It's not that I'm so smart, it's just that I stay with problems longer.

                                                                      --Albert Einstein

STRATEGY 5: LOOK FOR A PATTERN

One of the most famous patterns in mathematics is known as the Fibonacci series, named for a mathematician who lived in Italy in the 13h Century. Fibonacci introduced this pattern by posing a problem:

A pair of rabbits, one male and one female, are put into a pen. After two months they have two offspring, one male and one female. They continue to have an additional two offspring every month thereafter, always a pair, one male and one female. This pattern continues: after two months, every pair of rabbits start to reproduce and every month thereafter they have a pair of offspring. After one year, how many pairs will there be?

The solution produces a series of numbers now known as the Fibonacci series.

If you see the pattern that develops month by month, you can easily predict how many pairs there will be after 12 months (144) and after 13 months (233). Each succeeding number in the series is the sum of the previous two numbers.

When you see a pattern you can make a prediction, and that is the essence of the problem solving strategy: see the pattern, make a prediction.

Here is an example of a pattern and a prediction:

The Problem

In their biology class, Ayse and Mehmet learned how to count a population of yeast cells. Using a special counting microscope, they counted the cells every hour and entered their data in a table.

Their teacher told them that the population would stop growing and remain stable at about 500 cells. At what time would Ayse and Mehmet discover that the population had stopped growing?

The pattern

Ayse and Mehmet see a pattern in their data. The population doubles (approximately) every hour.

The prediction

At 13:00 there will be (approximately)150 cells; at 14:00 there will be (approximately) 300 cell. At 15:00 , if the teacher is correct, there will be (approximately) 500 cells. They will know the population has stopped growing at 16:00 if there are still (approximately) 500 cells.

Problems:

Problem 1: The Teacher's Circle Problem

Problem 2: Lockers

Problem 1: The Teacher's Circle Problem

Which student will be left standing? Can you predict the number of the student who will still be be standing for any number of students in the circle?

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Problem 2: Lockers

Here is a well-known problem that can be solved by looking for a pattern. At first it seems like a huge problem, but it can be solved by starting optimistically at the beginning and looking for a pattern. Then, when a pattern emerges, the solution to the rest of the problem is both interesting and predictable.

In a high school of 1000 students every student has a locker. Imagine that one student opens all the doors of all 1000 cupboards.

Then a second student starts at the second locker and closes every second door.

Then a third student starts at the third locker and changes the state of every third door (closes it if it was open or opens it if it was closed.)

Then a fourth student starts at the fourth locker and changes the state of every fourth door.

Then a fifth student starts at the fifth locker and changes the state of every fifth door.

And so on . . . After 1000 students have followed the same pattern, which doors will be open and which doors will be closed?

The patterns that emerge when this problem will be especially useful for mathematics teachers who are teaching their students about factors, multiples, prime numbers, and perfect squares. As students begin to understand the problem there will be opportunities for lively discussions in the classroom.

(More advanced students might want to undertake the challenge of making a spreadsheet to solve the problem.)

Solution -->

PROBLEM SOLVING STRATEGIES

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