Making a diagram is a simple technique that is a good starting point in teaching problem solving. "Doodling" while you think is useful! Information is continually documented, which helps to keep track of all discoveries and see patterns which might not be immediately obvious.

There are two diagrammatic techniques which are especially useful:

a) Scaling - where the precise information can be converted to a scale diagram. Solutions can be determined by the size, shape, amount etc. indicated by the resultant diagram.



b) Sketching - where the information is arranged visually to produce a pattern and a possible solution

In a game of Pacman there are 15 rows of 10 power points. Calculate the number of power points Pacman devours as he travels the following route from the start point:  3, right 2, up 6, right 3, up 4, right 1, down 9, left 6.
Describe how far away Pacman lands from where he started.

Teaching Strategies
Read all the problem through before commencing.
Do each instruction or step in isolation. Don't try to predict outcomes too prematurely.
Be specific as you answer the questions.

Working the Solution
3+2+6+3+4+1+9+6=34

Ends four power points from the start.

As the list suggests, this strategy arrives at a verifiable solution by hypothesizing possible answers, checking back to see which fits the problem, and modifying answers from the results of previous checks.

In this strategy the teacher's role is to guide students towards:
a) a probable or likely starting point.
b) working in the right direction with large enough gains to quickly solve the problem.
i.e. The guessing game: I'm thinking of a number between 1 and 1000. Can you guess it in 10 tries?
The most successful students achieve this by eliminating as many numbers as possible with each guess, e.g. 500 lower, 250 higher.

Billy has 56 marbles in his collection. If he had 14 more catseyes than bullets, how many of each marble did he have?

Teaching Strategies

Decide what you are trying to find out.
Read the question for clues of where to start, i.e.
a) In this question, the number of catseyes and bullets is 'given':
Number of catseyes = bullets + 14,  Number of bullets = catseyes - 14.
b) There are more catseyes than bullets.
As there are more catseyes than bullets, make the first guess about the number of catseyes, and make the estimated number higher than half of the marbles:
Half of 56 marbles = 28, therefore catseyes = 30?
catseyes = 30
bullets = 30-14=16
If there were 30 catseyes, the total number of marbles would be 46, 10 less than the actual total.
Halve the difference between the two totals, and add it to the previous guess:
30 catseyes + 5 = 35.
bullets = 35 - 14 = 21.
Total marbles equals 56.
Therefore, the second guess is correct. Billy has 35 catseyes and 21 bullets in his collection.

This strategy requires students to set out the information given on an orderly chart, graph or table. When set out, the data should suggest a pattern or part of a solution that can be completed by filling in the missing information.
The teacher's role is to assist students in the creation of the most appropriate headings to display all the information provided.

Jane and Ben go to the same dancing centre that operates 7 days a week. Jane goes every third day while Ben attends every second day. If they meet at the school on Sunday, when would they meet again?

Teaching Strategies

Read all the problems, be aware of unstated factors (e.g. operates only during the weekdays?)
Work out which way to present the table - days or visits.
Create symbols for the information and fill in the table.
Answer: They would meet again on Saturday.

While similar to the previous technique (using a table or chart), this technique is normally used when there is far more information to present. The information needs to be set out more systematically to present the array of probable solutions. Students need to be encouraged and allowed to create lists and jottings, rather than carrying all the permutations in their heads.

When 5 friends meet they each shake hands with each other. How many handshakes are exchanged?

Teaching Strategies

Read the problem carefully. Imagine what will happen.
Ascertain how many variables there are, e.g. five people - but only 4 actions as you don't shake hands with yourself.
Look for repeats, 1 with 2 is the same as 2 with 1