Stories from the Classroom

Slopes and equations with Grade 9 students

Contributed by: ML Thompson, KCVI;
Nathalie Sinclair, Queen's

The Setting: November 2nd at the Computer Lab at KCVI in Kingston, Ontario. There are 24 PCs. ML had just completed a unit with her students on slopes, parallel and perpendicular lines, and equations of lines. The students were to write their unit test the following day so ML wanted to use this period with Sketchpad in order to review concepts covered during the unit. None of the students have used Sketchpad before.

Principles of Mathematics, Grade 9, Academic (MPM1D) Analytic Geometry identify the properties of the slopes of line segments (e.g., direction, positive or negative rate of change, steepness, parallelism, perpendicularity) through investigations facilitated by graphing technology, where appropriate. Investigating Geometric Relationships pose questions about geometric relationships, test them, and communicate the findings, using appropriate language and mathematical forms (e.g., written explanations, diagrams, formulas, tables); confirm a statement about the relationships between geometric properties by illustrating the statement with examples, or deny the statement on the basis of a counter-example (e.g., confirm or deny the following statement: If a quadrilateral has perpendicular diagonals, then it is a square).

Day 1:

The students had never worked with The Geometer's Sketchpad before so we decide to start with the "Make a Messy Sketch" introduction. For about five minutes, we ask the students to make points, lines, circles, to move them around, to label them, and to play with the Undo/Redo functionality. This would have been a good time to ask them to move two or more objects at a time using the Shift key, but I forget to do this.

I ask the students to go under the Edit menu and select Label options where they can choose to have new objects automatically relabelled. This is often helpful when working without a projector and with novice Sketchpad users so that you can refer to specific objects by name. I then ask them open a new sketch, to draw a line, to select it, and to measure its slope. We discuss various positions of the line and the corresponding slope values. Sarah provides a nice explanation for why the slope of a vertical line is undefined. I now ask them to draw another line, and to measure its slope. We then drag the second line to make it parallel to the first line, noting that their slopes are equal. When I ask them to make the second line perpendicular to the first, they try to make it form a 90 angle with the first. I ask them how the slopes of perpendicular lines are related. After some thought (or silence!) Lucy remembers that "one slope is the negative reciprocal of the other." Together, we translate that into "if you multiply the two slopes, you should get -1." So they use the Sketchpad calculator to multiply the two slopes. Most of them have results close to but not equal to -1. They try to manoeuvre the lines a little and then I tell them that Sketchpad can help make perfect perpendicular lines. I ask them to select one of the points on their first line, and the line (now explaining to them how to select more than one object using the Shift key), and to construct a perpendicular line under the Construct menu. Again, they find the slopes of these lines, multiply them in the calculator and verify the property.

Since they now have three lines on their sketch, I initiate an exploration on the maximum number of intersections for a given number of lines (this fits in very well with the two objectives listed above under "Investigating Geometric Relationships"). As a class, we construct a table on the blackboard starting with 0 intersections for 1 one line, 1 for 2 lines, and 2 for 3 lines. I ask them to construct 4 and 5 lines so that they can fill in the table. Students come up to the board, quickly identifying 6 as the maximum numbers of intersections for 4 lines. One student conjectures 7 intersections for 5 lines but that is replaced by a 9 from a another student and then a 10 by a third student. The class agrees that 10 is the correct number. Kyle suggests a pattern for the table: you add 1, then 2, then 3, etc., to get from one entry to the next.

# of lines	max. # intersections
1	0
2	1
3	3
4	6
5	10
6	?


That lets them conjecture that 6 lines will produce a maximum of 15 interestions. Everyone tries this out on Sketchpad and we find no disagreements. We discuss the fact that this is a compelling relationship but how do we know that it will always extend. Hai Chi offers the following argument: every time you draw a new line, you cross the number of lines you had before and that's why you have to add that number to the previous total. The class accepts this argument. Though Cat counters that you could choose a line that wouldn't cross all the previous lines. Hai Chi says that it's hard to see the crossings on the sketches but because they are infinite in both directions, they actually will cross at some point. Cat corrects him by pointing out that you could draw a line that was parallel to one of the others. But Hai Chi reminds her that we are looking for the maximum number of intersections!

Now it's time to look at equations of lines (and back to the "Analytic Geometry" objectives). I ask the students to open a new sketch and to draw a line on it. I also ask them to Create axes under the Graph menu. Using the Hand tool to write text on the sketch, I ask them to write down their best guess of the equation of the line. The room falls silent as the students look for the y-intercept and count out the slope. Many need help to remind them how to write the equation of a line and others need help interpreting the axes correctly. Once they are happy with their conjectures, I ask them to switch seats with their neighbours "to get a second opinion." After they have discussed their guesses, I ask them to select their lines and, using the Measure menu, to find its equation. There are many aahhhs, especially for those who were quite close. If I were to do this activity again, I might first ask the students to position their lines so that they fell on integer points on the x- and y-axes. Those who erred were able to understand what they had done wrong, some just miscounted while others were confused by negative signs.

We have a class discussion of how we should go about finding the equation of the line on Sketchpad, some pointing out sneaky methods like using Sketchpad to calculate the slope first.

Next, I write two equations on the blackboard:

• y = 3x + 12,
• y = -¹/₃x - 4
and ask the students to construct these two lines on their sketches. Only a few students take the easy way out and draw random lines, measure the equations, and manoeuvre them into place. Most of the students first try to locate the y-intercept (we have to show them how to change the scaling in order for them to be able to see 12 on their sketches). And then they count out the required slope. The second line is more challenging; they have a more difficult time counting out the fractional slope. I suggest to some that they first get a ballpark idea of how the slope will look (what direction will it slope in, will it be a steep slope or not?) and then measure the their approximate slope using Sketchpad. This provides them with a slope in decimal form which they have difficulty reconciling with the fractional form. So we use the calculator to see what 1/3 would give in decimals. This allows them to create the second line.

With only about 10 minutes left in the 75 minute class, we do a quick recap of the strategy for drawing a specific line. Then we invite the students to look at some of the sample sketches in the presentation folder. They especially like the Life's a Highway sketch.