I start this lesson by telling the class that they will be watching a video demonstration of the relationship between the volume of a cylinder and that of a sphere. I say:
At the end of the video I would like you to write one complete sentence to summarize your observations about this relationship. So, please watch closely.
One very important thing: I show the video without audio. I don't what students to listen to the narrator:
I give the class a couple of minutes to write their sentence and revise it. Once they have, I ask that they share their written responses with their elbow partner. I will then replay the video. Before I do, however, I allow some time to discuss points of disagreement as a class. I often like to record the different observations on the board for all to see. Then, before sharing my explanation, I show the video one more time. It probably won't be long before I hear "I told you" or "you see, I was right" from the students in the class, (see Be Aware of Put Downs reflection). I then call on volunteers to share their responses with the entire class.
My experience is that after repeating the video, most students will see that the volume of the sphere volume is 2/3 of the volume of the cylinder. It is likely that we will work together to add more detail such as, "with the same height and the same diameter (or radius)." As students share there definitions I will ask out loud, "Is there any important information missing from Johnny's sentence?" Intervening in this way encourages careful proofreading and editing. And, it usually motivates a full and correct explanation from the class.
I end this activity by writing on the board: V(sphere) = 2/3(pi)(r)(h)
After watching the video twice, we'll take a stretching break. My students love it when I give them 60 seconds between activities to stretch their bodies or take a walk around the room. My students sometimes ask for the minute stretch themselves and I always give it to them. I know it helps their thinking, and I find it improves our student-teacher relationship.
Now that students have watched the video and figured out the relationship between the volume of a sphere and its surrounding cylinder of equal height and diameter, I form small homogeneous groups of twos or threes, and I hand each student a Speaking of Spheres exploration sheet for cooperative work.
The questions aim to help students work out the Sphere Volume formula for themselves. Once they understand the derivation of the formula, they apply it in a series of interesting and unusual problems. To make this work, I “give up the floor” to the students encouraging them to discuss and tune into each other’s opinions (see Assign a Monitor reflection)
For Questions 1-3 I expect students to see that the height of the cylinder is 2r and substituting this for h in the equation 2/3(pi)r2h yields the sphere volume formula (See Exploration answer sheet). For Question #4 I want students to find the amount of water in the balloon using the sphere volume formula, but I make sure at least one student finds two thirds of the cylinder volume. I want to compare answers produced by these two different approaches at the end of the lesson.
To close the lesson I divide the board into 5 sections, reunite the group, and ask volunteers to share their answers to one of the four exploration questions on the board.
For the 5th section I ask a student who found the volume of the sphere in Question 4b by taking 2/3 of the volume of the cylinder. Students will see that the answers to 4a and 4 b are the same, confirming the volume relation.
In this Homework assignment I have students re-visit our efforts to derive the sphere volume formula. Then, I have students apply the formula to ordinary problems where they are given the diameter or the volume of a sphere. The assignment is "short and sweet", simply returning to concepts and tasks similar to those seen in class.