I-MATH
Polygons
I-MATH
Polygons
In our textbook, the polygons chapter begins with the sum of the interior and exterior angles of a polygon in general, and then proceeds to regular polygons, and then to quadrilaterals and the parallelograms. The book states the definition of a polygon, and then two theorems about polygons, with brief algebraic proofs.
I prefer to begin the study of polygons by handing out a worksheet and having the students work together in collaborative groups to find first the numbers and then the "formula". We then discuss the words "inductive reasoning" and make a comparison to "deductive reasoning" as done in our two-column proofs.
Filling out the worksheet provides the students with an experience of inductive reasoning, and it is an activity that they enjoy. I wind up the class with a deductive proof of the same formula.
To experience this hands-on discovery project yourself, click on the link below and print the page - then forget what you know about the sum of the angles of a polygon, and "discover" it for yourself. I would like to know if you find it a more meaningful approach to the concept than just giving them the formula!
If you have access to the Geometer's Sketchpad software, there is
an activity that visually demonstrates the sum of the exterior angles
of a polygon. First, construct a polygon such as the one on the left
below. Then, click and hold down on the arrow tool (in the tool panel
at the left of the Sketchpad screen) then drag to the right to get
the "Dilate" tool. Then, using the dilate tool, click in the interior
of the polygon and drag slowly. If you imagine you are walking away
from the polygon, then it can appear to be "shrinking" into the
distance. Watch the exterior angles - not just one of them, but all
of them - what seems to happen to their sum as you move further and
further away from the polygon (by "shrinking" it)?
"Properties of Quadrilaterals" is a topic that contains many, many theorems, and a very large quantity of information. It is a trying task to memorize it all, and I believe that even if we can get our students to memorize all the properties, they will soon forget them. Fortunately, there is another way to approach this portion of the geometry course.
Each of the quadrilaterals, parallelogram, rhombus, rectangle, trapezoid, and isosceles trapezoid have special properties. Using Sketchpad, or compass and straightedge, students will benefit from interactive investigations, and can discover for themselves what properties each of the quadrilaterals has. A good way to do these investigations is to let the students work in groups of three or four on one computer, and use the files to test by dragging and observing, measuring when needed. In this way they can see for themselves what happens in each of the quadrilaterals - are the diagonals of a parallelogram congruent? Maybe, but maybe not always . . . they seem to be congruent in certain parallelograms . . . in a rhombus? And if so in a rectangle, then certainly also in a square! If you do not have the Sketchpad software, students can still investigate the properties using "paper folding", compass and straightedge constructions, or other hands-on methods. You will find many suggestions at the following web page: http://mathforum.org/~sanders/creativegeometry/2.4strawpolygons.htm
You may recall that we discussed this investigation before in I-MATH, in the chapter called Investigation. If you would like to review this investigation, and get a printable worksheet that students may use to investigate the properties of the quadrilaterals, click on the link below to go back to the I-MATH page on Investigations, which documents this project and has a link to the worksheet:
Having compiled a list of all their conjectures, the students should be encouraged to ask themselves just what it is about the rectangle that would make the diagonals congruent, and why the diagonal of a rhombus are congruent, to list just two examples, and you might want to ask them to write proofs, if yours is a proof-oriented course.
My students have done some beautiful geometric graphics using regular
polygons. An example of a graphic based on regular hexagons is shown
below, and the instructions for creating graphics like this are on
the link that follows:
There are many interesting applications of polygons, and your
students might find it valuable to do some exploration on the
internet. You might want to begin by visiting my Geometry Pages on
this topic: