This project hinges on a question that is not usually explored in the traditional geometry curriculum. But it is an interesting question, and one that challenges students to do some serious thinking on a topic for which they will probably not find answers in the textbook!

The questions I asked were these: "What would it take to prove two quadrilaterals congruent? How many pieces of information would you need? How do the conditions that determine similar triangles relate to the conditions for proving quadrilaterals congruent? What are the similarities and differences between proving triangles congruent and proving quadrilaterals congruent?"

My students worked in small groups of 3 or 4, exploring this idea. In the early years, before The Geometer's Sketchpad, we experiment with quadrilaterals made of drinking straws and elastic string. The way you create a quadrilateral this way is to cut the straws to the appropriate lengths, and thread the elastic through them, as shown below:

The elastic allows students to change the shape of the quadrilateral, within the limits of the lengths of the straw sides. Using this small model, they can convert their square into a parallelogram, as shown below:

Students will enjoy making these, and by manipulating them they will be able to easily observe certain properties of the quadrilaterals, and can explore the differences between the first and second quadrilateral above. For example, as you can see in the above diagram, the diagonals that were congruent in the first diagram are no longer congruent in the second. Therefore the students can clearly see that two quadrilaterals can have the same side lengths but not be congruent quadrilaterals.

My students enjoyed this project, and had some interesting comments to make, in their reflections. Mark had this to say: "As soon as I tried playing with my 'straw quad' I realized that it was going to take more than 3 pieces of information (like we used to prove triangles congruent) to prove quadrilaterals congruent. With 2 triangles, if the sides of one are congruent to the sides of the other, then it is immediately obvious that the triangles are congruent, but obviously this is not going to be true with quadrilaterals!"

Students can also make straw trapezoids (isosceles and non-isosceles), pentagons and hexagons, etc. The polygons become a bit unwieldy as the number of sides increase, but they are easily manipulated if placed horizontally on the table.

Julie wrote "Triangles have 3 sides and 3 angles - this is the least amount of sides and angles that a polygon can have. This property of triangles allows us to prove them congruent using only 3 parts of the triangle, like SAS etc. We cannot use the exact same shortcuts to prove quadrilaterals congruent, but we explored this, and found that you can use similar shortcuts for quadrilaterals but it takes more than 3 pieces of information. We thought it would probably take maybe 4 because a quad has 4 sides and 4 angles. It just seemed logical. In exploring this idea, it seemed like a good idea to draw one diagonal of the quadrilateral, which would break it into 2 triangles. Besides parallelograms, we explored other quadrilaterals too. Here are some of the diagrams we drew with trapezoids."

These straw polygons were used to make conjectures about the relations between the quadrilaterals, which were then verified by mathematical proof. The students drew diagrams for all of the other possibilities as well, and then wrote proofs for each of their conjectures formed by experimenting with their constructions. In summarizing their conclusions, Jeff, Ethan and Karen wrote "Triangles have 3 sides and 3 angles - this is the least amount of sides and angles that a shape can have. This property of triangles allows us to prove triangles congruent using only three parts of the triangle, like two sides and the angle between (SAS), etc. We can't do that with quadrilaterals, but we can use similar methods with quads: we figured out that the following shortcuts would work with quads: SASAS, ASASA, SSASA and AASAS.

The teacher can create and use these models to display and discuss the properties, but models work best when created and manipulated by the students themselves. It does take a bit of class time and require materials, but the effort is well worth the results, both in student participation and in learning.

Students and teachers will find more information on this and other projects at my web page Creative Geometry: http://mathforum.org/~sanders/creativegeometry/

In may of the assignments in this class, I asked the students to write reflections on what they learned, including how they felt about the process. They included their reflections in their portfolios, which they submitted at the end of each quarter. In her reflection on this project, Elena said "I chose this project for my portfolio because there was so much to talk about in our group, and so much you could say. I also liked it because it is not a topic that is usually included in geometry, so I felt like we were real mathematicians, exploring unknown territory instead of just doing what the book said. It was really hard getting all of our ideas down on paper, but it was kind of fun too. We really tried to make it understandable, and when I read it again, I'm really impressed with what we did! "

Shaleigh had this to say about the project: "This assignment was very confusing at first. There didn't seem to be any way to approach the question/problem! Since we couldn't see anything we could do at first, we decided to draw a couple of triangles and quadrilaterals so we could look at the differences. We got several workable ideas by doing this, and it started moving! It was really great to have other students' opinions; together we really took off. Our group worked really hard on this, and there were lots of ideas and opinions. It wasn't like only one person was doing all the work; we were all getting into it and coming up with 'what if' and 'what about that' and stuff. It was also kinda fun trying to make our work better, and explaining everything really clearly. I think a good (but friendly) argument is really a great way to figure things out."

David wrote the following reflection, with his feelings about this project: "This piece of work was very challenging. It challenged me by making me think about quadrilaterals and how they work, and to compare them with triangles. I then had to figure out how to explain what I discovered. It helped a lot that I could talk to the people in my group, and we argued a bit but it was a productive argument, not an angry one. It was interesting to hear what the others had to say, and I sure learned a lot by the discussion. Here's some of what we did: we started with squares and rectangles and made some preliminary conclusions based on regular figures. Then we realized we coundn't just look at regular polygons, we need to look at ones where the sides and angles weren't neccesarily congruent. Yikes! So much to do. But then we had a breakthough - we could draw a diagonal and break the quad into two triangles! We were off and running then. Another brainstorm - how will we know when we are finished? Answer: By making a chart listing all the possibilities!"

I really enjoyed the students' reflections on this project, especially the one that said "I chose this project to put in my portfolio because it was a very big effort. Our whole group worked hard on this and we practically fainted from exertion! But, even with all the panic, argument, and effort, this was a really fun assignment! When we finally finished it, we wanted to holler 'hooray'."

As a teacher, I can ask no more of my students than this effort and enthusiasm. It was hard work for all of us, but in the end, a job well done.

"No human investigation can be called real science if it cannot be demonstrated mathematically"

Leonardo da Vinci (1452-1519)