Day 2
Prism and Pyramid
• In what ways are these shapes alike?
• In what ways are these shapes different?
Distribute set
Sort into 2 groups
• Group 1: shapes that are polyhedra with
faces that are polygons.
• Group 2: shapes that are polyhedra with
faces that are not polygons.
• Return all the shapes that do not have all
polygons as faces to the plastic bag.
Analyze polyhedra
• There are 12 different polyhedra.
• Find the 12 different ones.
• Keep only those 12 and put the rest back
in the bag.
Name the shapes
• What are the names of the shapes?
• How are the names determined?
• What two classes of polyhedra are
represented in the set? Explain.
Analyzing Polyhedra
• All of the polyhedra have polygons for faces.
• Polyhedra have vertices and edges.
• The faces are rectangles, squares, triangles,
hexagons, and octagons.
• The polyhedra in the set are prisms and
pyramids.
• The polygons of a polyhedra always have three
or more edges.
• Prisms are polyhedra that have two bases which
are congruent. The other faces are rectangles.
• Pyramids have a base that is any polygon. The
other faces are triangles.
Investigate
• Polyhedra can be analyzed in many
different ways.
• One way is to compare the number of
faces, vertices, and edges.
• In your group analyze the faces, vertices,
and edges of the prisms and pyramids.
• What did you discover about the
faces, vertices, and edges of the
shapes?
• In what ways are the faces of the
shapes alike? Different?
• What are some other mathematical
names we can use to describe the
faces?
• In what ways are the vertices of the
shapes alike? Different?
• What are some other mathematical
names we can use to describe
vertices?
• In what ways are the edges of the
shapes alike? Different?
• What are some other mathematical
names we can use to describe
edges?
Faces
• Prisms have 5 or more faces.
• Pyramids have 4 or more faces.
• The faces of polyhedra are always
polygons.
Vertices
• Vertices are points where edges meet.
Edges
• The faces of polyhedra always have 3 or
more edges.
• The edges of polyhedra are always line
segments.
• The endpoints of the edges are called
vertices.
• There are always more edges than faces
or vertices.
Journal
• “Analyzing Polyhedra”
• Work within your group fill in table.
What you notice
• What patterns do you notice going across
in the rows of the table, between the
number of faces, vertices, and edges?
• In what way do the number of faces,
vertices, and edges relate to one another
for any given shape?
Did you notice this?
• The number of edges is always greater.
• The number of faces and vertices is
always fewer than the number of edges.
• If you add the first two columns, you will
have 2 or more than the number of edges.
• You can add 2 to the edges and you will
have the sum of the faces and vertices.
Journal
• Add the words “Number Rule” to the table
in the fifth column.
• Try and write algebraic rules for finding
faces, edges, and vertices.
Rename
•
•
•
•
Name the following variables:
F= number of faces
E= number of edges
V= number of vertices
• How can you use these variables to write
an algebraic rule that relates to the
number of faces, vertices, and edges to
each other?
Euler’s Formula
• Leonhard Euler discovered that the
number of faces and vertices of polyhedra,
when added together, were always two
more than the number of edges.
• F+V=2+E
• It can be written in other ways…know
any?
• V+F=E+2
• V+F–E=2
Let’s investigate
• If a polyhedron has 5 faces and 5 vertices,
how many edges does it have?
• How do you know?
• F+V=2+E
• 5+5=2+E
• 10 = 2 + E
-2
8=E
-2
Your Turn
• If a polyhedron has 8 vertices and 12
edges, how many faces does it have?
• Explain how you know.
• Faces = 6
Journal
• In your journal draw either a pentagonal
prism or pentagonal pyramid.
• Write a description of the shape.
• Test Euler’s Formula using the shape.
Share with classmates.
• Did you hear any ideas that you want to
add to your journal entry?
• Did you hear anything that makes you
want to change something in your journal
entry?
• Review chart “Analyzing Polyhedra”
Analyzing Polyhedra
• All of the polyhedra have polygons for faces.
• Polyhedra have vertices and edges.
• The faces are rectangles, squares, triangles,
hexagons, and octagons.
• The polyhedra in the set are prisms and
pyramids.
• The polygons of a polyhedra always have three
or more edges.
• Prisms are polyhedra that have two bases which
are congruent. The other faces are rectangles.
• Pyramids have a base that is any polygon. The
other faces are triangles.
• Is there anything on the chart that should
be changed?
• Are there any ideas to add to the chart?
• Do you have any questions about
polyhedra?
• Make changes, add new ideas, and add
questions to the chart.
Questions about polyhedra
• Do other shapes besides polyhedra work
for Euler’s Formula?
• Do the types of polygons used in a shape
make a difference in the number of faces,
edges, and vertices?
• What is the greatest number of faces a
polygon can have?
Wrap up
• Think about our school…
• What combinations of
polyhedra were used in the
building’s design?
• Discuss with group.