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 -2 8 = E

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.