Engage:

 Take a sheet of paper and join the top and bottom edges to form a 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.

   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. Mark each cylinder now to avoid confusion later. All of this can be done before class to save time, or at the beginning to enforce that you are using the same sized paper for each cylinder.

   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.

Time: 3-5 minutes

Students should respond to the questions by quickly deliberating with classmates before recording their answer and explanation.

Explore:

  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, r has more effect than h [because r is squared], and therefore the cylinder with the greater radius will have the greater volume.)

Time 5-7 minutes

Now have the students divide into groups of 2-3 with all the appropriate equipment. Ask the students to write as much as they can about the cylinders as they can figure out. This should include at least measurements of the rectangle (paper) used to form the cylinder as well as measurements of the cylinder. These include length, width, height, and circumference. Also, some connections should be drawn between the length/width and height/circumference.

Tell the students to construct their own cylinders A and B. Next, have the students construct other cylinders using the whole paper. The students can cut the paper, but have to use the entire sheet. Make two cylinders with the new dimensions and use cylinders A and B as an example to form cylinders from this new shape. Make sure data on the length/width of the new rectangle is recorded.

Perform the same experiment between A and B on all 4 cylinders until they can be ranked from largest volume to smallest.

Time 20-25 minutes

What is the one dimension that most greatly affects the ability to hold more of the fill?

Students will observe the class experiment.

Students will participate in the discussion by volunteering answers and critically thinking about other students’ responses and in turn asking more questions.

Students divide into their groups either at the teacher’s discretion or by pairing off. They will then perform their own version of the experiment. Each student should keep record of the measurements they find.

The groups should work to construct their own cylinders and find the measurements of the new shape.

The groups will perform their own experiments and rank the cylinders in the described fashion.

Explain:

Bring the students back together and keep them in their given groups. Have one student from each group put up their data concerning their cylinders on the board.

Have all the students study the data and to determine a relationship among the data. The class should come to the conclusion that the taller the cylinder, the less volume and the shorter, the more volume.

Can we write this in mathematical language that will help us confirm our observations? What formulas relate to this problem?"

C = 2pi*r or pi*d [circumference of a circle]
A = b*h [Area of rectangle]
V = pi*r²*h [Volume of cylinder]

So if our ultimate goal is to calculate the volume, then the formula we will need to use is V = pr²h. How are we going to find r and h? That is the challenge.

Find h. (assuming regular sized paper and cylinder A, h=11)

Find r. (same assumption: r is approx 1.5282)

"Now we have r and h and we are ready to find the volume. Let's put them both into the volume formula,

V = pi*r²*h       Using substitution,
V = pi(1.35282)²*(11) in³
V = 63.2442 in³

Time 20-25 min

What dimension of our original rectangle is related to the height of the cylinder?

What dimension of our original rectangle can we use to find r?

What did this relate to on the cylinder? (circumference)

How can we find r?

Students once again participate in the class discussion.  All students are expected to try to study the data with their group and come up with responses to the questions asked to the class.

Hopefully, students will recall formulas for circumference and area when asked. They will perhaps need more prompting for volume of a cylinder or even flat out require you tell them.

Students will think with their groups and volunteer any answers they come up with and defend them if questioned by the rest of the class. Move to the next measurement/calculation when the whole class is satisfied.

Students will make the calculations as you do.

Extend / Elaborate:

The students will be given several theoretical dimensions and asked what kind of cylinder would have to be constructed to meet the criteria. For instance, find a cylinder with a surface area of 93.5 in² and volume of 300 in³.

Time: 10-12 minutes

What was the easiest way to increase the volume?

Students work either in groups or alone, depending on your preference, to find some of these theoretical cylinders. Any uncompleted cylinders will be completed for discussion the next day.

  Evaluate:

Evaluation will be ongoing throughout the lesson. Special attention should be paid during Explain section of the lesson. The extension here is pretty close to an easy evaluation.  An additional, simple worksheet involving the volumes of cylinders can be handed out for completion to further check for individual understanding.

Time NA

Students complete any additional work. They are also expected to ask any questions during the lesson to further illuminate any problem areas.