The Geometer’s Sketchpad The van Hiele model of geometric thought outlines the hierarchy of levels through which students progress as they develop of geometric ideas. The model clarifies many of the shortcomings in traditional instruction and offers ways to improve it. Pierre van Hiele and his wife, Dina van Hiele-Geldof, focused on getting students to the appropriate level to be successful in high school Geometry. Dr. Mufid Abudiab Mrs. Marcia Venzon Mrs. Fatma Abudiab 11/15/2018 GEAR UP/STAR - Summer Math Institute

GEAR UP/STAR - Summer Math Institute Outline Introduction - Terms and Definitions - Geometer's Sketchpad (GSP) - Basic Skills in GSP Activities on Angles Formed by Parallel Lines and Transversal Types of Triangles Quadrilaterals Transformation Animation 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Geometer's Sketchpad software system for creating, exploring, and analyzing a wide range of mathematics. helps students explore topics from geometry and mathematical ideas in algebra, trigonometry, calculus, and other areas. provides teachers an environment with which to present mathematical concepts, model classroom questions, and encourage student conjecturing. Only through the use of a common language, can we develop the network of structures and relations which help produce good definitions. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Geometer's Sketchpad (cont.) help researchers and other mathematics enthusiasts pose “what if?” thought experiments, discover new results, and create high-quality mathematical illustrations for use in activities and assignments, reports and publications. Can be used to construct interactive mathematical models ranging from basic investigations about shape and number to advanced, animated illustrations of complex systems. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Geometer's Sketchpad (cont.) GSP allows students to create and manipulate figures that enables them to - Visualize and produce many examples - examine properties of figures - look for patterns, and - make conjectures. During the use of GSP Students experience the joy of discovery, the confidence that comes with success, the framework for a pattern of independent learning. The development of geometric ideas progresses through a hierarchy of levels. The research of Pierre van Hiele and his wife, Dina van Hiele-Geldof, clearly shows that students first learn to recognize whole shapes then to analyze the properties of a shape. Later they see relationships between the shapes and make simple deductions. Only after these levels have been attained can they create deductive proofs. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Geometer's Sketchpad (cont.) Sketchpad’s Menu Structure Document Tools Objects Object Relationships: parents and children Path Objects Points Segments Rays Lines The hierarchy for learning geometry described by the van Hieles parallels Piaget’s stages of cognitive development. One should note that the van Hiele model is based on instruction, whereas Piaget’s model is not. The van Hiele model supports Vygotsky’s notion of the “zone of proximal development” which is the “distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers.” (Vygotsky, 1978, p. 85-86) 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Geometer's Sketchpad (cont.) Sketchpad’s Menu Structure (cont.) Circles and Arcs Polygons and other Interiors Measurements, Calculations, and Parameters Coordinates Systems and Axes Functions and Function Plots Language at the Visual Level serves to make possible communication for the whole group about the structures that students observe. The vocabulary representing the figures helps in describing the figures. Any misconceptions identified may be clarified by the use of appropriate language. The language of the next level, e.g., congruence, will not be understood by students who are at the Visual Level. 11/15/2018 GEAR UP/STAR - Summer Math Institute

GEAR UP/STAR - Summer Math Institute Geometer's Sketchpad 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Basic Skills in GSP Constructing and Naming of Points, lines, segments, rays Angles Circles Midpoints Perpendicular lines parallel lines In this example, the term “rotation” is introduced. Basic Skills in GSP 11/15/2018 GEAR UP/STAR - Summer Math Institute

Introduction: Basic Skills in GSP Measuring - length of a segment - size of an angle - radius of a circle - circumference and area of a circle - dimensions, areas of other geometrical shapes like …. Trapezoid, kite, parallelogram, rhombus, rectangle, square 11/15/2018 GEAR UP/STAR - Summer Math Institute

Exploring Angles Formed by Parallel Lines and Transversal Activity 1: Exploring Angles Formed by Parallel Lines and Transversal Exploring Angles Formed by Parallel Lines and Transversal 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 2: More on Angles- Conjectures based on Exploration The congruent, parallel translation vectors become the defining property behind a translation. The symbols used to describe the vectors, parallel and congruent, both on the figure and in written geometric language, become a part of the formal language of the Descriptive Level. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 3: Exploring Types of Triangles The line of reflection bisects the parallel segments which connect corresponding points on the pre-image and image. This property describes and defines a reflection. The congruency symbols, the perpendicular symbol, and the corresponding symbols used in written descriptions, become part of the language of the Descriptive Level. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 4: Pythagorean Theorem- Visual Demonstration The language of the Relational Level is based on ordering arguments which may have their origins at the Descriptive Level. For example, a figure may be described by an exhaustive list of properties at the Descriptive Level. At the Relational Level it is possible to select one or two properties of the figure to determine whether these are sufficient to define the figure. The language is more abstract with its causal, logical and other relations of the structure. A student at the Relational Level is able to determine relationships among figures, and to arrange arguments in an order in which each statement except the first one is the outcome of previous statements. 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 5: Exploring Quadrilaterals 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 6: Transformation Transformations on the Coordinate Grid 11/15/2018 GEAR UP/STAR - Summer Math Institute

Activity 7: Constructions & Animation Constructing Tessellations by Translations Dueling Pinwheels 11/15/2018 GEAR UP/STAR - Summer Math Institute