Stories from the Classroom

Three days bridging from slopes to medians with Grade 9 students

Contributed by: Shirley Baird, KCVI;
Nathalie Sinclair, Queen's

The Setting: October 24th at the Computer Lab at KCVI in Kingston, Ontario. There are 24 PCs (one per student, but not all in working condition...). Shirley had just completed a unit with her students on slopes and was about to start with the product rule for perpendicular lines. But all of a sudden the computer lab became available and it was 'now or never' for Sketchpad. We had planned on starting with some triangle properties but Shirley didn't want the students to experience such a big disconnect between their work in class and the Sketchpad activities. So this is what we did...

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 illustrate and explain the properties of the interior and the exterior angles of triangles and quadrilaterals, and of angles related to parallel lines; determine the properties of the sides and the diagonals of polygons (e.g., the diagonals in quadrilaterals, the diagonals of regular pentagons, the figure that results from joining the midpoints of sides of quadrilaterals) through investigation; determine the properties of angle bisectors, medians, and altitudes in various types of triangles through investigation; 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 have never worked with The Geometer's Sketchpad before so we decided to start quite simply. and work with some ideas that they are already familiar with from their work on slopes in the classroom. I ask each student to draw a (long) line segment on their sketches and to measure its slope by first selecting the line then chossing Slope under the Measure menu. We then discuss positionings that give rise to negative, positive, zero, and undefined slopes while they manoeuvre the endpoints of their line segments around to watch the changing value of the slope. We try to explain why the slope becomes undefined when it is a vertical line by watching it grow larger and larger as it approaches the vertical.

I then ask the students to draw another line segment that intersects with their first line segment and to measure its slope as well. We discuss how many angles have now been formed and how they are related. I ask them to drag their line segments so that they are perpendicular to each other. The students do this by eye-balling, so I challenge them to figure out how they could prove that their lines were really perpendicular. Many suggest that we could measure one (or all) of the angles and check whether they meaure 90 degrees. So we do this by selecting the three points that define the angle and choosing Angle under the Measure menu and although some of the students have perfect (!) perpendicular lines, others are a few decimals places off. I then ask them how they could figure out whether they were perpendicular if they weren't able to measure the angles. Not many suggestions. I ask those who have perpendicular lines whether the slope measures provide any clues. Still no suggestions. So i tell them that i have a trick: I ask them to find the product of the slopes of the lines using the Sketchpad calculator (Select Calculate under the Measure menu and click on the values of the slopes in your sketch--they will appear in the window on the calculator). Some are suprised to see that the product = -1. Of course, many students have configured their lines to be horizontal and vertical, in which case the product is undefined, so we ask them to drag their lines into an oblique position. They then drag their line segments around to check whether the product of -1 is particular to perpendicular lines or not. We then talk about what would happen if you took the product of two parallel lines--how big and how small could the product get? Now, since we already have two line segments drawn, I ask them to draw a third that would be parallel to one of the first two lines segments. This allows us to see how Sketchpad can construct parallel lines and also allows us to discuss opposite, inside-outside, and transversal angles. And of course, the students tell me that they have already learned this last year.

Having spent some time with perpendicular and parallel lines, we can now segue into some geometry. I ask the students to make a rectangle on their sketches. Of course, everyone freehand draws a rectangle with the segment tool. I ask those that think they have a rectangle to put their hands up and I show them how I can "ruin" their rectangles by dragging a vertex. So then they want to know how to draw a rectangle that I can't ruin. I ask them to think about the special properties of a rectangle and most of them mention either its parallel or perpendicular features. With a little help, they use both perpendicular and parallel lines to construct rectangles. They are proud when I can't ruin them! They also want to know how to get rid of the long lines and replace them with segments. So we show them how to hide lines and construct segments between the vertices of the rectangle.

Now that everyone has a rectangle on their sketches, we talk about other properties of the rectangle, how it relates to the square, etc. I then ask them what they think the sum of the interior angles of the rectangle is. They tell me that's easy since each angle measures 90 degrees. One student points out that a rectangle is two triangles and since the interior angles of a triangle add up 180 degrees, then the interior angles of a rectangle should add up to 360 degrees. I ask them whether they think that holds for all quadrilaterals. Opinions are divided. So we investigate. Some students draw (not construct) special quadrilaterals like, others just draw an irregular quadrilateral. When they add up the angles, they are convinced the sum is always 360 degrees, except for special cases: the quadrilaterals that are 'poked in' or 'caved in,' that is, concave quadrilaterals.

At this point, half the computers crash so we ask one half of the class to team up with someone in the other half... I then ask them whether they could guess what the interior angles of a pentagon would add up to. Some students investigate the pentagon while others make hexagons, heptagons, or octogons. They notice that each time you move up one side from the triangle, you have to add another 180 degrees. The bell rings just before we are able to express this idea algebraically.

Day 2, October 25th, 2000

Ah, class starts so much more smoothly when they know how to login! We start where we ended last day, trying to express the relationship for the interior angles of a polygon. Following our reasoning from yesterday, we formulate the following expression: to get the sum of the interior angles of an n-gon, you add n-3 times 180 to 180. I tell them that in the textbook they write that the sum of the interior angles of an n-gon is n-2 times 180 and ask them whether we are saying the same thing or whether the textbook is wrong. Although some immediately announce with pride that the textbook is wrong, others explain why it's the same thing... but they prefer their own formulation.

Now we move on to look at the sum of exterior angles. We first define what we mean by exterior angle and then we hand out this sheet of instructions for the students to follow:

Today we are going to look at the exterior angles of a polygon. Follow these directions: 1. Select the ray tool (hold down the segment tool and move your mouse to the right when you see the ray tool appear). 2. Construct one ray, then starting at a point on your first ray, construct another. Keep doing this until you have four, five, or six sides. Make sure you construct your last segment so that it joins up with the starting point of your first ray. 3. Label your rays using the hand tool. 4. On each ray, outside your shape (say, a pentagon), construct a point. 5. Measure all the exterior angles. 6. Find their sum using the calculator.

I ask 1/3 of the class to investigate quadrilaterals, another 1/3 to investigate pentagons and the remaining third to investigate hexagons. They conjecture that we will find results similar to the interior angles result.

Many students have trouble correctly constructing the rays, neglecting to make the last one connect to the first. It might be worthwhile to make this clearer in the instructions or to display how to do this on an overhead projector. As students start obtaining the constant sum of 360 degrees (for convex polygons), I circulate and show them how to dilate their polygons in order to 'see' how the angles collapse into a circular arrangement, hence the 360 degree sum. In order to do this, "Select all" on your sketch, choose the dilate arrow, and drag to the center. The students liked to think of this process and being able to cut out all the angles and to rearrange them around a common vertex. A couple of students play with the idea that if you use a huge number for n, the polygon would look like a circle and they already know that a circle is 360°.

As all groups are completing, we discuss their results and compare them to the interior angle results.

We now move on to the second part of the class: investigation of the diagonals of a quadrilateral. We provide the students with these instructions:

Now we are going to look at the diagonals of different polygons, and especially the quadrilateral. Using your segment tool, draw a quadrilateral on a blank sketch. Use your segment tool to construct the diagonals of the quadrilateral. Using your quadrilateral, answer the following questions: 1. Which quadrilaterals have diagonals that are equal in length? 2. Which quadrilaterals have diagonals that bisect each other? 3. Can you find a quadrilateral that has one diagonal shorter than two of its sides? Can you ever find a diagonal that is shorter than all four of its sides?

Naturally, the students begin by trying to manoeuvre their quadrilaterals by dragging their vertices around to force the diagonals to be the same length. But this leads them to conjecture that the rectangle is a good candidate for question 1. With a little prompting, they construct rectangles as they had done yesterday and verify this conjecture. Some immediately think to check the properties of the diagonals of a parallelogram but they can't think of any other special quadrilaterals to test (apparently many of them have had little geometry before grade 9 and are unfamiliar with some geometric terms). So we brainstorm different types: trapezoid, rhombus, and kite. The students then construct and investigate some of these. It takes many students a little longer to construct the rectangle and to measure its lengths and angles so by the end of the period only a handful have progressed to question 3. Many students have also forgotten what bisection means and they rely mostly on visual proof rather than measuring the appropriate lengths.

Day 3: October 26^th

Today, Shirley wants to move on to the triangle investigation which is described in David, Leslie, and Nathalie's classroom story. Since we only have this one class left in the computer lab, we decide to do an abridged version. Since the students have little geometry vocabulary we constantly ask them to label their objects (like a centroid, a median, an orthocentre, a bisector, a perpendicular) using the hand tool. Instead of re-writing the same classroom story, we will just offer a few observations. The students responded very well to using pre-made sketches and liked to be able to "Show all hidden" in order to see how the mystery points P and Q had been constructed. As was the case with David and Leslie's class, these students were very intrigued by the behaviour of the orthocentre and were quick to point out when and why the orthocentre travelled outside of the triangle. Similarly, they were able to find relationships between the points P, Q, and R (centroid, orthocentre and circumcentre), and were particularly intrigued that Q and R were always on opposite sides of P.

I found that this class was a little less comfortable with Sketchpad and they also found it difficult to create their own mystery points. We have come up with two possible explanations for this. The first is that David and Leslie's class had watched me use Sketchpad on an overhead projector in the classroom a few times before they came into the computer lab and tried to use Sketchpad themselves. This might have given them a better sense of what is possible in Sketchpad and might have reduced the initial load of learning how to use Sketchpad. The second is that I also had an overhead projector in the computer lab for David and Leslie's class and often showed them the beginning of a construction or a few instructions before they began on their own. This meant that less time was spent attending to individual problems for the students who are less able to follow only verbal directions.