Mark scheme for Paper 2

Transcription

1 Ma KEY STAGE 3 ALL TIERS 2006 satspapers.g Mathematics tests Mark scheme f Paper 2 Tiers 3 5, 4 6, 5 7 and

2 satspapers.g 2006 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 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking infmation f questions is set out in the fm of tables, which start on page 12 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. The column indicates alternative acceptable responses, and provides details of specific types of response that are unacceptable. Other guidance, such as when follow through is allowed, is provided as necessary. Questions with a Using and applying mathematics element are identified in the mark scheme by an encircled U with a number that indicates the significance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. 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. The 2006 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at Edexcel. 2

3 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance General guidance 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 on the following two pages with the prescribed crect action. This is followed by further guidance relating to marking of questions that involve money, native numbers, algebra, time, codinates probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3

4 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance What if The pupil s response does not match closely any of the examples given. Markers should use their judgement in deciding whether the response cresponds with the statement of requirements given in the Crect response column. Refer also to the. 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 has made a conceptual err. 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 t 6 e 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 35 t 27; subtracting the smaller value from the larger in calculations such as to give the answer 21; increct signs when wking with native numbers. 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. If the iginal intention difficulty level of the question is not reduced, deduct one mark only. If the iginal intention difficulty level is reduced, do not award any marks f the question part. 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 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance What if 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 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. 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 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance Marking specific types of question Responses involving money F example: Accept 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 unit, p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units with given in the answer space, accept Given units amended with crossed out in the answer space, accept 320p 700p Do not accept Increct ambiguous indication of the amount 320, 320p 700p Ambiguous use of units outside the answer space with given in the answer space, do not accept 3.20p outside the answer space Increct placement of decimal points, spaces, etc increct use omission of 0 3.2, 3 200, 32 0, Responses involving native numbers F example: 2 Accept Do not accept To avoid penalising the err below me than once within each question, do not award the mark f the first occurrence of the err within each question. Where a question part carries me than one mark, only the final mark should be withheld. Increct notation 2 6

7 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving the use of algebra F example: 2 p n n p 2 2n n 2 n 2 Accept Take care! Do not accept Unambiguous use of a different case variable N used f n x used f n! Unconventional notation n t 2 2 t n n2 n p n f 2n n t n f n 2 n n 1 d 2 f n p 1n f 2 p n 2 p 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 p 2 = 32, 3 t 10 p 2 = 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 part carries me than one mark, only the final mark should be withheld. 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! Wds units used within equations expressions n tiles p 2 n cm p 2 Do not accept on their own. Igne if accompanying an acceptable response. Ambiguous letters used to indicate expressions n e n p 2 f n p 2 7

8 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving time A time interval F example: 2 hours 30 minutes Accept Take care! Do not accept Any unambiguous indication 2.5 (hours), 2h 30 Digital electronic time ie 2:30 Increct ambiguous time interval 2.3(h), 2.30, 2-30, 2h 3, 2.30min! The unit, hours and/ minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, crosses out the given unit, accept answers with crect units, unless the question has specifically asked f other units to be used. A specific time F example: 8:40am 17:20 Accept Any unambiguous, crect indication 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Unambiguous change to hour clock 17:20 as 5:20pm, 17:20pm Do not accept Increct time 8.4am, 8.40pm Increct placement of separats, spaces, etc increct use omission of 0 840, 8:4:0, 084, 84 Responses involving codinates F example: ( 5, 7 ) Accept Unconventional notation ( 05, 07 ) ( five, seven ) x y ( 5, 7 ) ( x e 5, y e 7 ) Do not accept Increct ambiguous notation ( 7, 5 ) y x ( 7, 5 ) ( 5x, 7y ) ( 5 x, 7 y ) ( x m 5, y m 7 ) 8

9 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only 7 F example: % 10 Accept Take care! Do not accept Equivalent decimals, fractions and 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 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 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 9

10 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 General guidance 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. A total of 121 marks is available in tier 5 7. Awarding levels The sum of the marks gained on paper 1, paper 2 and the mental mathematics paper determines the level awarded. Level threshold tables, which show the mark ranges f the award of different levels, will be available on the NAA website from Monday 19 June NAA will also send a copy to each school in July. Schools will be notified of pupils results by means of a marksheet, which will be returned to schools by the external marking agency with the pupils marked scripts. The marksheet will include pupils sces on the test papers and the levels awarded. 10

11 2006 KS3 Mathematics test mark scheme: Paper 2 satspapers.g BLANK PAGE 11

12 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 1 Crect response Matching 2m Matches all four sets of wds to the crect numbers, ie thirty-six 3006! Set of wds matched to me than one number F 2m, do not accept as a crect match three hundred and six 36 three thousand and six 306 three thousand and sixty 3600 three thousand six hundred 3060 Matches at least two sets of wds to the crect numbers 12

13 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 2 a 7 Crect response Pupil list b Huw Davies c Leroy Tayl Unambiguous indication, f part (b) Huw Davies 21/11/92, f part (c) Leroy LT 06/10/92 d Gives the crect date 07/01/93 7 Jan 93! Date given in different fm Accept only if unambiguous commonly used, accept 1/7/93 [US notation] Year not given 7 th January 13

14 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 3 Crect response Thinking angles a Indicates Angle d, ie b Gives a crect explanation It s a right angle It must be 90 Minimally acceptable explanation Right Quarter turn! Units increct omitted 90 C 90% 90 Condone Incomplete explanation It s a square angle It s a cner 14

15 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 4 Crect response Moving on a grid a Gives the crect direction South 1 1 S! Crect compass point(s) indicated, but indication of the number of squares to move increct omitted Penalise only the first occurrence, f parts (a) and (b) South 2 [f part (a)] then Nth 1 East 2 South 3 [f part (b)] Mark as 0, 1 b Gives all three crect directions in a crect der to fm a square Nth 1 East 1 South 1 1 S 1 E 1 N! F part (b), response uses additional directions but a square is still fmed West 1 [repeated] South 2 East 2 Nth 2 Condone 5 Crect response Cards a 2.60! Final zero omitted Provided this is the only err, penalise only the first occurrence b 6.10! Value given in pence without the cresponding change in units Provided this is the only err, penalise only the first occurrence c Gives a crect pair of codes in either der, ie C and D B and E Unambiguous indication, f C and D Digits 165 and 195 C and 1.95, f B and E Digits 125 and 235 U1 Gives a crect pair of codes, other than any previously credited 15

16 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 6 a 3 Crect response Tennis b Ed Unambiguous indication E c Gives a crect explanation that one person cannot play against themselves You can t play against yourself It s where each person is matched with themselves, so there is no game It s Ann v Ann, Bob v Bob etc and that s impossible There are five people so only four possible games each U1 Minimally acceptable explanation It s a person matched with themselves It s Ann v Ann There are only four possible games each Incomplete explanation There can t be a game They didn t play It s impossible 16

17 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 7 Crect response Joining points a Joins only four points to make a square! Lines not ruled accurate Accept provided the pupil s intention is clear! Points crectly indicated but line(s) increct omitted Penalise only the first occurrence b Joins only three points to make an equilateral triangle c Joins only three points to make an isosceles triangle! Equilateral triangle made f part (c) Accept provided a set of three points other than one credited f part (b) is used 17

18 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Mirr lines 2m Reflects the triangle crectly in both mirr lines, completing the triangles in all three quadrants crectly, ie! Lines not ruled accurate Accept provided the pupil s intention is clear mirr line Completes the triangles in any two of the three quadrants crectly Makes an err in the position of one triangle, and follows through crectly if the increct image may have been used f further reflection mirr line F, err in the ientation of a reflected triangle mirr line mirr line Makes an err in the position of one vertex, but still draws a right-angled triangle with the hypotenuse in the crect ientation, and follows through crectly if the increct image may have been used f further reflection mirr line mirr line 18

19 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, a a 20, 28 Crect response Using rules 36, , 14 equivalent 2! First new term f each sequence crect, with second terms all increct omitted Mark as 0, 0, 1 b b Indicates No and gives a crect explanation The most common crect explanations: Show that the rule does not wk f the third term It doesn t wk f the second two numbers, 22 m 8 e 14 not 18 If it was subtract 8, the last number would be 14 It s 22 m 4 e 18, not 22 m 8 22 m 18 e 4 not 8 Minimally acceptable explanation 22 m 8 e 14 When you take away 8, it should be should be 14 The third number should be m 8 18 It s 22 m 4 18 to 22 is 4 Incomplete increct explanation 18 is wrong It should be 14 It doesn t wk f 22 and 18 You subtract a different number the second time 8 m 22 e m 8 e 15 State what the crect rule could be It should be divide by 2, then add 7 The rule is add 14 then halve it You take away half as much each time Minimally acceptable explanation d 2 p 7 It s take away 8, then take away 4 m8 and m4 You halve what you subtract U1 Incomplete increct explanation You subtract a different number each time You subtract 4 The rule is subtract 4 Take away half 19

20 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Cough mixture 2m Gives a crect justification that shows implies there is not enough cough mixture The most common crect justifications: Refer to the amount needed f 5 days Adult: 10 t 4 t 5 e 200 Children: 5 t 4 t 5 e 100 but there is only p 100 e 300, so no You need 300ml You need 60ml f each of the 5 days, and the bottle only holds 250ml You need 50ml me 250 m 40 m 40 m 40 m 40 m 40 e m 20 m 20 e 10, so the child will not have enough f the last 3 days Refer to how long the bottle will last how many doses it will provide Each day they need 60ml so there is only enough f just over 4 days It will last about 4 days They need 15ml each time, but 250 d 15 < 20 t 15 There is only enough f 16 doses, but they need 20 Minimally acceptable justification 200, 100 so no (10 p 5) t 20 > needed 50 too little 250 m 200 e m 20 e m 20 e 10, so there s not enough Only 4 days F 2m, incomplete justification 200, m m 200 e 50, 50 m 20 e 30, 30 m 20 e 10 4 days It will run out 20

21 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Cough mixture (cont) Shows implies a crect method f the amount needed f 5 days, f how long the bottle will last, with not me than one err 300 seen [no decision] 300, there is enough [increct decision] 200, 100 [no decision] m50 seen [no decision] 4 days [no decision] 16 doses [no decision] 10 t 4 e 40, 5 t 4 e 30 (err), 40 p 30 e 70, 70 t 5 e 350, not enough [computational err] 10 p 5 = 15, 15 t 4 t 4 (err) e 240 needed so there is enough [err in number of days as 4] 10 p 5 then t 5, so yes [err in number of doses per day as 1] 75, so yes [err in number of doses per day as 1] Shows implies a crect method f finding the amount f one adult f 5 days 10 t 4 t 5 40, 40, 40, 40, , with no evidence of an increct method Shows implies a crect method f finding the amount f one child f 5 days 5 t 4 t 5 20, 20, 20, 20, , with no evidence of an increct method U1 Shows implies a crect method f finding the total amount needed per day 60, with no evidence of an increct method 10 p 5 then t 4 40, 20 21

22 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Wking with areas Draws a rectangle of area 6cm 2! Lines not ruled accurate Accept provided the pupil s intention is clear Grid lines used as side(s) of shape, f the first mark Draws a rectangle of area 4cm 2! Draws shapes f both grids with crect areas that are not rectangles Provided the given shapes are not repeated, mark as 0, 1! Shows implies the totals 6 and 4, but shapes are increct omitted 6, 4 seen Rectangles transposed but otherwise crect Mark as 0, 1 22

23 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, a a Whale b b Seal Crect response Unambiguous indication, f part (a) W 365 Prnancy c c Dolphin Crect response Missing numbers

24 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Hexagons 2m Indicates only the three hexagons, ie F 2m, unambiguous indication f a hexagon, f not a hexagon Indicates only two of the three hexagons with no other errs, ie (err) (err) (err) 15 8 a a 400 b b 430 Crect response Sponsed swim! Zeros given after the decimal point Condone two zeros, f part (a) accept Penalise only the first occurrence of one zero, f parts (a) and (b) Mark as 0, 1 24

25 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Cat food 1 a a a equivalent probability 4 1 b b b equivalent probability 3! Probability rounded Accept 0.33 better, percentage equivalents c c c 0.3 equivalent probability 25

26 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Wine gums a a a 3m Completes all three rows of the table crectly, ie can cannot Ravi Sita Tina ! Inaccurate reading of bar charts f Ravi and Tina Accept values in the following ranges provided the total f the row is crect, accept Ravi 35 ± 1 15 ± 1 Tina 100 ± ± 4, within a response using only percentages, accept Ravi 70 ± 2 30 ± 2 2m Completes two rows of the table crectly Completes one column of the table crectly Completes the table with the two columns transposed but otherwise crect Tina 50 ± 2 50 ± 2! Increct units inserted Igne Completes either the row f Ravi the row f Tina crectly Completes the table using crect percentages from the bar charts, ie can cannot Ravi Sita Tina

27 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Wine gums (cont) b b b Explains that Tina used the largest sample size The me tests you do, the me reliable the results Tina asked me people than the others 200 is bigger than Minimally acceptable explanation Me tests Me people Me wine gums 200 is bigger She asked 200 and the others asked [comparison implicit] She asked twice as many people as Sita [comparison with Ravi implicit]! Irrelevant infmation claim It was 50/50 Hers were me evenly split She asked a wider range of people Igne if accompanying a crect response U1 Incomplete increct explanation Me She asked 200 people [no comparison] Her results are me reliable as it was half and half Crect response Values 2m Gives all three crect values in the crect positions, ie 18, 30 and 100 Gives two crect values in the crect positions Shows all three values 18, 30 and 100, even if their positions are increct Shows crect substitutions, interpreting the addition, multiplication and squaring crectly, but fails to process processes increctly 8 p 10, 3 t 10, 10 t 10 seen! Increct notation, f the value of 8 p k 18k Withhold 1 mark only f the first occurrence 27

28 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Thinking triangularly 3m Gives all four crect responses, including examples f the two true statements false Unambiguous indication of true and false f true, f false! True example(s) drawn crectly but indication of true omitted Condone, provided the examples show unambiguously that the statement is true true true false! Angles in the triangles not marked marked increctly Igne! Triangles not drawn accurately Accept provided the pupil s intention is clear, f the first true example accept 2m Gives any three crect responses, including a crect example f any true statement Gives crect responses f the two true statements, including crect examples, but leaves the spaces f the false statements blank! Acute obtuse angles look like right angles Do not accept if the angles are 90 ±1 Otherwise, condone U1 Gives a crect response f one of the true statements, including a crect example Crectly labels all four statements true false but examples f the true statements are increct omitted! Example(s) given alongside false As these may be trials, igne 28

29 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Toilet rolls 3m 2m Indicates the pack of 6 toilet rolls and gives a crect justification, based on a pair of comparable values The 6-pack costs 1.25 f 3 rolls, but the 9-pack costs 1.30 f 3 rolls 3.9(0) d 9 e 0.43( ) 2.5(0) d 6 e 0.41( ) F 9 rolls we have 3.90 and 2.50 d 2 t 3 e rolls: 390 d 3 t 2 e 260, ie 10p me The 3 extra toilet rolls in the 9-pack cost 1.40, but in the 6-pack 3 rolls cost 1.25 If the 9-pack were decreased by 3 rolls its price should go down by 1.30, but the difference is 1.40 so it s a better reduction 3 extra rolls cost 1.40 so 12 rolls using the large pack is 3.90 p 1.40 e 5.30, whereas 2.50 p 2.50 f the small pack is only 5.00 Shows a crect pair of comparable values but makes either an increct no decision Attempts to find a pair of comparable values, making not me than one computational rounding err, then follows through to make their crect decision The 6-pack is 1.30 (err) f 3 rolls and so is the 9-pack, so they are the same The 9-pack is 3.90 but should be 2.50 d 6 t 9 e 0.41(rounding err) t 9 e 3.69 so 6-pack is cheaper F 3m, no decision F 3m, crect decision and any pair of comparable values shown Note that common pairs (in pounds) are: 1.3 and 1.25 (per 3 rolls) 0.43( ) and 0.41( ) 0.42 (per 1 roll) (3.9 and) 3.75 (per 9 rolls) 2.6 (and 2.5) (per 6 rolls) 7.8 and 7.5 (per 18 rolls) 15.6 and 15 (per 36 rolls) 23.4 and 22.5 (per 54 rolls) 1.4 and 1.25 [ 1.3] (3 extra rolls) 2.3( ) and 2.4 (rolls per pound)! Comparison is per 9 rolls per 6 rolls but the given price is not restated Condone, f 3m accept The 6-pack, because 9 rolls should be 3.75! Units omitted, increct inconsistent Condone provided the pupil s intention is clear, f 3m accept The 6-pack, because 3.9(0) d 9 e (0) d 6 e 42! Additional increct wking Igne U1 Shows, implies by a crect value, a crect method to calculate at least one value f comparison, even if there are computational rounding errs Shows the difference in price f 3, 6, 9 18 rolls, even if the comparable values the methods to calculate them are not shown The 6-pack is 5p cheaper The big pack is 10p me 15p difference 30p less Note that common calculations are: 3.9 d d 2 (per 3 rolls) 3.9 d d 6 (per 1 roll) 2.5 d 2 t 3 (per 9 rolls) 3.9 d 3 t 2 (per 6 rolls) 3.9 t t 3 (per 18 rolls) 3.9 t t 6 (per 36 rolls) 3.9 t t 9 (per 54 rolls) 3.9 m d 2 [ 3.9 d 3] (3 extra rolls) 9 d d 2.5 (rolls per pound) 29

30 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Woodpeckers a a a Gives all three crect values in the crect der, ie b b b 1 : 3 Equivalent ratio 1 : : Crect response Changing equivalent 20cm 0.12 equivalent 30

31 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Four angles 3m Gives all four crect angles, ie a e 110 b e 70 c e 50 d e 130 Angles indicated on the diagram 2m Gives any three crect angles Gives all four values 110, 70, 50 and 130, but in the increct der U1 Gives any two crect angles Shows three of the angles 110, 70, 50 and 130, but with the links to each letter increct omitted Gives four different angles (ie no two of the angles are equal) that sum to a a a 5 Crect response Balancing b b b 35! Answers to parts (a) and (b) transposed but otherwise crect Mark as 0, 1 31

32 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Five cubes Draws a crect view of the shape from above using the square grid, in either ientation Internal lines omitted! Throughout the question, lines not ruled accurate Accept provided the pupil s intention is clear 2m Draws a crect view of the shape using the isometric grid, in either crect ientation F 2m, internal lines omitted, f 2m accept! Their shape takes the given view as a view from below rather than from above Condone, f 2m accept Shows a shape drawn on the isometric grid that takes the given view as a view from one side rather than from above The only err is to omit some external lines to show some hidden lines! Their shape takes the given view as a view from one side rather than from above F 2m, accept only if this err was penalised f the first mark Mark as 0, 1, 1! Hidden lines shown F 2m, accept provided they are clearly indicated as hidden lines, f 2m accept then Shape with me than 5 cubes drawn 32

33 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response nth term a a a Gives a crect expression 4n p 2 4n p 1 p 1! Unsimplified expression unconventional notation, f part (a) 4 t n p 2 n4 p 2 Condone b b b Gives a crect expression 3n p 3 3(n p 1) 1 (6n p 6) 2 (6n p 6) d 2 6n 6 p 2 2 Necessary brackets omitted, f part (b) 6n p 6 d 2, f part (c) 2 t 5n m 3 c c c Gives a crect expression 10n m 6 2(5n m 3) (5n m 3) t 2 33

34 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Enlargement Indicates the crect centre of enlargement f the first diagram, ie! Centre of enlargement indicated only by intersection of construction lines Accept provided there is no ambiguity! Inaccurate indication Accept provided their indication is within 2mm of the crect position! Increct construction lines shown Igne Indicates the crect centre of enlargement f the second diagram, ie 34

35 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Err a a 120! Increct use of % sign Igne 84 b b 2m Gives two crect percentages that sum to ! Values rounded F 2m, accept percentages crectly rounded to two me significant figures, provided they sum to 100 Note to markers: Crect percentages are Gives one crect percentage even if truncated, ie 38 better, 61 better Shows implies a crect method f both percentages 80 d d 206 Digits 38( ) ( 39) and 61( ) 35

36 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Tomatoes a a a Gives a value between 7.2 and 7.5 inclusive, equivalent b b b Indicates A and gives a crect explanation The most common crect explanations: Use the trend line f type A It is closest to the line f type A (3.2, 5.8) is close to (3, 6) which is on line A Type A have smaller diameters with bigger heights than the other types F A, the height is about double the diameter, and that s roughly true f this one Minimally acceptable explanation It s closest to that line The line goes through (3, 6) which is very close It is closest to type A [with point crectly plotted on graph] Type A have small diameters with big heights F A, height is bigger than diameter A tomatoes are thin but tall Incomplete increct explanation It is closest to type A It s in the A section F A, the height is double the diameter The graph shows it It is on A s line Type A tomatoes have small diameters Refer to the diameters of type B being consistently larger than 3.2cm, to the heights of type A differing from their diameters It s between the lines f A and B, but all the type Bs have diameters between 6 and 7 It s too far from the type C line so it s A B, and the A ones don t have similar diameters and heights Minimally acceptable explanation B tomatoes have bigger diameters A tomatoes have diameters that are not roughly equal to their heights Incomplete explanation It could be A B but it s me like A 36

37 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Tomatoes (cont) c c c Indicates B and gives a crect explanation The most common crect explanations: Refer to the position of its line on the graph B s graph is closest to y e x ( h e d) The line f B is closest to the line drawn [line h e d crectly indicated on graph] Minimally acceptable explanation B s line is about 45 through the middle It goes through (0, 0) then when d goes up by 1, so does h The x and y ( h and d) codinates are nearly equal Incomplete increct explanation B s line is at about 45 B s line is a diagonal through the middle The graph shows it B has h e 2 and d e 2 U1 Refer to the dimensions of the tomatoes The height and the diameter of a sphere are equal and that s also roughly true f B The height and diameter of B are both around 6 A tomatoes are too tall f their diameter, but C tomatoes are too fat f their height Minimally acceptable explanation Same height and diameter h and d are closest The two values are nearly equal The others are either too tall and thin too sht and wide d d 2m Gives the value 22.4( ) 22.5! F 2m, answer of Do not accept unless a crect method a me accurate value is seen Shows implies a crect method with not me than one computational rounding err 3.14 t d 6 1 π t π d 6 e 0.52 (premature rounding), 0.52 t t 3.5 e 22.3 Answer of 22 23, with no crect method me accurate value F, no indication of multiplication 1 π π F, conceptual err 1 t π t 7 t

38 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Expressions 2m 8x p 31 Shows implies the four crect terms resulting from multiplying out the brackets, even if there is increct further wking 5x, 10, 21, 3x 5x p 10 and 21 p 3x 5x p 31 p 3x 8x p 10 p 21 Multiplies out both sets of brackets with not me than one err, then follows through using their expansion to give a fully simplified expression 5x p 10 p 27 (err) p 3x e 8x p 37 F, incomplete processing in constant terms, f the first expression 5x p 5 t 2 p 3 t 7 p 3x 2m x 2 p 7x p 10! Expression equated to zero Condone Shows implies the four crect terms resulting from multiplying out the brackets, even if there is increct further wking x 2, 2x, 5x, 10 x t x p 5x and 2 t x p 10 The only err in an otherwise crect and simplified expression is to give an increct but non-zero constant term, to leave incomplete processing in the crect constant term x 2 p 2x p 5x p 7 (err) = x 2 p 7x p 7 x 2 p 7x p 2 t 5 x t x p 7 t x p 2 t 5 38

39 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Marking overlay available Crect response Tracking elephants 2m Uses compasses to draw two arcs centred on A and B within the tolerances as shown on the overlay, and indicates the crect rion Draws two arcs centred on A and B within the tolerances as shown on the overlay, even if compasses are not used, and/ an increct no rion is indicated Indicates the crect rion f their arcs centred on A and B, even if they are outside the tolerance as shown on the overlay The only err is that the two arcs are centred on the increct vertices of the square! Arcs extended Igne! Extra arcs drawn Igne provided there is no ambiguity! F, follow through Accept unambiguous indication of a crect rion fmed by an attempt at two symmetrical arcs sets of lines centred on A and B, even if inaccurately drawn, accept R R R Do not accept follow through from only one arc line, from non-symmetrical arcs lines 39

40 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Algebra grids 3m Completes all three grids crectly, ie 6x 3x + 1 2x 4x x + 1 2x! Unconventional notation, f 6x x6 6 t x, f 8x 2 8 t x t x Withhold 1 mark only f the first occurrence 8x 2 5x 2x(x + 1) 2x 2 + 2x 5x Unsimplified expression(s) and/ incomplete processing, f 6x 2x p 4x, f 8x t 4 t x 2x 3x 3x 2x 6x 2 6x 2 2m Completes the first two grids crectly Completes the third grid crectly and gives any two crect entries in the first two grids Completes the third grid crectly, gives any one crect entry in the first grid, makes an err in the right-hand entry of the second grid, but follows through crectly to give their product Gives any two crect entries in the first two grids Completes the third grid crectly U1 Gives any one crect entry in the first grid, makes an err in the right-hand entry of the second grid, but follows through crectly to give their product 40

41 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Four kites 2m 115 Shows the value Shows the value 90, provided there is no evidence that this value has been assigned to angle k Shows implies a complete crect method with not me than one computational err 1 ( 320 m ) 180 m 45 m m 4 t 40 8 Fms a crect equation involving k, even if the 90 angle has not been found 2k e 360 m 40 m x 1 (k e ) 160 m x Crect response Volume of 100 Gives a crect pair of positive values such that x 2 y e 100 x e 2, y e 25 x e 1, y e 100 x e 5, y e 4 x e 10, y e 1 x e 4, y e 6.25 Gives a different crect pair of positive values from any credited f the first mark! Value(s) rounded Accept x as (100 d their y) y as 100 d their x 2 to 3 s.f. better, accept x e 3.16, y e 10 x e 3, y e 11.1 Native value of x! F both marks, values of x and y transposed, but otherwise crect Mark as 0, 1 41

42 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Bias 2m Indicates the coin is not biased ( Not biased No ) and gives a crect justification Of the 200 trials, 110 are heads 110 e < t 200 e > 110 The mean number of heads is t 0.56 e 11.2, 11 < p 3 p 1 p 1 p 2 p 2 p 1 m 1 p 0 p 1 e 10, 10 d 200 e 5%, so it s 55% which is less than 56% Minimally acceptable justification 55% , , 11.2! Response assumes the pupil has already concluded the coin is biased Condone, f 2m accept 55%, so her conclusion is wrong! Irrelevant infmation 7 of the 10 sets of results were less than 11.2 Igne if accompanying a crect response, otherwise do not accept U1 Shows a crect estimate of probability based on all 200 results, even if it is written unconventionally, but makes an increct no decision (%) out of 200 Shows the values 110 and 112, 11 and 11.2, but makes an increct no decision Shows implies a crect method with not me than one computational err, then follows through to make their crect decision 5 p 6.5 p 5.5 p 5.5 p p 5.5 so not biased 10 p 13 p 11 p p 11 e 114 (err), > 0.56 so biased F 2m, incomplete increct justification They add up to 110 The mean is t 20 e 11.2 Median e 11 and 11 <

43 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Area A 2m 45, with no evidence of an increct method Increct method 3 t (5 p 10) Shows implies that the width of B is 6 15 t 2 d 5 e 6 C is 5 by 3, so B is 5 by 6 B is 5 t 6 6 crectly marked on diagram The width of A must be 9 Shows implies a complete crect method with not me than one computational err 5 t (15 m (15 t 2 d 5)) 75 m 15 t 2 15 t 8 m 15 m 30 m t 2 e 30, 30 d 5 e 5 (err), 15 m 5 e 10, 10 t 5 e 50! Increct units inserted Igne! F, dimension of 6 f B within increct wking As this could represent the height rather than the width, do not accept, do not accept B is 6 by 10 U1 43

44 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Field voles a Gives a value between 0.65 and 0.68 inclusive equivalent probability 660 [0.66] 1000 b Gives a value between 0.5 and 0.61 inclusive equivalent probability 160 [ ] [ ] [ ]

45 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 16 Crect response Films 2m 168 Shows implies a complete crect method with not me than one computational err 24 t t 60 t 24 d 25 d m t 175 d d d 25 e 6 (err), 175 m 6 e 169 Shows implies that the difference in the number of minutes is 7, even if there is increct no further wking 175 t 60 e 10500, d 25 e 420, 420 d 60 e d 25 e 7, 175 p (err) 7 e 182! F, value of taken to imply a difference of 7 minutes Accept only if a crect method f finding either is seen Otherwise, do not accept, accept 175 d 25 e 7 [without sight of 175 d 24] 175 p 175 d 25 e 182, do not accept 175 d 24 e t 175 e

46 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 17 Crect response Equations of lines a Gives the equation of a straight line, other than y e x p 1, that passes through (0, 1) y e 2x p 1 y emx p 1 y p x e 1 3y p 3x e 3 y e 1 x e 0! Throughout the question, unsimplified equation unconventional notation, f part (a) y = 2 t x p 1 y = x p x p 1 Condone Same equation as the given line, but rearranged y m x e 1 y e x p 2 m 1 2y = 2x p 2 Gives a crect equation, other than one previously credited Same equation as the given line one previously credited, but rearranged b Gives the equation of a straight line that is parallel to x p y e 5 x p y e 3 y emx p 6 Same equation as the given line, but rearranged 2x p 2y e 10 y e 5 m x 46

47 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 18 Crect response Households 3m 1.6 F 3m, equivalent fractions decimals 2m Shows the value 98.4, 98.3(...) 98 Shows implies a crect method even if there are rounding truncation errs t 2.34 t m t 2.34 e m e ( m 2.34) t t (...) m e 21.(...), (...) Gives an answer that rounds truncates to 1.6, is equivalent to 1.6 Shows the digits 16( ) Shows the number of people who did live in households million 49.1 million 49.0( ) million Shows the number of people who did not live in households 0.8( ) million Shows the number of households there would have been if every person had lived in one 21.3( ) million F, million omitted! Value of 49 (million) given as the number of people who did live in households F, do not accept unless a crect method a me accurate value is seen 47

48 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 19 Crect response 2m Gives both crect surface areas, ie 88 and 104 Cuboid Gives one crect surface area Shows the values 22 and 26 Shows a complete crect method with not me than one computational err 24 d 6 e 4, (4 t 6 p 2 t 1) t 4 and (2 t 6 p 2 t 3 p 2 t 2) t 4 24 t 6 e 144, 144 m 14 t 4 and 144 m 10 t 4 24 d 6 e 3 (err) Answers: 66 and t 6 e 124 (err) 124 m 14 t 4 e m 10 t 4 e 84! F, other wking shown As these may be trials, igne U1 The only err is to take 24 as the area of one face of each small cube, ie gives the answers 528 and

49 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 20 Crect response Five points 3m 9 2m Shows implies a complete crect method with not me than one err EA : EC is 6 : 4 e 3 : 2, 40 m 10 AC is e 15, 2 3 t m 10 e 30, 1 BCE and ADE similar, ratio 1 : 1, p 1 p 1 p 1 e 5, d 5 e 6, 6 t 1 2 (40 m 4 m 6) d 2 e 16 (err), 16 d 5 e 3.2, 3.2 t 3 e Shows implies that EA ( ED) is of AC 5 ( BD) EA : EC is 6 : 4 1 BCE and ADE similar, ratio 1 : : 2 2 : 3 equivalent ratio seen 3 equivalent seen 5 d 5 t 3 equivalent seen 18, 12 seen U1 Shows implies that the length of AC ( BD) is m seen AE ( DE) e 10, EC ( EB) e 5 [increct but total 15] 49

50 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 21 2m 1 36 Crect response equivalent probability Shows implies a complete crect method, even if values are rounded truncated t t t t t 6 6 ( )3 1 t t Shows implies a crect method to find the total number of possible outcomes t 6 t 6 1 ( ) 6 3 Three dice! F 2m, values rounded truncated F 2m, accept 0.03, ( ), the percentage equivalents F 2m, do not accept 0.02 unless a crect method a me accurate value is seen 1 F, accept ( ) f, 6 the percentage equivalents F, do not accept 0.2 f me accurate value is seen 1 6 unless a Shows a crect method that uses explicitly the fact that, in this case, the outcome of one dice is irrelevant It doesn t matter what you throw on the first dice, but the other two dice must 1 match it, so it s then

51 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 22 Crect response 2 2m 2 equivalent 3 Population of Wales! F 2m, value rounded truncated Accept better, provided there is no evidence of an increct method Do not accept 2.6 unless a crect method a me accurate value is seen Shows implies that 3 million represents 3 t 8 d m d 9 3 e 112.5% Shows the digits ( ), with no evidence of an increct method 9 8! F 2m, million repeated, f 2m accept F 2m, evidence of an increct method 3 d 8 t 7 which is about , so

52 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 23 Crect response Leaning tower of Pisa 2m Gives a complete crect explanation The most common crect explanations: Use 5.5 and 56m to show that 5.2m cannot be crect sin 5.5 t 56 = 5.3( ) [ 5.4] Use 5.5 and 5.2m to show that 56m cannot be crect 5.2 e 54.(...) sin 5.5 Use 5.2m and 56m to show that 5.5 cannot be crect sin 1 ( ) e 5.3(...) 5.2 d 56 e 0.092( ) [ 0.093] but sin 5.5 e 0.095( ) [ 0.096] F 2m, minimally acceptable explanation sin 5.5 t sin sin 1 ( ) d 56 sin 5.5 F 2m, crect explanation using the vertical height (56 2 m ) e 55.7( ) [ 55.8] tan 5.5 e 0.096( ), but 5.2 d 55.7 ( ) e 0.093( ) 56cos 5.5 e 55.( ) [ 56], but 5.2 d tan 5.5 e 54.( ) F 2m, crect explanation using angle of 84.5 cos 84.5 t 56 e 5.3( )! F 2m, other redundant increct wking Igne alongside crect wking, f 2m accept sin 5.5 t 56 e 5.3 not 5.2, e 0.09 F 2m, explanation is based on scale drawing Shows a crect trigonometric ratio involving two of the three values given 5.2 tan 5.5 e h cos 5.5 e h d 56 F, crect ratio using angle of 84.5 F, incomplete but unambiguous notation 5.2 sin e 56 U1! F, their ratio uses all three values sin 5.5 e 5.2 d 56 Condone 52

53 2006 KS3 Mathematics test mark scheme: Paper 2 satspapers.g BLANK PAGE 53

54 2006 KS3 Mathematics test mark scheme: Paper 2 satspapers.g BLANK PAGE 54

55 satspapers.g 2006 KS3 Mathematics test mark scheme: Paper 2 Index Index to mark schemes Tier Question Page Matching 12 2 Pupil list 13 3 Thinking angles 14 4 Moving on a grid 15 5 Cards 15 6 Tennis 16 7 Joining points Mirr lines Using rules Cough mixture Wking with areas Prnancy Missing numbers Hexagons Sponsed swim Cat food Wine gums Values Thinking triangularly Toilet rolls Woodpeckers Changing Four angles Balancing Five cubes nth term Enlargement Err Tomatoes Expressions Tracking elephants Algebra grids Four kites Volume of Bias Area A Field voles Films Equations of lines Households Cuboid Five points Three dice Population of Wales Leaning tower of Pisa 52 55

56 satspapers.g EARLY YEARS NATIONAL CURRICULUM 5 16 GCSE GNVQ GCE A LEVEL First published in 2006 Qualifications and Curriculum Authity 2006 NVQ 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 83 Piccadilly London W1J 8QA Further teacher packs may be purchased (f any purpose other than statuty assessment) by contacting: QCA Orderline, PO Box 29, Nwich NR3 1GN tel: ; fax: derline@qca.g.uk Order ref: QCA/06/