CHAPTER 20 Geometric Thinking and Geometric Concepts

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 0 to 1 Move from Level 1 to 2 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. • 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.

10. Try this oneActivity 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 CategoriesTry this oneActivity 20.10 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 20.15 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 Explore line symmetry with dot grids. • Line symmetry-reflectional or mirror symmetry • All of these or none of these?

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