Coloured pegs and pegboards are useful resources that are often overlooked.

Learning about numerical and algebraic ideas and relationships can be stimulated and aided in an enjoyable way by creating and exploring arrangements of coloured pegs placed in pegboards. This is true for learners of all ages and at all stages, from primary through secondary to university level.

Pegs may be placed so as to:

create and continue patterns

satisfy particular conditions

modify arrangements in particular ways.

Coloured pegs and pegboards are attractive materials. When using them learners are able to set their own rules and conditions, experiment freely, make and test conjectures, generalise and reason. It is easy to shift ‘to and fro’ between arranging and thinking about pegs on pegboards and making and studying coloured marks on dotty paper. Learners can quickly and easily record ‘transient’ peg arrangements on dotty paper.

The possibilities for explorations and activities are endless – as are the kinds of examples and phenomena that learners may create and investigate. In this Focus on… only a few ideas are briefly suggested.

How many different arrangements of three pegs placed next to each other are possible?

The arrangements that are possible depend on our decisions about the meaning of ‘next to’ and our criteria for sameness. What different decisions lead to each of these sets of possibilities?

Under each of these three different sets of assumptions what will be the possible arrangements of four pegs, of five pegs…?

Make a shape with some pegs, and then place pegs in all the holes next to its boundary. Continue to ‘grow’ new shapes in this way. For example:

Count the pegs on the boundary of the original, innermost, shape (4). Count the pegs that were added at the next stage (8). Continue to count the pegs added at each stage. What is happening numerically? Describe the number sequence that is being generated?

Make ‘filled in’ rectangles:

Ask yourself questions about composite and prime numbers.

Explore some number sequences that you can generate, such as the sequence of squares, 1, 4, 9, 16…

What happens when you explore ‘tilted’ rectangles?

Look for number patterns, and generalise!

Make triangular arrangements of pegs:

If we assume that the first triangle in the sequence of triangles of the left-hand kind in the diagram above consists of one peg, the number of pegs in the nth triangle is the sum of the first n whole numbers. What is it in the sequence of triangles of the right-hand kind? How can this situation be generalised further?

Suppose any arrangement of three pegs may be the first triangular arrangement in a sequence of peg triangles…

...what happens in various different cases?

How many moves are required to change a square arrangement of four pegs...

...to this triangular arrangement?

And how many moves to change this...

...to this?

Investigate number sequences generated by continuing in this way.

The black pegs in this board make a pattern of parallelograms:

Each row has a peg in every sixth hole, and every row slips two places to the right.

If, instead, we put a peg in every tenth hole of each row, and make every row slip three places to the right, we create a pattern of squares.

Investigate in general placing a peg in every nth hole of each row and slipping each row m places to the right. For which values of n and m do we create squares? What has Pythagoras’ Theorem got to do with this?

Place a special peg (a white peg in this diagram) in a hole. Working outwards from the white peg, place a black peg in every hole that is ‘visible’ from the white peg – a hole is ‘visible’ from the white peg if there is no peg already on the straight line joining it to the white peg. Place a green peg in every hole that is ‘obscured’:

As the pattern is symmetrical, we might look at just one quadrant:

We find that, if we regard the white peg as the origin, the coordinates (a, b) of every black peg, and of no green peg, are relatively prime – and so the fraction b/a that we could associate with each black peg is a fraction in its lowest terms. By going further with these ideas we can discover Farey Sequences in the coordinates of the peg positions.

The Farey Sequence, F_{n}, is the sequence in increasing size order of all the irreducible fractions with denominator no greater than n that are no greater than one.

The arrangement of ‘visible’ pegs can help learners explore the number of terms in each Farey sequence. Generalising leads to Euler’s Totient Function.

When using pegs and pegboards to help you explore ideas and situations printed sheets representing arrangements of pegboard holes are useful.

You will also find some excellent suggestions in Starting Points by Banwell, Saunders and Tahta.

After an extensive search I can find on the web only one – rather ‘babyish looking’ – free interactive pegboard. Click on the black and white peg dog!

You will find a better interactive pegboard in Oral and Mental Starters Y7, ISBN 978 0 340 88360 0, which was originally published by Hodder Education. It is number 22 on the disc, which unfortunately is not one of the three interactive environments in this free Y7 interactive demonstration.