Geometer’s Sketchpad Tour 7: Algebra Anyone?
Adapted by Geoff Hagopian from Key Curriculum Press’ The Geometer’s Sketchpad – Learning Guide.
Geometer’s Sketchpad has many tools for exploring the relations of algebra with other branches of mathematics such as algebra, trigonometry and calculus. Here you’ll find some explanation for the algebra features.
Ideas in this Tour:
· Creating an xy-coordinate system in Geometer’s Sketchpad.
· Defining and plotting functions.
· Plotting x and y measurements as an (x,y) point in the plane.
· Constructing a locus of points.
Using Geometer’s Sketchpad you can plot simple functions and measure coordinates.
1. In a new sketch choose Define Coordinate System from the Graph menu.
2. Using the Point tool, create a point somewhere not on an axis.
3. With the point selected, choose Coordinates from the Measure menu.
4. Now plot a simple equation, say, y = x, by choosing Plot New Function from the Graph menu.
5. Click x on the dialog box keypad and click OK. Voila, the graph of y = x.
6. Select both the independent point and the line y= x and choose Merge Point to Function Plot on the Edit menu. Drag the point (1,0) to scale the axes so you see something like what’s shown here:
Plotting a Family of Curves with Parameters
Here you’ll animated parameters slope/intercept parameters for a family of lines.
7. Choose New Parameter from the Graph menu. Enter m for the name and 2 for the value.
8. Similarly, create parameter b with value -1.
9. Select the function f(x) = x (the equation may be to the left of the plot on the sketchpad page) and choose Edit Function. Either right click and choose Edit Function from the pop-up menu, or simple double click on the equation.
10. Redefine the function with the formula f(x) = m.x + b. Enter the parameter values by clicking on them in the sketch itself. You will have something like this:
11. Deselect all objects select parameter m and choose Animate Parameter from the Display menu. You can use the Motion Controller to adjust speed and direction.
12. Stop the animation, right click on m and choose Properties. Adjust these to match the those shown at right. Click OK.
13. Do Animate Parameter from the Display menu again.
Functions in a Circle
Here you’ll construct a circle whose radius varies continuously along some straight line and construct functions related to the properties of the circle.
14. Create a new sketch and use the Ray tool to construct a horizontal ray. Holding down the shift key helps keep the ray horizontal.
15. With the ray selected choose Point On Ray from the Construct menu.
16. Deselect all and then select the endpoint of the Ray and the point constructed on the Ray, in that order. Choose Circle by Center+Point from the Construct menu. You can also use the Compass tool for this. You’ll have something like what’s shown to the right:
17. Selecting the circle (circumference) you can measure the radius, circumference and area from the Measure menu.
18. To plot how these quantities change in relation to one another, select, in order, the radius and the circumference measurements and then choose Plot As (x,y) from the Graph menu. A coordinate system appears and the point with coordinates (radius, circumference) is plotted.
19. Choose Rectangular Grid from the Graph | Grid Form submenu. Drag the vertical axis unit point up and down to find the plotted point from the previous step.
20. Select the plotted point and choose Trace Plotted Point from the Display menu. Drag the circle radius our and in, observing the patter of the traced points. Is it a straight line with slop 2pi extending up to the right from zero? Anytime you want to erase the trace, right click on the page and choose Erase Traces from the context menu.
21. Deselect all, and then choose in order the plotted point and the radius point. Choose Locus from the Construct menu. You may want to do an Erase Traces (ctrl+B) from the Display menu. Instead of disjoint blobby points, you see a continuous smooth line.
22. Repeat steps 18-21 except to explore the relation between radius and area or circumference and area.
Looking back, the key to getting started with coordinate geometry in GSP is to use the Define Coordinate System command from the Graph menu. The type and scale of the coordinate system created depends on your selections, as shown in this tableK
Square coordinate system centered on the selected point with default unit scale.
Square coordinate system centered on the selected circle with unit scale defined by the circle’s radius.
One defining distance
Define Unit Distance
Square coordinate system centered on a default origin with unit scaling determined by the defining distance.
One point and one defining distance
Define Unit Distance
Square coordinate system centered on the selected point with unit scaling determined by the defining distance
Two defining distances
Define Unit Distances
Rectangular coordinate system centered on a default origin, with horizontal unit scaling determined by the first selected distance and vertical units scaling determined by the second selected distance.
One point and two defining distances
Define Unit Distances
Rectangular coordinate system centered on the selected point, with horizontal unit scaling determined by the first selected distance and vertical units scaling determined by the second selected distance.
Nothing or anything other than the above
Define Coordinate System
Square coordinate system with default origin and unit scaling.