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3 E 3 Minimal: Materials Paper (grid, construction, blank) Pencils ( both colored and graphite) Straightedge (Ruler) Scissors Transparent tape Compass Ruler Cereal boxes or other boxes that can be cut by participants, one for each participant. They should be empty. Compass (or Mira) Protractor or angle ruler Optimal list includes Pattern blocks Computer with: Internet connection The Geometer s Sketchpad or Cabri

5 E 5 Exploring Quadrilaterals: Sides and Angles Participant Page Use the hand out from Appendix A and/or the Web page, to explore the properties of these quadrilaterals What are some attributes about the sides of each figure? How do they relate to one another? Are they perpendicular? Parallel? When? Always? Are they congruent? Always? How do the angles relate to each other? Place this information down in the table below. Characteristics of Quadrilaterals Name of Polygon Relationship of opposite sides Relationship of adjacent sides Relationship of angles Quadrilateral Rhombus Parallelogram Square Rectangle Trapezoid Construct (on paper or with Sketchpad) the quadrilaterals that satisfy the list below: Any Quadrilateral A Parallelogram A Rectangle A Square A Rhombus A Trapezoid

7 Exploring Quadrilaterals: Sides, Angles, and Diagonals E 7 Participant Page Each drawing at the right is the beginning of a quadrilateral. Finish each figure. Draw the diagonals, measure them and fill in the table below. Quadrilaterals and Their Diagonals Diagonals Bisect Each Other? Diagonals Congruent? Diagonals Perpendicular? Diagonals Bisect Opposite Angles? Quadrilateral Rhombus Parallelogram Square Rectangle Trapezoid Kite Isosceles Trapezoid

9 E 9 Drawing Quadrilaterals from the Diagonals Participant Page Place points on the intersecting segments at the left so that they form the vertices of the indicated quadrilateral. Rectangle Square Parallelogram Rhombus Kite Trapezoid If it is not possible to make the labeled quadrilateral, write impossible on the axes. Isosceles Trapezoid Generic Quadrilateral Any quadrilateral List the shapes that you could not draw above. Draw a set of diagonals below that could be used to create these shapes. Why were you unable to create these shapes?

12 E 12 Exploring Quadrilaterals in Three Dimensions Instructor Page Cutting up cereal boxes Pass out empty boxes of cereal. Have the participants cut their boxes according to the instructions on the participant sheet. They should sketch their nets and share the nets and boxes with another group to confirm their work. Can the same box have more than one net? YES In a cereal box, which sides must be the same sized rectangles? Opposite sides, top and bottom and side panels. Pass out the net for the cube in Appendix C. Have participants cut out the net and fold it to form the cube. Appendix D has graph paper that may help with sketching nets. Pass out the set of nets in Appendix E, and have participants pick which nets they think will fold to form a cube. It is fine for them to cut the nets out and fold them to check their predictions. The answers are below. A cube has 8 vertices, 12 edges, and 6 faces. Circled numbers go with those nets that will form cubes Answers Terms to know: Face: Sides of a polyhedron Edge: Segment where two faces meet Vertices: The points where three or more edges meet Extension 1 Have participants cut out the six pieces of their cereal box net. Have them find as many arrangements of the six pieces as they can that will fold back into the original box. When they find an arrangement, they should sketch the net of this arrangement. Have students imagine that these nets were to be placed on a sheet of cardboard to be cut and folded as cereal boxes. Determine which of their nets would require the least amount of cardboard waste when cut out.

13 E 13 Exploring Quadrilaterals in Three Dimensions Participant Page Cutting up cereal boxes Your instructor will give you a cereal box. Think of how to cut it so that it will lay flat in one piece on a table top. Then cut the box along the edges and lay it out flat on the table top. If you make mistakes, you can correct them with tape. In the space below, make a sketch of the flattened box, including the folds. It will look something like the figure above. This sketch is called a net. A net is a pattern for a three-dimensional figure. If you cut out the net and fold on the edges, it will form a solid. Label your net and your box and give them to a different group to confirm that the unfolded box does match the net that was drawn. If it does not, correct the net. Can the same box have more than one net? In a cereal box, which sides must be the same sized rectangles? You instructor will pass out the net for a cube. Cut it out and fold it together to make the cube. Count the vertices, edges, and faces and put the values in the lines below. # vertices on a cube # of edges on a cube # of faces on a cube

14 E 14 Visualizing Cubes and Stacks of Cubes Instructor Page Discussion: Each section of the Sears Tower is a tower itself. We could build a model of the Sears Tower using blocks. Ask participants to think about how they could draw a picture of a building and have it appear three-dimensional. It is often helpful to use a special paper to draw such sketches. It is called isometric dot paper. This activity should take between 10 minutes and half an hour. Blank sheets of isometric dot paper are available in Appendix F. Use overhead copy of the isometric dot paper to sketch one cube and have participants try it themselves. Note: The cube needs to be oriented with one vertical edge centered and the sketch should show the horizontal edges as segments on the diagonal, all parallel.. Give participants about 10 minutes to attempt to replicate the drawings on the isometric dot paper. They may build the figure with blocks to assist in the drawing.. Challenge (Find the pattern): Give participants an extra sheet of isometric dot paper and have them draw one cube, then a cube made up of two smaller cubes on each edge. How many cubes are hidden from view? Make a 3 x 3 x 3 cube. How many are hidden from view? In a 4 x 4 x 4? A 5 x 5 x 5? Keep track of the numbers that are hidden in a table like the one below. The answers are in red. Predict how many will be hidden in the drawing of a 50 x 50 x 50 cube. Be careful not to count the same cube twice! Number of cubes on each side Total number of cubes Number of cubes hidden from view To Draw a Cube Drawing a cube on regular paper is easy. 1. Draw a square 2. Draw a second square the same size as the first which overlaps the first at the lower left. The midpoints of the sides of the second square match the midpoints of the sides of the first square (points E and F). 3. Connect the vertices. Hidden lines should be dashed.

15 E 15 Visualizing Cubes and Stacks of Cubes Participant Page Below are drawings of several stacks of cubes. Use a pencil and the isometric dot paper below to draw the stacks without the cubes that have stars.

16 E 16 Euler s formula Instructor Page Larger versions of the nets on the participant page are in Appendix G. Have participants cut out the nets and fold them into polyhedrons. Have participants create a table like the one below and fill in the values by counting the faces, vertices, and edges for each polyhedron. Euler s Formula tells us that, for any polyhedron, there is a relationship between the number of faces, edges, and vertices. We can see this by looking at the table below. One way to write the formula is V+ F - E = 2. In other words, the number of vertices plus the number of faces minus the edges is always 2. Notice in the table that the numbers for both triangular prisms are the same. It does not matter that they look somewhat different. Also, the cube is merely a special case of the rectangular prism. So the number of faces, edges and vertices are the same for both. Illustration of Euler s Formula Name of Polyhedron # of Faces # of Vertices # of Edges A. Tetrahedron B. Hexagonal Prism C. Triangular Prism D. Rectangular Prism (Box) E. Triangular Prism Cube Internet Resource: A more general view of the Euler Relationship can be found on the NCTM Student Math Notes site at There is a.pdf file called Let s Be Discrete at (requires Adobe Acrobat Reader)

17 E 17 Euler s Formula Participant Page A Cut out the nets on this page or the larger ones that your instructor gives you. All but one have quadrilaterals for faces. We include a tetrahedron as an illustration of the fact that Euler s Formula holds for all polyhedrons. B Count the Vertices, Faces and Edges of each polyhedron. Put the values in a table and look for a pattern. C E D

19 E 19 Appendix A Various Quadrilaterals

21 Appendix A (continued) E 21 Rhombi Quadrilaterals Parallelograms

23 E 23 Appendix A (continued) Squares Trapezoids Rectangles

25 Appendix B E 25 Quadrilateral Diagonals Below are a set of diagonals for quadrilaterals Draw the sides of the quadrilaterals and fill in the table below with the most specific name for the quadrilateral. ( 1 ( 2 ( 3 ( 4 ( 5 (

27 E 27 Appendix C Net for a Cube

29 E 29 Appendix D

31 E 31 Appendix E Which Nets Fold to a Cube? Circle the number of each net that can be folded to form a cube. Check your predictions by cutting out the larger versions of the nets and folding them to see if they form cubes.

33 E 33 Appendix E (continued) Larger versions of the Nets 1 2 3

35 E 35 Appendix E (continued) 4 6 5

37 E 37 Appendix E (continued) 7 8

39 E 39 Appendix F Isometric Dot Paper

41 E 41 Appendix G

43 Appendix G (continued) E 43

45 Appendix G (continued) E 45

47 E 47 Annotated References: Internet Resources and Text Resources that are useful for exploring this geometry unit. Korbus, Linda Perspective PowerPoint 1999: Hinsdale South High School, Hinsdale, IL. A Powerpoint presentation focusing on Perspective drawing. The teacher presents famous art images using perspective and she demonstrates how to make a drawing in twopoint perspective. This presentation is available at the Java Perspective site listed below. Serra, Michael. (1997). Discovering geometry: An inductive approach (Second ed.). Berkeley, CA: Key Curriculum Press. This textbook has many wonderful activities for exploring geometry. - National Council of Teachers of Mathematics. NCTM. See Principles and standards for school mathematics, as well as the Navigations and the on-line Illuminations. Java Perspective All the electronic files associated with drawings in perspective. They are interactive, web-based, and have downloadable documents and sketches. James P. Dildine s Java Geometry Additional Java sketches that any student or teacher can access from the Web Webmath Web-based interactive mathematics resources (Dictionary, Formulas, etc ). Math.com Web-based interactive mathematics resources (Dictionary, Formulas, etc ) Key Curriculum Press Textbook and Software Publisher specializing in The Geometer s Sketchpad and references for GSP.

48 E 48 questions and comments to Geometry

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