1. Pythagoras, proof, and problem posing
Math 653 – November 7, 2012

2. Outline for Today’s Class
Talk about Inquiry Project progress
Finish talking about proofs of the Pythagorean Theorem
Talk a bit about the process of problem posing
Explore some extensions and alternatives to the Pythagorean Theorem and/or Pythagorean Triples

3. Problem Posing & Pythagoras
For the next 40 minutes, and again next week, we are going to try out the procedures outlined in The Art of Problem Posing to further analyze the Pythagorean Theorem and its “cousins.”
Problem posing is simply a way to practice the art of thinking of extensions and other new interesting questions related to a topic.
Our goal is to gain some new strategies that you can use in the context of your inquiry projects and classroom teaching.

4. Problem Posing – Informal
Let’s start by asking ourselves the question, “What new math questions could arise as we start to think about the Pythagorean Theorem”?

5. Problem Posing
The Art of Problem Posing outlines five steps to pose and solve new mathematics problems:
Level 0. Choosing a Starting Point
Level 1. Listing Attributes
Level 2. What-If-Not
Level 3. Question Asking or Problem Posing
Level 4. Analyzing the Problem
We will go through these steps, but a bit informally.

6. Level 1. Listing Attributes
In this step, we “notice” things about our topic under consideration (in this case, the Pythagorean Theorem).
Just as expert noticing in the classroom (the ability to easily see and interpret students’ actions and responses to fluently manage class discussions, etc) is a skill that doesn’t develop overnight, noticing attributes of a mathematical idea can be a non-trivial task.
Let’s start by brainstorming. What can we say about the Pythagorean Theorem? What are its attributes?

7. Level 2. What-if-Not
Now, for each of the attributes we developed previously, we can list several alternatives to that attribute.
In other words, we are looking for ways to make that attribute “not true” in a specific sense.
Let’s try it.

8. Level 3. Asking Questions.
Once we have discovered variations on each of the attributes, we can think about potential questions to ask about each of the “What If Nots” or alternatives.

9. Level 4. Analyzing the Question
Pick a question and problem-solve. What did you learn about the new question, original question, or additional pertinent questions?

11. Some particularly fruitful explorations
What about if we use different shapes, rather than squares, constructed on the sides of a right triangle?
What about if we consider the value of for different kinds of triangles?
What about if we consider this as a direction for making Pythagorean Triples? What can we say about these triples?
What about if we ask for additional integers to be involved, e.g.:
A right triangle with all three sides integers and the altitude (to the hypotenuse) also an integer
Triangles with integer sides and a 60 degree angle
Triangles with integer sides and integer coordinates

12. Inquiry Projects
For the last half hour of class, please meet with your inquiry project support groups and talk about your progress over the past couple of weeks.
I will set a timer for 8 minutes for each person to share and get feedback. I would recommend spending 3-4 minutes giving a brief overview of your work, then 4-5 minutes engaging in conversation.
Learn from each other!