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Mathematics tests Ma KEY STAGE 3 ALL TIERS 2001 Mark scheme for Paper 1 Tiers 3–5, 4–6, 5–7 and 6–8 KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 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 and the extension paper mark schemes are printed in separate booklets. 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 11 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 and the minimum acceptable. 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. 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 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, coordinates, algebra or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 3 2001 KS3 Mathematics Test Mark Scheme: Paper 1 General guidance What if … The pupil’s response does not match closely any of the examples given. 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. The pupil has responded in a non-standard way. 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. The pupil has made a conceptual error. 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. The correct answer is in the wrong place. 4 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. 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. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 The final answer is wrong but the correct answer is shown in the working. General guidance 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. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 5 2001 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 6 Sourced from SATs-Papers.co.uk 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 http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 General guidance Responses involving coordinates For example: ( 5, 7 ) Accept Do not accept Unambiguous but unconventional notation eg ( 05, 07 ) ( five, seven ) x y ( 5, 7 ) ( x e5, y e7 ) Incorrect or ambiguous notation eg ( 7, 5 ) ( 5x, 7y ) ( x5, y7 ) ( 5x, 7y ) Responses involving the use of algebra For example: 2 p n n p 2 2n Accept The unambiguous use of a different case eg N used for n Unconventional notation for multiplication eg n t 2 or 2 t n or n2 or n p n for 2n n t n for n2 Multiplication by 1 or 0 eg 2 p 1n for 2 p n 2 p 0n for 2 Words used to precede or follow equations or expressions eg t e n p 2 tiles or tiles e t e n p 2 for t e n p 2 Unambiguous letters used to indicate expressions eg t e n p 2 for n p 2 Embedded values given when solving equations eg 3 t 10 p 2 e 32 for 3x p 2 e 32 Sourced from SATs-Papers.co.uk Take care ! Do not accept ! Words or units used within equations or expressions should be ignored if accompanied by an acceptable response, but should not be accepted on their own eg do not accept n tiles p 2 n cm p 2 Change of variable eg x used for n Ambiguous letters used to indicate expressions eg n e n p 2 However, to avoid penalising any of the three types of error above 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. Embedded values that are then contradicted eg for 3x p 2 e 32, 3 x 10 p 2 e 32, x e 5 http://www.SATs-Papers.co.uk 7 2001 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 Accept A correct probability that is correctly expressed as a decimal, fraction or percentage. Equivalent decimals, fractions or percentages eg 70 35 0.700, , , 70.0% 100 50 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 Take care ! Do not accept The following four categories of error should be ignored if accompanied by an acceptable response, but should not be accepted on their own. ! A probability that is incorrectly expressed eg 7 in 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. However, each of the three types of error above should not be penalised 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 expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 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, 4–6, 5–7 and 6–8. The extension paper carries 42 marks. Awarding levels The sum of the marks gained on paper 1, paper 2 and the mental arithmetic 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 Friday 22 June 2001. 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. The 2001 key stage 3 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 9 2001 KS3 Mathematics Test Mark Scheme: Paper 1 BLANK PAGE 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 3–5 only Tier & Question Multiplication Table 3-5 4-6 5-7 6-8 1 Correct response a 1m 22 1m 12 1m 24 t 11 or 22 t 12 1m b 312 1m The other pair as shown above. Additional guidance Numbers in a pair in either order ! Other pairs of factors of 264 eg 44 t 6 88 t 3 Penalise the first occurrence only. N N Tier & Question Number Cards 3-5 4-6 5-7 6-8 2 a Correct response 1m 60 , 10 (either order) and 40 , 30 (either order) b 1m 60 , 6 , 4 (any order) c 1m 1m Numbers or operations other than those specified eg, for part (b) N 100 – 60 p 30 eg, for part (d) N 3 t 30 p – 20 Repeated values eg, for part (b) N 30, 30, 10 100 , 30 (either order) d Additional guidance 10 , 40 in the correct order only Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 11 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier & Question Computation 3-5 4-6 5-7 6-8 3 1 a Tiers 3–5, 4–6 Correct response 1m 65 1m 13 1m 36 1m 7 1m 1725 1m 569 b 1m 43 c 1m Additional guidance 14 Tier & Question What’s the Point? 3-5 4-6 5-7 6-8 4 2 Correct response a a 1m (5, 2) b b 1m Additional guidance (2, 1) 12 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier & Question Temperature 3-5 4-6 5-7 6-8 5 3 a Tiers 3–5, 4–6 Correct response 1m Additional guidance ! Values incorrectly or not labelled Indicates 7ºC Accept if unambiguous. b 1m Indicates –5ºC a 1m 5 d b 1m 11 c Temperatures shown as negative eg N –5 N –11 Tier & Question Twenty-seven 3-5 4-6 5-7 6-8 6 4 a Correct response 1 2 1m 1m b 54 108 1m b 1m Two numbers that multiply to make 27 eg I 3t9 I 54 t 0.5 1m Two numbers that divide to make 27, in the correct order eg I 27 d 1 I 54 d 2 Equivalent fractions or decimals 123 1m a 1 Additional guidance Sourced from SATs-Papers.co.uk Follow through as 2 t their incorrect 54 Values given that are not exact eg N 81 t 0.33 Follow through from their incorrect part (a) eg, from a quarter of 107 (error) e 27, allow N 107 d 4 e 27 eg, from 50% of 52 (error) e 27, allow N 52 d 2 e 27 http://www.SATs-Papers.co.uk 13 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Clocks 3-5 4-6 5-7 6-8 7 5 a a Correct response 1m A different time to 09:15 with both hours and minutes as multiples of 3 eg I 09:18 I 12:12 I 12:15 I 6:15 I 03:12 Additional guidance Zero as a multiple of 3 eg N N N 00:12 15:00 00:00 Minutes written without the leading zero eg N 12:9 Impossible time eg N 12:60 b b 1m 2 or 8 ! Answer of the form 2 p (a multiple of 6) Accept provided it is less than 60 eg N 14 N 20 N 26 Specific time, rather than time interval eg N 12:00 Tier & Question Folding and Cutting 3-5 4-6 5-7 6-8 8 6 a a Correct response 1m Additional guidance Correct diagram, ie – – – – b b 1m Correct diagram, ie – – – – c c 1m Correct diagram, ie – 14 – – – Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Motorway 3-5 4-6 5-7 6-8 9 7 Correct response a a 1m 22 b b 1m 5 Additional guidance ! More than one junction indicated Accept if unambiguous eg, accept N 1 to 5 eg, do not accept N 4 to 5 c c 1m 70 Tier & Question Using Brackets 3-5 4-6 5-7 6-8 10 8 Correct response a a 1m 18 and 10 (correct order) b b 1m 60 c 1m (4 p 5 p 1) t 5 Additional guidance c ! Multiple brackets Accept if the brackets are paired and unambiguous, even if redundant eg, for part (c), accept N ((4 p 5) p 1) t 5 However, if multiple brackets are not paired, but there is no further ambiguity, penalise the first occurrence only eg, for part (c) N (4 p 5) p 1) t 5 eg, for part (d) N 4 p (5 p 1) t 5) Mark as 0, 1 d d 1m 4 p (5 p 1) t 5 ! Change in order of numbers eg, for part (d) N (1 p 5) t 5 p 4 Ignore if alongside a correct response, otherwise do not accept Calculations separated into a series of operations eg, for part (c) N (4 p 5) e (9 p 1) e (10 t 5) e 50 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 15 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Box 3-5 4-6 5-7 6-8 11 9 1 Correct response 1m Additional guidance Two sections drawn on each of N, S, E and W, even if incorrect. 1m partial flaps at least 1.5cm in length eg Both E and W completely correct. 1m Outer rectangle(s) on E and W shown as Both N and S completely correct. N ! Lines not ruled and/or not accurate 6 t 1.5 both 1 t 3 6t1 W both 1 t 3 E 6t1 6 t 1.5 S Accept if the pupil’s intention is clear and if the 1.5 width is closer to 1.5 than 1 or 2 ! Flaps drawn Ignore if unambiguous. ! Fold lines not shown within sections For the second and third marks, accept if the overall dimensions of the sections representing E/W (or N/S) are correct eg, mark the following as 0, 1, 0 6t2 (incorrect) each 2t3 6t2 (incorrect) ! Extra lines shown within sections that have overall correct dimensions Penalise one mark only eg, mark the following as 0, 1, 1 6 t 2.5 2t3 2t3 6 t 2.5 16 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Fractions 3-5 4-6 5-7 6-8 12 10 4 a a a Correct response 1m Additional guidance Both placed correctly, and labelled, ie ! Arrows, or other indication, inaccurate Accept only if unambiguous. ! No labelling 1 3 b b b 5 6 2m All three correct, ie 1, 24, and 4 or 1m Accept only if both are correct and no surplus arrows are indicated. Any two correct. Tier & Question Crisps 3-5 4-6 5-7 6-8 13 11 2 a a a Correct response Additional guidance Table not interpreted eg N 5 c c c Plain 1m b b b 1m 1 or equivalent probability. 10 1m 1 or equivalent probability. 8 Rounded values All correct, ie plain vinegar chicken cheese The value for cheese left blank 2m or 1m ie 0.12 or 0.13 or 12% or 13% 7 3 2 0 At least two correct, and the total sums to 12 eg I plain 7 vinegar 3 chicken 1 cheese 1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 17 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier & Question Sunshine 3-5 4-6 5-7 6-8 14 12 3 a a a Tiers 3–5, 4–6, 5–7 Correct response 1m Additional guidance 28, ie b b b 1m Indicates ‘not possible to tell’, ie c c c 1m Indicates month B and gives a correct explanation eg I B has more ‘more than 8 hours’ days. I A has a greater number of less than 4 hours. I B is probably summer as it had lots of days with more than eight. A only had a few so it’s probably winter. Number of days in the month specified eg N 25 written in the ‘not possible to tell’ box. ! Explanation does not explicitly compare the months Accept provided box B is indicated eg N Box B ticked and the explanation as B has lots of days with more than 8 hours of sunshine. No interpretation eg N There’s a big piece on the pie chart. 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Shapes 3-5 4-6 5-7 6-8 15 13 5 a a a b b b Correct response 1m 1m Correct simplified fraction, ie 1 3 40 Additional guidance ! Shaded area consistently expressed as a simplified fraction or percentage of the unshaded area eg, for part (a) N 1 2 eg, for part (b) N 66 2 (or 67) 3 Mark as 0, 1 c c c 1m Chooses shape A and gives a correct explanation. The most common correct explanations: State that A has one quarter shaded (or equivalent percentage or decimal) but that B has less eg I I I 1 1 A has shaded, B is less than as it has 4 4 only one leg shaded. A is a quarter, but B needs an extra quarter square to make it up to a quarter. Diagram showing how the shaded part of B needs to be changed to make it 25% Use the fraction 3 to compare to A 13 eg I B has 3 shaded. If they were the same 13 then it would have I A is 3 12 3 3 , B is 12 13 ! Percentage value for B Accept between 23 and 24 inclusive, or accept between 20 and 25 exclusive if the approximate nature of the percentage is specified. Minimally acceptable explanation eg 1 shaded, B has less. 4 N A has N Part of one square on shape B should be shaded. Incomplete explanation eg N A has a quarter shaded, B hasn’t. N B has a bit less shaded than A. Minimally acceptable explanation eg N A has 1 3 , B has 4 13 Incomplete explanation eg N B has 3 13 Incorrect statement eg N A has 1 3 shaded, B has 3 13 Focus on four equal parts in both shapes eg I A has 4 equal parts, B has 4 equal and another square in the middle. I If you shade 3 more like the part on B you wouldn’t fill the shape. If you do it on A you do fill it. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 19 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Trip 3-5 4-6 5-7 6-8 16 14 6 a a a Correct response 2m or 1m Additional guidance 12168 Shows a complete correct method with not more than one computational error eg I 234 52 11700 468 12268 (error) Conceptual error eg N 234 52 1170 468 1638 I 2 1 1 3 1 4 1 0 0 2 5 0 0 0 20 1m 6 8 7 (error) b b b 4 6 5 2 8 Answer 11768 13 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Glasses 3-5 4-6 5-7 6-8 15 7 1 Correct response a a a 2m or 1m Additional guidance Both values correct, ie 36 and 324, in either order. One correct value or Both values sum to 360, but none are 0, 90 or 180 b b b 1m Indicates ‘not possible to tell’, ie Tier & Question Bags 3-5 4-6 5-7 6-8 16 8 2 Correct response a a a 1m b b b 2m or 1m Both correct eg I Barry as a p 2, Cindy as 4a Both correct, ie Ali as b – 2, Cindy as 4(b – 2) One correct Additional guidance Equivalent expressions eg, for Cindy N apapapa Variable or brackets omitted eg, for a p 2 N p2 eg, for 4(b – 2) N t4–2 N 4b – 2 N b–2t4 ! Follow through from incorrect expression given for Ali Accept provided the expression is of at least two terms. c c c 1m Indicates – – c 4 p 2, ie – – Sourced from SATs-Papers.co.uk Follow through from part (b) – http://www.SATs-Papers.co.uk 21 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Finding Angles 3-5 4-6 5-7 6-8 17 9 3 Correct response Additional guidance a a a 1m A correct angle of 75 indicated. Extra lines added to the diagram to create an angle of 75 b b b 2m 50 Follow through from an incorrectly marked 75 in the lower triangle or 1m c c 1m Shows a correct method eg I (180 – 80) d 2 I 100 d 2 Correct expression or equation with x as the subject eg I 180 – y I x e 180 – y ! Units inserted Ignore eg, accept N x e 180º – y ! Correct equations in (c) and (d) but with x d d 1m e 22 Correct expression or equation with x as the subject eg I 180 – t – w I x e 180 – (t p w) Correct explanation eg I 180 – y e 180 – (t p w), so y e t p w I x e 180 – y, x e 180 – (t p w), so y e t p w I x p t p w e 180 and x p y e 180, so y e t p w e 1m Sourced from SATs-Papers.co.uk not the subject eg N x p y e 180 and x p t p w e 180 Mark as 0, 1 Spurious explanation eg N y e 180 – x, x e 180 – t – w, so y e t p w http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Statistics 3-5 4-6 5-7 6-8 18 10 4 Correct response 2m Identifies the three numbers as 5, 5, 14 or 1m Additional guidance Any order Identifies two of the numbers as 5 or Gives three numbers that sum to 24, or otherwise indicates the sum of the three is 24 Tier & Question Lambs 3-5 4-6 5-7 6-8 19 11 5 Correct response 2m or 1m Additional guidance 104 Shows a complete correct method with not more than one computational error eg I 80 t 1.3 I 30% of 80 e 24, 24 t 2 e 48 twins, 48 p 56 I 30% of 80 e 26 (error), 26 t 2 e 52 80 – 26 e 54, and 52 p 54 e 106 Incorrect method for calculating 30% of 80 eg N 100 d 30 t 80 or The only error is to double the number of sheep having single lambs rather than the number of sheep having two lambs eg I 56 t 2 p 24 e 136 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 23 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Tiles 3-5 4-6 5-7 6-8 20 12 6 Correct response 2m Additional guidance Gives a correct justification. The most common correct justifications: Show the areas are 90 and 54, and justifies that 90 : 54 simplifies to 5 : 3 either by showing correct divisors or by showing at least one intermediate correct ratio eg I Area of black is 90, grey is 54 90 d 18 e 5, 54 d 18 e 3 I 90 d 2 e 45, 45 d 9 e 5 54 d 2 e 27, 27 d 9 e 3 I 90 : 54 e 45 : 27 e 5 : 3 Focus on proportion eg I If the 3 by 3 square is one unit, there are 10 black and 6 grey in total; 10 : 6 e 5 : 3 or 1m 24 Shows the values 90 and 54 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Thinking Equations 3-5 4-6 5-7 6-8 21 13 7 Correct response a a a 1m Additional guidance Incomplete processing eg 2 N b b b 2m or 1m 2m or 1m 6 3 0.5 or equivalent fraction or decimal. Incorrect method that leads to a correct answer Simplifies correctly to 2 terms eg I 6y e 3 I –6y e –3 I 6y – 3 e 0 I ye3d6 For 1m, incomplete equation other than y e 3 d 6 written without the subject eg, accept N 3d6 eg, do not accept N –3 N e 6y – 3 1 or equivalent fraction or decimal. 3 For 2m, decimal rounded to 0.33 or better Shows 6y p 12 e 14 or equivalent equation or Indicates y is 0.3 Tier & Question Comparing Powers 3-5 4-6 5-7 6-8 14 8 Correct response a a 1m 4 Additional guidance ! Answers incorrectly evaluated 3 (or 81) The answers do not need to be evaluated, but if they are they should be correct. 1m b b 1m 4 3 5 7 2 (or 32) and 2 (or 128), either order Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 25 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Evens or Odds 3-5 4-6 5-7 6-8 15 9 Correct response a a 2m All correct, ie even or 1m b b 1m Additional guidance odd even even Any 3 correct. Minimally acceptable explanation Indicates not possible to tell, ie eg N and gives a correct explanation eg I When m is odd, m p 1 is even. If m p 1 is a multiple of 4, even d 2 remains even, but if it is not a multiple of 4, even d 2 becomes odd. mp1 I If m e 5, then is odd, 2 but if m e 7, I mp1 is even. 2 As m increases consecutively, N Odd p 1 is even, when you divide it it might be odd or even. Some even numbers divide by 2 to get even results and some divide to give an odd result. Incomplete explanation eg N Because we don’t know the value of m N Using different odd numbers you can get odds or evens. mp1 will be 2 odd, even, odd, even and so on. 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Computer Game 3-5 4-6 5-7 6-8 16 10 Correct response a a 1m 7 b b 2m Additional guidance 30 or 1m Shows a correct method eg I 12 d 0.4 I 12 d 4 t 10 I 12 is 40%, so 6 is 20%, so 18 is 60% 18 p 12 or Indicates that 0.4 represents 12 c c 1m No, with a correct explanation. The most common correct explanations: Indicate that the 200 games is only a sample eg I Random variation means you wouldn’t expect exactly 130 games out of 200 I She only played 200 but the manufacturer would have played it lots more and taken an average. Minimally acceptable explanation Show that 0.62 approximates to 0.65 eg Minimally acceptable explanation I I I eg N N Incorrect, incomplete or irrelevant explanation eg N She needs to play 6 more times. N 100 times is 0.62 so no. N 0.65 > 0.5 and she won more than half of her games. N She could be really bad at the game. eg 124 e 0.62, that’s close to 0.65 200 N Expect 130, got 124; near enough. 0.65 t 200 e 130; that’s close enough. N N N N N N N Sourced from SATs-Papers.co.uk If she did it again it could be different. She should play more games. The manufacturer just gave an average. It’s more or less correct. Probability is what is expected to happen not what actually happens. In real life, things don’t always work out as they should. 124 is about what you’d expect. 0.65 is only an approximation. It’s close to 0.65 0.65 is a bit more than half and 124 is a bit more than a half of 200 http://www.SATs-Papers.co.uk 27 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Graphing 3-5 4-6 5-7 6-8 17 11 Correct response Additional guidance 2 a a 1m F (or y e x ) b b 1m C (or x e –5) c F (or y e x ) c 1m d d 2m or 1m e e 3m or 2m 2 D (or x p y e 10) and E (or y e 2x p 1), either order. At least one correct with not more than one incorrect or omitted. (1, 3) and (–3, –5), either order. Draws the line y e 2x p 1 correctly, of length at least to intersect the quadratic curve twice. ! Line inaccurate Accept provided the pupil’s intention is clear. or Draws an incorrect line but through (0, 1), or with a gradient of 2, then follows through correctly to give their two points of intersection. or ! Plots a correct set of points but does not join them with a line Accept if sufficient to identify the points of intersection. Factorises the quadratic formed from 2 4 – x e 2x p 1 eg I 0 e (x p 3)(x – 1) or 1m Identifies one correct point even if the second point is incorrect or omitted eg I (1, 3) or Draws an incorrect line but through (0, 1), or with a gradient of 2, then follows through correctly to give one of their points of intersection, even if the second point is incorrect or omitted. or 2 Equates the equations 4 – x and 2x p 1 eg 2 I 4 – x e 2x p 1 2 I x p 2x – 3 e 0 28 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6–8 only Tier & Question True or False 3-5 4-6 5-7 6-8 12 Correct response 3m Additional guidance All five rows correct, ie or 2m Any four rows correct. or 1m Any two or three rows correct. Tier & Question Congruent 3-5 4-6 5-7 6-8 13 a 1m Correct response Indicates A and C and gives a correct explanation eg I Corresponding sides are equal. I SSS Additional guidance Minimally acceptable explanation eg N N N They have the same length sides. A could fit exactly onto C A is a mirror image of C Incomplete or incorrect explanation eg N They have the same perimeter. N Same length sides but different shapes. b 1m Indicates B and E and gives a correct explanation eg I The corresponding sides are in the same ratio. I The sides in E are 1.5 times bigger. I E is an enlargement of B Sourced from SATs-Papers.co.uk Minimally acceptable explanation eg N N N The sides are in the same proportion. Same shape, different size. All sides increase the same. http://www.SATs-Papers.co.uk 29 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier & Question Tier 6–8 only Thomas the Tank Engine 3-5 4-6 5-7 6-8 14 Correct response a 3m Both estimates, x, within the ranges as below, with a correct method shown either on the graph or through calculations Old: 7 ≤ x < 8, or 2m Additional guidance New: 6 ≤ x < 7 One estimate correct within the ranges above, with a correct method shown. or Both estimates correct within the ranges above, but with partial or no method shown. or 1m Correct method shown on the graph for one of the medians eg I Markings on the y-axis, or the graph, indicating 29 or 29.5, or 34 or 34.5 I Markings on the x-axis, or the graph, corresponding to the y-values of 29 or 29.5, or 34 or 34.5 Incorrect method eg N For old version, median shown on the graph as the 35th value. or One estimate correct within the range above, but with no method shown. b 1m 30 Makes a correct comment eg I The new version has shorter sentences. I There are more words per sentence in the old version. Sourced from SATs-Papers.co.uk Follow through from part (a) ‘Words’ used to stand for ‘words per sentence’ Given that the context and question refer to words per sentence, accept such an abbreviation eg N There are more words in the old version. http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier & Question Thomas the Tank Engine (cont) 3-5 4-6 5-7 6-8 14 c Tier 6–8 only Correct response 2m or 1m Additional guidance Gives a value in the range 20 to 30 inclusive, with no incorrect method shown. Gives a value in the range 70 to 80 inclusive. or Shows or implies 14 14 or an estimate of 58 58 eg I I I 14 d 58 14 d 56 15 60 or Shows evidence of using the wrong graph, and gives a value in the range 12 to 20 inclusive. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 31 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6–8 only Tier & Question Scores 3-5 4-6 5-7 6-8 15 Correct response a 1m Gives a correct justification eg I I b 2m or 1m 1 3 ( ) 2 1 1 1 t t 2 2 2 Additional guidance All 8 possibilities listed correctly, or a tree diagram shown Minimally acceptable explanation eg N 1 1 1 , , 2 4 8 3 or equivalent probability. 8 Shows a correct method eg 1 2 2 I ( ) t I 1 t3 2 1 t3 8 Tier & Question Writing Numbers 3-5 4-6 5-7 6-8 16 Correct response a 1m 4 t 10 –4 or 4 t 10 – 04 Additional guidance ! Unconventional index from notation eg, for part (a) –3 N 0.4 t 10 4 N 4 d 10 Penalise the first occurrence only. Incorrect notation eg, for part (a) –4 N 4 b 1m 4 t 10 –5 ! Follow through from part (a) Accept provided the power is negative –5 eg, for part (a) as 2 t 10 –6 N 2 t 10 c 2m or 1m 4.4 t 10 Digits 44 seen eg I 4400 I 0.00044 I 32 –4 Digits 44 seen from an incorrect method eg –4 –5 –9 N 4 t 10 p 4 t 10 e 44 t 10 44 100000 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 1 Index Index to mark schemes Tier 3-5 4-6 5-7 6-8 Question Page 1 Multiplication Table 11 2 Number Cards 11 3 1 Computation 12 4 2 What’s the Point? 12 5 3 Temperature 13 6 4 Twenty-seven 13 7 5 Clocks 14 8 6 Folding and Cutting 14 9 7 Motorway 15 10 8 Using Brackets 15 11 9 1 Box 16 12 10 4 Fractions 17 13 11 2 Crisps 17 14 12 3 Sunshine 18 15 13 5 Shapes 19 16 14 6 Trip 20 15 7 1 Glasses 21 16 8 2 Bags 21 17 9 3 Finding Angles 22 18 10 4 Statistics 23 19 11 5 Lambs 23 20 12 6 Tiles 24 21 13 7 Thinking Equations 25 14 8 Comparing Powers 25 15 9 Evens or Odds 26 16 10 Computer Game 27 17 11 Graphing 28 12 True or False 29 13 Congruent 29 14 Thomas the Tank Engine 30 15 Scores 32 16 Writing Numbers 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 33 EARLY YEARS NATIONAL CURRICULUM 5 –16 GCSE GNVQ GCE A LEVEL NVQ First published in 2001 © Qualifications and Curriculum Authority 2001 Reproduction, storage, adaptation or translation, in any form or by any means, of OTHER VOCATIONAL QUALIFICATIONS 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/ Further teacher packs may be purchased (for any purpose other than statutory assessment) by contacting: QCA Publications, PO Box 99, Sudbury, Suffolk CO10 2SN (tel: 01787 884444; fax: 01787 312950) SourcedOrderSATs-Papers.co.uk from ref: QCA/01/660 http://www.SATs-Papers.co.uk 00-6651/11