How to Solve Problems

1
How to Solve Problems

• Alison Kiddle

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How to Solve It

• How to Solve It (1945) is a small volume by
  mathematician George Pólya, suggesting the
  following steps when solving a mathematical
  problem
• First, you have to understand the problem
• After understanding, then make a plan
• Carry out the plan
• Look back on your work.

3
Understanding the problem

• Restate the problem in your own words
• What area of mathematics is this?
• What exactly am I being asked to do?
• What do I know? What do I need to find out? What
  am I uncertain about?

4
Devising a plan

• Work out the first few steps before leaping in!
• Have I seen something like it before?
• Is there a diagram I could draw to help?
• Is there another way of representing?

5
Carrying out the plan

• Im STUCK! How do I get UNSTUCK?
• Work backwards
• Subgoals
• Simpler problem, special cases
• More general case
• Act it out
• Guess and check
• Has the plan failed? When is it time to abandon
  the plan and try something else?

6
Looking back

• Have I answered the question?
• Does it make sense?
• Have I fully solved the problem?
• Is my solution consistent?

7
Control

• Am I getting stressed?
• Am I spending too long on this?
• Is my plan working?
• Could I move on to something else and come back
  to this later?
• Am I focussing on the problem?

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Solving an unseen problem

• On the next slide is a problem that I have only
  seen very briefly. I will attempt to solve it,
  and talk through my mental process.
• Try to note down any problem solving techniques
  you notice me using.

9
The picture shows four squares on the sides of a
quadrilateral. Prove that the if the diagonals of
the quadrilateral intersect at right-angles then
the sum of the areas of one pair of opposite
squares equals the sum of the areas of the other
two squares.