Developing Geometric Thinking and Spatial Sense Chapter Fourteen Dodecagon Pentagon Nonagon

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.

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.

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.

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 http://uk.ixl.com/math/year-1

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

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 http://britton.disted.camosun.bc.ca/mug_torus_morph.gif

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

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.

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

Three-Dimensional Shapes Semiregular polyhedra Truncated and stellated polyhedra Semi-regular Polyhedron of 62 Faces. http://www.literka.addr.com/hexsqr14.htm stellated polyhedra

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

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

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.

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

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

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

Triangles Triangles can be classified by angles and sides Sides Equilateral – all sides equal Isosceles – two sides equal Scalene – no sides equal http://www.mathsisfun.com/triangle.html

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° http:// http://www.mathsisfun.com/triangle.html

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

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

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. http://britton.disted.camosun.bc.ca/sun.jpg

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

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

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

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

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

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.

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

Triangle

Rectangle

Trapezoid

Parallelogram

Square

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.

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

Rectangle

Trapazoid

Paralellogram

Triangle

Square

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

Rectangle

Parallelogram

Trapezoid

Triangle

Square

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