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XY Encounter

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Teacher's Guide: Part 1: Conclusion

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Map of Teacher's Guide	Part 1	Part 2	Part 3	Part 4
	Introduction	Introduction	Introduction	Introduction
	Pause Point 1	Pause Point 1	Pause Point 1	Pause Point 1
	Pause Point 2	Conclusion	Pause Point 2	Conclusion
	Pause Point 3		Conclusion
	Conclusion

Mathematically Speaking:

When Max and Keisha first plot the points and connect them, the graphs do not intersect. Then Keisha hits the “extend” button and the graph continues in a line. In effect, the graphs are represented by lines, and lines extend in both directions without end.

Since a graph is represented by a line, and that line can be extended, you may wish to discuss the four quadrants. At this point, you may introduce locating and naming points using integers as ordered pairs.

Coordinate grids have four quadrants, (Figure 1) labeled with Roman numerals in a counter clock-wise direction. Most students this age have only worked in Quadrant I.

Figure 1

Introduce students to the fact that the x-axis and the y-axis are like integer number lines.

On the x-axis, numbers to the right of 0 are positive and to the left negative. On the y-axis, numbers above 0 are positive and numbers below 0 are negative (Figure 2).

Figure 2

Students should always read the x-value first and then the y-value. That is why it is an ordered pair.

So, (+3,+4) is in Quadrant I, (-3,+4) is in Quadrant II, (-3,-4) is in Quadrant III, and (+3,-4) is in Quadrant IV. Many students need practice in naming ordered pairs and plotting points.

Max’s map only showed Quadrant I. If all four quadrants were shown, the map would show a larger region. The graph would look like Figure 3.

Figure 3

Each value in Max’s Table 1 would be on the line in Quadrant 1. Have students try different points to see if they would be in the extended table or fit Max’s rule: (x, x+1).

For example: Pick a point on the graph, Ex. (9,10), and ask students if Max’s rule fits (Figure 3). Would this value be on the table if the table continued? How about (100,101)? Would this value be on the extended table? [Yes. The rule: (x, x+1) still holds.] Where would these points fall on the extended graph? [Allow students to predict.]

If the pattern (rule) in each table is followed, students will encounter other values including negative numbers.

What happens when x = 0? [ Y = 1, the values would be on the table and it still fits the rule.] Also negative integers will be used in naming ordered pairs. Is (-3,-2) on the line? [Yes, the values would also be listed in Table 1 when extended, and it satisfies the rule.]

You might explore how the rest of the points were determined in order to extend the line. [By substituting a value for x, the value of y can be determined by adding 1 to the x-value, (x, x + 1), and entering the value in the table or graphing it as an ordered pair. The computer can do this rapidly.]

Related/Extension Activities

1. Line Positions
   When Max and Keisha first plot the points and draw their lines, the lines do not intersect – until they extend the lines. Discuss the geometric property that lines extend infinitely in both directions.

   Since all lines extend infinitely in both directions, will two lines always intersect at some point? Can the students give examples of lines that will never intersect? You’ll find this is an excellent entrée into the concept of parallel lines, which will never intersect because they never move closer together or farther apart. The distance between parallel lines is always equal.

   Another situation in which the lines will never intersect involves two lines on different planes (Figure 4).

   
   Figure 4

2. Coordinate Grids
   Use a coordinate grid, which involves graphing in all four quadrants, to create a map of a fictitious city in which the x-axis is labeled “Main Street” and the y-axis is labeled “Broadway.”
   1. Label the streets along the x-axis, beginning at the origin, as: East 1st St., East 2nd St., and so on. To the left of the origin, along the x-axis, label the streets: West 1st St., West 2nd St., and so on.
   2. Label the streets along the y-axis, above the origin: North 1st Ave., North 2nd Ave., and so on. Below the origin, label the y-axis: South 1st Ave., South 2nd Ave., etc.
   3. Ask students if a house on the corner of North 2nd Ave. and East 6th St. is the same house as the one on the corner of North 6th Ave and East 2nd Street.
      To establish the idea of integers, sometimes called “directional numbers,” ask questions like:
   4. What would happen if we eliminated the directional labels east and west or north and south?
   5. How could you tell the difference between the numbers to the right of the axis and those to the left?
   6. How could you tell the difference between the numbers above the axis and those below it?