Problem Posing: What – If – Not? Glenn Waddell, Jr.

Why?: I have struggled to find a way to TEACH my teachers to create rich problems.

The Art of Problem Posing By Stephen I. Brown & Marion I. Walter

An example:

Level 1: Attributes Level 2: What-If-Not Pot is a cylinder Pot has a diameter of 14 inches There is sauce in all but 2 inches of the pot Meatballs are spherical Meatballs are 2 inches in diameter Any others? Level 2: What-If-Not What if attribute 2 is not 14 but 18? What if attribute 4 is not spherical but square? Etc.

Now you try! The problem: x2 + 6x + 5 = 0 Level 1: Attributes The exponent is an integer The degree is 2 There are 3 terms on left The solutions are integers The middle term > last term There is an equal sign There are two plus signs

The problem: x2 + 6x + 5 = 0 Level 2: What-If-Not What if attribute 4 is not known? x2 + ?x + 5 = 0

x2 + ?x + 5 = 0 Can you give 4 solutions that are integers? Can you give 4 solutions that are NOT integers? Can you give 4 solutions that are not real? How does the graph reflect the different solutions? Can we generalize something about the different solutions?

The problem: x2 + 6x + 5 = 0 Level 2: What-If-Not What if attribute 2 is a 4? What else needs to change to keep the pattern? x4 + 6x2 + 5 = 0

x4 + 6x2 + 5 = 0 Can we substitute something to simplify the problem? What if the exponents are not integers, but rational? (attribute 1) x1/4 + 6x1/2 + 5 = 0

What – If – Not Substitution of values (what if it is not an x, but a 3?) Exploring Conics (what if it is not a +, but a -?) Exploring rational expressions / equations Truly creates a spirit of “this is a game” and wraps up with “What did you learn about ________?”