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Mathematics tests Ma KEY STAGE 3 ALL TIERS 2003 2003 Mark scheme for Paper 1 Tiers 3–5, 4–6, 5–7 and 6–8 3 3 K E Y S TA G E K E Y S TA G E 3 K E Y S TA G E Sourced from SATs-Papers.co.uk K E Y S TA E G http://www.SATs-Papers.co.uk 2003 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 10 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. The 2003 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 2003 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. 3 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 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 2003 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 2003 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 ✓ 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 eg £3.2, £3 200, £32 0, £3-2-0, £7.0 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 2003 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 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 t 10 p 2 e 32, x e 5 7 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 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 ✓ Take care ! Do not accept ✓ A correct probability that is correctly expressed as a decimal, fraction or percentage. ✓ Equivalent decimals, fractions or 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 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 2003 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 tiers 3–5, 4–6 and 6–8. A total of 122 marks is available in tier 5–7. 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, 23 June 2003. 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. 9 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Pictogram 3-5 4-6 5-7 6-8 1 Correct response a 1m Draws two circles Additional guidance ✓ Circles not shaded ! Circles inaccurate in size and/or shape Accept provided the pupil’s intention is clear b 1m 2 Tier & Question Missing numbers 3-5 4-6 5-7 6-8 2 Correct response 1m Gives any three numbers that add to 15 eg ■ 5p6p4 ■ 5p5p5 1m Gives any two numbers that divide to give 15 eg ■ 30 d 2 ■ 15 d 1 1m Gives any three numbers that combine as shown to give 15 eg ■ 2t6p3 ✓ Throughout the question, use of fractions, decimals, negatives or zeros Gives any two numbers that multiply to 15 eg ■ 3t5 ■ 1 t 15 1m Additional guidance Incorrect order eg ◆ 2 d 30 ✓ Brackets inserted to change order of operations eg ◆ 3 t (1 p 4) Incorrect order of operations eg ◆ 3 t 1 p 4 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Scales 3-5 4-6 5-7 6-8 3 a Correct response 1m 60 Additional guidance ✓ Value between 59 and 61 inclusive ! Units given Ignore b 1m Indicates the correct position eg ■ ✓ Unambiguous indication eg ◆ ! Follow through Accept follow through from part (a), provided their (a) is not 0, 50 or 100 ! Position not indicated accurately Accept within ± 2mm 11 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Prices 3-5 4-6 5-7 6-8 4 a Correct response 1m Indicates a correct amount in pounds or pence and gives the correct units eg ■ 75p ■ £0.75 1m b Indicates one eraser ! Units incorrect or omitted Penalise only the first occurrence eg ◆ 75 (units omitted) 1.05p (units incorrect) Mark as 0, 1 Indicates a correct amount in pounds or pence and gives the correct units eg ■ £1.05 ■ 105p 1m Additional guidance 1m Indicates a correct way, other than two rulers eg ■ 4 pencils ■ 3 erasers ■ 1 eraser and 1 green pen ■ 1 ruler and 2 pencils U1 1m ✓ Quantity of one implicit but not specified eg, for the third mark in part (a) ◆ Eraser eg, for part (b) ◆ Ruler and two pencils ✓ Unambiguous indication eg, for the third mark in part (a) ◆ E ◆ Rubber eg, for part (b) ◆ R and 2P Indicates a correct way, other than one previously credited U1 1m Indicates a correct way, other than one previously credited U1 12 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Clock 3-5 4-6 5-7 6-8 5 a Correct response 1m Indicates only the two correct clocks eg ■ ✓ Additional guidance ! Indication other than ticks eg ◆ used Accept provided unambiguous ✓ b 1m 5:15 or 05:15 ✓ Superfluous indication of morning eg ◆ 5:15 am Time incorrect eg ◆ 5:15 pm ◆ 17:15 c 1m 17:15 ! Follow through Accept follow through as 12 hours later than their (b), even if their (b) was 17:15, provided this is written as a possible time eg, from part (b) as 03:26, accept ◆ 15:26 ✓ Superfluous indication of evening eg ◆ 17:15 pm Time incorrect or not using 24 hour clock eg ◆ 17:15 am ◆ 5:15 pm 13 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Calculations 3-5 4-6 5-7 6-8 6 3 Correct response a 1m 72 b 1m 22 c 1m 97 1m 26 1m 1256 1m Additional guidance 4348 Tier & Question Chains 3-5 4-6 5-7 6-8 7 1 Correct response a a 1m Gives both correct values correctly positioned, ie 20 and 320 b b 1m Gives both correct values correctly positioned, Additional guidance ie 5 and 2 1 or equivalent 2 1 2 ✓ For 2 , 5 2 14 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Puzzling out 3-5 4-6 5-7 6-8 8 2 Correct response 2m or 1m Additional guidance Indicates the numbers 1, 3, 5, 7, 9 in any order Indicates any five numbers that are less than 10 eg ■ 0, 2, 4, 6, 8 ■ 7, 7, 1, 2, 6 ■ m4, m4, m4, m4, m1.5 or Indicates any five odd numbers eg ■ 7, 7, 15, 13, 9 U1 Tier & Question Wind chill 3-5 4-6 5-7 6-8 9 4 Correct response 1m m19 Additional guidance ! Incorrect notation for negative numbers eg 19m Penalise only the first occurrence ◆ 1m 16 1m m22 –16 given for 16 15 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Throwing dice 3-5 4-6 5-7 6-8 10 5 a a Correct response 2m Indicates only the five points with positive integer coordinates whose sum is 6 eg Additional guidance ! Point(s) not indicated accurately Accept in parts (a) and (b) provided the pupil’s intention is clear ■ ! Additional points indicated that assume zero to be on the dice eg ◆ (0, 6) and/or (6, 0) indicated If this is the only error, mark as 1, 0 ! Additional points with non-integer coordinates whose sum is 6 indicated eg or 1m ◆ Indicates at least four correct points with no incorrect points or Indicates all five correct points with not more than one incorrect point If this is the only error, mark as 1, 0 b b 2m Indicates only the six points with positive integer coordinates such that y e x eg ■ ! Additional point indicated that assumes zero to be on the dice eg ◆ (0, 0) indicated If this error has been penalised in part (a), condone If this is the only error and it has not been penalised in part (a), mark as 1, 0 ! Additional points with non-integer coordinates such that y = x indicated eg ◆ or 1m Indicates at least five correct points with no incorrect points or Indicates all six correct points with not more than one incorrect point If this error has been penalised in part (a), condone If this is the only error and it has not been penalised in part (a), mark as 1, 0 16 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Throwing dice (cont) 3-5 4-6 5-7 6-8 10 5 c c Correct response 1m Completes the sentence to give a correct rule eg ■ One less than the number on the red dice ■ Red m 1 ■ Needing 1 added to get the number on the red dice Additional guidance ✓ Minimally acceptable rule eg ◆ ◆ 1 below the other dice The number below the red dice ✓ Rule expressed algebraically eg ◆ ◆ berm1 rm1 ! Rule that does not use the given starting phrase Accept only if unambiguous eg, accept ◆ Red e blue p 1 eg, do not accept ◆ 1 more on the red Ambiguous rule eg ◆ m1 ◆ 1 below ◆ A number below the red dice ◆ The number lower than the red dice ◆ Followed by the number on the red dice Incomplete rule eg ◆ Less than the number on the red dice Rule not generalised Do not accept rules only shown through particular numerical examples eg ◆ 2 m 1 e 1, 3 m 2 e 1, 4 m 3 e 1 etc 17 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Perimeter and area 3-5 4-6 5-7 6-8 11 6 a a Correct response 1m Indicates No and gives a correct explanation Additional guidance ! Units given Ignore The most common correct explanations: Quantify the areas eg ■ The area of the hexagon is 6 but the triangle is only 4 ■ The hexagon has two more triangles ✓ Minimally acceptable explanation eg ◆ ◆ 6 and 4 2 more Incomplete or incorrect explanation eg ◆ The hexagon is 6 ◆ The hexagon has 5 triangles, the triangle has 4 ◆ The hexagon has one more shape in it than the triangle Interpret ‘area’ eg ■ Different amount of space inside ■ Different numbers of triangles ✓ Minimally acceptable explanation eg ◆ ◆ ◆ Count the triangles They have different numbers of shapes One has less triangles ! Inaccurate description of shapes in an otherwise correct explanation Condone eg, accept ◆ They have a different number of squares inside Incomplete explanation that does not interpret area eg ◆ Different sizes ◆ Different numbers of dots ◆ Different Identify which shape has the bigger area eg ■ The area of the hexagon is greater ■ The triangle has a smaller area ✓ Minimally acceptable explanation eg ◆ ◆ ◆ U1 The hexagon is bigger The triangle is smaller The triangle has only 4 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Perimeter and area (cont) 3-5 4-6 5-7 6-8 11 6 b b Tiers 3–5, 4–6 Correct response 1m Indicates Yes and gives a correct explanation Additional guidance ! Units given Ignore The most common correct explanations: Quantify the perimeters eg ■ The perimeter of both is 6 ■ They both have 6 along the sides ! Perimeters measured Accept values between 8.4cm and 9.6cm inclusive, even if units are not given ✓ Minimally acceptable explanation eg ◆ ◆ 6 and 6 Both 6 sides Incorrect explanation eg ◆ They have 5 lines round the sides Interpret ‘perimeter’ eg ■ Both have the same distance around the edges ■ The number along the sides is the same ■ Each side of the triangle = two sides of the hexagon ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ ◆ ◆ Same length edges Same amount of triangle sides Same number of dots Same number of points Dots to dots is the same Same number of sides between the dots I counted around them Incomplete or ambiguous explanation eg ◆ They are the same size ◆ They are both the same ◆ They take up the same number of squares ◆ Same number of sides ◆ I counted the sides ◆ I measured them ! Responses to parts (a) and (b) transposed U1 but otherwise completely correct, even if there is incorrect use of words ‘area’ and ‘perimeter’ Do not award the mark for part (a), but award the mark for part (b) 19 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6 Weighing 3-5 4-6 5-7 6-8 12 7 Correct response 2m 1.2 or 1m Additional guidance ✓ Equivalent fractions and decimals Shows 2.4 or Shows the digits 12 or Shows or implies a complete correct method, with not more than one error eg ■ (5 m 2.6) d 2 ■ 5 m 2.6 then d 2 ■ 5 m 2.6 e 3.4 (error), 3.4 d 2 e 1.7 ! For 1m, necessary brackets omitted As this is a level 4 mark, condone eg, accept for 1m ◆ 5 m 2.6 d 2 For 1m, incorrect order of subtraction eg ◆ 2.6 m 5 then d 2 U1 20 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Patterns 3-5 4-6 5-7 6-8 13 8 2 a a Correct response 1m Shows two rectangles in a pattern with two lines of symmetry eg ■ Additional guidance ! Lines of symmetry drawn Ignore ! Rectangles not shaded Accept only if unambiguous ! Edges of rectangles not explicit Pupils may use the edge of the grid or not show an edge when the rectangles are adjacent. Accept only if unambiguous ■ ! Rectangles placed within the grid but covering only parts of squares Accept provided the pupil’s intention is clear eg, for the first mark, accept ◆ b b a 1m Shows two rectangles in a pattern with only one line of symmetry eg ■ ! Rectangles placed with parts or all outside the grid Accept provided the pupil’s intention is clear eg, for the third mark, accept ◆ ■ ◆ c c b 1m Shows two rectangles in a pattern with rotation symmetry of order 2 eg ■ ! Rectangles overlapping Accept only if unambiguous eg, for the third mark, accept ◆ ■ ! Incorrect size of rectangles Do not treat as a misread, ie do not accept Grid not taken to be part of the pattern 21 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Simplifying 3-5 4-6 5-7 6-8 14 9 3 Correct response Additional guidance 1m 8k p 7 Use of multiplication sign in simplified expressions eg, for the first mark ◆ 8 t k p 7 1m 2k p 5 Partially simplified expressions Tier & Question Car parking 3-5 4-6 5-7 6-8 15 10 1 Correct response 2m Indicates the remaining five combinations in any order, with no duplicates and none incorrect eg ■ 10p 20p 50p 7 0 0 5 1 0 3 2 0 2 0 1 1 3 0 0 1 1 Additional guidance ✓ Zeros omitted ! Amounts given rather than numbers of coins Accept provided the number of each type of coin is unambiguously implied eg, for the combination 2 0 1, accept ◆ 10p 10p 10p 20p 50p 50p eg, for the combination 2 0 1, do not accept ◆ or 1m 10p 20p 20p 50p 50p Indicates at least four correct combinations, with not more than one duplicated, incorrect or omitted 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 Thinking fractions 3-5 4-6 5-7 6-8 16 11 4 Correct response 1m 40 1m 150 1m Additional guidance 30 Tier & Question 17 12 5 a a a b b b Moving C Marking overlay available 3-5 4-6 5-7 6-8 Correct response 1m 1m Additional guidance Gives correct coordinates eg ■ (6, any value except 6 or 1) ■ (4, 5) ■ (8, 5) ■ (4, m3) ■ (8, m3) ! Use of overlay Gives correct coordinates, ie (4, 5) or (8, 5) or (6, 3) or (4, m3) or (8, m3) or (6, m1) ✓ Same correct position used for part (b) as As there is an infinite number of correct coordinates, a marking overlay is available for use if pupils give non-integer coordinates. Accept coordinates of any point that lies exactly on the straight line or on one of the circles, provided their point is neither (6, 6) nor on the same straight line as A and B for part (a) 23 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 Shoe sizes 3-5 4-6 5-7 6-8 18 13 6 Correct response a a a 1m 6 b b b 1m Additional guidance 2 U1 Tier & Question Marking overlay available 3-5 4-6 5-7 6-8 19 14 7 Correct response 2m or 1m Constructs a completed triangle with the vertices in the regions indicated, and arcs within the tolerance, shown on the overlay Construction Additional guidance ! Longer arcs drawn than are shown on the overlay Ignore inaccuracies in sections of arcs extending beyond those shown on the overlay Draws a completed triangle with the vertices in the regions indicated on the overlay, with either no arcs or incorrect arcs or Draws arcs that are within the tolerance shown on the overlay, even if there is an incorrect or no completed triangle 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Travel to work 3-5 4-6 5-7 6-8 20 15 8 a a a Correct response 2m or 1m Additional guidance £ 729(.00) Shows the digits 729 eg ■ 72900 ■ 72.90 or Shows a complete correct method with not more than one computational error, but with the decimal point correctly positioned eg ■ 20 t 45 e 900 16 t 45 e 8 t 90 e 720 720 p 9 ■ 1620 45 64800 8100 73900 (error) so £ 739 ■ b b b 1m 1 6 2 0 2 0 0 0 0 4 4 4 8 0 3 1 0 7 0 5 5 0 0 0 (error) 1 9 0 0 Conceptual error eg ◆ 1620 45 6480 8100 14580 so £ 145.80 So £ 719 £ 14 25 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7, 6–8 Solving 3-5 4-6 5-7 6-8 21 16 9 1 Correct response 1m Additional guidance ! Throughout the question, incorrect notation 2 eg, as an answer for the first mark ◆ ket2 Withhold one mark only for the first occurrence 1m 2 1 or equivalent 2 2m 4 1 or equivalent 2 or 1m 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 p 4 e 13 ■ 3t e t p 9 ■ 3t m t e 13 m 4 ■ 2t e 9 ! Method used is trial and improvement Note that no partial credit can be given m1 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Shapes 3-5 4-6 5-7 6-8 17 10 2 Correct response 3m All four angles correct and correctly positioned, ie Additional guidance ✓ Units omitted ! Units incorrect eg ◆ 50% Withhold one mark only for the first occurrence or 2m At least three angles correct and correctly positioned or ✓ Follow through For 2m or 1m, follow through for 47° as 360 m sum of their other three angles or 97 m their 50 All four correct angles shown but identification of which angle is which size is not clear or 1m At least two angles correct and correctly positioned 27 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Mixed numbers 3-5 4-6 5-7 6-8 18 11 3 Correct response a a a 1m Gives the value 1 Additional guidance 4 9 or or equivalent 5 5 fraction or decimal 1m Indicates the correct position on the number line, ie ! Indication inaccurate Accept provided the pupil’s intention is clear ! Follow through Accept provided their incorrect value for the addition is between 0 and 2, but is not an integer eg, from 12 for the first mark, accept 15 ◆ b b b 1m 20 ✓ Answer given as a fraction eg ◆ 60 3 ✓ Answer repeats sixths eg ◆ 20 6 28 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Mixed numbers (cont) 3-5 4-6 5-7 6-8 18 11 3 c c c Correct response 2m Additional guidance ✓ Answer given as a fraction 4 eg ◆ 4 1 ◆ 60 15 ! Follow through For 2m or 1m, accept their (b) d 5 provided their (b) is a positive integer or 1m Shows a complete correct method with not more than one computational error eg ■ 20 d 5 10 6 60 t e e 3 (error) 3 5 15 Conceptual error eg ◆ 20 d 5 e 4 6 ◆ 20 5 4 d e 6 6 6 ◆ 10 6 16 t e 3 5 15 ◆ ■ 1 6 6 t e 10 5 50 ■ ■ 1 7 7 6 42 3 e (error), t e 3 3 3 5 15 ◆ 3 1 6 6 t e3 3 5 15 29 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Areas algebraically 3-5 4-6 5-7 6-8 19 12 4 Correct response a a a 1m 1m Additional guidance Gives a correct simplified expression for the area eg ■ 15ab ■ 15 t a t b Gives a correct simplified expression for the perimeter eg ■ 6a p 10b ■ 2(3a p 5b) ■ 6 t a p 10 t b ■ 2 t (3a p 5b) ! Partially simplified or unsimplified expressions eg, for the area ◆ 3a5b eg, for the perimeter ◆ 2(3a) p 2(5b) ◆ 2 t (3 t a p 5 t b) If both expressions are correct but are partially simplified or unsimplified, mark as 0, 1, provided neither has subsequently been incorrectly simplified ! Expressions transposed but otherwise correct and simplified Mark as 0, 1 b b b 1m Gives both correct dimensions in either order, ie 4a and 3a ! Correct dimensions embedded Accept provided both the area and perimeter have been considered eg, accept ◆ 12a2 e 3a t 4a 14a e 2(3a p 4a) ! Dimensions labelled as length or width Ignore 30 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Arranging 3-5 4-6 5-7 6-8 20 13 5 Correct response a a a 1m Additional guidance Gives a correct arrangement using each of the numbers 1 to 6 once only, ie for each 3 digit number: hundreds tens units 5 or 6 2 or 4 1 or 3 eg ■ ■ 543 p 621 641 p 523 U1 1m Gives a correct arrangement using each of the numbers 1 to 6 once only, ie for each 3 digit number: hundreds tens units 2 or 5 1 or 3 4 or 6 eg ■ ■ 514 p 236 216 p 534 U1 b b b 1m 536 m 421 or 356 m 241 Incorrect order of subtraction eg ◆ 421 m 536 U1 31 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Lines on a square 3-5 4-6 5-7 6-8 21 14 6 Correct response a a 2m or 1m b b 1m Matches all three equations correctly, ie Additional guidance Any equation matched more than once Matches any two equations correctly Gives a correct equation eg ■ xe1 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Lines on a square (cont) 3-5 4-6 5-7 6-8 21 14 6 c c Tiers 4–6, 5–7, 6–8 Correct response 1m Indicates No and gives a correct explanation Additional guidance ✓ Minimally acceptable explanation eg The most common correct explanations: ◆ ◆ ◆ Give a correct equation of the line through E and G eg ■ It should be y e x Refer to gradients eg ■ Gradient of EG is 1 but the gradient of y e mx is m1 ■ Gradient of EG is positive but the gradient of y e mx is negative ■ EG is the wrong diagonal for a negative gradient EG slopes up but y e mx slopes down At E (or G) y is not mx If you put in E’s coordinates, it doesn’t work Incomplete explanation eg ◆ Gradient of EG is 1 ◆ y e mx slopes down ◆ 1 ≠ m1 ◆ x e 1, y e 1 ◆ ◆ ◆ x y 0 0 1 1 2 2 3 3 Because each point has the same numbers (1, 1), (2, 2) etc If you put in coordinates, it doesn’t work Give a counter-example eg ■ The point (1, 1) is on the line, but 1 ≠ m1 ■ (1, m1) works for y e mx, but is not on the line ■ E is (1, 1), but that’s x e y ■ At (1, 1), x and y are equal ■ At (1, 1), x and y have the same sign ■ x 1 2 y m1 m2 these points are not on EG Identify the line with equation y e mx eg ■ y e mx is the other diagonal through H and F ■ y e mx does not exist in the first quadrant ■ y e mx marked on graph U1 33 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Scatter graphs 3-5 4-6 5-7 6-8 22 15 7 Correct response a a a 1m Indicates a positive correlation eg ■ There is positive correlation between diameter and height ■ As diameter increases, height increases ■ Higher trees have wider trunks ■ Bigger trees are fatter ■ They both increase together Additional guidance ✓ Minimally acceptable response eg ◆ Big trees have big diameters Incomplete response eg ◆ It’s positive ◆ Big trees have big heights ◆ Higher trees are bigger Incorrect reference to proportion eg ◆ It’s directly proportional b b b 1m Gives a correct explanation The most common correct explanations: Refer to the trend in the data eg ■ It would be too far away from the other points ■ It would be an outlier ■ It doesn’t fit the general trend ■ It would be a long way from the line of best fit ■ This diameter is far too big for the height ■ It is too small to have such a big diameter ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ ◆ It’s on its own on the graph It doesn’t fit the correlation It doesn’t fit the pattern It doesn’t have the same relationship The diameter in cm is bigger than the height in m The diameter is big but the height is small Incomplete or incorrect explanation eg ◆ It’s different from the others ◆ It’s on its own ◆ It doesn’t fit the graph ◆ Poplar trees are tall and thin ◆ It would not be on the line of best fit ◆ It’s not in the same range ◆ The diameter is too big ◆ Poplar trees don’t have diameters bigger than their height ◆ For poplars, diameter p 1 e height Give a value for the height or diameter if the tree were a poplar eg ■ If it was a poplar you would expect it to be about 6 metres high ■ Poplars that are 3m high are only about 2cm in diameter c c c 1m d d 2m ! Height for diameter of 5cm given Accept values in the range 5.5m to 7m inclusive ! Diameter for height of 3m given Accept values in the range 1cm to 2.3cm inclusive Indicates a value between 4 and 5.2 inclusive Indicates that all four statements are false ! Indication other than ticks Accept only if unambiguous or 1m Makes three correct decisions 34 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Winning ticket 3-5 4-6 5-7 6-8 16 8 Correct response a a 1m Gives a correct probability eg ■ ■ ■ ■ b b 1m 75 245 15 49 0.306(...) 31% Gives a correct probability eg ■ 3 245 Additional guidance ! Answer of 0.3(0) or 30% Accept provided a correct method or a more accurate value is seen Incorrect method eg 1 ◆ 3 colours so e 0.3 3 ! Follow through Accept follow through from an incorrect total number of tickets seen in part (a), provided their total is not 3 or 100 eg, from ◆ 75 for part (a), accept 255 3 255 ! Decimal or percentage value Accept 0.01 or 0.012(…), or the equivalent percentages, provided no incorrect method is seen c c 1m 1 or equivalent probability 3 ! Decimal or percentage value Accept 0.33 or better, or the equivalent percentages 35 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Journeys 3-5 4-6 5-7 6-8 17 9 a a 1m Correct response Indicates 60 miles per hour or 1 mile per minute Additional guidance ✓ Equivalent fractions or decimals ! Throughout the question, pupils unambiguously working in miles per hour or miles per minute but some or all of the units are omitted At least 2 of the 3 correct values (ie 60, 30, 40 or 1, 1 2 , ) must be shown. 2 3 If correct units are shown once within the question condone other omissions. Otherwise withhold only the first mark that would otherwise have been awarded eg, for parts (a), (b) and (c) respectively ◆ 60, 30mph, 40 Mark as 1; 1; 1, 1 ◆ 50 (error), 30, 40 Mark as 0; 0; 1, 1 ◆ 60, 30, 45 (error) Mark as 0; 1; 0, 0 ◆ 1, 1 , 1.5 (error) 2 Mark as 0; 1; 0, 0 Note that the marking of these last two examples assumes there is no working shown that might allow partial credit for part (c) b b 1m Indicates 30 miles per hour or 1 mile per minute 2 ! Unconventional expression or abbreviation of units Accept provided unambiguous and expressed as a unitary rate, including the use of m for miles eg, for part (a), accept ◆ 1m per min ◆ 60 miles each hour If ambiguous, mark as if units omitted eg, for part (b) ◆ 0.5 mpm ! Incorrect units eg, for part (a) ◆ 1 mph Mark as if units omitted ! Rate of travel expressed as time per unit distance eg ◆ 1 minute per mile (part (a)) 2 minutes per mile (part (b)) 1.5 minutes per mile (part (c)) If the values shown are as above and correct units are shown at least once, mark as 0; 0; 1, 1 Otherwise do not accept Non-unitary rates eg, for part (b) ◆ 1 mile every two minutes 36 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Journeys (cont) 3-5 4-6 5-7 6-8 17 9 c c Tiers 5–7, 6–8 Correct response 2m Indicates 40 miles per hour or 2 mile per minute 3 Additional guidance Note to markers: Apply the additional guidance from parts (a) and (b) to part (c) ! Answer given in miles per minute and rounded or truncated In this context, accept rounding or truncation to two or more decimal places. For 2m, do not accept an answer of 0.6 or 0.7 unless a correct method or a more accurate value is seen or 1m Shows or implies a complete correct method for calculating the speed in miles per hour or miles per minute eg ■ 20 d 30 t 60 ■ 20 t 2 ■ 20 d 30 or For 1m, method leading to rate of travel as time per distance eg ◆ 30 d 20 For 1m, incomplete method eg ◆ 20 miles in 30 minutes, so 2 miles per 3 minutes Indicates a correct speed in miles per minute, rounded or truncated to one decimal place eg ■ 0.6 miles per minute ■ 0.7 miles per minute 37 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Different ways 3-5 4-6 5-7 6-8 18 10 Correct response a a 3m Additional guidance ✓ Unambiguous indication Indicates correct decisions for all six statements, ie eg T ✓ F F ✓ T F ✓ T ✓ for True and for False F Indicates correct decisions for five statements Indicates the correct decision for at least one of the false statements, and makes at least three other correct decisions eg ■ T ✓ F ✓ T F (error) ✓ T F ✓ T F ✓ T b b 1m ◆ F ✓ T or 1m T ✓ T or 2m ✓ F F T ✓ F (error) Gives a correct explanation The most common correct explanations: Refer explicitly or implicitly to the more efficient method of algebraic manipulation suggested by part (a) eg ■ You can solve the equation quickly by doing the same thing to both sides of the equation ■ 1 Because 2x e 3, x e 1 2 Refer to the time taken or the number of trials needed eg ■ Takes a long time ■ You need to try lots of values ■ It is inefficient U1 ✓ Minimally acceptable explanation eg ◆ ◆ ◆ Using algebra is better Decimal answers can be hard to find It can be inaccurate ! Incomplete explanation eg There is a better method The answer is always one value ◆ Easy to make a mistake ◆ It is inaccurate ◆ It’s like guess work ◆ You could go on forever Do not accept unless alongside a correct explanation ◆ ◆ Refer to the difficulty of finding an exact answer eg ■ If there are lots of decimal places it might be hard to be exact 38 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Locus of points Marking overlay available 3-5 4-6 5-7 6-8 19 11 Correct response a a 1m b b 1m Additional guidance Indicates the two points of intersection of the circles with radius 4cm, within the tolerance as shown on the overlay Draws a straight line that fulfils the following three conditions below: 1. Ruled 2. Within the tolerance as shown on the overlay 3. Extended to at least both the 6cm circles as shown on the overlay Tier & Question ! More than two points indicated Ignore additional points that are equidistant from A and B, as these may be working for part (b). Ignore other additional points provided it is clear which two points are the pupil’s answer for part (a) Shading, or additional lines or curves marked Line indicated by a series of points or shown dashed or dotted Evens or odds 3-5 4-6 5-7 6-8 20 12 Correct response a a 1m Indicates (x m 2) 2 is even, ie Additional guidance ✓ Minimally acceptable explanation ✓ and gives a correct explanation eg ■ Subtracting an even number keeps it even, and any even number multiplied by any even number stays even ■ (x m 2) is even, an even number squared is even ■ m 2 and it stays even, t by itself and it’s still even ■ Even m even e even, even t even e even ■ Even m even e even, and even multiplied by even or odd is even U1 b b 1m Indicates (x m 1) (x p 1) is odd, ie ✓ and gives a correct explanation eg ■ Adding or subtracting 1 makes it odd, and odd t odd e odd ■ (x m 1) is odd, (x p 1) is odd, so multiplied together the answer is odd ■ Even m odd e odd, even p odd e odd, odd t odd e odd ■ x2 m 1 is one less than an even number 2 ■ It’s x m 1 and the difference between an even square and an odd square is always odd eg, for part (a) ◆ Even t even is even eg, for part (b) ◆ Odd t odd is odd ◆ It’s x2 m 1 and x2 is even ◆ The difference between an even square and an odd square is always odd ! Explanation lacks generality eg, for part (a) ◆ If x e 6, (x m 2) 2 = 16 which is even Do not accept unless alongside a correct explanation Incomplete explanation eg, for part (a) ◆ (x m 2) is even eg, for part (b) ◆ Adding or subtracting 1 makes it odd U1 39 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Straight line 3-5 4-6 5-7 6-8 13 Correct response a 1m Additional guidance Gives a correct explanation eg ■ Right-angled triangle drawn on graph, with correct dimensions which are then used for height d base 5 ■ Change in y e 10 Change in x ■ 6 right, 3 up, so 3 d 6 ■ Half a square up for every one square along ■ (10, 6) is on the line, so 6 e 10m p 1 10m e 5 m e 0.5 ✓ Correct description of method eg ◆ You draw a right-angled triangle on the line, then you divide height by base ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ Right-angled triangle drawn on graph, with correct dimensions labelled y 5 = x 10 6 right, 3 up Half a square up each time ! Units given Ignore Incomplete explanation eg ◆ ◆ b 2m 5 10 It’s 1 on the y-axis, m2 on the x-axis, and 1 d 2 e 0.5 Gives a correct equation eg 1 ■ ye xp1 2 ■ ■ or 1m y e 0.5x p 1 2y m x e 2 1 x, or equivalent, seen 2 eg ■ 1 xp1 2 or Shows understanding that the equation is of the form y e mx p c and that m e 1 and c e 1, 2 even if there are subsequent errors eg ■ y e mx p c y e 0.5 p 1 (error) c 1m Gives any equation equivalent to y e where c is any value other than 1 eg 1 ■ ye xm7 2 ■ ■ y e 0.5x p 5 2y e x 1 x p c, 2 ! Follow through from part (b) Accept an equation of a line parallel to their (b), provided the equations contain both the variables x and y eg, from y e 2x p 1 for part (b), accept ◆ y e 2x p 4 40 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Theme park 3-5 4-6 5-7 6-8 14 Correct response a 1m Gives a value between 27.5 and 28.5 inclusive b 2m Additional guidance Gives a value between 16 and 18 inclusive or 1m Identifies both the upper quartile age as a value between 37.5 and 38.5 inclusive and the lower quartile age as a value between 20.5 and 21.5 inclusive or On the graph, indicates on the x-axis two correct points corresponding to one value between 37.5 and 38.5 inclusive, and one value between 20.5 and 21.5 inclusive, even if the values are not stated or are stated incorrectly c 1m ! More than two points indicated on Gives a correct statement of comparison, interpreting the data eg ■ On average, younger people went to the theme park ■ There was not as much variation of age at the theme park ■ Older people tended to go to the flower show ■ More people from different age groups went to the flower show ✓ Follow through from incorrect values for the x-axis Ignore a point intended to correspond to the median. Otherwise do not accept parts (a) or (b) ✓ Minimally acceptable statement eg ◆ ◆ ◆ The average age was higher at the flower show Younger people went to the theme park The theme park had people closer to each other in age ! Comparison with the theme park is implicit Given the wording of the question, condone eg, accept ◆ Older people tended to go No interpretation of the median or IQR within the context given eg ◆ The median age in the theme park is lower ◆ The range was higher at the flower show Incorrect statement eg ◆ People at the theme park were between 20 and 40 but people at the flower show were between 30 and 60 ◆ Young people went to the theme park but did not go to the flower show 41 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Inequality 3-5 4-6 5-7 6-8 15 Correct response a 2m or 1m ! Solution not given in terms of y y < m2 eg, for the first inequality ◆ my > 2 Mark as 1, 0 Shows or implies a correct first step of algebraic manipulation eg 2y p 7 ■ <1 3 ■ 2(2y p 7) < 6 4y p 14 <2 ■ 2y p 7 < 3 ■ 3 or Identifies the value m2 even if the inequality sign is incorrect or omitted eg ■ y > m2 ■ y e m2 ■ m2 2m or 1m Additional guidance ! Incomplete processing eg, for the first inequality ◆ y< m4 2 Mark as 1, 0 ✓ For 1m, algebraic manipulation of the corresponding equation, rather than the inequality eg, for the first inequality 2y p 7 ◆ e1 3 eg, for the second inequality ◆ 7 m 2y e 3 ✓ For 1m, value embedded eg, for the first inequality ◆ 2 t m2 p 7 < 3 y<2 Shows or implies a correct first step of algebraic manipulation eg 7 m 2y 1 ■ > 12 ■ ■ ■ ■ 4 4(7 m 2y) > 12 28 m 8y >1 12 7 m 2y > 3 7 m 2y >1 3 or Identifies the value 2 even if the inequality sign is incorrect or omitted eg ■ y>2 ■ ye2 ■ 2 42 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Inequality (cont) 3-5 4-6 5-7 6-8 15 Correct response b 1m Additional guidance Gives a correct explanation The most common correct explanations: Give the correct solution eg ■ m3 < y < 3 ■ It’s only true if y is between m3 and 3 ■ y must be bigger than m3 as well ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ Give a counter-example eg ■ m4 is less than 3 but 16 > 9 ■ If you square minus 8 the answer isn’t less than 9 Numbers less than m3 don’t work (m5) 2 > 9 m4 t m4 e 16 It’s not true for m7 Incomplete explanation eg ◆ When you square a negative number, the answer is positive ◆ √9 e ± 3 U1 43 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Angle proof 3-5 4-6 5-7 6-8 16 Correct response a 1m Gives a correct explanation eg ■ BO and OA are radii, so triangle OBA is isosceles, so ∠ABO e ∠BAO. The same is true of CO and OB, so ∠BCO e ∠OBC ■ Both triangles are isosceles because two of their sides are radii of the circle ■ Triangle AOB is isosceles because O is the centre and A and B are on the circumference, and so is triangle BOC Additional guidance ✓ Minimally acceptable explanation eg ◆ BO and OA are radii, so ∠ABO e ∠BAO. The same is true of CO and OB ! Explanation correct but only refers to one of ∠ABO or ∠CBO As the explanations are essentially the same for both angles, condone Incomplete explanation that does not explain why the triangles are isosceles eg ◆ OC e OB, so ∠OCB e ∠OBC, and the same for ∠ABO ◆ Both triangles are isosceles U1 b 1m Gives a correct proof eg ■ x p x p y p y e 180 2x p 2y e 180 so x p y e 90 e ∠CBA ✓ Minimally acceptable justification eg ◆ x p x p y p y e 180, so x p y e 90 U1 44 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Computer game 3-5 4-6 5-7 6-8 17 Correct response a 1m Indicates W, L, L Additional guidance ✓ Unambiguous indication eg ◆ b 1m 45 1, 0, 0 ! Qualifier used eg About 45 Accept, even though the exact value can be determined from the information given ◆ c 1m 0.6 (± 0.02) or equivalent probability ✓ Qualifier used eg ◆ d 1m Joins (100, 0.6) to (120, 0.5) using a curve with negative but increasing gradient, or by marking and joining any number of individual points on such a curve eg ■ About 60% ! Point (120, 0.5) not accurately marked Accept if on line x e 120 and within ± 1mm of correct height ! Part(s) of curve with incorrect gradient Condone use of a straight line, ruled or unruled, ie However do not accept other incorrect gradients eg ◆ (decreasing gradient) ◆ (gradient stops increasing) ◆ (gradient positive in parts) 45 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Index Index to mark schemes Tier Question 3–5 4–6 5–7 6–8 Page 1 Pictogram 10 2 Missing numbers 10 3 Scales 11 4 Prices 12 5 Clock 13 6 3 Calculations 14 7 1 Chains 14 8 2 Puzzling out 15 9 4 Wind chill 15 10 5 Throwing dice 16 11 6 Perimeter and area 18 12 7 Weighing 20 13 8 2 Patterns 21 14 9 3 Simplifying 22 15 10 1 Car parking 22 16 11 4 Thinking fractions 23 17 12 5 Moving C 23 18 13 6 Shoe sizes 24 19 14 7 Construction 24 20 15 8 Travel to work 25 21 16 9 1 Solving 26 17 10 2 Shapes 27 18 11 3 Mixed numbers 28 19 12 4 Areas algebraically 30 20 13 5 Arranging 31 21 14 6 Lines on a square 32 22 15 7 Scatter graphs 34 16 8 Winning ticket 35 17 9 Journeys 36 18 10 Different ways 38 46 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2003 KS3 Mathematics test mark scheme: Paper 1 Index Index to mark schemes Tier Question 3–5 4–6 5–7 6–8 Page 19 11 Locus of points 39 20 12 Evens or odds 39 13 Straight line 40 14 Theme park 41 15 Inequality 42 16 Angle proof 44 17 Computer game 45 47 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk EARLY YEARS NATIONAL CURRICULUM 5 –16 GCSE GNVQ GCE A LEVEL NVQ First published in 2003 © Qualifications and Curriculum Authority 2003 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/03/963 254655 http://www.SATs-Papers.co.uk