Chapter 9 - Midpoints of the Quadrilaterals

Having learned all the properties of the quadrilaterals that are usually taught in math books, my students went on to explore the quadrilaterals in greater depth. We had previously explored the midpoints of triangles, and now extended that exploration into polygons with more than three sides. In most years, we spent our time with just the qaudrilaterals, but occasionally explored other polygons, such as pentagons, octagons etc. In some years, we used Sketchpad, which allowed them to drag points on their constructed quads, in their explorations. At other times, we would do this project using paper models, which the students constructed using paper, pencil and scissors; in this case they would fold their models to test for congruence, bisectors, and symmetry.

The instructions to the students were as follows:

Construct the following polygons based solely on their definition; for example, a quadrilateral is a polygon with four sides. Here is an example of what your quadrilateral might look like:

Once you have constructed this quadrilateral, construct the midpoints of all four sides; then connect all four midpoints. What kind of figure is formed?

After making your conjectures, draw a diagram, write the "Given" and the "Prove" statements, and prove your conjecture.

Does a relationship exist between the number of lines of symmetry of the original quadrilateral and those of the midpoint quadrilateral? What about rotational symmetry?

Make some generalizations about what you have discovered. Your generalizations and connections should answer these questions: Did some (different) quadrilaterals give you the same midpoint figure? Which ones? Why? What property or properties of the original figures was/were most influential (important) in determining the type of midpoint quadrilateral produced?

In my class, if time permitted, I offered an "extra credit" project in which the students continued the process of connecting midpoints, forming smaller and smaller quadrilaterals to create some very interesting geometric graphics such as the one below:

The students were absolutely thrilled, when the National Council of Teachers of Mathematics published some of their geometric graphics! The poster book, entitled Geometric Designs, contains 6 outstanding graphics, with a detailed explanation for each one, complete with the construction steps for each one.You can see these beautiful designs at the following web address:

http://mathforum.org/~sanders/geometry/NCTMposters.html

The students loved these graphic design projects. They were surprised and delighted that Geometry could be so beautiful, and so interesting! Their enthusiasm for geometry in general was kindled, and the mathematics seemed more creative and useful to them.

"This project helped us understand what kinds of properties of quadrilaterals can lead to other conclusions. For example, if the diagonals of a quadrilateral are congruent, then the figure formed by connecting its midpoints is a rhombus. Instead of having these results told to us by the teacher or from the textbook, we got to figure things out by ourselves. Not only is this more convincing, but we have the satisfaction of knowing that we can prove conclusions on our own. Through this project, we were also able to put what we learned so far into our proofs, especially the Midline Theorem. I'm proud of this project because it shows that our whole group worked together for a good result. We each wrote different proofs, but we made sure to discuss what we came up with so we caught each other's mistakes. Doing all of this work really made me remember the theorems when the test came around, too!" Jenna

Remy had some very interesting observations, which she wrote in her reflections: "While I was doing this project, I noticed that some different quadrilaterals had the same midpoint figure. For example, the midpoint figure of parallelograms and trapezoids are both parallelograms. Likewise, the midpoint figure of both rhombi and kites is a rectangle. And if you join the midpoints of rectangles and isosceles trapezoids, you get a midpoint figure that is a rhombus.

Different quadrilaterals give you similar midpoint figures because their diagonals intersect in similar ways. Take the rectangle and the isosceles trapezoid,. Their diagonals are both congruent, so the sides of their midpoint figure are also congruent. Now look at the rhombus and the kite. Their diagonals intersect because the midsegments are parallel to the diagonals joining the endpoints of the two segments it is the midpoint of. So, if the diagonals intersect at 90 degree angles, then so will the midsegments forming rectangles. So the most important of the properties used in determining the midpoint figure was tghe way the diagonals intersected. And also, the handy little theorem that all midpoint figures of a quadrilateral were parallelograms were parallelograms too.

While I was working on this project I became very well acquainted with the intricate workings of the theorem saying that the midpoint figure of a quadrilateral is a parallelogram - I will never forget it! This project also burned the properties of the special quadrilaterals into my brain and it also helped me practice the way I do proofs. I also feel a great sense of accomplishment when this project was completed! Out of all the projects we did, this was my favorite one."

And Angela also wrote comments along the same lines as Remy's. Angela said "I chose this assignment for my portfolio because I learned so much from it. As with other projects we have done, it was easier for me to understand and remember everything because I had discovered it myself!"

Matthew had the following comments to make: "In terms of how much I learned, this project on the midpoint figures of various quadrilaterals was undoubtedly one of the best. While I had already been familiar with many of the theorems in this course, I had never before encountered or even heard of these midpoint theorems; they weren't even in the textbook! From start to finish I can honestly say I learned a great deal, more than I initially thought I would. I found it challenging as well, and I believe that no one person in our group dominated the thinking. We were able to bounce ideas off one another and confirm or reject possible solutions as they came up. This project only reinforces what the quadrilaterals project first instilled in me, that learning by discovery is a very informative, stimulating, and enjoyable experience."

"A man is like a fraction whose numerator is what he is and whose denominator is what he thinks of himself. The larger the denominator the smaller the fraction."

Tolstoy, Count Lev Nikolgevich (1828-1920)