About the authors

During his long and distinguished career, David Hillard spent more than 45 years teaching mathematics in two preparatory schools. From 1980 to his retirement, he was associated with the Common Entrance Examination at 11+, 12+ and 13+ levels in the role of adviser, assessor or setter, playing a significant role in the revision of the syllabus in 2003 when the present format of the examination was introduced.

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 © David Hillard and Serena Alexander 2014

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

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Printed in the UK

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

ISBN: 978 1 4718 2939 0
eISBN: 978 1 4718 3166 9

Introduction

This book provides a complete set of answers to Galore Park Maths Book 5

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 Introducing investigations

This chapter covers the following elements of the National Curriculum for Year 5:

Number: Number and place value

•  Programme of study (statutory requirements)

    Pupils should be taught to:

    •  solve number problems and practical problems that involve all of the above

•  Notes and guidance (non-statutory)

    •  Pupils should recognise and describe linear number sequences … and find the term-to-term rule.

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

•  Investigating number (patterns and finding a rule)

•  Inverse functions

Exercise 1.1

  1  (a)  Add 2

      (b)  Multiply by 3

      (c)  Multiply by 2, add 3

      (d)  Multiply by 3, subtract 1

      (e)  Multiply by 4, add 1

      (f)  Multiply by 2, subtract 2

  2  (a)  a = 10, b = 15

      (b)  c =21, d = 12

      (c)  e = 21, f = 25

      (d)  g = 44, h = 20

      (e)  i = 41, j = 25

      (f)  k = 30, l = 50

Exercise 1.2

  1  (a)  

      (b)  

      (c)  19 (pattern number)

      (d)  39 (pattern number + 1)

      (e)  83 (41 + 42)

      (f)  70

      (g)  50

      (h)  72

  2  (a)  

      (b)  

      (c)  No. of lines = 5 × pattern number + 1

      (d)  No. of dots = 3 × pattern number + 2

      (e)  51 ((5 × 10) + 1)

      (f)  38 ((3 × 12) + 2)

      (g)  20 ((101 − 1) ÷ 5)

      (h)  49 ((149 − 2) ÷ 3)

2 Roman numerals

This chapter covers the following elements of the National Curriculum for Year 5:

Number: Number and place value

•  Programme of study (statutory requirements)

    •  read Roman numerals to 1000 (M) and recognise years written in Roman numerals

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

•  Convert numbers to Roman numerals to over 2000

•  Roman addition

Notes

The chapter starts with a practical exercise in which pupils use their hands to demonstrate the numbers. This use of hands is the traditional explanation of the origin of Roman numerals.

Exercise 2.1

1–7  Practical

Exercise 2.2

  1  2

  2  3

  3  6

  4  8

  5  12

  6  20

  7  51

  8  61

  9  163

10  166

11  110

12  151

Exercise 2.3

  1  4

  2  14

  3  9

  4  59

  5  29

  6  45

  7  96

  8  248

  9  159

10  99

11  195

12  149

Exercise 2.4

  1  III

  2  VII

  3  XXII

  4  XV

  5  XXIX

  6  LVI

  7  XXXVIII

  8  XLIX

  9  LXII

10  LXXXIV

11  XCII

12  CLXIV

13  XCIV

14  XLVII

15  LXXXIX

16  LXXII

17  LXIX

18  XXXV

19  CXC

20  CXCIV

Exercise 2.5

  1  600

  2  1100

  3  1200

  4  509

  5  804

  6  1500

  7  1609

  8  1094

  9  1594

10  190

11  1699

12  1964

Exercise 2.6

  1  8

  2  24

  3  559

  4  944

  5  1001

  6  38

  7  96

  8  448

  9  159

10  99

11  17

12  1546

Exercise 2.7

  1  XLV

  2  CXXXII

  3  DI

  4  MCXI

  5  CDXXIV

  6  LXVI

  7  MLXVI

  8  DCCCXLIX

  9  MCMLXV

10  MMXIV

11  CXCIX

12  CMXIX

13  CDXLIV

14  CMXLIX

15  MDCCXLV

16  CDLXIV

17  MDCCCXII

18  CMLXIX

19  MDVIII

20  MCMXCIX

Exercise 2.8

  1  1830, date of building

  2  1890, date of building

  3  1922, date of building

  4  1910, memorial to Queen Victoria

  5  13 May 1981, date of attempted assassination of Pope John Paul II

  6  1897, date of building

  7  52, gate number

  8  4 July 1776, declaration of independence (on Statue of Liberty)

  9  No. 7, Roman Colosseum

Exercise 2.9: Summary exercise

  1  8

  2  66

  3  19

  4  1605

  5  505

  6  2015

  7  1151

  8  509

  9  1049

10  1984

11  IX

12  XVII

13  CXII

14  CCCIV

15  DXLII

16  DCCXXXV

17  MCXIV

18  MDVI

19  MCMXXXIX

20  MMXCIX

21  27 miles

22  1838, founding of the school