Goals of the Session:
- Provide a hands-on opportunity to examine the cylinder problem
- Overview the videopaper, Encouraging Mathematical Thinking
- Discuss the BRAP project as a model for Professional Development
1) Initial Experiment:
Form two cylinders from a rectangular piece of paper, one by joining the long sides, one by joining the short sides. Which of these cylinders will have greater volume, or will they hold the same amount?
See it in action
Was your prediction correct? Do the two cylinders hold the same amount? Why
or why not? Can we explain why they don't?
2) Second Experiment:
Are there any other cylinders that we can make from this same sheet of paper?" Which of them would hold the most? Write down your predictions."
How many other cylinders could we make from a rectangle with these same
dimensions? Do you see any pattern that relates the size of the cylinder
and the amounts they hold? Do your results confirm what we discovered with
our physical models?
3) Why we chose the Cylinder problem?
-The lesson involves multiple stages of investigation: prediction, testing, rejection or extension of hypotheses, discovering and exploring the underlying mathematics, and making generalizations and proving results.
-It involves a number of mathematical concepts and problem-solving approaches that can lead to rich discourse in the classroom.
Students interpret a graph in
Judith Koenig's class
Reasoning for predictions in Susan
-Students are engaged by the mathematical demonstration afforded by this investigation, the results of which seem counterintuitive to many of them.
-The lesson also offers multiple entry points for a variety of levels, from elementary school through calculus.
-The problem has real life applications.
Jon Basden's real life example
Explaining predictions is an important part of the process
- It eliminates thoughtless guessing and helps students think about what might happen and the mathematical relations inherent in the situation;
- Our experience suggests that the initial struggle students go through in formulating their reasons for a prediction -- whether naive or correct -- enriches their
experience and motivates them to continue.
5) Useful links: