The following situation illustrates the power of "good" questions to generate a useful investigation, providing the student with plenty of scope to develop investigational skills and providing the teacher with the opportunity to input useful techniques where appropriate.

The Situation :

The Questions :

Class discussion would be useful at this point for generating questions. Possible questions are:

  • What "patterns" are there. eg in the distribution of odd numbers, even numbers, multiples of 3, prime numbers , square numbers ?
  • Are there formulae that will give the sequences of numbers in the rows, columns or diagonals ?

Let's choose to explore the distribution of prime numbers in the spiral.

Student Activity

Draw the spiral, or use Tasksheet 1, and highlight the primes.

Class discussion can then take place on these initial results with the aim of particularising or focusing down from the general question. After this discussion Tasksheet 2 should be issued.

Student Activity Tasksheet 2

Are your initial conjectures ( eg spotted patterns etc ) still holding up on the larger diagram ? Which conjectures have held? Which have fallen? Record any new "interesting" observations based on Tasksheet 2. Record any questions that you think are worth investigating. Try not to ask vague questions. Try to be precise.

This last activity should produce a host of observations. Among the many possibilities are: long rows of primes in some of the diagonals; rows, columns and diagonals are sometimes free of primes. Thse observations give rise to the following questions: Which rows, columns and diagonals are free of primes? Will they always be "prime-free" if they were extended? Why are they prime-free? Can a formula be found for a long row of primes? Let's choose to particularise and focus on one question:

Student Activity

Group work to investigate this question.

During this group work activity students are expected to progress towards an answer through the following stages:

Experimenting -> Organising -> Generalising -> Checking (or Proving)

Here is a brief outline of three different approaches to this question:

1st approach:

This approach is dependent on recognition of square numbers. During the "Asking the question" stage when students were becoming familiar with the spiral it may have been observed that the square numbers were confined to only two diagonals in the spiral, the odd squares along one diagonal and the even squares along the other. The sequence of numbers now under investigation lie side by side with the even squares:

This discovery can be organised in a table ready for the Generalisation Stage:

16 - 1
42 - 1
36 - 1
62 - 1
64 - 1
82 - 1
100 - 1
102 - 1

This pattern can now be generalised. The n th term is :

( 2n + 2 )2 - 1

Since this is a "difference of two squares" it can be factorised as follows:

( 2n + 2 - 1 )( 2n + 2 + 1)


( 2n + 1 )( 2n + 3 )

This may now be checked for n = 1, 2, 3, 4, ... Since this n th term expression always factorises there will be no primes in the diagonal.

2nd approach:

Lack of primes implies that the numbers in this diagonal factorise. It seems worthwhile therefore to investigate the factors of the numbers in this sequence:

This pattern can be generalised to give n th term = ( 2n + 1 )( 2n + 3 ). Again checking is essential using larger examples.

3rd approach:

The initial experimenting may produce a difference table for the sequence:

From this table it becomes clear that the "2nd differences" are constant. Progress now depends on whether the Differences Method has been taught:

Constant 2nd difference --> n th term is quadratic

Once obtained, does the quadratic factorise? If it does then there will be no primes along this diagonal.

The Proof :

It is important that the need for justification is discussed. Class discussion, led by the teacher should consider for each approach questions similar to the following

Student Activity

Will the terms always be one less than a square number? Why?

Why should the regular pattern in the factors continue?

Would you expect the 2nd differences always to be constant if you extended the difference table? How do you know?

The nature of the build up of the spiral should be considered and its regularity discussed. The aim is to discourage blind acceptance of technique. The method may work here but not in another situation.

The Extension :

Now that the spiral is familiar and useful techniques have been developed there are many more explorations that can be made.

Student Activity

Group work with each group responsible for formulating and tackling a further question based on Tasksheet 2.

Possible questions are:

  • For other apparently prime-free rows, columns or diagonals can formulae be found and the lack of primes explained?
  • Using the pattern of primes in the spiral can a quadratic formula ( n th term: an 2+bn +c ) be found that produces 5 prime values for n = 1, 2, 3, 4, 5 ? What about a formulae producing 6 or more primes?
  • "First Quadrant"

Can a formula be found, in terms of x and y , that gives the correct number at position ( x , y ) ?

  • In the "First Quadrant" the columns in positions x = 1, 5, 13, 25 ... appear to be prime-free. Does this pattern continue and can it be explained? Do similar patterns appear in the other "Quadrants"?