Geometry: 3D to 2D and Back Again

1
Concrete Representations of Geometric Concepts
National Council of Teachers of Mathematics 2007
Annual Meeting and Exposition Atlanta, Georgia
-- March 22, 2007
Dr. Nancy F. Matthews Montgomery Central Middle
School -- Cunningham, TN nancy.matthews_at_cmcss.net
Kay D. Haralson Austin Peay State University --
Clarksville, TN haralsonk_at_apsu.edu
2
Which Came First?Solids or Plane Figures

• Some ideas developed from the article
• Dear Verity, Now Im Getting Into Shape!
• by Deirdre Dempsey and John Marshall. Phi
  Delta Kappan, April 2002.

3
Dear Verity, Now Im Getting Into Shape!

• Francis W. Parker (mid 1980s)
• Concept of structured play
• Too much emphasis on drill and memorization
• Encouraged teaching by using childrens natural
  curiosity about the world
• Suggested learning arithmetic by studying objects
  and their properties

4
Concrete versus Abstract

• According to Piaget, children learn from the
  concrete first, then abstract.
• Are squares, rectangles, and triangles concrete?
• Are cylinders, cubes, and prisms concrete?

5
Mathematical Building Blocks

• What are the shapes of childrens blocks?
• Are they squares, triangles, and rectangles?
• Why not teach young children the mathematical
  names of familiar solids?
• Preschool anecdote.

6

• What is the shape of this object?
• Is it a square?
• How many corners does it have?
• Is there a problem with attribute blocks?

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Solidsto Plane Figures
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Use Solids to Draw Plane Figures

• Consider a lesson beginning with the concrete
  (cylinders and prisms) and progressing to the
  abstract (plane figures).
• Show students a rectangular prism.
• Discuss its featurescorners and faces.
• Give some examples of objects in the room.
• Have students name examples from home.
• Repeat with other 3-D shapes.

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Footprints

• Obtain geometric solids.
• Collect various objects that illustrate geometric
  solids and make a good footprint.
• Cut geometric solids from wooden dowels and
  pieces of wood or pieces of foam.
• Buy geometric solids with the footprint stamp on
  one end.

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Footprints

• 2. Have students use a stamp pad or a marker and
  a solid to make footprints on paper or draw
  around the solid. Footprints should be placed
  randomly with different orientations.
• 3. Have students draw lines to all footprints
  that are alike. Students should begin to realize
  that since the same foot made all the
  footprints, they are actually all the same.

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Footprints

• 4. Tell students to use 2 or more different
  solids to make footprints.
• Have students draw line connecting all footprints
  that are the same.
• Exercise may be repeated with various solids,
  several at a time.
• Students will begin to learn that geometric
  figures can be the same even though they are
  oriented differently.

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Connect all the figures like A with a red line.
Connect all the figures like B with a purple
line.
A
B
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Nets to Solids
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Do These Nets form Cubes?For each net, write
your prediction on the net. Cut out the net and
fold. Did it make a cube? Were your
predictions correct? Sort the nets into cubes
and not cubes. Write a generalization for
predicting which nets make cubes.
15
Test your generalization on the following
nets.Predict, then cut and fold.
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Do These Nets form Triangular Prisms?1. For
each net, write your prediction on the net. 2.
Cut out the net and fold. Did it make a
Triangular Prism? 3. Were your predictions
correct? 4. Sort the nets into Triangular
Prisms and not Triangular Prisms. 5. Write a
generalization for predicting which nets make
Triangular Prisms.
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Do These Nets Form Prisms?1. For each
net, write your prediction on the net. 2. Cut
out the net and fold. Did it make a Triangular
Prism? 3. Were your predictions correct? Sort
the nets into Prisms and not Prisms. 4. Write
a generalization for predicting which nets
make Prisms.
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Do These Nets form Pyramids?1. For each net,
write your prediction on the net. 2. Cut out
the net and fold. Did it make a pyramid? 3.
Were your predictions correct? Sort the nets
into pyramids and not pyramids. 4. Write a
generalization for predicting which nets make
pyramids.
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Number Cubes
3
20
Number Cubes
5
3
21
Number Cubes
5
6
3
22
Number Cubes
5
6
3
Now you try!
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1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
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1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
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1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
D
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1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
D
27
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
28
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
29
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
D
30
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
A
D
31
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
A
D
E
32
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
A
D
E
A
33
1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
A
D
E
A
B
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1. The net below is folded to form a cube.2.
Predict how the other 2 sides be labeled when
placed as shown?3. Be sure the letters are in
the correct orientation.
E
A
C
D
A
A
D
E
A
C
B
35
This activity demonstrates abstract
manipulations used on standardized testing (at
the 8th grade level in Tennessee).
E
A
C
D
A
A
D
E
A
C
D
B
36
Concrete Representations of Geometric Concepts
Thanks for your time and attention. If you have
questions, you may contact us through
email. Worksheets and PowerPoint presentation
are available on the web at www.apsu.edu/haralson
k
Dr. Nancy F. Matthews Montgomery Central Middle
School -- Cunningham, TN nancy.matthews_at_cmcss.net
Kay D. Haralson Austin Peay State University --
Clarksville, TN haralsonk_at_apsu.edu