Concrete Representations of Geometric Concepts

Concrete Representationsof 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@cmcss.net Kay D. Haralson Austin Peay State University -- Clarksville, TN haralsonk@apsu.edu

Which Came First?Solids or Plane Figures Some ideas developed from the article: Dear Verity, Now I’m Getting Into Shape! by Deirdre Dempsey and John Marshall. Phi Delta Kappan, April 2002.

Dear Verity, Now I’m Getting Into Shape! • Francis W. Parker (mid 1980s) • Concept of structured play • Too much emphasis on drill and memorization • Encouraged teaching by using children’s natural curiosity about the world • Suggested learning arithmetic by studying objects and their properties

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?

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

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

Solidsto Plane Figures

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 features—corners and faces. • Give some examples of objects in the room. • Have students name examples from home. • Repeat with other 3-D shapes.

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.

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.

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.

Connect all the figures like A with a red line. Connect all the figures like B with a purple line. A B

Nets to Solids

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.

Test your generalization on the following nets.Predict, then cut and fold.

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.

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.

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.

Number Cubes 3

Number Cubes 5 3

Number Cubes 5 6 3

Number Cubes 5 6 3 Now you try!

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.

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

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

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

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

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

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

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

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

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

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

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

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

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/haralsonk Dr. Nancy F. Matthews Montgomery Central Middle School -- Cunningham, TN nancy.matthews@cmcss.net Kay D. Haralson Austin Peay State University -- Clarksville, TN haralsonk@apsu.edu

Concrete Representations of Geometric Concepts