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 ONeal
• (joneal_at_ngcsu.edu)

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Topics Addressed

• Nets
• Prisms
• Pyramids
• Cylinders
• Cones
• Surface Area of Cylinders

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Nets

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

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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.

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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.

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Nets for a Cube

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

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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.

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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.

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Another Possible Solution

• Are there others?

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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

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Nets for a Square Pyramid

• Square pyramid
• Pentahedron - Base is a square and faces are
  triangles

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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)

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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/pyra
  mid2.html (Spanish version available)

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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
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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

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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.

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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

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Surface Area of a Cylinder

• Closed cylinder
• Surface Area 2Base area Rectangle area
• 2Area 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

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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)

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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.

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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?

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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

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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?

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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.

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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.

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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?

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Alike or Different?

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

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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?

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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

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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_ma
  instream/classroom/units/foil_fun/home_e.php