You must have Geometer's Sketchpad (version 4) to open these files. All files were produced using a PC. You are welcome to use the sketches for non-profit educational purposes. Please do not sell them. The newest sketches are on top, meaning that the topics are mixed up. The sketches can be used for individual exploration or in front of a classroom. An interactive whiteboard (e.g. Smartboard, Promethean) can enhance the experience. The dropdown menus may be helpful. Enjoy.

This sketch explores Euler's method for approximating a solution to a differential equation. It includes a brief explanation of the concept and a simple example of the calculations, as well as a depiction of the method with a variable interval and a variable number of iterations. The slope field cannot be adjusted very easily (to reflect alternate differential equations) but the axes can be differently scaled to explore a variety of scenarios. The software does not allow a function of two variables, so the construction is not easily generalized. The featured slope field goes with the differential equation dy/dx=x^2+y. Keywords: euler, euler's method, euler's approximation, slope field, differential equation, local linearity, solution curve

download GSP file:    [Calculus_Euler.gsp]
 download ZIP file:    [Calculus_Euler.zip]

This sketch allows a brief look at the mechanism of polar integration. The concept is essentially identical to integration with Riemann sums in Cartesian coordinates, but with wedges instead of rectangles. The polar setting brings up some novel facets of the concept of integration. For example, the integral area can overlap itself (see the page of this sketch dealing with the Archimedes Spiral). The construction was tricky, but I recommend it for the experience with the polar coordinates. The theta-interval and the number of sub-intervals are both easily changed to highlight the simplicity and precision of this process as could not be conveyed by a stationary picture. Keywords: polar integration, polar integral, polar calculus, limacon, cardioid

download GSP file:    [Calculus_Polar_Integration.gsp]
 download ZIP file:    [Calculus_Polar_Integration.zip]
    Updated: April 16, 2008 to include additional explanations.

The Pizza Theorem: If a circular pizza is cut by four straight cuts into eight slices whose tips have the same angle and come to the same point somewhere on the surface of the pizza, then the sum of the areas of alternating slices is equal to half the area of the pizza. This sketch gives a 'visual proof' of this neat idea. I have used it as part of a discussion of mathematical proof. While this is not a proof, it is a very convincing display and it can certainly offer some amount of insight into the problem. The evidence put forth by this sketch is more accessible and interesting to most people than any analytical approach. The sketch is dynamic in that the 'center' can be moved and the rays can be rotated to allow the exploration of different scenarios. The proof relies on the tilting or shearing of triangles, which is briefly explained on an additional page of the sketch. Keywords: gsp presentation, animation, pizza theorem, visual proof, shearing

download GSP file:    [Geometry_Pizza_Theorem.gsp]
 download ZIP file:    [Geometry_Pizza_Theorem.zip]

This sketch gives some accompanying visuals for a discussion about significance testing. This could be particularly valuable to students finding it difficult to grasp the quantities described by a Type I Error (rejecting the Null Hypothesis when it is correct) or a Type II Error (Accepting the Null Hypothesis when a particular alternative is correct). The sketch allows the student to change the value of alpha and the value of standard deviation sigma, and thereby discover the costs of controlling either type of error. This sketch includes the confidence interval sheet from the "Normal Distribution and Z-scores" sketch. Keywords: normal distribution, normal curve, mu, sigma, mean, standard deviation, z-score, probability, density curve, standardize, standard normal, normal random variable, confidence interval, CI, z*, statistics, statistically significant, p-value, alpha, one-sided, two-sided, z-test, null hypothesis, H0, Ha, alternative hypothesis, type i error, type 1 error, type ii error, type 2 error, power

download GSP file:    [Statistics_Ztest.gsp]
 download ZIP file:    [Statistics_Ztest.zip]
    Updated: Feb 29, 2008 to include corrected sample size (n) slider.

This will only be useful to a few people out there, but it may be worth putting here. The first of the three tricks in this sketch finds the maximum (or minimum) of two numbers, providing an invaluable link to discrete mathematics. I have used the max/min idea to form a dynamically scaled residual plot and a confidence interval that does not fold over on itself. The other two tricks are less elegant, but they improve the usability of Sketchpad by avoiding an error message and providing a button that erases traces. Keywords: sketchpad tricks, geometers sketchpad, geometer's, geometers', GSP, erase traces, maximum, minimum

download GSP file:    [GSP_Cheap_Tricks.gsp]
 download ZIP file:    [GSP_Cheap_Tricks.zip]

This sketch can be used to discuss various aspects of the normal curve. Normal curves have a specific nature to them that is brilliantly captured by Sketchpad. Every combination of Mean (mu) and Standard Deviation (sigma) yields a graph with the same area, and this sketch provides a look at the changes that take place in order to satisfy such a constraint. The calculations of area were made using trapezoidal estimates and summed iteratively as described in Sketchpad's sample calculus sketches. There are four pages: Normal Distribution with sliders, Standard Normal Distribution with z-scores, Confidence Intervals, and Standard Normal Distribution with a calculation of the shaded region corresponding to P( a < Z < b ). Keywords: normal distribution, normal curve, N(0,1), mu, sigma, mean, standard deviation, z-score, probability, density curve, standardize, standard normal, normal random variable, confidence interval, CI, z*, statistics

download GSP file:    [Statistics_Normal_Distribution.gsp]
 download ZIP file:    [Statistics_Normal_Distribution.zip]
   Updated: Feb 9, 2008 to include confidence intervals

There are three levels of visualization to be gleaned from this illustration. The segments themselves give a low-resolution picture of the family of curves which satisfy the differential equation. Secondly, the solution curves can be viewed with the click of a button. Finally, a set of tracing points can be sent along their respective curves, illustrating the concept of initial values. The sketch is simple, but it covers the basics in a way that any visual learner will appreciate. The sketch works nicely when dy/dx is a function of x, but if dy/dx is implicitly defined, the construction is more complex and the differential equation is not easily changed. Keywords: slope fields, direction fields, anti-derivative, anti-differentiation, integral curves, solution curves, family of solutions, differential equations, dy/dx

download GSP file:    [Calculus_Slope_Field.gsp]
 download ZIP file:    [Calculus_Slope_Field.zip]

This sketch shows the six trigonometric functions (Sine, Cosine, Tangent, Cosecant, Secant and Cotangent) as distances on a unit circle diagram. The angle is defined by the positive x-axis and a point on the circle that can be moved manually or animated. The unit circle is depicted adjacent to a Cartesian coordinate system of the same scale, for an easy visual translation between two important representations of trigonometric functions. Individual functions can be hidden or shown separately. Keywords: trigonometry, trig functions, sine, sin, cosine, cos, tangent, tan, cosecant, csc, secant, sec, cotangent, cot, geometric, radians, degrees, unit circle diagram, period, animation

download GSP file:    [Trigonometry_UnitCircle_SixFunctions.gsp]
 download ZIP file:    [Trigonometry_UnitCircle_SixFunctions.zip]

This sketch shows the tangent line to a function at a moveable point a. The slope of the tangent line is depicted as a length, which reveals the visual connection to the derivative f'(x). As a is moved around, the slope/derivative can be observed with or without tracing. The sketch includes linear, quadratic, cubic, quartic, and sine functions, but the function can be redefined easily on any of the pages. Keywords: calculus, derivative, differentiation, slope, gradient, rate

download GSP file:    [Calculus_Derivative.gsp]
 download ZIP file:    [Calculus_Derivative.zip]

This sketch explores the idea of an integral as an area beneath a curve. Riemann Sums are brought to life as the number of sub-intervals on a user-defined (a,b) interval can be varied between 1 and 1000 using a sliding parameter indicator. Comparisons can be made between right sums, left sums, midpoint, and trapezoid techniques. The user can see the errors diminish as the sum of simply calculated areas approaches the exact area beneath a curve. The sketch includes a tool for shading the region between two curves, whose arguments are 2 points followed by two functions. Keywords: calculus, Riemann sum, right sums, RRAM, left sums, LRAM, midpoint, MRAM, trapezoid, trapezium, converge, area, integral, integration, subintervals, definite integral

download GSP file:    [Calculus_Integral.gsp]
 download ZIP file:    [Calculus_Integral.zip]

Newton's Method for root approximation can go unappreciated by people who get bogged down in a repeated calculation toward an inexact (and sometimes elusive) solution. By outsourcing the calculations to GSP, this sketch offers a new look at a truly impressive little trick. The sketch looks at the premise of the algorithm while drawing attention to its strengths and limitations. Keywords: calculus, Newton, Newton's method, approximation, root, zero, function, iterative, converge, diverge, local linearity

download GSP file:    [Calculus_Newton.gsp]
 download ZIP file:    [Calculus_Newton.zip]

This sketch shows the polynomial which represents the partial sum of the infinite series for a function. The sketch allows quick and continuous movement between each of the first ten polynomials centered at a (Taylor) or 0 (Maclaurin). The function can be changed, but Sketchpad calculates ten derivatives for each new function, which may be time consuming for some functions. By adding one term at a time, the user can watch as the approximating polynomial clings to curve of the original function and increases in its ability to describe the shape. Keywords: calculus, Taylor, Maclaurin, series, infinite series, partial sum, polynomial, approximation, degree, derivative, power series

download GSP file:    [Calculus_Taylor_Maclaurin.gsp]
 download ZIP file:    [Calculus_Taylor_Maclaurin.zip]

This sketch uses sliding parameter indicators to explore the effects of each parameter in a general function. The Sinusoid and Vertex Form Parabola allow the user to see vertical and horizontal dilations and translations, while the Parabola in Standard Form and Factored Form reveals different effects due to parameters. While only two functions are represented here, the sketch could be adapted by a savvy user to include additional families of functions. The underlying principles apply to a general understanding of functions, and this type of sketch uses some of the most valuable aspects of the program. Keywords: pre-calculus, function, parabola, sinusoid, parameters, phase shift, horizontal shift, vertical shift, dilation, translation, period, family of equations, general form, factored form, vertex form

download GSP file:    [Function_Transformations.gsp]
 download ZIP file:    [Function_Transformations.zip]

A group of n arbitrary points on the x-y plane uniquely defines a polynomial function of degree n-1 (provided that no two points share the same x-value). While other sketches explore the curve's response to a change in parameters, this tactile picture offers what is probably a more aesthetic understanding of the nature of the curve. The math behind such a picture is a learning experience in itself, as it (perhaps unexpectedly) involves the calculation of inverse matrices. There are tools included to construct a cubic with four mouse-clicks and a parabola with three, and an additional version of each tool that includes the function equation. Keywords: polynomial, cubic curve, parabola, line, function, regression, linear algebra, matrices, system of equations, inverse matrix

download GSP file:    [Polynomials_Through_Points.gsp]
 download ZIP file:    [Polynomials_Through_Points.zip]

This sketch has been my greatest GSP challenge to date, and it has expanded my impression of what GSP might be used to show. Seven independent points on a scatterplot can be manually moved around or sent to predetermined arrangements. The user can enhance an understanding of correlation and the implication of changes that is elsewhere not granted in such a dynamic and continuous setting. The sketch focuses on a visual examination of concepts whose calculations are otherwise prohibitively time-consuming and distracting. Keywords: scatterplot, LSRL, least squares regression line, correlation, residuals, residual plot, transformation of nonlinear data, modeling, statistics

download GSP file:    [Statistics_LSRL.gsp]
 download ZIP file:    [Statistics_LSRL.zip]