Lesson Title: Geometry Virtual Tour
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
Lesson Objective(s): What mathematical skill(s) and understanding(s) will be developed?
G.MG.A.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a
tree trunk or a human torso as a cylinder).
G.GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
MP.2 Reason abstractly and quantitatively
MP.3 Construct viable arguments and critique the reasoning of others
MP.4 Model with mathematics
Lesson Launch Notes: Exactly how will you use
the first five minutes of the lesson?
In your own words record the process or write the
formula for determining the volume of following
three-dimensional figures:
1.
2.
3.
4.
5.
6.
7.
8.
Square Prism
Rectangular Prism
Cylinder
Cone
Square Pyramid
Triangular Pyramid
Sphere
Hemisphere
How would you approach determining the volume
for a three-dimensional figure if you did not know
the formula for its volume? Generalize the process
for determining volume into a statement that
applies to any three- dimensional figure.
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 the volume would be a
useful calculation.
List 1 way that you will remember the how to
determine the volume for any three-dimensional
figure.
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 groups of four. Have groups complete the Seven Wonders of the Geometric
World Virtual Tour. (Look for evidence of MP 4)
2. Have groups prepare a 5 minute presentation in the final step of the activity. Prepare to have
computers available if groups choose to create a PowerPoint or Glog for their presentation.
3. Monitor group progress and provide assistance as required. (Look for evidence of MP 2, MP3, and MP
4)
4. Following the presentations, use the follow-up questions below to facilitate a class discussion involving
the volumes of three-dimensional. (Look for evidence of MP 2, MP3, and MP 4)
 What is the most commonly shaped building in cities? Why do you think that buildings use this
shape? (Possible responses: The most likely answer is right rectangular/square prisms. This figure
has a base that that is easily adapted to the street design in major cities. It has also been used
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Geometry Virtual Tour
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
frequently to design and erect some of the world’s tallest buildings. Finally, the right
rectangular/square prism is a figure that gives the office building perhaps the most usable space.)
 The Pentagon is the world’s largest office building in terms of floor area with 6,500,000 square
feet. The Pentagon has 2 floors underground and 3 floors above ground. With this in mind why do
we not find more pentagonal shaped buildings in cities? (Possible responses: A pentagonal
building creates floor space that may be non-usable since it does not use right angles in the exterior
walls. The Pentagon has a very large base area and is not very tall. This may be impractical in a
crowded city. Buildings in crowded cities need to fully use the space between streets. In cities
these regions between streets are often rectangles or squares. Another example of a pentagonal
building is the World Trade Center in Baltimore. It has a much smaller base area, but has 30
floors.)
 What does the term “best use of space” mean to you? How did your understanding of the meaning
“best use of space” influence your choice of the three-dimensional figure? (Possible response A:
The best use of space may indicate a design that is easily adaptable to traditional office spaces with
right angle corners. Designs with bases that are circular or not squares or rectangles may not be the
“best use of space”. The best use of space might also indicate a design putting a premium on the
floor area of the base such as a pyramid. Possible response B: If we are given a scenario where a
100 ft tall building can be erected using a base that cannot exceed a square with 100 ft sides then
the following can be deduced: A right square prism will have the greatest volume at 1,000,000
cubic feet, any other figure will have less volume as the base would need to be inscribed into the
square will 100 foot sides. For example a right cylinder with these restrictions has a volume of
785,000 cubic feet and a right square pyramid has a volume of 333,333 cubic feet. Therefore if we
want to maximize the space or volume that can be created in an urban setting then a prism with a
square or rectangular base becomes the figure with the “best use of space”.)
5. Evaluate the presentations based on correctly finding the volume, finding the number of times the
classroom space would fit into a building, and the effort the group put into making the presentation
informative for the class. The comparisons, contrasts, and “best use of space” questions are designed
to be starting points for classroom discussions and student discourse. (Look for evidence of MP 2,
MP3, and MP 4)
Note: Due to the group presentations this lesson may need multiple days to complete. This activity could
be completed online with students searching for their own examples of buildings that model threedimensional figures. Instead of giving the seven buildings ask the groups to search the Internet for
examples of famous buildings that model a rectangular prism, cylinder, sphere, square pyramid, etc.
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 student group work and student responses during
the class discussion. Students will be able to correctly derive the formula for calculating the volume of a
three-dimensional figure and generalize volume for any three-dimensional figure. Additionally, students
will be able to make comparisons about relative volume between figures.
Notes and Nuances: Vocabulary, connections, common mistakes, typical misconceptions, etc.
Volume – the amount of three-dimensional space a figure occupies or can hold. The figure’s capacity.
Height or Altitude – the length of a segment measured perpendicularly between the two bases or between
HCPSS Secondary Mathematics Office (v2); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: Geometry Virtual Tour
Date:
Teacher(s):
Course: Common Core Geometry
Start/end times:
the base and vertex of a three-dimensional figure.
Base Area – the area of the base of a three-dimensional figure.
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?
Calculator
Seven Wonders of the Geometric World Virtual
Tour Worksheet
Computers/LCD Projector
Poster Paper, Pens, etc.
Select a variety of practice problems that reinforce
the concept of volume and the application volume.
Lesson Reflections: What questions, connected to the lesson objectives and evidence of success, will you
use to reflect on the effectiveness of this lesson?
Are students able to derive the formula for calculating volume of a given three-dimensional figure?
Are students able to apply the formula for calculating volume to a variety of situations?
How did the worksheet assist students in the making decisions based on the relative volume of three-dimensional
figures?
How can this lesson be refined to make in more meaningful for students?
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.