About the author

Serena Alexander has taught mathematics since 1987, originally in both maintained and independent senior schools. From 1999 she taught at St Paul’s School for Boys, where she was Head of Mathematics at their Preparatory School, Colet Court, before moving first to Newton Prep as Deputy Head and more recently to Devonshire House. She is an ISI inspector and helps to run regular mathematics conferences for prep school teachers. She has a passion for maths and expects her pupils to feel the same way. After a lesson or two with her, they normally do!

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Published by Galore Park Publishing Ltd
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Text copyright © Serena Alexander 2014

The right of Serena Alexander to be identified as the author of this Work has been asserted by her in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988.

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A catalogue record for this title is available from the British Library.

ISBN: 978 1 47182937 6
eISBN: 978 1 47183188 1

Introduction

This book provides a complete set of answers to Galore Park Maths Year 6.

For some questions and activities, pupils are asked to copy diagrams from the book. They may find tracing paper helpful when doing this. Such activities are also supported by separate worksheets which are at the back of this book. These worksheets may be photocopied.

Setting out work

Pupils are individuals and no two will present work in exactly the same way. However, it is important to set basic ground rules for presentation, as keeping their calculations neat and clear will help pupils to avoid mistakes. It will also enable you to see what they are doing.

For simple arithmetic:

For more complex problems, encourage the ‘saucepan’ shape to setting out chains of calculations.

1 Investigations with numbers

This chapter covers the following elements of the National Curriculum for Year 6:

Algebra

•  Programme of study (statutory requirements)

    Pupils should be taught to:

    •  use simple formulae expressed in words

    •  generate and describe linear number sequences.

•  Notes and guidance (non-statutory)

    •  Pupils should be introduced to the use of symbols and letters to represent variables and unknowns in mathematical situations that they already understand, such as generalisations of number patterns.

This chapter also covers the following elements not mentioned in the National Curriculum:

•  Investigations, triangular numbers

•  Discovering a formula from a pattern of numbers

The following can be used as cross-curricular links:

•  Mystic roses and architecture

ICT can be used in the following ways:

•  Pupils could use a spreadsheet to explore their number patterns.

Notes

•  This chapter is intended to give some experience of mathematical investigation, to revise square numbers and introduce triangular numbers. In addition, it introduces the concept of developing a simple formula from the patterns generated in an investigation.

•  The stunning patterns generated from the handshake investigation can be explored further as they become ‘mystic roses’. These link in to architecture, so there is plenty to capture the interest of your pupils.

•  Consider ways of building up patterns of triangle and square numbers, as these can be used for classroom displays. Coloured or foil dots can be very popular.

Exercise 1.1

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Exercise 1.2

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  6  Multiple n by (n + 1)

  7   n(n + 1)

Exercise 1.3

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  6  The sum of two consecutive triangular numbers is a square number.

Exercise 1.4

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2  (a)–(b)

    (c)  Square numbers

    (d)  Triangular numbers

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  4  The sequence shows that every square number is a sum of consecutive odd numbers.

      It also shows that the sum of a pair of consecutive triangular numbers is a square number.

      The nth number in the pattern will have a total of n² circles.

      When n is odd that will be the sum of the nth triangular number of black circles and the (n − 1)th triangular number of red circles.

      When n is even that will be the sum of the (n − 1)th triangular number of black circles and the nth triangular number of red circles.