# Implementing the 6th Grade GPS via Folding Geometric Shapes

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

Topics Addressed Nets Surface Area of Cylinders Prisms Pyramids
Cones Surface Area of Cylinders

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

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.

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.

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

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.

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.

Another Possible Solution
Are there others?

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

Nets for a Square Pyramid
Pentahedron - Base is a square and faces are triangles

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)

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)

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”

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

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.

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

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

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)

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.

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?

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

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?

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.

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.

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?

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

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?

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