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1 LEVEL 6 TESTS MARK SCHEMES Ma MATHEMATICS MARK SCHEMES LEVEL 6 TESTS Mark schemes Qualifications and Curriculum Development Agency 2011

2 We want our website and publications to be widely accessible, so please contact us if we're not meeting your needs. Qualifications and Curriculum Telephone Development Agency Textphone Butts Road Fax Earlsdon Park Coventry CV1 3BH Qualifications and Curriculum Development Agency 2011 Reproduction, stage translation, in any fm by any means, of this publication is prohibited without pri written permission of the publisher, unless within the terms of 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. Page 2 of 28

3 Contents Introduction 5 General guidance 5 Mark scheme f paper 1 12 Mark scheme f paper 2 18 Level threshold infmation 25 Page 3 of 28

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5 Introduction This booklet contains the mark schemes f papers 1 and 2 of the Optional level 6 tests in mathematics. Each mark scheme was devised after trialling the tests with pupils and contains examples of some frequently occurring crect and increct answers given in the trials. Each mark scheme indicates the criteria against which judgements should be made. The last section of this booklet provides infmation about interpreting the sces from the tests. General guidance The structure of the mark schemes The marking infmation f each question is set out in the fm of tables, which start on page 12 of this booklet. The Q column on the left-hand side of each table provides a quick reference to the question number. The mark column indicates the total number of marks available f each question part. On some occasions the symbol U1 may be shown in the mark column. The U indicates that there is a Using and applying mathematics element in the question. The number, 1, shows the number of marks attributed to using and applying mathematics in this question. The crect response column may include 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 examples of some different types of crect response. The additional guidance column indicates alternative acceptable responses, and provides details of specific types of response which are unacceptable. Other guidance, such as the range of acceptable answers, when follow-through is allowed, is provided as necessary. Applying the mark schemes In der to ensure consistency of marking, the most frequent procedural queries are listed on pages 6 and 7 along with the action the marker will take. This is followed by further guidance relating to the marking of questions that involve money, time and other measures, codinates, probability and algebra. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. Page 5 of 28

6 What if The pupil s response is numerically equivalent to the answer in the mark scheme. The pupil s response does not match closely any of the examples given. The pupil has responded in a non-standard way. There appears to be a misreading affecting the wking. No answer is given in the expected place, but the crect answer is given elsewhere. Marking procedure Markers should award the mark unless the mark scheme states otherwise. Markers should use their judgement in deciding whether the response cresponds with the statement of the requirements given in the crect response column. Reference will also be made to the additional guidance and, if there is still uncertainty, markers should consult the supervising marker. 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, should be accepted. Provided there is no ambiguity, condone the continental practice of using a comma f a decimal point. 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. 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. The response in the answer box is wrong, but the crect answer is shown in the wking. Where appropriate, detailed guidance will be given in the mark scheme, which markers should follow. If no guidance is given, markers will need to examine each case to decide whether: the increct answer is due to a transcription err in questions not testing accuracy, the crect answer has been given but then rounded truncated the pupil has continued to give redundant extra wking which does not contradict wk already done the pupil has continued to give redundant extra wking which does contradict wk already done. If so, the mark should be awarded. If so, the mark should be awarded. If so, the mark should be awarded. If so, the mark should not be awarded. Where a question part covers me than one mark, only the final mark should be withheld. Page 6 of 28

7 What if The pupil s answer is crect but the wrong wking is shown. The pupil has made a conceptual err. The crect response has been crossed out and not replaced. Me than one answer is given. The pupil s answer crectly follows through from earlier increct wk. The answer is crect but, in a later part of the question, the pupil has contradicted this response. The pupil s accuracy is marginal accding to the overlay provided. The pupil has drawn lines which do not meet at the crect point. Marking procedure A crect response should always be marked as crect unless the mark scheme states otherwise. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, err. A computational err is a slip such as writing 4 6 = 18 in an otherwise crect long multiplication. A conceptual err is a me serious misunderstanding of the relevant mathematics; when such an err is seen no method marks may be awarded. Examples of conceptual errs are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating subtracting the smaller value from the larger in calculations such as to give the answer 21 increct signs when wking with native numbers. Any lible crossed-out wk that has not been replaced should be marked accding to the mark scheme. If the wk is replaced, then crossed-out wk should not be considered. If all answers are crect ( a range of answers is 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. 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. 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. Overlays can never be 100% accurate. However, provided the answer is within, touches, the boundaries given, the mark(s) should be awarded. Markers should interpret the phrase slight inaccuracies in drawing to mean within on a circle of radius 2mm with centre at the crect point. within the circle accepted on the circle accepted outside the circle not accepted Page 7 of 28

8 Marking specific types of question Responses involving money Where the sign is given f example: 3.20, 7 Where the p sign is given f example: 40p p Where no sign is given f example: 3.20, 40p Accept Any unambiguous indication of the crect amount, 3.20p 3 20 pence , :20 320p with sign crossed out 40p Any unambiguous indication of the crect amount, 0.40p.40p 0.40 with p sign crossed out p 40p p Any unambiguous indication of the crect amount in p as shown above Do not accept Increct placement of pounds pence, p Increct placement of decimal point, increct use omission of 0, Increct ambiguous use of pounds pence, 0.40p 40p Omission of final zero, Page 8 of 28

9 Responses involving time A time interval f example: 2 hours 30 minutes A specific time f example: 8:40am, 17:20 Accept 2 hours 30 minutes Any unambiguous, crect indication, hours 2.5 hours 2h 30 2h 30 min 2 30 Digital electronic time, ie 2:30 8:40am 8:40 twenty to nine Any unambiguous, crect indication, ,40 Unambiguous change to hour clock, 17:20 as 5:20pm 17:20pm Do not accept Increct ambiguous time interval, 2.3 hours 2.3h 2h min , Increct time, 8.4am 8.40pm Increct placement of separats, spaces, etc increct use omission of 0, 840 8:4: Responses involving measures Where units are given ( kg, m, l) f example: 8.6kg kg Accept 8.6kg Any unambiguous indication of the crect measurement, 8.60kg kg 8kg 600g Do not accept Increct ambiguous use of units, 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 crect answer using the unit they have chosen, unless otherwise indicated in the mark scheme. Page 9 of 28

10 Responses involving codinates F example: (5, 7) Accept Unconventional notation, (05, 07) (five, seven) x y (5, 7) ( x = 5, y = 7) Do not accept Increct ambiguous notation, (7, 5) y x (7, 5) (5x, 7y ) (5 x, 7 y ) (x - 5, y -7) Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only. F example: % 10 Accept Equivalent decimals, fractions and percentages, % Take care! Do not accept The first four caties of err below should be igned if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of err below 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 crectly expressed in one acceptable fm which is then increctly converted, but is still less than 1 and greater than 0, = 18 25! A probability that is increctly expressed, 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 inters in the numerat and/ denominat.! A probability expressed as a ratio, 7:10, 7:3, 7 to 10 A probability greater than 1 less than 0 Page 10 of 28

11 Responses involving the use of algebra F example: 2 + n n + 2 2n n 2 n 2 Accept Unambiguous use of a different case variable, N used f n x used f n Take care! Do not accept! Unconventional notation, n 2, 2 n, n2 n + n f 2n n n f n 2 n 2 f n n 2 + 1n f 2 + n 2 + 0n f 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, within an explanation general wking. Embedded values given when solving equations, in solving 3x + 2 = 32, = 32 f x = 10 To avoid penalising the two types of err below me than once within each question, do not award the mark f the first occurrence of each type within each question. Where a question carries me than one mark, only the final mark should be withheld. Wds used to precede follow equations expressions, t = n + 2 tiles tiles = t = n + 2 f t = n + 2 Unambiguous letters used to indicate expressions, t = n + 2 f n + 2! Wds units used within equations expressions, n tiles + 2 n cm + 2 Do not accept on their own. Igne if accompanying an acceptable response. Ambiguous letters used to indicate expressions, n = n + 2 f n + 2 Page 11 of 28

12 Paper 1 Q Mark Crect response Additional guidance 1 2m Completes all 8 entries of the table crectly, ie do wear glasses do not wear glasses Total boys girls Total Completes at least four entries crectly U2 2 2m Indicates crect codinates f both points, ie A as (7, 13) and B as (17, 13) Indicates crect codinates f one point Transposes the responses, ie A as (17, 13) and B as (7, 13) U2 3 2m The only err is to indicate increct, but consistent, y dinates, provided y > 3 A as (7, 12) and B as (17, 12) 16 8 Answer of 17 with 50 3 equivalent seen (the only err is to fail to subtract 1 at the start) Answer of 17 without 50 3 equivalent seen U1 4 Shows understanding of a crect method even if there are computational errs = 12 1 : 3 28% Equivalent fractions decimals Page 12 of 28

13 Paper 1 Q Mark Crect response Additional guidance 5 2m Gives all three crect values, ie a = 16, b = 8, c = 6 Gives at least one crect value Gives three values that satisfy the second and third equations a = 18, b = 6, c = 8 (satisfies a + b = 24 and b + c = 14 : note that a c = 10) 6 Indicates the crect area Unambiguous indication Indicates the crect area Unambiguous indication Page 13 of 28

14 Paper 1 Q Mark Crect response Additional guidance 7 Gives a crect description f B that shows implies the link between the two variables The me computers a person has in their home, the fewer hours they are likely to spend watching television There is native crelation between the number of hours watched and the number of computers in the home If you have lots of computers you don t tend to watch TV much Minimally acceptable description Me computers, less watching Fewer computers, me TV Me television, less computers Less TV, me computers Native crelation! Number of hours watching interpreted increctly as number of televisions Condone, f the first mark accept The me computers people have, the fewer TVs they have Gives a crect description f C that states implies that the two variables are not linked How much television a person watches is independent of the number of mobile phones they have There is no crelation between the number of hours watched and the number of phones Time watching is not dependent on the amount of mobiles People with lots of mobile phones don t necessarily watch any me than those with just one Incomplete description If you have one computer you watch me TV Minimally acceptable description Mobiles don t affect watching No crelation Not connected No relationship No link No pattern It s random Me less phones won t affect hours Number of mobiles doesn t affect the situation Someone watching 1 hour of TV might have as many mobiles as someone who watches 8 hours [generality implied] How much is watched depends on the person not on their mobile phones Incomplete description There is a range of numbers of mobile phones and the number of hours spent watching TV It doesn t make much difference! Description of graph s appearance Accept alongside a crect response, f C accept It s all spread out so there is no link, f C do not accept It s all spread out Page 14 of 28

15 Paper 1 Q Mark Crect response Additional guidance 8 2m Indicates No and gives a crect explanation The angles are not the same size A rular pentagon looks like this, with its angles all the same size All the angles should be 108 It doesn t have rotation symmetry It s got me sides than a square so all its angles should be obtuse, but they re not 60º Shows that the 150º angle can be split into 90º and 60º Divides the pentagon vertically and shows that half a is 30º Draws triangles to show a rectangle, labelling the non-right angles on at least one side crectly 60!! Minimally acceptable explanation Different angles A rular pentagon doesn t have right angles in it A rular one can t have 150 angles It doesn t look the same when it s turned Not all the angles are obtuse Increct angle size f a rular pentagon given Condone alongside a crect response, accept The angles are different, they should be 60 (err, but all equal implied) The angles should all be 70 (err), do not accept The 90 angles should be 60 (does not imply the angles should all be the same) Incomplete explanation Not the same It has two right angles Two angles are the same A rular pentagon looks like this A rular pentagon doesn t have any vertical lines Indicates Yes, no decision made, but explanation clearly crect Condone provided the explanation is me than minimal 30 Shows implies that the angle sum of a pentagon is 540º Page 15 of 28

16 Paper 1 Q Mark Crect response Additional guidance 9 Indicates D then Indicates B Gives a crect equation y = 4 y 4 = 0 Line not drawn increct Follow-through from their increct line 10 Joins dots to make a triangle that has only one side of 4cm and only one angle of 45.! Lengths angles shown on their triangle(s) Igne, even if increct U1 Dots not used U1 Gives four numbers that sum to 16 and have a range of 4, ie 1, 5, 5, 5 2, 2, 6, 6 2, 3, 5, 6 2, 4, 4, 6 3, 3, 3, 7 Numbers given in any der 3, 7, 3, 3 Decimals fractions used 1.5, 4, 5, 5.5 Page 16 of 28

17 Paper 1 Q Mark Crect response Additional guidance U1 14 U1 Indicates the answer could be positive native and gives a crect explanation A positive multiplied by 5 gives a native answer, but a native multiplied by 5 gives a positive answer Positive numbers will become native, native numbers will become positive If the number is 10 the answer will be 50, which is native, but if the number is 10, the answer is 50, ie positive! Minimally acceptable explanation 10 becomes native, but 10 becomes positive +ve F ve ve F +ve 5 3 = 15, 5 3 = 15 Incomplete explanation 5 3 = 15 The iginal number could be positive native so the answer could be positive native Makes an increct decision, no decision made, but explanation clearly crect Condone provided the explanation is me than minimal Page 17 of 28

18 Paper 2 Q Mark Crect response Additional guidance 1 Shows the crect rotation, ie! Lines not ruled accurate Accept slight inaccuracies in drawing (see general guidance) Equivalent fractions decimals U1 3 2m Completes all three rows crectly, ie Completes two rows crectly 4 2m 26 Answer of 26 Shows implies a complete method with not me than one computational err rounding err 35 x = 860 (err) = = 27.2 Answer = 27 ( x 24.75) = seen Shows the crect total f the trees, ie Shows the crect change, ie 8.75! Answer of 27 without a crect method shown implied Method used f 12.5 is repeated subtraction Do not accept as a crect method Page 18 of 28

19 Paper 2 Q Mark Crect response Additional guidance 5 U1 6 3m Indicates Nik and gives a crect explanation 1 sandwich, 2 apples and 1 banana is missing from the graph and that is what Nik had in his lunch box The graph shows the crect number of fruit bars and Nik is the only one who does not have a fruit bar in his lunch box so his must be the missing one The totals from the table are 7, 6, 5, 6, and from the graph 6, 4, 4, 6, and the difference is Nik Completes the drawing accding to the following conditions, with a tolerance of 3mm in each case the circle has a diameter of 8cm the highest point at which the circle crosses the central vertical line is 3cm from the top of the answer box the lowest point at which the circle crosses the central vertical line is 7cm from the bottom of the answer box!! Minimally acceptable explanation 1 sandwich, 2 apples, 1 banana Because the number of fruit bars is crect 1 banana missing 7, 6, 5, 6 and 6, 4, 4, 6 seen Increct incomplete explanation 1 sandwich, 2 apples There are 6 fruit bars 2 apples are missing Flag constructed upside down Shading increct omitted, additional lines drawn Condone, provided the response is unambiguous Compasses not used F pupils who meet one me of the conditions without using compasses, deduct ONE mark 3cm 8cm 7cm 2m Any two of the three conditions given above are crect Any one of the three conditions given above is crect Page 19 of 28

20 Paper 2 Q Mark Crect response Additional guidance 7 3m equivalent 2m Shows implies a complete crect method with not me than one computational err The most common crect methods: Find the total area of the trapezia and divide by 8 ( ) = 94 (err) 94 8 = Squaring evaluated as 2 ( ) 8 = (24 12) 8 Find the dimensions of a trapezium and use the fmula component parts 1 (3 + 6) (3 3) 2 The only err is to wk with 4 congruent trapezia (not 8), but crectly finds the area of one of them (144 36) 4 = 27 F 2m, 27 seen with no method = 9, 9 3 = 27 Shows implies a crect method to find the total area of the trapezia ( ) seen! Brackets omitted F, condone, accept = U1 Show the parallel sides of the trapezium are 3(cm) and 6(cm), and the height is 3(cm) Diagram marked crectly Page 20 of 28

21 Paper 2 Q Mark Crect response Additional guidance 8 The first 3 multiples of 10 add to 60 The first 4 multiples of 6 add to 60 The first 2 multiples of 20 add to 60 U1 The first 1 multiples of 60 add to 60 Page 21 of 28

22 Paper 2 Q Mark Crect response Additional guidance 9 2m 16.8p 17p equivalent Shows the digits ! Money See general guidance on page 8 Shows a complete crect method with not me than one computational rounding err (0) = 5.6 6p (premature rounding) 3 = 18 Page 22 of 28

23 Paper 2 Q Mark Crect response Additional guidance 10 Gives crect infmation f x = 4 4, too big 4, too high 4, too much above 1 Gives crect infmation f x = , too big Gives a logical value f the next trial, and justifies their decision 3.2, because I know it is between 3 and , it is halfway between 3 and 3 and a half 3.3 because it is bigger than 3 which was too small but smaller than 3.5 which was too big 3.4, it has to be smaller than 3.5 (that it is greater than 3 is implicit)!!!! Incomplete infmation that does not link to the value of 1 4, too increct In both the first and second answers, shows crect values but omits gives increct further infmation 4, too small 1.75, too Do not award the first mark, but award the second mark Value rounded Accept 1.8 Do not accept 1.7 Logical values Accept any of the following: Also accept any value between 3.3 and 3.4 provided their justification shows why the solution is between these values, accept (since a further trial has clearly taken place) 3.35, 3.3 is too small 3.302, because is just over 1, do not accept 3.35, because I know it is between 3 and 3.5 Minimally acceptable justification 3.2, 3.5 is too big Incomplete justification 3.3, it gets closer to because it is at an appropriate interval F the third part, follow-through If their calculation in the second part f x = 3.5 was too small, accept x = 3.6, 3.7, 3.75, alongside an explanation comparable with those given in the mark scheme Page 23 of 28

24 Paper 2 Q Mark Crect response Additional guidance ± 1 then 80 ± ± 1 U1 12 Describes the key features of the infmation U1 2m are available, one from each of the caties A and B below: Caty A States that the rate the mass of the dog increases slows as it gets older They get heavier in their first few months but as they get older their weight doesn t go up as much Caty B Makes an observation that links the infmation in the bar chart to the adult mass It reaches adult size after the first year A dog is about half grown when it is 4 months old! Minimally acceptable explanation, f caty A Grows quickly then me slowly After a few months the amount it increases by gets smaller [accept any value from 4 8 months inclusive within this type of response] They start by gaining about 5kg per month but this gets less and less, f caty B Doesn t get any fatter after it is a year old They stop at 12 months At 6 months, it s me than half-sized, f both caties (ie 2m) It grows quickly then slowly until 12 months when it stops Values given As this question is assessing understanding of infmation presented graphically, condone increct numbers f caty A, but do not accept f caty B, f caty A, accept They increase by about 10kg per month but not as much as they get older, f caty B, do not accept A dog is about half grown after half a year Incomplete explanation, f caty A Dogs get heavier as they get older [doesn t say how rate of change alters], f caty B A German Shepherd stops growing when it reaches 35kg [no link to 12 months] It grows quickly then slowly until 12 months [gains caty A mark but no link to full weight being reached f caty B] Page 24 of 28

25 Level threshold infmation This section provides infmation about interpreting the sces from the Optonal level 6 tests in mathematics. In der to make use of the infmation in this section, you should administer the tests accding to the guidance given in the test administrats guide. The guide can be downloaded from the NCA Tools website at: It is particularly imptant that you observe the time limits given in the test instructions, and mark questions strictly accding to the mark scheme. If not, the infmation derived from this section cannot be used reliably. The table below gives an indication of the national curriculum level f pupils, based on their sce in the test. In der to use this infmation, the total sces on paper 1 and paper 2 should be added together. Mathematics test (maximum mark 50) Sce Outcome 0-24 marks Level 6 not achieved marks Level 6 achieved Page 25 of 28

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