This page is dedicated to the spirit of open exploration with Geometer's Sketchpad. This activity is the first one I do using GSP without giving my PST a ready made sketch to interact with. During my course, I will cover the different classifications of triangles, properties of quadrilaterals, and some general ideas about polygons. These activities give my PST an opportunity to see some of these ideas first hand, to discover [or more likely remember] some facts about polygons, give them a chance to get acquainted with using GSP and it's tools, and give me a platform for talking about the difference between drawing and constructing objects. I have found that having a discussion about the more formal mathematics of polygons is more fruitful after they have had an opportunity to engage in some of the property challenges that are found on this page.
My classroom approach to this lesson is to give a hand out with the property challenges on it and let my students choose the activities are most interesting to them to examine. I would suggest that before opening these up to your class that you spend some time yourself working on them in GSP. I have found that many of the same issues and thoughts that occurred to me when doing these are the same as comments brought forth by the PST. This list below is not meant to be exhaustive in any way, and I encourage people to go beyond this list into other areas that they think are interesting as well.
Activities and Challenges
Triangles
1. Make five different triangles. How are they different?
2. Construct the following: a right scalene triangle, an obtuse
isosceles triangle, and equilateral triangle.
3. Find as many triangle combinations using the three angle and side classifications as you can. Are all of them possible? If not, which ones cannot be made?
Quadrilaterals
1. Make some different 4-sided shapes. How are they different?
2. Make a 4-sided shape with two opposite sides the same length
but not parallel.
3. Make some 4-sided shapes with three equal sides.
4. Make some 4-sided shapes with two pairs of equal sides.
Polygons
1. Draw some 5-sided shapes. Make one of them have four congruent
sides
2. Draw some 6-sided shapes. Make some of them have 1 pair, 2
pair, 3 pair, or no pairs of parallel sides.
3. Make some shapes with square corners. Can you make one with
3 sides? 4 sides? Five, six or seven sides?
4. Make a shape with one or more lines of symmetry or with rotational
symmetry.
Discussion and Recap
I try to encourage my PST to engage in as any different activities as possible. Hopefully there will be enough diversity in choices in your class to get most of the challenges covered. Due to time constraints, I have not required more than an informal oral summary of their findings to be shared during our discussions. Another option is to have the students do write-ups of their explorations similar to ones done in Jim Wilson's EMAT 6680 class, or in the Intermath courses he helped designed.
I like to use these explorations before having a formal discussion of triangle classification and the properties of quadrilaterals. Hopefully when discussing the different combinations of triangles classifications that are possible, not only will my PST be able to say which combinations are not possible, but will be able to give some sort of geometric reason for why they are not. During this discussion we will be able to get ideas concerning the sum of the interior angles of a triangle as well as the notion that larger angles of a triangle are opposite larger sides. I hope that the exploration with quadrilaterals will help my PST to understand the inherent properties of all parallelograms. When we discus the classification of quadrilaterals, we will refer back to some of their sketches to examine the properties of parallelograms, rhombi, rectangles, squares, and trapezoids. I have found that having dynamic sketches to manipulate and measure helps them to understand how certain constructions cause certain characteristics. For example, that having all right angles in a quadrilateral goes hand in hand with having congruent diagonals.
There are a lot of different geometric topics that can be accessed and explored through these types of activities. If you have the time in your classroom,whether it is one of content or methods, I highly recommend this approach as one that allows PST to make sense and organize their geometric knowledge.