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
√ Chapter 5: Exploring Transformations and Tessellations
Principles and Standards recommendations on transformations
Introducing transformations through body motions
Line dancing and partner dancing
Paper Doll stick figures in GSP
Reflections of a triangle
Challenge: Reflections about two mirrors
Burning Tent Problem
Dilations by fixed ratio
Introduction to Dynamic Transformations
Dynamic Rotations, Dilations and Custom Transformations
Dynamic custom translations
Investigation of transformation isometries
Reflections across two intersecting lines (rotation)
Reflections across two parallel lines (translation)
Glide Reflection
Challenge: Find the Glide Reflection equivalent to three reflections about
arbitrary, non-parallel lines
From Transformations to Tessellations
Tessellations using any triangle, any quadrilateral
Using vector translation, reflections and rotations to tessellate the plane
Translation tessellation based on a parallelogram with altered sides
Rotation tessellation based on any triangle (two sides have point-symmetry)
Tessellations based on the techniques developed by M. C. Escher
Reflecting on connections between art and mathematics
Part II: Going beyond geometry
√ Chapter 6: Number and Operations
Historical Development of Some Computing Tools (slide rule)
Constructing a product of two numbers on a number line using dilation
(connecting with transformational geometry)
Representing arithmetic operations dynamically on a number line
Investigating arithmetic relations dynamically: Mystery Machines
Investigating number patterns, factors and multiples (DYGL sketches)
Some explorations with discreet mathematics
√ Chapter 7: Functions
Dynamic algebraic functions using transformations
Constructing a linear function through dilation and translation
Constructing powers of x through dilation
Comparing Functions on parallel number lines
Dynagraphs: Input-output model for functions using parallel number lines
Composing functions using a sequence of Dynagraphs
From Dynagraphs to coordinate axes
Using multiple coordinate systems to investigate operations on functions
Iterative functions
(√) Chapter 8: Trigonometry of the unit circle with GSP
Using the GSP coordinate system to define a unit circle
Constructing a right triangle in the unit circle with radius as hypotenuse
Defining the trigonometric ratios using this right triangle
Angle measure as arc-length: unwrapping the circle along the x-axis
Wrapping the x-axis around the circle: extending the domain for angle
measurement
Connecting linear and circular motions: the locus of the sine function
Constructing similar right triangles in the unit circle to generate loci for all of the
trigonometric functions
What determines the period of each function? What determines the range?
Reflecting on the pedagogical implications of these explorations
(√) Chapter 9: Data Representations and Mathematical Modeling
Generating data through GSP simulations
The Box problem -- linking GSP simulation to measurements and graphing
The Cardboard Tube Telescope
Fitting functions to data
Line of best fit and least squared differences
Linearizing quadratic data
Exponential Growth
Linearizing exponential data
Modeling a basketball throw
√ Chapter 10: The Dynamic Geometry of Calculus
Secant lines and Tangents to functions
Investigating the relation between secant and tangent for quadratic functions
Does the relation hold for higher-order functions?
Tangents to sine and cosine functions
Area under a function curve
Lower Riemann sum
Upper Riemann sum
Mid Riemann sum
Trapezoidal sum
Varying the parameter of the x-term for area under a quadratic
Lower bound = -Upper bound (see Quadratic Integral.2 sketch)
Reflecting on the connections between geometry and calculus
Part III: Extensions into advanced mathematical topics
(√) Chapter 11: Investigations in the Complex Plane
Modeling complex numbers with polar coordinates: z = r(cos ø + i.sin ø)
Constructing the product of two complex numbers
Using the product construction to verify De Moirve’s Theorem for zn
Constructing zn using similar triangles and iteration
Using the construction of zn to find an nth root of a complex number
Constructing all n nth roots of a complex number
Reflecting on the connections between geometry and complex analysis
(√) Chapter 12: Inversion in a Circle
Folding everything outside the circle into the circle (and visa-versa): the circle
inversion transformation
All lines transform to circles but not all circles transform to lines!
Investigating inversion of polygonal figures
An investigation from antiquity: constructing circles in the Arbelos of Archimedes
through inversion
An investigation from modern times: creating airfoils from circles – the Joukowski
Transformation
Reflecting on the pedagogical implications of these explorations
Chapter 13: Conic Sections
Modeling the full cone
Loci definitions of conics
Paper Folding to generate the envelope of each conic function
Simulating paper folding with GSP
Generating all conics through simulated folding with an inverted circle
Conics through 5 points
The eccentricity of conics
Conics and Optics
Conics and planetary motions
Reflecting on the pedagogical implications of these explorations
Chapter 14: Recursion and Self-Similarity
Building Fractals through Recursive Transformations
Koch Snowflake Curve
Sierpinski Gasket
Fractal trees
From chaotic growth come Golden Trees
Reflecting on connections between geometry and nature
Chapter 15: Putting it all together: Designing an Integrated Instructional Unit
Dynamic Geometry as a catalyst for integrating the study of mathematics
Reflect back on the investigations in the second half of this book: you were
doing geometry, algebra, trigonometry, calculus and even some complex
analysis.
What are the implications for the high-school mathematics curriculum?
What are the implications for teaching mathematics?
What are the implications for assessing students’ learning?
Final group project: Design your own integrated teaching unit using GSP
Design an investigation that integrates at least two areas of mathematics
Plan the investigation to cover 3 or 4 class periods
Describe the mathematics involved in your investigation
Describe how you would assess your students’ investigations
A Possible Bibliography
Exploring Algebra with The Geometer’s Sketchpad, D. Bennett (1999). Key Curriculum
Press, Emeryville, CA.
Exploring Geometry with The Geometer’s Sketchpad, S. Chanan, E. Bergofsky & D.
Bennett (2002). Key Curriculum Press, Emeryville, CA.
Exploring Calculus with The Geometer’s Sketchpad, C. Clements, R. Pantozzi & S.
Steketee (2003). Key Curriculum Press, Emeryville, CA.
Exploring Conic Sections with The Geometer’s Sketchpad, D. Scher (2003). Key
Curriculum Press, Emeryville, CA.
Build-a-Book Geometry, C. C. Healy (1993). Key Curriculum Press, Berkeley, CA.
Construction and Investigation of Golden Trapezoids, N. Oppong (1997). Mathematics
and Computer Education, 31(3), 230-236.
Discovering Geometry, (Third Edition) M. Serra (2004). Key Curriculum Press,
Berkeley, CA.
Geometry Turned On, J. King & D. Schattsschneider (Eds.) (1997). MAA.
Principles and Standards for School Mathematics, NCTM (2000).
Pythagoras Plugged In, D. Bennett (1995). Key Curriculum Press, Berkeley, CA.
Dynagraph, E. P. Goldenberg (1990). Draft paper. Education Development Center,
Newton MA.
Making Connections with Geometry, E. P. Goldenberg, A. A. Cuoco & J. Mark (1992).
Education Development Center, Newton MA.
Reconnecting Geometry: A Role for Technology, A. A. Cuoco & E. P. Goldenberg
(1992). Education Development Center, Newton MA.
Rethinking Proof with the Geometer’s Sketchpad, M. de Villiers (1999). Key Curriculum
Press, Berkeley, CA.
Parameter Effects and Solving Linear Equations in Dynamic, Linked, Multiple
Representation Environments. Lin, P-P. and Hsieh, C-J. (1993). The Mathematics
Educator, 4, 1, 25-33.
Abstracts of papers from the International Working Group on the Role of Geometry in
General Education. Seventh International Congress on Mathematics Education
(ICME-7), Quebec, Canada.
Implications of Using Dynamic Geometry Technology for Teaching and Learning. J.
Olive (2000). Plenary paper in Saraiva, M., Matos, J., Coelho, I. (Eds.) Ensino e
Aprendizagem de Geometria. Lisboa: SPCE.
Technology and School Mathematics, J. Olive (1992). Reform of School Mathematics
in the United States, special issue of the International Journal for Educational
Research, Chapter 7.
Visions of Symmetry, Works of M. C. Escher, D. Schattschneider (1993). Dale
Seymour Publications, Palo Alto, CA.
Fractals Everywhere, M. Barnsley (1988). Academic Press, Inc., Boston, MA.
The Fractal Geometry of Nature, B. B. Mandelbrot (1983). W. H. Freeman & Co., New
York, NY.