Transforming Mathematics with the Geometer s Sketchpad A GSP Companion for Mathematics Education Dr John Olive and Dr Nicholas Oppong Department of Mathematics and Science Education University of Georgia revised 02 28 08 The NCTM Principles and Standards call for an approach to teaching mathematics that emphasizes student explorations and conjecturing and makes use of modern technological tools The Geometer s Sketchpad has been heralded as the most powerful educational software available for geometry learning This proposed GSP companion for mathematics education courses for pre service and in service middle and high school teachers extends the capabilities of GSP beyond geometry It makes full use of the most powerful aspects of the Sketchpad and incorporates the recommendations of the Principles and Standards for teaching Mathematics Through their experiences with the activities in this manual and the selected readings students will have an opportunity to question and transform the current teaching of mathematics make connections within and outside of mathematics and learn to make effective use of the most up to date technology Book Outline Checked items indicate a draft manuscript has been completed and sent A check in parentheses indicates that this chapter is in preparation but not yet completed Preface and Acknowledgements Introduction to Mathematics using Dynamic Geometry software Recommendations for using dynamic geometry software from the NCTM Principles and Standards for teaching math in grades 7 12 Our View of Mathematics and Learning What does it mean to learn mathematics How this companion is organized and suggestions for use with different courses Part I Starting with Euclidean Geometry familiar territory for GSP users Part II Going beyond geometry explorations with number data algebra trigonometry and calculus Part III Extensions into advanced mathematical topics the complex plane inversion in a circle conic sections and fractal geometry To the Student Activity 0 1 Assignment 0 1 Reflection Question Why Use Dynamic Geometry Software Van Hiele levels of thought in geometry Activity 0 2 Assignment 0 2 Going beyond geometry Using GSP 4 in other mathematical domains Implications for Teaching and Learning Mathematics in School Part I Starting with Euclidean Geometry Chapter 1 From Euclidean Tools to Dynamic Construction Tools Euclid s construction tools Compass and straightedge Duplicating a segment using compass and straight edge Using compass and straightedge to carry out the basic constructions from the GSP Construct menu Midpoint of a segment perpendicular to a line through a point a line parallel to a given line through a given point an angle bisector Euclid s construction of an equilateral triangle Duplicating an angle using compass and straight edge The Geometer s Sketchpad Construction Tools Using the GSP free hand tools to carry out all of the Euclidean constructions Euclid s construction of an equilateral triangle using the freehand tools Duplicating a segment using Circle by Center and Radius from the GSP Construction menu Duplicating an angle in GSP Using the Construction Tools to Construct Different Triangles Given three free points create different kinds of triangles Given two free points which special triangles can you construct Duplicating a triangle SAS SSS ASA SSA Reflecting on Euclidean and Sketchpad construction tools Chapter 2 Exploring Quadrilaterals Given four free points create different kinds of quadrilaterals Given three free points which special quadrilaterals can you construct Given two free points which special quadrilaterals can you construct Starting with the diagonals of a quadrilateral Given two segments that intersect construct the quadrilateral for which these segments are the diagonals investigate the relations between the diagonals that create each of the special quadrilaterals Investigating midpoint quadrilaterals What kind of quadrilateral The ratio of the areas of the midpoint quadrilateral and its parent quadrilateral Classifying Quadrilaterals Quadrilaterals on a circle another class of quadrilaterals Cyclic quadrilaterals Quadrilaterals with an inscribed circle Investigating the symmetry of special quadrilaterals Constructing similar quadrilaterals Using parallel lines to construct similar quadrilaterals Does it work for any other polygon Using projection lines to construct similar polygons Constructing congruent Quadrilaterals Using triangles to duplicate quadrilaterals and other polygons Construction Problems for Special Quadrilaterals Investigating Golden Quadrilaterals The Golden Rectangle The Golden Parallelogram Golden Trapezoids Reflecting on the pedagogical implications of these dynamic explorations Chapter 3 Exploring Centers Balance Points and Loci Exploring centers of triangles When three lines meet in a single point Constructing the Incenter Constructing the Circumcenter Constructing the Orthocenter Constructing the Centroid Relations among the centers constructing the Euler line Balancing cardboard triangles and quadrilaterals Using coordinates to find the balance point of a triangle Do coordinates work for the balance point of a quadrilateral Challenge Construct the balance point of any convex pentagon and any convex polygon The Power Plant Problem another center of a triangle Locus of the Orthocenter of a triangle Challenge Find the Focus and Directrix of the orthocenter parabola Parabolas from paper folding Paper folding and other conics Reflecting on the interplay of technologies GSP balancing paper folding Reflecting on the pedagogical implications of these explorations Chapter 4 Investigating the Pythagorean Theorem An introduction to the Pythagorean Theorem Creating a script tool for a square Constructing squares on the sides of any triangle Measuring the areas of the squares Calculating the sum of the areas of the two smaller squares Varying the triangle and making it right Repeating the investigation with equilateral triangles on the sides of the right triangle Repeating the investigation with regular pentagons on the sides of the right triangle Repeating the investigation with semi circles on the sides of the right triangle Generalizing the theorem Two Dissection Proofs of the PythagoreanTheorem The simplest dissection Start with a square of side length a b Euclid s Proof Further reading and explorations Pythagoras Plugged In by Dan Bennett Reflecting on the pedagogical implications of these explorations