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Generalisation in growing patterns

The sequence of square numbers is 1, 4, 9, 16, 25… Students often find it difficult to find a rule for this sequence.

The numbers in the sequence of squares grow by the numbers 3, 5, 7, 9…. This looks like a sequence of odd numbers. 

Ask students: Is it? If so, why?

The explanation can be found by linking the sequence to a growing pattern of square arrays.

The number of dots on each side of a square is one more than on the square before.

Four arrays of dots: a single dot, a square made of 4 dots (2 by 2), a square made of 9 dots (3 by 3) and a square made of 16 dots (4 by 4).

A growing pattern of square arrays.

The dots that are added at each stage can now be identified.

Four arrays of dots as in the previous diagram. The extra dots that were added to each square to get the next square are highlighted.

How the pattern grows.

The added dots form an inverted L-shaped pattern.

The number of dots in these Ls increases by two each time, one extra dot being added at each end of the L. Because the number of dots in the first L is odd, the number of dots in all the Ls must therefore be odd.

This argument shows why an odd number of dots is added at every stage.


Ambitious students can be encouraged to go further by answering the following questions.

  • The 10th shape in the given pattern is a 10 \(\times\) 10 square. How big is the L shape that is added to this square to make the next square?
  • What is the 10th number in the sequence 3, 5, 7…?


In tasks like these students are beginning to go beyond specific shapes and numbers to look for general relationships. The activity Let's Have a Party! is another example of looking for a generalisation.

Growing polygons

A fun way of learning the names of some polygons is to use a patterning approach.

Growing fractions

How can you make fractions by folding?

Odds and evens

In this digital learning object, odd and even numbers are represented by different chimes. Create growing patterns that produce a particular pattern of even and odd numbers. Play them together and make music.