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The Geometer's Sketchpad Activities (version 4)
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.
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.
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.
Using problem solving skills, formal geometry, and The Geometer’s
Sketchpad, students will model and solve two realworld 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 realworld scenarios.
Using problem solving skills, formal geometry, and The Geometer’s
Sketchpad, students will model and solve a realworld 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.
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 fourthyear preservice
teacher at the University of Virginia in Spring 1998.
Students explore and construct several different types of tessellations.
The first part of the activity uses only regular polygons to create
pure and semipure tessellations; therefore, you will need prerecorded
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 nonregular polygon and
"Escherlike" tessellations.
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), 428434.
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.
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), 504508.
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.
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.
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.
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.
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).
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.
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 xvalues 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.
