G3-33 Building Pyramids

G3-33 Building Pyramids Goal: Students will build skeletons of pyramids and describe properties of pyramids. Prior Knowledge Required: Polygons: triangles, quadrilaterals, pentagons, hexagons Vocabulary: edge, vertex, vertices, face, pyramid, skeleton, base, triangular, rectangular, pentagonal, hexagonal Start with a riddle: You have 6 toothpicks. Make 4 triangles with them. The toothpicks must touch each other only at the ends. Let your students try to solve the riddle using toothpicks and modelling clay to hold the toothpicks together at the vertices of the triangles. The answer, of course, is the triangular pyramid. You might give your students the hint that the solution is three-dimensional. Sketch a rectangular pyramid on the board and shade the base. Ask volunteers to mark the edges and the vertices (count them and make a tally chart). Write the words base, edge, vertex, and vertices on the board. Give your students modelling clay and toothpicks. Show them how to make a pyramid. Start with a base, then add an edge to each vertex of the base and join the edges at a point. The students should make triangular, square, and pentagonal pyramids. Then let them fill in the chart and answer the questions on the worksheet. After finishing the worksheet, they may check their prediction for the hexagonal pyramid by making one. Tell your students that the shapes they have built are called skeletons of pyramids. You might write the following equation on the board: Skeleton = Edges + Vertices. As animal skeletons are covered with flesh and skin, the skeleton of a pyramid can be covered with paper, glass, or other substances and will have faces. Show a pyramid (with faces) and write the word faces on the board as well. Assessment: Add a row to the chart on the worksheet for a pyramid with a heptagonal (7-sided) base, and fill it in. Activity: Build skeletons of pyramids using marshmallows and toothpicks or straws. Extensions 1. How many faces, edges, and vertices would a pyramid with a 10-sided base have? 2. Ask your students to bring to class pyramids or pictures of pyramids (e.g., Egypt, Mexico, Japan, entrance to Louvre in Paris) that they can find at home. You can use these pyramids in lesson G3-38: Edges, Vertices and Faces. 3. PROJECT: Ask your students to learn about a pyramidal structure and give a presentation about it what was the structure used for, when and where was it built, why does it have the pyramidal form? Geometry Teacher s Guide Workbook 3:2 24

G3-34 Building Prisms Goal: Students will build skeletons of prisms and describe properties of prisms. Prior Knowledge Required: Polygons: triangles, quadrilaterals, rectangles, pentagons, hexagons Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, triangular, rectangular, pentagonal, hexagonal Give each student several pattern block triangles and ask them to place the triangles on the table one on top of the other, aligning the sides. Ask your students what shape the stack has. What does this shape look like from above? (triangle) What does it look like from the side? (rectangle) Explain that mathematicians call this 3-D shape a triangular prism. Let your students build prisms from other pattern blocks. Point out the faces, the edges, and the vertices. Explain that the shape that was used to build the prism (triangle, for example) is called the base. Sketch a prism on the board and shade the bases. Ask volunteers to mark the edges and the vertices (count them and make a tally chart). Write the words base, faces, edges, vertex, and vertices on the board. Give your students modelling clay and toothpicks. Show them how to make a prism. First make two copies of the base, and then join each vertex on one base to a vertex on the other base with an edge. Students should make triangular prisms, pentagonal prisms, and a cube. Let them fill in the chart and answer the questions on the worksheet. After finishing the worksheet, they may check their prediction for the hexagonal prism by making one. ASK: What have you built? (skeletons of prisms) What do we call the bones of your skeletons? (edges) What do the skeletons need to become prisms? (faces) Assessment: Add a row to the chart on the worksheet for the prism with a heptagonal (7-sided) base, and fill it in. Geometry Teacher s Guide Workbook 3:2 25

G3-35 Edges, Vertices and Faces Goal: Students will identify vertices, edges (including hidden edges), and faces (side, back, front, top, and bottom) in the drawings of 3-D shapes. Prior Knowledge Required: Count edges of polygons Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape Remind your students that lots of 3-D shapes in the world around us are either pyramids or prisms. As an example, you might show them a photo of the pyramids in Egypt. Hold up a 3-D shape and draw a picture of the shape on the board. Write 3-D shape next to your drawing. Ask volunteers to identify the edges, the faces, and the vertices on the shape itself and on the drawing, and write the terms edge, face, and vertex on the board. Remind your students that the plural of vertex is vertices. Your students will need the skeletons of the cubes they made during the last two lessons. Give each student 2 squares made of paper and 4 squares made of transparent material. Ask them to add the paper (non-transparent) squares as the bottom face and the back face; the transparent squares as the top, front, and side faces. It is a good idea to show the students how to add faces on a larger model before they work on their own models. Add the faces one at a time, emphasizing the position and name of each one. ASK: Which edges of the cube do you see only through the transparent paper. If the transparent faces were made of paper, would you see these edges? (no) The edges that would be invisible if all the faces were nontransparent are called the hidden edges. On a two-dimensional drawing of a cube these hidden edges are marked with dotted lines. Extension: Ask students to hold or place their cubes in various positions and to look at them from different angles (on the table, on the floor seen from above, slightly above their heads, and so on.) Ask students to describe what the faces look like when seen from different angles (they look like a square, a parallelogram, etc.). The outline of the shape itself can look like a square, a rectangle, a hexagon, a trapezoid, and a rhombus. Geometry Teacher s Guide Workbook 3:2 26

G3-36 Pyramid Nets Goal: Students will build pyramids from nets. Prior Knowledge Required: Count edges of polygons Polygons: triangle, square, pentagon, hexagon Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape Show your students nets for hexagonal and square pyramids. ASK: What do these drawings have in common? What is different? Let your students count the faces in the net. What shape do they have? How many shapes of each kind? If your students do not mention that each net has one face that is different from the others, point that out and ask what this face is called. (the base) Ask your students what the net for a triangular pyramid might look like. Ask students to complete the worksheet for this lesson. They can use the nets on the worksheet to answer Question 1. Alternatively, you can give them copies of pyramid nets (see BLM section) to cut and fold. Let your students count the edges, the vertices, and the faces of the shapes they make. Draw a net for a triangular pyramid on the board and ask your students to count the number of edges on the net. Is it the same as the number of edges in the 3-D pyramid? Let the students explain. (There are nine edges on the net and only six edges in the shape. The six edges on the outside of the net are glued in pairs, so they produce three edges in the 3-D shape. The three internal edges of the net together with the other three produce six edges in the pyramid.) Repeat with a square pyramid. As a challenge, draw a net of a pentagonal pyramid and ask your students to tell from the net how many edges are in the 3-D shape. Extensions: 1. ASK: What happens if you cut one of the side faces in a pyramid net and try to glue it at some other place? Students might actually cut off one of the triangles and reattach it to another edge. You might draw the following examples on the board. Will the net still fold into a pyramid if the face is glued in this position? no yes no 2. Describe the nets for different shapes. Describe the shape of each face and count the number of faces of a given shape. Draw a freehand sketch of all the faces that make up a particular 3-D shape. For example, the parts of a square prism are and the net is. Geometry Teacher s Guide Workbook 3:2 27

G3-37 Prism Nets Goal: Students will build prisms from nets. Prior Knowledge Required: Count edges of polygons Polygons: triangle, square, pentagon, hexagon Vocabulary: edge, vertex, vertices, face, prism, skeleton, base, 3-D shape Repeat the previous lesson for prism nets. Activities: 1. After your students have constructed the pyramids and prisms from the nets on worksheets G3-36 and G3-37, or the BLM section, ask them to sketch the nets from memory. They can test their nets by making the shapes. You might also ask students to sketch and test nets for pentagonal and hexagonal pyramids and prisms. 2. Give students square or pentagonal pyramids and ask them to trace the faces on a piece of paper, so that they create a net. Ask them to cut out the nets they have drawn. Let them cut off faces of the net and reattach the faces at different places. Will the new net fold into the same pyramid? Which edges are places where you would want to re-glue the faces and which are not? Repeat this exercise with a prism. This activity is important because it lets students explore various ways to create nets for the same solid rather than memorizing a single net shape. Geometry Teacher s Guide Workbook 3:2 28

Nets for 3-D Shapes (continued) Cube Triangular Prism BLACKLINE MASTERS Workbook 3 - Geometry, Part 2 11