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1 EARLY YEARS NATIONAL CURRICULUM 5 16 GCSE GNVQ GCE A LEVEL NVQ First published in 000 Qualifications and Curriculum Authity 000 Reproduction, stage, adaptation translation, in any fm by any means, of this publication is prohibited without pri written permission of the publisher, unless within the terms of licences issued by the Copyright Licensing Agency. Excerpts may be reproduced f the purpose of research, private study, criticism review, by educational institutions solely f educational purposes, without permission, provided full acknowledgement is given. OTHER VOCATIONAL QUALIFICATIONS Produced in Great Britain by the Qualifications and Curriculum Authity under the authity and superintendence of the Controller of Her Majesty s Stationery Office and Queen s Printer of Acts of Parliament. The Qualifications and Curriculum Authity is an exempt charity under Schedule of the Charities Act Qualifications and Curriculum Authity 9 Bolton Street London W1Y 7PD Further teacher packs may be purchased (f any purpose other than statuty assessment) by contacting: QCA Publications, PO Box 99, Sudbury, Suffolk CO10 SN (tel: ; fax: ) Order ref: QCA/00/

2 Ma KEY STAGE 3 ALL TIERS 000 Mathematics tests Mark scheme f Paper 1 Tiers 3-5, 4-6, 5-7 and 6-8 EY STAGE 3 AGE 3 KEY STAGE 3 AGE 3 KEY STAGE 3 KEY EY STAGE 3 KEY STAGE 3 KE Y STAGE TAGE 3 KEY ST KEY STAGE 3 KEY GE 3 KEY ST AGE 3 KE GE 3 KEY S STAGE 3 K TAGE 3 K KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY STAGE 3 KEY GE 3 KEY STAGE 3 KEY STA STAGE 3 KEY ST E 3 KEY STAG GE 3 KEY ST 3 KEY STAG GE 3 KEY ST KEY STAG GE 3 KEY S S 3 KEY ST 3 KEY TAGE 3 KE KEY STAG AGE 3 KEY 3 KEY STAGE 3 KEY STAGE Y STAGE 3 KEY STAG STAGE 3

3 000 KS3 Mathematics Test Mark Scheme: Paper 1 Index Index to mark schemes Tier Question Page 1 Milk Shakes 11 Three Coins 11 3 Time 11 4 Sums 1 5 Calendar Sixty-fives 13 7 Spinners Shapes Birthdays Rainfall Angles Prism Race Percentages A Museum Simplify 1 7 Puzzle Coloured Cubes Perimeters Equations Rounding Mast Factising Operations Expressions Values 8 1 Similar Triangles 9 13 Powers Expansion Isosceles 3

4 000 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 infm teachers. This booklet contains the mark scheme f paper 1 at all tiers. The paper and the extension paper mark schemes are printed in separate booklets. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking infmation f questions is set out in the fm 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 f that question part. The Crect response column usually includes two types of infmation: a statement of the requirements f the award of each mark, with an indication of whether credit can be given f crect wking, and whether the marks are independent cumulative; examples of some different types of crect response, including the most common and the minimum acceptable. The column indicates alternative acceptable responses, and provides details of specific types of response which are unacceptable. Other guidance, such as when follow through is allowed, is provided as necessary. F 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. 3

5 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction Using the mark schemes Answers that are numerically equivalent algebraically equivalent are acceptable unless the mark scheme states otherwise. In der to ensure consistency of marking, the most frequent procedural queries are listed below with the prescribed crect action. Unless otherwise specified in the mark scheme, markers will apply the following guidelines in all cases. 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 cresponds with the statement of requirements given in the Crect response column. Refer also to the additional guidance, and if still uncertain contact the supervising marker. The pupil has responded in a non-standard way. Calculations, fmulae and written responses do not have to be set out in any particular fmat. Pupils may provide evidence in any fm as long as its meaning can be understood. Diagrams, symbols wds are acceptable f explanations f indicating a response. Any crect method of setting out wking, however idiosyncratic, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma f a decimal point. The pupil s accuracy is marginal accding to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, touches, the boundaries given, the mark(s) should be awarded. The pupil s answer crectly follows through from earlier increct wk. 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 crect response an acceptable follow through response should be marked as crect. There appears to be a misreading affecting the wking. This is when the pupil misreads the infmation given in the question and uses different infmation without altering the iginal intention difficulty level of the question. F each misread that occurs, deduct one mark only. The crect answer is in the wrong place. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a wd number response is expected, a pupil may meet the requirement by annotating a graph labelling a diagram elsewhere in the question. 4

6 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction The final answer is wrong but the crect answer is shown in the wking. 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 increct answer is due to a transcription err; If so, award the mark. in questions not testing accuracy, the crect answer has been given but then rounded truncated; If so, award the mark. the pupil has continued to give redundant extra wking which does not contradict wk already done; If so, award the mark. the pupil has continued, in the same part of the question, to give redundant extra wking which does contradict wk already done. If so, do not award the mark. Where a question part carries me than one mark, only the final mark should be withheld. The pupil s answer is crect but the wrong wking is seen. A crect response should always be marked as crect unless the mark scheme states otherwise. The crect response has been crossed ( rubbed) out and not replaced. Mark, accding to the mark scheme, any lible crossed ( rubbed) out wk that has not been replaced. Me than one answer is given. If all answers given are crect ( a range of answers are given, all of which are crect), the mark should be awarded unless prohibited by the mark scheme. If both crect and increct responses are given, no mark should be awarded. The answer is crect but, in a later part of the question, the pupil has contradicted this response. A mark given f one part should not be disallowed f wking answers given in a different part, unless the mark scheme specifically states otherwise. 5

7 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction General guidance Throughout the marking of the key stage 3 mathematics tests, the following general guidelines should be observed unless specific instructions to the contrary are given. This guidance reflects decisions made to ensure fairness and consistency of marking. Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only. Accept Take care! Do not accept F example: 0.7 A crect probability that is crectly expressed as a decimal, fraction percentage. Equivalent decimals, fractions percentages ,,, 70.0% A probability crectly expressed in one acceptable fm which is then increctly converted, but is still less than 1 and greater than 0 70 e The following four caties of err should be igned if accompanied by an acceptable response, but should not be accepted on their own.! A probability that is increctly expressed 7 in 10, 7 out of 10, 7 from 10! A probability expressed as a percentage without a percentage sign.! A fraction with other than inters in the numerat and/ denominat. However, each of the three types of err above should not be penalised me than once within each question. Do not award the mark f the first occurrence of each type of err unaccompanied by an acceptable response. Where a question part carries me than one mark, only the final mark should be withheld.! A probability expressed as a ratio 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 less than 0 6

8 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction Responses involving money Accept Do not accept F example: Any unambiguous indication of the crect amount 3.0(p), 3 0, 3,0, 3 pounds 0, 3-0, 3 0 pence, 3:0, 7.00 The sign is usually already printed in the answer space. Where the pupil writes an answer other than in the answer space, crosses out the sign, accept an answer with crect units in pounds and/ pence 30p 700p Increct ambiguous use of pounds pence 30, 30p 700p, p not in answer space. Increct placement of decimal points, spaces, etc increct use omission of 0 3., 3 00, 3 0, Responses involving the use of algebra Accept Take care! Do not accept F example: p n n p n The unambiguous use of a different case N used f n Unconventional notation f multiplication n t t n n n p n f n, n t n f n Multiplication by 1 0 p 1n f p n, p 0n f Wds used to precede follow equations expressions t e n p tiles tiles e t e n p f t e n p Unambiguous letters used to indicate expressions t e n p f n p Embedded values given when solving equations 3 t 10 p e 3 f 3x p e 3! Wds units used within equations expressions should be igned if accompanied by an acceptable response, but should not be accepted on their own do not accept n tiles p n cm p Change of variable x used f n Ambiguous letters used to indicate expressions n e n p However, to avoid penalising any of the three types of err above me than once within each question, do not award the mark f the first occurrence of each type within each question. Where a question part carries me than one mark, only the final mark should be withheld. Embedded values that are then contradicted f 3x p e 3, 3 t 10 p e 3, x e 5 7

9 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction Responses involving time Accept Take care! Do not accept A time interval F example: hours 30 min Any unambiguous indication.5 (hours), h 30 Digital electronic time ie :30 Note that :30 is accepted f h 30m because it is a common electronic expression ( the time interval shown on an oven timer). Increct ambiguous time interval.3(h),.30, -30, h 3,.30min! The time unit, hours minutes, is usually printed in the answer space. Where the pupil writes an answer other than in the answer space, crosses out the given unit, accept an answer with crect units in hours minutes, unless the question has asked f a specific unit to be used. A specific time F example: 8.40am Any unambiguous, crect indication 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Increct time 8.4am, 8.40pm Increct placement of diviss, spaces, etc increct use omission of 0 840, 8:4:0, 084, 84 Responses involving co-dinates Accept Do not accept F example: (5,7) Unambiguous but unconventional notation ( 05, 07 ) ( five, seven ) x y (5,7) ( x e 5, y e 7) Increct ambiguous notation ( 7, 5 ) (5x, 7y ) ( x5, y7 ) (5 x,7 y ) 8

10 000 KS3 Mathematics Test Mark Scheme: Paper 1 Introduction Recding marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 a 0 entered in each marking space. Where m can be split into gained and lost, with no explicit der, then this will be recded by the marker as 1 0 The total marks awarded f 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 recded on the front of the test paper. A total of 10 marks is available in each of tiers 3-5, 4-6 and 6-8, and a total of 11 marks in tier 5-7. The extension paper carries 41 marks. Awarding levels The sum of the marks gained on paper 1, paper and the mental arithmetic paper determines the level awarded. A copy of the level threshold tables which show the mark ranges f the award of different levels will be sent to each school by QCA in July 000. 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 sces on the test papers and the levels awarded. The 000 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 9

11 000 KS3 Mathematics Test Mark Scheme: Paper 1 BLANK PAGE 10

12 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 3-5 only 1 Crect response Milk Shakes a 9 Incomplete processing, f part (b) 10 p 1 b c 11 d 4 Crect response Three Coins 3m m All 6 entries crect (see below) Any 4 5 crect entries Any 3 crect entries 1p p 14p 10 15p p p p (ie not possible) Coins in any der Trials f 18p that are unambiguously rejected 10p crossed out.! Entries made f 10p, 11p and 13p Igne.! Increct response alongside crect response Igne any increct response. No decision taken f 18p Entry f 18p left blank. 3 Crect response Time a 7:55 b 33 c 14:0 11

13 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 3-5 only 4 Crect response Sums Crect response Calendar a Tuesday Unambiguous abbreviation Tues Tu Ambiguous abbreviation that could refer to Thursday T b 30 (th) Unambiguous indication Marking of diagram. c 1 1

14 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, a 35 Crect response Sixty-fives 6 a b b m 1040 Shows a reasonably efficient crect method,! Method uses long multiplication even if there are processing errs Accept, provided there is not me than one 8 t 65, then doubled. processing err. 10 t 65 add 6 t p 650 Added 4 of them, to of them, to 10 of them. Their 1 t 65 p t 8 50 p 50! Method uses repeated addition from 650 Accept provided there is not me than one computational err. The crect intermediate values are: Method uses repeated addition without building on from 650, better Adding 65 sixteen times. Adding 130 eight times. Adding 60 four times.! Follow through from part (a) F m, where a pupil has used, unambiguously, an increct answer from 1 t 65 0 t 65 to calculate 16 t 65, allow follow through. F example, suppose their 1 t 65 e 73, accept f m 73 p 60 e 983 However, it must be clear from the wking that this increct value has been used, hence in this example do not accept 983 without wking. 13

15 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, Crect response Spinners a a Spinner A with a crect reason The most common crect reasons refer to: Fewer numbers / sections / sides on A A, less digits. B has me numbers. A has one piece missing. A has only 5 sides, not 6 A because B has 6 triangles. Minimally acceptable reason There s 5 not 6 Gaps on A are bigger than on B! Use of bigger smaller without qualification Igne, accept A, less numbers and A is smaller than B, do not accept A is smaller ( bigger) than B Greater probability on A 1 1 The probability is not 5 6 A has a bigger probability. A crect probability but expressed in wds At this level, accept Spinner A is 1 out of 5! A crect probability expressed as a ratio In parts (a) and (b), penalise the first occurrence only In (a), 1 to 5 ( 1 to 4), with reason In (b), 1 to 6 ( 1 to 5), with reason Mark as 0, 1! Use of chance with a descript As this restates the question asked, do not accept unless accompanied by a crect response. Increct probability Do not accept, even if accompanied by a crect response. Bigger angles on A Each triangle on B has a smaller angle in the middle. Minimally acceptable reason A s triangles are wider.! Reference to fce of spin Igne, accept Depends how hard you spin it, but A has a 1 in 5 chance. 14

16 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, Crect response Spinners (cont) b b Doesn t matter with a crect reason The most common crect reasons refer to: Same numbers / sections / sides Both numbered 1 to 6 Same amount of numbers. Same shape. Same probability Both 1 6 C is same probability as B Same chance. Minimally acceptable reason Same numbers. Both have only one 3 Same gaps.! Use of different size without qualification Igne, do not accept It doesn t matter even though they are different sizes. Doesn t matter if one is bigger than the other.! A crect probability increctly expressed Mark as part (a) of this question, accept Both spinners are 1 in 6 Same angles Both 60 Minimally acceptable reason The triangles are the same width. c c m Two 3s and three 4s in any der. Partially crect, ie exactly two 3s seen. Exactly three 4s seen. 15

17 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, Crect response Shapes a a Both crect, ie b b ( 5, 7 ) Co-dinates of A given alongside B (3,3, 5,7) Label included within the co-dinates (B5, 7) c c Crect place Place identified by crect co-dinates! Label rather than point identified Accept any indication, cross B, provided it is nearer to the crect co-dinate than to any other co-dinate with inter values. d d Crect place 16

18 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, a a 1 Crect response Birthdays b b ! Follow through as 010 m (a) Accept provided their (a) > 1 and is not a multiple of 10 c c Follow through as part (b) p 6 Crect birth date month given March a a Tuesday Friday Crect response Any unambiguous indication 0.8 f Tuesday 0.05 f Friday Rainfall b b 0.5 c c 1.5 Their number of cm converted to mm 17

19 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, 4-6, Crect response Marking overlay available Angles a a a Q Unambiguous indication Second. b b b m Angle within ±, unambiguous, using an end point of the given line, starting again, cutting the given line Angle within ± 4, unambiguous Using end point starting again: crect angle drawn ±, but reflex angle indicated Cutting the given line: crect angle drawn ±, but no, increct, indication of which angle to choose Draws the supplement to ± 18

20 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, 4-6, Crect response Angles (cont) c c 38 with explanation that focuses on most frequent Me of them. Mode. Most. Twice as many said 38 as the others put together. 10 people aren t likely to be wrong. Minimally acceptable explanation 10 people measured it. Crect response accompanied by an increct, irrelevant ambiguous statement Me people chose it and it s the middle number. It s the average, there s me of them. Use of average without a crect response It s the average. c 135 with crect explanation The most common crect explanations are: 135 being the mode / median / mean when 45 removed 10 said about 135, only half of that said 45 so it s is too different, 135 most likely. Discard 45, then 135 is the average. 45 is too different, then 135 is in the middle. That 45 is due to increct reading of the scale The 45s are errs, 135 most likely. 135 is the mode. The 45s are probably reading the scale increctly. 135 lies by 45 which is the wrong side. Minimally acceptable explanation referring to is too different to the others, so , fewer people but 45 is nothing like the rest.! Minimally acceptable explanation that does not refer to 45 The explanation must justify why 135 chosen rather than , accept Most of them are around is between 134 and 136 Misinterpreting the table other increct ambiguous response Only 1 said 45, 3 said , and 135 is in the middle of them so it s 135 Out of 3, 4 and 5, 4 is the middle so 135 The average is 3 but there are answers with 3 so I chose 135 as it s in-between. Most of them are 135 (no reference to 45 around 135) 135 is the middle of the table (no reference to 45) 19

21 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, 4-6, Crect response Prism a a a Crect edge, ie Unambiguous indication use of different labels, f edge A b b b Crect edge, ie Ambiguous indication that could refer to me than one edge, f edge A! Both edges identified crectly but no indication of which is which Mark as 0, 1 c c c Both crect, ie! Different symbols Accept if unambiguous, but do not accept use of letters A and/ B Crect response Race m 3 and 100 and Maria and 6 Times within ± 0.1 minutes Any two three of these crect Distance within ± 5 metres 1.5, equivalent! Igne name (already assessed) 0

22 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, 4-6, Crect response Percentages A (.00) 35 5(.00) a a a m 88 Crect response Museum Digits 88 seen Complete crect method, with not me than one computational err 40 t Answer t 1 e 40, and 40 t 0p e 40 d 5 e 46 so the answer is 86 Conceptual err t 1 t b b b m 500 Gives an answer of 5 followed only by one me zeros

23 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 3-5, 4-6, 5-7, Crect response Simplify 7 p 5t Expression not simplified 3b p 17 4d p 3 4m 1 7 a a a (p) 5 and m 3 Crect response Either der Puzzle b b b m 5 and (p) 3! Answers to parts (a) and (b) reversed Mark as 0, 1 c c c m 4 and m d d d m 5

24 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4-6, 5-7, Crect response Coloured Cubes a a a 4, equivalent probability 5 b b b 4 c c c 8! Crect digit expressed as a proption of the total cubes Mark only the first such occurrence as increct, mark the following as 0, 1 then 1, (part b) (part c) (part d) 0 If this is seen f the first time in part (d), mark part (d) as 1, 0 Answer to part (c) refers to part (b) 4 me. d d d m 7 3 Shows of 0 is (green) 3 1 e 5 0 1B 1Y 13 seen with no evidence of an increct method. 3

25 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4-6, 5-7, Crect response Perimeters a a a Both crect, ie b b b m Crect simplified expression 4n p 10 4(n p.5) (n p 5) (n p 5) t Crect expression seen, even if terms are not collected together 5 p n p n p n p n p 5 n p n p 10 An otherwise crect simplified expression, of 4n p k, (k 10) 4n p (10 d ) 4(n p 5) (n p ) c c c m Crect simplified expression 40 p n 1 (0 p n)! Increct wking follows crect response, f part (a) 4n p 10 e 40n Mark as 1, 0! Crect answer preceded by increct wking If the intended answer is unambiguous, igne preceding wk, f part (a), accept 5 p n t e 10 p 4n! An otherwise crect simplified expression with the only err being that the brackets are omitted, f part (b) n p 5 t Mark as 1, 0! Value f n substituted into an otherwise creditwthy response Igne. Crect expression seen, even if terms are not collected together 10 t 4 p n p n p n (n d p 0) An otherwise crect simplified expression, of n p k, (k 40 but is a multiple of 10) (n d p 10) d d d cm 4

26 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 4-6, 5-7, Crect response Equations a a m All crect, ie E A C D B At least 3 crect b b No with crect explanation The most common crect explanations are: Substituting x e 10, y e 10, both, into the equation 10 t 10 m 5 The line would go through (7.5, 10) The line would go through (10, 15) When x e 10, y e 15 3 Comparing y e x m 5 to y e x m 5, identifying the common y-intercept and different gradient It goes through m 5 on the y-axis, but it s 3 steeper than y e x m 5, so it can t go through (10, 10) No. The intercept is the same as line D but it s a different slope. Minimally acceptable explanation It would go through about (8, 10) t 10 m 5 e 15 Minimally acceptable explanation Line D goes through (10, 10) so y e x m 5 can t. Infmal description of the y-intercept It cuts at m5 but different gradient to D Starts at m5 then different slope to D Goes through m5, gradient not like D Interpreting the gradient and y-intercept It goes through m 5 on the y-axis, and the gradient of takes it up f every 1 along, so no. Showing a minimum of crect points on the line y e x m 5, and a convincing reason Crect line drawn, crossing y e 10 Two crect points shown that are either side of the line y e 10 The line goes through (6, 7) and (7, 9) and if you continue it, it will miss (10, 10)! Increct co-dinates alongside crect ones Igne if the crect line is drawn on the graph, the crect points are otherwise identified. If no identification is possible, accept one me increct provided there are at least 3 crect and there is a convincing reason. c 1 10 Equivalent fractions and decimals 1 x 10 5

27 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5-7, Crect response Rounding a a 8 b b 16 c c d d Answer in the range 8 to 34 inclusive 1 7 Crect response Marking overlay available Mast Perpendicular bisect between A and C, at least cm in length. Their line, when extended, must all be within the tolerance of the rion shown on the overlay. F all marks, accept freehand within the tolerance defined Line and/ curve represented, unambiguously, by a series of points Arc, of at least the length defined on the overlay, within the tolerance shown.! Arc inaccurate beyond the rion defined on the overlay Igne the arc drawn outside the rion on the overlay. Complete crect rion following through from their attempt at an arc from B with either an attempted perpendicular bisect an arc from A Rion incomplete Not enclosed at one end. Point(s) rather than rion identified. 6

28 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5-7, Crect response Factising a a Crect two expressions Any indication b b Crect expression c c 7(y p ) y expressed as 1y d d m y (3y m 1) F m, 6y 1 (y m ) m y (1 m 3y) 3 Any of the following partial simplifications F, the equivalent partial simplification (3y 3 m y ) with the term outside the bracket native y(3y m y) y(6y m y) F, the only err is in the sign in y (6y m ) the last bracket omitted y (3y p 1) 14 9 Crect response Operations a a m 0.05 and 0.1 chosen, and answer F m, 0.05 chosen twice (answer 0.005) Both cards crect but answer increct omitted The crect answer to any two decimal values 0.1 t 0. e 0.0 b b 10 d 0.1 7

29 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tiers 5-7, a a 3x p 10 Crect response Expressions b b Any three expressions that add to 1x 4x m 5, 4x, 4x p 5 4x, 4x, 4x x p 4, x m 3, 9x m 1 One two numerical values given 1x, 7, m7 Expressions that are not simplified 4 t x, 6x m x, x p x c c m 4x p (x p 1) Crect total of the expressions seen inferred, even if followed by increct wking 1x p 6 x p 5x p 5x e 1x, 3 p m9 p 1 e 6 Complete crect method with not me than one computational err x p 5x p 5x e 1x, 3 m 9 p 1 e 3 so mean is 4x p Crect response Values a a n Crect responses that are then evaluated b b n Evaluation without indication of crect response, f part (c) ( m ) c c n 8

30 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6-8 only 1 a m 15 Crect response Similar Triangles Crect ratio identified P 10 e e P 1 b 1 Crect scale fact seen 1.5 t 1 d 8 c Yes with crect justification Angles same, so yes. Minimally acceptable justification Yes, with 40 and 80 shown crectly on diagram. Crect angles found but no decision taken rarding similarity Note that finding the missing angles is a lower level skill. What is being assessed here is applying that knowledge within the context of similarity, hence a decision must be taken. Crect angles found but explanation is incomplete increct Missing angles are 40 and 80, and 40 is half of 80, so yes. 9

31 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6-8 only 13 Crect response Powers a Crect explanation showing both aspects below: Shows all three component parts as 7 7 t t 7 t t 7 t 7 t 7 t 7 and Shows how the component parts are linked, either through multiplication through addition of the powers (must be stronger than a restatement of the given 49 t 343 e 16807) 49 t 343 e 7 t 7 3 e t 7 3 e e (7 t 7) t (7 t 7 t 7) 49 e 7 t 7; 343 e 7 t 7 t 7 (7 t 7) t (7 t 7 t 7) e It s 7 t 7 3, add the powers you get 7 5 Crect explanation showing the powers add to 5 The powers are and 3, and p 3 e 5 7 e 49, 7 3 e 343, p 3 e 5 Component parts not all shown 49 t 343 e 7 t e 7 t 7 t 7 t 7 t 7 No indication that the component parts need to be multiplied the powers need to be added 49 e 7, 343 e 7 3, e and 7 3 e e 7 t 7; 343 e 7 t 7 t 7 7 t 7 t 7 t 7 t 7 e Shows 7 p 7 3 e 7 5 alongside a crect response 49 e 7, 343 e 7 3, 7 p 7 3 e 7 5, powers add to 5! Unconventional notation showing powers are added Accept only if clearly intended to refer to powers 7 p 7 3 e 7 5 b 7 Incomplete processing 7 8 d 7 7 (part b) 7 6 d 7 4 (part c) c 49 7 Increct processing 7 e 14 (part c) d 1! Other digits preceding 1 Igne. 30

32 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6-8 only 14 Crect response Expansion a Crect explanation Any crectly evaluated counter-example. It s y p 3y p 3y p 9 There has to be a y-term in it. Counter-example not evaluated If y e 5, you can see they are not equal. y e 10, (10 p 3) 10 p 9 Increct expansion The expression need not be expanded, but if it is then all terms must be crect. Incomplete explanation Because it s squaring the whole answer. It s (y p 3)(y p 3) not (y t y) p 9 It s not y p 3, it s (y p 3) b Crect simplified expansion y p 7y p 10 m y m 1y p 36 Crect expansion, but not simplified, simplified increctly y m 6y m 6y p 36 seen m 6y m y m 40 Crect expansion, but not simplified, simplified increctly 6y m 16y p 15y m 40 seen F, condone positive sign f 15y omitted 6y m 16y 15y m 40 seen 31

33 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6-8 only 15 Crect response Isosceles a Crect justification The most common crect justifications are: Calculations shown on the diagram The following may be helpful: Wking with triangle ACB The other angles in triangle ACB are 16 and 148, and 180 m 16 m 148 e 16 Wking with triangles ABD and ACD 180 m 3 BAD e e 74, 180 m 64 CAD e e 58; 74 m 58 e 16 CAD e 58, if a e 16 then 58 p 16 p 3 p 16 p 58 e 180, so true. In triangle ACD there s another 116, in triangle ABD there s another 148, 148 m 116 e 3, 3 d e 16 Wking with quadrilateral ACDB 180 p 116 p 3 p a e 360, a e 360 m 38, a e 3 Subtraction shown in increct der 64 m 180 e 116 Computational err 180 m 64 e 68, 74 m 68 e 16 Incomplete response 74 and 58 calculated, but no indication that 74 m 58 is needed. Spurious wking 3 d 16 (unsuppted) Small triangle 64, 58, 58; big triangle 3, 74, 74 and 90 m 74 e 16 3

34 000 KS3 Mathematics Test Mark Scheme: Paper 1 Tier 6-8 only 15 Crect response Isosceles (cont) b m Crect proof The most common crect proofs are: Wking with triangle ACB m (180 m x p x) e x Value substituted f x without suppting algebraic proof F m, incomplete wking (180 m x) m (180 m x) e x Wking with triangles ABD and ACD BAD m CAD e 1 1 (180 m x) m (180 m x) 1 e 90 m x m 90 p x 1 e x Wking with quadrilateral ACDB Other angle at C is 360 m x, so a p x p 360 m x e 360 a m x e 0 a e x a e x d 1 (180 m x) seen 1 (180 m x) seen Showing reflex angle at C is 360 m x 33