Introduction to Geometer's Sketchpad (GSP)
Introduction
Geometer's Sketchpad (GSP) is a new computer program that was developed in the last five years. The program allows students to create and explore through geometric proofs. New developments and updates to the program are also allowing calculus and algebra teachers to integrate this technology into the classroom.
There are many educational implications for integrating GSP into a mathematics classroom. As discussed in Olive (2002), "The traditional approach to geometry instruction based on definitions, theorems, and their proofs makes little sense in an environment in which real-time phenomena can be explored. An inductive approach based on experimentation, observation, data recording, and conjecturing, in which (as deVilliers proposes in this volume, chap. 15) proof takes on the role of explanation rather than verification, would be much more appropriate." A constructivist teacher may agree that this is a better way of teaching geometry because students make connections and relationships on their own. Geometer's Sketchpad also allows students to work collaboratively by sharing their own individual ideas and findings with peers and their teachers.
Unit Objectives
- Demonstrate an understanding of the capabilities of GSP in a mathematics classroom
- Demonstrate the ability to create points, lines, and circles in GSP
- Demonstrate the ability to create an equilateral triangle in GSP
- Demonstrate the ability to create a square in GSP
- Demonstrate an understanding of possible misconceptions students may have using GSP
Lesson Activities
Introduction to GSP
1. Open the GSP Program (for information on obtaining a trial visit: Integrating Technology in Mathematics)
2. Notice below what each button at the left of the screen will do:
- The first row indicates three versions of arrows. The first is used to click and select a point or line and move it in any direction. The second arrow is used only to rotate points and lines. The third arrow is used only to move points together or further away to create shorter or longer lines.
- The second row is a button that is used to create points.
- The third row is a button that is used to create circles.
- The fourth row indicates three versions of lines. The first is used to create a line segment, the second is used to create a ray and the third is used to create a line.
- The fifth row is a button that is used to insert text.
- The last is used to create tools. This is a more advanced topic and allows buttons to be created as well as java scripts.
3. Using these buttons and tools insert circles, lines, and points.
Notice that if you want to connect two lines when you put the arrow over one endpoint it is highlighted blue. This is how you know they are connected and not near each other.
Notice that if you want to put a point ON a line the line will be highlighted blue. This is how you know that the point will be on the line and not near it.
REMEMBER: LEARN THROUGH EXPLORATION. TRY NEW THINGS AND EXPLORE THE TOOLS THAT THE PROGRAM GIVES YOU BEFORE YOU MOVE ON!
Creating an equilateral triangle
Now that you have learned how to create points, lines, and circles you are ready for a more difficult construction: an equilateral triangle. Before using a step-by-step walkthrough think about how to develop a square. Think about what it means for a shape to be an equilateral triangle. How many sides does it have? What do we know about the angles and the side lengths?
How do you know if you actually made an equilateral triangle? If you select an arrow and attempt to move any point, line or circle the shape will remain an equilateral triangle.
Once you have explored the program and attempted to create a square on your own. Click below for a step-by-step walkthrough.
Now, think about what discussions can happen as a result of students making this construction.
1) How do we know that the sides are equal length? (Think about the definition of radii of a circle)
2) How do we know that the angles are equal? (Think about the fact that equal sides relate to equal angles)
Creating a square
Lets continue with more complicated constructions and develop a square. Think about what it means for a shape to be a square. How many sides does it have? What do we know about the angles and the side lengths?
Notice: You will need to use extra tools provided under "CONSTRUCT" in the toolbar.
How do you know if you actually made a square? If you select an arrow and attempt to move any point, line or circle the shape will remain a square.
Once you have explored the program and attempted to create a square on your own. Click below for a step-by-step walkthrough.
Now, think about what discussions can happen as a result of students making this construction.
1) How do we know that the sides are equal length? (Think about the definition of radii of a circle)
2) How do we know that the angles are 90 degrees? (Think about what it means to be perpendicular)
Summary
Notice that if a teacher has students develop these constructions, either by exploration or step-by-step tutorials, they are able to connect these shapes to a deeper understanding of geometry. For example, when constructing a square the teacher can bring up topics such as what type of angles perpendicular angles make. It is these connections that constructing, rather than drawing, that programs such as Geometer's Sketchpad assist students in making.
References:
Olive, J. (1998). Opportunities to explore and integrate mathematics with the Geometer’s Sketchpad. In R. Lehrer & D. Chazan (Eds.) Designing Learning Environments for Developing Understanding of Geometry and Space, 395418. Hillsdale, NJ: Lawrence Erlbaum Associates.
Back to: Mathematical Exploration with Geometer's Sketchpad (GSP)