Dynamic Geometry Module Lesson 1: What Is Dynamic Geometry?
The term "dynamic" in mathematics refers to the ideas of motion and change. Dynamic Geometry is a new term coined in response to the new software packages such as Sketchpad and Cabri. These products act as a sort of electronic ruler and compass. What really sets a Sketchpad sketch apart from one that you might have made on a sheet of paper with the good old tools of our high school days is not just the accuracy of the constructions. It is the fact that the software remembers relationships among the various components of a construction --- that point P is the midpoint of the segment AB, that circle c has center O and goes through point X etc.
So, for instance, when you construct the circumcircle of a triangle (Open the file ex1_1.gsp found in the Examples folder in the download) you can move any of the vertices of the triangle and see that the circle still circumscribes the triangle. Try this by clicking on a vertex, holding down the mouse, and dragging the vertex around. Notice that you can make the center, O of the circumcircle either inside, outside, or on an edge of the triangle. Can you find a triangle where the circumcenter is at a vertex? What can you say about the triangles where the circumcenter is on one of the edges? These are the kinds of questions you can have students explore when they can see different instances of the same construction just by moving the initial objects (in this case the three vertices of the triangle).
Points equidistant from a pair of points
Let's illustrate how we might guide a student to a simple discovery. Suppose we are trying to find the locus of all point X that are equidistant from points A and B.
We can start with a simple special case. Find all the points that are a given distance from both A and B. The points at distance d from A form a circle (center A and radius d) and the This means that the points at distance d from B form a circle (center B and radius d). Thus the points at distance d from both A and B will be the points of intersection of these two circles.
To illustrate this with a Sketchpad sketch (See file ex1_2.gsp in the Examples folder), you need the points A and B along with a segment whose length we can use for the distance d. On the sketch:
- Select point A and the segment d,
- Construct the Circle By Center+Radius,
- Repeat this process, constructing the circle center B with radius d,
- Select both these circles,
- Construct the Point At Intersection of these circles.
Now we are in the position to ask, "What happens if we change the distance d?" To answer this:
- Select the two points of intersection of the circles drawn previously,
- Choose Trace Points from the Display menu,
- Drag one end (either X or Y) of the segment to change the distance d.
When you finish dragging the endpoint around, Sketchpad will connect the traced points with a "curve." This curve looks for all the world like a straight line, or perhaps two pieces of a straight line. What is the relationship between this line and the segment AB? To make sure that this is not a coincidence, move either A, or B, or both, then repeat the process of dragging an endpoint of the segment. Do you still appear to get a straight line?
At this point you could construct the perpendicular bisector of AB and check try dragging an endpoint again. You should see the traced points lying along this perpendicular bisector. In some sense, this is a proof of the result:
The locus of points equidistant from the two distinct points A and B is the perpendicular bisector of the segment AB.
You, of course, could proceed with a proof of this fact. That can be illustrated by other Sketchpad sketches, but the software can't do everything!!
At this point, let us examine a more sophisticated version (See file ex1_3.gsp) of the sketch we just used, and see how it was constructed. Notice that, in this version:
- when X or Y are moved, the segment remains horizontal,
- there is a button you can double click to animate the changing of the distance d --- clicking anywhere in the sketch will halt the animation,
- the circles are drawn as dashed lines,
- the intersection points are traced in red, so the line connecting them at the end of the animation is also in red.
These are all quite simple enhancements. Try these steps on a new sketch.
- First choose the line tool and draw a horizontal line. (Try holding down the shift key while you draw the line if you have trouble making it horizontal.)
- Select the two points that define the line and Hide them.
- Place the point X on this line near the left end and draw a segment from X to another point on the line about 3/4 of the way across the screen.
- Select the original line and Hide it.
- Place a point Y on the segment showing and construct the segment XY.
- Label this segment d.
- Place the points A and B and construct the circles with centers at these points and with radius d.
- Construct the intersection points and change their color to dark red.
- Select the point Y and the longer of the two segments. Then choose Action Button / Animation ... from the Edit menu and click OK. This will produce an animation button. Drag this to the location you would like.
- Select the hand tool then double click on the animation button. This gives you a chance to edit the message in the button. You are limited to 33 characters.
- Select the longer of the two segments and its right endpoint and Hide them.
Three of the four sketches for this section are actually from the examples that are distributed with Sketchpad itself. The first sketch for this section (See file ex1_4a.gsp) illustrates the formula for the area of a parallelogram. We are assuming that students believe the formula for the area of a rectangle. Area is actually a hard thing to define precisely and the formula for the area of a rectangle is often taken as one of the axioms of area.
This sketch presents a framework for experimentation and discovery, but it does not give any proof of the result it is looking for. The next sketch (See file ex1_4b.gsp) addresses the same issue, the formula for the area of a parallelogram, but it actually includes a proof a text box beneath the sketch. Obviously there are times when one approach is preferred over the other and vice versa.
The two sketches also vary somewhat in the number of objects that are hidden or shown. Remember that if you leave a point or a segment visible, the user can drag it. This movement may affect the way the sketch appears in a manner that does not contribute to its overall effect. The next sketch (See file ex1_5.gsp) illustrates the formula for the area of a trapezoid. The final sketch (See file ex1_6.gsp) illustrates the formula for the area of a triangle. You should experiment with all of these examples. Notice how nicely the sketches behave when the indicated point is dragged. The parallel lines don't move and just the right concept is illustrated. What happens if you move some of the other points in the sketches? These may seem like very simple sketches. Indeed, they are not too complicated, but it does take some practice to get them to behave as nicely as these do. You should try to replicate these sketches. If you are having trouble getting them to work properly, you might try going to the original and using Show All Hidden from the Display menu. This will show you what other objects were constructed and then hidden to make the sketch. It is not always easy to see what is going on, but it may help. If you have messed up the original, don't worry. Just close it without saving and open it again in its original condition! Be sure to contact us at email@example.com if you have trouble here. The techniques you discover here will be very helpful in creating good sketches of your own later.
Complete the exercises in Lesson 1 Homework and submit them to the MTL handin system.