Implementing the 6 th Grade GPS via Folding Geometric Shapes

1. Implementing the 6th Grade GPS via Folding Geometric Shapes Presented by Judy O’Neal (joneal@ngcsu.edu)

2. Topics Addressed • Nets • Prisms • Pyramids • Cylinders • Cones • Surface Area of Cylinders

3. Nets • A net is a two-dimensional figure that, when folded, forms a three-dimensional figure.

4. Identical Nets • Two nets are identical if they are congruent; that is, they are the same if you can rotate or flip one of them and it looks just like the other.

5. Nets for a Cube • A net for a cube can be drawn by tracing faces of a cube as it is rolled forward, backward, and sideways. • Using centimeter grid paper (downloadable), draw all possible nets for a cube.

6. Nets for a Cube • There are a total of 11 distinct (different) nets for a cube.

7. Nets for a Cube • Cut out a copy of the net below from centimeter grid paper (downloadable). • Write the letters M,A,T,H,I, and E on the net so that when you fold it, you can read the words MATH around its side in one direction and TIME around its side in the other direction. • You will be able to orient all of the letters except one to be right-side up.

8. Nets for a Rectangular Prism • One net for the yellow rectangular prism is illustrated below. Roll a rectangular prism on a piece of paper or on centimeter grid paper and trace to create another net.

9. Another Possible Solution • Are there others?

10. Regular pyramid Tetrahedron - All faces are triangles Find the third net for a regular pyramid (tetrahedron) Hint – Pattern block trapezoid and triangle Nets for a Regular Pyramid

11. Nets for a Square Pyramid • Square pyramid • Pentahedron - Base is a square and faces are triangles

12. Nets for a Square Pyramid • Which of the following are nets of a square pyramid? • Are these nets distinct? • Are there other distinct nets? (No)

13. Great Pyramid at Giza • Construct a scale model from net to geometric solid (downloadable*) • Materials per student: • 8.5” by 11” sheet of paper • Scissors • Ruler (inches) • Black, red, and blue markers • Tape *http://www.mathforum.com/alejandre/mathfair/pyramid2.html(Spanish version available)

14. Great Pyramid at Giza Directions • Fold one corner of the paper to the opposite side. Cut off the extra rectangle. The result is an 8½" square sheet of paper. • Fold the paper in half and in half again. Open the paper and mark the midpoint of each side. Draw a line connecting opposite midpoints. 4 ¼” 8 ½”

15. More Great Pyramid Directions • Measure 3¼ inches out from the center on each of the four lines. Draw a red line from each corner of the paper to each point you just marked. Cut along these red lines to see what to throw away. • Draw the blue lines as shown

16. Great Pyramid at Giza Scale Model • Print your name along the based of one of the sides of the pyramid. • Fold along the lines and tape edges together.

17. Nets for a Cylinder • Closed cylinder (top and bottom included) • Rectangle and two congruent circles • What relationship must exist between the rectangle and the circles? • Are other nets possible? • Open cylinder - Any rectangular piece of paper

18. Surface Area of a Cylinder • Closed cylinder • Surface Area = 2*Base area + Rectangle area • 2*Area of base (circle) = 2*r2 • Area of rectangle = Circle circumference * height = 2rh • Surface Area of Closed Cylinder = (2r2 + 2rh) sq units • Open cylinder • Surface Area = Area of rectangle • Surface Area of Open Cylinder = 2rh sq units

19. Building a Cylinder • Construct a net for a cylinder and form a geometric solid • Materials per student: • 3 pieces of 8½” by 11” paper • Scissors • Tape • Compass • Ruler (inches)

20. Building a Cylinder Directions • Roll one piece of paper to form an open cylinder. • Questions for students: • What size circles are needed for the top and bottom? • How long should the diameter or radius of each circle be? • Using your compass and ruler, draw two circles to fit the top and bottom of the open cylinder. Cut out both circles. • Tape the circles to the opened cylinder.

21. Can Label Investigation • An intern at a manufacturing plant is given the job of estimating how much could be saved by only covering part of a can with a label. The can is 5.5 inches tall with diameter of 3 inches. The management suggests that 1 inch at the top and bottom be left uncovered. If the label costs 4 cents/in2, how much would be saved?

22. Nets for a Cone • Closed cone (top or bottom included) • Circle and a sector of a larger but related circle • Circumference of the (smaller) circle must equal the length of the arc of the given sector (from the larger circle). • Open cone (party hat or ice cream sugar cone) • Circular sector

23. Cone Investigation • Cut 3 identical sectors from 3 congruent circles or use 3 identical party hats with 2 of them slit open. • Cut a slice from the center of one of the opened cones to its base. • Cut a different size slice from another cone. • Roll the 3 different sectors into a cone and secure with tape. Questions for Students: • If you take a larger sector of the same circle, how is the cone changed? What if you take a smaller sector? • What can be said about the radii of each of the 3 circles?

24. Cone Investigation continued • A larger sector would increase the area of the base and decrease the height of the cone. • A smaller sector would decrease the area of the base and increase the height. • All the radii of the same circle are the same length.

25. Making Your Own Cone Investigation • When making a cone from an 8.5” by 11” piece of paper, what is the maximum height? Explain your thinking and illustrate with a drawing.

26. Creating Nets from Shapes • In small groups students create nets for triangular (regular) pyramids (downloadable isometric dot paper), square pyramids, rectangular prisms, cylinders, cones, and triangular prisms. • Materials needed – Geometric solids, paper (plain or centimeter grid), tape or glue Questions for students: • How many vertices does your net need? • How many edges does your net need? • How many faces does your net need? • Is more than one net possible?

27. Alike or Different? • Explain how cones and cylinders are alike and different. • In what ways are right prisms and regular pyramids alike? different?

28. Nets for Similar Cubes Using Centimeter Cubes • Individually or in pairs, students build three similar cubes and create nets • Materials: • Centimeter cubes • Centimeter grid paper Questions for Students • What is the surface area of each cube? • How does the scale factor affect the surface area?

29. GPS Addressed • M6M4 • Find the surface area of cylinders using manipulatives and constructing nets • Compute the surface area of cylinders using formulae • Solve application problems involving surface area of cylinders • M6A2 • Use manipulatives or draw pictures to solve problems involving proportional relationships • M6G2 • Compare and contrast right prisms and pyramids • Compare and contrast cylinders and cones • Construct nets for prisms, cylinders, pyramids, and cones • M6P3 • Organize and consolidate their mathematical thinking through communication • Use the language of mathematics to express mathematical ideas precisely

30. Websites for Additional Exploration • Equivalent Nets for Rectangular Prisms http://www.wrightgroup.com/download/cp/g7_geometry.pdf • Nets http://www.eduplace.com/state/nc/hmm/tools/6.html • ESOL On-Line Foil Fun http://www.tki.org.nz/r/esol/esolonline/primary_mainstream/classroom/units/foil_fun/home_e.php