Posing problems with What-if-Not questions
What if not? questions are so important to investigations that an example
might be useful. This example comes from Steven I. Brown and Marion
Walter, in their book Problem Posing: Reflections and Applications.
The Pythagorean Theorem says that if the edges of a right triangle have
lengths a, b, and c, with c the length of the
hypotenuse, then a2 + b2 = c2.
This theorem statement has several attributes.
- It's about edges.
- It's about triangles.
- It's about right triangles.
- It says something about squares of numbers.
- It says something about the sum of two numbers.
- It says something about equality of numbers.
For each of these attributes you can ask "What if not?" and related "What
if?" questions.
- What if not edges? What if areas?
- What if not triangles? What if squares? What if quadrilaterals?
- What if not right triangles? What if acute triangles? What if obtuse
triangles?
- What if not squares of numbers? What if cubes of numbers?
- What if not the sum of two numbers? What if the sum of three numbers?
What if the difference of two numbers?
- What if not equality of numbers? What if there were a less than (<) instead of equals (=)?
Some of these questions don't lead to deep problems for rich
investigations. But some of them do. For example, the question about sums
of cubes of numbers, when asked about whole numbers, led to significant
historical developments in algebra.
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