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General
This tool can be used to explore basic linear and exponential
functions numerically and symbolically. The tool has several pages
with directions to help students to develop the data and functions and it
can be used as an exercise or as a project.
Percentage Investigation
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Applications of Quadratic Functions
These tools are designed to be used as
assignments/projects. Each tool has several pages on which the user
is guided through the process of developing and then applying a quadratic
function. These tools have user-manipulated scrollbars as well as
buttons that provide hints to help the user to develop and then apply the
quadratic functions.
Monthly Rental
Revenue
Human Cannonball
(My
personal favorite!)
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Statistics
These tools can be used as demonstrations or as guided
learning tools. Each tools has a graph with user controlled
scrollbars that allow the user to manipulate a point and to view the
affect of the value on the various statistics.
The Distribution tools allow the user to explore
rectangular/uniform distributions, skewed-left and skewed-right
distributions, and Normal distributions. Since these tools use the
Analysis ToolPak, you must enable this MS Excel Add-In. To do this,
open MS Excel, left-click the Tools menu and move to and left-click the
Add-Ins entry on this menu to open the Add-Ins dialog box; left-click the
check-box next to Analysis ToolPak and left-click the OK button. The
Analysis ToolPak will be enabled unless the computer on which you are
working does not have a full version of MS Excel installed. To view a
demonstration regarding enabling the Analysis ToolPak Add-In, select
MS Excel 2003 or earlier or
MS Excel 2007.
The following tools allow the user to examine specific distribution types.
In order to generate a new distribution, press SHIFT and F9 simultaneously.
Rectangular /Uniform
Skewed
Normal
The following tools allow the user to examine random
distributions, the first without classification and the second with
classification. In order to generate a new distribution, press SHIFT
and F9 simultaneously.
Random Distributions
Random Distributions with Classification
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The Exploring the Mean tool allows the user to explore how the mean of a
data set as the user changes one data value using a scrollbar.
Exploring the Mean
The Exploring the Mean and the Median tool allows the user
to explore how the mean and the median of a data set change as the user
changes one data value using a scrollbar.
Exploring the Mean and the
Median
The Five-Number Summary tool allows the user
to explore how to determine the five-number summary for data sets having
twenty-six, twenty-seven, twenty-eight, or twenty-nine data values; the user
can generate new data sets of the selected size and display the five-number
summary for the data. Since this tool uses the Analysis ToolPak, you must enable this MS Excel Add-In. To do this,
open MS Excel, left-click the Tools menu and move to and left-click the
Add-Ins entry on this menu to open the Add-Ins dialog box; left-click the
check-box next to Analysis ToolPak and left-click the OK button. The
Analysis ToolPak will be enabled unless the computer on which you are
working does not have a full version of MS Excel installed. To view a
demonstration regarding enabling the Analysis ToolPak Add-In, select
MS Excel 2003 or earlier or
MS Excel 2007.
Five-Number Summary
The Five-Number Summary with Graphs tool allows the user
to explore how to determine the five-number summary for data sets having
twenty-six, twenty-seven, twenty-eight, or twenty-nine data values,
displaying the histogram and the Box plot for the data when the
five-number summary is displayed; the user
can generate new data sets of the selected size and display the five-number
summary for the data. Since this tool uses the Analysis ToolPak, you must enable this MS Excel Add-In. To do this,
open MS Excel, left-click the Tools menu and move to and left-click the
Add-Ins entry on this menu to open the Add-Ins dialog box; left-click the
check-box next to Analysis ToolPak and left-click the OK button. The
Analysis ToolPak will be enabled unless the computer on which you are
working does not have a full version of MS Excel installed. To view a
demonstration regarding enabling the Analysis ToolPak Add-In, select
MS Excel 2003 or earlier or
MS Excel 2007.
Five-Number Summary with Graphs
The Outlier Investigation tool allows the user to
explore how various statistics for a data set change as well as when a
data set has and does not have outliers as the user changes one data value
using a scrollbar.
Outlier
Investigation
The Linear Regression Investigation tool allows the
user to explore linear regression as the user changes one point using a
scrollbar.
Linear Regression
Investigation
The Coin Toss Investigation tool allows the user to explore
probability for a fair coin's landing head-side up or tail-side up when
the coin is tossed 1000 times, 5000 times, 10,000 times, and 25,000
times. Use SHIFT F9 to re-calculate an individual sheet.
Note: This tool takes time to load and requires the Analysis
ToolPak. To view a
demonstration regarding enabling the Analysis ToolPak Add-In, select
MS Excel 2003 or earlier or
MS Excel 2007.
Coin Toss
(Coin_Toss.zip)
The Fair Die Investigation tool allows the user to explore
probability for a fair die's landing to display a 1, 2, 3, 4, 5, or 6 on
the top face when
the die is rolled 1000 times, 5000 times, 10,000 times, and 25,000
times. Use SHIFT F9 to re-calculate an individual sheet.
Note: This tool takes time to load and requires the Analysis ToolPak. To view a
demonstration regarding enabling the Analysis ToolPak Add-In, select
MS Excel 2003 or earlier or
MS Excel 2007.
Fair Die
(Fair_Die.zip)
The Normal Comparison tool allows the user to compare a
normal curve with mean and standard deviation set by the user with
scrollbars to the standard normal curve.
Normal
Comparison
The Left/Right and Between Standard Normal Distribution
tools allow the user to explore the area below the standard normal curve
to the left/right a user-set value or between two user-set values; the
user sets the values using scrollbars.
Standard Normal Distribution
Left/Right
Between
The Probability Density Function Comparison tool allows the user to compare a
graphs for the probability density function for the standard normal
distribution and the probability density function for the t-distribution The
number of degrees of freedom for the t-distribution are set using either a
scrollbar or direct entry.
Probability Density Function Comparison
The Confidence Intervals tools allows the user to explore
confidence intervals for user-set values. If the population mean and
population standard deviation are known then use the Confidence Intervals -
Normal tool, and if the sample mean and sample standard deviation are known
then use the Confidence Intervals - t-Distribution tool. For each, the
sample size and level of confidence must be entered; the level of confidence
may be entered using scrollbars or
entered directly.
Confidence Intervals - Normal
Confidence Intervals - t-Distribution
The Hypothesis Testing tools allow the user to explore
1-sample hypothesis testing for user-set values. Provided
the sample-size check has been performed, the Hypothesis Testing for
Population Mean Tool can be used to explore to hypothesis testing for
the population mean; for this tool, the population standard deviation is
unknown. Provided the sample size checks have been
performed, the Hypothesis Testing for Population Proportion can be used
to explore hypothesis testing for the population proportion.
Hypothesis Testing for Population Mean
Hypothesis Testing for Population Proportion
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Slope Graph Exploration
These calculus tools allow the user to explore the slope
graph for a function graphically and numerically.
The Graphical tool allows the user to explore the slope graph using a line
segment that is tangent to the curve. There are pages that allow the user to explore
functions based on their graph attributes, how the function is changing
and the curvature; the page labels are increasing, decreasing, concave up,
and concave down (increasing concave up, decreasing concave up,
increasing concave down, and decreasing concave down curves are included
on these page).
Graphical
The Graphical and Numerical tool allows the user to
explore the slope graph using a line segment that is tangent to the curve;
the slope of the line segment is provided. There are pages that
allow the user to explore functions based on their graph attributes, how
the function is changing and the curvature; the page labels are
increasing, decreasing, concave up, and concave down (increasing concave
up, decreasing concave up, increasing concave down, and decreasing
concave down curves are included on these page).
Graphical
and Numerical
This tool allows the user to explore the slope graphs
for the functions y = x2, y = x3, and y = (1/3)x3
+ x2 + 3x - 25 graphically using a line segment tangent to the
curve and numerically using the slope. This tool has buttons that
can be used to provide hints about the slope graph as well as the graph of
the slope graph.
For y = x2,
y = x3, and y = (1/3)x3 + x2 + 3x - 25
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Exploring Graphs of Functions
These tools can be used in General Mathematics, Basic
Algebra, College Algebra, or Precalculus courses. Each of these tools
allows the user to graphically explore functions using user-manipulated
scrollbars.
The Standard Form for the Equation of a Line tools allow the user to explore
the equation of a line as (s)he changes the coefficients of the line
expressed using standard form, ax + by = c; the use controls the
coefficients of the line using scrollbars.
Standard Form
for the Equation of a Line
The Comparing Slopes of Lines tool allows the user to explore
and two compare two lines and the equations of these lines, each expressed
using
standard form, ax + by = c, as (s)he changes the coefficients of these lines
using scrollbars; the graphs of the lines as well as the equations of the
lines are displayed as the user changes the coefficients.
Comparing Slopes of
Lines
The Intersecting Lines tool allows the user to explore the
relationship between two lines. The coefficients of the two lines,
specified using standard form, are controlled using scrollbars; there is a
scrollbar for each coefficient. The user is asked to determine if the
lines are parallel, distinct or the same, or perpendicular. If the
lines are not parallel then the user can determine the point at which the
lines intersect; if the lines intersect then the coordinates for the point
are given to five decimal places. Hints are provided for determining
if the lines are distinct parallel lines, if the lines are the same, if the
lines are perpendicular, and if the lines intersect; the hints can be reset.
Once the user has made her/his determination if the lines are parallel,
distinct or the same, or perpendicular as well as for the point, if any, at
which the lines intersect then the user can enter her/his response using a
option button to determine if her/his response is correct; these option
buttons can be reset for additional testing.
Intersecting Lines
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The Linear and
Quadratic Functions tool allows the user (i) to compare a line in
slope-intercept form, y = ax + b, to the line y = x in order to explore the
values of the slope, a, and the y-coordinate of the y-intercept, b, (ii) to
explore the values of the coefficients a and b in the quadratic function y =
ax2 + b while comparing the graph of this function to the
graph of the function y = x2, and (iii) to explore the the
coefficients and the graph for the quadratic function in standard form y =
a(x - h)2 + k.
Linear and
Quadratic Functions
These Linear and Quadratic tools allow the user to
change the graph displayed from a Linear function y = Ax + B to a
Quadratic function
y = Ax2 + B using either an option button or a command button.
Each tool displays the function equation as well as the function type and
allows the user to change the coefficients, A and B, for the function in
order to explore the effect of theses coefficients on the graph of the
function.
Option Button:
Linear and
Quadratic
Command Buttons:
Linear and
Quadratic
The Polynomial Functions tool allows the user to
explore cubic, quadratic, linear, and constant functions. The workbook
opens on the cubic function page. There are pages for cubic,
quadratic, and linear functions. The tool allows the user to explore
how polynomial functions degrade from one type to another as the leading
coefficient of the polynomial is set to zero (0).
Polynomial Functions
The Exponential and Logarithmic Functions tool allows
the user to explore (i) exponential functions of the form y = A(Bx),
(ii) logarithmic functions of the form y = A logB x, each for
various values of the coefficient, A, and the base, B, and (iii) to
explore the relationship between the functions y = Ax and y = logA
x.
Exponential and
Logarithmic Functions
The Trigonometric Functions tool allows the user to
explore the graphs of the cosine, sine, tangent, cotangent, secant, and
cosecant functions. The initial values for the function parameters
have been chosen to make the initial graphs more "interesting" for the user.
Trigonometric Functions
These tools allow the user to explore the trigonometric
functions and the unit circle and to generate the graphs of the
trigonometric funtions.
Cosine, Sine, and Tangent on Unit Circle
Tangent, Cotangent, Secant, and Cosecant on Unit Circle
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Exploring Sequences and Series
These tools can be used to explore series and sequences.
The P-Series Exploration tool allows the user to
explore p-series for p = 1/2, p = 1 (the harmonic series), and p = 2 both
graphically and numerically. The user controls the number of terms
in the partial sum using a scrollbar. The partial sum can be
explored graphically and numerically.
A P-Series
Exploration
The Sequence and Series Exploration for 1/n tool
allows the user to explore (i) the sequence {1/n} and (ii) the harmonic
series graphically and numerically. The user controls the value of n
using a scrollbar. In the harmonic series exploration, the user can
view the number of terms in the partial sum, the value of the current term
being added to the partial sum, and the value of the partial sum.
Sequence
and Series Exploration for 1/n
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Differential Equations
These tools can be used to explore differential equations.
The Pendulum Equation tool allows the user to explore
the simple pendulum equation θ'' + (g/L) θ' = 0 on the interval [0,50]
with the values of L, t, and the initial values of θ and θ' changed using
scrollbars.
Pendulum
The Mass-Spring tools allow the user to explore the
equation m y'' + b y' + ky = 0 for the mass-spring system with the values of
m, b, k, and the initial values of y and y' changed using scrollbars.
Mass-Spring interval [0, 50]
Mass-Spring
interval [0, 100]
Mass-Spring t-change 1/10
Mass-Spring t-change 1/100
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Balancing Chemical Equations
Examination of chemical equations provides a wonderful
application of systems of linear equations. One factor that makes
this application so useful is that in balancing a chemical equation the
lowest positive integer coefficients are desired. So, not only
must one determine the coefficients that balance the chemical equation
(satisfy the system of equations) but also make sure that these
coefficients are relatively prime.
This tool can be used to explore/balance a chemical
equation for which the reactant molecules are Ammonia and Oxygen and the
products molecules are Nitric Oxide and Water.
Balance
Cyanogen Gas, C2N2, has been found
in outer space. Cyanogen Gas can react with Fluorine to form
Tetrafluoride, CF4, and Nitrogen Trifluoride, NF3.
Cyanogen Gas
Nitrogen Oxide, NO, can be removed from smoke stack
emissions if it is reacted with ammonia, NH3.
Nitrogen
Oxide in Smoke Stack Emissions
This reaction is used to launch astronauts into space:
Hydrazine, N2H4, is combined with Dinitrogen
Tetraoxide, N2O4 to produce gaseous Nitrogen, N2,
and water
Space Launch
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