Much of this class was spent working on a large activity Circles, Cylinders, Rectangles, and Parallelograms. This exercise involved finding different cross-sections of cylinders in order to represent them in terms of familiar shapes. For example, a cylindrical tube could be cut lengthwise and unrolled into a rectangle. The activity was a great way to explore the different relationships between 2-D and 3-D shapes, and for students to estimate and communicate their thoughts on each step of the process.
It was interesting how our measured circumference using a string was 14cm, while our calculated value was only 12.56cm. There were a couple of possible reasons for our low calculated value, as we had used a diameter of 4cm, which I think was slightly low. The diameter was measured from the ink footprint of the tube, however after holding the tube in the ink pad long enough to be able to make a stamp, the cardboard was soft and warped when pressed against the page. As a result, the edge of the tube folded inward slightly and our stamp measured less than we had predicted it would. Another reason the values may have differed is that it is difficult to wrap a string around the tube without compressing it slightly into an ellipse.
We cut the tubes and laid them flat into rectangles and then used our l x w algorithm to find the area. While we are working away from going straight to standard algorithms, I would assume that by the time students are doing cylinders this relationship would be well understood. The area of the paper tube was compared to the area of the sheet metal tube to find that the area was 100 times greater for the metal tube. A student might find this confusing, as we were told that the paper tube was 1/10 of the size, and we might want to simply multiply the area of the paper tube by 10. I used a whiteboard to draw out a grid for this part of the problem. If a square was to be placed on this grid, we would be able to find the area by using l x w. Making the square ten times bigger results in the length and width being 10l and 10w in terms of the original square. We can see from such a diagram that we will end up with 100 times the area, and this concept can be extended to volume, which in this case would be 1000 times greater for the metal vs. paper tube.
The last part of the exercise (the OH NO!! moment) stated that we received metal in the wrong size. That we only had a length of 80cm to work with instead of 100cm. We then explored cutting a toilet paper tube along the spiral edge, resulting in a parallelogram. This was not something I had ever done before and was a great way to visualize the relationship between the two shapes.
The activity of measuring the sizes of different lakes was a great way to compare how shapes may differ even if they share the same value for perimeter or area. This was related to our opening exercise when we were to find two gardens where the perimeters were the same but the areas differed by 6. To do this I created a chart of dimensions and started to pick out areas that differed by six. I then wrote small notes next to these numbers for their perimeters and started to pair them off. I found that for this exercise there was a pattern where if you moved diagonally you could predict the next two cages that would share these properties. Two cages that would share the desired properties would be 1x6 and 3x4. The next two would be 2x7 and 4x5. Each number increases by one each time, and you could continue this pattern for all fences.
My favourite activity, and not just because it had marshmallows, was comparing volumes of two tubes. Each tube was made from the same size sheet of paper, one length-wise and one width-wise, and we filled each with marshmallows. The wider tube was then emptied and placed around the narrower but taller tube, then the narrow tube was lifted so that its contents went into the wide tube. The marshmallows that had previously filled the narrow tube did not fill the wide tube, so we were able to see that the volume of the wide tube was bigger. Reflecting on why this might be, considering each was made from the same size of paper, indicated that volume increased more with width as the radius was greater. Looking at the equation for volume of a cylinder (pi)hr^2 shows that doubling the height will double the volume, but doubling the radius will quadruple the volume. This was a great exercise for demonstrating this principle, and something I would definitely try with my class sometime.
