Dynamic Visualization of PreCalculus and Calculus Concepts with Geometer's Sketchpad
for AMATYC Workshop
November 5, 1998
Martin Flashman
Department of Mathematics
Humboldt State University
flashman@axe.humboldt.edu
http://flashman.neocities.org
What to do when you have
- an idea on how to do something: Try It!
Abstract: Traditional approaches to pre-calculus and calculus emphasize graphs and functions in static visualizations. Those attending this presentation will discover how Geometer's Sketchpad's dynamic features allow the instructor and the student to create an active environment for exploring functions, graphs, roots, extremes, differentiability, tangent/vector fields, and integration.

1. Introduction.
1. Who are we?
2. Getting familiar with Geometer's Sketchpad layout.
3. Overview of this workshop.
2. Coordinates and Graphs from Formulae.
1. Use the Graph menu to Create Axes.
2. Plotting Points
1. Select point plotting tool and plot points with mouse.
1. Plot free point in the plane.
2. Plot point on X-axis.
2. Select select and translate tool.
1. Select the X-axis with your mouse.
2. Use the Construct menu to construct a point on the object (the X-axis).
3. Select the Label tool.
1. Label one of the points you have constructed on the X -axis.
2. By "double clicking" on the point, change the label of the point to x.
3. (optional) Change the font and size of the label.
3. Finding Coordinates and Calculating with Measurements.
1. Select a point (or two- hold the shift key and use the mouse left button).
2. Use the Measure menu to obtain the coordinate(s) of the selected point(s).
3. Use the Measure menu to calculate the first coordinate of a point on the X-axis. After the screen calculator appears, click on coordinates of the point. Then select OK on the screen calculator or hit the enter key.

4.

[Quicky: Triple click with the left mouse button on the coordinates of the point. Select OK on the screen calculator or hit the enter key.]

5. Use the Measure menu to calculate other values as functions of the first coordinate of the point on the X-axis from the following list:
1. x2
2. x3 - 4x
3. x2 - 3x + 2
4. ex
5. ln(x)
6. sin(x)
6. Exercise: By selecting and dragging the point on the X axis, estimate the value(s) of x for which each of these functions individually has its value equal to
1. 0
2. .5
3. 1
7. Optional: Find the coordinates of a free point in the plane.Use those coordinates to calculuate following values as functions of both of the coordinates of that point.
1. x + y
2. x2 + y2
3. x2 - y2
4. ex ey - ex+y
5. ln(x2 + y2 )
6. sin(x) cos(y)
4. Plotting points from measurements and calculations.
1. Select a pair of measurements, x and f (x) of part C.4, or your own favorite. Using the Measure menu Tabulate this pair. By moving the point on the x-axis, and then "double clicking" on the table, you can add other pairs to the table. (Note here undo/redo on the Edit menu)
2. Select the table of pairs you have made in D.2. Plot the points that correspond to these pairs by using Plot table data from the Graph menu.
3. Select a pair of measurements, x and f (x) [ in that order, remember to hold the shift key down to select the second point] of part C.4, or your own favorite.Using the Graph menu plot a single point from the selected pair of measurements.
5. Playing with controlled points.
1. Display menu features-Trace and Animate - These will be discussed in the presentation.
2. [Hold down the shift key and ] Select the point on the x-axis, the x-axis, and the point (x, f (x)). Using the Construct menu, the graph of the function f can be created using Locus.

3. Using the graph of a function for investigating the derivative and the integral.
1. The derivative.
1. Plot points on the locus.
2. Construct a line (segment) between two distinct points on the locus.
3. Use the measurement menu to find the slope of the line.
4. By moving one point on the curve closer to the locus point for the curve estimate the slope of the curve (the derivative) at that point.
5. Plot the estimated derivative point (x , f '(x)) and then use that point and the Locus construction to graph the estimated derivative. Use the display menu to change the color of the derivative locus.
6. Exercises for the derivative:
1. Notice the usual features of the relation between the derivative and the shape of the curve.
2. For one of the functions in the previous exercises, create points with coordinates (1,0) and (1, f (x)) and construct a thick line segment between these two points. Similarly construct a thick line segment between (2,0) and (1, f '(x)). Describe the motion of these segments as you move the original pont along the x - axis.
3. Plot the second derivative for one of the functions in the previous exercises and examine its relation to the graph of the function.
7. Some Bells and Whistles: Action Buttons and other Edit and Display features:
1. From the Edit menu create Action buttons that hide/show some of the figures constructed so far.
2. From the Edit menu create Action buttons that animates the point on the x axis that controls f (x) for some of the figures constructed so far.
2. The Definite Integral.
1. Plot and mark numerous points on the graph (locus) of a selected function.
2. Plot points on the x axis vertically above or below the far left and far right points on the curve. [Select the point and the x-axis, then the construct perpendicular line using the Construct menu.]
3. Use the points from the previous two constructions to Construct a polygon interior. Measure the Area of this polygon which is an estimate for the definite integral of the function.
4. Measure the first coordinate of the right point of the estimating polygon. Construct (using Locus) the graph of the areas of the polygons as a function of the right hand endpoint for the region.
3. Vectors and Vector Fields -Just a start.
1. Using and making Scripts.
1. From the File menu choose Open. Go to the samples sub-directory, then scripts, and finally utilities. Select arrowopn.gss and open that script.
2. Construct and mark two points on the sketchpad. Now (1) use the STEP button, (2) use the PLAY button, and (3) use the FAST button in the Script window to construct an open arrow (1) step by step, (2) all at once but seeing the steps, and (3) all at once- but FAST!
3. [This will be discussed in the presentation.] Use the Display menu to change the Preferences for "More" to determine a Script Tool Directory. Now use the Script Tool to construct an open arrow.
4. Construct a free point in the plane. Find its coordinates. Now use those coordinates (x, y) to construct the point (x + y, y - 2x). Construct an open arrow with tail (x, y) and head (x + y, y - 2x). You have constructed the vector ( y, - 2x).
5. Select the tail of the vector and move it in the plane. Notice how the vector changes its length and direction.
6. Exercise: By placing a trace on the tail of the vector constructed in part d, trace an integral curve for the vector field determined by this vector that passes through the point (2,1).

4. Graphs from measurements of Sketches.
1. Draw a sketch of a problem figure to be measured.

2.

Keep the figure dynamic.

Be sure you can control the parameters that interest you.

1. Problem: Given a line l and two points A and B not on the line, determine the position of a point P on the line where the total of the distances between P and the points A and B will be smallest.
2. Problem: Given a line l and three points A, B, and C not on the line, determine the position of a point P on the line where the total of the distances between P and the points A, B, and C will be smallest.
3. Problem: Find the area for the rectangle of largest area that can be inscribed in a given right triangle using the right angle vertex of the triangle for one of the vertices of the rectangle.
4. Problem: Find the area for the rectangle of largest area that can be inscribed in a given right triangle using the hypotenuse for one side of the rectangle.
5. More challenging: Given a length AB, determine the dimensions of an isosceles triangle of maximal area that will be enclosed by using the length AB to determine the total length of two sides of the triangle.
6. More challenging: Given a length AB, determine the dimensions of a rectangle of maximal area that will be enclosed by a using the length AB to determine the toal length of three sides of the rectangle.
3. Optional- label some/all of your objects. Write text.
1. You can show and change labels.Use the Text tool (the hand) to show the label of an object. This tool also is used to change the label, write text on the sketch, or change the label on any button, etc.
2. Use labels for objects that makes sense.
4. Use the Measure menu to measure the objects you wish compare.
1. Notice units that appear. ( You can change the units by going to the Display menu and using the Preferences.)
5. Use the Graph menu to plot a point as (x,y) with the coordinates determined by two of the measurements you have selected. (Select the value of the controlling variable first for the first coordinate of the point.)
6. Play with the controlling measurement on the figure.
1. Tables -Measure (Note again here undo/redo)
2. Plot from Tables -Graph
7. Playing with controlled points.
1. Display menu features-Trace and Animate
2. Construct the graph of a function (x, f (x)) using Locus.