seriously my students took it, and after they finished, they all agreed it was helpful.

To set up a classroom for a speed dating session, divide all the desks into pairs facing each other. I put mine in a large circle.

At each pair of desks, place a problem for the partners to solve.  There are 18 students in the class, so I wrote up 9 problems. If there an odd number of students, the teacher can fill in and give whoever the solo student is some hints to solve her problem.

Give the students a set amount of time to work together on each problem. I used a timer app on my tablet that I projected to the front of the room.

At the end of the allotted time, the students in the inside circle move to their left, and the students in the outside circle move to their left. The focused interaction during the speed dating session was amazing! Every student worked with every other student in the class.

Afterwards, we went through each problem to make sure everyone understood how to solve them. Then, time for the quiz.