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Ma Mathematics tests KEY STAGE 3 for Paper 1 Tiers 3–5, 4–6, 5–7 and 6–8 5–7 and 6–8 2009 ALL TIERS Mark scheme National curriculum assessments Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Introduction Introduction 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 each question 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 Using and applying mathematics (UAM) element are identiﬁed in the mark scheme by the symbol U1 . The number 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 2009 key stage 3 mathematics tests and mark schemes were developed by the Test Development Team at Pearson Research and Assessment. 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 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 speciﬁcally to the marking of questions that involve money, negative numbers, time, measures, coordinates, probability or algebra. Unless otherwise speciﬁed in the mark scheme, markers should apply the following guidelines in all cases. Recording marks awarded on the test paper All questions, even those not attempted by the pupil, should 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 should be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper should 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. 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 NAA website www.naa.org.uk/tests from April 2009. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 3 2009 KS3 Mathematics test mark scheme: Paper 1 General guidance What if… Marking procedure The pupil’s response is numerically or algebraically equivalent to the answer in the mark scheme. Markers should award the mark unless the mark scheme states otherwise. 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 the 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, should be accepted. Provided there is no ambiguity, condone the continental practice of using a comma for a decimal point. 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 without altering the original intention or difﬁculty level of the question. For each misread that occurs, deduct one mark only. No answer is given in the expected place, but the correct answer is given elsewhere. 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. 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 Sourced from SATs-Papers.co.uk 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. 4 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 shown. 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. http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 What if… General guidance Marking procedure 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 × 6 = 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 × 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 correct response has been crossed or rubbed out and not replaced. Any legible crossed or rubbed out work that has not been replaced should be marked according to the mark scheme. If the work is replaced, then crossed or rubbed out work should not be considered. More than one answer is given. 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. 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. The answer is correct but, in a later part of the question, the pupil has contradicted this response. 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. 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 has drawn lines which do not meet at the correct point. Markers should interpret the phrase ‘lines not accurate’ to mean meeting within or on a circle of radius 2mm with centre at the correct point. within the circle accepted Sourced from SATs-Papers.co.uk on the circle accepted outside the circle not accepted http://www.SATs-Papers.co.uk 5 2009 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving money Accept Where the £ sign is given for example: £3.20, £7 Where the p sign is given for example: 40p Where no sign is given for example: £3.20, 40p £3.20 £7 £7.00 Any unambiguous indication of the correct amount, eg £3.20p £3 20 pence £3 20 £3,20 £3-20 £3:20 320p with £ sign crossed out 40p Any unambiguous indication of the correct amount, eg £0.40p £.40p £0.40 with p sign crossed out Do not accept Incorrect placement of pounds or pence, eg £320 £320p Incorrect placement of decimal point, or incorrect use or omission of 0, eg £3.2 £3 200 £32 0 £3-2-0 Incorrect or ambiguous use of pounds or pence, eg 0.40p £40p £3.20 320p 40p £0.40 Any unambiguous indication of the correct amount in £ or p as shown above Omission of ﬁnal zero, eg 3.2 0.4 At levels 3 and 4 only also accept omission of units, eg 3.20 320 40 0.40 Responses involving negative numbers Accept For example: –2 Do not accept To avoid penalising the error below more than once within each question, do not award the mark for the ﬁrst occurence 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 2– 6 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving time Accept A time interval for example: 2 hours 30 minutes 2 hours 30 minutes Any unambiguous, correct indication, eg 1 2 2 hours 2.5 hours 2h 30 2h 30 min 2 30 Digital electronic time, ie 2:30 A speciﬁc time for example: 8:40am, 17:20 Do not accept Incorrect or ambiguous time interval, eg 2.3 hours 2.3h 2h 3 2.30 min 2.30 2-30 2,30 2.3 8:40am 8:40 twenty to nine Any unambiguous, correct indication, eg 08.40 8.40 0840 8 40 8-40 8,40 Unambiguous change to 12 or 24 hour clock, eg 17:20 as 5:20pm or 17:20pm Incorrect time, eg 8.4am 8.40pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0, eg 840 8:4:0 8.4 084 84 Responses involving measures Accept Where units are given (eg kg, m, l) for example: 8.6kg 8.6kg Any unambiguous indication of the correct measurement, eg 8.60kg 8.6000kg 8kg 600g Do not accept Incorrect or ambiguous use of units, eg 8600kg Note If a pupil leaves the answer box empty but writes the answer elsewhere on the page, then that answer must be consistent with the units given in the answer box and the conditions listed above. If a pupil changes the unit given in the answer box, then their answer must be equivalent to the correct answer, using the unit they have chosen, unless otherwise indicated in the mark scheme. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 7 2009 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving coordinates Accept For example: (5, 7) Do not accept Unconventional notation, eg (05, 07) Incorrect or ambiguous notation, eg (7, 5) (ﬁve, seven) x y (5, 7) (x = 5, y=7) y x (7, 5) (5 x , 7 y ) x y (5 , 7 ) ( x − 5, y − 7) Responses involving probability Accept A numerical probability should be expressed as a decimal, fraction or percentage only. for example: 0.7 7 10 Equivalent decimals, fractions and percentages, eg 0.700 70 100 35 50 70% 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 = 100 25 ! Take care Do not accept 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 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving the use of algebra Accept For example: 2+n n+2 ! Take care Unambiguous use of a different case or variable, eg N used for n x used for n ! Unconventional notation, eg 2n n 2 n2 Do not accept n × 2, or 2 × n, or n 2 or n + n for 2n n × n for n2 n ÷ 2 for n or 1 n 2 2 2 + 1n for 2 + n 2 + 0 n 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 3 x + 2 = 32, 3 × 10 + 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 + 2 tiles or tiles = t = n + 2 for t = n + 2 ! Words or units used within equations or expressions, eg n tiles + 2 n cm + 2 Do not accept on their own. Ignore if accompanying an acceptable response. Unambiguous letters used to indicate expressions, eg Ambiguous letters used to indicate expressions, eg t = n + 2 for n + 2 n = n + 2 for n + 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 9 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Circle totals 3–5 4–6 5–7 6–8 1 Mark Correct response 2m Additional guidance Completes the diagram correctly, ie 25 40 20 30 45 or 1m Gives two correct values U1 10 Sourced from SATs-Papers.co.uk 35 10 45 ! For 1m, follow-through from their 25 Accept follow-through for their 30 as 55 – their 25 http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Dishes 3–5 4–6 5–7 6–8 2 Mark Correct response a 1m £11 b 2m or 1m Additional guidance £2.50 Gives the answer 2.5 or 250 or Shows the value 7.5(0) or 750 or Shows or implies a complete correct method with not more than one computational error eg • 1.50 + 2.50 + 3.50 = 7.00 (error) Answer given as 3 c 1m Gives a correct pair of colours, in any order, ie Green and Orange or Blue and Red 1m Unambiguous indication of colour eg • G and O • B and R ! Response gives costs rather than colours Withhold 1 mark only for the ﬁrst occurrence. Allow costs given in pence eg • 1.50 and 3(.00) 2(.00) and 2.50 • 150 and 300 200 and 250 Mark as 0, 1 Gives a correct pair of colours, other than any previously credited U1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 11 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Five squares 3–5 4–6 5–7 6–8 Mark Correct response 3 Additional guidance a 1m Draws the correct line of symmetry, ie ! Line not ruled, accurate or extended Accept lines of at least 3 diagonals in length provided the pupil’s intention is clear b 1m Completes the diagram correctly, ie ! Squares not shaded Accept provided indication of squares is unambiguous Tier & Question Javelin 3–5 4–6 5–7 6–8 4 Mark Correct response a 1m 16 to 18 inclusive b 1m 4 c 1m Additional guidance 17 to 19 inclusive 12 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Digit cards 3–5 4–6 5–7 6–8 Mark Correct response 5 1m Gives four of the digits to make a correct calculation eg • 7 + 8 = 15 • 5 + 6 = 11 • 9 + 9 = 18 1m Gives four of the digits to make a correct calculation eg • 6 × 7 = 42 • 7 × 5 = 35 • 9 × 9 = 81 1m 1m U1 Gives ﬁve of the digits to make a correct calculation eg • 23 – 4 = 19 • 67 – 5 = 62 • 24 – 2 = 22 Additional guidance ! Zero used at the end of a number eg, for the ﬁrst mark • 2 + 8 = 10 Penalise only the ﬁrst occurrence Zero used or card left blank at the beginning of a two-digit number eg, for the second mark, do not accept • 2 × 3 = 06 Card left blank at the end of a number eg, for the third mark, do not accept • 2–1=1 Extra digit inserted eg, for the fourth mark, do not accept • 36 ÷ 2 = 18 Gives four of the digits to make a correct calculation eg • 14 ÷ 2 = 7 • 24 ÷ 4 = 6 • 36 ÷ 6 = 6 Tier & Question Heights 3–5 4–6 5–7 6–8 6 Mark Correct response a 1m Indicates 1.8 metres, ie b 1m Additional guidance Indicates 7 metres, ie Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 13 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Change 3–5 4–6 5–7 6–8 Mark Correct response 7 Additional guidance a 1m 3 b 2m Completes all three rows of the table correctly in any order eg • Number of 20p coins Number of 10p coins 2 3 0 2 2 2 2 1 4 2 or 1m Number of 50p coins 0 Cell that should contain zero left blank 6 Completes two rows of the table correctly Tier & Question Doctors 3–5 4–6 5–7 6–8 Mark Correct response Additional guidance 8 1 a a 1m Gives a value between 49 and 53 inclusive b b 1m Gives a value between 23 and 27 inclusive Value qualiﬁed eg, for part (a) • About 50 c 1m Gives a possible reason eg • They might think their doctor’s treatment is sometimes very good, but not at other times • They might not think that any of the possible answers is what they think • They don’t have a doctor • They might not want to comment • They could be worried about giving an opinion • They may have only ever had one doctor • They don’t always see the same doctor Minimally acceptable reason eg • Could be sometimes one category and sometimes another • They may not like the choices • If they’re not sure • They don’t see their doctor very often • They have just got a new doctor • Not relevant • They don’t want to answer • They can’t tell what is meant by good c Incomplete reason eg • They don’t know 14 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Using tens 3–5 4–6 5–7 6–8 9 Mark Correct response 2 1m ÷ 10 1m ÷ 10 – 10 1m + 10 Additional guidance ! Correct operation indicated, but 10 omitted eg, for the ﬁrst mark • ÷ Penalise only the ﬁrst occurrence ÷ 10 Tier & Question Card shape 3–5 4–6 5–7 6–8 Mark Correct response 10 3 2m Indicates only the three correct shapes, ie Additional guidance Unambiguous indication eg • or 1m Indicates any two of the correct shapes with the third incorrect or omitted for yes and for no ! For 1 mark, response indicates only the three shapes showing the grey side of the shape, eg or Indicates the three correct shapes with not more than one other incorrect Condone Tier & Question Number lines 3–5 4–6 5–7 6–8 11 4 Mark Correct response 1m Gives both the values 2 and 8 in the correct positions 1m Gives the value – 4 in the correct position 1m Gives the value (+)6 in the correct position Additional guidance Sourced from SATs-Papers.co.uk ! Follow-through from their –4 Accept the sum of their –4 and 10 provided their –4 is a negative number http://www.SATs-Papers.co.uk 15 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Rhombus grid 3–5 4–6 5–7 6–8 12 5 a Mark Correct response a 1m 12 b b 1m Draws a correct triangle eg Additional guidance • ! Lines not ruled or accurate, or triangle not shaded Accept provided the pupil’s intention is clear ! Vertices of triangle not on the intersections of the grid Accept vertices within 2mm of the intersections of the grid ! Other shapes drawn As these may be trials, ignore • • • 16 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Missing digits 3–5 4–6 5–7 6–8 Mark Correct response 13 6 1m Additional guidance Completes the second calculation correctly, ie 1 7 × 3 = 5 1 ! Both digits placed in the same box eg • 17 × 3 = 5 1 Condone 1m Completes the third calculation correctly, ie 1 4 × 3 = 4 2 1 5 × 3 = 4 5 1 6 × 3 = 4 8 or or U1 Tier & Question Clocks 3–5 4–6 5–7 6–8 14 7 a Mark Correct response a 1m 10am b b 1m 6pm Sourced from SATs-Papers.co.uk Additional guidance ! Indication of am or pm incorrect or omitted Condone omission of am or pm but do not accept incorrect times eg, for part (a) accept • 10 (o’clock) eg, for part (a) do not accept • 10pm • 22:00 eg, for part (b) accept • 6 (o’clock) • 18:00 eg, for part (b) do not accept • 6am • 06:00 http://www.SATs-Papers.co.uk 17 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Sum of 80 3–5 4–6 5–7 6–8 Mark Correct response 15 8 1m Indicates Set A and gives a correct explanation eg • A = 74 and 80 – 74 = 6 B = 90 and 90 – 80 = 10 • A is –3, –2, –1, (0) and B is +1, +2, +3, +4, so A is only 6 less than 80, but B is 10 more U1 Additional guidance Minimally acceptable explanation eg • 6 and 10 seen • 74 and 90 seen • (–)3, (–)2, (–)1, (0) and 1, 2, 3, 4 seen Incomplete or incorrect explanation eg • A adds up to 74 • B is 10 more than 80 • A adds up to 74, B adds up to 110 • 17, 18 and 19 are all under 20 so A is smaller Tier & Question Number chains 3–5 4–6 5–7 6–8 16 9 a Mark Correct response a 1m Gives the values 14 and 41 in the correct positions b b 1m Shows a correct rule eg • ×3 • Multiply by 3 • Triple • × 3 then + 0 Additional guidance Minimally acceptable rule eg • Add the number 3 times • Add on double itself • Double then add the number • It’s the next power of 3 • 3× ! Rule embedded or shown in working Accept provided a correct rule is shown explicitly, even if an incorrect value for the next number in the chain is shown on the answer line eg, accept • 81 × 3 seen • (4 – 1) × 81 eg, do not accept • 81 + 81 + 81 • 81 × 2 + 81 Incomplete or incorrect rule eg • 3 • +54 • 3n 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Making 1 3–5 4–6 5–7 6–8 Mark Correct response 17 10 1 a a a 2m Additional guidance Joins all four pairs of numbers correctly, ie 0.1 0.99 0.11 0.9 0.01 0.999 0.91 0.89 0.001 0.09 or 1m Joins at least two pairs of numbers correctly 2m b b b Number matched to more than one other For 2m or 1m, do not accept as a correct match Joins all four pairs of numbers correctly, ie 1 2 0.5 4 0.25 1 0.1 0.05 or 1m 20 10 Joins at least two pairs of numbers correctly Tier & Question T-shirts 3–5 4–6 5–7 6–8 18 11 2 Mark Correct response Additional guidance a 1m 1 5 or equivalent probability b b b 1m 2 3 or equivalent probability ! Value rounded Accept 0.66(…) or 0.67 or the percentage equivalents c 1m 1 3 or equivalent probability ! Value rounded Accept 0.33(…) or the percentage equivalent a a c c Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 19 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Water 3–5 4–6 5–7 6–8 Mark Correct response 19 12 3 1m Additional guidance Indicates the value 500 on the jug, ie millilitres 1000 Unambiguous indication ! Inaccurate indication Accept provided the pupil’s intention is clear 800 600 400 200 U1 Tier & Question Boxes 3–5 4–6 5–7 6–8 Mark Correct response 20 13 4 2m or 1m U1 Additional guidance 90 Shows or implies a complete correct method with not more than one computational error eg • 72 ÷ 4 = 16 (error) 72 + 16 = 88 • 72 ÷ 4 = 18 18 × 5 = 80 (error) Tier & Question Percentages 3–5 4–6 5–7 6–8 21 14 5 a Additional guidance a 1m 18 ! Throughout the question, incorrect use of % sign eg • 18% 54% Penalise only the ﬁrst occurrence b b b 1m 54 ! For part (b) follow-through Accept follow-through as their (a) × 3, or as 36 + their (a) provided the result is less than 360 20 a Mark Correct response Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Number grids 3–5 4–6 5–7 6–8 Mark Correct response 22 15 6 1m Additional guidance Completes the ﬁrst grid correctly, ie 22 13 35 4 17 52 1m Completes the second grid correctly, ie 7 1 8 3 4 12 U1 Tier & Question Angles in a triangle 3–5 4–6 5–7 6–8 23 16 7 Mark Correct response 3m or 2m Additional guidance Gives all three correct angles, ie x = 90 y = 20 z = 20 Gives any two correct angles or or 1m Gives x = 90 and y = z, provided this value is < 90 and > 0 Gives any one correct angle or Gives y = z, provided this value is < 90 and > 0 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 21 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7, 6–8 Tier & Question Finding b 3–5 4–6 5–7 6–8 Mark Correct response 24 17 8 2m or 1m Additional guidance 2 Shows or implies that a = 5 and shows the intention to substitute this value into the second equation eg • 5 + 7 = 10 + b • b = 12 – 10 or Shows a complete correct method with not more than one computational error eg • b = 11 – 6 + 7 – 10 • a = 11 – 6 = 6 (error) 6 + 7 = 10 + b b=3 Conceptual error eg • a = 11 + 6 = 17 Tier & Question Matching 3–5 4–6 5–7 6–8 18 9 1 Mark Correct response 1m Additional guidance Instruction on the left matched to more than one instruction on the right Matches both instructions on the left to the equivalent instruction on the right, ie Subtract 0 1 Add 0 Add 2 Add 2 Subtract 2 Subtract 2 Add –2 1 Subtract –2 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Oak leaves 3–5 4–6 5–7 6–8 19 10 2 Mark Correct response 1m Gives a correct reason from one of the ﬁve categories below that states or implies the problem, or suggests an improvement The most common correct reasons: Category 1: Refer to the number of leaves in the sample being too small eg, problem • The sample is too small • Those 10 leaves might all be diseased eg, improvement • They should pick more than 10 Category 2: Refer to the number of trees in the sample being too small eg, problem • One oak tree might be different from others • May be something wrong with that tree eg, improvement • They should use more than one tree Category 3: Refer to the conditions in which the tree is growing being too uniform eg, problem • Different conditions may affect the leaves on other trees • The soil might be very bad in that area eg, improvement • They should choose trees in different areas Category 4: Refer to the area of the tree from which the leaves are picked being too small eg, problem • The leaves on higher branches might be different • Those branches may not get enough light eg, improvement • They need leaves from all over the tree Additional guidance Minimally acceptable reason eg, problem • Too small • Only 10 • Not enough • Just one • Same growing conditions for the tree • Other branches might be different • Only the lowest branches eg, improvement • 100 is better • More than one • Need different areas • Use other branches • Collect at other times ! For the ﬁrst or the second reason, more than one reason given within one response Do not accept a correct response accompanied by an incorrect response from the same category. Otherwise ignore irrelevant or incorrect further responses. If two correct reasons from different categories are given in one response space, both marks should be awarded eg • They need more trees from more areas Mark as 1, 1 Incomplete reason that repeats the information given with no further explanation eg • They are taking 10 leaves • They are using one oak tree • They are taking them from one part of the tree Category 5: Refer to the period for picking the sample being too short eg, problem • The leaves may be different at different times of year • It may be winter eg, improvement • They should collect throughout the year 1m Gives a correct reason from a different category from one already credited U1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 23 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Missing lengths 3–5 4–6 5–7 6–8 20 11 3 Mark Correct response 2m or 1m Additional guidance Gives both correct lengths, ie x = 10 and y = 3.9 or equivalent Gives y = 3.9 or equivalent or Gives the two values transposed, ie x = 3.9 or equivalent and y = 10 or Shows a complete correct method with not more than one computational error eg • x = 10, 10 – 6.1 = 4.9 (error) • 4 × 6.1 = 24.4, 40 – 24.4 = 16.6 (error) 16.6 ÷ 4 = 4.15, 4.15 + 6.1 = 10.25 • 40 ÷ 4 = 20 (error) 20 – 6.1 = 13.9 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Counters 3–5 4–6 5–7 6–8 21 12 4 a a a Mark Correct response 2m or 1m Gives the value 3, with no evidence of an incorrect method Shows or implies a correct equation for the bags and 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 • 6y + 1 = 4y + 7 6y – 4y = 7 – 1 • –2y + 7 = 1 • 6y – 6 = 4y • 2y = 6 2m Gives an answer of 4.(…) ! Method used is trial and improvement Note that no partial credit can be given 5 or 1m b b b Additional guidance or U1 ! Method used is trial and improvement Note that no partial credit can be given Shows or implies a correct inequality using the expressions for the bags eg • 4k > k + 12 • 3k > 12 • k>4 Tier & Question Prize money 3–5 4–6 5–7 6–8 22 13 5 Mark Correct response 2m £ 490 000 or 1m Additional guidance £ 490k Shows the value 980 000 or Shows a complete correct method with not more than one error eg • 1 000 000 – 20 000 = 98 000 (error), 98 000 ÷ 2 = 49 000 For 1m, one million taken to be 100 000 eg • 100 000 – 20 000 = 80 000, 80 000 ÷ 2 = 40 000 For 1m, computational error that simpliﬁes the division eg • 1 000 000 – 20 000 = 800 000, 800 000 ÷ 2 = 400 000 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 25 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Correlation 3–5 4–6 5–7 6–8 23 14 6 a a a Mark Correct response 1m Additional guidance Indicates B and gives a correct explanation The most common correct explanations: Refer to the ‘slope’ or ‘gradient’ of the points eg • The points make a pattern that is sloping upwards from left to right • The line of best ﬁt would have a positive gradient Minimally acceptable explanation eg • It slopes upwards • It goes up • It’s like this Incomplete explanation eg • It slopes the positive way Describe the relationship between the two variables eg • As the value on the x-axis increases, so does the value on the y-axis Minimally acceptable explanation eg • As one amount gets bigger, so does the other • It could be the higher the temperature, the more ice creams are sold Incomplete explanation eg • They both increase • It goes from the left-hand corner • It is slanted towards the right b b b 1m Indicates A and gives a correct explanation The most common correct explanations: Refer to the points being closer to a line of best ﬁt eg • The points are practically in a straight line, so the correlation is very strong • If you drew the line of best ﬁt, the points in A would all be close to it but many would be further away in B Minimally acceptable explanation eg • They are closer to one line • In B they are less bunched together in a line Refer to the ‘line’ or sloping pattern being clearer to see eg • You can see the pattern of a very clear, almost straight line • In B you can see a pattern sloping upward, but it’s not as clear Minimally acceptable explanation eg • They are in a straight line • The pattern sloping downwards is clear • In B the line is less easy to see • B’s points are sloping upwards, but not as deﬁnitely as in A Incomplete explanation eg • The points are closer together • In B they are more spread out Incomplete explanation eg • The pattern is clearer • They are in a line 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Shape rules 3–5 4–6 5–7 6–8 24 15 7 Mark Correct response 2m Additional guidance Completes all three rules correctly, ie H = N + 1 A = H × 2 A = 2N + 2 ! Throughout the question, unconventional notation eg, for the ﬁrst rule • 1N+1 Condone ! Throughout the question, words used instead of letters eg, for the second rule • A = Height × 2 Penalise only the ﬁrst occurrence ! For the second rule, N + 1 used Accept provided there is no ambiguity eg, accept • (N + 1) × 2 eg, do not accept • N+1×2 For the third rule, 2H used or 1m Completes two rules correctly Tier & Question Fortieths 3–5 4–6 5–7 6–8 25 16 8 Mark Correct response Additional guidance 1m 0.775 Equivalent fractions 1m 0.575 Follow-through as their value for the ﬁrst mark – 0.2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 27 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Expressions 3–5 4–6 5–7 6–8 26 17 9 a a a Mark Correct response 1m Indicates 2n must be even and gives a correct explanation eg • Any whole number multiplied by two gives a number in the two times table, so is even • Odd × 2 = even, even × 2 = even • 2 × odd is odd + odd = even 2 × even is even + even = even • All multiples of 2 are even • Halving an odd number does not give a whole number Minimally acceptable explanation eg • × 2 gives even • Doubling any number gives even • All the numbers in the 2 times table are even Indicates 3n could be odd or even and gives a correct explanation eg • 3 × 1 = 3 which is odd, but 3 × 2 = 6 which is even • Odd × 3 = odd, even × 3 = even • Multiples of 3 can be odd or even • An even or odd number can have a factor of 3 Minimally acceptable explanation eg • 3 × 1 = 3, 3 × 2 = 6 • If n is 5 you get odd, if n is 6, you get even • 3 × some numbers = odd, but 3 × some numbers = even • Because 3 goes into both odd and even numbers • In the 3 times table there are odd and even numbers U1 b b b 1m U1 28 Additional guidance Sourced from SATs-Papers.co.uk Incomplete explanation eg • 2 × 1 = 2 which is even, and 2 × 2 = 4 which is also even • Even × even is even Even × odd is even • Because when you add two odd numbers together you always make an even • Because 2 is even Incomplete explanation eg • 3n is sometimes odd and sometimes even • Even × odd gives even odd × odd gives odd • 3, 6, 9, 12, 15….. http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Ratio 3–5 4–6 5–7 6–8 27 18 10 a Mark Correct response a 1m 8 b b 1m Gives a correct number of black beads and white beads such that: the number of black beads is (3n – 1) and the number of white beads is (2n – 3), provided n ≥ 2 eg • 5 black beads, 1 white bead • 8 black beads, 3 white beads • 11 black beads, 5 white beads Additional guidance Markers may ﬁnd the following list of correct examples helpful: White 5 1 8 3 11 5 14 7 17 9 20 U1 Black 11 Tier & Question Powers 3–5 4–6 5–7 6–8 28 19 11 Mark Correct response 1m Gives a correct justiﬁcation that the difference between 32 and 33 is 18 eg • 32 = 9, 33 = 27, and 27 – 9 = 18 • 33 – 32 = 32(3 – 1) =9×2 = 18 Sourced from SATs-Papers.co.uk Additional guidance Minimally acceptable justiﬁcation eg • 27 – 9 • 9 + 18 = 27 Incomplete or incorrect justiﬁcation eg • 32 = 9, 33 = 27 • 33 – 32 = 18 • 9 – 27 = 18 http://www.SATs-Papers.co.uk 29 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Sorting primes 3–5 4–6 5–7 6–8 20 12 Mark Correct response 1m Identiﬁes a value, n, such that n is prime, and shows that 2n + 1 is not prime to demonstrate that the statement is incorrect eg • 7 is a prime number, but 2 × 7 + 1 = 15, and 15 is not a prime number • 13 is prime, but 27 is not Additional guidance Minimally acceptable response eg • 7, 15 • 2 × 13 + 1 = 27 Incomplete or incorrect response eg • 2 × n is even, even + 1 is odd and not all odd numbers are prime ! More than one example given Accept provided a counter example is clearly identiﬁed eg, accept • 11 gives 23 13 gives 27 so this one eg, do not accept • 11 gives 23 13 gives 27 Markers may ﬁnd the following list of correct examples helpful (n < 100): n 35 19 39 31 63 37 75 43 87 47 95 59 119 61 123 67 135 71 143 73 147 79 Sourced from SATs-Papers.co.uk 27 17 30 15 13 U1 2n + 1 7 159 97 195 http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Score 3–5 4–6 5–7 6–8 21 13 a a Mark Correct response 2m or 1m Additional guidance 11 Shows the values 56 and 45 or Gives an answer of 9 [the points gained in round 5] b b 1m Gives a response that states or implies that Derek gained the same number of points in each round eg • He got the same number of points in each round • To keep the gradient the same, an equal number needs to be added each time • For every round going across, the line must have gone up the graph in equal steps U1 Sourced from SATs-Papers.co.uk Minimally acceptable response eg • Same • Equal • No change • The total increases by the same number in each round • He gained 10 points each round Incomplete or incorrect response eg • He gets about the same number of points in each round • It increases by the same number in each round • His points were consistent • A steady increase • He gets maximum points each round • The line could be horizontal http://www.SATs-Papers.co.uk 31 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Rhombus 3–5 4–6 5–7 6–8 22 14 Mark Correct response 2m or 1m Additional guidance 24 Shows a correct method with not more than one computational error The most common correct methods: Calculate the area of the rhombus as half the area of the rectangle eg • 1 (6 × 8) 2 • 48 ÷ 2 Work with 2 or 4 triangles eg • Area of one little triangle is half of 3 × 4, there are 4 little triangles so × 4 • (6 × 4) ÷ 2 = 14 (error), 14 × 2 = 28 • 8 triangles altogether, so one is 48 ÷ 8 = 7 (error), 4 shaded so 4 × 7 = 28 • Area of rectangle: 6 × 8 = 48, Conceptual error eg • Area of triangle given as base × height Area of white triangle: 1 × 3 × 4 = 6 2 4 × 6 = 18 (error), answer 30 1m 32 Shows the correct unit for their area or method eg • 24cm2 • 2400mm2 Sourced from SATs-Papers.co.uk ! Area incorrect or omitted, but units given If the mark(s) for the correct area have not been awarded, condone cm2 seen for the third mark http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Sums and products 3–5 4–6 5–7 6–8 23 15 a Mark Correct response a 1m Gives a pair of values with a negative sum and a positive product, ie where a and b are both negative eg • –2 and –1 • –9 and –10 • –0.5 and – 2 5 • –3 and –3 b b 1m Additional guidance Gives a pair of values with a positive sum and a negative product, ie where a is positive, b is negative and |a| > |b| eg • 2 and –1 • –9 and 10 • 0.5 and – 2 5 Tier & Question Mean 3–5 4–6 5–7 6–8 24 16 Mark Correct response 2m or 1m Additional guidance 16 Shows the value 66 or Shows or implies a complete correct method with not more than one computational error eg • 6 × 11 – 5 × 10 • 5 × 10 = 50, 6 × 11 = 65 (error) so 15 U1 Sourced from SATs-Papers.co.uk ! For 1m, method uses arbitrary values with a mean of 10 for the original ﬁve numbers Condone eg, for 1m accept • 8 + 9 + 10 + 11 + 12 = 49 (error) 6 × 11 – 49 = 17 For 1m, error is in the number of values in the set after one is added eg • 5 × 11 = 55, 55 – 50 = 5 http://www.SATs-Papers.co.uk 33 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Simultaneous 3–5 4–6 5–7 6–8 25 17 Mark Correct response 3m or 2m or 1m Gives both x = 5 and y = 5 or equivalent and 2 shows or implies a complete correct method for solving algebraically eg • 2x = 10, x = 5 and y = 5 2 • 3x + 18y = 60 3x + 6y = 30 12y = 30, so y = 2.5 and x = 5 • 30 – 3x = 20 – x 10 = 2x, x = 5 and y = 2.5 • 3(20 – 6y) + 6y = 30 60 – 18y + 6y = 30 30 = 12y, y = 2.5 and x = 5 Additional guidance Method used is trial and improvement Gives either x = 5 or y = 5 or equivalent and 2 shows or implies a correct method for solving algebraically for that variable eg • 2x = 10, x = 5 • 3x + 18y = 60 3x + 6y = 30 12y = 30, so y = 2.5 • 30 – 3x = 20 – x 10 = 2x, x = 5 • 3(20 – 6y) + 6y = 30 60 – 18y + 6y = 30 30 = 12y, y = 2.5 Subtracts the two given equations to eliminate y, or forms two correct equations that would allow elimination of x eg • 2x = 10 • 3x + 18y = 60 3x + 6y = 30 or Attempts to solve by substitution and forms a correct equation in only one variable eg • 3(20 – 6y) + 6y = 30 • x + 30 – 3x = 20 34 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Shape 3–5 4–6 5–7 6–8 26 18 Mark Correct response 2m or 1m Additional guidance 200, with no evidence of an incorrect method Shows or implies that a = 5 or Shows or implies that the area of one rectangle is 50 or Shows a complete correct method with not more than one computational error eg • 16a = 80, so a = 6 (error) 6 × 12 = 72, 72 × 4 = 288 ! Error made in coefﬁcient of a Follow-through with this value provided 12 coefﬁcient of a 20 eg • 12a (error) = 80, so a = 6.6 6.6 × 13.2 × 4 = 348 Tier & Question Circle shapes 3–5 4–6 5–7 6–8 27 19 Mark Correct response Additional guidance 1m Gives the correct expression for area A, ie Area A = y + 3w ! Throughout the question, unconventional notation or unsimpliﬁed expressions Condone eg, for Area A, accept • y+3×w • y+w+w+w eg, for Area B, accept • 1y + 1w • y + 3w – 2w 1m Gives the correct expression for area B, ie Area B = y + w ! Answers for Area A and Area B transposed but otherwise correct Mark as 0, 1 U1 Sourced from SATs-Papers.co.uk ! Answers for Area A and Area B correct followed by incorrect further processing Mark as 0, 1 http://www.SATs-Papers.co.uk 35 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question False 3–5 4–6 5–7 6–8 28 20 a Mark Correct response 2m Additional guidance Gives a correct explanation The most common correct explanations: Give a correct counter example eg • When j = 2 and k = 3, ( j + k )2 = 25, but j2 + k2= 13 • If j is 2 and k is 3, (2 + 3)(2 + 3) ≠ 22 + 32 Minimally acceptable explanation eg • When j = 2 and k = 3 you get 25 and 13 Give the correct expansion of the expression eg • ( j + k )( j + k ) = j2 + 2jk + k2, not j2+ k2 • It should be j2 + jk + jk + k2 • jk + jk has been missed out so it should be j2+ 2jk + k2 Minimally acceptable explanation eg • j2+ 2jk + k2 • j2+ jk + jk + k2 • 2jk is missing Incomplete explanation eg • When j = 2 and k = 3 you get different answers for each side, so it can’t be right ! Correct expression equated to zero eg • j2+ 2jk + k2 = 0 Condone Incomplete or incorrect explanation eg • ( j + k )( j + k ) ≠ j2+ k2 • ( j + k )2 = j2+ jk + k2 Address the misconception eg • Both things in the ﬁrst brackets should be multiplied by both things in the second brackets, but the pupil has done j × j and k × k Minimally acceptable explanation eg • The pupil hasn’t multiplied the j by the k or the k by the j • There should be a jk term • It should have been like this: ( j + k)( j + k) Incomplete explanation eg • There should be 3 terms in the answer • The jks should be added • You have to multiply everything in the second brackets by everything in the ﬁrst brackets • The pupil hasn’t multiplied the ﬁrst set of brackets by the second set properly • You don’t square j and k, you square the answer of ( j + k ) 36 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question False (cont) 3–5 4–6 5–7 6–8 28 20 Mark Correct response or 1m Shows a complete correct method with not more than one computational error when substituting values eg • If j = 2 and k = 3 (j + k)2 = (2 + 3)2 = 20 (error), j2 + k2= 4 + 9 = 13 Additional guidance Conceptual error eg • 32 = 6 or U1 b 1m U1 Shows or implies the four correct terms resulting from multiplying out the brackets, even if there is incorrect further working eg • j2, jk, jk, k2 • j×j+j×k+j×k+k×k Gives a correct counter example eg • j=0 • k=0 • Either j or k is zero • Both j and k are zero • It doesn’t work if k is nought Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 37 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 6–8 only Tier & Question Dice probability 3–5 4–6 5–7 6–8 21 Mark Correct response 2m or 1m 3 4 Additional guidance or equivalent probability Shows or implies the number of possible outcomes where the product is a multiple of 3 eg • × 3 4 5 6 3 9 12 15 18 4 12 16 20 24 5 15 20 25 30 6 18 24 30 36 3 4 5 6 • 3 4 5 6 • 3 × 3, 3 × 4, 3 × 5, 3 × 6, 4 × 3, 4 × 6, 5 × 3, 5 × 6, 6 × 3, 6 × 4, 6 × 5, 6 × 6 or Shows a complete correct method but makes not more than two errors in identifying multiples of 3, then follows through to give their correct probability • × 3 4 5 6 3 9 12 15 18 4 12 16 20 24 5 15 20 25 30 6 18 24 30 36 U2 38 Sourced from SATs-Papers.co.uk so 11 16 http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Solving 3–5 4–6 5–7 6–8 22 Mark Correct response 2m Additional guidance Gives y = 20 and shows or implies a correct ﬁrst step of algebraic manipulation that either removes the denominator or removes the brackets eg • 5(3y – 4) = 14y • 5(3y – 4) = 2y × 7 • 15y – 20 =7 2y 5 × 3y – 5 × 4 =7 2y • 15y – 20 = 14y • y – 20 = 0 • or 1m 2m Shows or implies a correct ﬁrst step of algebraic manipulation that either removes the denominator or removes the brackets, even if there are other errors Gives x = 5 and x = –5, in either order and shows or implies the correct expansion of ( x + 4)( x – 4) eg • x2 + 4x – 4x – 16 x • x –4 4 x2 4x –4x –16 • x2 – 16 • x2 = 25 or 1m Shows or implies the correct expansion of ( x + 4)( x – 4), even if there are other errors Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 39 2009 KS3 Mathematics test mark scheme: Paper 1 Tiers 6–8 only Tier & Question 23 a Distance from school Marking overlay available 3–5 4–6 5–7 6–8 Mark Correct response 2m or 1m Draws a correct graph within the tolerance as shown on the overlay that fulﬁls the following conditions: 1. All four points marked correctly, ie (2, 19), (3, 25), (4, 28) and (5, 29) 2. All points joined with a series of straight lines Additional guidance ! For 2m or 1m, points joined with a curve Condone Shows or implies the values 19, 25, 28 and 29 eg • Fulﬁls condition 1 only • Marks the points (1.5, 19), (2.5, 25), (3.5, 28) and (4.5, 29) [ie uses midpoints of each range as x-coordinates] or Marks and joins at least three points correctly or Makes an error in marking one of the points, but follows through correctly for later points, and joins all their points ! Follow-through For 1m, accept the following values as follow-through: Distance 2 1m 6 + their 19 28 3 + their 25 5 Gives a value between 1.4 and 1.6 inclusive none 25 4 b 19 3 U2 Cf f-t 29 1 + their 28 Equivalent fractions or decimals or Follows through from an incorrect total to give the correct median for their graph ! Follow-through Follow-through can only be given for an increasing graph which reaches (5, y) Tier & Question 24 Mark Correct response 1m Gives B as (1, –1) Additional guidance Gives A as (0, –2) 1m 40 Coordinates Marking overlay available 3–5 4–6 5–7 6–8 Sourced from SATs-Papers.co.uk ! Answers for A and B transposed but otherwise completely correct If this is the only error, ie gives A as (1, –1) and gives B as (0, –2), mark as 0, 1 http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Similar triangles 3–5 4–6 5–7 6–8 25 Mark Correct response 2m or 1m 3, with no evidence of accurate or scale drawing Additional guidance For 2m or 1m, evidence of accurate or scale drawing, with no other method Shows or implies the ratio 4 : 10 eg • 0.4 or equivalent seen • 2.5 or equivalent seen • 2 : 5 = ? : 71 2 • 7.5 ÷ 10 × 4 • 0.75 × 4 • 30 ÷ 10 Tier & Question Regions 3–5 4–6 5–7 6–8 26 a Mark Correct response 1m Additional guidance Gives the four correct letters, ie A, B, G and H, in any order b 1m Gives the four correct letters, ie B, C, D and E, in any order c 1m Gives the four correct letters, ie A, B, E and F, in any order Tier & Question Average speed 3–5 4–6 5–7 6–8 27 Mark Correct response 2m Gives a correct justiﬁcation that the average speed is 20km per hour eg • 1km at 15km/h takes 60 ÷ 15 = 4 minutes, 1km at 30km/h takes 60 ÷ 30 = 2 minutes, 2km in 6 minutes = 20km in 60 minutes = 20km per hour 1 1 3 • 15 + 30 = 30 1 = 10, 1 2km in 10 hour = 20km in 1 hour or 1m Shows or implies that the journey time up the hill was 4 minutes or equivalent, and the journey time down the hill was 2 minutes or equivalent eg • 4, 2 seen Additional guidance For 2m, minimally acceptable justiﬁcation eg • 4 + 2 = 6 mins for 2km 1 1 1 • 15 + 30 = 10 for 2km For 2m, incomplete justiﬁcation eg • 1km at 15km per hour takes 60 ÷ 15 = 4 mins, 1km at 30km per hour takes 60 ÷ 30 = 2 mins • 6 mins for 2km, so it’s 60 mins for 20km which is 20km per hour ! For 1m, total of 6 minutes or equivalent seen As the total of 6 minutes can be calculated from the given 20km per hour, do not accept as implying 4 minutes and 2 minutes unless a correct method is also seen 1 1 • 15, 30 seen • 60 ÷ 15, 60 ÷ 30 seen Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 41 2009 KS3 Mathematics test mark scheme: Paper 1 Index Index to mark schemes Tier 3–5 4 –6 Question 5–7 Page 6–8 1 Circle totals 10 2 Dishes 11 3 Five squares 12 4 Javelin 12 5 Digit cards 13 6 Heights 13 7 Change 14 8 Doctors 14 9 2 Using tens 15 10 3 Card shape 15 11 4 Number lines 15 12 5 Rhombus grid 16 13 6 Missing digits 17 14 7 Clocks 17 15 8 Sum of 80 18 16 9 Number chains 18 17 10 1 Making 1 19 18 11 2 T-shirts 19 19 12 3 Water 20 20 13 4 Boxes 20 21 14 5 Percentages 20 22 15 6 Number grids 21 23 16 7 Angles in a triangle 21 24 42 1 17 8 Finding b 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 Index Question Page 4 –6 5–7 6–8 18 9 1 Matching 22 19 10 2 Oak leaves 23 20 11 3 Missing lengths 24 21 12 4 Counters 25 22 13 5 Prize money 25 23 14 6 Correlation 26 24 15 7 Shape rules 27 25 16 8 Fortieths 27 26 17 9 Expressions 28 27 18 10 Ratio 29 28 19 11 Powers 29 20 12 Sorting primes 30 21 13 Score 31 22 14 Rhombus 32 23 15 Sums and products 33 24 16 Mean 33 25 17 Simultaneous 34 26 18 Shape 35 27 19 Circle shapes 35 28 20 False 36 21 Dice probability 38 22 Solving 39 23 Distance from school 40 24 Coordinates 40 25 Similar triangles 41 26 Regions 41 27 Average speed 41 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 43 QCA wishes to make its publications widely accessible. Please contact us if you have any specific accessibility requirements. 29 Bolton Street London W1J 8BT Telephone: 08700 60 60 40 Minicom: 020 7509 6546 Fax: 020 7509 5908 Email: tests@naa.org.uk Website: www.naa.org.uk/tests First published 2009 © Qualifications and Curriculum Authority 2009 ISBN 1-84721-699-1 Reproduction, storage, adaptation or translation, in any form or by any means, of this publication is prohibited without prior written permission of the publisher, unless within the terms of licences issued by the Copyright Licensing Agency. Excerpts may be reproduced for the purpose of research, private study, criticism or review, or by educational institutions solely for educational purposes, without permission, providing full acknowledgement is given. Printed in Great Britain by the Qualifications and Curriculum Authority under the authority and superintendence of the Controller of Her Majesty’s Stationery Office and Queen’s Printer of Acts of Parliament. The Qualifications and Curriculum Authority is an exempt charity under Schedule 2 of the Charities Act 1993. Qualifications and Curriculum Authority 83 Piccadilly London W1J 8QA www.qca.org.uk Sourced from SATs-Papers.co.uk For more copies: QCA Orderline, PO Box 29, Norwich NR3 1GN www.qca.org.uk/orderline email: orderline@qca.org.uk Tel: 08700 60 60 15 Fax: 08700 60 60 17 QCA/09/3783 http://www.SATs-Papers.co.uk290015