 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum            Session 3, Part C:
Definitions and Proof

In This Part: Definitions | Understanding Definitions | Dividing Polygons into Triangles
Triangles in Convex Polygons

People go about understanding mathematical definitions in different ways; the steps they take may vary. Here's one way:

 Step 1. Read the definition more than once. Step 2. Identify what "things" the definition is talking about. Step 3. Generate a test case. Step 4. Determine if the example fits the definition. Step 5. Find examples that do not fit the definition. Step 6. Try to generalize the examples to create an image of the full concept. (In the case of convex figures, you might think about curved figures, three-dimensional figures, etc.)

Convex, like many other mathematical ideas, has several different, but equivalent definitions. Not every proposed definition will work, though, and some definitions are better than others -- they may be more clear, use fewer words, or be easier to test. Problem C2 Which of these definitions work for convex polygons? A polygon is convex if and only if...

 a. all diagonals lie in the interior of the polygon. b. the perimeter is larger than the length of the longest diagonal. c. every diagonal is longer than every side. d. the perimeter of the polygon is the shortest path that encloses the entire shape. e. the largest interior angle is adjacent to the longest side. f. none of the lines that contain the sides of the polygon pass through its interior. g. every interior angle is less than 180°. h. the polygon is not concave.   Video Segment In this video segment, Ric and Michelle discuss whether a given statement about polygons works for convex polygons, then share their ideas with the whole group. Watch this video segment after you have completed Problem C2. How did Ric and Michelle test to see if their given statement about polygons works for convex polygons? If you are using a VCR, you can find this segment on the session video approximately 12 minutes and 48 seconds after the Annenberg Media logo.      Session 3: Index | Notes | Solutions | Video