1. CHAPTER 20 Geometric Thinking and Geometric Concepts
Elementary and Middle School Mathematics Teaching Developmentally
Ninth Edition
Van de Walle, Karp and Bay-Williams
Developed by E. Todd Brown /Professor Emeritus University of Louisville

2. Big Ideas
What makes shapes alike and different can be determined by geometric properties.
Transformations provide a significant way to think about the ways properties change or do not change when a shape is moved in a pane or space.
Shapes can be described in terms of their location in a plane or in space.
Three-dimensional shapes can be seen from different viewpoints which help us understand relationships between two- and three- dimensional figures and mentally change the position and size of shapes.

3. Geometry Goals for Students
Spatial sense- an intuition about shapes and the relationships between them
Familiarity with geometric descriptions of objects and position
Mentally visualize objects and spatial relationships
Appreciate geometric forms in art, nature, and architecture
Geometric content goals
Shapes and properties
Transformations
Locations
Visualization

4. The Development of Geometric Thinking
The van Hiele levels of geometric thought
Level 0: Visualization
objects of thought are shapes and what they look like
objects of thought are classes or groupings of shapes that can be alike
Level 1: Analysis
objects of thought are classes of shapes rather than individual shapes
properties of shapes

5. Characteristics of the van Hiele Levels

6. Van Hiele Level 0 Visualization
Recognize and name figures based on the global visual characteristics.
Sort and classify shapes based on their appearance.
Able to to see how shapes are alike and different to work toward classification.

7. Van Hiele Level 1 Analysis
Consider all shapes within a class.
Able to talk about the properties of all rectangles.
Focus on what makes a rectangle a rectangle.
If a shape belongs to a particular class it has the corresponding properties of that class.

8. Van Hiele Level 2 Informal Deduction
Think about properties of geometric objects without focusing on one particular object (shape).
Engage in “if then” thinking- if all four angles are right angles the shape must be a rectangle.
3. Can classify shapes with a minimum set of defining characteristics. 4. Observations go beyond properties and focus on logical arguments about the properties. 5. Inclusion of informal logical reasoning.

9. Implications for Instruction
Move from Level 1 to 2
Move from Level 0 to 1
Focus on properties of figure rather than on simple identification. Challenge student to test ideas about shapes using a variety of examples from a particular category. Provide ample opportunities to draw, build, make, put together, and take apart shapes in both 2 and 2 Dimension.
Challenge students to explore and test examples. Encourage the making and testing of hypotheses or conjectures. Examine properties of shapes to determine necessary and sufficient conditions for a shape to be a particular shape. Use the language of informal deduction Encourage students to attempt informal proofs.

10. Try this one Activity 20.5 What’s My Shape
Materials- glue double sets of 2-D shapes on card stock, glue one of each shape in a file folder to make “secret shape” folders- other shapes can be placed on table for reference Directions- Designate one student as the leader and they hold the “secret shape” folder Other students ask yes or no questions to find out the shape that matches the shape in the folder Group can eliminate shapes by turning over the shapes placed on the table for reference. (they cannot point to a shape and ask “is it this one?”)

11. Composing and Decomposing Shapes
Tangram Puzzles
Mosaic Puzzle
Geoboards

12. Categories of 2 Dimensional Shapes

13. Categories of 3 Dimensional Shapes

14. Applying Definitions and Categories Try this one Activity 20
Applying Definitions and Categories Try this one Activity Mystery Definition
Students develop ideas and definitions based on their own concept development. Students who struggle may need hints like angle size, congruent sides. Contrast student ideas with the conventional definition for that shape.

15. Investigations, Conjectures and Development of Proof
Activity True or False? Materials- set of true/false statements If it is a square, then it is a rhombus. All squares are rectangles. Some parallelograms are rectangles. All parallelograms have congruent diagonals.
If it has exactly two lines of symmetry, it must be a quadrilateral. If it is a cylinder, then it is a prism. All prisms have a plane of symmetry. All pyramids have square bases. If a prism has a plane of symmetry, then it is a right prism.

16. Investigations, Conjectures and Development of Proof cont.
Directions- Ask students to determine if the statements are true or false. Students have to prepare an argument to support their decision. Students can challenge their classmates.
Activity supports logical reasoning and not restricted to quadrilaterals.

17. The Pythagorean Relationship
Explored in 8th grade and warrants in-depth conceptual investigation.
The two large squares together are the proof without words. Can you supply the words?

18. Transformations
Line symmetry-reflectional or mirror symmetry All of these or none of these?
Explore line symmetry with dot grids.

19. Rigid Motions Translation requires a direction and a distance.
Reflection requires a line of reflection (object flipped across the line of reflection)
Rotation requires a center of rotation and a degree of rotation.

20. Using Transformations, Symmetries and Location

21. Visualization- “geometry done with the mind’s eye”
Two- dimensional Can you remember? Pentominoes
Three- dimensional Face Matching Building views