Wave Projects - Mathematica Lessons for the
Classroom
|
Compiled by
Carol Castellon and Debra Woods
To learn how to create your own Mathematica projects like these,
visit:
Math Teacher Link - Professional
Development Courses for Teachers
- 3Dimensions.nb
By Roberta Jones
This
lesson introduces the idea of the relationship between a system of
three equation with three variables and its graph in three-space.
- Changing
Functions By Brian Zecher
This lesson illustrates three types of function
transformations: reflections, translations and dilations. By
looking at the graphs of several functions before and after certain
variables are added, the student will be able to see how the graphs are
affected by the changes. This should help them graph various
functions in the future.
- DWPlanes(94)
By Debra Woods
This project shows the three possibilities for the
intersections of three planes by drawing the planes, and animating a
rotation to show several views.
- InverseFunctions
By Eric Hofelich
This module has been created for students enrolled
in College Algebra that are seeking further insight of Inverse
Functions.
- Parabola.def
By Debra Woods
The purpose of this lesson is to generate a parabola by
using the definition of a parabola.
- ParabolaPlotter(94maxi).nb
By Marion Harding
This is a very good program that defines a parabola, shows
graphically through animation that the focus and directrix are the same
distance from each point on the parabola, and plots any parabola
through input of the coefficients, a, b, and c . Finally, the
program will find the vertex, focus and directrix for a parabola, given
the coefficients a, b, and c.
- parabolas(11/94)
By Debra Woods
Introduction to parabolas and the effects that the
coefficients have on the plots..
- parabroots.nb
By Ann Webbink
When studying the parabola, many students become confused
about what is happening when the roots are no longer real roots.
This program will attempt to show that there are always two roots to
the quadratic equation, whether the roots be real, one root of
multiplicity two, or two imaginary roots.
- PlaneIntersect
By Debra Woods
In this lesson, given the equations of any three
planes in three variables, the program will solve the system, and
display graphically the solution.
- RationalFunctions
By Robin Sitton
This project is designed as an introduction to graphing of
rational functions for my average to lower level Algebra 2
students. My goal is for my students to use this guided
exploration to try and find the patterns that result in vertical and
horizontal asymptotes. I intentionally did not cover asymptotes
that are not vertical or horizontal. In my design of this
project, I tried to pick clear examples that would be easy for my
students to see a pattern. I also tried to use several different
methods of graphing in Mathematica in the hopes that my students might
see these graphs in a different view. I think that the animated
graphs will certainly allow them to see what happens to functions as
the values of "a" changes. I tried to implement lots of color
just for fun and also tried to give the necessary directions for
executing cells and animating.
- ReflectionsandTrans
Author Unknown
Lesson showing how functions reflect across
different axes, how translations work and how dilations work.
- Relations&Functions(93)
By Shirley Barrett
All the sciences using mathematics employ it essentially to
study relationships. Physicists, engineers, chemists, biologists,
economists, psychologists, sociologists, and other social scientists,
seek to find connections among the various elements of their chosen
fields and to arrive at a clearer understanding of why these
elements behave as they do. In this lesson, you will examine one
of the central ideas in the mathematical study of relationships.
- roots
By Mark Hampton
A graphical Investigation of the complex
- slope
By Cheryl Hobneck
An introduction to the slopes of lines
- Slopeintercept(92)
By Don Anglen
Description: This lesson solves and graphs
linear equations, explains the slope-intercept form of a line, and
explains some Mathematica language.
- Areas(94maxi)
By Don Anglen
his lesson graphs areas between curves. It finds areas between
curves, also. The Mathematica language is discussed as
well. This program is somewhat of an extension of the Area under
curve (94) program. There are some bugs in the last part of this
one. Let me know if this is useful, or what improvements can be
made.
- Area under curve
(94) By Don Anglen
This lesson graphs trapezoids under a parabola to
compare areas. It explains some Mathematica
language,also. The commands used are explained. This may be
a good first program to look at for new users. The program could
use a little sprucing up. I look forward to your comments.
- Calculus graphing
(92) Author Unknown
This file contains three lessons on graphing. The first
is titled Lesson 29 - translations. The student is given a plot
and asked to change the code to produce the translations. I would
suggest a possible change by plotting both the original untranslated
plot and the translated plot together in different colors. The
second lesson is titled Lesson 39 Rational Functions II. This
lesson discusses x-intercepts and asymptotes. The third
lesson is called Lesson 44, Factors of Polynomial
Functions. It covers the turning point theorem and the average
value of roots.
- CalculusProblem(92)
By Shirley Barrett
This
is a calculus problem concerning "on the job"
teaming. As workers are teamed together, they tend to
interfere with one another.
- Calculus Stuff
(92) Byu Jenna Caldwell
This file contains lots of different calculus
demonstrations. There is not much text that explains anything,
but the graphics are good and the animations are great. The first
animation is a saddle in 3-space. The second is called limiting
rectangles. It animates the rectangle approximation of area under
the curve y = Sqrt[x] as the number of rectangles increases. The
area under the curve demo allows the user to alter the picture of
rectangles under the curve. There is some impressive code in this
section. It is worth looking into. Keep me informed as to
whether it is easy for you to use. The section titled Derivative
stuff shows a curve and the tangent lines at various points and then
shows the function f and f', f'', and f'''. My only suggestion is
to include some comments. The last section is called Funky
Fonts. It does some probability (not using Mathematica and it
uses funky font characters as variable symbols.
- Cycloid
By Bruce Lewis
This module begins by defining the cycloid curve.
An animation will visually show what was described in the cycloid's
definition. The student will be asked to find out what happens to
the graph of the cycloid when changes are made to the constants in its
equation. The definition of a catenary will then be given. The
student will be asked to find out what happens to the graph of the
catenary when changes are made to its constant. We will then run
an animation that will view a box being rolled across a catenary-shaped
floor. Next, the definition of a roulette will then be
given. Finally, the student will be given some future ideas to
explore.
- Cycloids(93)
Author unknown
There is no written explanation in this
file. However, there are two extremely nice animations of
cycloids. Worth looking at. A very good visual explanation
of how cycloids are generated.
- DE-Harm.Motion(93)
By Debra Woods
In this lesson, we will discuss one application of the use of
second order linear differential equations with constant
coefficients. a y'' + b y' + c y = f(x)
The application of interest in this case is harmonic motion. We
will limit our discussion to free vibrations, that is, when f(x) =
0. The focus is on using Euler's formula to transform the
solution equation with imaginary exponents into the CiS form.
- Derivative of
tangent(92) By Jo Anne Kenyon
This program uses the definition and graphing to
find the derivative of the tangent function. It shows the average
growth for smaller and smaller values of h. It might be nice to
animate it someday. Then the program shows that the derivative of
Tan[x] is Sec^2[x].
- DWDirectionFields(94)
By Debra Woods
This project is a demonstration to show the visual behavior of
solutions of differential equations of the form dy/dx = f(x, y).
It creates a direction field, isoclines, and shows possible solutions
to the differential equation.
- DWLocalCoord(94)
By Debra Woods
This code demonstrates the tangent, normal,
binormal system (local coordinate system) as it moves along a circular
helix.
- Ellipsoids(93)
Author Unknown
This program introduces the equation of an ellipsoid, the plot of an
ellipsoid, it's cross sections and some exercises about
ellipsoids. In the section on cross sections, it also includes
the three traces.
- Equation of Tan line
(92) By David Lystila
In this program, examples of various curves
are graphed along with their tangent lines at different points.
The tangent line is then animated moving along the curve. The
equation of the tangent line is also given for each point. There
are many different examples in this program, including some functions
with infinite discontinuities.
- IncFunctions.windchill
By Richard Monke
Applications of the derivative using windchill as an example. A
very nice lesson.
- Limits.nb
by Gordon McClarren
The goal of this lesson is to become familiar with
limits. We know that some functions have values for which they
are undefined. What happens as the function gets close to these
values? This can be investigated by looking at the graphs of the
functions, or making a table of values near the undefined point. Since
limits are important part of Calculus, Mathematica has a limit
function. To find the slope of the tangent line to a curve, the
limit of the secant will be investigated as the two points approach
each other. This limit will be defined as the derivative of a
function.
- Limits(93)
By Debbie J. Barnett
This project is intended to be used near the beginning of a
traditional course in introductory Calculus. Because the limit is
in a sense the glue that holds all of the calculus together, it is
important that students have a clear understanding of this topic.
This project is designed to provide the student with an intuitive feel
for the limit by presenting a few common functions along with their
graphs and tables of function values. Examples include functions
whose limits do not exist for certain values of the independent
variable. Mathematica is used to calculate limits so that
students may verify their intuitive guesses as to what the limit should
be in each example and animation is used to provide a visual feel for
what is taking place in the function values as the independent variable
moves closer to a particular value from either direction.
- Limits(94)
Author Unknown
This program begins with a very interesting, and
pretty example of using limits of the areas of polygons as the number
of sides increase to show that the area is approaching the area of a
circle. The example is animated. The program then proceeds
with a rocket trajectory problem. This has nothing to do
with limits, and should probably be separated into its own program.
- NumericIntegration
By Bob Sompolski
Euler's method, modified Euler's method and Runge-Kutta
integration
- Polyno&Deriv(93)
By Daniel C. Jones
The purpose of this lesson is to show the
relationship between the first derivative and the graph of the
polynomial function. Mathematica will allow us to plot the
function, the derivative, and the slope of the graph at several
discrete points.
- series/convergence
By Jim Stewart
A lesson showing sequences, series, alternating series and the
Ratio test.
- TrapezoidRule(92)
By John Luker
The lesson successively draws the curve, shows the
area, breaks the region into trapezoids, computes the area of one
trapezoid, sums the areas of all the trapezoids and then computes the
area using NIntegrate. I would say we should try for a better
color in the graphics. This is an excellent program, worth
looking at.
- conic
sections(94) By Rick Meyer
This notebook shows a right double cone being intersected by
planes at various angles, thus showing the conic sections produced.
- Conics(92)
By Marian Robinson
This lesson treats conics as graphs of a general
second degree equation in two variables and relates each of them to
their respective standard equation form through the concept of
locus. Each conic illustrated becomes an integral part of the
design above. Other treatment of conics mentioned will be
developed at a later time. Some parts of this program are
incomplete.
- Ellipse(93)
By Shirley Treadway
This notebook demonstrates the definition of an
ellipse through animation. It is worth looking at.
- Ellipsoids(93)
Author unknown
Introduction to ellipsoids and their cross-sections
- GraphConics
By Jim Mann
This notebook is intended to give a working idea
concerning three conic sections: circle, parabola, and
ellipse. This notebook is well thought out and ties the ideas
together. Color needs to be added to the section on circles.
- Limacon
By Ginnie Teuscher
The Limaçon (pronounced lee muh SOHN) was first studied
and named by Etienne Pascal(father of Blaise Pascal). The word
limaçon comes from theLatin word meaning "a snail."
The general forms of the equations that graph as a limaçon are:
r= a + b cosθ r=a - b
cosθ r=a + b sinθ r=a -
b sinθ We will explore the graphs for various values of a and
b. Can we draw some generalizations to connect the value of the
ratio a to different shapes of the limaçon?
- Polar
Intersections By Debra Woods
This notebook provides a demonstration of the
possible intersections of polar graphs. Animation is used to
demonstrate why certain point(s) of intersection may not be solved for
algebraically.
- rect2polar
By Kathy Hammad
An exploration that introduces rectangle and polar graphs.
Includes lots of nice examples.
- Rose
Curves(92) By Marcella Cremer
The following notebook allows students to explore
Rose Curves by changing constants in the equations: r = a
Sin(nt), r = a Cos(nt). After running the animations and looking
at the graphs, the students should be able to generalize how the
constants change the graph of the curve. This notebook is a
complete treatment of roses, including, stretching, rotating, and
translating.
- CompoundInterest
By Steve Johnson
Simple verses compound intereset, computing interest and continuous
compounding of interest.
- GeometricDiscovery(93)
By Pat McLaughlin
In beginning calculus it is common to show how to
evaluate ∫ adx directly from the definition of the definite integral by
using the formula [n(n+1)(2n+1)]/6 for the sum of the squares of the
first n integers, a. In most texts this formula is presented
without consideration of its genesis. The formula is usually
proved using mathematical induction, but induction really does little
to explain how the formula was discovered in the first place; in fact,
students often object that there is a certain "circularity" to the
process since they are being asked to prove something that they have no
real reason to predict might be true. This project shows that there is
a simple geometric technique for demonstrating the origin of the
formula. This technique can even be used by pre-algebra students
to find the sum of the first n squares for any particular n even if
they have not yet learned to use a variable. For students who do
understand variables, working with the this technique and Mathematica
can provide a nice exercise in algebraic identities.
- OverlappingTriangles(94)
Author Unknown
This program only looks at the triangles in a picture that
overlap. The code to this program is not there, so modifications
cannot be made.
- Parallelogram
Area(94) Author Unknown
This is a very good visual demonstration that
graphically shows that the area of a parallelogram is base times height
by animation. The animation slides a triangular chunk from one
side to fill the void on the other side of the parallelogram, thus
forming a rectangle.
The
Penrose tilings are a class of beautiful nonperiodic tilings, which
lately have been useful in helping scientist studying crystalography to
understand quasicrystals structure. Tilings of the plane have been
studied for centuries because of decorative and mathematical use.
Before going any further with Penrose tilings, we need some
definitions, and background. A set T of regions tiles the plane
if the entire plane can be covered by copies of regions in T so
that these copies have no interior points in common. A periodic tiling
is one in which the region tiles the plane by translation. M.C. Escher
is famous for his artistic periodic tiling.
- PlaneSections
By Joyce McGiles
Drawing pyramids, prisms, and plane sections
- Rotation,vol,SurfAr(93)
By Sandy Emerick
This program rotates an equilateral triangle about
it's vertex and through animation shows the solid that is formed.
- LinearProg(94)
Author Unknown
This notebook provides the steps one should take in solving a
linear programming problem along with an example using the steps
given. Could be improved if the graphics are done in color.
- Motion(94)
By Patrick Meyer
This miniproject determines the acceleration, the
velocity, and the total time it takes for an object to move without
friction down a straight ramp of a given length and a given inclination
angle. It then uses straight ramp segments to approximate a parabolic
ramp, and determines the approximate length of the curved ramp and the
approximate time it takes the object to reach the bottom of the ramp.
- CCDescriptive
Stat By Carol Castellon
This project is an illustration of the built-in functions of
the Descriptive Statistics package. This package is automatically
loaded when most other statistical packages are used. This
project also provides an explaination of the measures used in
descriptive statistics.
- CCRegression
By Carol Castellon
This project takes an arbitrary set of bivariate
data (paired data) and looks at vaious ways to "fit" a model to the
data. Models which were fitted include linear, quadratic, cubic,
trigonometric, logrithmic, and a sum of non-integral-power
ploynomial-like functions. Part 1 of this project uses the "Fit"
command and allows the user to compare the resulting models on the
scatter plot visually. The statistical packages are not used in
this section of the project. Part 2 of this project uses the
Linear Regression Package and the "Regress" command to obtain a
statistical analysis of the models, and uses this analysis to compare
three different models. Part 3 of this project uses the
NonLinear Regression Package to obtain several non-linear models and
alter the method used for the fit.
- DataAnalysis
Author unknown
This is a data set of actual data from XXXXXX High
School. Theses are student admission scores from students
selected for admission from the pool of applicants in Spring
1990. This program provides an analysis of the predicted GPA's of
these students.
- Markov
By Teresa Anderson
This unit is designed for use in any Algebra class where
students are familiar with matrices, matrix multiplication, and
properties of matrices. Markov chains use probability theory and
matrix multiplication to model a variety of situations.
- AmpPeriodPhase
By Rachel Cuttriss
The
lesson presented here is to assist students in correctly identifying
the period, amplitude, and phase shifts in the sine
function. The student will be asked to compare graphs and
describe the changes that take place and what causes the changes.
- Dom.,Range(trig)(92)
By Jenna Caldwell
This lesson serves to utilize Mathematica's
graphing capabilities to help students visualize the results of
changing a function's domain and range in various ways. The
intended audience spans over several years, from perhaps middle school
to undergraduate mathematics. This is not a formal treatment of
domain and range concepts, since they are only treated here as
intervals.
- generateSinefromUnitCir(93)
By Joe Darschewski
This project introduces parametric equations showing their
relationship to the unit circle and the Sine and Cosine
functions. Code shows an excellent animation of how the sine plot
relates to the unit circle.
- GraphingSine(94)
By Wayne Brown
This is an excellent program that makes use of
animation to demonstrate the effects of the coefficients in determining
amplitude, period and phase shift. Worth looking into.
- PolarGraphs(94)
This notbook contains a demonstration to show to the
students. It will show why a graph of two polar equations may
have intersections that are not solutions to the system of
equations. The example here is: r = sin 2t
r = 1
- SineFunction/sound
By Paul Wittmer
"Sine Function Exploration" using graphics and
sound capabilities for an introduction to functions dealing
with sin( ). Sine Function Exploration can be used to
provide advanced students a course component or it can provide a first
exposure to trigonometry.
- Sinusoids2
By Henry Brink
General sinusoid graphs, Writing equations for
sinusoids, problems to solve using sinusoids
- Sinusoids3
By Rob Mason
This exercise deals with the graphs of sinusoidal waves. Students
will learn the general form of the time independent sinusoidal
function. They will explore the manner in which changes in the
general form alter the graph of the function. Next, the students
will also be introduced to the form of a time-dependent sinusoidal
function. They will animate graphs to observe how a
time-dependent wave moves along its axis. Finally, the students
will investigate how both time-independent and time-dependent waves can
be combined into composite waves. They will then be given the
opportunity to further explore these ideas with both periodic and
non-periodic functions.
- Trig
Fcns(94)
Author
Unknown
The subject matter of trigonometry is based upon
six trigonometric functions. In this section we will introduce
the first three--the sine, cosine and tangent functions (abbreviated
sin, cos and tan). This program uses animation to relate the
functions to the unit circle. In my opinion, this is the better
of the notebooks in this section that demonstrate that point.
- TrigIdSound
By Brian Pasero
An introduction to trig identities and a play with Mathematica's sound
capabilities.
- trigsum
By Debra Woods
The purpose of this lesson is to investigate the sum of sine
and cosine functions each with the same period, and to derive the
relationship between this sum and a single sine function.
D sin(Bx) + F cos(Bx) = A sin(Bx + C)
- Waves-sin and
cos(92) By Kathleen Smith
This notebook is a look at the Sine and Cosine waves and the
parameters that determine the shape of the graphs.
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10/18/04