CLASSROOM PROJECTS with ENVIRONMENTAL APPLICATIONS
Faculty
Classroom Guide
This project is funded by
The National Science Foundation #
DUE-9952568
The
With additional support from
Written by
Professor of Mathematics,
All
rights reserved. Manual may be printed
for classroom use only. No part of this guidebook may be reproduced, in any
form or by any means, without the permission in writing from the authors.
The Earth Math project
provides you with a new way to integrate environmental science data and real
life problems into your classroom.
Whether you are familiar with the authors of the Earth Algebra texts or
new to this project, this faculty classroom guide is designed to help you in
several ways. You will want to use the
guide because you…
want to become familiar with the Earth Math web site and
tools.
are preparing for instruction with the Earth Math
modules.
need to take note of the ways you might want to use the
Earth Math modules as you teach your class.
want to learn to use the Earth Math demonstration
materials, modules, and applets.
wish to use the guide as a reference throughout the term
and beyond.
We would suggest you print this guide to have a hard
copy reference to the site materials and uses.
It will provide you a map to the materials, descriptions, and
directions.
Notes to the Instructor
The content of this handbook reflects
the third version of the Earth Math modules.
This guidebook will be updated to reflect the changes made to the
modules and applets as the project evolves.
There may be minor differences in the directions given in this guidebook
compared to the web site as the project is continued. Faculty designed this guidebook for faculty
and we welcome your suggestions and comments, so that we may improve the
training manual. Send all suggestions
and comments, as well as ask assistance when you need it, to:
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Please
contact |
Table of Contents
Welcome to the Earth Math Projects
Notes to the Instructor
History of the Earth Math Project
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Chapter
1 |
Home Page and Site Map Instructor Materials …………………………..… 1-1 Course Materials ……………….………………... 1-3 Evaluation ..………………………………...…..... 1-7 Button Links ….………………………………….. 1-8 |
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Chapter 2 Using
Earth Math Modules |
Learning to Use Earth Math Modules Starting the Modules …………………………..… 2-1 Website Windows
………………………………... 2-3 Using Review Topics
.…………………..….…..... 2-7 Using the Applets ………………………..…..…... 2-9 |
Chapter 3 |
Instruction Preparation Introductory Module: Your First
Module ……… 3-1 Modules Available ………………………………… 3-11 Review Topics
……………………………………… 3-16 Tool Chest
…..………………………………………. 3-31 |
Chapter 4 |
Inquiry in the Math Classroom Introduction to Inquiry-based Instruction ……..
4-1 Good Questions
………………………………….... 4-4 Assessment and Adaptation
……………………... 4-5 Web Sites and Bibliography ……………………… 4-7 |
History of the Earth Math Projects
The Earth Math Projects are an exciting
extension of the previous mathematics textbooks and materials produced with
applications to Environmental Science.
Over the past 12
years, the Earth Math Projects have produced unique materials that teach
mathematics through the study of environmental issues and problems. The materials are appropriate for use in
mathematics courses ranging from beginning algebra through calculus.
All Projects are
directed by Dr.
Three textbooks have
resulted from these projects, Earth
Algebra (algebra), Addison/Wesley, Earth
Angles (pre-calculus), Addison/Wesley, and Earth Studies (applied calculus), Kendall/Hunt. The applications
are based on real environmental problems that effect students’ lives and
therefore generate more interest in the use of mathematics as a tool that can
be used to analyze real situations. The
materials have been tested and formally evaluated and have shown that the use
of mathematics to study real problems that are interesting to students
significantly improves interest in, understanding and appreciation of the role
of mathematics in science and society.
As a spin-off,
special class notes based on water availability for a growing community were
developed for
Current Project Activities
The National Science
Foundation and the U.S. Dept. of Education, FIPSE have funded Schaufele and
Zumoff, for a curriculum development project which was an extension of the work
described above. The goal is to produce versatile, technology-intensive
materials for classroom use and teacher training. Reform-based applications are
incorporated into platform-independent software to make them accessible to
anyone with a computer. The versatility of these materials will allow an
instructor to use them regardless of textbook or underlying curricula.
Environmental applications
from the previous projects and new applications are designed for courses from
beginning algebra through calculus. The project features an inquiry-based
format, web-based interactive materials, seamless interface with
state-of-the-art technology, use of real data, interesting applications of
mathematical concepts, and flexibility of classroom use.
Materials
Each module, or
study, is presented on the Web (also available on CD-Rom) and can be used as an
in-class or lab project that illustrates real applications of mathematics to
students. Students are able to access data provided on the website or download
relevant data from the Internet.
Software applets are written that can perform mathematical operations reflecting
concepts learned in the classroom or be used for demonstration purposes. These
tools are available from the Web and packaged on the CD. Review topics of related prerequisite skills
are linked to the modules to provide a quick, easy view of content necessary to
complete the study.
Resources
Resources for the
faculty using the Earth Math modules include links to data, other related
projects, references, announcements of user activities, as well as contact with
authors and others involved in the project consortium.
Chapter
1 Web Site Specifics
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Home
Page and Site Map This
section of the guidebook gives an overview of infor-mation
regarding the web site, using the web site buttons, and all its materials. |
FAQ
What does the web site include? Becoming Familiar with the
Earth Math Web Site Instructor
Materials ……………………… 1-1 Course
Materials ………………………… 1-4 Evaluation …………….…………………… 1-8 Button Links ..…..………………………….
1-9 |
Instructors using Earth Math modules
will find a description of the Earth Math homepage and the site for easy
reference and use. In this first
chapter, the instructor materials and course materials will be discussed, as
well as the site evaluation and links for resources and use in the classroom. The next chapter will describe the content of
the modules and the use of each of the site tools.
Starting the Modules
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Standard
Formatting of the Earth Math Modules Introduction A
standard format and methods are used in the analysis of the data and solution
of problems that are presented in Earth Studies. The Studies will all be presented in a
somewhat structured format that is demonstrated in the Introductory Module
described in depth in Chapter Three. The Earth Studies Introductory Module
demonstrates the basic format and how to integrate the website
materials. We recommend you use the
introduction modules as a starting point for students (and instructors) to
learn to use the Earth Studies. The
standard format should make the implementation of any of the modules easy –
use all or any part of a module, along with the appropriate applets and
review. Click on the Course level page
and module name. The page opened will
be written in the standard format, with components as follows. |
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Study Components Each
Study will have three components:
The
text will be presented in four sections:
answers to questions raised regarding the
issue. It is in this section that the
applets will be used.
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The Tool
Chest will be the place where students can access the applets that will
perform the mathematics. Here students
can enter their data or other quantitative information and use the applets to
analyze data and construct mathematical models. The results should answer the questions
involved in the issue. A complete
listing of the Tool Chest Applets can be found in Chapter 3, Tool Chest. |
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Journal The journal
will be the place students record their answers to questions from the text
and include the explanations of the mathematics they used. Student’s journals should be written as a
file in the word processing package on their computer; this will be the
document they turn in to you as instructor. Instructions
on selection highlighting, copy and paste of selected text and saving the
journal files should be given to students as they begin Earth Studies. It is appropriate to do this while students
work with the demonstration modules or the first module you use, if you wish
students to use a journal. |
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Using the Website Go to the
website address: http://earthmath.kennesaw.edu to the site
homepage. All Earth Math materials
available can be accessed through this address. At the Earth Math homepage, click on the
study level and scroll down to the module desired and choose the study by
clicking on the name. This will
automatically open the beginning page of the study and you are ready to
begin. You can see this is the same as
the previous image, in the CD-Rom instructions above. |
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Using the CD-Rom The CD-Rom is provided for you to
have an alternative to the website, to have a backup if you are having
problems with internet access. Contact
a project team member to request a CD-Rom version of Earth Math. Insert the disk in the drive and open
My Computer. Double-click the CD-Rom
drive (usually D:) and you will see the contents of
the CD. |
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Double-click on index
and it will open the first page of Earth Math Study materials. Once into the homepage, scroll down the page and click on the
study that you want to use and it will link automatically to all windows and
materials that are associated with that module. |
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These materials are the same as those on the website and can be
used when no internet access is available.
One caution – remember you will not have access to the websites that
are linked to the CD if you don’t have internet access, but much of the
material can be used – including the applets. |
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Using the Windows If a student clicks on Algebra with Equations they will bring up the course module listing for this level. Notice the menu map at the top of this window, which allows the student to go back to the homepage. They can then click on the Predicting Streamflow Module, to start from the beginning, or click on a particular part of the module to pick up where they left off. |
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A
listing of the available modules is included on the course level main page. Each
link is designed to go directly to that module or part of the module. Notice
the link to the Home page in blue, as well as the standard menu links on the
left of the page. You can
access any of the materials from these buttons. |
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If the student clicks on the
Predicting Streamflow Part 1 Temperature link, they
will bring up the Streamflow Temperature page, as
shown below. |
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The
student then will click on the first link, Part 1: Temperature, to begin work
in this module. They may
also access the other parts from here. Notice they can also go back to the Streamflow course main page, the Algebra with Equations
page or Home page from the top menu.
They may also access the other parts of the Streamflow
module from this top menu. The
student then will begin work on Part 1: Temperature, beginning with the
Comprehension questions, including the use of the applets. |
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Using
the Review Topics
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Review
Topics The topics provided are
for the review of the mathematical concepts and terms included in the
study. These concepts are assumed
prerequisite concepts and skills required for the successful completion of
the study. Students may access these
topics to find general information and formulas for working with the
data. Interactive applets are included
for students to explore the impact of changes on symbolic expressions, graphs,
etc. Example problems and solutions
are included to assist the student’s review of prerequisite material.
The review topics are listed on the left at the beginning of a module study, with the Tool Chest. Only those topics necessary for successfully completing that particular module of study are included. For example, when opening the Algebra with Functions Passenger Car Gasoline Consumption module, use the scroll bar to follow down the left side of the display which will reveal the following list of Menu Topics for this module. |
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When a student clicks on the topic
name, the topic opens in a new window, to be used along with the study
module. For example, if the Linear
Functions topic is opened, the following window includes embedded applets and
complete examples. |
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The student can review the material ‘just in time’ at the time needed, including working examples within the applets. The review topics include the solutions to the examples with descriptions and definitions, as well as appropriate formulas. |
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Review Topics may be accessed from
the standard menu on the left of the page by clicking on the button link
under Instructor’s Materials. The student may select to review
prerequisite material at a point that is appropriate in order to work
successfully on the Earth Math study. |
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Review Topics
The Review Topics
button is a link to prerequisite review that students can use at any
time. Embedded within the modules of
study are topics given in the left menu of the page, appropriate for each study. Students can return to the homepage to use
a topic they’ve forgotten at any time.
Available topics are listed and a description of each follows. |
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Available Review Topics ALGEBRA TOPICS § Solving Equations Graphically 1. Slope 2. Intercepts Precalculus Topics § Trigonometric Functions: Sine & Cosine § Period, Phase Shift and Amplitude of Trigonometric Functions Calculus Topics |
For
example, if a student clicks on the linear functions topic, a window begins
with the definition and an applet for reviewing the visual concepts. The general formula for slope and intercept
are defined and include examples for the student to review problems
appropriate for the topic and level.
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The applets are embedded in the
Introductory Module and some Review Topics for ease of use but may be
accessed from the standard toolbar on the left in any window. Students can choose to follow the study
step-by-step, or you may have them do selected parts of the study. You can also assign additional problems for
exploration using the applets. Internet data sites are linked within
the study, marked by the blue highlight.
Students can click on these links to obtain more data and information. |
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Using an Applet
The
Tool Chest is a set of mathematical tools for students to use in their
study. Each applet has a window with
specific directions for use below the applet window. By clicking on the applet name in the tool
chest the student brings up the applet and directions for use. Applets accessed by using a study or review
topic that has the applet embedded, do not contain the directions listed with
the applet necessarily. Most of the
directions for the applets are self-explanatory and there are groups of
applets that function in the same manner, with the various models.
To begin to use an applet, open the
applet from the Tool Chest, for example the Linear Regression Applet. Notice the window frames for entering the x
and y coordinates of the data set, the CLEAR, PLOT and ANALYZE buttons for
use. Enter the data set first, than
click on PLOT. The applets will open
in another window so the window may be resized and used with the module
window. |
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You will need to enter the X coordinates and
then the Y coordinates in the Data window.
You then click PLOT and the coordinates are
plotted in the top right window. When you click ANALYZE the ‘best fit’ linear
regression model and the resulting function is given in the lower results
window. You may clear the data to enter a new data set
by using the clear button. Students can practice entering data and using
the applet buttons, noting the resulting change from additional or new
information. |
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You will notice the graph displayed in the window on the top right and the model in the RESULTS window. |
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The linear regression applet will plot
the data points on the coordinate graph of the applet. Students will see the
data points plot in the applet. This applet result is from the
Introductory module. |
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Click ANALYZE and the best fit regression model for the entered data
set will be displayed along with the data set, with the symbolic model and
error in the RESULTS window, as shown below.
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The linear regression applet will plot
the regression line over the data points. Students will find the data model
and error in the lower window. |
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Tool Chest
The Tool Chest is
a link to applets that students can use at any time. Embedded within the modules of study and
review topics are the links to these applets, given in the left menu of the
page. Students can return to the
homepage to use an applet to graph or calculate a value at any time. Available applets are listed and a
description of each follows. A
complete set of directions for each applet are given in Chapter 3, Applets. |
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Available Applets COMPUTATION APPLETS · Math Pad INSTRUCTIONAL APPLETS · Defining Trig. Functions: Aine & Cosine · Quadratic Functions Standard Form EXPERIMENTATION APPLETS |
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example, if a student clicks on the linear regression applet, a window for
calculating the linear regression from a data set is opened. The student can then enter data and plot
the data points, then calculate the linear regression equation. The description of each applet is given in
the section on applets.
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Help
assistance is provided for each applet of the Tool Chest. Most applets are self explanatory, with
buttons and clear displays. There are
several notes regarding the use of the Math Pad and Plot Solve to consider. |
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Applet Descriptions The
descriptions below briefly identify the function of the applets
available. The list follows the same
sequence as the website TOOL CHEST. |
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Computation Applets EXPONENTIAL REGRESSION APPLET This applet has
two functions: plotting and finding the exponential function which best
approximates the user supplied data.
Using an appropriate data set, this applet
plots and data and will find the best fit exponential function for that
data. The best fit model is displayed
on the graph with the data set for comparison. The symbolic model and error is displayed
in a results window. |
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LINEAR REGRESSION APPLET This applet has
two functions: plotting and finding the linear function which best
approximates the user supplied data.
Entering an appropriate data set, the user plots the data and finds
the best fit linear function. The
function graph is plotted with the data set.
The symbolic model and associated error is displayed in the results. |
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JAVA Math Pad This applet
allows the user to evaluate mathematical expressions and functions. A
mathematical expression is made of numbers, variables, built-in and user
defined functions together with operations. It has an open screen and several
buttons. For further
information, click on the scroll down menu, under "Help Topics" to
get more detailed help on how to use this applet. You need to enter expressions including all math
symbols ( , /, ^ , etc.) in order to use the math pad. For example: Enter function
f 1(x) = 2*x Enter f1(3)
Gives answer 6.0 This will allow
students to define up to ten functions. |
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Plotting Applet The purpose of
this applet is to plot functions. It can plot up to 10 different functions at
the same time. It can also be used to solve equations numerically. Changes in window parameters and function
information, using zoom and trace buttons, can allow the user to define and
view different functions on the same graph. |
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QUADRATIC REGRESSION APPLET This applet has
two functions: plotting and finding the quadratic function which best
approximates the user supplied data. Entering an appropriate data set, the
user plots the data and finds the best fit quadratic function. The function graph is plotted with the data
set. The symbolic model and associated
error is displayed in the results. |
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Instructional
Applets The The purpose of
this applet is to illustrate the relation between the unit circle and how the
trigonometric functions are defined.
As you drag a point around the unit circle, the applet displays the
coordinates and angle direction.
Choosing sine or cosine, the function’s value is displayed as the
length of the segment on the unit circle, as well as the corresponding angle. |
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The purpose of this applet is to illustrate the relationship of the slope of the tangent to the graph of y = f(x) at x = a, that can be approximated for a certain value of h where the secant line becomes the tangent to the graph of y = f(x) at x = a. This applet illustrates the concept that the slope of the secant approaches the slope of the tangent, allowing the user to enter a function, select the points (a, f(a)) and (a+h, f(a+h)), to change h so that the point (a+h, f(a+h)) gets closer to the point (a, f(a)). |
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Role of a and b in y = ab^x The purpose of
this applet is to illustrate how the values of a and
b affect the exponential function y = a b^x. Changes made to the general form
exponential function, using a scroll bar, will display the affect of the
coefficient and base on the resulting graph. |
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This applet illustrates the fact that the derivative of a function y = f(x) at x = a, denoted f'(a), is the slope of the tangent to the graph of y = f(x) a x = a. The user can plot a function, and select points on the graph. As the points are selected, the tangent to the graph at the chosen point is drawn, if desired, its slope is displayed and plotted as a new point. If the user selects enough points, the graph of the derivative will become evident. |
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Role of a, b, k and c in y = a
+ b cos(k(x - c)) The purpose of
this applet is to illustrate how the values of a, b, c and k affect the graph
of the cosine function. Plotting an appropriate data set supplied by the
user, the cosine function coefficients and constants selected are then
entered and tested by graphing with the data set to view the fit of the
function. |
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Role of a, b, k and c in y = a
+ b sin(k(x - c)) The purpose of
this applet is to illustrate how the values of a, b, c and k affect the graph
of the sine function. Plotting an appropriate data set supplied by the user,
the sine function coefficients and constants selected are then entered and
tested by graphing with the data set to view the fit of the function. |
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Understanding the Slope of a
Line The purpose of
this applet is to understand the notion of "the slope of a
line". By selecting two points
on the graph, the point coordinates and quotient rise/run is displayed, as well
as the triangle symbolizing the relationship.
Point positions can be changed and the information is updated. |
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ROLE of m and b in y = m x + b The purpose of
this applet is to illustrate how the values of m and b affect the line y = m
x + b. Using a scroll bar to change
the slope m and/or intercept b, the line position moves to reflect those
changes. |
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Role of a, b and c in y = a
x^2 + b x + c The purpose of
this applet is to illustrate how the values of a, b and c affect the parabola
y = a x^2 + b x + c. Using a scroll bar to change
the coefficients and constant, the parabola’s position moves to reflect those
changes. |
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Role of a, h and k in y = a (x
- h)^2+ k The purpose of
this applet is to illustrate how the values of a, h and k affect the parabola
y = a( x - h)^2 +k. Using a scroll bar to change the coefficients and
constant in the graphing form of a quadratic, the parabola’s position moves
to reflect those changes. |
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The The purpose of
this applet is to illustrate the relation between a point on the unit circle,
its coordinates and the angle between the x-axis and the line through the
origin and the point. By dragging a
point around the unit circle, the coordinates of the point and the angle
values will be displayed in either degrees or radians. |
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Sums and Approximating Areas This
applet allows you to experiment with Riemann sums
and approximating the area between the x-axis, the graph of y = f(x),
the vertical lines x = a and x = b. The
user specifies what kind of Riemann sum is to be used, the applet draws the graph of the function, the
rectangles corresponding to the partition, and the kind of Riemann sum being used. |
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Experimentation Applets FITTING TRIGONOMETRIC (COSINE
FUNCTION) DATA APPLET This applet has
two functions: First, it can be used to plot user supplied data. It can also
be used to test if a user supplied trigonometric (cosine) function (a
function of the form y = a + b cos(k (x - c)) ) fits the given data by plotting the
function. Plotting an appropriate data
set supplied by the user, the cosine function coefficients selected are then
entered and tested by graphing with the data set to view the fit of the
function. |
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FITTING EXPONENTIAL DATA
APPLET This applet has
two functions: First, it can be used to plot user supplied data. It can also
be used to test if a user supplied exponential function (a function of the
form y = a (b^x) ) fits the
given data by plotting the function. Plotting an appropriate data set
supplied by the user, the exponential function coefficients selected are then
entered and tested by graphing with the data set to view the fit of the
function. |
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FITTING LINEAR DATA APPLET This applet has
two functions: First, it can be used to plot user supplied data. It can also
be used to test if a user supplied linear function (a function of the form y
= m x + b) fits the given data by plotting the function. Plotting an
appropriate data set supplied by the user, the linear function coefficient
and constant selected are then entered and tested by graphing with the data
set to view the fit of the function. |
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FITTING QUADRATIC DATA APPLET This applet has
two functions: First, it can be used to plot user supplied data. It can also
be used to test if a user supplied quadratic function (a function of the form
y = a x^2 + b x + c) fits the given
data by plotting the function.
Plotting an appropriate data set supplied by the user, the quadratic
function coefficients and constant selected are then entered and tested by
graphing with the data set to view the fit of the function. |
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FITTING TRIGONOMETRIC (SINE FUNCTION) DATA APPLET This applet has
two functions: First, it can be used to plot user supplied data. It can also
be used to test if a user supplied trigonometric (sine) function (a function
of the form y = a + b sin(k (x - c)) )
fits the given data by plotting the function. Plotting an appropriate data
set supplied by the user, the sine function coefficients selected are then
entered and tested by graphing with the data set to view the fit of the
function. |
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Chapter
3 Instruction With
Earth Math
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Instruction
Preparation This
section of the guidebook gives you directions on how to get started and help you
prepare for the classes in which you will use the Earth Math modules. |
FAQ How do I teach students to
begin? Web Site Assistance To The Student …………………………………….. 3-1 Introduction
Module: Your First Module ……… 3-2 Modules
Available ………………………………… 3-8 Review
Topics ……………………………………… 3-13 Applets ………………………………………………. 3-22 |
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This
last chapter of the guide will assist you in understanding how to begin
working with the Earth Math Studies.
The Introductory Module material can be used to instruct students in
the use of the modules, integrating the applet tools and review topics. The Introduction to Applets contains step
by step instructions in the use of two applets. |
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Introductory
Material
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Introductory
Module
World Population
In this module, the student is introduced to the format and methods
that will be used in the analysis of data and solution of problems that are
presented in Earth Studies. The Studies will all be presented in a
somewhat structured format that will be outlined in this module.
Applets, or little programs, designed to be used with each Study; these will
perform the mathematical operations needed to analyze data and build
mathematical models. Also, links are included throughout each study
that will take students to explanations of mathematical terms and
methods. Here you will be guided through a study on population that
will illustrate the most common methods used in all of the Earth Math Studies
and will also illustrate the use of the applets. This demonstration
module can be used as a preliminary study to learn the use of the modules and
tools provided in the Earth Studies.
First, we outline the general structure for all the Studies. |
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Each Study will have three components: ·
Text - which will provide information about a particular issue and ask
questions; ·
Menu - lists in two parts: Tool Chest - contain the applets to perform the mathematical
operations Review Topics - contain links to explanations of relevant
mathematical topics and examples; ·
Journal - which will be the place where students can record thoughts,
answers, and mathematical solutions.
This is kept on a word processor and should be open as students record
the answers to questions. |
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The Text
will be presented in four sections: ·
Comprehension section - will provide information about the issue
and ask pertinent questions for students to think about and express their own
ideas regarding the issue. ·
The Acquisition section - will provide more detailed
information, relevant data, or direct students to a website to acquire more
information or data. ·
The Application section - will ask specific questions
which will require mathematical analysis that will lead to a mathematical
model. The model should be able to provide answers to questions raised
regarding the issue. It is in this section that the applets will be
used. ·
The Reflection section - will ask questions regarding
the reasonableness or validity of a student’s model. |
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The
Tool Chest will be
the place where students can access the applets that will perform the
mathematics. Here they can enter data or other quantitative information
and use the applets to analyze data and construct mathematical models.
The results should answer the questions involved in the issue. |
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The Journal
will be the place students record their answers to questions from the text
and include the explanations of the mathematics used. The journal
should be written as a file in the word processing package on the computer;
this will be the document turned in to you as instructor. |
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Your First Module |
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EARTH
STUDY: WORLD POPULATION |
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Comprehension
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In
this section, questions are posed that are designed to stimulate thoughts and
observations regarding population increase/decrease, reasons for studying
population change, and methods that might be used. At this point,
students should open a blank document in their word processor and record the
answers to these questions. Copy each question into the journal (Ctrl-C to copy
and Ctrl-V to paste) and type their response below the question in a
different color or font. Be sure to remind them to write in
complete sentences and express their ideas so that others can
understand. Students should save their work and keep this window open
throughout this study so they can easily record other questions and answers. Each of the applets is embedded in this
first module and an additional applet display follows with the correct
result. The applet tools are embedded in this first Introductory Module only. Questions
A. Do
you think that the population of the World is increasing or decreasing? B. What
have you noticed recently that led you to your answer to A? C.
How do you think population change in the world might influence your
life? D.
How do you think mathematics can be used to study population change? E. What are
some reasons for studying population change? |
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Acquisition
In
this section, students will be given information about population increase,
real data is provided, and links to websites that contain more information
and updated data are also included. Estimates of world
population before the twentieth century vary widely, but most sources put the
number of people in the world in 1750 at about 750 million. By
mid-nineteenth century the population reached 1 billion and until 1930 the
growth was never more than 1%; however, since 1950 the increase has never
fallen below 1.6%. Current patterns of population growth and the
accompanying changes in consumption are placing increasing stresses on
ecosystems through environmental degradation, deforestation, loss of biological
diversity, over harvesting, and accumulation of toxic wastes. |
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The Table below
gives the United Nations estimates of the world population (in billions)
every five years from 1950 -1995. Students producing a model for
prediction of population will use this table of information.
World Watch 1996 In this section, objectives and
assumptions are stated; these will determine the mathematical model for
population. Students will use relevant mathematical tools that are
provided, the applets, to perform an analysis on the data and build an
appropriate model. Students will develop a model for world measure of
population growth. Applets can be accessed by clicking “Tool Chest” on the Menu to the left of the screen
in the demonstration module. Instructors and students should follow the steps outlined after each question to familiarize themselves with the techniques used and the applets available. Record answers in the student journal. |
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The objectives and assumptions are
listed below. Objectives
To determine: 1. a
linear model for world population; 2. the annual rate of change of world population. Assumption
The current
trends for world population continue. |
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Application
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Linear Models
In this part, students will find a linear
function to approximate data for world population and use this to • estimate
annual population growth, • estimate the
population for years other than those in the data set, and • forecast
future population size. (Note
the link to a Topic for review, if the student needs to review linear
functions.) (Students
should round off to three places for this work.) Problem
Set
The applets needed to
complete this problem are provided in the text for this demonstration module.
Usually, students will need to follow the following steps. ·
Scroll down to “Tool Chest” on the applet menu to the
left of the screen. ·
Click “linear regression” (since we are looking for a
linear function). The linear regression
applet will open in a separate screen. Students will enter the data in the
applet. 1. Plot
the points corresponding to the data in Table 1. The first coordinate
is year; denote this by t with t = 0 in year 2000. The second
coordinate is population in billions. Enter the data
provided in Table 1; enter the t value for each year (t = -10 for 1990, t =
-5 for 1995, etc.) in the “x” column, then enter the population figures in
the “y” column. Click
“Plot”. |
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The linear regression applet will plot their data points
on the coordinate graph of the applet. Students will see the data points plot
in the applet. |
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2. Determine
the linear regression
function that best fits these data; call this function P(t).
Graph the function P(t) on the same coordinate
system as the plot of the data points.
Click “Analyze”. The screen shown below should appear. |
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The linear regression
applet will plot the regression line over the data points. Students will find
the data model and error in the lower window. |
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Students should obtain the linear function P(t)
= 0.072t + 5.952. Here, the variable x has been replaced by t, and the
variable y has been replaced by the function notation P(t);
coefficients are rounded to three places according to instructions. The
graph is shown on the applet screen. 3.
What is the slope of P(t)? Students can give a verbal interpretation of
the answer; make sure they identify the units clearly. The slope is
m = 0.072; this means that the world population is growing at the rate of
0.072 billion (72,000,000) people each year. (There is not an applet
for this!) Use the
function P(t) to answer the following questions. 4.
What is the annual population growth? The annual
population growth is 72,000,000 people (see #3) The next
computations can be done using the "Java Math Pad" applet in the Tools Menu. Usually,
students will need to follow these steps. Scroll down
to “Tool Chest” on the menu to the left of the screen. Click “Java Math Pad”. This applet performs mathematical evaluations. 5.
How much will the population grow in 10 years? Six months? One week? Type 10*72000000 then press enter. Your answer will appear in scientific notation,7.2E8 which is 720,000,000 people. Or, you can type
10*.072, press enter. Here your answer will be .72 billion people. Type .5*72000000 then press enter. Type
(1/52)*72000000 then press enter. In 10 years
the population will grow by 10x72,000,000 =
720,000,000 people; in 6 months, 0.5x72,000,000 = 36,000,000 people; in one
week, (1/52)x72,000,000 = 1,384,615 people. |
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The screen shown below should appear.
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The Java Math Pad
applet will perform mathematical evaluations. Students will find the value of
basic computation and evaluation of expressions with this applet. |
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6.
Estimate the population in the year 2005. In the year 2005,
t = 5. You can continue using the “Java Math Pad”. Type in P(t) = 0.072*t + 5.952, then “Enter”. Next type P(5), then “Enter”. You should see the value of the
function P at t = 5:
P(5) = 6.312 (rounded to 3 decimal places). In the year 2005,
there will be approximately 6.312 billion (6,312,000,000) people in the
World. 7. Predict
when the population will reach 8,000,000,000. To make this
prediction, you can solve the linear equation
P(t) = 8. Students can do this prediction by hand algebraically
fairly easily but we illustrate an applet that will instead do this
graphically. This will make a
student’s work easier for more complicated equations. The graphical
solution will be the intersection of the graphs of P(t)
and the horizontal line y = 8. Scroll down the menu to the “Tool Chest”, then
click Plot-Solve. (First, read the "help" section.) Enter the first
function P(t), then enter the second function as the
constant 8. Click "plot". Then click on the screen to see a red
point on the graph of one of the functions. Click on "right" or
"left" to move the point to the intersection of the two graphs. You
may want to zoom in for more accuracy. The coordinates of the point can be
read on the screen. |
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The
screen shown below should appear.
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The Plot-Solve applet
will plot functions, given window parameters and the expression that defines
the function. Students will find the applet will plot several functions on
the same coordinate plane and can use zoom and trace to find the
intersections of the given functions. |
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In the year 2028 (t = 28.4), there will be
approximately 8 billion people in the World. |
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Reflection
In this section, students will examine
their model, solutions, and implications. Students will think about the
validity and reasonableness of their answers. How closely does the function
fit the data? Relevant questions are provided although students may
raise questions of their own. Record student answers to these questions and thoughts of
their own in the student journal. Students return their
journals to you upon completion of the module. Questions A.
Do you think that a linear function is good to
use for this study? Are there other functions that you think might
provide a better model? Why? B.
How long do you think this model will be
accurate; i.e., what is a reasonable domain for the
function? C.
How do you think the predicted increase
in world population might affect future life in the World? In the |
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MODULES
AVAILABLE
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Introductory Studies |
Tutorial Demonstration Studies
The following
demonstration studies are available to both the instructor and the student
for use while learning to use the Earth Studies modules. A demonstration study, a practice study,
and an applet demonstration are included in the learning modules. |
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This first note to the student is a discussion of
the student perception and real application of mathematics to modeling. It includes the relevance of the topic and
important issues presented in the modules. |
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This sample study on World population is designed to assist the instructor in working with the Earth Math materials, both for their own reference and to use with the students as a first, directed and step-by-step introduction to the format of the modules. All components of the study have directions for using the data and applets. A practice study is included for use after students complete the demo module. |
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This module is a demonstration in the use of the
applets. Specific directions to the
use of the applets in the Tool Chest menu are given, using the Java Math Pad
and the Plot-Solve applets as an example. |
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This sample study on |
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Earth Studies |
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Earth
studies are provided for different levels of mathematical study. Levels of study are for foundation
mathematics Algebra with Equations, intermediate level Algebra with
Functions, Pre-calculus and Calculus. |
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Levels Available These Modules are appropriate for foundation mathematics: introductory and intermediate level algebra. They emphasize linear, quadratic, piecewise, and prediction from these models. Fuel
Economy This is a study of the
fuel economy rates for two different vehicles and the cost of fuel for each
over a given time period. A comparison
of the miles driven and fuel used for each vehicle leads to the cost analysis
of the vehicle driven. |
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Predicting Streamflow This
study looks at streamflow rates for the foundation
algebra level. The three parts
together make up a complete study on predicting streamflow.
However, each part is designed so that it can be done independently. Relevant
answers from preceding parts are included in each part as needed. Temperature
In this module we will construct a model for average temperatures for spring
through fall. Precipitation In this module we will construct a model
which gives the rate of precipitation in a certain region of the country. Snowmelt
In this module, we will determine the streamflow
for a river. It will first be necessary to find out how much snow
contributes to the flow, then the volume of water distributed over the
watershed for the river. |
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These Modules are appropriate for college algebra level
mathematics. They emphasize linear,
quadratic, piecewise, and prediction from these models. Carbon
Dioxide Concentration (linear) The study is in two parts. In the first
part you will model atmospheric carbon dioxide concentration. In the second
part you will use this model to predict the impact of changes in
concentration on global temperature and ocean level. Carbon Dioxide Concentration This study will model the CO2 concentration
with a linear model for the last forty years and predict the future levels. Impact:
Temperature This module examines the possible impact of changes in
atmospheric carbon dioxide concentration on the global temperature. CO2
Emission from Passenger Vehicles This Earth Math Study is written in
three parts. The three parts together make up a complete study on predicting
carbon dioxide emission from passenger vehicles. However, each part is
designed so that it can be done independently. Relevant answers from
preceding parts are included in each part as needed. Passenger
Car Gasoline Consumption In this
first part you will study the factors that affect passenger car gasoline consumption
including the number of vehicles, annual mileage, and average fuel
efficiency. CO2
Emission from Passenger Cars In the second part you will model gasoline consumption
for passenger vehicles in the Reducing
Emissions from Passenger Cars In
part three you devise a plan to decrease emissions. Streamflow The three parts together make up a complete study
on predicting streamflow. However, each part is
designed so that it can be done independently. Relevant answers from
preceding parts are included in each part as needed. Temperature
This module is the first of three
that are designed to lead to the prediction of streamflow
for a river in a particular region. Precipitation
In this module we will construct a
model which gives the rate of precipitation in a certain region of the
country. Snow
Melt and Streamflow In this module, we will
determine the streamflow for a river. Population
This sample study on World
population is designed to construct a linear model and predict the world
population values. It explores the
patterns in slope and the linear function, as well as the accuracy of
prediction. |
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These Modules are appropriate for
pre-calculus level mathematics. They
emphasize rate of change, exponential, trigonometric functions, and
prediction from these models. These studies are designed to cover the same
data and questions that the Algebra version does but it assumes the student
has completed the foundation courses of Algebra. Carbon
Dioxide Concentration (Exponential) This study is in two parts. In the
first part you will model atmospheric carbon dioxide concentration. In the
second part you will use this model to predict the impact of changes in
concentration on global temperature and ocean level. Concentration
This study will model the CO2 concentration with a linear model for the last
forty years and predict the future levels. Impact This module examines the possible impact of
changes in atmospheric carbon dioxide concentration on the global temperature. Population
This sample study on World population is designed to construct an exponential
model and use rate of change to predict the world population values. Streamflow The three
parts together make up a complete study on predicting streamflow.
However, each part is designed so that it can be done independently. Temperature
This module is the first of three that are designed to lead to the prediction
of streamflow for a river in a particular
region. It looks at how temperature
and amount of precipitation affect the amount of water in a river. Precipitation
This module is the second of three that are designed to lead to the
prediction of streamflow for a river in a
particular region. A model for average
precipitation for the region is constructed using the previous model for
temperature. Snow Melt
and Streamflow This module is the third of
three that are designed to lead to the prediction of streamflow
for a river in a particular region. Results from the previous parts are
combined with area to compute monthly streamflow
figures. The Gas-Mileage
Bill and The Arctic National Wildlife Refuge This
module explores the efficiency of automobile gasoline consumption and the
feasibility of drilling in the Artic National Wildlife Refuge to relieve this
nation’s dependency on foreign oil.
Students use earlier explorations on gasoline consumption to extend
this into a prediction equation for oil consumption. Water
Consumption in the This
module explores the water consumption in the |
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These
Modules are appropriate for calculus level mathematics. They emphasize rate of change, integrals,
derivatives, demand functions and prediction from these models. World Coal
Supply In this module we will study availability and world use of coal.
In particular, we will determine the current usage rate and how long the coal
supply will last at this rate of consumption. World
Grain The three parts together make up a complete study on world grain
supply and demand. However, each part is designed so that it can be done
independently. Relevant answers from preceding parts are included in each
part as needed. Grain Supply
This part of the study looks at the area of the land cultivated and the world
grain supply yield. Supply
versus Demand This part looks at
the worldwide demand for grain and when the demand will equal the supply. Per
Capita Supply This part derives a function that describes the per capita
grain supply. Students predict how
much grain will actually be available per person. World
Population The world population
growth rate is revisited as the rate of change described as a percent of the
population. Students see what affect the change in growth rate has on the
population and prediction. In
this study we will examine oil supply and gasoline demand in a small
(fictional) community. In connection with this we analyze the economics
involved in the sale of oil and the purchase of gasoline. Specifically, we
will be concerned with cost, revenue and profit associated with the
production and sale of a product. Cost of oil This
part explores the factors affecting the cost of a barrel of oil. Fixed, variable and marginal costs are included
in the discussion to develop a cost function. Revenue
and profit Students examine the
revenue and profit for a company and use a cost function to see how the
selling price influences the profit. Supply curve Students examine the
relationship between selling price of oil and the amount of oil the company
is willing to extract. Supply,
demand and equilibrium Now we assume that currently the market
is in equilibrium. You have derived the oil supply curve (step three),
and will determine the current quantity of oil demand, the current
price, and the demand equation The
Gas-Mileage Bill and The Arctic National Wildlife Refuge This
module explores the efficiency of automobile gasoline consumption and the
feasibility of drilling in the Artic National Wildlife Refuge to relieve this
nation’s dependency on foreign oil.
Students use earlier explorations on gasoline consumption to extend
this into a prediction equation for oil consumption. Water Consumption in United States This module explores the
water consumption in the This study is the
investigation of available resources for a growing village. It is divided into three parts: Population;
Food, Coal and Electricity; and Water.
Models are used to predict the natural and human consumption of
resources for the village, as well as when the water source will be totally
depleted. Other Modules The
authors have other planned modules that will be available on-line in the
future. Keep watching the site for
updates. |
changes
as the values of the coefficients m and b change. Note any useful
observations in your journal.
.
The
solution is given below. 
Figure
2
can
be factored, then we
that
has real solutions can be solved by the quadratic formula
is
called the 
,
the equation has two real solutions.
,
the equation has no real solutions.
,
we see that 
Substituting
these values into the quadratic formula we get
Substituting
these values into the quadratic formula, we get
.
.
Round your answers to three decimal places.
and
the right-hand side, 
on
the same coordinate axes. See Figure 1.
(accurate
to one decimal place).
,
the range is the set of all real numbers except 0.
for
all 
. 
.
,
the radian measure of an angle. Also, given any real 
.
Tool Chest
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