Mark scheme for Paper 2

Transcription

1 Ma KEY STAGE 3 ALL TIERS 2000 Mathematics tests Mark scheme f Paper 2 Tiers 3-5, 4-6, 5-7 and 6-8 KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST

2 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Index Index to mark schemes Tier Question Page 1 Menu 10 2 Making Shapes 11 3 Calculations 12 4 Sun Cream 12 5 Rulers 13 6 Measuring Shapes Tokens Temperatures Coaches Cereal Drawing Huts Canteen Percentages B Sign Teachers Lift Sces Polygons Hedging Area Pitch Dots Families Triangles Counters Satellites Homewk Pyramid Pots 35 2

3 2000 KS3 Mathematics Test Mark Scheme: Paper 2 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 2 at all tiers. The paper 1 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 10 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available 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

4 2000 KS3 Mathematics Test Mark Scheme: Paper 2 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

5 2000 KS3 Mathematics Test Mark Scheme: Paper 2 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

6 2000 KS3 Mathematics Test Mark Scheme: Paper 2 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

7 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Introduction Responses involving money Accept Do not accept F example: Any unambiguous indication of the crect amount 3.20(p), 3 20, 3,20, 3 pounds 20, 3-20, 3 20 pence, 3:20, 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 320p 700p Increct ambiguous use of pounds pence 320, 320p 700p, p not in answer space. Increct placement of decimal points, spaces, etc increct use omission of 0 3.2, 3 200, 32 0, Responses involving the use of algebra Accept Take care! Do not accept F example: 2 p n n p 2 2n The unambiguous use of a different case N used f n Unconventional notation f multiplication n t 2 2 t n n2 n p n f 2n, n t n f n 2 Multiplication by p 1n f 2 p n, 2 p 0n f 2 Wds used to precede follow equations expressions t e n p 2 tiles tiles e t e n p 2 f t e n p 2 Unambiguous letters used to indicate expressions t e n p 2 f n p 2 Embedded values given when solving equations 3 t 10 p 2 e 32 f 3x p 2 e 32! 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 2 n cm p 2 Change of variable x used f n Ambiguous letters used to indicate expressions n e n p 2 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 2 e 32, 3 t 10 p 2 e 32, x e 5 7

8 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Introduction Responses involving time Accept Take care! Do not accept A time interval F example: 2 hours 30 min Any unambiguous indication 2.5 (hours), 2h 30 Digital electronic time ie 2:30 Note that 2:30 is accepted f 2h 30m because it is a common electronic expression ( the time interval shown on an oven timer). Increct ambiguous time interval 2.3(h), 2.30, 2-30, 2h 3, 2.30min! 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

9 2000 KS3 Mathematics Test Mark Scheme: Paper 2 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 2m 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 120 marks is available in each of tiers 3-5, 4-6 and 6-8, and a total of 121 marks in tier 5-7. The extension paper carries 41 marks. Awarding levels The sum of the marks gained on paper 1, paper 2 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 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 2000 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 9

10 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3-5 only 1 Crect response Menu a ! Follow through from an increct total Allow provided the total is me than 1, and is not an intral number of pounds. b 16 10

11 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3-5 only 2 Marking overlay available Crect response Making Shapes a F part (a), the example repeated b A line through a vertex and a side! Line not ruled but intention clear In parts (a) to (d), penalise only the first occurrence.! Line not drawn accurately Accept lines to a vertex to ± 2mm. However, lines to a side must be unambiguously to a side rather than to a vertex, hence do not accept lines that are within ± 2mm of a vertex. c A line through opposite sides d e Four congruent squares, each midpoint and line within ± 2mm, and ruled, ie 11

12 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3-5 only 3 a 662 Crect response Calculations b 6000 c 483 d 56 Answer 56 4 a 8 Crect response Sun Cream b UK Unambiguous indication Blackpool. A warm place. Medium Abbreviation M 12

13 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3-5 only 5 Crect response Rulers a 120! Increct units Igne. b 11.60! Both money answers omit final zero Mark as 0, 1 c 2.90 d 5! Increct units Igne. 6 a 190 ± 1 Crect response Measuring b Crect place identified Within ± 2mm Any unambiguous identification Scale redrawn using an easier numbering system 13

14 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, a a Area 5 Crect response Shapes Perimeter 12 b b Any shape of area 6cm 2 Shape connected at vertices Accept if unambiguous c c Crect perimeter Note: If the pupil uses whole squares, aligned with complete edges touching, the perimeter is 10, cm.! Follow through from increct shape using whole squares Allow provided the area > 4 cm 2 and the shape is not a copy of the diagram in (a).! Follow through from shape using diagonals Allow measuring, ± 2mm, but do not allow answers rounded to the nearest centimetre unless a me accurate value is seen. Follow through from shape with an enclosed space d d 7 e e Explains that the diagonals of the grid are greater than 1 Minimally acceptable explanation Because the lines go through the middle of a square. Diagonal measured as 1.3 to 1.5 cm inclusive Perimeter measured as 9 to 10 cm inclusive Partial response I measured the perimeter. 14

15 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, Crect response Tokens a a Crect explanation focusing on me gold 4 gold and only 1 silver. Not as many silver. Gold to silver is 4 to 1 Explains there would need to be an equal amount of each colour There s not the same number of gold and silver. Only one silver. There should be 4 Minimally acceptable explanation Better/Me chance of getting gold. Only one silver. Crect probability expressed in wds At this level, accept It s a 1 in 5 chance of getting silver. Increct infmation, even if accompanying a crect response Me gold, it s a 1 in 4 chance of getting silver. Me gold, so she must take out a gold. Infmation restated with no indication of me gold 4 gold and 1 silver. Use of even f equal b b 3 Gold and silver inserted in the crect proptions 2 gold, 5 silver. c c At least one of 5, 6, 7 8 Any unambiguous indication Tokens drawn. A crect range Me than A crect value expressed as a ratio fraction of Not quantified Me gold than silver. 15

16 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, a a 77 Crect response Temperatures 80 b b 32 and 30 in the crect der. c c 2m Shows both rules give a value of t 1.8 p 32 e 50, 10 t 2 p 30 e 50 Minimally acceptable response 50, seen! Increct units Igne Crect response Coaches a a 2m (..) seen 58 shown as a minimum 58 me d 52 seen b b c c 8.12! Follow through as their (a) t 420 If their answer to (a) is not an inter, accept their (a) rounded truncated, and accept the answer then rounded truncated to the nearest penny. Follow through from their part (b) ie (b) d 3000 Answer from their (b) rounded truncated to the nearest penny 16

17 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, 4-6, Crect response Cereal a a a 1, equivalent probability 4 1, equivalent probability 2 Crect response accompanied by description of the probability Igne the description, accept 25%, that s fairly likely. b b b 0, equivalent probability 2, equivalent probability 3 Probability of zero expressed in wds as a fraction, even if the denominat is increct, as a ratio None. Impossible : a a a Crect response Drawing Freehand Accept if the pupil s intention is clear. Hidden lines made visible b b b Internal lines omitted, f part (c) c c c External edges omitted, f part (d) d d d! Shading Igne. 17

18 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, 4-6, Crect response Huts a a a 2m 33 Crect method 4 t 8 p 1 F, method is repeated addition with not me than one computational err 13 p 4 p 4 p 4 p 4 p 4 17, 21, 25, 29, 32 b b b 2m 20 Crect method 80 d 4 seen c c c Crect expression of m e 5h p 1 18

19 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, 4-6, Crect response Canteen Gives a crect explanation. The most common crect explanations are: Explaining events are not equally likely Not many people wk in the canteen. They might not be equal chances. The probability is different f each group. There are different amounts of pupils and teachers. The number of pupils is me than one third. The probability needs to be out of all the pupils, teachers and canteen staff. Explaining the statement implies equal numbers of pupils, teachers and canteen staff It would be true if there were 20 pupils, 20 teachers and 20 dinner people. Giving a counter-example Suppose there were 190 pupils, 8 teachers and 2 canteen staff. The probability would not be a third.! Explanation infers exact quantities required Accept if accompanied by a crect response, accept Each probability is different. You need to know the numbers in each group., do not accept You need to know the exact numbers in each group. Increct statement, even if accompanied by a crect response It s not equal chances, the probability is 1 divided by the whole school. It depends on how many children there are. If there were 10 children the probability would be 0.1 Incomplete ambiguous statement Me pupils. There is me than 1 pupil, 1 teacher and 1 canteen staff. Me than 3 people. There are 3 choices but there s me than 3 papers in the box. It depends on how many pupils, teachers and canteen staff there are Crect response Percentages B ! Redundant % sign 2.12% Penalise first occurrence only. 25p expressed as a fraction of a pound Crect response Sign 8 Answer between 8 and 8.1 inclusive 19

20 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3-5, 4-6, Crect response Teachers a a a 20 to 23 inclusive b b b 35 to 39 inclusive c c c to inclusive d d d Indicates second statement with a crect justification. The most common crect justifications are: Comparing at least one caty f both males and females There are me year old females than males. Me males are over 50 Females start with bigger slices but then it changes. The striped part is longer on the females than the males. Me black and striped part f females than males. Comparing caties by using percentages within the inclusive ranges shown below, reference to the approximate value shown in brackets (do not accept approximation without indication that it is only an estimate) 13% of men are age 20-29, but it s me like 20% f women. 50% of men will be over 50, but only about 40% of women. males females 50p (40) (20) (10) (10) (20) 39 less (30) less (60) 40 me (70) me (90) (80)! Response does not refer to the chart Accept only if accompanied by a crect response There are me year old females than males and there are me young female teachers in my school.! Different caties compared f males and females Accept only if one implies the other, accept 50% of males are 50p, but over 50% of females are younger than 50, do not accept 50% of males are 50p, about 40% of females are younger than 30! Use of young old without caties specified Accept only if justification implies which caties are being compared, accept 13% men, 23% women are young. Also accept young to refer to the first, the first two, the first three caties, accept 62% females young, only 50% males. No comparison 50% of males will be 50p Increct statement Less females in each group. Female teachers most likely to be Ambiguous response with caties gender not identified There are me 50p but less Female is higher on the chart. 20

21 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4-6, 5-7, Crect response Lift a a a Ground flo (0) and 12, either der b b b 60 ± 2 c c c A line from (80, 22) to (125, 0) that has no positive gradients. Extends the hizontal line at flo 22 befe descending Accept provided the descent takes 45 seconds with no further stops. Line from (75, 22) to (120, 0) Line not ruled but intention clear Parts of the line show acceleration and deceleration a a a 6 Crect response Sces b b b 1 and 5, either der c c c 2m Any set of three numbers that total 9 and have a range of 4 1, 3, 5 1.5, 2, 5.5 Their three numbers total 9 Their three numbers have a range of 4 21

22 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4-6, 5-7, Crect response Polygons a a a Crect explanation The most common crect explanations refer to: The angles in a triangle summing to 180 Each triangle is 180 and 180 t 2 e 360 The crect use of a relevant fmula such as 180(n m 2) (2n m 4) right angles 180(4 m 2) 2 t 4 m 4 e 4, and 4 t 90 e 360 Minimally acceptable explanation 2 t 180 Each triangle is 180 Explanation lacks generality Specific quadrilaterals used as examples. 4 t 90 e 360 No evidence given Because all 4-sided shapes have 360 Incomplete use of external angles If you turn all the way round the shape you turn 360 Use of cners Cut the cners and put them together and it makes a complete turn. b b b 540 c c c 2m 900! Follow through as part (b) p 360 Allow, provided (b) > 360 Crect method 5 t t (7 m 2) 360 p 360 p p 540! Throughout the question, the only err is to use an increct, but consistent, value f the number of drees in a triangle Mark part (c) as 1, 0 provided (c) >

23 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4-6, 5-7, a a a 2m 78.(0..) Crect response Hedging Digits 650(00) d digits 833 Digits 78(0..) seen b b b 2m (..) 89 Digits 2437(5) d digits 2751(15)! Answer 88 Accept provided there is no evidence of an increct method. An otherwise crect response, with the decimal point omitted increctly placed. c c c Privet, with digits 17(..) and 13(..) seen. Privet, by comparing unit prices f both plants One privet is 1.3, so 125 would cost 162.5, so privet is cheaper. 5 privet cost 6.50, 5 beech cost 8.50 so beech is me expensive. Privet, d 125 is bigger than 45.5 d 35 4 t 35 e 140 and 4 t 45.5 e 182, so me privet f less money than beech. Privet, by using ratio to compare prices (condone the use of rounded/truncated values) 125 d 35 e 3.57, t 45.5 e so privet cheaper. 125 d 35 e 3.57, 212 d 3.57 e 59.4, privet. Use of rounded truncated values f and/ and 45 used, resulting in rounded truncated f beech, and rounded truncated f privet. Use of rounded truncated values f intermediate values! Conclusion not shown Accept only if prices are crect and identified with the crect plant, accept Privet 1.3, beech 1.7, do not accept Digits 17(..) and 13(..) seen without linking to relevant plants.! Plants per pound calculated Accept only if the crect interpretation is shown, accept You d get beech plants with one pound, and privet plants f one pound so privet is cheaper., do not accept 125 d e 0.588, 35 d 45.5 e 0.769, so privet is cheaper. 23

24 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4-6, 5-7, Crect response Area a a a 2m 452 Crect method π t 12 2 π t 12 t (..) 144π Use of mm 2 as evidence of t 12mm 2 b b b 226 Follow through as part (a) d 2! Answer not rounded to the nearest mm 2 Accept if their answer to part (a) was 452.(..) 144π, ie this err has already been penalised. c c c 2m (..) Crect method their (b) (72π)! F 2m, follow through as (their b) Accept answers rounded truncated, provided there is no evidence of an increct method.! Method is trial and improvement Do not penalise as an increct method, but do not credit as a crect method. 24

25 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Pitch a a 92.5 Use of extra zero(s), f part (a) b b 49.5 Use of centimetres, f part (b) 49m 50cm c c 2m Calculates 3000 d 2(their a p their b) Crect method seen, ie 3000 d 2(their a p their b) Uses their (a) and their (b) rounded to at least 1 d.p., with a crect follow through answer. Uses the values 93 and 50 to give an answer of (..) 286 t 11 e 3146, so 11! Answer given with little no wking F 2m, accept (..) F 2m, accept 11 with sight of t (92.5 p 49.5) Do not accept 11 with no relevant wking.! Answer rounded truncated Accept to the nearest inter, rounded to at least 1 d.p. Do not accept a truncated value unless a me accurate value is seen. Method is trial and improvement F 2m, accept trials of 2(their a p their b) leading to a crect solution. The minimum amount of wking is their solution t 2(their a p their b) > t 284 is me than 3000! Use of 93 50, with one of their (a) (b) F, accept their crect answer with a crect method seen.! Use of 1km rather than 3km Only accept as a valid method if subsequently multiplied by 3 Use of 300 f

26 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Dots a a 2m All points with x 0 and y 0 identified Crect rion identified, f part (a) All points with native co-dinates identified, even if some all of the points with zero dinates are omitted, and no increct points. Not me than three values omitted.! Points identified with alternative notation Accept if unambiguous, f example in (b) crosses with a square around.! Points in parts (a) and (b) identified using the same notation If both parts are completely accurate, mark as 1, 1 in (a) and 1 in (b). If the only err is that some all of the points with x e 0 and y e 0 are omitted, mark as 1, 0 in (a) and 1 in (b). b b All points with x p y 4 identified! All points other than (2, 1) identified using the same notation Mark as 1, 0 in (a) and 1 in (b).! Remaining points f which x > y identified using a different notation Igne. c c ( 2, 1 ) 26

27 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Families a a Shows a crect total f each bar 1 t 1 p 2 t n p 5 t 3 p 6 t 4 p 3 t 5 1, 2n, 15, 24, 15 Fully explains 2n with reference to the context There are 2n children in the families that have 2 children. There are 2 children in n families.! Crect wking f each bar shown, but subsequent wking marred by computational err Accept provided there is not me than one such slip 1 t 1 e 1 5 t 3 e 5 6 t 4 e 24 3 t 5 e 15 2 t n e 2n Fully explains 55 1 p 15 p 24 p 15 e 55 b b Crect expression 15 p n Expression not simplified 1 p n p 5 p 6 p 3 c c 2m 10 Crect equation fmed 55 p 2n 3 e their (b) F, follow through from part (b), even if non-algebraic 27

28 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Triangles a a Crect explanation The most common crect explanations are: Showing implying that 6 2 is added to 8 2, with either 10 2 the use of 6 2 p 8 2 e p 8 2 e 100, 100 e is 100, and 100 e 64 p 36 AB 2 p BC 2 e 100, e 10 AB 2 p BC 2 e AC 2 Answer found through scale drawing Do not accept in any part of this question. Incomplete explanation that does not refer to either 10 2 r AB 2 p BC 2 e p 64 e 100 e 10 Referring to the 3, 4, 5 triangle Each side is double 3, 4, 5 It s the 3, 4, 5 triangle. The 3, 4, 5 triangle must have a right angle. b b 2m r (..) Complete crect method (6 2 p 10 2 ) 136 e (increct) Minimally acceptable explanation 6, 8, 10 triangle. Incomplete explanation Because of Pythagas. If it wasn t 10 it wouldn t be rightangled. a 2 p b 2 e c 2 (without linking to the diagram).! Answer Accept only if a valid method, me accurate response, seen. Use of tangent to find an angle, then crect use of sine cosine Partial Method If Pythagas is used, the square root must be seen implied. Do not accept AD 2 e 136 as sufficient. 28

29 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6-8 only 9 c 3m 6 5.9(..) Crect response Triangles (cont) 2m Both angles crect, ie x as (..), and y as (..) Complete crect method tan m tan cos m cos 1 (10 d their b) One angle crect. Crect trigonometric ratios f both angles identified tan y e 0.6, tan x e 0.75 tan e 6 d 10, tan e 6 d 8! Increct evaluation of angles F 2m, accept provided crect trigonometric ratios are shown, and the angles are subtracted tan x e 6 d 8, x e 43 tan y e 6 d 10, y e m 34 e 9 Misunderstanding of trigonometric ratio tan x e 6 d 8, x e

30 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Note that as there are many alternative crect justifications, f ease of use this mark scheme shows caties of response, with and 0m responses shown alongside. Counters 2m Chooses A; justifies using fractions of totals of 30 and 26 then converting to fractions that have common denominats e, but e, choose A e, e which is less, so A common numerats e, B is, A me chance e, e so A Unconventional fractions used In this context, accept, f 2m A is, B is so A A because > A because e, > Converts to a fm that enables comparison, but makes an increct no conclusion e, but e so choose B Shows a complete crect method with only one computational err, then chooses the crect bag f their calculation e e but B is, so choose A e e but B is, so choose B Chooses bag A and partially justifies by cancelling both fractions crectly, even if not to their simplest fm e, e, so A A because e, B is Fractions not cancelled A is, B is, so A

31 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5-7, Crect response Counters (cont) 2m 2m 2m Using totals of 30 and 26 then converting to decimals percentages e 0.4, e 0.38, so A > 0.38(..), A best. A is 40%, B is 38%, choose A Comparing an amount of counters If A had 26 counters it would be 12 t 26 e 10.4, that s less so choose A 30 A: red is 40%, 40% f B would be only 10.4 red, that s smaller. B is 38% red, 38% of 30 e 11.4 so A best. Using ratio of red : yellow in the fm of a : b The ratio must enable comparison 12 : 18 would be the same as 10 : 15 but B has an extra yellow, so it must be bag A In A, red to yellow is 1 : 1.5, in B it s 1 : 1.6 so I d choose A Ratio R to Y is : 1 f A, : 1 f B, so A is better.! Decimals percentages rounded truncated Accept provided a comparison can still be made.! Partial credit F, mark as f the previous page ie Crect method, with increct no conclusion. Crect method, with one computational err, followed by their crect conclusion. 2m Using totals 18 and 16 ( 30 d 12 and 26 d 10) Note that although these can be crect methods, many pupils apply these numbers with no understanding, hence this caty is marked as follows. F 2m, crect interpretation must be shown and the ratio must enable comparison e, e If there were 144 yellow there would be 96 red in A but only 90 in B, so choose A Red 12 e Yellow 18 e 0.67, but Red 10 e Yellow 16 e 0.625, so A Ratio red over yellow is e e, f B it s e e, so A d 12 e 2.5, so f every red ball there are 2.5 balls in bag A. It s 2.6 f B so choose A! F this caty only, chooses A, but no interpretation shown Accept f only e, e so A d 12 e 2.5, 26 d 10 e 2.6, A Increct interpretation shown 12 2 A: Probability of red is e B: Probability of red is e, so A d 18 e 66% chance of getting red, 62.5% f B so A 30 d 12 e 2.5 red, 26 d 10 e 2.6 red, A 31

32 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6-8 only 11 Crect response Satellites a 2m 5.82 t 10 6 Digits 582(..) seen m 5.82 t 10 6 Shows and , then makes no me than one computational err when subtracting, then crectly converts their answer into standard fm m e e 5.72 t 10 6 Answers rounded, f part (a) 6 t 10 6, f part (b) 2.5 t 10 7 F (a), minimises A and maximises B F 2m, accept t 10 6 F, accept digits 576(..) seen. F (a), minimises 5.82 F 2m, accept t 10 6 F, accept digits 5815 seen.! Unconventional standard fm notation Penalise the first occurrence only. b 2m t 10 7 Digits 247(..) 248 seen F (b), maximises A and maximises B F 2m, accept 2.48(..) t 10 7 F, accept digits 248(..) seen. Shows 2.4 t 10 7 Shows and , then makes no me than one computational err when adding, then crectly converts their answer into standard fm p e e t

33 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6-8 only 12 Crect response Homewk a 2m Complete crect method, using mid-points , d 50 6 t 15 p 14 t 45 p 21 t 75 p 9 t 105, then d p 630 p 1575 p 945 is 3240, and 64.8 t 50 e 3240! 3240 seen with no wking As this could come from 64.8 t 50, allow 1 mark only. Showing that at least 2 mid-points are multiplied by the frequency, even if the others are increct omitted. An otherwise complete crect method with clear intent to use the mid-points, but inaccurate values used d 50 Mid-points used increctly 15 p 45 p 75 p 105 e 240, d 50 e 4.8, and 4.8 t 13.5 e 64.8 b 2m 67 ± 1 F 2m, 67 seen then rounded to 70 Crect method seen implied Vertical line seen. Crect marking on the x-axis. Crect point identified on the graph and a value of between 63 and 70 inclusive given. The hizontal line only seen c 2m 4 46 seen Value between 3.5 and 4.5 The crect hizontal line shown implied on the graph, with the scale misinterpreted but leading to a value of less than 10 33

34 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6-8 only 13 a 50 Crect response Pyramid Inaccurate answer due to rounding of one third, in part (b) 9.06 b c 2m 2.5 equivalent Complete crect method 1 25 e.b e b b e 4 (25 t 3 d 12) 1 (6 ) 4 d Crect fmula V e V e m m2 m 2 Fmula not explicit f V 6V e m 3 m e 3 (6V) Fmula omits V m

35 2000 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6-8 only 14 Crect response Pots a , equivalent probability % b 2m , equivalent probability Crect method (0.03 t 0.97) t equivalent! 0.06 Do not accept unless a crect method, a me accurate value, is seen (1), equivalent probability, seen. c Yes, with justification The most likely justifications involve: The use of t 80 e t 80 e 2.4 Yes, since only 2.4 will crack. Comparing probabilities Yes, because 5 in 80 is me than 3 in 100 Increct use of t 75 e m 2.25 e 77.75, so that s enough. The use of 75, with a crect explanation interpreting the calculation 0.03 t 75 e 2.25 so that s only 72 made, but there s enough f 5 me pots and with such a low probability you wouldn t expect me than one to crack t 75 e 2.25, that gives you enough f 77 pots t 75 e 72.75, but with the clay f 5 me pots you are going to break one so you ll have enough. The use of 100 As 3 broke in every 100, there will be enough. As 3 in every 100 break, it s not likely 5 will break. Yes, because only 3 will break. Yes, only about 2 will break. 35

36 EARLY YEARS NATIONAL CURRICULUM 5 16 GCSE GNVQ GCE A LEVEL NVQ First published in 2000 Qualifications and Curriculum Authity 2000 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 2 of the Charities Act Qualifications and Curriculum Authity 29 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 2SN (tel: ; fax: ) Order ref: QCA/00/