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Ma Mathematics tests KEY STAGE 3 for Paper 1 Tiers 3–5, 4–6, 5–7 and 6–8 2007 ALL TIERS Mark scheme National curriculum assessments Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 1 http://www.SATs-Papers.co.uk 19/12/06 2:32:09 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Introduction Introduction The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to inform teachers. This booklet contains the mark scheme for paper 1 at all tiers. The paper 2 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identiﬁer irrespective of tier. The structure of the mark schemes The marking information for questions is set out in the form of tables, which start on page 11 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part and the total number of marks available for that question part. The Correct response column usually includes two types of information: a statement of the requirements for the award of each mark, with an indication of whether credit can be given for correct working, and whether the marks are independent or cumulative examples of some different types of correct response, including the most common. The Additional guidance column indicates alternative acceptable responses, and provides details of speciﬁc types of response that are unacceptable. Other guidance, such as when ‘follow through’ is allowed, is provided as necessary. Questions with a UAM element are identiﬁed in the mark scheme by an encircled U with a number that indicates the signiﬁcance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet. The 2007 key stage 3 mathematics tests and mark schemes were developed by the Test Development team at Edexcel. 2 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 2 http://www.SATs-Papers.co.uk 19/12/06 2:32:09 pm 2007 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, algebra, time, coordinates or probability. Unless otherwise speciﬁed in the mark scheme, markers should apply the following guidelines in all cases. Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 3 http://www.SATs-Papers.co.uk 3 19/12/06 2:32:09 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance What if … The pupil’s response does not match closely any of the examples given. Markers should use their judgement in deciding whether the response corresponds with the statement of requirements given in the Correct response column. Refer also to the Additional guidance. The pupil has responded in a non-standard way. Calculations, formulae and written responses do not have to be set out in any particular format. Pupils may provide evidence in any form as long as its meaning can be understood. Diagrams, symbols or words are acceptable for explanations or for indicating a response. Any correct method of setting out working, however idiosyncratic, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma for a decimal point. The pupil has made a conceptual error. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, error. A computational error is a ‘slip’ such as writing 4 × 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 pupil’s accuracy is marginal according to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, or touches, the boundaries given, the mark(s) should be awarded. The pupil’s answer correctly follows through from earlier incorrect work. Follow through marks may be awarded only when speciﬁcally stated in the mark scheme, but should not be allowed if the difﬁculty level of the question has been lowered. Either the correct response or an acceptable follow through response should be marked as correct. There appears to be a misreading affecting the working. The correct answer is in the wrong place. 4 This is when the pupil misreads the information given in the question and uses different information. If the original intention or difﬁculty level of the question is not reduced, deduct one mark only. If the original intention or difﬁculty level is reduced, do not award any marks for the question part. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question. Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 4 http://www.SATs-Papers.co.uk 19/12/06 2:32:10 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance What if … Marking procedure 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 If so, award the mark. in questions not testing accuracy, the correct answer has been given but then rounded or truncated If so, award the mark. the pupil has continued to give redundant extra working which does not contradict work already done 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. If so, do not award the mark. Where a question part carries more than one mark, only the ﬁnal mark should be withheld. The pupil’s answer is correct but the wrong working is seen. A correct response should always be marked as correct unless the mark scheme states otherwise. The correct response has been crossed or rubbed out and not replaced. Mark, according to the mark scheme, any legible crossed or rubbed out work that has not been replaced. More than one answer is given. The answer is correct but, in a later part of the question, the pupil has contradicted this response. Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 5 If all answers given are correct or a range of answers is given, all of which are correct, the mark should be awarded unless prohibited by the mark scheme. If both correct and incorrect responses are given, no mark should be awarded. A mark given for one part should not be disallowed for working or answers given in a different part, unless the mark scheme speciﬁcally states otherwise. http://www.SATs-Papers.co.uk 5 19/12/06 2:32:10 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance Marking speciﬁc types of question Responses involving money For example: £ 3.20 £7 Accept Do not accept Any unambiguous indication of the correct amount eg £ 3.20(p), £ 3 20, £3,20, 3 pounds 20, £3-20, £ 3 20 pence, £3:20, £ 7.00 Incorrect or ambiguous indication of the amount eg £320, £320p or £700p The unit, £ or p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units eg with £ given in the answer space, accept 3.20 7 or 7.00 Ambiguous use of units outside the answer space eg with £ given in the answer space, do not accept 3.20p outside the answer space Given units amended eg with £ crossed out in the answer space, accept 320p 700p Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 eg £3.2, £3 200, £32 0, £3-2-0 £7.0 Responses involving negative numbers For example: –2 Accept Do not accept To avoid penalising the error below more than once within each question, do not award the mark for the ﬁrst occurrence of the error within each question. Where a question part carries more than one mark, only the ﬁnal mark should be withheld. Incorrect notation eg 2 – 6 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 6 http://www.SATs-Papers.co.uk 19/12/06 2:32:10 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving the use of algebra For example: 2+n n+2 2n n n2 2 Take care ! Do not accept Accept Unambiguous use of a different case or variable eg N used for n x used for n ! Unconventional notation eg n × 2 or 2 × n or n2 or n + n for 2n n × n for n2 n ÷ 2 for n or 1 n 2 2 2 + 1n for 2 + n 2 + 0n for 2 Within a question that demands simpliﬁcation, do not accept as part of a ﬁnal answer involving algebra. Accept within a method when awarding partial credit, or within an explanation or general working. Embedded values given when solving equations eg in solving 3x + 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 Unambiguous letters used to indicate expressions eg t = n + 2 for n + 2 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 7 ! 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. Ambiguous letters used to indicate expressions eg n = n + 2 for n + 2 http://www.SATs-Papers.co.uk 7 19/12/06 2:32:11 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving time A time interval For example: 2 hours 30 minutes Accept Take care ! Do not accept Any unambiguous indication eg 2.5 (hours), 2h 30 Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min ! Digital electronic time ie 2:30 A speciﬁc time For example: 8:40am The unit, hours and/or minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, or crosses out the given unit, accept answers with correct units, unless the question has speciﬁcally asked for other units to be used. 17:20 Accept Do not accept Any unambiguous, correct indication eg 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Incorrect time eg 8.4am, 8.40pm Unambiguous change to 12 or 24 hour clock eg 17:20 as 5:20pm, 17:20pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0 eg 840, 8:4:0, 084, 84 Responses involving coordinates For example: ( 5, 7 ) Accept Do not accept Unconventional notation eg ( 05, 07 ) ( ﬁve, seven ) Incorrect or ambiguous notation eg ( 7, 5 ) x y ( 5, 7 ) ( x = 5, y = 7 ) 8 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 8 y x ( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x – 5, y – 7 ) http://www.SATs-Papers.co.uk 19/12/06 2:32:11 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction or percentage only. 7 For example: 0.7 70% 10 Accept Take care ! Do not accept Equivalent decimals, fractions and percentages 70 35 , 70.0% eg 0.700, , 100 50 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 correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 70 = 18 eg 100 25 A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 9 http://www.SATs-Papers.co.uk 9 19/12/06 2:32:12 pm 2007 KS3 Mathematics test mark scheme: Paper 1 General guidance Recording marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1 0 The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper. A total of 120 marks is available in each of tiers 3–5, 4–6, 5–7 and 6–8. 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 Monday 25 June 2007. QCA will also send a copy to each school in July. Schools will be notiﬁed of pupils’ results by means of a marksheet, which will be returned to schools by the external marking agency with the pupils’ marked scripts. The marksheet will include pupils’ scores on the test papers and the levels awarded. 10 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 10 http://www.SATs-Papers.co.uk 19/12/06 2:32:12 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 only Tier & Question Euros 3-5 4-6 5-7 6-8 1 Correct response 1m 305 1m 1005 1m 1030 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 11 Additional guidance http://www.SATs-Papers.co.uk 11 19/12/06 2:32:12 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 3–5 only Number line 3-5 4-6 5-7 6-8 2 Correct response 2m Additional guidance Gives all three correct values in the correct positions, ie 1 –2 –5 or 1m Gives at least two correct values in the correct positions Tier & Question ! For 1m, follow through Accept as their previous incorrect value – 3, provided their previous incorrect value < 3 eg, for 1m accept 1 –3 (error) –6 2 (error) –1 –4 36 times table 3-5 4-6 5-7 6-8 3 Correct response 1m 36 1m 5 1m Additional guidance 396 U1 12 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 12 http://www.SATs-Papers.co.uk 19/12/06 2:32:12 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 3–5 only Feeding times 3-5 4-6 5-7 6-8 4 Correct response a 1m 12:35(pm) Additional guidance ! Throughout the question, omission of pm Condone eg, for part (a) accept 12:35 Throughout the question, incorrect times eg, for part (a) 12:35am Unambiguous indication eg T b 1m Tigers c 1m 30 d 1m Gives three correct times, ie Elephants at Otters at Seals at Use of the 24 hour clock 2:15(pm) 1:00(pm) 4:00(pm) or Elephants at Otters at Seals at 3:20(pm) 1:00(pm) 4:00(pm) U1 Tier & Question Work out 3-5 4-6 5-7 6-8 5 Correct response 1m 121 1m 7 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 13 Additional guidance http://www.SATs-Papers.co.uk 13 19/12/06 2:32:13 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 3–5 only Dollars 3-5 4-6 5-7 6-8 6 Correct response a 1m 12 b 1m 5 c 1m Additional guidance 40 Tier & Question Divisibility 3-5 4-6 5-7 6-8 7 Correct response a 1m Indicates only 15, 20 and 30, ie b 1m Indicates only 12, 15 and 30, ie c 1m Indicates only 15 and 30, ie Additional guidance 14 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 14 ! Follow through For part (c), accept follow through as all values indicated that were also indicated in both parts (a) and (b), provided this is not all or none of the boxes http://www.SATs-Papers.co.uk 19/12/06 2:32:13 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6 Populations 3-5 4-6 5-7 6-8 8 1 Correct response a a 1m Harrogate b b 2m Additional guidance Unambiguous indication eg, for part (a) H Completes all three of the sentences correctly, ie Ash Vale London 100 or 1m Completes two of the sentences correctly Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 15 http://www.SATs-Papers.co.uk 15 19/12/06 2:32:14 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Number grids 3-5 4-6 5-7 6-8 9 2 Correct response 3m Additional guidance Completes all three grids correctly, ie 11 4 12 7 9 3 28 27 10 10 6 4 or 4 24 6 24 or 2m Completes the ﬁrst two grids correctly or Completes the third grid correctly and gives any two correct entries in the ﬁrst two grids or Completes the third grid correctly, gives any one correct entry in the ﬁrst grid, makes an error in the right hand entry of the second grid, but follows through correctly to give their product or 1m Gives any two correct entries in the ﬁrst two grids or Completes the third grid correctly or Gives any one correct entry in the ﬁrst grid, makes an error in the right hand entry of the second grid, but follows through correctly to give their product U1 16 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 16 http://www.SATs-Papers.co.uk 19/12/06 2:32:14 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6 Tier & Question Babies 3-5 4-6 5-7 6-8 10 3 Correct response a a Additional guidance 1m 1m Gives a value greater than 52.5, but less than 53.5 ! The two values in part (a) transposed but otherwise correct Mark as 0, 1 1m b b Gives a value greater than 62.5, but less than 63.5 Indicates the correct point at (34, 72) ! Inaccurate indication Accept any unambiguous indication within 2mm of the correct intersection of the grid Tier & Question Cards for fractions 3-5 4-6 5-7 6-8 11 4 Correct response Additional guidance 2 4 or 6 12 a a 1m Gives either b b 1m Gives a correct fraction between 1 and 1, ie 2 10 4 6 6 8 8 , , , , or 12 6 8 10 10 12 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 17 http://www.SATs-Papers.co.uk 17 19/12/06 2:32:14 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6 Pentagon perimeter 3-5 4-6 5-7 6-8 12 5 Correct response 1m Shows or implies that the length of one side is 5.4cm 0.2cm eg 5.4 0.2 seen 54 2 seen Answer of 26, 26.5, 27, 27.5 or 28 with no evidence of incorrect working Answer of 260, 265, 270, 275 or 280 with no evidence of incorrect working Additional guidance Markers may ﬁnd the following values useful: Side length 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Perimeter 25.5 26 26.5 27 27.5 28 28.5 For the ﬁrst mark, more than one value for the side length is given 1m 1m 18 Shows or implies a correct method for ﬁnding their perimeter eg Intention to multiply their side length by 5 Intention to add all 5 of their side lengths Answer of 26, 26.5, 27, 27.5 or 28 with no evidence of incorrect working Answer of 260, 265, 270, 275 or 280 with no evidence of incorrect working 5.4 seen, answer of 25.20 (error) 5.4 seen, answer of 45 (error) 5 (error) seen, answer of 25 For the second mark, follow through from their side length(s) Gives the correct value for their perimeter eg Answer of 26, 26.5, 27, 27.5 or 28 with no evidence of incorrect working Answer of 260, 265, 270, 275 or 280 with units amended to mm, and no evidence of incorrect working ! For the third mark, follow through from their side length(s) Accept follow through provided their calculation involves ‘bridging the units’ eg, from a side length of 5.7 accept 28.5 eg, from a side length of 5.1 do not accept 25.5 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 18 For the second mark, response does not show or imply an unambiguous or clear method for ﬁnding their perimeter http://www.SATs-Papers.co.uk 19/12/06 2:32:15 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6 Multiple and factor 3-5 4-6 5-7 6-8 13 6 Correct response a a 1m Gives two different 3-digit multiples of 4 eg 100 104 132 236 b b 1m Gives a 2-digit factor of 100, ie Additional guidance 10, 20, 25 or 50 1m Gives a different 2-digit factor of 100 from any credited for the ﬁrst mark of part (b) ! Factors of 100 given that do not have 2 digits eg, for both marks in part (b) 100 1 2 5 Mark as 0, 1 ! 2-digit factors of 100 given within factor pairs eg, for both marks in part (b) 10 × 10 20 × 5 25 × 4 50 × 2 Mark as 0, 1 Negative factors Tier & Question Same number 3-5 4-6 5-7 6-8 14 7 Correct response a a 1m 7.5 b b 1m Additional guidance Gives the value 1.5 in each of the three boxes Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 19 Equivalent fractions and decimals http://www.SATs-Papers.co.uk 19 19/12/06 2:32:15 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 Squares 3-5 4-6 5-7 6-8 15 8 Correct response 1m Indicates No and gives a correct explanation eg A rectangle is four-sided and also has four right angles Additional guidance Minimally acceptable explanation eg Rectangle ! Response names rectangles, but also implies there are other examples Condone provided there are no other incorrect examples named eg, accept Many shapes have four right angles, for example a rectangle eg, do not accept A rhombus and a rectangle must have four right angles Incomplete explanation eg There is another four-sided shape with four right angles U1 Tier & Question x=8 3-5 4-6 5-7 6-8 16 9 1 Correct response a a a 1m Indicates only 40, ie b b b 1m Indicates only 16, ie c 1m Additional guidance Indicates only 64, ie 20 c c Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 20 http://www.SATs-Papers.co.uk 19/12/06 2:32:16 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Grid 3-5 4-6 5-7 6-8 17 10 2 Correct response 3m Additional guidance Gives all six correct values in the table, then gives the correct answer to the multiplication, ie × 100 40 3 30 3000 1200 90 6 600 240 18 Answer: 5148 or 2m Gives at least four correct values in the table, even if the answer to the multiplication is incorrect or omitted or Gives at least three correct values in the table, then follows through to add correctly their six values that would give the answer to the multiplication or Gives the answer to the multiplication as 5148, even if the grid has not been used or has incorrect entries or 1m Gives at least three correct values in the table, even if the answer to the multiplication is incorrect or omitted or Gives at least two correct values in the table, then follows through to add correctly their six values that would give the answer to the multiplication U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 21 http://www.SATs-Papers.co.uk 21 19/12/06 2:32:16 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 3–5, 4–6, 5–7 Tier & Question Sweets 3-5 4-6 5-7 6-8 18 11 3 Correct response a a a 1m 1 or equivalent probability 20 b b b 1m Indicates green Additional guidance Unambiguous indication of colour eg G Tier & Question Completing calculations 3-5 4-6 5-7 6-8 19 12 4 Correct response 1m 1m 22 Gives any two values that add to give –8 eg –4 + –4 5 + –13 0 + –8 Fractions, decimals or negatives Given operation changed eg, for the ﬁrst mark 0–8 Gives any two values that subtract to give –8, in the correct order eg 2 – 10 –5 – 3 –12 – –4 0–8 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 22 Additional guidance http://www.SATs-Papers.co.uk 19/12/06 2:32:16 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 Rectangles 3-5 4-6 5-7 6-8 20 13 5 Correct response 2m Gives three different pairs of positive numbers with a product of 24 eg 1 24 2 12 3 8 4 1.5 5 Additional guidance Two correct values repeated in reverse order 6 16 4.8 or 1m Gives two different pairs of positive numbers with a product of 24 Tier & Question Percentages 3-5 4-6 5-7 6-8 21 14 6 a a a Correct response 2m Gives all three correct values, ie 40 4 0.8 Additional guidance Throughout the question, equivalent fractions and decimals ! Throughout the question, incorrect use of percentage sign Condone or 1m b b b Gives any two correct values ! For 1m, follow through Accept follow through as their 1st value ÷ 10 and/or their 2nd value ÷ 5 eg, for 1m accept 30 (error) 3 0.6 40 0.4 (error) 0.08 1m 44.8 ! Follow through as the sum of their three values from part (a) Accept provided the sum is less than 80 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 23 http://www.SATs-Papers.co.uk 23 19/12/06 2:32:17 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 3–5, 4–6, 5–7 Positive and negative 3-5 4-6 5-7 6-8 22 15 7 Correct response a a a 1m 18 b b b 1m 2 c 1m Additional guidance Indicates the equation y = x2, ie c c Tier & Question Barrels 3-5 4-6 5-7 6-8 23 16 8 Correct response 2m Gives all three correct fractions in their simplest form, ie 1 4 1 3 2 3 Additional guidance Equivalent decimals or 1m Gives any two correct fractions in their simplest form or Shows a correct ﬁrst step of simpliﬁcation for all three correct fractions, even if there is incorrect subsequent simpliﬁcation eg 3 12 9 27 18 27 24 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 24 ! For 1m, follow through Accept their third fraction as their second fraction × 2, provided all three fractions are given in their simplest form eg, for 1m accept 1 4 1 (error) 4 1 2 http://www.SATs-Papers.co.uk 19/12/06 2:32:17 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Speed Marking overlay available 3-5 4-6 5-7 6-8 17 9 1 a a a 1m b b b 1m Correct response Draws a straight line on the graph joining the points (0, 0) and (60, 30) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 30 km/hour Draws a straight line on the graph joining the points (0, 0) and (30, 60) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 120 km/hour Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 25 Additional guidance Unambiguous labelling eg, for 30 km/hour 30 ! Labels omitted or incorrect For two correct lines of full length with labels omitted, mark as 0, 1 Do not accept incorrect labels ! Lines not of full length For two correct lines at least 5cm long but not of full length, mark as 0, 1 Do not accept lines less than 5cm long http://www.SATs-Papers.co.uk 25 19/12/06 2:32:18 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Grey and black designs 3-5 4-6 5-7 6-8 18 10 2 Correct response Additional guidance a a a 1m 25 Equivalent fractions or decimals b b b 2m 3:2 For 2m, correct ratio given in the form n : 1 or 1 : n eg 1.5 : 1 2 1: 3 or 1m Gives the ratio 3 : 2 but includes words, letters or symbols eg 3 grey : 2 black g=3:b=2 ! For 1m, incorrect use of percentage sign Condone only within the ratio 3 : 2, ie for 1m accept 3% : 2% or Shows a correct ratio even if not in its simplest form, or there is incorrect further working eg 60 : 40 6:4 1.5 : 1 2 1: 3 or Gives the ratio 2 : 3 26 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 26 http://www.SATs-Papers.co.uk 19/12/06 2:32:18 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Counter probabilities 3-5 4-6 5-7 6-8 19 11 3 Correct response a a a 2m Additional guidance Completes the table with the three correct values in the correct positions, ie Colour of counters Number of counters Probability Red 6 2 5 Blue 3 6 Incorrect notation eg 1 2.5 1 5 Green Equivalent probabilities 2 5 or 1m Gives at least one correct value in the correct position U1 b b b 1m Indicates that the probability has decreased, ie U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 27 http://www.SATs-Papers.co.uk 27 19/12/06 2:32:18 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Three straight lines 3-5 4-6 5-7 6-8 20 12 4 Correct response 1m Additional guidance Gives a = 50 and gives a correct reason eg Angle a is on a straight line with 130, so a = 180 – 130 a is supplementary with 130, so a + 130 = 180 The angle vertically opposite 130 is 130, 360 – (130 + 130) = 100, (angles at a point) 100 a is = 50 (also vertically opposite) 2 Minimally acceptable reason eg On a straight line Supplementary Opposite angles and angles at a point Informal reason without the correct geometrical property identiﬁed eg 180 – 130 360 – 260 2 Incomplete reason eg It is adjacent to the 130° angle U1 1m Gives b = 60 and gives a correct reason eg Angle b is vertically opposite the 60° angle, so it is also 60° The angle on a straight line with b is 120, so b is 360 – 120 – 120 – 60 (angles at a point) Minimally acceptable reason eg Opposite Angles on a straight line and angles at a point Informal reason without the correct geometrical property identiﬁed eg b is equal to the 60° angle next to it Incomplete reason eg It is the same as the 60° angle U1 28 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 28 http://www.SATs-Papers.co.uk 19/12/06 2:32:19 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Three straight lines (cont) 3-5 4-6 5-7 6-8 20 12 4 Correct response 1m Additional guidance Gives c = 70 and gives a correct reason eg There are 180° in a triangle, so c = 180 – 50 – 60 The exterior angle of a triangle is equal to the sum of the two opposite interior angles, so c = 130 – 60 Minimally acceptable reason eg Angles in a triangle Exterior angle = sum of two opposite interior angles We’ve already got 50 and 60 in the triangle ! Follow through Accept as 180 – (their a + b), alongside a correct reason referring to angles in a triangle, or as 130 – their b alongside a correct reason referring to an exterior angle of a triangle Informal reason without the correct geometrical property identiﬁed eg 180 – (a + b) 130 – b Incomplete reason eg It is in a triangle All the inside angles add up to 180° U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 29 http://www.SATs-Papers.co.uk 29 19/12/06 2:32:19 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 4–6, 5–7, 6–8 Tier & Question Ninths 3-5 4-6 5-7 6-8 21 13 5 Correct response a a a 1m b b b 1m Indicates 11%, ie c 2 9 Additional guidance 1 1 and Indicates , ie 100 10 c c 1m Equivalent fractions Equivalent decimals or percentages Incorrect notation eg 1 4.5 Tier & Question Equation 3-5 4-6 5-7 6-8 22 14 6 Correct response 2m Additional guidance 1 or equivalent 2 or 1m 30 Shows or implies a correct ﬁrst step of algebraic manipulation that removes the brackets eg 2 × 2n + 2 × 5 = 12 4n + 10 = 12 2n + 5 = 6 4n = 2 2n = 1 2÷4 1÷2 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 30 http://www.SATs-Papers.co.uk 19/12/06 2:32:20 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Circle working 3-5 4-6 5-7 6-8 23 15 7 Correct response 1m Additional guidance Gives a correct explanation The most common correct explanations: Show the correct working eg It should be × 16 not × 8 Needs to be × radius2, not × diameter Minimally acceptable explanation eg 16 42 4×4 r2 r2 Incomplete explanation eg The 8 is wrong Address the misconception eg He is ﬁnding the circumference not the area He is using 2 r, not r2 He has done 4 × 2 instead of 42 Minimally acceptable explanation eg Circumference It’s not 2 r [or d] He didn’t square the 4 He didn’t square the radius ! Use of ‘perimeter’ for ‘circumference’ Condone Incomplete explanation eg He used the wrong formula He used the diameter He hasn’t used the radius He doubled the radius Show that his working gives an incorrect answer eg He gets 25.(…), but it should be 50.(…) His answer is half as big as it should be Minimally acceptable explanation eg 50, not 25 It should be his answer × 2 Incomplete explanation eg 50 His answer is too small U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 31 http://www.SATs-Papers.co.uk 31 19/12/06 2:32:20 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 4–6, 5–7, 6–8 Thinking fractions 3-5 4-6 5-7 6-8 24 16 8 Correct response 1m 6 1m Additional guidance 12 Tier & Question Cube 3-5 4-6 5-7 6-8 25 17 9 Correct response a 1m 54x2 b 2m Gives a correct, simpliﬁed expression, ie 27x3 or (3x)3 Additional guidance Unsimpliﬁed expression or unconventional notation eg 9x2 × 6 9x2 + 9x2 + 9x2 + 9x2 + 9x2 + 9x2 54xx ! Unsimpliﬁed expression or unconventional notation eg 33 × x3 27 × x × x × x For 2m, do not accept For 1m, condone or 1m 32 Shows or implies a correct method for ﬁnding the volume of the cube with not more than one error eg 3x × 3x × 3x 9x2 × 3x 9 × 3 × x3 ( 9x2 )3 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 32 Conceptual error eg 9x2 = 9x (error) 9x × 9x × 9x = 729x3 http://www.SATs-Papers.co.uk 19/12/06 2:32:21 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 5–7, 6–8 Random numbers 3-5 4-6 5-7 6-8 18 10 Correct response Additional guidance a a 1m 98 Key not interpreted eg 9|8 b b 1m Gives a correct explanation that refers to the ordering of some or all of the numbers eg He could ﬁnd the row with the 28th number in it, then put that row in order to get the right one If the row for the 50s is put in order, it is easier He could put the numbers in each row in order of size Minimally acceptable explanation eg Order the 5 row The 6th line needs to go in order Put it in order Incomplete or incorrect explanation eg He should rearrange the diagram He needs to put the middle row in order ! Explanation refers to producing a cumulative frequency chart As this is not the simplest method available, do not accept without a complete explanation of method U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 33 http://www.SATs-Papers.co.uk 33 19/12/06 2:32:21 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 5–7, 6–8 Geometric mean 3-5 4-6 5-7 6-8 19 11 Correct response a a 1m 90 b b 1m Indicates Yes and gives a correct explanation eg –2 × 8 = –16, but you can’t ﬁnd the square root of a negative number Multiplying negative by positive gives a negative, but negative is impossible The numbers must be both positive or both negative for a positive product, as you can’t ﬁnd the square root of a minus number Additional guidance ! Indicates No Accept provided their explanation refers to imaginary numbers eg, accept You can use an imaginary number 4i eg, do not accept –4 Minimally acceptable explanation eg You can’t ﬁnd the square root of a minus number You can’t ﬁnd the square root of –16 Incomplete or incorrect explanation eg You can’t ﬁnd the square root When you do –16 the calculator would say ‘error’ –2 × 8 = –16, which is not a square number It doesn’t work with negative numbers The numbers must be both positive or both negative –2 × 8 = –18, but then you can’t do –18 U1 34 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 34 http://www.SATs-Papers.co.uk 19/12/06 2:32:21 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Sequences 3-5 4-6 5-7 6-8 20 12 Correct response a a 2m Additional guidance Matches all four nth term rules correctly, ie 4n 4, 7, 12, 19, … ( n + 1)2 4, 8, 12, 16, … n2 + 3 4, 9, 16, 25, … n ( n + 3) 4, 10, 18, 28, ! Rule matched to more than one sequence For 2m or 1m, do not accept as a correct match … or 1m b b 2m Matches at least two nth term rules correctly 4, 11, 30 and 67, in the correct order or 1m Gives at least two of the four correct terms, even if their positions are incorrect or Shows the values 1, 8, 27 and 64 or Shows a complete correct method for all four terms provided the ‘cubed’ has been interpreted, even if there is further incorrect working eg 1×1×1+3 2×2×2+3 3×3×3+3 4×4×4+3 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 35 http://www.SATs-Papers.co.uk 35 19/12/06 2:32:22 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 5–7, 6–8 Rhombus 3-5 4-6 5-7 6-8 21 13 Correct response 2m Additional guidance 145 or 1m Shows the value 35 or Shows a complete correct method with not more than one computational error eg 180 – 110 = 70, 180 – 70 ÷ 2 (540 – 110 – 70 – 70) ÷ 2 180 – 110 = 70 70 ÷ 2 = 25 (error) 180 – 25 = 155 Tier & Question Counters 3-5 4-6 5-7 6-8 22 14 Correct response 2m Additional guidance 5 or 1m Shows or implies that half the counters are green or that half are red or black eg 1 1 2 1 1– – =1– – 3 6 6 6 3 = 6 3 Green is 6 1 1 1 + = 3 6 2 Total = 30 Red + black = 15 or Shows or implies a complete correct method with not more than one computational error eg 15 × 2 ÷ 6 15 ÷ 3 U1 36 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 36 http://www.SATs-Papers.co.uk 19/12/06 2:32:22 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Marking overlay available Tiers 5–7, 6–8 Fence plan 3-5 4-6 5-7 6-8 23 15 Correct response 2m Completes the perpendicular bisector, fulﬁlling the four conditions below: 1. Ruled 2. Within the tolerance as shown on the overlay 3. Touching or crossing both roads 4. Evidence of correct construction arcs that are centred on A and B, are of equal radii, and show at least one intersection Additional guidance ! Use of construction arcs on the overlay Note that these are to give a visual guide as to whether the correct centres have been used, and do not indicate tolerance Spurious construction arcs Do not accept arcs drawn without compasses or arcs that do not show a distinct intersection, eg arcs that just touch ! Perpendicular bisector is not a solid line Condone provided the pupil’s intention is clear or 1m Completes the perpendicular bisector with all of conditions 1 to 3 fulﬁlled or Fulﬁls condition 4, even if the perpendicular bisector is incorrect or omitted Tier & Question Powers 3-5 4-6 5-7 6-8 24 16 Correct response 1m 12 1m 4 Additional guidance ! Answer embedded in working Accept provided there is no ambiguity and any statements made are correct eg, for the ﬁrst mark accept 512 [shown in working] Otherwise, penalise only the ﬁrst occurrence eg, for the ﬁrst and second marks m = 512 n = 54 Mark as 0, 1 ! Incomplete processing Penalise only the ﬁrst occurrence eg, for the ﬁrst and second marks 8+4 8–4 Mark as 0, 1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 37 http://www.SATs-Papers.co.uk 37 19/12/06 2:32:23 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tiers 5–7, 6–8 Dissection 3-5 4-6 5-7 6-8 25 17 Correct response 2m Additional guidance Gives both correct pairs of dimensions eg A: 6cm by 8cm (either order) B: 2cm by 8cm (either order) or 1m Gives one correct pair of dimensions, even if assigned to the incorrect rectangle eg 6cm by 8cm (either order) seen 2cm by 8cm (either order) seen or Shows or implies that the ratio of the shorter side of A to the shorter side of B is also 3 : 1, even if there is incomplete processing or other incorrect working eg 6, 2 seen as shorter sides 48 ÷ 8, 16 ÷ 8 3 × 8x = 8y Answer of A: 3cm by 4cm B: 1cm by 4cm The side of the square is 6cm (error), so A is 4.5cm wide and B is 1.5cm wide 38 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 38 48, 16 seen without further processing http://www.SATs-Papers.co.uk 19/12/06 2:32:23 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tiers 5–7, 6–8 Tier & Question Coins 3-5 4-6 5-7 6-8 26 18 Correct response 2m Additional guidance Gives the correct set of eight coins, ie 1p, 1p, 2p, 2p, 2p, 2p, 5p, 10p in any order or 1m Shows the set of four coins 1p, 2p, 2p, 20p or Shows evidence of understanding that the total number of coins must be 4x and the number of 1p coins must be x, where x 1 eg 1p, 1p, 2p, 2p, 2p, 5p, 5p, 10p It could be three 1p coins and twelve coins altogether For every four coins there is one 1p and three that are not 1p For 1m, number of coins is not a multiple of 4, even if total value is 25p eg 1p, 1p, 2p, 2p, 2p, 2p, 5p, 5p, 5p U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 39 http://www.SATs-Papers.co.uk 39 19/12/06 2:32:23 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 6–8 only Increases by 3 3-5 4-6 5-7 6-8 19 Correct response 2m Additional guidance Gives all three correct values, ie 3 6 9 or 1m Gives two correct values ! Incomplete processing Withhold only 1m for the ﬁrst occurrence eg, for 1m accept 3 2×3 3×3 ! For 1m, follow through For the second value, accept their ﬁrst value × 2, provided this does not give a value of 0 or 2 For the third value, accept their ﬁrst value × 3 or 3 their second value × , 2 provided this does not give a value of 0 or 3 40 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 40 http://www.SATs-Papers.co.uk 19/12/06 2:32:24 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Perimeters 3-5 4-6 5-7 6-8 20 a 1m Correct response Draws the correct triangle in any orientation eg Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Side lengths labelled Ignore, even if incorrect b 1m Draws a correct shape in any orientation, ie or or or Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 41 http://www.SATs-Papers.co.uk 41 19/12/06 2:32:24 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 6–8 only More powers 3-5 4-6 5-7 6-8 21 Correct response 1m 100 1m Additional guidance 6 Tier & Question Threes 3-5 4-6 5-7 6-8 22 Correct response a 1m Additional guidance Indicates Odd and gives a correct explanation eg Odd × odd = odd and since 3 is odd, it doesn’t matter how many times you do 3 × 3 × 3… the answer will still be odd 3 is odd, so 3 × 3 gives an odd answer 9, then 9 × 3 gives an odd answer 27 and so on Minimally acceptable explanation that states or implies that 3 is odd eg Odd × odd = odd and 3 is odd It’s 3 × 3 × 3… and odd × odd = odd 3 is odd, so multiplying it by itself over and over again will always give an odd answer Any power of an odd number is odd, eg 32 = 9 Incomplete or incorrect explanation eg Odd × odd = odd You get 32 = 9, 33 = 27 etc and they always come out to be odd It’s 3 multiplied by itself 100 times It can’t be even Whenever 3 has a power the answer is always odd Because 3 is an odd number Multiplying by 3 always gives an odd answer U1 b 1m 42 Indicates only the value 3200, ie Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 42 http://www.SATs-Papers.co.uk 19/12/06 2:32:25 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Tan 35 3-5 4-6 5-7 6-8 23 Correct response Additional guidance a 1m 7 b 1m Gives the value 70 with a correct method, including evidence that the height of the triangle has been taken as 7 eg 20 × 7 = 70 2 10 × 7 = 70 20 × 3.5 = 70 tan 35 × 10 × 10 Method used is accurate or scale drawing Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 43 ! Follow through Accept follow through as 10 × their (a) instead of 70 and their (a) instead of 7 eg, with their (a) as 0.07 accept 20 × 0.07 = 0.7 2 http://www.SATs-Papers.co.uk 43 19/12/06 2:32:25 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Tests 3-5 4-6 5-7 6-8 24 a 1m Correct response Indicates False and gives a correct explanation eg The median was about 44.5 The median is at the 2500th value and when you read the graph down from that value you can see it is greater than 40 Only 1750 pupils got up to 38 marks and you need 2500 for the median About 1750 pupils scored 38 or less which is the 35th percentile Up to 38 is only 1750 pupils and that’s less than half Additional guidance ! Range of values For the median on paper 1, accept 44 to 45 inclusive For the position of the median, accept 2500 or 2500.5 For a value corresponding to a mark of 38, accept 1700 to 1800 inclusive, or 34% to 36% inclusive Minimally acceptable explanation eg 44 to 45 inclusive seen Correct value for the median on paper 1 marked on x-axis The 2500th mark is bigger than 38 1750 and 2500 seen 1750 and 35% seen Incomplete explanation eg The 2500th value is not 38 38 is not in the middle of the cumulative frequency 38 is too small to be the median Most pupils scored more than 38 b 1m Indicates True and gives a correct explanation eg The LQ is about 33.5 The UQ is about 56.5 56.5 – 33.5 = 23 or Indicates either True or False and gives evidence that the inter-quartile range is between 22 and 24 inclusive, excluding 23 eg The LQ is about 33 The UQ is about 57 57 – 33 = 24 ! Range of values For the lower quartile on paper 1, accept 33 to 34 inclusive For the upper quartile on paper 1, accept 56 to 57 inclusive For the position of the lower and upper quartiles, accept 1250 or 1250.25 and 3750 or 3750.75 respectively Minimally acceptable explanation eg Correct values for the lower and upper quartiles on paper 1 marked on x-axis 33 to 34 inclusive and 56 to 57 inclusive seen From the 1250th to the 3750th marks is about 23 Incomplete explanation eg The lower quartile taken away from the upper quartile gives 23 [no indication of quartiles on graph] 44 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 44 http://www.SATs-Papers.co.uk 19/12/06 2:32:25 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 6–8 only Tests (cont) 3-5 4-6 5-7 6-8 24 Correct response c 1m Additional guidance ! Range of values For the median on paper 1, accept 44 to 45 The most common correct explanations: inclusive For the median on paper 2, accept 51 to 52 inclusive Use values from the graph For any other values on the x-axis, eg accept the correct values 0.5 The median on paper 1 is 44.5, the median on For corresponding values on the y-axis, paper 2 is 51.5, so paper 1 is harder accept the correct values 50 About 850 pupils got less than 30 marks on paper 1 but only about 250 did on paper 2 Minimally acceptable explanation About 400 pupils got more than 65 marks on eg paper 1, but about 600 did on paper 2 The median on paper 1 is lower than the median on paper 2 More people got lower marks [paper 1 implied] Use or interpret the relative positions of Fewer people got lower marks on the lines paper 2 eg More people got better marks on paper 2 The graph for paper 2 is always lower The line for paper 1 is higher The dotted line is always on the right of the other line The marks on paper 2 were higher Incomplete or incorrect explanation eg Paper 2 was easier Everybody’s score is higher in paper 2 than in paper 1 Indicates False and gives a correct explanation U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 45 http://www.SATs-Papers.co.uk 45 19/12/06 2:32:26 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier & Question Tier 6–8 only Circle angle 3-5 4-6 5-7 6-8 25 Correct response 2m Additional guidance 50, with no evidence of an incorrect method or 1m Shows or implies that angle ABC is 90° or Shows a complete correct method with not more than one computational error eg 180 – 110 – 30 = 40, 90 – 40 180 – 70 – 60 110 + 30 – 90 Tier & Question Inequalities 3-5 4-6 5-7 6-8 26 Correct response 1m Gives two non-zero values x then y such that | x | > | y | and either x < 0 or y < 0 eg 2 ÷ –1 –10 1m ÷ 4 Gives two non-zero values x then y such that | x | < | y | and either x < 0 or y < 0 eg –1 ÷ 2 4 46 ÷ –10 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 46 Additional guidance http://www.SATs-Papers.co.uk 19/12/06 2:32:26 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 6–8 only Tier & Question Drawing triangles 3-5 4-6 5-7 6-8 27 Correct response 1m Additional guidance Indicates No and gives a correct explanation The most common correct explanations: Show how the two triangles could be different eg One could draw: 60° 20° 5cm But the other one could draw: Minimally acceptable explanation eg The 5cm side could be in different places compared to the angles ! Diagrams inaccurate Accept provided the response makes the position of the 5cm sides relative to the angles clear eg, accept 60° 60 20° 5cm There are 3 different triangles possible because the 5cm side could be opposite the 20°, the 60° or the other angle One could have those angles on either side of the 5cm line, but the other could have those angles on either side of a different line Reason generally from known facts eg For them to be congruent, you need to know that the positions of the angles and sides correspond in the two triangles It needs to be AA corresponding side, but we don’t know the order in the two triangles 60 20 5 or 20 5 Incomplete explanation eg They could have drawn three different triangles The angles are the same but the lengths may be different Minimally acceptable explanation eg You don’t know if there are corresponding sides Incomplete explanation eg You don’t know the arrangement of the side and angles, so you can’t be sure U1 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 47 http://www.SATs-Papers.co.uk 47 19/12/06 2:32:27 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Index Index to mark schemes Tier 3–5 4–6 Question 5–7 Page 6–8 1 Euros 11 2 Number line 12 3 36 times table 12 4 Feeding times 13 5 Work out 13 6 Dollars 14 7 Divisibility 14 8 Populations 15 9 2 Number grids 16 10 3 Babies 17 11 4 Cards for fractions 17 12 5 Pentagon perimeter 18 13 6 Multiple and factor 19 14 7 Same number 19 15 8 Squares 20 16 9 1 x=8 20 17 10 2 Grid 21 18 11 3 Sweets 22 19 12 4 Completing calculations 22 20 13 5 Rectangles 23 21 14 6 Percentages 23 22 15 7 Positive and negative 24 23 48 1 16 8 Barrels 24 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 48 http://www.SATs-Papers.co.uk 19/12/06 2:32:28 pm 2007 KS3 Mathematics test mark scheme: Paper 1 Tier 3–5 Index Question Page 4–6 5–7 6–8 17 9 1 Speed 25 18 10 2 Grey and black designs 26 19 11 3 Counter probabilities 27 20 12 4 Three straight lines 28 21 13 5 Ninths 30 22 14 6 Equation 30 23 15 7 Circle working 31 24 16 8 Thinking fractions 32 25 17 9 Cube 32 18 10 Random numbers 33 19 11 Geometric mean 34 20 12 Sequences 35 21 13 Rhombus 36 22 14 Counters 36 23 15 Fence plan 37 24 16 Powers 37 25 17 Dissection 38 26 18 Coins 39 19 Increases by 3 40 20 Perimeters 41 21 More powers 42 22 Threes 42 23 Tan 35 43 24 Tests 44 25 Circle angle 46 26 Inequalities 46 27 Drawing triangles 47 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 49 http://www.SATs-Papers.co.uk 49 19/12/06 2:32:28 pm 2007 KS3 Mathematics test mark scheme: Paper 1 BLANK PAGE 50 Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 50 http://www.SATs-Papers.co.uk 19/12/06 2:32:29 pm 2007 KS3 Mathematics test mark scheme: Paper 1 BLANK PAGE Sourced from SATs-Papers.co.uk 275668_MS_P1.09.indd 51 http://www.SATs-Papers.co.uk 51 19/12/06 2:32:29 pm 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 2007 © Qualifications and Curriculum Authority 2007 ISBN 1-85838-875-9 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 275668_MS_P1.09.indd 52 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/06/2778 http://www.SATs-Papers.co.uk 275668 19/12/06 2:32:29 pm