Getting Started with Interactive Geometry

Stephen Arnold

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Begin with a point.
What could be simpler or more enticing?

Welcome to the elegant and beautiful world of interactive geometry, in which the static images of the past make way for strange flexible creations which offer insight into many of the most fascinating parts of mathematics.

Be warned, however. As teachers, it lies within our power to make the extraordinary appear mundane.

Some approach the blank screen of interactive geometry with a recipe book of step-by-step instructions, ensuring that they "cover the work required". This will most certainly guarantee boredom and rapid dissatisfaction on the parts of both students and teacher.

The secret to effective use of technology in the classroom has always been the same. Learning to use the new tools is about learning to ask new questions.

This is the challenge of open-ended tools. They make trivial much that we previously devoted great amounts of classroom time to. Too often, "the first instinct of educators is to couple the technology to their old methods of instruction." (Papert, S. [1980] Mindstorms: Children, computers and powerful ideas. Brighton, UK: Harvester Press). Like Seymour Papert, our "vision should be of something much grander".

The few simple activities which follow may offer some guidance in this new approach to the teaching of geometry. The questions being asked are simple yet rich. They invite students to engage with important ideas and the role of the teacher shifts from being the source of knowledge to being co-learner. Our primary responsibility, however, remains unchanged. We need to ensure that students take from the learning experience at least what we aimed for them to achieve. Hence we need to have them reflect upon what they discover, record it in their own words, make meaning of their experience, and compare with others in order to deepen and validate their new understandings.

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Learning Activity G1: Getting Started
(CabriJr™)
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1. Begin with a point. Choose POINT from the drop-down F2 (WINDOW) menu and simply click wherever you wish to place a point. Now add three more.
2. Choose QUAD from the same menu and then click once on each of the points you have created. You may save time by clicking twice on the third point and the quadrilateral closes off automatically.

3. Press the green ALPHA key when close to one of the points and watch the pencil change into a hand, allowing you to drag this point (and the rest of the quadrilateral) around. You have not created a quadrilateral, you have created many possible quadrilaterals!

4. Press F3 (ZOOM) and choose MIDPOINT. Click once on each of the opposite corners (vertex points) of the quadrilateral. This constructs the midpoint of each of the two diagonals.
5. Now again choose a point, press ALPHA, and drag it around trying to make these two midpoints coincide.
6. What shape have you formed? Check with others: did everyone produce the same shape? Try this again using another quadrilateral until you are sure about your result.
7. Write down what you have observed and what this tells you about quadrilaterals.
8. A theorem is a statement in mathematics that may be true, but needs to be proved. Work with a partner or a small group to write down a theorem that you have discovered from this activity. How might such a statement be proved to be always true!

Learning Activity G2: How Square!
(GeoMaster™)

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In how many ways can you construct a square?

AIM: To encourage students to explore the various menus and features of their interactive geometry package while investigating properties of a square.

METHOD: After checking that students all know what a square is, give them the question above and let them loose! You may choose to offer a reward for the student or group with the most different methods.

PROCESS: Most students will begin with a segment, representing the first side of the square. Some methods may include constructing a perpendicular line through one end point, then using a circle to mark the position on this perpendicular, which corresponds to the length of the first segment.
Alternatively, try rotating a copy of the first segment around one end through 90 degrees.

If you are using GeoMaster™, why not go straight to the Regular Polygon command on the DRAW menu? (It begins by default with 6 sides, but pressing + and - changes the number of sides).
A special prize should go to anyone who thinks in terms of diagonals! Begin with the diameter of a circle. Create another equal diameter perpendicular to the first. The four intersection points with the circle form a square! Any more?

Learning Activity G3: What is your angle?
(CabriJr™)

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AIM: To develop students' understanding of angles both acute and obtuse by introducing an important circle property to which they will return later.

METHOD: This simple activity not only introduces Thales' Theorem (the angle on the circumference of a semi-circle is always a right angle) it breaks students from the usual perception of angle by deliberately not constructing the arms.

PROCESS: Begin once again with a point. Anywhere.
Then construct a segment (independent of the original point), put in the midpoint and use this as the center of a circle with the segment as diameter.
Finally, measure the angle from one end point of the diameter, to the free point, then to the other diameter end point.
You may now begin to move the point around the screen, observing the way that the angle changes in relation to the position of the circle.
Record and discuss observations.
Later, you may choose to allow students to create the triangle defined by the three points and make explicit the nature of the angle. Some students will be quite uncomfortable until this is done. Talk about angles.

Learning Activity G4: All shapes and sizes
(CabriJr™)

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Once again, begin with a point. Place it in the bottom left corner of the screen, and construct two short segments coming out from this point as shown. The angle they make is not important at first, but we will now measure it.

To measure this angle we need to click in turn on each of the three points which define it. Make sure that the central point (your original point) is the second point that you click on. So, choose F5 (GRAPH): Measure: Angle. Click on the last point you created, then the middle (or vertex point) then the third point. Place the angle somewhere out of the way, and then press ENTER to leave it there.

Now move to the other side of the screen and create another segment as shown. We are going to rotate this segment using the angle we have just created as the angle of rotation.

Choose F4 (TRACE): Rotation, then move down a little in order to select the segment you have just created (as shown). Move up to click on the end point of the segment &n; this will be the center of rotation. Finally, move across and again click on the three points making up our angle (do this in the same order that you just used to measure the angle). A new segment will appear, rotated by the number of degrees that you have chosen.

Now choose F4: Rotation again, select the new segment, and rotate it about its free end point by the same angle. Repeat this until you have closed your figure. Keep creating sides, or press CLEAR to leave Rotation mode.

Move to the top point of your angle, press ALPHA to grab it and move it carefully to the right, until the angle reads 60 degrees, or close to it. What do you observe? Now move to the left until the angle is close to 120 degrees. You may be able to adjust this angle more accurately by changing the length of the angle arm slightly (moving up and down).

Try moving until the angle is about 150 degrees, then continue to rotate the last side until you form a closed figure once more. Investigate the many regular polygons you have created!

Some other Investigations

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1. Begin with a quadrilateral: any quadrilateral.
Construct the midpoint of each of the four sides and join these as shown.
What may be observed about the new quadrilateral formed.
Will this always be true?

2. Once more, begin with a quadrilateral.
Construct a square on each side as shown.
Find the midpoint of each square and join these four points.
What may be observed about the quadrilateral formed?
Will this always be true?

What would happen if regular shapes other than squares were used (such as equilateral triangles)?
Did you know that the famous general, Napoleon Bonaparte, was an amateur mathematician. He is remembered for his work related to a simpler version of this figure (with equilateral triangles on each side of any triangle) and the general result is known as Napoleon's Theorem.

3. We all know about the center of a circle. But how many centers has a triangle?
Join each vertex of any triangle to the midpoint of the opposite side. These three segments are called medians and each meets in a special point called the centroid. This point is the center of gravity for the triangle: you should be able to balance the triangle on a pin at this point. Cut a triangle out of cardboard and try this.

Taking the perpendicular bisector of each side (as shown) yields three lines which all meet in a single point, called the circumcentre. Try to find out more about this special point.

See what other centers you can discover and then research these to find out more about them.

Challenge: If Australia is drawn using a series of straight lines (turning it into a polygon) would it be possible to find the centre of gravity of this shape?

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