Don’t let the name of the program—The Geometer’s
Sketchpad has a host of tools for exploring algebra, trigonometry, and calculus,
both symbolically (with equations) and graphically. In this tour, you’ll sample
several of Sketchpad’s algebra features.
What You Will Learn
• How to create an x-y coordinate system and measure the coordinates of
• How to define and plot functions
• How to plot two measurements as an (x, y) point in the plane
• How to construct a locus
A Simple Plot in the Coordinate (x-y) Plane
Presenter: Allow 10 minutes for the first two sections.
At the heart of “visual” algebra is the x-y (or Cartesian,
or coordinate) plane. In this
section, you’ll define a coordinate plane, measure a point’s coordinates on it, and
plot a simple function.
1. Open a new sketch and choose Define Coordinate System
2. Using the Point tool, create a point somewhere other than on an axis.
3. With the point still selected, choose Coordinates from the Measure menu.
Drag the point to see how the coordinates change.
|Actually, you can plot
a function without
first creating a
|Now that there’s a
coordinate system, let’s plot a
simple equation on it: y = x.
4. Choose Plot New Function from the Graph menu.
5. Click on x in the Function Calculator (or type x
from your keyboard). Click OK.
6. Select the plot you just created and the
independent point (whose coordinates you
measured). Then choose Merge Point To Function
Plot from the Edit menu. Drag the point along the
plot and observe its coordinates.
|You may notice that ->
the plot’s domain is
restricted. To change
the domain, drag
the arrows at either
end of the plot.
|7. Drag the unit point—the point at
(1, 0)—to the right and left and observe
how the scale of the coordinate system changes.
8. Release the unit point when the x-axis goes from about –10 to 10.
Plotting a Family of Curves with Parameters
Plotting one particular equation, y = x, is all well and
good. But the real power of
Sketchpad comes when you plot families of equations, such as the family of lines
of the form y = mx + b. You’ll start by defining parameters m and b and editing
the existing function equation to include the new parameters. Then you’ll animate
the parameters to see a dynamic representation of this family of lines.
|A parameter is a ->
type of variable
that takes on a
|9. Choose New Parameter
from the Graph
menu. Enter m for Name and 2 for Value
and click OK.
10. Use the same technique to create a
parameter b with the value –1.
The New Parameter dialog box (Mac)
|Edit Function ->
appears as Edit
Edit Plotted Point
when one of those
types of objects is
|11. Select the function equation f (x) = x
the equation itself, not its plot) and choose
Edit Function from the Edit menu.
12. Edit the function to be f (x) = m ! x + b. (Click on the parameters m and b in
the sketch to enter them. Use * from the Function Calculator or keyboard
[Shift+8] for multiplication.) Click OK.
13. Change m and b (by double-clicking them) to explore several different
graphs in the form y = mx + b, such as y = 5x + 2, y = –1x – 7, and y = 0.5x.
You can learn a lot by changing the parameters manually,
as you did in the
previous step. But it can be especially revealing to watch the plot as its
parameters change smoothly or in steps.
14. Deselect all objects. Then select the parameter
equation for m and choose
Animate Parameter from the Display menu. What happens?
15. Press the Stop button to stop the
16. Select m and choose Properties from the
Edit menu. Go to the Parameter panel
and change the settings so they resemble
those at right. Click OK.
17. Again choose Animate Parameter.
18. Continue experimenting with parameter
animation. You might try, among other
things, animating both parameters
simultaneously or tracing the line as it
moves in the plane.
The Parameter Properties dialog box
Presenter: Stop here and answer questions. You may wish to
y = mx + b using sliders instead of parameters (perhaps using sliders from the sample
sketch Sliders.gsp). Allow about 15 minutes for the remainder of the activity.
Functions in a Circle
How does the radius of a circle relate to its
circumference? To its area? These are
examples of geometric relationships that can also be thought of as functions and
You’ll start by constructing a circle whose radius adjusts continuously along a
19. In a new sketch, use the Ray tool to construct a horizontal ray.
20. With the ray still selected, choose Point On Ray from the Construct menu.
21. With the Arrow tool, click in blank space to deselect all objects. Then select,
in order, the ray’s endpoint and the point constructed in the previous step.
Choose Circle By Center+Point from the Construct menu.
|In each case, use ->
the Arrow tool to
select the circle (and
nothing else), then
command in the
|22. Measure the circle’s
radius, circumference, and area.
23. Drag the circle’s radius point and watch the
Next we’ll explore these measurements by plotting.
24. Select, in order, the radius measurement and the
circumference measurement. Choose Plot As (x, y) from the Graph menu.
Can you see your point? If not, it might be off the screen.
|In a rectangular ->
grid, the x- and
y-axes can be scaled
|25. Choose Rectangular Grid from the Graph | Grid
Form submenu. Drag the
new unit point at (0, 1) down until you can see the plotted point from the
|Choose Erase ->
Traces from the
Display menu to
erase this trace at
|26. Select the plotted point and choose Trace
Plotted Point from the Display
menu. Drag the radius point and observe the trace.
In situations such as this (where you trace something as a point moves along a
path) you can often get a smoother picture by creating a locus.
|A Sketchpad locus is ->
a sample of possible
locations of the
selected object. To
change the number
of samples plotted,
select the locus,
from the Edit menu,
and go to the
|27. Select the plotted point and the
then choose Locus from the Construct menu.
What is the slope of the line containing this
28. Repeat steps 24 and 27, except this time
explore the relationship between the radius
and area measurements. How do the two
curves compare? For what radius do the
numerical values of a circle’s circumference
and area equal each other (ignoring units)?
• Graph pairs of parallel lines and show that their slopes are the same.
• Graph pairs of perpendicular lines and show that their slopes are negative
• Graph a parabola of the form y = a(x – h)^2 + k, using parameters for a, h, and k.