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Ma Mathematics tests KEY STAGE 3 for Paper 1 2 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 2 Introduction Introduction This booklet contains the mark scheme for paper 2 at all tiers. The paper 1 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 2 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 2 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 2 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 2 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 2 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 2 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 2 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 2 Tier 3–5 only Tier & Question School shop 3–5 4–6 5–7 6–8 Mark Correct response 1 a 1m 19 b 1m Friday c 1m Ruler Additional guidance U1 Unambiguous indication eg, for part (b) • F eg, for part (c) • R Tier & Question Missing numbers 3–5 4–6 5–7 6–8 Mark Correct response 2 1m 26 1m Additional guidance 3 Tier & Question Parcels 3–5 4–6 5–7 6–8 3 Mark Correct response 2m or 1m Additional guidance £ 6.10 Gives the answer 6.1 or Shows the value 3.9(0) or 390 or Shows a complete correct method with not more than one computational error eg • 1.3(0) × 3 = 3.6(0) (error), 10 – 3.6(0) = 6.4(0) 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Joining 3–5 4–6 5–7 6–8 4 Mark Correct response 2m Draws all three different sized squares, in any order eg • Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Internal lines shown Only the outline of any shape drawn should be considered Vertices of square do not use the pins or 1m Draws any two of the three different sized squares U1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 11 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Spinner 3–5 4–6 5–7 6–8 Mark Correct response 5 2m Makes all four correct decisions, ie True or 1m False Additional guidance Unambiguous indication eg • for True, for False Makes three correct decisions Tier & Question Fractions 3–5 4–6 5–7 6–8 6 a Mark Correct response 1m Indicates any three squares Additional guidance ! Squares not shaded Accept any unambiguous indication ! Part squares indicated Accept provided the pupil’s intention is clear b 12 1m 3 5 or equivalent Sourced from SATs-Papers.co.uk Equivalent decimals http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Number of sides 3–5 4–6 5–7 6–8 Mark Correct response 7 2m Additional guidance ! Shape matched to more than one number For 2m or 1m, do not accept as a correct match Matches all three shapes correctly, ie 2 ! Extra shapes added Ignore extra shapes and any lines drawn from them 3 Triangle 4 Hexagon 5 Octagon 6 Quadrilateral 7 8 or 1m Matches any two shapes correctly Tier & Question Grid 3–5 4–6 5–7 6–8 8 Mark Correct response 2m Additional guidance Completes the grid correctly, ie 2 1 1 2 3 3 or 1m 3 1 2 Completes both the middle column and the bottom row correctly U2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 13 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Digital 3–5 4–6 5–7 6–8 Mark Correct response 9 Additional guidance 1m 10:45 ! Indication of am or pm Condone either am or pm shown or implied eg, accept • 10:45 am • 22:45 1m Gives a correct description of the time in words eg • Five past ten ! Words and numbers used in description Condone, provided the time has been interpreted correctly eg, accept • 5 past 10 ‘Digital time’ described in words eg • Ten O ﬁve Description of time incorrect or using numbers eg • Ten ﬁve • 10 5 Tier & Question Reﬂecting 3–5 4–6 5–7 6–8 10 1 Mark Correct response 1m Reﬂects the shape correctly in the mirror line, ie Additional guidance ! Throughout the question, lines not ruled or accurate Accept provided the pupil’s intention is clear ! Throughout the question, extra lines drawn Accept provided the pupil’s intention is clear 1m 14 Reflects the shape correctly in the mirror line, ie Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Test 3–5 4–6 5–7 6–8 Mark Correct response 12 2 a a 1m C b b 1m Additional guidance 21 U1 Tier & Question Rounding 3–5 4–6 5–7 6–8 Mark Correct response 13 3 a a 1m 3000 1m b b 2700 1m Gives a value greater than or equal to 795 but less than 805 Additional guidance 800 Tier & Question Castle 3–5 4–6 5–7 6–8 11 4 Mark Correct response 2m £ 5(.00) or 1m Additional guidance Shows the value 22(.00) or Shows or implies a complete correct method with not more than one computational error eg • 12.00 (error) + 9 = 21.00 Answer given as 4.00 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 15 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Baby 3–5 4–6 5–7 6–8 Mark Correct response 14 5 Additional guidance a 1m 4 Any reference to extra days b b 1m 9 Any reference to extra weeks or days a Tier & Question Count on 3–5 4–6 5–7 6–8 15 6 a Mark Correct response a 1m 27 b b 2m Additional guidance 1 or 1m Shows or implies that the size of two steps is 4 eg • 4 –3 Shows steps of unequal size 4 ? 5 • –3 + 4 or U1 16 Shows or implies that the size of one step is 2 eg • The gaps are 2 • –3 + 2 • Second number is –1 • Fourth number is 3 • –3 to 5 is 8, 8 ÷ 4 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Shoe sizes 3–5 4–6 5–7 6–8 16 7 a Mark Correct response a 1m 12 b b 1m 3 c 1m Additional guidance Indicates Both the same and gives a correct explanation c The most common correct explanations: Use given values eg • Range of boys is 4, range of girls is 4 • 8 – 4 is the same as 9 – 5 • 5 to 9 = 4 to 8 Minimally acceptable explanation eg • 4, 4 • 8 – 4, 9 – 5 • Both 4 ! Ambiguous notation eg • 4 – 8, 5 – 9 Condone Reason generally about spread eg • Boys cover 5 sizes, girls cover 5 sizes Minimally acceptable explanation eg • Both have the same number of sizes ! Explanation implies references to the number of blank sizes eg • Boys have one blank, girls have one blank • Because the girls didn’t have size 9 and the boys didn’t have size 4 Condone U1 Sourced from SATs-Papers.co.uk Ambiguous or incorrect explanation eg • 5 in each • They both have a range of ﬁve sizes • Girls: 4, 5, 6, 7, 8 Boys: 5, 6, 7, 8, 9 http://www.SATs-Papers.co.uk 17 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Finding x and y 3–5 4–6 5–7 6–8 Mark Correct response 17 8 1m 652 1m Additional guidance 442 Tier & Question Seventy 3–5 4–6 5–7 6–8 18 9 Mark Correct response 1m Indicates No and gives a correct explanation that shows or implies at least one odd factor eg • Factors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70, so some are odd and some are even • There are four odd factors and four even factors of 70 • It could be 1 (odd) • 5 × 14 = 70 • 70 ÷ 2 = 35 • 70 is even, but 1 is odd and goes into everything U1 18 Sourced from SATs-Papers.co.uk Additional guidance Minimally acceptable explanation eg • 1, 2, 5, 7, 10, 14, 35 and 70 • 7 ! Incomplete list of factors given Condone, provided none is incorrect and at least one odd factor is shown eg, accept • The factors of 70 are 1, 2, 5 and 7 Incomplete or incorrect explanation eg • 70 has some odd and some even factors • 70 is a factor of 1 • All factors of 70 are odd http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Units 3–5 4–6 5–7 6–8 Mark Correct response 20 10 1 2m Additional guidance Completes all ﬁve rows of the table correctly, ie L A V M cm l miles g m2 oz or 1m Completes at least three rows of the table correctly Tier & Question Rainforest 3–5 4–6 5–7 6–8 19 11 2 a a Mark Correct response a 1m 27 b b b 1m 175 to 185 inclusive c 1m Additional guidance Indicates January, ie c c Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 19 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Doughnuts 3–5 4–6 5–7 6–8 21 12 3 Mark Correct response 2m Indicates shop A and gives a correct justiﬁcation, based on correctly calculating a pair of comparable values eg • At shop A: 2 × 5 = 10, at shop B: 3.5(0) × 3 = 10.5(0) • 3.5 × 3 – 2 × 5 = 0.5 • 2 ÷ 3 = 0.6(…), 3.50 ÷ 5 = 0.7 • For £1 you get 11 doughnuts or 13 doughnuts 2 7 • You pay £1.50 extra for 2 more doughnuts, but at shop A they’re less than 75p each so shop A must be a better deal or 1m Shows a correct pair of comparable values but makes either an incorrect or no decision or Additional guidance For 2m, no decision For 2m, correct decision and any pair of comparable values shown Note that common pairs (in pounds) are: 10 and 10.5(0) (per 15 doughnuts) 0.6(…) and 0.7(0) (per 1 doughnut) 2 and 2.1(0) (per 3 doughnuts) 3.3(…) and 3.5(0) (per 5 doughnuts) 1.5 and 1.4(…) (doughnuts per pound) ! For 2m or 1m, comparison is per 3 doughnuts or per 5 doughnuts but the given price is not restated Condone eg, for 2m accept • At shop B, 3 doughnuts would be £2.10 ! Additional incorrect working Ignore Shows a complete correct method for ﬁnding a pair of comparable values with not more than one computational or rounding error, and follows through to make their correct decision eg • 5 × 2, 3 × 3.50, shop A indicated • 2 ÷ 3 = 0.75 (error), 3.50 ÷ 5 = 0.7, shop B indicated or Makes a correct decision but the justiﬁcation uses only the difference between a pair of comparable values eg • A doughnut is 3.3(…)p cheaper at shop A 20 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Stopping distances 3–5 4–6 5–7 6–8 Mark Correct response 22 13 4 a a a 1m Draws a bar for the stopping distance for 40mph of length 9cm, ie Additional guidance ! Bar incorrectly positioned Condone if bar is drawn correctly one line above or below the position shown 20mph 40mph b b b 1m 18 ! Follow-through Allow follow-through as 2 × the length of their bar in (a), provided the result is greater than 12 Tier & Question Mark Correct response 23 14 5 Rotate 180 Marking overlay available 3–5 4–6 5–7 6–8 2m or 1m Draws the correct shape with all four vertices within the tolerances as shown on the overlay Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear Shows at least three vertices within the tolerances as shown on the overlay or Shows a correct shape in the correct orientation, with all four vertices within the tolerances as shown on the overlay, but in an incorrect position on the grid Tier & Question Value 3–5 4–6 5–7 6–8 24 15 6 Mark Correct response 1m 196 1m Incomplete processing 4 1m Additional guidance 1225 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 21 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question 12 cubes 3–5 4–6 5–7 6–8 25 16 7 Mark Correct response 2m Draws a 1 by 1 by 12, a 1 by 3 by 4 or a 2 by 2 by 3 cuboid, using the isometric grid eg • Additional guidance 1 by 2 by 6 cuboid repeated For 2m or 1m, internal lines omitted eg, for 2m accept • • • or 1m The only error is to omit some external lines or to show some hidden lines eg • • ! For 2m or 1m, hidden lines shown For 2m, accept provided they are clearly indicated as hidden lines eg, for 2m accept • ! Lines not ruled Accept provided the pupil’s intention is clear ! Drawing not accurate Accept vertices within 2mm of the dots of the grid Isometric grid not used correctly eg • or Correctly draws a possible 3-D shape made from 12 cubes that is not a cuboid, using the isometric grid eg ! Other shapes drawn As these could be trials, ignore • 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Cost of delivery 3–5 4–6 5–7 6–8 26 17 8 a a a Mark Correct response Gives both correct values correctly positioned, ie 7 then 20 1m b b b 1m Range given instead of 20 eg • 0 − 20 Gives a correct value with the correct unit for that value eg • 25p • 25 pence • £0.25 1m Additional guidance 18.25 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 23 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Shape area 3–5 4–6 5–7 6–8 27 18 9 a a a Mark Correct response 1m Gives a shape with area a + 2b, ie 16 small triangles eg • Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear Internal lines shown eg, for part (a) • • Shape formed with triangles and trapezium not joined side to side eg, for part (a) • • ! Other shapes drawn As these could be trials, ignore b b b 1m Gives a shape with area a – b, ie 4 small triangles eg • • U1 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Midpoints 3–5 4–6 5–7 6–8 19 10 1 Mark Correct response Additional guidance 1m Gives P as (30, 35) 1m Gives Q as (42, 0) ! Answers for P and Q transposed but otherwise completely correct If this is the only error, ie gives P as (42, 0) and gives Q as (30, 35), mark as 0, 1 1m Gives R as (42, 35) ! Follow-through for R as (their x coordinate of Q, their y coordinate of P) Allow follow-through provided their coordinates for P, Q and R are different U1 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 25 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Rainfall 3–5 4–6 5–7 6–8 20 11 2 Mark Correct response 2m or 1m Indicates place A and gives a correct justiﬁcation eg • 10 × 8 + 20 × 4 = 160cm 5 × 10 + 50 × 2 = 150cm • (80 + 80) ÷ 12 = 13.(…)cm per month (50 + 100) ÷ 12 = 12.5cm per month • (80 + 80) ÷ 2 = 80cm per 6 months (50 + 100) ÷ 2 = 75cm per 6 months Additional guidance For 2m, minimally acceptable justiﬁcation eg • 160, 150 seen • 80, 80 and 50, 100 seen • 10 × 8 + 20 × 4 > 5 × 10 + 50 × 2 • 13.(…), 12.5 seen Gives a correct justiﬁcation, even if the decision is incorrect or omitted or U1 Shows a complete correct method with not more than one computational error, and follows through to make their correct decision eg • 10 × 8 + 20 × 4 = 120 (error) 5 × 10 + 50 × 2 = 150, so place B Tier & Question Thinking distances 3–5 4–6 5–7 6–8 21 12 3 a a a Mark Correct response 2m 315 or 1m Additional guidance Shows the value 245 or Shows a complete correct method, in which the ‘squared’ has been correctly interpreted, with not more than one computational error eg × • 70 + 70 20 70 • 702 = 4900, 4900 ÷ 20 = 2450 (error), 70 + 2450 = 2520 b b b 26 1m 50 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Two shapes 3–5 4–6 5–7 6–8 22 13 4 Mark Correct response 2m 60 or 1m Additional guidance Shows the value 6 or U1 Shows a complete correct method with not more than one computational error eg • 72 ÷ 12 = 8 (error), 10 × 8 = 80 Tier & Question Recycling 3–5 4–6 5–7 6–8 23 14 5 Mark Correct response 2m 760 000 or 1m Additional guidance Shows the value 1 240 000 or Shows a complete correct method with not more than one error eg • 2 000 000 × 0.38 • 38 ÷ 100 × 2 × 1 000 000 • 2 million = 20 000 000 (error) 20 000 000 × 0.38 = 7 600 000 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 27 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Shapes on a grid 3–5 4–6 5–7 6–8 24 15 6 a a a Mark Correct response 1m Additional guidance Gives a correct explanation The most common correct explanations: Refer to the sum of the angles in a triangle eg • The angles are equal and add up to 180, so 180 ÷ 3 = 60 • Angles in a triangle add up to 180, the three angles are equal so 60 + 60 + 60 = 180 Minimally acceptable explanation eg • 180 ÷ 3 • 60 × 3 = 180 • The angles are the same and add up to 180 Incomplete explanation eg • The three angles add up to 180 • Angles in a triangle add up to 180 • The three angles are equal • 60 × 3 • It’s an equilateral triangle Refer to the sum of angles at a point eg • You can see that six of the triangles ﬁt together at a point, so 360 ÷ 6 = 60 • 60 60 60 60 60 60 Total: 360 b b b 2m or 1m Incomplete explanation eg • Six of the angles add up to 360 • Angles at a point add up to 360 • 60 × 6 Gives all three correct angles, ie a = 60, b = 120 and c = 240 Gives two correct angles U1 28 Minimally acceptable explanation eg • 360 ÷ 6 • 60 × 6 = 360 Sourced from SATs-Papers.co.uk ! For 1m, follow-through Provided their b is obtuse, accept c as 2 × their b or 360 – their b http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Five bags 3–5 4–6 5–7 6–8 25 16 7 Mark Correct response 2m Additional guidance ! For 2m or 1m, bag matched to more than one probability Do not accept as a correct match Matches all four bags to the correct probabilities, ie A B 1 3 C 1 2 D 5 11 E or 1m 1 4 2 5 Matches two of the four bags to the correct probabilities Tier & Question Bicycles 3–5 4–6 5–7 6–8 26 17 8 Mark Correct response 2m or 1m Additional guidance 184 Shows or implies a complete correct method with not more than one computational error eg • 46 × 8 2 • 46 × 4 • 46 ÷ 2 = 22 (error), 22 × 8 = 176 U1 • 2 is 46, so 4 is 82 (error), 8 8 so total is 164 • Digits 184 seen Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 29 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Eggs 3–5 4–6 5–7 6–8 18 9 Mark Correct response 2m or 1m Additional guidance Indicates the grade is medium and shows or implies a correct method for calculating the mass of the egg that interprets the ‘cubed’ correctly, even if a ﬁnal value is not shown eg • Value between 60 and 60.12 inclusive seen • 5.5 × 5.5 × 5.5 × π ÷ 10 × 1.15 • 166.375 × π × 0.115 Makes an incorrect or no decision about the grade of the egg, but shows or implies a correct method for calculating the mass of the egg that interprets the ‘cubed’ correctly, even if a ﬁnal value is not shown eg • 5.5 × 5.5 × 5.5 × π ÷ 10 × 1.15 • 522.7 ÷ 10 × 1.15 or Shows or implies a correct method for calculating the mass of the egg that interprets the ‘cubed’ correctly, with not more than one computational or rounding error, and makes their correct decision for the grade of the egg eg • 5.53 × 3 (error) ÷ 10 × 1.15 = 57.(…), so medium • 5.53 = 166 (premature rounding), 166 × 3.14 × 0.115 = 59.9(…), so medium 30 Sourced from SATs-Papers.co.uk For 1m, ﬁnal value and decision not shown within a method containing a computational or rounding error For 1m, conceptual error eg • 5.53 = 16.5, 16.5 × π ÷ 10 × 1.15 = 5.9(…) or 6, so small http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Ring size 3–5 4–6 5–7 6–8 19 10 a Mark Correct response Additional guidance a 1m 57 or 57.1(…) or 57.2 b b 2m Indicates size 6 and gives a correct justiﬁcation eg • 51 ÷ π = 16.2(…) • 51 ÷ 3.14 = 16.2(…) • 16.5 × π = 51.8(…) or 52 15.7 × π = 49.(…) • 16.5 × π = 51.8(…) and 15.7 × π < 51 For 2m, minimally acceptable justiﬁcation eg • 16.2(…) • 49.(…) and 51.8(…) or 52 seen Shows a correct justiﬁcation but makes an incorrect or no decision For 1m, incorrect or no justiﬁcation alongside a correct decision eg • 51 ÷ 3 = 17, so size 6 • Because the circumference of a size 6 is 51 or 1m or Indicates size 6 and gives an incomplete justiﬁcation eg • 51 ÷ π • 51 ÷ 3.14 • 51.8(…) or 52 • 49.(…) • 15.7 × π < 51 Sourced from SATs-Papers.co.uk For 2m, incomplete justiﬁcation eg • 51 ÷ π • 51 ÷ 3.14 • 16.5 × π = 51.8(…) http://www.SATs-Papers.co.uk 31 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Missing power 3–5 4–6 5–7 6–8 20 11 Mark Correct response 2m Shows correct working and gives the value of x as 3 eg • 35+ 102 = 343 7 × 7 × 7 = 343 • 35 = 243, 102 = 100 343 ÷ 7 = 49, 49 ÷ 7 = 7 • 343 7 Additional guidance ! Value embedded Accept provided there is no ambiguity and correct working is shown eg, for 2m accept • 73 shown in correct working eg, for 2m do not accept • 73 on the answer line, even with correct working 49 7 7 3 or 1m • 7 = 343 Gives the value of x as 3, even if working is incomplete or omitted or Shows the value 343 or Shows the values 243 and 100 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question School size 3–5 4–6 5–7 6–8 21 12 Mark Correct response 2m or 1m Shows that the average number of pupils in a secondary school is about four times as many as the average number in a primary school eg • Primary school: 4 069 385 ÷ 17 642 = 230.(…) (or 231) Secondary school: 3 315 805 ÷ 3385 = 979.(…) (or 980) 979 ÷ 230 = 4.2(…) • 17 642 ÷ 3 385 = 5.2(…) 4 069 385 ÷ 3 315 805 = 1.2(…) 5.2(…) ÷ 1.2(…) = 4.2(…) or 4.3 recurring Additional guidance For 2m, minimally acceptable justiﬁcation eg • 979.(…) ÷ 230.(…) • 980 ÷ 231 • 4 100 000 ÷ 18 000 ÷ (3 300 000 ÷ 3000) • 980, 4 × 230 = 920 • 231, 980 ÷ 4 = 245 Shows the values 230.(…) (or 231) and 979.(…) (or 980) or U1 Shows the intention to divide the total number of pupils by the number of schools for both categories using any reasonably rounded values eg • 4 069 385 ÷ 17 642, 3 315 805 ÷ 3385 • 4 100 000 ÷ 18 000, 3 300 000 ÷ 3000 • 4 000 000 ÷ 18 000, 3 000 000 ÷ 3000 Tier & Question Container 3–5 4–6 5–7 6–8 22 13 Mark Correct response 2m or 1m Additional guidance 15 Shows or implies a complete correct method with not more than one error eg • 12000 ÷ 800 • 12000 ÷ (40 × 20) • Shows the digits 15 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 33 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question nth term expressions 3–5 4–6 5–7 6–8 23 14 Mark Correct response 3m Additional guidance Completes all three rows of the table correctly, ie Expression … nth term expression? 5n No n + 11 No 11n – 6 Yes 38 Yes 32 2 n (6 – n) or 2m or 1m Unambiguous indication of ‘Yes’ or ‘No’ … 4th term Space for 4th term left blank for the expression n + 11 Completes two rows of the table correctly Completes either the row for the expression 11n – 6 or the row for the expression n2(6 – n) correctly or Completes the middle column of the table correctly, even if the right-hand column is incorrect or omitted 34 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Exam 3–5 4–6 5–7 6–8 24 15 Mark Correct response 2m or 1m 90 Additional guidance For 2m, reference to 100 marks eg • 90 out of 100 90 • 100 Shows or implies a complete correct method eg • Total mark on 6 units must be 80 × 6 = 480 Total so far = 78 × 5 = 390 480 – 390 • 80 × 6 – 78 × 5 • 80 – 78 = 2, 2 × 5 = 10, 80 + 10 or Shows the value 480 or 390 or Shows or implies a complete correct method with not more than one computational error eg • Total mark on 6 units must be 80 × 6 = 420 (error) 420 – 78 × 5 = 30 • 80 × 6 – 78 × 5 • 80 – 78 = 2, 2 × 5 = 10, 80 + 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 35 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Equations 3–5 4–6 5–7 6–8 25 16 a Mark Correct response a 1m b b 1m Additional guidance Indicates both correct equations, ie Gives two pairs of coordinates for which y = x + 1 and gives a correct equation eg • (3, 4) and (0, 1) y=x+1 • (1, 2) and (2, 3) x=y–1 • (–2, –1) and 1, 11 2 2 y–x=1 ! Unconventional notation eg, for y = x + 1 • y1 = 1 × x + 1 Condone Tier & Question House sales 3–5 4–6 5–7 6–8 26 17 a Mark Correct response 1m 75 000 b b 1m 33 1 3 c 36 a 1m Additional guidance 64 000 Sourced from SATs-Papers.co.uk ! Value rounded Accept 33 or better http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Standard form 3–5 4–6 5–7 6–8 18 Mark Correct response 1m Indicates 2 × 108 and 2.5 × 108, in either order Additional guidance Unambiguous indication eg, for part (a) • 200 000 000 and 250 000 000 Tier & Question Greater 3–5 4–6 5–7 6–8 19 Mark Correct response a 1m d, by 7 b 1m Additional guidance f, by 1 Tier & Question Three years old 3–5 4–6 5–7 6–8 20 Mark Correct response 2m or 1m Additional guidance 558 000 558 thousand Shows the value 557 551.(…) For 1m, conceptual error eg • 0.02 × 546 400 + 546 400 = 557 328 or Shows a complete correct method with not more than one computational or rounding error, even if their value is not rounded to the nearest thousand eg • 546 400 ÷ 98 × 100 • 550 000 (premature rounding) ÷ 0.98 = 561 224 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 37 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 6–8 only Tier & Question Height 3–5 4–6 5–7 6–8 21 Mark Correct response 2m 7.2 or 7.18(…) Additional guidance Method used is accurate or scale drawing ! For 2m, answer of 7 Do not accept unless a correct method or a more accurate value is seen or 1m ! Units given Ignore Shows or implies a correct trigonometric ratio eg • 17 × sin 25 h • 17 = sin 25 • 17 × cos 65 Tier & Question Fewest men 3–5 4–6 5–7 6–8 22 Mark Correct response 2m or 1m Additional guidance 115 Shows the digits 114(…) or 115 or Shows a complete correct method with not more than one computational or rounding error eg • 100 ÷ 87 × 100 • 100 ÷ 87 = 1.1 (premature rounding), 1.1 × 100 = 110 Tier & Question Daisies 3–5 4–6 5–7 6–8 23 Mark Correct response a 32 b 1m 7 c 38 1m 1m Additional guidance 25 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Using Pythagoras 3–5 4–6 5–7 6–8 24 Mark Correct response 2m or 1m U1 Additional guidance 20.8(…) ! Value of 20 or 21 Do not accept unless a correct method or a more accurate value is seen Shows or implies a correct method, using Pythagoras’ theorem, for calculating the length of the missing side of the right-angled triangle with a hypotenuse of 22cm eg • x2 = 222 – 202 • x = √84 • √(222 – 202) • √84 • 2√21 • 9.165 ! For 1m, value rounded or truncated Accept 9.1(…) or 9.2 Do not accept 9 unless a correct method or a more accurate value is seen Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 39 2009 KS3 Mathematics test mark scheme: Paper 2 Tiers 6–8 only Tier & Question Booklet 3–5 4–6 5–7 6–8 25 Mark Correct response 3m or 2m Additional guidance 60 Shows a correct value for the mass of the booklet eg • 59.8752 ! For 2m, value rounded or truncated Accept 59, 59.8(…) or 59.9 or or 1m 40 Shows or implies a correct method with not more than one error or omission, and follows through to give their value correct to 2 signiﬁcant ﬁgures, provided some rounding is required eg • 297 × 420 × 6 = 748 440, 748 440 × 80 = 59 875 200, so 60 000 000 [failure to convert to m2] • 297 × 420 ÷ 1000 (error) = 124.74, 124.74 × 6 × 80 = 59 875.2, so 60 000 [incorrect conversion to m2] • 0.297 × 0.42 × 80 = 10 [failure to ﬁnd mass of 6 pages] Shows or implies a correct method with not more than one error or omission, even if their value is not given correct to 2 signiﬁcant ﬁgures eg • 0.297 × 0.42 × 6 × 80 • 297 × 420 × 6 = 748 440, 748 440 × 80 (error) = 59 875 200 • 0.297 × 0.42 × 80 = 9.9(…) Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Hemisphere 3–5 4–6 5–7 6–8 26 Mark Correct response 2m Additional guidance Completes the table correctly with two fully simpliﬁed expressions, ie Radius Surface area r or 1m Volume 2 πr 3 3 3πr 2 Gives one correct and fully simpliﬁed expression or Gives both correct, unsimpliﬁed expressions eg • Radius r Volume 4 πr 3 ÷ 3 2 Surface area 2πr 2 + πr 2 • Radius Volume Surface area r 4 πr 3 6 3πr × r Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 41 2009 KS3 Mathematics test mark scheme: Paper 2 Index Index to mark schemes Tier 3–5 4 –6 Question 5–7 Page 6–8 1 School shop 10 2 Missing numbers 10 3 Parcels 10 4 Joining 11 5 Spinner 12 6 Fractions 12 7 Number of sides 13 8 Grid 13 9 Digital 14 10 Reﬂecting 14 12 2 Test 15 13 3 Rounding 15 11 4 Castle 15 14 5 Baby 16 15 6 Count on 16 16 7 Shoe sizes 17 17 8 Finding x and y 18 18 9 Seventy 18 20 10 1 Units 19 19 11 2 Rainforest 19 21 12 3 Doughnuts 20 22 13 4 Stopping distances 21 23 14 5 Rotate 180 21 24 15 6 Value 21 25 16 7 12 cubes 22 26 17 8 Cost of delivery 23 27 42 1 18 9 Shape area 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2009 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 Index Question Page 4 –6 5–7 6–8 19 10 1 Midpoints 25 20 11 2 Rainfall 26 21 12 3 Thinking distances 26 22 13 4 Two shapes 27 23 14 5 Recycling 27 24 15 6 Shapes on a grid 28 25 16 7 Five bags 29 26 17 8 Bicycles 29 18 9 Eggs 30 19 10 Ring size 31 20 11 Missing power 32 21 12 School size 33 22 13 Container 33 23 14 n th term expressions 34 24 15 Exam 35 25 16 Equations 36 26 17 House sales 36 18 Standard form 37 19 Greater 37 20 Three years old 37 21 Height 38 22 Fewest men 38 23 Daisies 38 24 Using Pythagoras 39 25 Booklet 40 26 Hemisphere 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.uk290016