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The Product of the Coordinates Is Constant
This activity allows students to explore situations in which points with a constant product of x-coordinate and y-coordinate are graphed. With TI-Navigators display, students can determine that a curve is formed from such points. This curve is in quadrants 1 and 3 if the product is positive or in quadrants 2 and 4 if the product is negative. The Learning Check enables the teacher to get immediate feedback from the students, thus giving opportunities to correct any errors in understanding.
Before the Activity create a Learning Check that has the questions:
If a graph has points for which the product of the x-coordinate and the y-coordinate is 24, what kind of curve is it?
( Line
( Curve in one quadrant
( Curve in two quadrants
If a curve has points for which the product of the x-coordinate and the y-coordinate is 24, its graph has either an x-intercept of 24 and a y-intercept of 1 or an x-intercept of 24 and a y-intercept of 1.
( True
( False
If a curve has points for which the product of the x-coordinate and the y-coordinate is 24, its graph passes through which quadrant(s)?
( Quadrant 1
( Quadrant 2
( Quadrant 3
( Quadrant 4
Start TI-Navigator and begin the class.
For this activity, we want the students to have a free moving cursor without marking any points on the plane. We also want the students to be able to see the coordinates of their points as they move their cursors around the screen.
1. To begin Activity Center on the computer, select the Activity Center icon from the toolbar or from the Tools tab.
You may also select Activity Center from the Tools menu (or Alt+A).
2. On the calculator, choose 1: ACTIVITY CENTER from the TI-NAVIGATOR HOME screen.
Calculators will display the message WAITING FOR TEACHER.
3. If the window settings have been previously saved, load that settings file. If not perform the following steps.
a. In Activity Center, it may be helpful to edit the window settings.
Choose the Edit Window Settings icon or choose Edit Window Settings from the Edit menu.
Use the following friendly window settings to make each calculator pixel count as 0.5 unit: Xmin=-23.5, Xmax=23.5, Ymin=-15.5, Ymax=15.5, Xscl=5, Yscl=5.
Press OK when done.
b. Select Points from the dropdown menu next to Contribute.
c. Then, select the Configure button to specify the configuration settings.
Number of points per student: 0.
Select the Display coordinates option.
Same Step Size for X and Y = 0.5.
Press OK when done.
Note: The settings for this lesson are now complete, and we can save them to be used at any time. Select File>Save>Save Activity Settings. Give the settings file an appropriate name, and save it in a convenient location on the computer.
4. Press Start Activity icon to begin.
5. Make the projected computer screen hidden from the students by switching the input to the document camera, covering the projector lens, switching to a different application on the computer, or some other means. Instruct the students to move their cursors to a point where the x-coordinate and the y-coordinate have a product of 12.
6. Ask the students for their opinions about what will be formed by all the cursors.
7. Reveal the screen so the students can see that their cursors all lie on a curve which is entirely in the first quadrant or the third quadrant.
Note: It may be that some points dont lie on the same line as all the other points. If so, do the following.
Select the List-Graph Tab, and study the numbers in the lists to see if they follow the verbal rule. Note the coordinates of any point that doesn't line up.
Use the Select Arrow to highlight that outlying point for discussion. Discuss with your students how to make the correction, and then edit either coordinate as necessary. Note how the change is live.
8. Repeat steps 5 through 7, except changing the number for the product of the coordinates. If you use a negative number for the product of the coordinates, the graph lies in quadrants 2 or 4. Dont use 24, as it is the one used in the Learning Check.
9. Send a Learning Check file to the class to check for their understanding. The first question is a multiple choice one. (They should select curve in two quadrants.)
If a graph has points for which the product of the x-coordinate and the y-coordinate is 24, what kind of curve is it?
( Line
( Curve in one quadrant
( Curve in two quadrants
The second question is a true-false type. (They should select false.)
If a curve has points for which the product of the x-coordinate and the y-coordinate is 24, its graph has either an x-intercept of 24 and a y-intercept of 1 or an x-intercept of 24 and a y-intercept of 1.
( True
( False
The third question is a multiple choice one for which there is more than one correct answer. (They should select quadrant 2 and quadrant 4.)
If a curve has points for which the product of the x-coordinate and the y-coordinate is 24, its graph passes through which quadrant(s)?
( Quadrant 1
( Quadrant 2
( Quadrant 3
( Quadrant 4
10. Collect the answer files from the classroom. Display the slide show of student responses and discuss the results.
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