Visualization and Spatial Reasoning

Essential content for elementary teachers

Become familiar with projections, cross-sections, and decompositions of common two- and three-dimensional geometric shapes.3
Construct three-dimensional geometric shapes from two-dimensional shapes and use a two-dimensional representation for three-dimensional shapes.
Create and recognize a geometric shape from different perspectives (top, bottom, side).
Use appropriate software (such as the Geometer's Sketchpad®) for showing representations of geometric shapes.

Essential content for students grades K-3

Specify locations and describe spatial relationships using representational systems.4
1. Describe relative positions in space; interpret and apply ideas about relative position.
Use visualization and spatial reasoning to solve problems.5
1. Use spatial visualization to create mental images of geometric shapes.
2. Identify and draw geometric shapes from different perspectives.
3. Recognize geometric shapes and structures in their environment.
4. Build and draw geometric objects from mental images.

Essential content for students grades 4-5

Maintain and expand on concepts introduced in primary grades.
Specify locations and describe spatial relationships using representational systems.6
1. Use appropriate geometric vocabulary to describe location, movement, and relative position.
Use visualization and spatial reasoning to solve problems.7
1. Identify and build a three-dimensional object from two-dimensional representations.
2. Draw a two-dimensional representation of a three-dimensional object.
3. Recognize that geometric ideas and relationships can be applied to other disciplines and can be related to problems in everyday life.

See Appendix A for a list of sample state standards for mathematics.

Instructional activities and lesson plans for visualization and spatial reasoning

The activities and lesson plans listed here (by grade level) are described more extensively in Instructional Activities and Lesson Plans, where they are listed alphabetically.

Activity	Description
Creating Tilings grades K-5	Have students create tilings with pattern block pieces and draw their tilings on triangle dot paper. Have upper-level students discuss why certain shapes tile and others do not.
Draw a Shape from Memory grades K-5	Show students a shape for a few seconds and then have them try to draw the shape from memory. Start with simpler shapes and then draw more complex ones.
Geoboard Polygons grades 1-4	Have students duplicate shapes on a geoboard and then break them into smaller shapes. For an irregular shape, have students find as many triangles, squares, or rectangles within the shape as possible.
Creating Tessellations grades 1-5	Have students use index cards to create a shape that will tessellate the plane. Display their tessellations on a bulletin board. This activity can be connected to the art of M. C. Esher.
Origami grades 2-5	Use origami to explore spatial relations and to investigate shapes and their properties. Some of the concepts to explore are similarity, congruence, and classification of triangles. As students do the folding, introduce mathematical terms such as diagonal and midpoint.
Build a Shape grades 3-5	Have students use cubes to build a shape and then draw the shapes on graph paper, showing how they would look from the front, back, side, top, and bottom.
Creating Nets grades 4-5	Have students build shapes with wooden cubes then create nets to wrap the shapes. Use the nets to discuss surface area.
Creating Pentominoes grades 4-5	Have students work together, using square tiles, to find the 12 different pentomino shapes. Have them record results on graph paper or square dot paper. Have students use transformations to verify uniqueness of shapes.
Pentomino Boxes grades 4-5	Have students try to visualize which of the 12 pentomino pieces can be folded to make a box without a top. Have students make the pentomino pieces on one-inch square grid paper and fold to confirm their conjectures.

Lesson Plan	Description
Tiling the Plane grades 2-3	Students use pattern blocks and triangle paper to develop an understanding of tessellations while reviewing names of geometric shapes.
Rep Tiles grade 3	Students develop a deeper understanding of similarity and how perimeter changes as a result of increase in size. They create "rep tiles" using four pattern blocks ("rep-4 tiles").
Visualizing Multiplication grades 3-5	Students use an area model to do multiplication.
Rep Tiles grades 4-5	Similar to the grade 3 lesson above, with added element of how area changes as a result of an increase in size.
Investigating Nets and Polyhedra grade 5	Students explore nets for three-dimensional shapes.
Tiling the Plane grade 5	Similar to the grades 2-3 lesson above, and addressing more complex tessellations.

Conference Board of the Mathematical Sciences, The Mathematical Education of Teachers, part 1. (Washington, D.C.: Mathematical Association of America, 2000), p. 80.
National Council of Teachers of Mathematics, Principles and Standards for School Mathematics, (Reston, Va.: National Council of Teachers of Mathematics), p. 396
Ibid.
Ibid.
Ibid.