I display the warm-up problem with an LCD projector using the slide show for this lesson. This classroom routine follows our Team Warm-up format. Students work the problem in their learning journals, and one member of each team writes the team’s answer on the white
board at the front of the classroom. While the students work, I complete administrative tasks.
The warm-up problem for today’s lesson asks students to find the area of a section of a sphere. The area is 144*pi.
BOLOs (things to be on the lookout for): Are students confusing the radius of the circular section with the radius of the sphere? Are students applying the Pythagorean Theorem correctly? (Often, students will simply add the squares of two side lengths without regard to which sides are legs and which is the hypotenuse.) Do students see that, when using the Pythagorean Theorem, they can save work by staying in the domain of area? (MP7) (They can omit the step of taking the square root of the difference in squared side lengths in order to find the radius (length). Since the area of the circle is proportional to the square of the radius, they simply must multiply the difference of squares by pi.) Are students writing exact answers (as multiples of pi), or are they approximating pi with a decimal fraction? (Students often want to get a decimal answer rather than write an exact value in terms of pi. While this is not wrong, it often makes it harder to see relationships between quantities. In the next activity, I use friendly numbers so that the relationship between areas is easy to discern. Those relationships are harder to see if
students work with decimal fractions.)
Purpose: The lesson opener gives students a chance to work in their cooperative learning teams practicing a skill they will need in the next activity. The activity allows me to assess whether the class is ready, and provide additional instruction when necessary.
Following the warm-up problem, I use the slide show to display the agenda and learning targets for the lesson. I tell the class that today they will be using Cavalieri’s Principle to find the relationship between the volumes of spheres, cones, and cylinders. The relationship is surprisingly simple! In the second part of the lesson, we will test our reasoning by hanging foam slices of a cylinder, cone, and hemisphere from a lever to see if they really balance.
This activity helps students deduce the relationship between the volumes of a hemisphere, cone, and cylinder using Cavalieri's Principle.
To encourage student engagement, the activity is conducted in cooperative learning groups using our Team Jigsaw routine. Members of a team divide up the work, and every team member must contribute for the team to succeed. For this activity, I allow students to choose their own roles: Area Finders find the areas of circular sections of one of the three solids; the Connection Maker enters the values obtained by the other team members into a table to make them easier to compare.
Using the slide show, I introduce the three solids we will be comparing. A cylinder whose radius is the same as its altitude, call it R. A cone with radius and altitude of R. A hemisphere with radius R. The cylinder circumscribes the cone and hemisphere. To make things more concrete, we will choose a value for R, something easy to work with: 10 centimeters.
I give the students 15 minutes to work and display a digital timer. As students work, I circulate around the classroom. I expect most teams to see the relationship between quantities with little help, but many students will need help writing general algebraic expressions for the radius and area of a section (MP2). At the end of the time limit, I bring the class together to review what they have discovered. I use student work, which I “buy” from students with bonus points and display on the overhead projector.
BOLOs: Same as for the opener. Notice that I have chosen “friendly” numbers for radius R and for the heights of the cutting planes so that students can see
the relationship between areas more easily. If a team is working ahead, I challenge the to compare the areas of sections taken at a different height. The numbers may be messy, but the relationship will hold.
Before class: Reproduce the How Are They Related Activity handout, 1 copy for each team. Cut into half-sheets.
I perform a demonstration designed to make the lesson memorable for
students. With the help of student volunteers, I hang discs of closed-cell foam padding on either end of a lever (a collapsible tent pole about 16 feet long) to see if sections of a hemisphere and cone together can be balanced against a section of a cylinder. I cut the circles out of camping mats to the radii of circular sections of the three solids, 5 sections of each solid at 5
different heights. A simple stand and a set of foam spacers holds the sections at their correct heights, so that students can visualize the solids from which the sections are taken. This is a dramatization, not a real test. The student assigned to hold the tent pole always makes it balance, compensating for small differences in the weights of the discs.
Before class: If you are interested in making a set of discs and a stand yourself, the cost of materials was about $40. The accompanying videos show how.
The purpose of this activity is to help students see that the volume of a hemisphere is 2/3 the volume of a cylinder whose base and altitude are the same as the radius of the sphere (the circumscribing cylinder). From here, it is a short leap to see that the volume of a sphere is 4/3 the volume of the same (circumscribing) cylinder. (The second problem in this set is intended to help students make that connection. However, there is no particular reason to know the relationship between the volumes of sphere and cylinder at this point. The purpose of the next lesson is for students consolidate their understanding of the relationship that governs the volumes of spheres, cylinders, hemispheres, and cones.
I pose the following question: The relationship between the volumes of this hemisphere, cone, and cylinder is truly remarkable, but how can we use it? For example, how could we use it to find the volume of any particular hemisphere?
Using diagrams in the slide show, I lead the class in a discussion and derivation of
the relationship between the volumes of a hemisphere and cylinder:
Vhemisphere = 2/3 x Vcylinder.
- We have found a relationship between the volumes of these three solids, but how can we use
it? Does it apply to other sets of spheres, cones, and cylinders?
- Do you think that the radius we used (10 cm) makes a difference, or would this work for sets of solids of a different size? (MP3) How can you tell?
- What was the purpose of writing algebraic equations for the areas of the sections of the different solids? (MP3)
- Suppose you want to find the volume of this hemisphere (e.g. radius 2 m). How would you go about it?
- What special constraint did we place on the dimensions of the solids at the start? (MP3)
- Involving a hemisphere, cylinder, and a cone just to find the volume of a hemisphere seems complicated. How can we take the cone out of the equation?
Next, I distribute the first of the set of two problems in the Applying Volume Relationships resource and challenge students to use the relationship we derived. This activity is conducted using the Rally Coach cooperative learning structure, a classroom routine. More information on how I use Rally Coach can be found in my Strategies folder.
Some students may want to know the formulas for the volume of a hemisphere or a sphere. Although I deliberately emphasize relationships (ratios) at this point, I do not discourage students from learning the formulas. Instead, I point them down the path to deriving the formulas on their own. If a student wants to use formula he or she has looked up in a reference, I show them how to see the “universal” volume formula V = Bh in the equation.
While some students look for formulas, others will actually draw a circumscribing cylinder around
the hemisphere or sphere given in a problem. This shows me that they understand the concepts. It also shows that teaching relationships and methods is an effective alternative to teaching formulas. I see this approach as a form of differentiation.
Before class: Reproduce the Applying the Relationship Activity handout, 1 copy per 2 students. Cut into half-sheets.
The homework provided with this lesson consists of three problems. The first problem asks students to find the volume of a hemisphere and a sphere. The second problem asks students to explain why a composite shape (a hemisphere ‘scooped out’ of a cylinder) has the same volume as a cone. The third problem asks students to apply their understanding to a real-world problem.
I display the Lesson Close prompt using the slide show for this lesson. Students write their answers in their learning journals. I remind students to turn in their work from the How Are They Related? activity as an exit slip. After class, I look over the students’ work to see who is getting this and what problems others are having.