1. Developing Geometric Thinking and Spatial Sense
Chapter Fourteen
Dodecagon
Pentagon
Nonagon

2. The van Hiele Levels of Geometric Thought
Description
0 – Visualization
Children recognize shapes by their global, holistic appearance.
1 – Analysis
Children observe the component parts of figures but are unable to explain the relationships between properties within a shape or among shapes.
2 – Informal Deduction
Children deduce properties of figures and express interrelationships both within and between figures.
3 – Formal Deduction
Children create formal deductive proofs.
4 – Rigor
Children rigorously compare different axiomatic systems.

3. Connecting van Hiele Levels to Elementary School Children
Most children at the elementary level are at the visualization or analysis level of thought.
Some middle school children are at the informal deduction.
Students who successfully complete a typical high school geometry course reach the formal deduction level.
The goal is to have children at the informal deduction level or above by the end of middle school.

4. Comments on the Levels of Thought
Levels are not age dependent, but are related to the experiences a child has had.
Levels are sequential.
Experience is key in helping children move from one level to the next.
For learning to take place, language must match the child’s level of understanding.
It is difficult for two people who are at different levels to communicate effectively.

5. Learning about Topology
Topology is the study of the properties of figures that stay the same even under distortions, except tearing or cutting.
Place and Order – Describing where something is located in the environment or in pictures. Focus on the following types of words
Is the pillow Inside or Outside the box?
Where is the picture?
Above, Below, Under,
Between, Behind or
After

6. Learning about Topology
Topology is the study of the properties of figures that stay the same even under distortions, except tearing or cutting.
Maze
Network – decision points and paths
One route to the centre is A -> B -> D -> K -> I -> M.

7. Learning about Topology
Topology is the study of the properties of figures that stay the same even under distortions, except tearing or cutting.
Distortion of Figures

8. Learning about Euclidean Geometry
Three-Dimensional Shapes
Polyhedra – three-dimensional shapes with faces consisting of polygons, that is, plane figures with three, four, five, or more straight sides.
Edge
Vertices
Face
curriculumsupport.education.nsw.gov.au

9. Learning about Euclidean Geometry
Three-Dimensional Shapes
Regular polyhedra – a regular polyhedron is one whose faces consist of the same kind of regular congruent polygons with the same number of edges meeting at each vertex of the figure.

10. Three-Dimensional Shapes
There are five regular polyhedra
Shape
Type and Number of Faces
Tetrahedron
4 equilateral triangles
Octahedron
8 equilateral triangles
Icosahedron
20 equilateral triangles
Hexahedron
6 squares
Dodecahedron
12 regular pentagons

11. Three-Dimensional Shapes
Semiregular polyhedra
Truncated and stellated polyhedra
Semi-regular Polyhedron of 62 Faces.
stellated polyhedra

12. Other Three-Dimensional Shapes
Discovering Euler’s Rule: Examining relationships between faces, vertices, and edges What relationship do you notice among the shapes?
Polyhedron
Faces
Vertices
Edges
Tetrahedron 4 6
Cube 8 12
Octahedron
wikipedia.org

13. Other Three-Dimensional Shapes
Discovering Euler’s Rule: Examining relationships between faces, vertices, and edges What relationship do you notice among the shapes?
Polyhedron
Faces
Vertices
Edges
Triangular Prism 5 6 9
Square Pyramid 8

14. Learning about Two-Dimensional Figures
Polygons – two-dimensional figures with straight line segments
Convex – interior angles are all less than 180°; any two points in a figure can be connected by a line segment that will be completely within the figure and all diagonals will remain inside the figure.

15. Learning about Two-Dimensional Figures
Polygons – two-dimensional figures with straight line segments
Concave – a geometric shape is concave if it has any line segment that joins two interior points outside the figure.
wikipedia.org

16. Convex Polygons Number of Sides Name of Polygon 3 Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon

17. Convex Polygons Number of Sides Name of Polygon 8 Octagon 9 Nonagon 10
Decagon 11
Hendecagon 12
Dodecagon

18. Triangles Triangles can be classified by angles and sides Sides
Equilateral – all sides equal
Isosceles – two sides equal
Scalene – no sides equal

19. Triangles Triangles can be classified by angles and sides Angles
Right – one angle is equal to 90°
Acute - all angles are less than 90°
Obtuse – one angle is greater than 90°

20. Quadrilaterals Quadrilateral Definition Parallelogram
A quadrilateral with two pairs of parallel sides
Rectangle
A parallelogram with 90-degree angles
Square
A rectangle with equal sides

21. Quadrilaterals Quadrilateral Definition Trapezoid
A quadrilateral with at least one pair of parallel sides
Isosceles Trapezoid
A trapezoid with two nonparallel sides
Kite
A quadrilateral with two nonparallel sides equal
Rhombus
A parallelogram with all sides equal
Trapezoid

22. Learning about Symmetry
Symmetry – when a figure is bisected into two congruent parts, every point on one side of the bisection line will have a reflective point on the other side of the bisection line.

23. Learning about Symmetry
Plane symmetry – a three-dimensional shape has plane symmetry if a plane passing through the figure bisects it such that every point of the figure on one side of the plane has a reflection image on the other side of the plane.
Magic-squares.net
Feko.info

24. Learning about Symmetry
Rotational symmetry – when a figure is rotated about a point for an amount less than 360°, and the rotated shape matches the original shape.
math.kendallhunt.com
mathexpression.com

25. Rotational Symmetry of a Square
Some three- and two-dimensional shapes and figures have rotational symmetry.

26. Transformation Geometry
Translation – a movement along a straight line
Slides Flips
Turns
learningideasgradesk-8.blogspot.com

27. Transformation Geometry
Reflections – the movement of a figure about a line outside the figure, on a side of the figure, or intersecting with a vertex
Rotation – the movement of a figure around a point.
art.unt.edu
intmath.com
mathsisfun.com

28. Developing Spatial Sense
Spatial sense involves both visualization and orientation factors
Spatial visualization – the ability to mentally picture how objects appear under some rigid motion or other transformation
Orientation – the ability to note positions of objects under different orientations.

29. Task 1
Using the three small pieces (two small triangles and the medium size triangle) create these five basic geometric shapes.
Square
Trapezoid
Parallelogram
Rectangle
Triangle

30. Triangle

31. Rectangle

32. Trapezoid

33. Parallelogram

34. Square

35. Explanation Task 1 Linear Relationships
The hypotenuse of the small triangle is congruent to the leg of the medium size triangle.
The hypotenuse of the medium sized triangle is congruent to twice the length of the leg of the small triangle.
The two small triangles are congruent because:
The legs of both triangles are congruent.
The hypotenuse of both triangles are congruent.
The angles of both triangles are congruent.

36. Task 2
Using the five small pieces (two small triangles, medium size triangle, rhombus, parallelogram) create these five basic geometric shapes.
Square
Trapezoid
Parallelogram
Rectangle
Triangle

37. Rectangle

38. Trapazoid

39. Paralellogram

40. Triangle

41. Square

42. Task 3
Using all seven tan pieces create these five basic geometric shapes.
Square
Trapezoid
Parallelogram
Rectangle
Triangle

43. Rectangle

44. Parallelogram

45. Trapezoid

46. Triangle

47. Square

48. Connecting the Tasks Working with Three Small Pieces
Identifying Linear Relationships
Examining Transformations
Working with Five Small Pieces
Application of Linear Relationship Identification
Strengthening Language Descriptions of Transformations
Working with Seven Pieces
Similar Task to Three Small Pieces
Introduce concept of Ratio and Proportion
4. Examining Area is Another Lesson