Lesson Title: Cavalieri’s principle
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
G.GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle,
volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal
limit arguments.
*This lesson plan will focus only on Cavalieri’s principle
MP 2:
MP 3:
MP 4:
MP 5:
Reason abstractly and quantitatively
Construct viable arguments and critique the reasoning of other.
Model with mathematics
Use appropriate tools strategically
Lesson Launch Notes: Exactly how will you use
the first five minutes of the lesson?
In your own words define the following geometric
concept or write down the formula associated with
the concept:
1. Volume (any figure)
2. Volume of a Cylinder
3. Volume of a Prism
Lesson Closure Notes: Exactly what summary
activity, questions, and discussion will close the
lesson and provide a foreshadowing of tomorrow?
List the questions.
321 Feedback:
Have each student record their responses to the
following:
Record 3 facts that you learned today.
Give 2 examples of where you can see the application
Cavalieri’s principle.
List 1 way that you can use one of the formulas from
today’s activity.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities,
investigations, problems, questions, or tasks will students be working on during the lesson?
1. Divide the class into five groups. Set up five stations with congruent objects that can be stacked to
form a cylinder or prism. (Examples: coins, paper, sticky note pads, a “Slinky”, “Legos”, textbooks,
etc.)
2. Have the group create a conjecture concerning the relationship between the volume of a right
prism/cylinder and the volume of a corresponding oblique prism/cylinder.
3. Assign each group a starting station. Using the manipulatives at the station the group needs take the
necessary measurements to calculate the volume of the objects stacked as a right figure and as an
oblique figure.
4. Rotate the groups every five to seven minutes. Move about the room and monitor student progress.
5. At the conclusion of the rotations give fifteen to twenty minutes for the groups to compile their results.
Groups need to validate or find a counterexample for the conjecture they created before beginning the
station rotations.
6. Have the groups turn in their arguments before having a class discussion concerning the results of the
visits to the stations.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Cavalieri’s principle
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
Discussion Questions:
1. Using all of the information that we have gathered today what is the conclusion that is described in
Cavalieri’s principle?
2. The following excerpt provides background information on Cavalieri’s principle and how it relates to
higher mathematics.
Cavalieri's principle, sometimes called the method of indivisibles, named after Bonaventura
Cavalier, is as follows:
 2-dimensional case: Suppose two regions in a plane are included between two parallel lines
in that plane. If every line parallel to these two lines intersects both regions in line segments
of equal length, then the two regions have equal areas.
 3-dimensional case: Suppose two regions in three-space (solids) are included between two
parallel planes. If every plane parallel to these two planes intersects both regions in crosssections of equal area, then the two regions have equal volumes.
Today Cavalieri's principle is seen as an early step towards integral calculus. Cavalieri's principle
grew out of the ancient Greek method of exhaustion, which used limits but did not use
infinitesimals.
http://en.wikipedia.org/wiki/Cavalieri%27s_principle
3. The stackable manipulatives can be thought of in terms of the 2 dimensional planes they represent. The
planes are stacked to create the 3 dimensional figure and form the volume. Each plane has an area and
when planes are stacked there are h planes (h is the height). Therefore in general terms the volume is
the base area times the number of planes or V=(Base Area)(height). Another way to look at this is to
imagine taking a cross section or slice of the 3 dimensional figure. The cross section must be parallel
to the base and will therefore be congruent to the base. There will again be h cross sections of the 3
dimensional figure so the volume formula is the same.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how
will I measure student mastery? That is, deliberate consideration of what performances will convince you
(and any outside observer) that your students have developed a deepened (and conceptual) understanding.
Success will be determined through an examination of group’s responses. If groups can finish the task,
successfully complete the calculations and verify the formulas then the lesson will be a success. Success
will further be demonstrated in the verbal responses to the formative assessment closure prompt at the end
of class.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Volume – an amount of space on the interior of a 3 dimensional figure. Simply put, how much a 3
dimensional figure can hold in the interior?
Cavalieri’s Principle – If two solids of equal altitude and the sections made by planes parallel to and at
the same distance from their respective bases are always equal, then the volumes of the two solids are
equal.
Cross Section - the intersection of a body in 3-dimensional figure with a plane. More plainly, when cutting
an object into slices one gets many parallel cross-sections.
Resources: What materials or resources are
essential for students to successfully complete the
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Cavalieri’s principle
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
lesson tasks or activities?
completion of the lesson?
Ruler, a length of string, calculator, protractors,
sets of stackable manipulatives
Assign a problem set from the appropriate textbook
section
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you
use to reflect on the effectiveness of this lesson?
How did the student groups perform making the calculations? Were the students actively involved in the
task? Did any student groups struggle to discover the relationship between the volumes of right and
oblique cylinders and prisms?
What revisions, modifications, or improvements can be made to make the task more engaging and
cognitively demanding? Was the lesson effective for students in developing their understanding of volume
and Cavalieri’s Principle, as well as modeling these concepts?
Can students explain the rationale for using a dissection argument to verify measurement formulas in 2 and
3 dimensions?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.