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Ma KEY STAGE 2 Mathematics tests Mark schemes LEVELS Test A, test B and mental mathematics test 2005 3– 5 2005 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk First published in 2005 © Qualifications and Curriculum Authority 2005 Reproduction, storage, adaptation or translation, in any form or by any means, of this publication is prohibited without prior written permission of the publisher, unless within the terms of licences issued by the Copyright Licensing Agency. Excerpts may be reproduced for the purpose of research, private study, criticism or review, or by educational institutions solely for educational purposes, without permission, provided full acknowledgement is given. Produced in Great Britain by the Qualifications and Curriculum Authority under the authority and superintendence of the Controller of Her Majesty’s Stationery Office and Queen’s Printer of Acts of Parliament. The Qualifications and Curriculum Authority is an exempt charity under Schedule 2 of the Charities Act 1993. Qualifications and Curriculum Authority 83 Piccadilly London W1J 8QA www.qca.org.uk/ Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk Marking the mathematics tests As in 2004, external markers, employed by the external marking agencies under contract to QCA, will mark the test papers. The markers will follow the mark schemes in this booklet, which is supplied to teachers for information. This booklet contains the mark schemes for the levels 3–5 tests A, B and mental mathematics. Level threshold tables will be available on the QCA website on 20 June 2005 (www.qca.org.uk/). General guidance The structure of the mark schemes The marking information for each question is set out in the form of tables, which start on page 6 of this booklet. The ‘question’ column on the left-hand side of each table provides a quick reference to the question number and the question part. The ‘mark’ column indicates the total number of marks available for each question part. On some occasions the symbol U1 may be shown in the mark column. The ‘U’ indicates that there is a ‘Using and Applying Mathematics’ element in the question. The number, 1, shows the number of marks attributed to using and applying mathematics in this question. The ‘requirement’ column may include two types of information: a statement of the requirements for the award of each mark, with an indication of whether credit can be given for correct working; examples of some different types of correct response. The ‘additional guidance’ column indicates alternative acceptable responses, and provides details of specific types of response which are unacceptable. Other guidance, such as the range of acceptable answers, is provided as necessary. Additionally, for the mental mathematics test, general guidance on marking is given on page 18, together with a ‘quick reference’ mark scheme. Applying the mark schemes In order to ensure consistency of marking, the most frequent procedural queries are listed on pages 2 and 3 with the action the marker will take. This is followed by further guidance on pages 4 and 5 relating to the marking of questions that involve money, time and other measures. Unless otherwise specified in the mark scheme, markers will apply the following guidelines in all cases. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 1 What if ... Marking procedure The child’s response is numerically or algebraically equivalent to the answer in the mark scheme. Markers will award the mark unless the mark scheme states otherwise. The child’s response does not match closely any of the examples given. Markers will use their judgement in deciding whether the response corresponds with the statement of the requirements given in the ‘requirement’ column. Reference will also be made to the additional guidance and, if there is still uncertainty, markers will contact the supervising marker. The child has responded in a non-standard way. Calculations, formulae and written responses do not have to be set out in any particular format. Children may provide evidence in any form as long as its meaning can be understood. Diagrams, symbols or words are acceptable for explanations or for indicating a response. Any correct method of setting out working, however idiosyncratic, will be accepted. There appears to be a misreading affecting the working. This is when the child misreads the information given in the question and uses different information without altering the original intention or difficulty level of the question. For each misread that occurs, one mark only will be deducted. In one-mark questions – 0 marks are awarded. In two-mark questions that have a method mark – 1 mark will be awarded if the correct method is correctly implemented with the misread numbers. No answer is given in the expected place, but the correct answer is given elsewhere. Where a child has shown understanding of the question, the mark(s) will be given. In particular, where a word or number response is expected, a child may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question. The response in the answer box is wrong, but the correct answer is shown in the working. Where appropriate, detailed guidance will be given in the mark scheme, which markers will follow. If no guidance is given, markers will examine each case to decide whether: the incorrect answer is due to a transcription error; the child has continued to give redundant extra working which does not contradict work already done; Sourced from SATs-Papers.co.uk If so, the mark will be awarded. the child has continued to give redundant extra working which does contradict work already done. 2 If so, the mark will be awarded. If so, the mark will not be awarded. http://www.SATs-Papers.co.uk What if ... Marking procedure The child’s answer is correct but the wrong working is shown. A correct response will always be marked as correct. The correct response has been crossed out and not replaced. Any legible crossed-out work that has not been replaced will be marked according to the mark scheme. If the work is replaced, then crossed-out work will not be considered. More than one answer is given. If all answers are correct (or a range of answers is given, all of which are correct), the mark will be awarded unless prohibited by the mark scheme. If both correct and incorrect responses are given, no mark will be awarded. The answer is correct but, in a later part of the question, the child has contradicted this response. A mark given for one part will not be disallowed for working or answers given in a different part, unless the mark scheme specifically states otherwise. The child has drawn lines which do not meet at the correct point. Markers will interpret the phrase ‘slight inaccuracies in drawing’ to mean ‘within or on a circle of radius 2mm with centre at the correct point’. within the circle accepted on the circle accepted outside the circle not accepted Recording marks awarded on the test paper In the shaded margin there is a mark box for each question part. For the written tests, the number of marks gained on each double page will be written in the total box at the bottom of the right-hand page. For all of the tests, the total number of marks gained on each paper will be recorded on the front of the test paper, and on the mark sheet. All questions in the tests, even those not attempted by the child, will be marked with a ‘1’ or ‘0’ entered in the mark box. A two-mark question which is correct has ‘1’ entered in both mark boxes. A two-mark question which is incorrect, but which has sufficient evidence of working or method as required by the mark scheme, will have ‘1’ entered in the first mark box and ‘0’ in the second. Otherwise ‘0’ will be entered in both mark boxes. Test A carries a total of 40 marks. Test B also carries a total of 40 marks. The mental mathematics test carries a total of 20 marks. The 2005 key stage 2 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 3 Marking specific types of question – summary of additional guidance Responses involving money Accept Do not accept Where the £ sign is given £3.20 for example: £3.20, £7 Any unambiguous indication of the correct amount, eg £3.20p £3 20 pence £3 20 £3,20 £3-20 £3:20 £ Where the p sign is given for example: 40p £7 £7.00 Incorrect placement of pounds or pence, eg £320 £320p Incorrect placement of decimal point, or incorrect use or omission of 0, eg £3.2 £3 200 £32 0 £3-2-0 40p Any unambiguous indication of the correct amount, eg £0.40p Incorrect or ambiguous use of pounds or pence, eg 0.40p £40p p Where no sign is given £3.20 320p for example: £3.20, 40p Any unambiguous indication of the correct amount, eg £3.20p £0.40p £3 20 pence £.40p £3 20 £.40 £3,20 40 £3-20 0.40 £3:20 3.20 320 3 pounds 20 4 Sourced from SATs-Papers.co.uk 40p £0.40 Incorrect or ambiguous use of pounds or pence, eg £320 £40 £320p £40p £3.2 0.4 3.20p http://www.SATs-Papers.co.uk Responses involving time Accept Do not accept A time interval 2 hours 30 minutes for example: 2 hours 30 minutes 1 2 2 hours Any unambiguous indication, eg 2.5 hours 2h 30 2h 30 min Digital electronic time, ie 2:30 A specific time for example: 8:40am, 17:20 Incorrect or ambiguous time interval, eg 2.30 2-30 2,30 2.3 2.3 hours 2.3h 2h 3 2.30 min 8:40am 8:40 twenty to nine Any unambiguous, correct indication, eg 08.40 8.40 0840 8 40 8-40 8,40 Unambiguous change to 12 or 24 hour clock, eg 17:20 as 5:20pm or 17:20pm Incorrect time, eg 8.4am 8.40pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0, eg 840 8:4:0 8.4 084 84 Responses involving measures Accept Where units are given (eg kg, m, l) for example: 8.6kg kg Do not accept 8.6kg Any unambiguous indication of the correct measurement, eg 8.60kg 8.6000kg 8kg 600g Incorrect or ambiguous use of units, eg 8600kg Note If a child leaves the answer box empty but writes the answer elsewhere on the page, then that answer must be consistent with the units given in the answer box and the conditions listed above. If a child changes the unit given in the answer box, then their answer must be equivalent to the correct answer using the unit they have chosen, unless otherwise indicated in the mark scheme. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 5 Test A questions 1–3 Question 1 Requirement Mark Award TWO marks for the four lines drawn as shown: 100 170 200 70 250 330 300 Do not award any marks if two or more incorrect lines are drawn. Lines need not touch the boxes, provided the intention is clear. 180 150 Up to 2m Additional guidance 220 If the answer is incorrect, award ONE mark for three correct lines drawn AND not more than one incorrect line drawn. 2 One of the following triples: 11, 12, 17 13, 18, 19 11, 13, 16 14, 17, 19 11, 14, 15 Accept alternative unambiguous indications, eg ticks, crosses. 15, 16, 19 12, 13, 15 1m 15, 17, 18 Do not award the mark if fewer or more than three numbers are circled. 12 10 8 6 4 2 0 blue green pink red 3a 1m 3b 6 Vertical axis completed correctly. Horizontal axis completed correctly. 1m Sourced from SATs-Papers.co.uk Accept abbreviations or recognisable misspellings. http://www.SATs-Papers.co.uk Test A questions 4–9 Question Requirement Mark Additional guidance 4a 30 minutes 1m The answer is a time interval (see page 5 for guidance). 4b 9:25 am 1m The answer is a specific time (see page 5 for guidance). 5 14 1m 6a B AND D 1m Both letters must be given. Letters may be given in either order. 6b C AND E 1m Both letters must be given. Letters may be given in either order. 7a Award TWO marks for the correct answer of £4.10 OR 410p Up to 2m If the answer is incorrect, award ONE mark for evidence of appropriate working, eg Accept for ONE mark £410 OR £410p as evidence of appropriate working. 4 t 60 = 240 Calculation must be performed for the award of ONE mark. 2 t 85 = 170 240 + 170 = wrong answer 7b £3.00 1m 8 1614 1m 9 Award TWO marks for all five numbers in any order as shown: 624, 642, 646, 662, 664 If the answer is incorrect, award ONE mark for: Up to 2m Ignore 626 or repeats of the five correct responses. U1 For ONE mark, ignore four-digit numbers. four out of five numbers correct and none incorrect OR five numbers correct and only one incorrect. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 7 Test A questions 10–13 Question 10 Requirement Mark Two cards ticked as shown: 0.01 12 9.9 1m Accept alternative unambiguous indications such as circling or a line joining the correct pair of cards. 1m Accept equivalent fractions. 1.01 9.09 11 0.11 Additional guidance 9.99 5 9 Award TWO marks for signs written in the order shown: Up to 2m < > = If the answer is incorrect, award ONE mark for two out of three signs correct. 13 Two more lines drawn as shown: 1m Accept slight inaccuracies in drawing (see page 3 for guidance). Do not accept lines drawn outside of the grid. Ignore line of symmetry if drawn. 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk Test A questions 14–17 Question 14 Requirement An explanation which recognises that the numbers of odd and even cards are not equal, eg Mark 1m U1 ‘Because there are more odds than evens’; Additional guidance No mark is awarded for circling ‘No’ alone. Do not accept vague or arbitrary explanations, eg ‘Because fair means half the time’; ‘Because there are fewer evens than odds’; ‘Because there are 7 cards’; ‘Because Sapna scores on more than half of the cards’; ‘Because there is an odd number of cards’; ‘Because there are only three even numbers’; ‘Because the game is unfair’; ‘Because Sapna will always win’. ‘Because Josh has 3 cards and Sapna has 4 cards’; If ‘Yes’ is circled but a correct, unambiguous explanation is given, then award the mark. ‘Because Sapna has more chances’. 15a Answer in the range 104 to 106 inclusive. 1m 15b 5 1m 16 Award TWO marks for diagrams ticked or crossed as shown: Up to 2m Accept alternative unambiguous indications such as Y or N. For TWO marks accept: If the answer is incorrect, award ONE mark for three diagrams ticked or crossed correctly. 17a Answer in the range 126mm to 128mm inclusive. 1m 17b Answer in the range 21 degrees to 23 degrees inclusive. 1m Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 9 Test A questions 18–20 Question 18 Requirement Mark Award TWO marks for boxes ticked and crossed as shown: Up to 2m Additional guidance Accept alternative unambiguous indications such as Y or N. For TWO marks, accept: If the answer is incorrect, award ONE mark for any three boxes correctly completed. 19 36 AND 9 1m 20 Award TWO marks for the correct answer of 5291 If the answer is incorrect, award ONE mark for evidence of appropriate working which contains no more than ONE arithmetical error, eg long multiplication algorithm such as 143 t 37 1001 4290 wrong answer Numbers may be given in either order. Up to 2m In all cases accept follow through of ONE error in working. Do not award any marks if: the error is in the place value, eg the omission of the zero when multiplying by three tens, 1001 + 429 the final (answer) line of digits is missing. 100 40 3 Variations on algorithms are acceptable, provided they represent viable and complete methods. 30 3000 1200 90 Calculation must be performed for the award of ONE mark. 7 700 280 21 grid method = wrong answer decomposition methods, eg 143 t 40 = 5720 143 t 3 = 429 5720 – 429 = wrong answer 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk Test A questions 21–24 Question 21 Requirement Mark Award TWO marks for boxes ticked and crossed as shown: Up to 2m Additional guidance Accept alternative unambiguous indications such as Y or N. For TWO marks, accept: If the answer is incorrect, award ONE mark for any three boxes correctly completed. 22 3 5 3 4 17 20 1m 9 10 Fractions must be written in the correct order for the award of the mark. Accept equivalent fractions or decimals. 23 Award TWO marks for the correct answer as shown: A = –80 B= Up to 2m Accept ‘minus 80’ Do not accept ‘80–’ 60 Answer need not be obtained for the award of ONE mark. If the answer is incorrect, award ONE mark for evidence of appropriate working, eg Accept for ONE mark: 140 ÷ 7 = 20 A = –80 AND B = wrong answer OR A = –80 AND B = blank OR A = 80 AND B = 60 OR A = 80 AND B = –60 OR A = 60 AND B = –80 24 Award TWO marks for the correct answer of 42 If the answer is incorrect, award ONE mark for evidence of appropriate working, eg Up to 2m U1 Calculation must be performed for the award of ONE mark. 18 – 10 = 8 10 + (4 t 8) = wrong answer OR 10, 18, 26, 34, wrong answer Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 11 Test B questions 1–3 Question 1 Requirement Amounts written in correct order as shown: £0.75 2 12 99p £2.05 £9 32 16 8 4 2 Additional guidance Accept use of equivalent units, eg 75p. Accept answers with missing or incorrect units. 1m 1 The correct shape ticked as follows: Sourced from SATs-Papers.co.uk 1m £10.50 Three numbers circled as shown: 64 3 Mark Do not award the mark if additional incorrect numbers are circled. Accept unambiguous alternatives, eg numbers ticked, crossed or underlined. 1m Accept alternative unambiguous indications of the correct shape, eg shape circled. http://www.SATs-Papers.co.uk Test B questions 4–8 Question 4a Requirement Mark Award TWO marks for the correct answer of £1.38 Up to 2m If the answer is incorrect, award ONE mark for evidence of appropriate method, eg Accept for ONE mark £138p OR £138 as evidence of an appropriate method. 1 78 + ( 2 t 1.20) 4b 5 Additional guidance Answer need not be obtained for the award of ONE mark. 6 1m Diagram completed as shown: 1m 6a 7 + 1 8 = 2 5 1m 6b 2 5 t 3 = 7 5 Accept slight inaccuracies in drawing (see page 3 for guidance). 1m U1 7 Award TWO marks for table completed correctly as shown: number of flat surfaces number of curved surfaces sphere 0 1 cone 1 1 cuboid 6 0 cylinder 2 Up to 2m 1 Accept a blank box for ‘0’. If the answer is incorrect, award ONE mark for two out of three rows completed correctly. 8a Answer in the range 340 to 360 inclusive. 1m 8b Answer in the range 240 to 260 inclusive. 1m Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 13 Test B questions 9–15 Question Requirement Mark 9a 955 in first box. 1m 9b 1010 in second box. 1m 10 Accept for TWO marks any arrangement using one of the following sets of eight numbers: Additional guidance Up to 2m Numbers may be written in any order. U1 1, 1, 1, 2, 2, 2, 3, 3 OR 1, 1, 1, 1, 2, 2, 2, 2 eg 3 2 1 1 2 2 3 1 If the answer is incorrect, award ONE mark for an arrangement such that: Do not accept answers that leave sections blank or include numbers other than 1, 2 or 3. the number of 1s and 2s is equal OR the the the the number number number number of of of of 3s is less than 2s AND 3s is less than 1s. 11 29 1m 12 12 1m 13 Sapna 8 Robbie 6 1m U1 14 Shape completed correctly, as shown: 1m Shape need not be completed accurately, provided the two correct triangles are identified unambiguously. 15a (0, 10) 1m Coordinates must be written in the correct order. Accept unambiguous answers written on the diagram. 15b 14 (10, 20) Sourced from SATs-Papers.co.uk 1m If the answer for 15a is (10, 0) AND the answer to 15b is (20, 10), award ONE mark only, in the 15b box. http://www.SATs-Papers.co.uk Test B questions 16–18 Question 16 Requirement Award TWO marks for all four factors, as shown: Mark Up to 2m 1, 2, 5, 10 Additional guidance Accept factors written in any order. All four factors and no incorrect numbers must be given for the award of TWO marks. If the answer is incorrect, award ONE mark for: three factors correct and none incorrect OR four factors correct and one incorrect. 17 3 2 t 3 2 = 1 0 2 4 1m Accept 32 U1 18a Answer in the range 14 to 16 inclusive. 1m 18b An explanation which recognises that the bar for tomato is shorter than the other two bars added together, eg 1m ‘Because there are 300 children altogether and only 135 chose tomato’; ‘Because 165 is more than 135’; ‘Because double 135 is 270 and there are more children than that altogether’; ‘Because half of 300 is 150’; ‘Because tomato is less than mushroom add chicken’. Sourced from SATs-Papers.co.uk U1 No mark is awarded for writing ‘No’ alone. Do not accept vague or arbitrary explanations, eg ‘Because most of the children chose tomato’; ’Because 135 children chose tomato’; ‘Because 75 + 135 + 90 = 300’. If ‘Yes’ is circled but a correct, unambiguous explanation is given then award the mark. http://www.SATs-Papers.co.uk 15 Test B questions 19–21 Question 19 Requirement Award TWO marks for the correct answer of 8 Mark Additional guidance Up to 2m If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg Answer need not be obtained for the award of ONE mark. 1+2+3=6 24 ÷ 6 = 4 4t2 OR 6 fruits 12 fruits 4 oranges 18 fruits 6 oranges 24 fruits 20 2 oranges wrong answer 21 16 and 9.4 x = 35° Sourced from SATs-Papers.co.uk 1m Accept numbers in either order. U1 7.4 Both numbers must be correct for the award of the mark. 1m http://www.SATs-Papers.co.uk Test B questions 22–25 Question 22 Requirement Mark Award TWO marks for all seven boxes completed correctly as shown: hockey rounders Total boys 22 28 27 26 49 54 U1 53 Total Up to 2m 50 girls Additional guidance 103 If the answer is incorrect, award ONE mark for five or six boxes completed correctly. 23a 18 1m Do not accept 18% 23b 200 1m Do not accept 200% If the answer for 23a is 18% AND the answer for 23b is 200%, award ONE mark only in the 23b box. 24 Award TWO marks for the correct answer of 26.8cm Up to 2m If the answer is incorrect, award ONE mark for evidence of an appropriate method, eg Award ONE mark for an answer of 53.6 OR for 53.6 shown with no evidence of an incorrect method. 85 ÷ 2 – 15.7 Answer need not be obtained for the award of ONE mark. OR 85 – (15.7 t 2) = wrong answer wrong answer ÷ 2 OR 85 – (15.7 t 2) = 53.6 25 Award TWO marks for the correct answer of 0.15 Up to 2m 3 Accept equivalent fractions, eg 20 If the answer is incorrect, award ONE mark for evidence of appropriate method, eg Accept for ONE mark 0.015 OR 15 OR 1.5 OR 150 as evidence of appropriate method. 45 – 12 = 33 Answer need not be obtained for the award of ONE mark. 33 ÷ 220 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 17 Mark scheme for the mental mathematics test Applying the mark scheme Please note that children will not be penalised if they record any information given in the question or show their working. Markers will ignore any annotation, even if in the answer space, and mark only the answer. Markers will accept an unambiguous answer written in the stimulus box, or elsewhere on the page. Full mark scheme information is given on page 20. In addition, a ‘quick reference’ mark scheme is provided on page 19. This is presented in a similar format to the children’s answer sheet. General guidance The general guidance for marking the written tests also applies to marking the mental mathematics test. In addition, the following principles apply. 1. Unless stated otherwise in the mark scheme, accept answers written in words, or a combination of words and figures. 2. Where units are specified, they are given on the answer sheet. Children are not penalised for writing in the units again. 3. Where answers are required to be ringed, do not accept if more than one answer is ringed, unless it is clear which is the child’s intended answer. Accept also any other way of indicating the correct answer, eg underlining. 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk Mental mathematics 2005 quick reference mark scheme Practice question 10 11 Time: 5 seconds 1 1.2 4 p 68 12 2 79 3 36 4 70 5 500 000 Accept an answer in the range 40° to 50° inclusive Do not accept 0.7, % 7 or equivalent 10 Words not acceptable Time: 10 seconds 6 2°C 8 9 14 15 –5°C 0°C 7 13 500 1250 ml 12 000 5°C –1°C Time: 15 seconds 16 17 60 1.15 19 Accept 5r2 or 2 71 18 50 cm 3 p remainder 2 Accept equivalent fractions Prices pen ruler pencil £1.20 £0.75 £0.55 65 Sourced from SATs-Papers.co.uk and 8 Accept numbers in either order p 20 273 http://www.SATs-Papers.co.uk 19 Mental mathematics questions 1–20 Question Requirement Mark Additional guidance 1 68 1m 2 79 1m 3 36 1m 4 70% 1m Do not accept 0.7 OR equivalent fractions. 5 500 000 1m Words not acceptable. 6 2°C 1m Accept any other way of indicating the answer, eg underlining. –5°C 0°C 5°C –1°C 7 10 OR Do not accept if more than one answer is indicated unless the child’s intention is clear. 7 50cm 1m 8 2 1m Accept 5 remainder 2 Accept remainder 2 9 1m 10 1.2 1m 11 4p 1m 12 Answer in the range 40 to 50 degrees inclusive. 1m 13 500 1m 14 1250ml 1m 15 12 000 1m 16 71 1m 17 60p 1m 18 1.15 1m Accept equivalent fractions. 19 3 AND 8 1m Answers may be written in either order. 20 20 65p 273 1m Sourced from SATs-Papers.co.uk Accept equivalent fractions. http://www.SATs-Papers.co.uk Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk EARLY YEARS NATIONAL CURRICULUM 5–16 GCSE GNVQ GCE A LEVEL NVQ OTHER VOCATIONAL QUALIFICATIONS For more information, contact: QCA key stage 2 team, 83 Piccadilly, London W1J 8QA For more copies, contact: QCA Publications, PO Box 29, Norwich NR3 1GN (telephone 08700 606015; fax 08700 606017) Order ref: QCA/05/1360 (mark schemes pack) Sourced from SATs-Papers.co.uk 264777 http://www.SATs-Papers.co.uk