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Developing Geometric Thinking and Spatial Sense. Chapter Fourteen. Dodecagon. Pentagon. Nonagon. The van Hiele Levels of Geometric Thought. Connecting van Hiele Levels to Elementary School Children.
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Developing Geometric Thinking and Spatial Sense Chapter Fourteen Dodecagon Pentagon Nonagon
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 Where is the picture? Above, Below, Under, Between, Behind or After http://uk.ixl.com/math/year-1 • 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?
Learning about Topology • Network – decision points and paths One route to the centre is A -> B -> D -> K -> I -> M. • 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/
Learning about Topology http://britton.disted.camosun.bc.ca/mug_torus_morph.gif • Topology is the study of the properties of figures that stay the same even under distortions, except tearing or cutting. • Distortion of Figures
Learning about Euclidean Geometry curriculumsupport.education.nsw.gov.au • 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
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
Three-Dimensional Shapes Semi-regular Polyhedron of 62 Faces. http://www.literka.addr.com/hexsqr14.htm stellated polyhedra Semiregular polyhedra Truncated and 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? 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?
Learning about Two-Dimensional Figures • 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. Polygons – two-dimensional figures with straight line segments
Learning about Two-Dimensional Figures wikipedia.org • 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.
Triangles http://www.mathsisfun.com/triangle.html • Triangles can be classified by angles and sides • Sides • Equilateral – all sides equal • Isosceles – two sides equal • Scalene – no sides equal
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 Trapezoid
Learning about Symmetry http://britton.disted.camosun.bc.ca/sun.jpg 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.
Learning about Symmetry Magic-squares.net Feko.info 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.
Learning about Symmetry math.kendallhunt.com mathexpression.com Rotational symmetry – when a figure is rotated about a point for an amount less than 360°, and the rotated shape matches the original shape.
Rotational Symmetry of a Square Some three- and two-dimensional shapes and figures have rotational symmetry.
Transformation Geometry learningideasgradesk-8.blogspot.com • Translation – a movement along a straight line Slides Flips Turns
Transformation Geometry art.unt.edu intmath.com mathsisfun.com 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.
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
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
Task 3 • Using all seven tan pieces create these five basic geometric shapes. • Square • Trapezoid • Parallelogram • Rectangle • Triangle
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
