Whether you view this as an SAT-type problem, a geometry challenge or just another investigation, I hope you will enjoy this in the true holiday spirit of giving! So Happy Chanukkah and Merry XMAS!
- 3-dimensional objects, spatials sense, volumes of cylinders
Although this investigation appears to be aimed at secondary students taking geometry, cylinders are introduced in middle school and even earlier. There's no reason why middle schoolers shouldn't be able to tackle this or perhaps a modified version. Are 7th and 8th graders expected to know the formula for the volume of a cylinder, or at least, the general form: Area of Base x Height?
In each figure, a 3x5 rectangle is rolled to form a cylindrical shape (a "can" without a top or bottom). In Fig. I, we "identify", i.e., paste edge AD onto edge BC. For Fig. II we reorient the same rectangle and paste edge DC onto edge AB. Even though these cylinders do not have a 'bottom', assume they are sitting on a flat surface and we will be "filling them up" to determine their volumes.
For the Instructor: Provide 3x5 index cards for the students. Students, working in pairs, should physically form each of the cylindrical shapes and refer to these while working the problem.
1) Without calculating, make a conjecture: Which cylinder would have the greater volume or would they be equal? (For instructor: Record the results of these conjectures on the board)
2) Compute the volumes of each cylinder - no calculators! Leave results in terms of π. Was your conjecture accurate?
3) Compute the ratio of the volume of cylinder I to the volume of cylinder II. Any surprises?
4) It wouldn't be an investigation if we didn't generalize! But this time, YOU have to write the generalization, state the conclusion and prove it! Remember, the true spirit of the holidays is to give, not only receive!!