1. Implementing the 6th Grade GPS via Folding Geometric Shapes
Presented by Judy O’Neal

2. Topics Addressed Nets Surface Area of Cylinders Prisms Pyramids
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. Nets for a Regular Pyramid
Tetrahedron - All faces are triangles
Find the third net for a regular pyramid (tetrahedron)
Hint – Pattern block trapezoid and triangle

11. Nets for a 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
* (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
Nets
ESOL On-Line Foil Fun