Encouraging Mathematical Thinking


 Abstract
 Introduction

 Discourse
 Interventions
 Decisions

 Cylinder Problem
  - Elementary
  - Middle School
  - High School
  - Calculus
 Lesson Reflections
 Student Predictions

 Project Reflections
 Conclusion

 References
 Acknowledgments
 Teacher Resources



Authors'
Biographies

Table of Contents


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Cylinder Problem: Elementary Level Lesson Plan



Objective:

Students will build a family of cylinders and discover the relation between the dimensions of the generating rectangle and the resulting pair of cylinders. They will also order the cylinders by the amount they hold, and draw a conclusion about the relation between the cylinder's dimensions and the amount it holds.

Materials:

8 1/2" by 11" sheets of paper for the class (transparencies work well for the initial experiment), tape, ruler, graph paper, fill material (birdseed, Rice Krispies, Cheerios, packing "peanuts," etc.).

Vocabulary:

Cylinder, rectangle, dimension, area, circumference, height, volume (optional).

Procedure:

Initial Experiment:

Take a sheet of paper and join the top and bottom edges to form a "base-less" cylinder. The edges should meet exactly, with no gaps or overlap. With another sheet of paper the same size and aligned the same way, join the left and right edges to make another cylinder.

Picture of cylinder A     Picture of cylinder B

Stand both cylinders on a table. One of the cylinders will be tall and narrow; the other will be short and stout. We will refer to the tall cylinder as cylinder A and the short one as cylinder B. Mark each cylinder now to avoid confusion later.

Now pose the following question to the class: "Do you think the two cylinders will hold the same amount? Or will one hold more than the other? If you think that one will hold more, which one will that be?" Have them record their predictions, with an explanation.

Place cylinder B in a large flat box with cylinder A inside it. Fill cylinder A. Ask for someone to restate his or her predictions and explanation. With flair, slowly lift cylinder A so that the filler material falls into cylinder B. (You might want to pause partway through, to allow them to think about their answers.) Since the filler material does not fill cylinder B, we can conclude that cylinder B holds more than cylinder A.

Ask the class: "Was your prediction correct? Do the two cylinders hold the same amount? Why or why not? Can we explain why they don't?" (Note to the teacher: because the volume of the cylinder equals pi*r2*h, the radius, r, has more effect [because r is squared] than the height, h, and therefore the cylinder with the greater radius will have the greater volume.)

Second Experiment:

"Let's go back and look at our original sheet of paper. We made two different cylinders from it. What geometric shape is the sheet of paper? (rectangle) What are its dimensions? (8.5" by 11".)

"What are the dimensions of the resulting cylinders? That is, what is the height and what is the circumference?" (The height of the cylinder is the length of the side of the paper rectangle that you taped, and the circumference is the length of the other side.)

"Are there any other cylinders that we can make from this same sheet of paper?" (Yes. There are many cylinders that can be made.)

"Let's try to make some other cylinders. If we fold a new sheet of paper lengthwise and cut it in half, we now get two pieces -- each measuring 4.25" by 11" -- which we can tape together to form a rectangle 4.25" by 22". We can repeat the process to create a second rectangle the same size. Now we can roll these rectangles into two different cylinders, one 4.25" high and another 22" high. We will label them cylinder C (4.25" high) and cylinder D (22" high)."

"Now we have four cylinders. Which of them would hold the most? Write down your predictions."

Test by filling. Have a student report the results.

Now have the students arrange the cylinders in order, by volume, from the cylinder that holds the least to the cylinder that holds the most. "Do you see any pattern that relates the size of the cylinder and the amounts they hold?" (As they get taller and narrower, they hold less, and as they get shorter and stouter, they hold more.)

"How many other cylinders could we make from a rectangle with these same dimensions?" (Theoretically, infinitely many. Cylinders could get taller and narrower and taller and narrower until they were infinitely tall and infinitely narrow, or they could get shorter and stouter and shorter and stouter until they were infinitely short and infinitely stout.)

"Tonight, for homework, you'll make two others." (Distribute the Family Math Activity sheet.)



Teacher Resource: You can download a completed table with all the calculations as an Excel spreadsheet here.

Assessment:

Evaluate the explanations from the Family Math Activity sheets.

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