Finding a Radius

I designed a class around the pedagogy of Project Based Learning this semester. As the school year passes by at mach speed, I have adapted certain activities and projects to fit my students’ needs. The result is a class based around providing tools, and tackling interesting tasks with them. Each set of problems (or unit) is capped with a large project.

We are in the midst of a surface area and volume unit. We have tackled the major solids and prisms. Netting, superimposing grids, converting units, analyzing packaging etc. Throughout the entire class, I have been highlighting the various “employable skills” that they are honing with their work. Estimation, problem solving, critical thinking, diagnostics, peer work, spatial reasoning and the like.

Today’s task was an innocent classic that ended up ballooning into some great peer discussion and problem solving. The class was introduced as follows:

• I set up table groups and we arranged chairs so they were facing each other
• Each group was assigned a cylinder of three tennis balls or racquetballs
• They were asked to use their knowledge to calculate three things:
  1. The volume of the container
  2. The volume of a single ball
  3. The total empty space in a sealed canister

They were given a ruler and a long leash. I sat back and watched as the various groups set off to work.

Of the five groups, two of them immediately began throwing the balls around the room. Once that calmed down, I simply circulated and waited for learning to make an appearance. It showed up… and brought a salad.

Students had very little trouble with the volume of the cylinder. The only real hang up was the accuracy of measurements. The potent learning occurred when students began to try and measure the radius of the ball. First off, every group decided to find diameter instead of radius. When I questioned them, they told me that it would be far too difficult to find the radius because they couldn’t find the center of the ball. I conceded to their logic, and allowed them to look for the diameter. The result was fantastic.

In all, there were six different methods for finding the radius of the ball.

1. Headphones
   • One group decided to hunt for the circumference and then use the formula to deduce the diameter. A girl brought out her headphones, surrounded the ball, transferred her bookmarked length to the ruler, and read her measurement. From there they used their knowledge of algebra to get their information.
2. Bendy Ruler
   • Another group decided to bend their flexible ruler around the ball and find the circumference. They too used their knowledge of algebra to then complete the calculations. The exciting thing was the fact that both these groups developed their strategy in isolation. They worked with what they knew, and developed creative methodologies.
3. Two Cellphones
   • A group decided to take their identical cellphones and stack them vertically against either side of the ball. They then measured the diameter as the distance between the two phones. One student watched the phones to make sure they were parallel (horizontally and vertically), another secured the ball, and another took the measurement. Teamwork and ingenuity on display.
4. Hovering Ruler
   • These students were suspiciously quiet for a long time. When I snuck a glace, they were all pouring over tennis balls with one eye closed. They had all decided to place their ruler on the top of the ball and look directly down from above. They were fairly confident that if three separate people got the same reading, it was accurate. Interesting use of the power of peers.
5. Can Measuring
   • One group simply measured the diameter of the canister that contained the balls. When I asked about the leftover space, they rolled the ball half way out, measured the displacement from the edge of the ball to the edge of the can, and subtracted the space. They were particularly proud of the efficiency of this method.
6. Slice and Dice
   • One group asked to cut the ball in half. I was not prepared for this method, and had not cleared that with the Phys. Ed. department from whom I borrowed the materials. I applauded them on their creativity, but asked them to find a separate method. (They eventually used “Can Measuring”)

When class was winding down, I pulled everyone back to the front and detailed the six different methods. As they were listed, I gave credit where it was due. We compared our calculations and analyzed error. It was the first time this semester where students truly took ownership of their mathematical strategy and debated amongst themselves.

I feel like this lesson is a perfect snapshot of what I want my math teaching to be. I had a plan, was well-prepared, and willing to roll with the student learning the entire time. My original learning objectives involved volume, displacement, and measurement; the actual learning was far more encompassing. From here, we can now create a perfect mathematical tennis ball container where the balls fit perfectly. What ratio between empty space and filled space do we see? Is it unexpected? What if we stacked cubes in a square based pyramid? Same result? different? etc.

Students need to see that within the strict confines of formulae, theorems, and conjectures exists ample room for originality.

NatBanting