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The Quotient of the Coordinates Is Constant
This activity allows students to explore situations in which points with a constant quotient of coordinates are graphed. With TI-Navigators display, students can determine that an oblique line is formed from such points. This oblique line always passes through the origin with a slope equal to either the constant quotient or its reciprocal. 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 line has points for which the quotient of the x-coordinate and the y-coordinate is 4, what kind of line is it?)
( Horizontal
( Oblique
( Vertical
If a line has points for which the quotient of the x-coordinate and the y-coordinate is 4, its graph has a y-intercept of ____ and a slope of ____.
If a line has points for which the quotient of the x-coordinate and the y-coordinate is 4, 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 quotient of the x-coordinate and the y-coordinate is 2.
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 lie on an oblique line through the origin with either a slope of 2 or a slope of EMBED Equation.DSMT4 .
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 difference between the coordinates. Dont use 4, 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 oblique.)
If a line has points for which the quotient of the x-coordinate and the y-coordinate is 4, what kind of line is it?
( Horizontal
( Oblique
( Vertical
The second question is a fill-in-the-blank type. (They should have 0 in the first blank and 4 or EMBED Equation.DSMT4 in the second.)
If a line has points for which the quotient of the x-coordinate and the y-coordinate is 4, its graph has a y-intercept of ____ and a slope of ____.
The third question is a multiple choice one for which there is more than one correct answer. (They should select quadrants 2 and 4.)
If a line has points for which the quotient of the x-coordinate and the y-coordinate is 4, 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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