1. An Introduction to Problem solving

2. Fruit Problem
There are three bags of fruit in front of you. One bag contains all apples, one bag contains all oranges, and one bag contains apples and oranges. Each bag is labeled with one of the labels: Apples, Oranges, or Apples & Oranges. However each bag is incorrectly labeled. Your task is to select one bag and reach in and grab one piece of fruit. Having done this and using the information above can you label each bag correctly?

3. What is Problem solving?
Problem solving has long been recognized as one of the hallmarks of mathematics.
“Solving a problem means finding a way out of difficulty, a way around an obstacle, attaining an aim which was not immediately attainable.” George Polya ( ).

4. Good Mathematical problem solving occurs when :
Students are presented with a situation that they understand but do not know how to proceed directly to a solution.
Students are interested in finding the solution and attempt to do so.
Students are required to use mathematical ideas to solve the problem.
Note: A reasonable amount of tension and discomfort improves problem-solving performance. Mathematical experience often determines whether situations are problems or exercises.

5. Some problems to consider
For the traffic light: Have students spell the word spot three times out lout. “S-P-O-T! S-P-O-T! S-P-O-T!” Now have them answer the question on paper “What do you do when you come to a green light?” Have them share the answer with a partner. If you answered “Stop,” you may be guilty of having formed a mind-set. You do not stop at a green light.
For the sheep: A shepherd had 36 sheep. All but 10 died. How many lived? Have students write the answer down and share with a partner. The answer is 10.

6. George Polya (1887 – 1995) Born in Hungary
Received his Ph.D. from the University of Budapest
Moved to the United States in 1940
After a brief stay at Brown University he joined the faculty at Stanford University
He focused on the vital importance of mathematics education
Published 10 books including How to Solve It (1945)
Developed the four-step problem-solving process

7. Four-step problem-solving process
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Look back
Before putting up the four parts, have the students work in pair to develop their own four-step problem-solving process. Have them share with entire class, then show Polya’s process.

8. Step one Understanding the problem
Can you state the problem in your own words?
What are you trying to find or do?
What are the unknowns?
What information do you obtain from the problem?
What information, if any, is missing or not needed?

9. Step Two Devising a plan (Some strategies you may find useful)
Look for a pattern.
Examine related problems and determine if the same technique can be used.
Examine a simpler problem to gain insight into the solution of the original problem.
Make a table or list.
Make a diagram.
Write an equation.
Use guess and check.
Work backward.
Identify a subgoal.
Use indirect reasoning.
Use direct reasoning.

10. Step three Carrying out the plan
Implement the strategy or strategies.
Check each step of the plan as you proceed.
Keep an accurate record of your work.
Implement the strategy of strategies in step 2 and perform any necessary actions or computations.
Check each step of the plan as you proceed. This may be intuitive checking or a formal proof of each step.
Keep an accurate record of your work. Label each step.

11. Looking Back Check the results in the original problem.
Interpret the solution in terms of the original problem.
Determine whether there is another method of finding the solution.
If possible, determine other related or more general problems for which the techniques will work.
Before posting the four points, have students reflect on what “looking back” means in relation to problem-solving. Have them list points for looking back.

12. The Great problem solver “The prince of Mathematics”
Carl Gauss ( ) is regarded as the greatest mathematician of the nineteenth century and one of the greatest mathematicians of all time.

13. Gauss’s Problem
When Carl Gauss was a child, his teacher required the students to find the sum of the first 100 natural numbers. The teacher expected this problem to keep the class occupied for some time. Gauss gave the correct answer almost immediately.
With a partner solve this problem. Be prepared to explain how you arrived at your answer.
The answer is 5050!

14. A Magic Square
Arrange the numbers 1 through 9 into a square subdivided into nine smaller squares like the one shown so that the sum of every row, column and main diagonal is the same. (The result is a magic square.)

15. Round-Robin
Sixteen people in a round-robin handball tournament played every person once. How many games were played?
Work with a partner to solve the problem. Be prepared to share your solution.
What strategy did you use?

16. Round Robin Problem “The Solution”
Sixteen people in a round-robin handball tournament played every person once. How many games were played?
Let’s look at some patterns that develop when we look at some simpler problems.
Let’s label the participants as: A, B, C, D, . . .

17. Round Robin Simpler Problems
Two Players
Three Players
Four
Players
Five
Six AB
AB AC
AB AC AD
AB AC AD AE
AB AC AD AE AF BC
BC BD
BC BD BE
BC BD BE BF CD
CD CE
CD CE CF DE
DE DF EF
Total Number
of Rounds 1 3 6 10 15

18. Round Robin Observation of pattern
So for 16 players it would be:
That is 120!
Remember the Gauss problem
That is 16×15 2 = 120

19. Round Robin General formula
If there were n players, how many tournaments would have to be played?
𝑛(𝑛−1) 2

20. Problems? . . .
"The problem is not that there are problems. The problem is expecting otherwise and thinking that having problems is a problem.“ Theodore Rubin
The best way to escape from a problem is to solve it.--Brendan Francis
Every problem contains within itself the seeds of its own solution.--Stanley Arnold
It isn't that they can't see the solution. It's that they can't see the problem.--G. K. Chesterton
Problems are to the mind what exercise is to the muscles, they toughen and make strong. - Norman Vincent Peale