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The Geometer's Sketchpad Activities (version 4)

Exploring Characteristics of Triangles

Students draw triangles and explore the classification of each using the measures of interior angles and lengths of sides. In the extension activities, the students further explore and generalize a formula for the sum of the interior angles of convex polygons.


Constructions of Isosceles and Equilateral Triangles

Students discuss the difference between drawing and constructing a triangle using The Geometer's Sketchpad. They will further develop constructions for both isosceles and equilateral triangles. In the extension activities, students will script the construction of a right triangle.


The Pythagorean Theorem

Students create both a visual and formal proof of the Pythagorean theorem, as well as view four additional geometric demonstrations of the theorem. These demonstrations are based on proofs by Bhaskara, Leonardo da Vinci, and Euclid. Students will value this activity more if they have already had some experience with the Pythagorean Theorem.


Exploring Centers of a Triangle: Part 1

Using problem solving skills, formal geometry, and The Geometerís Sketchpad, students will model and solve two real-world problems by constructing the incenter and circumcenter of triangles. Students will further explore relationships between the angle and side bisectors of arbitrary triangles. The author wishes to acknowledge Dr. Billie F. Risacher for providing the real-world scenarios.


Exploring Centers of a Triangle: Part 2

Using problem solving skills, formal geometry, and The Geometerís Sketchpad, students will model and solve a real-world problem by constructing the centroid of a triangle. Students will explore further relationships of the medians of a triangle and investigate how the centroid partitions each of the medians.


Determining the Uniqueness of a Circle

Are there a certain number of arbitrary points, such that there exists a unique circle through them? To explore and conjecture an answer to this question, students will construct circles passing through various numbers of given points. This activity was adapted from an activity developed by Sean Jones, a fourth-year pre-service teacher at the University of Virginia in Spring 1998.


Exploring and Creating Tessellations

Students explore and construct several different types of tessellations. The first part of the activity uses only regular polygons to create pure and semi-pure tessellations; therefore, you will need pre-recorded scripts for the construction of an equilateral triangle, a square, and a regular hexagon. The second part of the activity extends the notion of tessellations to create non-regular polygon and "Escher-like" tessellations.


Exploring Geometric Constructions of Parabolas

This activity is an introduction to geometric constructions of parabolas. Students investigate the properties and characteristics of parabolas. This activity has been adapted from the following article: Olmstead, E.A. (1998). Exploring the locus definition of the conic sections. Mathematics Teacher, 91(5), 428-434.


Exploring the Witch of Agnesi

Students construct the graph of the Witch of Agnesi, and investigate both its asymptotes and inflection points. Fermat studied this function in the seventeenth century. Students are introduced to the animate features of The Geometer's Sketchpad.


Exploring Infinite Series through Baravelle Spirals

This activity is an introduction to the concept of convergent infinite series using a recursive geometric construction. This activity has been adapted from the following article: Choppin, J. M. (1994). Spiral through recursion. Mathematics Teacher, 87(7), 504-508.


Exploring the Golden Rectangle

Students will construct a golden section and a golden rectangle and study their characteristics and connections with the Fibonacci series. They will import pictures from the internet and download them into the Sketchpad. Using their "golden" construction, they will discover the use of the golden rectangle in famous works of art. This exploration was adapted from an activity developed by Elizabeth Boiardi while she was a graduate student at the University of Virginia in the summer of 1998.


Exploring Trigonometric Functions

In the sixteenth century, right triangles were used to define the trigonometric functions that we are familiar with today. We will use a modified right triangle approach to define the trigonometric functions by placing one of the acute angles of a right triangle on a coordinate plane. Students geometrically construct and investigate the graphs of sine and cosine based upon the lengths of the sides of a reference triangle.

 

From Secant Lines to Tangent Lines

This activity demonstrates one of the many ways Sketchpad can be used in a calculus or math analysis class. Here, students explore the relationship between the slopes of secant lines to a curve and the slope of a tangent line to the curve using Sketchpad's dynamic features. They investigate the limit concept and express the relationship between the slopes of secant lines and slope of tangent lines as a limit statement.

 

Slopes and Derivatives

This activity demonstrates one of the many ways Sketchpad can be used in a calculus or math analysis class. Students manipulate a tangent line to a curve to investigate what it means for a curve to have slope. They explore the first derivative as a function which determines the slope of the curve for different values of x and express the derivative as a limit statement.

 

Graphing f '(x): Increasing, Decreasing, Concavity

This activity is designed to follow the activities From Secant Lines to Tangent Lines and Slopes and Derivatives, but it can easily stand on its own. Using a dynamic tangent line and the value of the tangent line's slope, students analyze the graph of a function, investigating relationships between the function's first derivative, absolute and local extrema, intervals of increasing and decreasing, concavity, and inflection points. As the lesson progresses, students create and refine a sketch of f '(x).

 

Sketches of the Formal Definition of the Limit of a Sequence

These sketches are designed to help students graphically visualize the concepts behind the formal definition of the limit of a sequence. Given a value for epsilon, students can manipulate N to find a value for N beyond which all further terms of the sequence lie within the distance epsilon from the limit. Various sequences are available for investigation such as a monotonically decreasing sequence, a damped oscillating sequence, a divergent oscillating sequence, and a constant sequence.

 

Sketches of the Formal Definition of the Limit of a Function

These sketches are designed to help students graphically visualize the concepts behind the formal definition of the limit of a function. Given a value for epsilon, students can manipulate delta to find a value for delta such that all x-values lying within distance delta from c (except x = c) correspond to f(x)-values lying within the distance epilson from the limit. Various functions are available for investigation such as a concave up quadratic function, a concave down quadratic function, a function with a removable discontinuity, a function with a removable jump discontinuity, and a constant function.

 

 

 

 

 

 

 

 

 

 


Send comments or questions to Beth Cory at blc4j@virginia.edu.

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Last modified on March 9, 2005.