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Implementing the 6 th Grade GPS via Folding Geometric Shapes. Presented by Judy O’Neal ([email protected]). Topics Addressed. Nets Prisms Pyramids Cylinders Cones Surface Area of Cylinders. Nets. A net is a two-dimensional figure that, when folded, forms a three-dimensional figure.

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Implementing the 6 th grade gps via folding geometric shapes

Implementing the 6th Grade GPS via Folding Geometric Shapes

Presented by Judy O’Neal

([email protected])

Topics addressed
Topics Addressed

  • Nets

    • Prisms

    • Pyramids

    • Cylinders

    • Cones

  • Surface Area of Cylinders


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

Identical nets
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
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 cube1
Nets for a Cube

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

Nets for a cube2
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
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
Another Possible Solution

  • Are there others?

Nets for a regular pyramid

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

Nets for a square pyramid
Nets for a Square Pyramid

  • Square pyramid

    • Pentahedron - Base is a square and faces are triangles

Nets for a square pyramid1
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
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

        * version available)

Great pyramid at giza directions
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
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
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
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
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
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
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
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
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
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
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.

Making your own cone investigation
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.

Creating nets from shapes
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
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
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
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

Websites for additional exploration
Websites for Additional Exploration

  • Equivalent Nets for Rectangular Prisms

  • Nets

  • ESOL On-Line Foil Fun