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EARLY YEARS NATIONAL CURRICULUM 5–16 GCSE GNVQ GCE A LEVEL NVQ 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, OTHER VOCATIONAL QUALIFICATIONS 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/ Further teacher packs may be purchased (for any purpose other than statutory assessment) by contacting: QCA Orderline, PO Box 29, Norwich NR3 1GN tel: 08700 606015; fax: 08700 606017 email: orderline@qca.org.uk SourcedOrder ref: QCA/05/1427 from SATs-Papers.co.uk 265511 http://www.SATs-Papers.co.uk Mathematics tests Ma KEY STAGE 3 ALL TIERS Tiers 3–5, 4–6, 5–7 and 6–8 2005 2005 Mark scheme for Paper 1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Introduction Introduction The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to inform teachers. This booklet contains the mark scheme for paper 1 at all tiers. The paper 2 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking information for questions is set out in the form of tables, which start on page 12 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available for that question part. The Correct response column usually includes 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, and whether the marks are independent or cumulative; examples of some different types of correct response, including the most common. The Additional guidance column indicates alternative acceptable responses, and provides details of specific types of response that are unacceptable. Other guidance, such as when ‘follow through’ is allowed, is provided as necessary. Questions with a UAM element are identified in the mark scheme by an encircled U with a number that indicates the significance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet. The 2005 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance General guidance Using the mark schemes Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise. In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance relating to marking of questions that involve money, time, algebra, coordinates, negative numbers or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance What if … The pupil’s response does not match closely any of the examples given. The pupil has responded in a non-standard way. The pupil has made a conceptual error. Markers should use their judgement in deciding whether the response corresponds with the statement of requirements given in the Correct response column. Refer also to the Additional guidance. Calculations, formulae and written responses do not have to be set out in any particular format. Pupils 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, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma for a decimal point. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, error. A computational error is a slip such as writing 4 t 6 e 18 in an otherwise correct long multiplication. A conceptual error is a more serious misunderstanding of the relevant mathematics; when such an error is seen no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 t 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers. The pupil’s accuracy is marginal according to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, or touches, the boundaries given, the mark(s) should be awarded. The pupil’s answer correctly follows through from earlier incorrect work. Follow through marks may be awarded only when specifically stated in the mark scheme, but should not be allowed if the difficulty level of the question has been lowered. Either the correct response or an acceptable follow through response should be marked as correct. There appears to be a misreading affecting the working. This is when the pupil misreads the information given in the question and uses different information. If the original intention or difficulty level of the question is not reduced, deduct one mark only. If the original intention or difficulty level is reduced, do not award any marks for the question part. The correct answer is in the wrong place. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question. 4 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance What if … The final answer is wrong but the correct answer is shown in the working. Where appropriate, detailed guidance will be given in the mark scheme and must be adhered to. If no guidance is given, markers will need to examine each case to decide whether: the incorrect answer is due to a transcription error; in questions not testing accuracy, the correct answer has been given but then rounded or truncated; More than one answer is given. The answer is correct but, in a later part of the question, the pupil has contradicted this response. If so, award the mark. the pupil has continued, in the same part of the question, to give redundant extra working which does contradict work already done. The correct response has been crossed or rubbed out and not replaced. If so, award the mark. the pupil has continued to give redundant extra working which does not contradict work already done; The pupil’s answer is correct but the wrong working is seen. If so, award the mark. If so, do not award the mark. Where a question part carries more than one mark, only the final mark should be withheld. A correct response should always be marked as correct unless the mark scheme states otherwise. Mark, according to the mark scheme, any legible crossed or rubbed out work that has not been replaced. If all answers given are correct or a range of answers is given, all of which are correct, the mark should be awarded unless prohibited by the mark scheme. If both correct and incorrect responses are given, no mark should be awarded. A mark given for one part should not be disallowed for working or answers given in a different part, unless the mark scheme specifically states otherwise. 5 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance Marking specific types of question Responses involving money For example: £3.20 £7 Accept ✓ Do not accept ✓ Any unambiguous indication of the correct amount eg £3.20(p), £3 20, £3,20, 3 pounds 20, £3-20, £3 20 pence, £3:20, £7.00 Incorrect or ambiguous use of pounds or pence eg £320, £320p or £700p, or 3.20 or 3.20p not in the answer space. Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 eg £3.2, £3 200, £32 0, £3-2-0, £7.0 ✓ The £ sign is usually already printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the £ sign, accept an answer with correct units in pounds and/or pence eg 320p, 700p Responses involving time A time interval For example: 2 hours 30 mins Accept ✓ Take care ! Do not accept ✓ Any unambiguous indication eg 2.5 (hours), 2h 30 ✓ Digital electronic time ie 2:30 Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min ! The time unit, hours or minutes, is usually printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the given unit, accept an answer with correct units in hours or minutes, unless the question has asked for a specific unit to be used. A specific time For example: 8.40am, 17:20 Accept ✓ ✓ Any unambiguous, correct indication eg 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 ✓ Unambiguous change to 12 or 24 hour clock eg 17:20 as 5:20pm, 17:20pm Do not accept 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, 084, 84 6 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving the use of algebra For example: 2 p n n p 2 2n n 2 Accept ✓ ✓ Unambiguous use of a different case or variable eg N used for n x used for n n2 Take care ! Do not accept ! Unconventional notation eg n t 2 or 2 t n or n2 or n p n for 2n n t n for n2 1 n d 2 for n or n 2 2 2 p 1n for 2 p n 2 p 0n for 2 Within a question that demands simplification, do not accept as part of a final answer involving algebra. Accept within a method when awarding partial credit, or within an explanation or general working. Embedded values given when solving equations eg in solving 3x p 2 = 32, 3 t 10 p 2 = 32 for x = 10 To avoid penalising the two types of error below more than once within each question, do not award the mark for the first occurrence of each type within each question. Where a question part carries more than one mark, only the final mark should be withheld. ✓ Words used to precede or follow equations or expressions eg t = n p 2 tiles or tiles = t = n p 2 for t = n p 2 ✓ Unambiguous letters used to indicate expressions eg t = n p 2 for n p 2 ! Words or units used within equations or expressions eg n tiles p 2 n cm p 2 Do not accept on their own. Ignore if accompanying an acceptable response. Ambiguous letters used to indicate expressions eg n = n p 2 for n p 2 7 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving coordinates For example: ( 5, 7 ) Accept ✓ ✓ Unconventional notation eg ( 05, 07 ) ( five, seven ) Do not accept Incorrect or ambiguous notation eg ( 7, 5 ) y x ( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x m 5, y m 7 ) x y ( 5, 7 ) ( x e5, y e7 ) Responses involving negative numbers For example: –2 Accept ✓ Do not accept To avoid penalising the error below more than once within each question, do not award the mark for the first occurrence of the error within each question. Where a question part carries more than one mark, only the final mark should be withheld. Incorrect notation eg 2m 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction or percentage only. For example: 0.7 or 7 10 or 70% Accept ✓ Take care ! Do not accept ✓ Equivalent decimals, fractions and percentages eg 0.700, 70 100 , 35 50 , 70.0% ✓ A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 eg 70 18 e 100 25 The first four categories of error below should be ignored if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of error below more than once within each question, do not award the mark for the first occurrence of each type of error unaccompanied by an acceptable response. Where a question part carries more than one mark, only the final mark should be withheld. ! A probability that is incorrectly expressed eg 7 in 10 7 over 10 7 out of 10 7 from 10 ! A probability expressed as a percentage without a percentage sign. ! A fraction with other than integers in the numerator and/or denominator. ! A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 9 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 General guidance Recording marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1 0 The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper. A total of 120 marks is available in each of tiers 3–5 and 4–6. A total of 121 marks is available in each of tiers 5–7 and 6–8. Awarding levels The sum of the marks gained on paper 1, paper 2 and the mental mathematics paper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the QCA website www.qca.org.uk/ from Monday 20 June 2005. QCA will also send a copy to each school in July. Schools will be notified of pupils’ results by means of a marksheet, which will be returned to schools by the external marking agency with the pupils’ marked scripts. The marksheet will include pupils’ scores on the test papers and the levels awarded. 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 BLANK PAGE 11 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 3–5 only Average heights 3-5 4-6 5-7 6-8 1 Correct response a 1m 133 b 1m Additional guidance 7 Tier & Question Making 24 3-5 4-6 5-7 6-8 2 Correct response 2m or 1m Gives three different correct pairs of numbers eg ■ 2 t 12 3t8 4t6 ■ 24 t 1 12 t 2 6t4 Additional guidance ✓ Fractions, decimals or negative numbers ! For 2m or 1m, correct pair of numbers repeated, but in reverse order Do not accept as a different correct pair Gives two different correct pairs of numbers 12 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Write a number 3-5 4-6 5-7 6-8 3 Correct response a 1m b 1m Gives a value that is greater than 1000, but less than 1100 eg ■ 1001 ■ 1099 Gives a decimal that is greater than 0, but less than 1 eg ■ 0.5 ■ 0.12 ■ Point two Tier & Question Additional guidance ✓ Fractions or decimals For part (a), number given in words For part (b), number given as a fraction 3-D shapes 3-5 4-6 5-7 6-8 4 a Tier 3–5 only Correct response 1m Indicates C b 1m Indicates A and E in either order c 1m Additional guidance ! Unambiguous indication Accept eg, for part (b) accept ◆ Cube and cuboid eg, for part (b) do not accept ◆ Square and rectangle 7 13 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Digits 3-5 4-6 5-7 6-8 5 a Correct response 1m Gives all four correct numbers, ie 537 573 735 753 b Additional guidance in any order 1m Identifies the smallest and the biggest numbers from their list (including the two given numbers), provided their list has at least four numbers 1m Correctly adds any numbers they identify, even if they are not from their list, provided their numbers each have at least three digits and the addition requires at least one ‘carry’ eg ■ 357 p 753 = 1110 ■ 537 p 753 = 1290 ■ 333 p 777 = 1110 ■ 357 p 375 p 537 p 573 p 735 p 753 = 3330 ✓ For both marks, follow through Markers may find the following sums using the values from a correct list useful: + 357 375 537 573 735 753 357 – 732 894 930 1092 1110 375 – 912 948 1110 1128 537 – 1110 1272 1290 573 – 1308 1326 or Gives the value 1110, without identifying their smallest and biggest numbers 14 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 735 – 1488 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Different shapes 3-5 4-6 5-7 6-8 6 Correct response 3m Gives all four different correct shapes in any orientations with none incorrect or duplicated eg Additional guidance ! Lines not ruled or accurate, shapes not shaded or internal lines omitted Accept provided the pupil’s intention is clear ■ ! For 3m, correct shapes duplicated even if orientation is different Condone duplication of the given shape, ie a 1 by 4 rectangle For 3m, do not accept other duplicates Squares not joined correctly side-to-side Do not accept as a correct shape eg ◆ or 2m or 1m ◆ Gives at least three different correct shapes, even if there are other incorrect or duplicated shapes Gives two different correct shapes, even if there are other incorrect or duplicated shapes U1 15 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Food and drink 3-5 4-6 5-7 6-8 7 Correct response a 1m £ 1.55 b 2m Indicates the correct item of food and the correct drink, ie Pizza and juice, in either order Additional guidance or 1m ✓ Unambiguous indication eg ◆ P, J Shows the digits 24(0) U1 Tier & Question Number lines 3-5 4-6 5-7 6-8 8 1 Correct response 1m 53 1m 17 1m –5 2m Additional guidance Gives both the values –9 and (p)3 in the correct positions or 1m Gives one correct value in the correct position or Gives both the values (p)3 and –9 but with the positions reversed 16 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Shapes 3-5 4-6 5-7 6-8 9 2 Correct response 2m Additional guidance Gives all three correct areas, ie 16 4 8 or 1m Gives any two correct areas ! For 1m, follow through Provided their 2nd < their 3rd < their 1st, accept the following: For their 2nd, accept follow through as their 1st d 4 For their 3rd, accept follow through as their 1st d 2 or their 2nd t 2 eg, for 1m accept ◆ 20 (error), 5, 10 ◆ 1 (error), 1 1 , 4 2 16, 2 (error), 4 eg, for 1m do not accept ◆ 16, 8 (error), 16 ◆ Tier & Question Computation 3-5 4-6 5-7 6-8 10 3 Correct response a a 1m 10.2 or equivalent b b 1m 9.5 or equivalent c c 1m 1270 d d 1m Additional guidance 57 17 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Scales 3-5 4-6 5-7 6-8 11 4 a a Correct response 1m 900 1m Additional guidance 200 ! Follow through Accept follow through as 1100 – their value for the first mark, provided this gives a positive value U1 b b 1m Indicates 1000, ie 1 10 100 1000 10 000 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Range of ages 3-5 4-6 5-7 6-8 12 5 a a b b Correct response 1m 1m Additional guidance Gives two ages with a difference of 7 years eg ■ 1 and 8 ■ 7 and 14 ■ 7 and 0 ■ 20 and 13 ! Ages given using part-years 0 ! Response given in words Accept provided the difference is 7 years eg, accept 1 ◆ 6 months and 7 2 Accept provided there is no ambiguity eg, accept ◆ Zero ◆ Nothing eg, do not accept ◆ No range ! Units amended Accept responses giving a short time interval eg, accept ◆ A few minutes ◆ A couple of hours 19 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Placing fractions 3-5 4-6 5-7 6-8 13 6 Correct response 2m Additional guidance Gives all four fractions in the correct positions, ie 0 0.5 1 8 1 3 1 3 5 ✓ Unambiguous indication of fractions 1 as a decimal, accept 0.33 or better 3 1 For as a decimal, accept 0.13 or better 8 For eg, for 2m accept 3 4 ◆ 0 0.5 1 8 1 3 1 12 20 15 20 ◆ 0 or 1m 0.5 0.125 0.33 1 0.6 0.75 Gives at least two fractions in the correct positions or Converts at least three of the four correct values into a form enabling comparison, even if the positions are incorrect and there are other errors eg 90 15 40 72 ■ At least three of: , , , 120 ■ ■ ■ ■ 120 120 120 At least three of: 0.75, 0.125, 0.33, 0.6 30 5 24 , , 40 40 40 18 3 8 , , 24 24 24 45 20 36 , , 60 60 60 20 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Survey results 3-5 4-6 5-7 6-8 14 7 a a Correct response 1m Draws a correct bar for Don’t know that indicates 9 people Additional guidance ! Bar not ruled, accurate or shaded Accept provided the pupil’s intention is clear, and the height of the bar is closer to 9 than to either 8 or 10 ! Bar incorrectly positioned or of an incorrect width Condone b b 1m Indicates 3 circles for Don’t know ! Circles not shaded or inaccurate in size Accept provided the pupil’s intention is clear ! Follow through from part (a) Accept the number of circles drawn as the height of their bar for Don’t know d 3 If this results in a part circle, condone any inaccuracy in their part circle U1 21 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Percentages 3-5 4-6 5-7 6-8 15 8 1 a a a Correct response 1m 7 Additional guidance ! For the first mark, ‘out of 10’ repeated eg ◆ 7 10 Condone 1m 1m Completes the sentence correctly with two values that are in the ratio 1 : 20 eg ■ 1 out of 20 ■ 5 out of 100 ■ 0.5 out of 10 ■ 10 out of 200 ■ 2.5 out of 50 1m b b b 50 Completes the sentence correctly, in a different way from one previously credited ! Follow through Accept as two values in the same ratio as their two values for the first mark, provided their first value < their second value eg, from their first mark as 1 out of 5 accept ◆ 2 out of 10 U1 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Rotating Marking overlay available 3-5 4-6 5-7 6-8 16 9 2 Correct response 2m or 1m Gives correct triangles for both grids with vertices within the tolerances as shown on the overlay, ie Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear Gives a correct triangle for either grid with vertices within the tolerances as shown on the overlay, even if the other is incorrect or omitted eg ■ (error) or Completes two rotations of 90˚ clockwise that do not use the given centre of rotation eg ■ or Fails to complete the first rotation correctly but draws a shape that is a triangle, then follows through to rotate their triangle correctly through 90˚ clockwise about the given centre of rotation eg ■ 23 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 What is my number? 3-5 4-6 5-7 6-8 17 10 3 Correct response 2m or 1m Additional guidance 21 Shows or implies that 2 t my number is 42 eg ■ 2 t my number e 357 m 315 e 42 ■ 2x e 42 ■ 42 d 2 or U1 Shows a complete correct method with not more than one computational error, even if their choice between alternative answers is incorrect or omitted eg ■ 15 t 10 e 150, 150 p 150 p 15 e 315, so it’s 10 p 10 p 1 ■ 357 m 170 m 170 m 17 m 17 (error) e 0, so it’s 10 p 10 p 1 p 1 e 22 ■ 1 (error) 1 15 315 ■ 21 5 (error) 1 15 315 17 357 Tier & Question Completing 3-5 4-6 5-7 6-8 18 11 4 Correct response 1m 32 1m 12 1m Gives a correct expression in x with a value of 48 when x is 8 eg ■ 6x ■ x p 40 ■ 3x p 24 Additional guidance ! For the first and second marks, incomplete processing Penalise only the first occurrence eg, for the first and second marks ◆ 4 t 8 48 d 4 Mark as 0, 1 ! For the third mark, unconventional notation Condone eg, for the third mark accept ◆ 6 t x ◆ x6 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Mean and median 3-5 4-6 5-7 6-8 19 12 5 a a a Correct response 1m Shows that the mean is 10 eg ■ 9 p 11 p 10 e 30, 30 d 3 ■ (9 p 11 p 10) d 3 ■ 10 is already 10, then 9 is 1 below and 11 is 1 above Additional guidance ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ 30 d 3 30 d 10 e 3 9 p 11 e 20, 20 d 2 Add one to 9 and take one off 11 10 is halfway between 9 and 11 ✓ Method described eg ◆ You add them up then divide by how many there are Incorrect statement eg ◆ 9 p 10 p 11 d 3 e 10 ◆ 3 d 30 e 10 1m Gives a correct explanation of why the median is 10 eg ■ 10 is the middle number when the numbers are in order ■ The median is the middle number when the numbers go from smallest to largest ✓ Minimally acceptable explanation eg ◆ ◆ ◆ It is the middle number It’s the middle largest It’s the second smallest ◆ 9 10 11 ◆ It is in between Incomplete or incorrect explanation eg ◆ 9 10 11 ◆ 10 is halfway between 9 and 11 b b b 1m Gives four values that total 40 and whose middle two numbers, when ordered, add to 20, with none of the values being 10 eg ■ 8 9 11 12 ■ 0 0 20 20 ■ 9 11 9 11 ■ 7 13 9 11 ✓ Fractions, decimals and negatives U1 25 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Angles 3-5 4-6 5-7 6-8 20 13 6 Correct response 1m Shows angle a as 50 1m Shows angle b as 130 Additional guidance ! For the second mark, follow through Accept follow through as 180 m their a, provided their a < 90 and is not 54 to 56 inclusive 1m Shows angle c as 20 ! For the third mark, follow through Accept follow through as 150 m their b or their a m 30, provided this gives a positive value Tier & Question Equations 3-5 4-6 5-7 6-8 21 14 7 Correct response 1m 5 Additional guidance ! Incorrect notation eg, for the first mark ◆ t5 Penalise only the first occurrence 1m 3 ! Incomplete processing eg, for the first mark ◆ 15 3 Penalise only the first occurrence 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7, 6–8 Long multiplication 3-5 4-6 5-7 6-8 22 15 8 Correct response 2m or 1m Additional guidance 8602 Shows a complete correct method with not more than one computational error eg ■ 3740 p 3740 p 374 t 3 = 7480 p 1122 ■ 300 70 4 6000 1400 80 900 210 12 so 6000 p 1400 p 80 p 900 p 210 p 12 20 3 ■ Conceptual error eg ◆ 374 23 1122 748 1870 374 23 1126 (error) 7480 8606 Tier & Question Midpoint 3-5 4-6 5-7 6-8 23 16 9 1 Correct response a a a 1m (60, 60) b b b 1m Additional guidance Gives M as (0, 100) 1m Gives N as (60, 0) ! Answers for M and N transposed but otherwise completely correct If this is the only error, ie gives M as (60, 0) and gives N as (0, 100), mark as 0, 1 ! x- and y-coordinates transposed but U1 otherwise correct for both M and N If this is the only error, ie gives M as (100, 0) and gives N as (0, 60), mark as 0, 1 27 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Square cut 3-5 4-6 5-7 6-8 17 10 2 Correct response 2m or 1m 42, with no evidence of an incorrect method Additional guidance Incorrect method eg ◆ 12 p 2 e 14, 14 t 3 e 42 Shows or implies that the square is a 9(cm) by 9(cm) square eg ■ 7 t 6 seen ■ 6 7 ■ Area of square = 81 or Shows or implies a correct method in which the only error is to use an incorrect value for the shorter horizontal side of rectangle A eg ■ 12 d 2 = 8 (error), 8 p 3 = 11 11 m 2 = 9, 8 t 9 = 72 ■ 4 (error) 5 U1 Answer: 20 28 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Making zero 3-5 4-6 5-7 6-8 18 11 3 Correct response a a a 1m Additional guidance Indicates only the second statement, ie ✓ b b b 1m Indicates that the other number is zero eg ■ 0 ■ Zero ✓ Minimally acceptable indication eg ◆ ◆ 0p0 Same ! Use of negative sign eg –0 Condone ◆ 1m Gives a correct pair of non-zero values that add to make zero eg ■ 1 and –1 ■ –45 and p45 ■ x and –x Operation changed eg ◆ 1 m 1 [‘and’ crossed out] U1 29 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Cuboid 3-5 4-6 5-7 6-8 19 12 4 Correct response 2m Draws a 1 by 3 by 4 cuboid in any orientation, using the isometric grid eg Additional guidance ✓ Some or all internal lines omitted eg ◆ ■ ■ ! Lines not ruled Accept provided the pupil’s intention is clear ! Drawing not accurate or 1m For 2m, accept vertices within 2mm of the dots of the grid For 1m, accept a less accurate drawing provided the pupil’s intention is clear Draws a correct view, using the isometric grid and maintaining three dimensions, but either omits one or more external lines or shows some hidden lines eg ■ ! Cuboid enlarged For 2m or 1m, accept provided a consistent scale factor has been used for all lengths, and any internal lines divide the cuboid into only 12 smaller cubes ! For 2m, hidden lines shown or Draws a view of a cuboid, using the isometric grid and with all external lines and no hidden lines shown, but with only one dimension incorrect, by not more than one unit eg Do not accept unless the lines are clearly identified as hidden lines eg, for 2m, accept ◆ ■ eg, for 2m, do not accept ◆ ■ For 2m, external lines omitted or Shows a 1 by 3 by 4 cuboid in any orientation, but does not use the isometric grid correctly eg ■ 30 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Dividing fractions 3-5 4-6 5-7 6-8 20 13 5 Correct response 3m Gives the first value as 2 and the second value as 6 Additional guidance ✓ For the second value 6 1 ! Eighths repeated or 2m Accept as the final answer for the first value eg, for the value 2 accept Gives an incorrect or omitted first value but correctly gives the second value as 6 or Gives an incorrect first value but follows through correctly for the second value as their first value t 3, provided their first value is a positive integer eg ■ first value: 4 second value: 12 ◆ 2 8 Do not accept as the final answer for the second value eg, for the value 6 do not accept ◆ 6 8 or Gives the correct first value and shows or implies a correct method for the second value with not more than one computational error eg ■ 2 t 3 ■ 3 d 0.5 ■ ■ ■ ■ 3 t 4 24 4 6 d 8 For 2m or 1m, conceptual error eg 6 1 5 ◆ d e 8 8 8 8 1 1 8 Answer of 6 or equivalent, with no 8 evidence of an incorrect method ■ or 1m Gives an incorrect or omitted first value but shows or implies a correct method for the second value with not more than one computational error ! For 1m, follow through For 1m, accept follow through as the intention to multiply their first value by 3 shown or implied eg, accept 4 ◆ first value: 8 second value: 1 ◆ first value: then 1 or equivalent 2 4 8 4 t 3 seen 8 31 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Solving an equation Refer to the new algebra general guidance 3-5 4-6 5-7 6-8 21 14 6 Correct response 2m 25 or equivalent 4 Additional guidance For 2m, 25 seen but with incorrect further 4 working eg 25 ◆ = 6.1 4 or 1m Shows or implies a correct first step of algebraic manipulation that either reduces the number of terms or collects variables on one side of the equation and numbers on the other eg ■ 2t e 25 m 2t ■ –25 p 2t e m2t ■ 2t p 2t e 100 m 75 ■ 75 p 4t e 100 ■ 4t e 25 ■ 25 d 4 seen Tier & Question ! Method used is trial and improvement Note that no partial credit can be given Angle p 3-5 4-6 5-7 6-8 22 15 7 Correct response 2m 140 or 1m Additional guidance Shows the value 110 or 220 or Shows or implies a complete correct method with not more than one computational error eg ■ 360 m 2 t (180 m 35 t 2) ■ 360 m (360 m 4 t 35) ■ 70 t 2 ■ 20 20 ■ p 180 – 40 35 p 35 e 80 (error), 180 m 80 e 100 360 m 100 t 2 e 160 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Speed bumps 3-5 4-6 5-7 6-8 23 16 8 Correct response a a 2m Additional guidance Completes both sentences correctly, with all four values in the correct positions, ie 46 ! Throughout the question, key not interpreted eg, for the value 46 ◆ 4|6 Penalise only the first occurrence 12 35 3 or 1m Gives at least two values in the correct positions or Shows the values 46, 12, 35 and 3, even if their positions are incorrect b b 1m Gives a correct justification eg ■ 38 m 28 e 10 ■ It falls from 38 to 28 ✓ Minimally acceptable justification eg ◆ 38 and 28 identified, with no evidence of an incorrect method ! Ambiguous notation eg ◆ 28 m 38 Condone Incomplete or incorrect justification eg ◆ The difference between the middle numbers before and after is 10 ◆ Indicates both values of 8 corresponding to the units of 38 and 28 on the diagram, but with no interpretation of the key ◆ Before the median was 39, after the median was 29, so it fell by 10 33 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Straight line graph Refer to the new algebra general guidance 3-5 4-6 5-7 6-8 24 17 9 Correct response a a a 1m Indicates that the y-coordinate is 146 Additional guidance ✓ Indication is within a pair of correct coordinates eg, for part (a) ◆ (50, 146) eg, for part (b) ◆ (18, 50) b b b 1m Indicates that the x-coordinate is 18 ! Answers to parts (a) and (b) transposed but otherwise correct Mark as 0, 1 c c 1m Indicates Yes and gives a correct explanation with no evidence of incorrect working eg ■ When x e m10, y e 3 t m10 m 4 e m30 m 4 e m34 ■ 3x m 4 e m34 3x e m30 x e m10 ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ m30 m 4 e m34 m30 d 3 e m10 When x e m10, 3x m 4 e m34 The second number is equal to the first number multiplied by 3, minus 4 Incomplete or incorrect explanation eg ◆ When x e m10, y e m34 ◆ 3x m 4 e m34 3x e m34 m 4 3x e m30 x e m10 34 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question 64 3-5 4-6 5-7 6-8 18 10 Correct response 3m Gives four different correct pairs of values for x and y eg ■ x e 64 ye1 xe8 ye3 xe2 ■ ye2 xe4 ye6 xe 1 64 y e m1 x e 4096 y e 1 2 x e √8 or 1m ye4 x e –8 or 2m Additional guidance ye2 Gives three different correct pairs of values for x and y, even if there are errors, omissions or repeats Gives two different correct pairs of values for x and y, even if there are errors, omissions or repeats 35 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Sixths 3-5 4-6 5-7 6-8 19 11 Correct response 1m Gives a correct justification The most common correct justifications: Additional guidance ! Response contains an incorrect statement Ignore alongside a correct response eg, accept ◆ State or imply that fact that 2 1 e and use the known 6 3 1 rounds to 33% 3 1 is 33 and 100 d 6 = 16 3 ✓ Minimally acceptable justification eg ◆ 1 is 33 3 eg ■ 2 1 e which is 33 to the nearest per cent 6 3 Show or imply that the percentage should be 33 by showing a more accurate percentage, or a correct method eg 1 ■ It’s 33 % so it rounds to 33 not 34 Incomplete justification eg ◆ It’s 33% not 34% ✓ Minimally acceptable justification eg ◆ ◆ ◆ 3 ■ ■ ■ 100 d 6 t 2 e 33.33, so 33 Double 16.7 is 33.4 34 t 3 e 102, but 33 t 3 e 99 which is closer to 100 33.3 100 d 6 t 2 gives 33 34 t 3 e 102 but 33 t 3 e 99 ! More accurate percentage rounded or truncated 1 , accept 33.3% or better 3 1 For , accept 16.7% or 16.66% or better 6 For Incomplete justification eg ◆ 100 d 6 t 2 ◆ 34 t 3 e 102 (or 33 t 3 e 99) Refer to the effect of the premature rounding, 1 for example by giving a possible value for in 6 the range 16.5 to 16.75 inclusive, or 17.25 to 17.5 inclusive, and shows or implies the 2 percentage for could be 33 or 35 6 eg ■ If 1 were 16.6%, it would be 17% to the 6 nearest per cent, but double 16.6 is 33.2 ■ 1 could be 17.4, but 17.4 t 2 e 34.8 6 ✓ Minimally acceptable justification eg ◆ ◆ 17% is rounded not exact, so when you double it, you double the error 17 is not exact, so it could be 33 or 35 Incomplete justification eg 1 2 ◆ rounds to 17, so could round to 33 6 ◆ ◆ 6 Keep adding 17 and you don’t get to 100 17 t 6 e 102 Incorrect justification that implies hypothetical values are the correct values eg 1 2 ◆ = 16.5% so = 33% 6 6 36 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Tyres 3-5 4-6 5-7 6-8 20 12 Correct response a a 1m 5 b b 1m Gives a value between 3500 and 5500 inclusive Additional guidance ! Incorrect units inserted eg ◆ 5000 miles Ignore 37 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 5–7, 6–8 Tier & Question 3-5 4-6 5-7 6-8 21 13 a a 1m Which triangles? Refer to the new algebra general guidance Correct response Additional guidance Indicates the correct triangle, ie ✓ and gives a correct equation linking a, b and c for the other triangle eg ■ a2 p c2 e b2 ■ b2 m a2 e c2 ■ b2 m c2 e a2 ■ b e √a2 p c2 b b 1m Indicates the correct triangle, ie ✓ and gives a correct explanation for the other triangle The most common correct explanations: State or imply that the third angle in the triangle on the right is not 90˚ eg ■ 180 m 75 m 25 e 80 not 90, so you can’t use Pythagoras’ Theorem ■ Angle C is not 90, so it’s not a right-angled triangle ✓ Minimally acceptable explanation Show that if the third angle in the triangle on the right were 90˚, the triangle would not be possible ■ If the missing angle is 90, the angles add up to 190 not 180 ✓ Minimally acceptable explanation eg ◆ ◆ ◆ Not 90 Not a right angle It only works when it’s right-angled Incomplete explanation that does not refer explicitly to 90˚ eg ◆ 180 m 75 m 25 e 80 ◆ You can’t use Pythagoras’ Theorem ◆ The angles are wrong eg ◆ ◆ ◆ ◆ If you put 90 in you don’t get 180 A right-angled triangle is impossible with those angles, they should make 180 The angles add up to 190 [right angle marked on right hand triangle] They add up to 100 not 90 Incomplete explanation that does not refer explicitly to 90˚ eg ◆ The angles would add up to 190, not 180 38 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Sweet peas 3-5 4-6 5-7 6-8 22 14 Correct response a a 1m 89 or equivalent probability 100 Additional guidance ! Unconventional notation, but equivalent value eg, for the first mark ◆ 1m 17 or equivalent probability 20 17.8 20 Condone ! Estimates transposed but otherwise correct Mark as 0, 1 b b 1m Indicates Ravi and gives a correct explanation that states or implies that he used more seeds eg ■ The more trials you have the more accurate your estimate of probability is likely to be ■ The number of seeds in each packet was the same but Ravi had more packets than Meg so he had a greater number of trials ■ There were more seeds to consider ■ 200 seeds is more than 100 seeds ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ More seeds More packets He tested more He had 200, not 100 Ravi had 10, Meg had 5 ! Irrelevant statement eg Ravi’s results were more accurate He had more chance of a bigger number germinating Ignore alongside a correct response, otherwise do not accept ◆ ◆ Incomplete, ambiguous or incorrect explanation eg ◆ More ◆ A bigger number 170 89 which is more than 200 100 ◆ Ravi’s = ◆ More of his seeds germinated He had 5 more seeds Meg’s numbers were more complicated and harder to work out ◆ ◆ U1 39 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question How many digits? 3-5 4-6 5-7 6-8 23 15 Correct response 2m Additional guidance Gives a correct response that satisfies the following four conditions: 1. Indicates the minimum is 4 2. Shows a correct justification for the minimum eg, for condition 2 ■ 100 t 10 e 1000 3. Indicates the maximum is 5 4. Shows a correct justification for the maximum eg, for condition 4 ■ 999 t 99 e 98 901 ■ 999 t 100 e 99 900, a 5-digit number and subtracting 999 does not change it from being a 5-digit number ■ 99 000 is just over the biggest possible so this must have the same number of digits ■ 100 t 1000 e 100 000, but this is the smallest possible 6-digit number, so 99 t 999 must have 5 digits ✓ Minimally acceptable justification for the minimum [condition 2] eg ◆ 1000 ◆ 100 t 10 (or 10 t 100) ✓ Minimally acceptable justification for the maximum [condition 4] eg ◆ 98 901 ◆ 999 99 8991 89910 ◆ ◆ ◆ ◆ ◆ 99 900 m 999 99 000 m 99 99 900 (or 99 000) is just over 100 000 1000 t 100 (or 100 t 1000) Incomplete or incorrect justification for the maximum [condition 4] eg ◆ 999 t 99 ◆ 99 900 m 99 ◆ 99 000 m 999 or 1m Gives a response that satisfies at least condition 4, even if condition 3 is not satisfied or Gives a response that satisfies condition 1, satisfies condition 4 with not more than one computational error, then follows through correctly to give their maximum Conceptual error eg ◆ 999 t 99 8991 8991 17982 ◆ 99 t 999 = 99 900 m 99 = 99 801 U2 40 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Refer to the new algebra general guidance 3-5 4-6 5-7 6-8 24 16 Correct response 3m Gives both x = Tiers 5–7, 6–8 Simultaneous Additional guidance 3 or equivalent and y = 5 2 and shows a complete correct method for solving algebraically eg ■ 4x p 3y e 21 4x p 2y e 16 so y e 5 1 2x p 5 e 8 so x e 1 Method used is trial and improvement 2 ■ ■ 4x p 3y e 21 6x p 3y e 24 so 2x e 3 therefore x e 1.5 and y e 5 2x p y e 8 2x p 2y e 13 3 so y e 5 and x e 2 ■ or 2m or 1m 4x p 3(8 m 2x) e 21 24 m 2x e 21 x e 1.5, so y e 5 Shows a complete correct method for solving algebraically with not more than one error eg ■ 4x p 3y e 21 4x p 2y e 16 so y e 4 (error) 2x p 4 e 8 so x e 2 ■ 4x p 3(8 m 2x) e 21 4x p 24 m 2x (error) e 21 2x e m3 x e m1.5 and y e 11 (or 9) ! Only error is to use the wrong operation, Forms two correct equations that would allow elimination of either x or y eg ■ 4x p 3y e 21 4x p 2y e 16 ■ 4x p 3y e 21 6x p 3y e 24 ! For 1m, equations subtracted without the spuriously eliminating either x or y eg ◆ 4x p 3y e 21 4x p 2y e 16 5y e 37, so y e 7.4 2x p 7.4 e 8 so x e 0.3 Mark as 1, 1, 0 second equation restated Accept eg, for 1m accept ◆ 2x p 2y e 13 seen or Attempts to solve by substitution and forms a correct equation in either x or y eg ■ 4x p 3(8 m 2x) e 21 ■ 8 m 2x e 21 m 4x 3 ■ 8 m y e 10.5 m 1.5y 41 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Angle bisector Marking overlay available 3-5 4-6 5-7 6-8 25 17 Correct response 2m Completes a correct angle bisector that fulfils all four of the following conditions: 1. Ruled 2. Within the tolerance as shown on the overlay, even if their line were to be extended 3. At least 3cm in length from A through the acute angle BAC 4. Evidence of correct construction arcs that are centred on two points on lines AB and AC equidistant from A, are of equal radii and have one point of intersection Additional guidance ! Use of construction arcs on the overlay Note that these are to give a visual guide as to whether a correct pair of centres has been used, and do not indicate tolerance ! Section of angle bisector extending from A through reflex angle BAC Accept if needed as part of the 3cm required, provided the section is within the tolerance as shown by the dashed lines on the overlay. Otherwise, ignore ! Extra arcs drawn Ignore Spurious construction arcs For 2m or 1m, do not accept arcs drawn without compasses, or arcs centred on points on the lines that are not equidistant from A or 1m Gives a response that fulfils condition 4, even if the angle bisector is incomplete, incorrect or omitted 42 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Star shapes 3-5 4-6 5-7 6-8 18 Correct response a 1m Additional guidance ! Units given 8 Condone responses of 8cm only b 1m Gives a different pair of dimensions in the ratio 5 : 2 or 2 : 5 eg ■ 2 and 5 (either order) ■ 10 and 25 (either order) ■ 1 and 2.5 (either order) ■ 12 and 30 (either order) Dimensions of either given diagram Do not accept value 6 and 15 (either order) or 8 and 20 (either order) Tier & Question Straight lines 3-5 4-6 5-7 6-8 19 Correct response a 1m 1m Additional guidance Gives A as (0, m8) Gives B as (2, 0) ! Answers for A and B transposed but otherwise completely correct If this is the only error, ie gives A as (2, 0) and gives B as (0, m8), mark as 0, 1 b 1m Gives a correct equation for the straight line eg ■ y e 2x ■ y m 2x e 0 y ■ xe ! Unconventional notation eg ye2tx y e 2x p 0 Condone ◆ ◆ 2 43 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Acorns 3-5 4-6 5-7 6-8 20 a 1m Correct response Gives a correct explanation The most common correct explanations: Show or imply that the median for group A is 26, and for group B is 29 eg ■ Median A m median B e 29 m 26 e3 ■ 26 p 3 e 29 and A is 26, B is 29 Additional guidance ! Median line referred to as the ‘middle’ or ‘centre’ Condone eg, accept ◆ The lines in the middle are at 26 and 29 ◆ The centre points of the boxes are 3mm apart ✓ Minimally acceptable explanation eg ◆ ◆ ◆ 26, 29 A is 29 m 3 B is 26 p 3 Incomplete explanation eg ◆ 29 m 3 ◆ 26 p 3 Indicate, in words or on the diagram, the locations of the medians for A and B eg ■ The vertical lines on the shaded part of the box plots represent the medians and they are 3mm apart on the graph ✓ Minimally acceptable explanation eg ◆ ◆ ◆ The lines in the shaded bit are 3 apart The lines in the boxes are the medians Arrows indicating both medians on the diagram Incomplete explanation eg ◆ The vertical lines are 3mm apart on the graph ◆ The lines for the medians are 3mm apart on the graph ! Throughout the question, incorrect units Condone eg, for part (a) accept ◆ The lines in the boxes are 3cm apart ! Throughout the question, ambiguous notation eg, for part (a) ◆ 26 m 29 eg, for part (b) ◆ 24 m 29 > 27 m 31 Condone 44 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Acorns (cont) 3-5 4-6 5-7 6-8 20 Correct response b 1m Indicates A and gives a correct explanation The most common correct explanations: Additional guidance ! Inter-quartile range referred to as ‘range’ Condone eg, accept ◆ Range for A e 5, range for B e 4 ◆ The boxes show the range and A’s is longer Show or imply that the inter-quartile range for A is 5 and for B is 4 eg ■ For A the IQ range is 29 m 24 e 5, for B the IQ range is 31 m 27 e 4 ■ The distance between 24 and 29 is greater than that between 27 and 31 ■ The IQR is 1mm bigger for group A Indicates, in words or on the diagram, the sizes of the inter-quartile ranges for A and B eg ■ The shaded box in A is longer than in B, so A has a bigger inter-quartile range ■ The box for group A covers 6 whole numbers, but for B only 5 c 1m ✓ Minimally acceptable explanation ✓ Minimally acceptable explanation eg ◆ ◆ ◆ 5, 4 29 m 24 > 31 m 27 1 more Incomplete or incorrect explanation eg ◆ 5 is the larger inter-quartile range ◆ 31 m 27 is less ◆ The inter-quartile range for A is 4cm and for B is 3.2cm [scale ignored] eg ◆ ◆ ◆ The box is bigger Distances between lower and upper quartiles for both A and B indicated It covers 6 numbers, the other covers 5 Gives a correct reason The most common correct reasons: Refer to possible differences in the conditions of the two samples eg ■ The two groups could have collected the samples at different times of year ■ Group A could have picked from one side of the tree and group B from the other side ■ One group could have picked from the tree, the other from the ground ■ Group B may have collected first and taken most of the larger ones ✓ Minimally acceptable reason Refer to possible differences in the sizes of the two samples eg ■ One group could have collected a much larger number of acorns than the other ■ One sample may be less representative as they didn’t collect enough ✓ Minimally acceptable reason U1 eg ◆ ◆ ◆ Different times Different areas of the tree B’s acorns may have had more sunlight Incomplete or incorrect reason eg ◆ Different areas ◆ They used different trees eg ◆ ◆ Different numbers of acorns You don’t know how many acorns Incomplete reason eg ◆ You don’t know how many ◆ One group could have spent longer ◆ There could have been more people to collect acorns in one of the groups 45 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Standard form 3-5 4-6 5-7 6-8 21 Correct response a 1m b 2m Additional guidance Gives a correct justification eg ■ (4 t 108) t (8 t 104) e (4 t 8) t (108 t 104) = 32 t 1012 = 3.2 t 1013 ■ 4 t 8 e 32, 8 p 4 e 12, so you get 32 t 1012 e 3.2 t 1013 ■ 400 000 000 t 80 000 e 32 000 000 000 000 = 3.2 t 1013 ✓ Minimally acceptable justification 5 t 103 ! Zero(s) given after the decimal point within eg ◆ ◆ ◆ 32 t 1012 4 t 8 t 1012 400 000 000 t 80 000 = 32 000 000 000 000 [12 zeros shown] Incomplete justification eg ◆ 32 000 000 000 000 = 3.2 t 1013 ◆ 400 000 000 t 80 000 = 3.2 t 1013 ◆ (4 t 8) t (108 t 104) = 3.2 t 1013 standard form notation Condone eg, for 2m accept ◆ 5.000 t 103 or 1m Shows a value equivalent to 5 t 103 eg ■ 5000 ■ 0.5 t 104 ■ 104 2 or Shows or implies a correct method that demonstrates understanding of how to process the indices and places the multiplication symbol correctly, with not more than one error eg ■ 4 d 8 t 10(8 – 4) ■ 4 t 108 d 8 t 104 e 2 (error) t 104 46 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Data sets 3-5 4-6 5-7 6-8 22 Correct response 2m Gives both correct values, ie median = 90 or 1m mean = 97 Additional guidance ! Incomplete processing Condone eg, for 2m accept ◆ median e 90 mean e 95 p 2 Gives one correct value or Shows the value 9700 U1 Tier & Question Marking overlay available 3-5 4-6 5-7 6-8 23 Correct response 2m Draws a correct rhombus that fulfils all three of the following conditions: 1. Ruled 2. Correct intersecting construction arcs for at least one vertex, using compasses at either 8cm and 10cm or 8cm and 8cm, within the tolerances as shown on the overlay 3. Vertices within the tolerances as shown on the overlay Drawing a rhombus Additional guidance ! Different orientations Markers should rotate and/or turn over the overlay as appropriate in order to check tolerances for construction arcs and/or vertices ! Arcs extended or extra arcs Ignore inaccuracies in sections of arcs extending beyond the tolerances as shown on the overlay, or arcs not indicated on the overlay, even if incorrect ! Spurious arcs or 1m Gives a response that fulfils either condition 2 or condition 3 Do not accept as correct arcs drawn without compasses 47 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question a and b Refer to the new algebra general guidance 3-5 4-6 5-7 6-8 24 Correct response 3m or 2m or 1m Gives a correct justification eg ■ b p b + 2 = b(b p 2) 2b p 2 e b2 p 2b 2 e b2 b e √2 which is not an integer, so a cannot be an integer either ■ 2a m 2 e a2 m 2a 2 a m 4a p 2 e 0 which doesn’t factorise, so a is not an integer Shows correct expressions for the sum and product of a and b using only one of the two variables eg ■ b p b p 2, b(b p 2) ■ 2a m 2, a2 m 2a Additional guidance ✓ Minimally acceptable justification eg ◆ 2b p 2 e b2 p 2b 2 e b2 ! Variables a and b transposed but justification otherwise completely correct eg ◆ a p a p 2 e a(a p 2) 2a p 2 = a2 p 2a 2 e a2 a e √2 Mark as 1, 1, 0 ! Numerical examples given Ignore Shows or implies the use of expressions for a and b involving only one of the two variables eg ■ b, b p 2 ■ a, a m 2 ■ 2b p 2 ■ a2 m 2a or Shows a different correct equation involving both the variables a and b eg ■ a p b e ab U3 48 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Refer to the new algebra general guidance 3-5 4-6 5-7 6-8 25 Correct response 2m or 1m Gives the value 10 and shows or implies a correct method for solving algebraically eg ■ 9C p 32 e 2C p 30 5 9C e 2C m 2 5 9C e 10C m 10 10 = C ■ 9C 2C m e 32 m 30 5 10C m 9C e2 5 C e2 5 Tier 6–8 only Temperature Additional guidance Method used is trial and improvement Shows or implies a correct first step of algebraic manipulation using a correct equation in terms of C, that either reduces the number of terms or collects unknowns on one side of the equation and numbers on the other eg ■ 9C p 2 e 2C 5 ■ 0.2C p 30 e 32 9C ■ 2C m e 32 m 30 5 C =2 ■ 5 ■ 2t5 49 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Index Index to mark schemes Tier 3–5 4–6 5–7 6–8 1 2 3 4 5 6 7 8 1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 1 16 9 2 17 10 3 18 11 4 19 12 5 20 13 6 21 14 7 22 15 8 23 16 9 1 17 10 2 18 11 3 19 12 4 20 13 5 21 14 6 22 15 7 23 16 8 24 17 9 18 10 19 11 20 12 21 13 Question Average heights Making 24 Write a number 3-D shapes Digits Different shapes Food and drink Number lines Shapes Computation Scales Range of ages Placing fractions Survey results Percentages Rotating What is my number? Completing Mean and median Angles Equations Long multiplication Midpoint Square cut Making zero Cuboid Dividing fractions Solving an equation Angle p Speed bumps Straight line graph 64 Sixths Tyres Which triangles? Page 12 12 13 13 14 15 16 16 17 17 18 19 20 21 22 23 24 24 25 26 26 27 27 28 29 30 31 32 32 33 34 35 36 37 38 50 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2005 KS3 Mathematics test mark scheme: Paper 1 Tier Question 3–5 4–6 5–7 6–8 Index Page 22 14 Sweet peas 39 23 24 15 16 How many digits? Simultaneous 40 41 25 17 18 Angle bisector Star shapes 42 43 19 Straight lines 43 20 21 Acorns Standard form 44 46 22 Data sets 47 23 Drawing a rhombus 47 24 25 a and b Temperature 48 49 51 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk