Lab 6 – Algebra and GSP
Section I — Trace and animation
1. Let’s make an ellipse! The geometric definition of an ellipse is that it is the locus of points the sum of whose distances from two given points (foci) is constant.
a. Draw a segment representing the sum of the distances. Draw two points for the foci.
b. Divide the segment into two unequal parts, representing the distances from the foci.
c. Now construct circles using the two lengths from the foci. Adjust the size of the two parts and observe!
2. We can now jazz things up a little.
a. Click on both of the points of intersection and choose “Trace” from the Display menu. Adjust the size of the two parts and observe!
b. Explore what happens when you move the two foci closer or further apart, or when you increase or decrease the total distance.
3. Now let’s automate the process. Click on the segment and the point on the segment (while holding shift) and choose “Action Button” from the Edit menu, selecting “Animation” from the pop-up menu. Double click the button to see what happens.
4. Consider drawing other conic sections using their definitions.
5. Consider how could animation speed up your explorations.
a. Example 1: Consider the relationship between area and perimeter. To do this, construct a triangle with a given height and base. That is, its vertex will be on a line parallel to the base. Measure the area and perimeter of the triangle. Then animate the vertex on the parallel line, and observe!
b. Find another sketch you have created where animation would be useful.
Section II — Coordinate geometry
6. Start GSP. Under Graph choose “Make axes”. Move the origin and the point beside it and observe what happens.
a. Draw a point. Then under Measure choose “Coordinates”. Move the point around and observe what happens.
b. Under Graph choose “Plot Points…” Add one or more fixed points, then click on Plot.
c. Again, move the origin and the point beside it and observe what happens.
7. Draw a line. Then, under Measure, choose “Equation”. Move the line around and notice what happens.
8. Now let’s do a dynamic investigation:
a. Draw a circle with its center on the y-axis. Then draw a line from the center to some point on the circle. Measure its equation.
b. Animate the movement of the point on the circle. Make one or more observations.
c. Animate the movement of the point on the y-axis. Make one or more observations.
d. Highlight the equation. Under Graph, change “Equation Form” for line to Standard Form, and explore what happens to its coefficients.
9. You can also do equations of circles. Draw a circle. Then, under Measure, choose “Equation”. Move the circle around and notice what happens. What sort of investigation might you set up?
Section II — Dynamic graphing
10. Graphing a relationship. Let’s create a graph comparing the area and radius of a circle.
a. Open a new window with a set of axes. Draw a circle. Then measure its area and radius.
b. Click on the area measurement. Then under the Graph menu, choose “Plot measurement…” Click on the button that says, “On the y-axis”, and then click OK. Move the circle around and note what happens.
c. Now click on the radius and again “Plot measurement” from the Graph menu. But this time, choose “On the x-axis”. Move the circle around and note what happens.
d. Click on the intersection of the horizontal and vertical measures, and trace that point. Change the dimensions of the circle and observe!
NOTE: Some versions of GSP let you combine parts (c) and (d): Click on the measurement for the x, and then the measurement for y. Then select “Plot(x,y)” under Graph
e. It might be nice to animate this. But to do that you would have to start over, this time constructing the circle so that one of its points is on a line. Otherwise, there is nothing to animate!
11. Graphing using a formula. We can also create a graph using a formula, let’s say y = 1/x. Here’s one approach:
a. Start with a fresh set of axes. Draw a point on the x-axis to control our x-value. Measure the Coordinates of the point (see 6(a)). “Calculate” the x-coordinate of that point — open the calculator, click on the ordered pair, then select the x-value from the pop-up menu. Slide the point back and forth on the x-axis and watch its x-value change.
b. To make this clearer, you may want to rename the x-coordinate to be simply x. Select the “finger” cursor, then double-click on the measurement. Select text format and change the name to x. You may also want to disable the units.
c. Now calculate the formula for y; in this case, y = 1/x. Use the calculator to enter the formula; click on the x-value to enter x..
d. As in #10 above, “Plot” the x-value on the x-axis and the formula value on the y-axis. Trace their point of intersection. You can then animate the original point along the x-axis for really cool results.
e. Try some other equations.
12. Setting up sliders to control constants. Let’s consider y=ax
a. Draw a segment and a point on the segment and measure its distance to the left endpoint. Following the method from 11(b), rename this value to be a. (Depending on your scale, you may also want to redefine the unit of measurement under Preferences…)
b. Now follow the method of 11(a-b) to set up your x-value.
c. Use the calculator to calculate y, as in 11(c). Then plot the x and y values, as in 11(d).
d. Look at different values for a. How does increasing a change the graph? What if a is less than 1?
e. Improving the performance of the slider:
- What if we wanted a to have negative values? Here is one trick: Subtract half the length of the segment from your previous measurement.
- You may also want to multiply or divide by a value in order to make the answer more reasonable. (In fact, you can add a second segment to control the size of the unit.)