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

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

3. The Art of Problem PosingBy
Stephen I. Brown
&
Marion I. Walter

5. An example:

6. 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.

7. 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

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

9. 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?

10. 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

11. 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

12. 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 ________?”