Modeling Solids of Known Cross-Sections

One of the hardest type of problem for calculus students to understand is calculating the volume of solids of known cross-sections. It’s hard, because they have difficulty visualizing it.

Last year, I went to the regional NCTM conference here in Nashville, TN, and one of the sessions I attended addressed this exact issue. Nina Chung Otterson was the presenter, and she teaches at The Hotchkiss School in Connecticut. She has her students cut cross-sections of different shapes and apply them to a base area enclosed by two parabolas, y = x^2 – 3 and y = 3 – x^2.

Here’s what the base area looks like, courtesy of’s online function grapher:

Base area

In her session, Nina Otterson provided templates that fit the given base area for different shapes: semicircles, squares, and equilateral triangles. I had my students cut the square templates diagonally for isosceles right triangles, and horizontally for rectangles. Students use the templates to cut out a cross-section that fits down the middle of the base area, and six others on each side.

Here are my students in action, cutting out the cross-sections:

Cutting Pieces 3 Cutting Pieces 1 Cutting Pieces 2

Here they are, taping the cross-sections onto the base area:

Building Solids 1 Building Solids 2

And here are the finished models:

Squares Equilateral Triangles Isosceles Right Triangles Rectangles Semicircles

Once they understood that the thickness of the paper was dx, it was very easy to set up the integrals to calculate the volumes of their models. I’ve never had students grasp the idea behind this type of volume as quickly and as easily as this group did. Building a model using actual cross-sections made all the difference!

When I do this activity next year, I think I’ll glue the base area to foamboard, and have students insert the cross-sections into slits cut into the foamboard. That way, they will stand up straighter and stay evenly spaced.

You can download the templates provided by Nina Otterson here.

Update: I used spray adhesive to glue the base area to some foamboard and cut slits in it with an Xacto knife. Then I carefully slid each cross-section into its appropriate slit. It worked great! Now each cross-section stands nicely spaced and vertical.

Final Version

Calculus for Geometry Students

In Geometry, we’re beginning a chapter on areas of polygons, and the first lesson is area of a rectangle. Pretty exciting, huh? My students are mostly ninth-graders, with a few tenth-graders, and I thought they might enjoy seeing how the area of a rectangle is used to estimate the area under a curve, i.e. a Riemann Sum.

I used a Geogebra activity created by Alex Kasantsidis to demonstrate a simple Riemann sum. We discussed how the sum of the rectangles can either overestimate or underestimate the area under the parabola, and how we can get a better approximation of the area by increasing the number of rectangles used.


Then, I had my students work through an activity (you can download it here) to estimate the area under the curve y = 12 – x^2 for x = -1 to x = 3 using eight rectangles. After averaging the left-hand and right-hand sums, they came up with 38.5. The actual area is 38.66…, so with only eight rectangles they achieved very good results!

What my students enjoyed even more, though, was the satisfaction of learning calculus-level mathematics. Hopefully, this activity allayed some of the apprehension they might have when they hear the word “calculus”.


NCTM Comes to Music City

The National Council of Teachers of Mathematics held a regional conference in Nashville, TN last week. Since I live in Music City, this was too good an opportunity to pass up! I’ve attended several NCTM conferences, both national and regional, and this was by far the best and most technologically savvy one.

To begin, I installed the conference app on my phone, which made planning and organizing my experience a breeze. It synced the sessions I wanted to see with my calendar, so I received reminders and updates for each one. The app also promoted interaction among attendees through its Activity Feed. We could also keep up with what was happening via Twitter using the hashtag #NCTMregionals.

The first session I went to was “Hands On Trigonometry”, by Elizabeth Petty. She led us through a lesson on the unit using paper plates and twizzlers to demonstrate what a radian is. We then folded the paper plates in half to create an x-axis, then thirds to mark the 30 and 60-degree points on the circle. It was an excellent example of using a hands-on activity to reinforce a difficult math concept.

The next session, “Keeping It Real: Authentic Real-World Math Lessons”, was presented by Ginny Stuckey of Mathalicious. Their mission is to develop lessons that use real-world data and situations to spur critical thinking in students. The lesson Ms. Stuckey demonstrated involved how municipal fines can quickly become impossible to pay off for low-income offenders.

Next up was one of the best sessions of the conference, “Using Manipulatives and Investigations to Teach Geometry”, by Christine Mikles. Ms. Mikles uses the CPM Geometry text, which is full of hands-on activities. Here’s an example of using two mirrors and a protractor to learn about central angles in regular polygons:

Polygon with mirrors

The last session I attended on Thursday was “Slices of Calculus”, by Nina Otterson. As a veteran calculus teacher, I was very excited to learn a new way to teach how to calculate the volumes of solids of known cross-sections. Ms. Otterson’s approach makes a lot of sense: she has her students learn what ratio of a square’s area is an equilateral triangle, an isosceles right triangle, and a semicircle. Then, they find the volume of the solid using a square, which is easy, and apply the appropriate ratio. Her students build models of each type of solid. Here’s one I built using equilateral triangles whose base is a pair of intersecting parabolas:

Solid with Triangle cross-sections

Friday’s first session was “The Math Department I’ve Always Wanted: Twitter As My PLC”, by Michael Felton. It was an excellent presentation on how math teachers can use Twitter to ask questions of other teachers, get great ideas for lessons, and get feedback on their own lessons. Michael is part of the “Math Twitter Blogosphere” (#MTBoS on Twitter), where some of the most innovative teaching is being developed today. If a teacher needs some ideas on how to teach practically any math topic in a creative way, he or she can find it at MTBoS.

Next, I went to a session sponsored by CPM (College Prep Mathematics): “CCSS Math Practices? Trust CPM’s 25 Years of Writing Experience”. This company is a nonprofit textbook publisher run by math teachers. Their series of books stresses the importance of hands-on explorations to teach math concepts. I was very impressed with the passion and excitement of the teachers who use these books.

Finally, I went to another session by Michael Felton, “Desmos and Modeling”. I’ve used Desmos‘ online grapher for several years now (I even did a presentation on it at a TAIS conference last year). They have moved far beyond a simple function grapher, though. Their site now includes all kinds of activities that teachers can use in the classroom. The one Michael demonstrated involved matching transformed sinusoidal curves, and it was a lot of fun. Go to to see all the fantastic lessons they offer. Teachers can also keep track of students’ progress as they work through the activity.

All in all, an excellent conference. It is fascinating to see how social media is transforming teaching, and how much teachers are trying to incorporate active learning into their lessons. I figure a conference is worthwhile if I can take home at least three good ideas/lessons/activities from it. After this one, I have more than dozen to try out with my students!


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:

Volume Formula

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:

Vol Example


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: