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 = 2 + E 10 = 2 + E -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.