Wednesday, March 28, 2012

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.


Thursday, March 8, 2012

Questionless Scavenger Hunt

My involvement with a provincial math executive presented me with an interesting task recently. Like most tasks, I turned to get some input from the strong contingent of math teacher tweeps.

I needed to develop an activity for 100-115 students in grades Seven to Eight. All I was told is that it should be about an hour and a half, and be active in nature. The students are taking part in a math contest in the morning, and it would be great to get the blood pumping. I turned these demands to twitter, and came up with some excellent options:
  • A building task where students need to develop a structurally sound building out of a certain amount of newspaper and tape.
  • The classic spaghetti and marshmallow tower
  • A very interesting activity where solutions to questions refer to coordinates of a grid overlayed on a map. The students would then be directed around the university campus using a Cartesian plane.
  • Another building that needed to be constructed on a budget. Differing materials range in price and efficiency.
I wanted the activity to include explicit mathematics. These students are all quite gifted, and I wanted them to be able to link an overarching mathematical concept to the activity. I felt that the building challenges did not offer that. The budget element did provide a very interesting twist, but was too difficult to set up on short notice.

I also didn't want the mathematical element to be in the form of explicit math questions. I wanted the math activity to be an interesting break from the formal testing that the students did the entire morning. This meant that the Cartesian grid option was eliminated. (Although I plan to develop this at a later date).

I was left with the scavenger hunt. Traditionally, scavenger hunts are a series of multiple choice questions. Each response moves the team, but they have no idea if they are right or wrong until they either finish the hunt or repeat a question. I began to draw this framework out; it became complicated quickly. In the diagram below, a squiggly line represents a correct answer and a solid line represents an incorrect one. There is one correct and three incorrect from each question. It would take six consecutive correct responses to complete the scavenger hunt.

It's a total mess. It did get students moving, but it still relied on the traditional "question-driven" hunt. I began to grow fond of the complexity that went into a hunt like this. A fairly simple concept has a fairly elaborate mathematical underpinning. Then the idea hit me, what if the framework was shown and the task was to decipher it?

Enter the idea of the "Questionless Scavenger Hunt". I created a map of a possible scavenger hunt with the possible interactions between "questions". I simplified the process by including 9 "questions" with 3 "responses" each.

The goal of the activity is now to use logic to determine which "question" fits in which slot on the map. Every question has it's possible destinations listed on a card. The cards were placed in a certain room on campus. The goal can then be re-defined as trying to determine which "room" goes where on the map.

I included 2 anchor rooms on the map. They will serve as the starting point (Room 1) and a reference point (Room 5) throughout the activity. Students are given the Room 1 card with the names of the three possible rooms they could be sent to. These three rooms are represented on the map by the rooms that have an arrow pointing to them from Room 1. From there, students need to move around rooms collecting cards of the possible connections from each room. To complete the hunt, they must be able to arrange the remaining 7 rooms on the map so all interconnections work.

Here are some pictures of the files that can be downloaded here.
The Scavenger Hunt Map (given to each student or group)
The cards that can be found in the respective rooms (Need to be cut and sorted)
Various strategies could be used to complete. Groups may just collect the cards from each room and then sit down and try to puzzle it out. Other groups may use a process of elimination. If they pick up a card from Room 3 and it does not list Room 1 as a possible destination, than any space that has an arrow pointing to Room 1 can't possibly be Room 3.

The result should be an energetic twist on a classic favourite. I am expecting students to come with correct answers. (These are very gifted students). I plan on challenging them to prove that their solution is unique. If students claim it is impossible, I will question them as to why. Every mathematical step needs to be justified.

The materials can be customized to your situation. Choose a starting room and replace "Room 1" with the name or number of that space. Do the same thing with 8 other rooms around your building. I altered my activity to fit the Arts building of the local university. If you wanted to implement it on a smaller scale, set up nine "stations" around the classroom. Put a stack of cards at each one, and students could move around the room looking for logical connections.

This was a great learning experience for me. I struggled with the possibility of creating a puzzle with a viable solution. I often learn the most mathematics when I am trying to create meaningful experiences for my students.

I would like to see a possible "Questionless Scavenger Hunt" with more than 9 stops. That is the challenge if you choose to accept it.