Teachers should read through the following activity ideas and make their own risk assessment for them before proceeding with them in the classroom.

Mathematics is the search for pattern. For children of primary age there are few places where this search can be more satisfyingly pursued than in the field of figurate numbers – numbers represented as geometrical shapes.

• In these activities, pupils are encouraged to construct and compare series of 2d shapes to find the following -

• Add a pair of consecutive natural numbers and you get an odd number; add the consecutive odd numbers and you get a square number;

• Add the consecutive natural numbers and you get a triangular number; add a pair of consecutive triangular numbers and you also get a square number.

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Number shapes investigation

Teacher notes

Resources needed for practical activity

• Multilink cubes in a range of colours • Multilink ‘sliders’ – these are simply sticks of cubes of different lengths

• Multiplication squares • Isometric paper • Squared ‘dotty’ paper

Introduction

This masterclass does not follow a Powerpoint presentation, but is presented as a series of short activities that encourage pupils, through building their own structures, to investigate the numbers associated with these shapes, with particular emphasis on how they appear on the multiplication square.

In this session, 2d shapes are explored. For further activities and more detailed explanations, please see the full Number Shapes package on the Ri website. This contains information about activities that can be used with pupils in KS3, as well as details of activities that can be done with the addition of some other equipment.

Activity one – (establishing that complicated shapes should be regarded as composites of a few simple ones)

• Ask the pupils to choose the first letter of their name and colour in dots on either the isometric paper or the squared ‘dotty’ paper.

• Discuss the example of L. What is the difficulty? (The foot and tail could be different lengths, so the L should be broken into two rectangles. On isometric paper, these can be represented as parallelograms.)

• Discuss the example of V (on isometric paper). What is the difficulty? (The triangle at the bottom could be very big, the parallelogram-shaped sides very small, so the V should be

broken into a triangle and two parallelograms. On squared paper, these can be represented as a triangle and two rectangles.)

Activity two – (showing the equivalence of certain of these simple shapes in terms of the numbers they represent)

• By using the multilink sliders, children can convert: non right-angled triangles into right-angled isosceles triangles; parallelograms into rectangles; rhombuses into squares.

Activity three – (showing that the natural numbers are of only two kinds: those which can be shown as a rectangle and those which cannot – the primes)

• List on the whiteboards half a dozen numbers, one of which is a prime. Pupils should then arrange multilink cubes into rectangles containing those numbers of cubes. The children will find they can show all the composite numbers as rectangles, perhaps some in more than one way, but the prime only as a row of unit width.

Activity four (in the same way that a triangle number is the sum of consecutive natural numbers, a square number is the sum of consecutive odd numbers)

• Build an odd number as a rectangle of width 2 units with a single unit on the end. Rebuild it by using the single unit as the corner of an ‘L’ and splitting the rectangle to provide an equal foot and tail.

• Added to a square, this piece completes a larger one – it is the ‘gnomon’ to a square. Grow a square from a single unit by adding successive gnomons – ie. successive odd numbers.

Activity five (to recognise where square numbers fall on the multiplication square)

• Ask pupils to shade in the squares containing the square

numbers. They should notice that these fall along the main diagonal. Discuss why this should be so.

• Where do prime numbers fall on the multiplication square? Why?

• What about numbers that are the product of two primes?

Activity six

If you have access to a floor tiled in a checkerboard pattern, this activity can be conducted as ‘people maths’ but, working on the table scale, make a 9x9 checkerboard with black and white multilink cubes with white cubes at the corners. These can be used as a basis for the following constructions

Arrangement one

• Notice that the blue cubes form the 4th square number and the red cubes, the 5th.

• Notice incidentally that the green cubes form a 4x5 rectangle with its long side aligned east-west and an identical rectangle with its long side aligned north-south.

• The children should check the arithmetic:

o The ‘red’ total = 25 o The ‘blue’ total = 16 o The ‘white = ‘red’ + ‘blue’ total = 41 o The ‘black’ = ‘green’ total = 40 o The grand total = 81 o The whole square = 9x9 = 81

Arrangement two

• Redistribute the colours like this.

• Looking at rows, notice that: the blue triangle is the sum of the first 4 odd numbers, therefore – as we are expecting – the 4th square.

• The red triangle is the sum of the first 5 odd numbers, therefore – as we are expecting – the 5th square.

• But notice that the blue and red triangles together form a square set at 45 degrees. This shows the 5th centred square number, the sum of the 4th and 5th squares.

• If time allows, it’s worth working up to the 9x9 square through the 1x1, 3x3, 5x5 and 7x7 sizes.

Activity seven – (we now establish a relation between square and triangular numbers)

• Build two consecutive triangular

numbers from multilink – preferably in their ‘right angled’ form – the same way round. Now give the smaller one a half-turn so that it complements the larger one to form a square.

Activity eight – (this leads to the formula for the nth triangular number – however expressed)

• Build two identical triangular numbers in their ‘right angled’ form. Performing the last experiment yields in this case not a square, but a rectangle. What is special about this rectangle? (Answer: the longer side is one unit greater than the shorter side.)

• Encourage pupils to recognise that we can find the total number of cubes in this composite of 2 nth triangle by multiplying n by n+1. Since this is the number of cubes needed for 2 such triangles, we need to halve this to find the number of cubes in each triangle.

Find out more In addition to the 2d examples given, studying figurate numbers provides a wealth of other interesting activities. Information about these can be found in the Magic Mathworks Number Shapes booklet. This can be downloaded from the Ri website, or obtained by emailing maths@ri.ac.uk.