This warm-up gives students a preview of the sub-problems they will solve as we work
through Archimedes’ mechanical method for finding the volume of a sphere. The
warm-up problem asks students to find the volumes of a small cylinder (2*pi), a large cylinder (8*pi) and a large cone (8/3*pi).
This Lesson Opener follows our Team Warm-up routine.
BOLOs: Do students recognize that doubling the radius of the base of a cylinder increases its volume by a factor of 4 (altitude being unchanged)? Do they see that the volume of the cone is
1/3 the volume of the large cylinder? If students are trying to find volumes using formulas, they will have trouble following Archimedes’ method. I always ask students to show their thinking when they write their solutions to warm-up problems, but in this case it is a good sign if students balk at showing work. I expect most students to be using relationships between solids rather than formulas (MP2), since this is the approach I have emphasized all along—what I call “heads up math”. I am ready to point out the advantages of this strategy, using the team answers on the board for examples (MP5).
- How is the large cylinder related to the small cylinder? What measurements are the same? Which are different?
- What happens to the area of a circle when you double the radius? How does that come into play here?
- Since we have found the volume of the large cylinder, what is the shortest route to finding the volume of the cone?
I display a press release announcing the Archimedes palimpsest exhibition at the
Walters Museum in Baltimore on the white board in the front of the class. I ask students to read the headline and first paragraph. I have already distributed paper copies (1 per 2 students), because I hope some students will be intrigued enough to read further.
I ask the class if anyone can explain what a palimpsest is and briefly tell the story of
the Archimedes Palimpsest. Point to highlight:
- The Archimedes Palimpsest may be the most important scientific manuscript ever to be
sold at auction, because it contains the writings of Archimedes, including two
books that had been lost for nearly 2000 years.
I tell the class that we are going to learn about the contents of one of those books, the Method, as I display the agenda and learning targets for the lesson. By the end of the lesson, we will all know what secret methods of Archimedes the manuscript revealed, and what this has to do with what Archimedes tomb.
I ask students to share what they know about Archimedes. Someone will say that he
shouted “Eureka!” (it is one of the quotations on the walls of my classroom), and another will have seen the Mythbusters episode about Archimedes’ heat ray. Playing on what students already
know, I tell the story of Archimedes Tomb with the help of a slide show. I like to ad lib when I tell stories about the history of mathematics, so I have to watch the clock. I want to move into section #3 no later than 15 minutes into the lesson. There are more slides in the slide show than are needed.
Points to highlight:
- Archimedes was the first physicist. He made an incredible number of discoveries in physics and mathematics. You will understand why someone might think he was an alien when you learn about his mysterious mechanical method for calculating the volume of a sphere.
Purpose: If it is not obvious, I am trying to motivate what will probably be a challenging topic. (I
expect a few of my students to be fans of the Dan Brown books or the spin-off movies.) But, I also want students to learn about the history of mathematics. I want them to see that math is a very personal, creative undertaking; that math is an ever-changing body of knowledge (not a comfortable topic for adolescent brains); and that mathematical thinking is rooted in history and
Provisioning: Reproduce the Walters Museum press release, about 1 copy for every 2 students.
I tell students that to understand Archimedes’ mechanical method for solving math problems, they must understand the Law of the Lever (which Archimedes also discovered). We have learned about the Law of the Lever in an earlier lesson [add hyperlink], so this section of the lesson is meant as a quick review. However, it should be possible for students to learn enough about the Law of the Lever (and the concept of a center of gravity) to appreciate his mechanical method.
Students will love this activity, but it is not absolutely essential to the learning targets for this lesson.
I tell the class that I want them bouncing ideas off each other as we review the Law of the Lever and work through Archimedes’ method (MP3), so we will be using our Team Checkpoints
routine, which is explained in my Strategies Folder. I briefly review the rules, using the slideshow for the lesson and appoint a student to act as scorekeeper.
To carry out this activity, I use an interactive on-line simulation. While the application is coming up, I write on the board at the front of the classroom: How far from the fulcrum does the weight have to be to balance the lever? Then, using the Balance Lab feature, I present 6 problems for students to solve.
As shown in Law of the Lever Review., I place a weight on the left arm of the lever. With the
supports in place, I then drag out one or more weights and drop them haphazardly on the right arm of the lever. Once each team has held up its answer, I move the right-side weights to the position chosen by the majority of teams and remove the supports. If the lever does not balance, I then move the weights to the correct position to balance the lever. At the end of each question, I ask if anyone can explain how they knew where to place the weights on the current problem (MP3).
If the class completes all the problems in the time limit, there are 6 team points possible in this section of the lesson.
Purpose: The better students understand the Law of the Lever and the concept of a center-of-gravity, the better they will understand and appreciate Archimedes’ method for finding the volume of the sphere. It is enough, however, that they recognize that if the lever balances with one object twice as far from the fulcrum as the other, the first object must have half the weight of the second. More on this in the Explaining the Math Resource for the lesson.
In this section, I use the Mechanical Method slide show to lead the class in a re-creation of Archimedes’ reasoning about the problem of the sphere and cylinder (MP3). The slide show uses animations to tell the story, freeing me to focus on students.
We continue using the Team Checkpoints routine (the checkpoint questions are built into the slide show) to encourage students to cross-check what they think Archimedes' was saying with one another as well as to allow me to check for understanding. There are 20 checkpoint questions in the presentation. Most of them are quick (I allow 15 seconds for teams to answer, but most will not need that much time). I describe the more difficult questions below.
If you want to reduce the number of checkpoint questions in the lesson, omit the repeated questions. In the presentation, the question, “How do the radius and area of the large cylinder compare to the radius and area of the small cylinder?” appears 5 times (each time a section of the large cylinder is taken). A second version of the presentation, Mechanical Method slide show (fewer checkpoints), omits this question after the first time.
I begin by displaying Archimedes proposition that the volume of the cylinder is 3/2 the volume of its inscribed sphere. I ask for a volunteer to explain what Archimedes is saying. I write on the board: “volume of cylinder = 1.5 volume of (inscribed) sphere’ and make sure that the class understands what ‘inscribed’ and ‘circumscribed’ mean.
BOLOs and Questioning Strategies:
- Are students having trouble seeing that the radius of a section of the cone is equal to the distance of the cutting plane from the apex? How did we find the radius of a section of cone in the last lesson, The Inscribed Cone? If you think of the sections of a cone as dilations of the base (as we have), what is the center of dilation? How can you use the properties of dilations to find the radius of the section? What is the ratio between the distance of a section from the apex and its radius?
- Do students show signs of confusion when the lesson goes from comparing areas (of sections of solids) to comparing weights, then volumes? (If they are, then good. They are paying attention!) I explain my thoughts on this issue in the Explaining the Math Resource for the lesson.)
- Do students have trouble finding the area of sections of the sphere taken at a distance of half the radius of the sphere from the fulcrum? (I expect most to need support here, so I have prepared the resource described below.)
I want students to work together to figure out this problem, but most will need a little help to get started. To get them going and yet leave them with a reasonable share of the work, I pass out clues in the form of the Slicing a Sphere Activity (MP1). This resource is designed to be reproduced and cut into half sheets.
Each student in a team gets a different view of the problem with the information presented in a slightly different way. I tell them that I want them to share information and work together to solve the problem, but every student must hand in his or her own work. The different views have descriptive titles to entice students to make the choice that is best suited to them. I encourage students who are strong in algebra to try View 2. I tell student that they should feel free to share ideas with others and do not necessarily have to use the information in the way it is presented on their paper.
Before Class: Reproduce the Slicing a Sphere Activity handout, 1 per team of students. Cut into half sheets.
The homework provided with this lesson consists of three problems. The first asks students to practice finding the areas of sections of different solids. The second asks students to use algebra to think about how the area of a section of a sphere is related to its distance from the center of the sphere. The third asks them to find the area of sections of a sphere in a real-world context.
I display the Lesson Close problem using the slide show for this lesson. Students work the problem in their learning journals.
I remind students to turn in their Slicing a Sphere work as an exit slip. After class, I look over the students’ work to see who is getting this and what problems others are having. Students will need to find the areas of sections of a sphere in the next lesson, where we use Cavalieri’s method.