Lesson Title: Cavalieri’s Principle (Volume)
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
Geometry Common Core Standards
G.GMD.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.
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. Circumference
2. Area of Circle
3. Volume of a Cylinder
4. Volume of a Prism (square base)
5. Right Cylinder or Prism
6. Oblique Cylinder of Prism
[UDL I: 2,3]
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 leaned today.
Give 2 examples of where you can see the application
Cavalieri’s Principle.
List 1 way that you will remember Cavalieri’s
Principle.
[UDL III: 9]
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities,
investigations, problems, questions, or tasks will students be working on during the lesson?
Procedure:
1. Divide the class into 5 groups. Set up 5 stations with objects that are congruent that can be stack to
form a cylinder or prism. (Examples: pennies, paper, sticky note pads, a “SLINKY”, “Legos”,
textbooks, etc)
[UDL III: 8]
2. Assign each group to start at a station. Using the manipulatives at the station the group needs to
measure (using the ruler and/or the string) the objects stacked to form a right cylinder/prism and
calculate the volume of the cylinder/prism. Using any of the tools available calculate the volume of
an oblique cylinder/prism constructed by stacking the same number of objects.
[UDL II: 4, UDL III: 8]
3. Rotate the groups every 5 minutes until they have been at every station. During the rotations move
about the room and monitor group progress.
[UDL II: 2]
4. Give time (approximately 15 to 20 minutes) for the groups to answer the following question: “How
are the volumes of a right and oblique cylinder or prism related? Use your calculations from the
stations in your group’s argument defending your conclusion.
[UDL II: 6, UDL III: 4]
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 (Volume)
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
5. Have the groups turn in their arguments before having a class discussion concerning the results of
the group’s visits to the stations.
6. In the discussion be sure to include the following:
a. the relationship between the volumes of right and oblique cylinders/prisms (Cavalieri’s
Principle).
b. the concept of a 2 dimensional cross section of a 3 dimensional figure (parallel to the base).
c. the implication of Cavalieri’s Principle in the real world (i.e. The Leaning Tower of Pisa has
the same volume whether it is leaning or not).
[UDL III: 7]
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 volume calculations, and discover Cavalieri’s Principle, then will have
developed a deeper conceptual understanding of volume. 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.
[UDL I: 2,3]
Resources: What materials or resources are
essential for students to successfully complete the
lesson tasks or activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Ruler, a length of string, calculator, 5 sets of
Assign a problem set from the appropriate textbook
stackable manipulatives
section
[UDL II: 4]
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you
use to reflect on the effectiveness of this lesson?
1. How did the student groups perform making the calculations?
2. Did any student groups struggle to discover the relationship between the volumes of right and
oblique cylinders and prisms?
3. Were the students actively involved in the task?
4. What revisions, modifications, or improvements can be made to make the task more engaging and
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 (Volume)
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
cognitively demanding?
5. Was the lesson effective for students in developing their understanding of volume and Cavalieri’s
Principle, as well as modeling these concepts?
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.