Mark scheme for Paper 1

Transcription

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

2 2006 KS3 Mathematics test mark scheme: Paper 1 Introduction Introduction The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to infm teachers. This booklet contains the mark scheme f paper 1 at all tiers. The paper 2 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking 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 2006 KS3 Mathematics test mark scheme: Paper 1 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 2006 KS3 Mathematics test mark scheme: Paper 1 General guidance What if The pupil s response does not match closely any of the examples given. Markers should use their judgement in deciding whether the response 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 2006 KS3 Mathematics test mark scheme: Paper 1 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 2006 KS3 Mathematics test mark scheme: Paper 1 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 2006 KS3 Mathematics test mark scheme: Paper 1 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 = n p 2 tiles tiles = t = n p 2 f t = n p 2 Unambiguous letters used to indicate expressions t = 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 = n p 2 f n p 2 7

8 2006 KS3 Mathematics test mark scheme: Paper 1 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 5, y 7 ) 8

9 2006 KS3 Mathematics test mark scheme: Paper 1 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 2006 KS3 Mathematics test mark scheme: Paper 1 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 1 BLANK PAGE 11

12 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 1 Crect response Line symmetry Draws only the crect line of symmetry on the first shape, ie! Lines not ruled, accurate solid Accept lines, even if dotted dashed, extending at least across the shaded area, provided the pupil s intention is clear Draws only the crect line of symmetry on the second shape, ie 12

13 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 2 Crect response Step sizes a Shows a crect way, other than add 8 then add 12, using exactly two steps add add Add 12 then add 8 Fractions, decimals natives! Operations omitted Condone, provided the directions of any arrows, if shown, are crect, accept 15 5! Arrows not shown not consistent with their numbers Condone, provided the directions of any arrows, if shown, are crect, accept 0 20 add 10 add b 5! Answer shown only on the diagram Accept provided there is no ambiguity c equivalent 2 8 Answer of 8 13

14 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 3 Crect response Temperature Indicates 8 on the second thermometer! Inaccurate indication Accept provided the indication is nearer to the crect inter than either of the neighbouring inters Indicates 2 on the third thermometer! Responses f the second and third thermometers transposed but otherwise crect Mark as 0, 1 4 Crect response Attending school 2m U1 Completes the graph crectly f all four days, ie Completes the graph crectly f two of the days The only err is that the intended widths of the bars are inconsistent Mon Tues Wed Thurs Fri Shows implies the values 2, 6 and 3! Bars not ruled accurate Accept provided the pupil s intention is clear and, f Friday, provided the height is nearer to 3 than to either 2 4! Bars drawn with widths different from that given Accept provided their intended widths are consistent! Mark inserted to indicate zero on Thursday Accept provided the pupil s intention is clear Values 2, 6 and 3 implied by bars of heights 2, 6 and 3 squares drawn 14

15 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 5 a 1.17 Crect response Lemonade! F parts (a), (b) and (c), costs given in pence without amendment of units Penalise only the first occurrence of the cost given in pence within a crect response, f the costs as 117, 110, 109 [with crect bottle sizes] Mark as 0, 1, 1, f the costs as 110, 109, 117 [with crect bottle sizes] Mark as 0, 0, 1 b Gives a complete crect response, ie 1 Two 1 litre bottles, cost A 1 litre bottle and a 2 litre, cost 1.09 U1 F parts (b) (c), unambiguous identification 1, f the two 1 litre bottles 2 One of the middle size, and another p and 55 c Gives a complete crect response that is different from one credited in part (b)! F parts (b) (c), uses three 1 litre bottles and gives the cost as 1.17 If their (a) is either , accept U1! F parts (b) and (c), crect costs given with increct no identification of bottle sizes 1.10, then 1.09 Mark as 0, 1 d 2m 13 Shows a complete crect method with not me than one computational err (err) p 10 e 14 Shows the value 3, with no evidence of an increct method f this value Conceptual err p 10 e t 2 e 10, 2 t 1 e p 2 e

16 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 6 a 83 Crect response Computation 185 b 37 c 62 16

17 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Gives five numbers less than 6 Crect response Spinners Numbers repeated on the same spinner! Numbers written outside the spinner on lines between the sections of the spinner Accept provided the pupil s intention is clear ! Zero Accept zero as an even number, but not as an odd number Accept zero as a multiple of 3! Native inters Accept numbers of the fm 2n as even, numbers of the fm 2n p 1 as odd and numbers of the fm 3n as multiples of 3, where n is an inter Gives five numbers with me even numbers than odd numbers ! Section(s) of the spinner left blank F the first mark, do not accept F the second and third marks, accept provided exactly five digits have been given and the statement is satisfied, f the second mark accept , f the third mark accept Gives five numbers that are not multiples of

18 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Crect response 3 4 equivalent 4 Adding three m2 9 3 a a 30 Crect response Changing numbers b b 1012 c c a a 1992 Crect response Unambiguous indication of year 92 Red Kites b b 1! Units given Igne c c 6 18

19 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Crect response Red Kites (cont) d d Gives a crect statement that shows implies that the number of nests has increased over the years The number of nests has increased a lot over the years The number of nests has nearly always gone up The ones with gs increased and the ones without decreased and then increased again They roughly doubled each year, except from 1994 to 1995 when the number didn t change Minimally acceptable statement Increased Got bigger They multiplied Me with gs in They went 2, 4, 9, 21, 21, 39! Statement states implies that the number of nests increased every one of the years Condone! Statement confuses nests with birds gs but is otherwise crect Condone Incomplete increct statement The ones without gs decreased and increased They roughly doubled each year They went 2, 4, 9, 21, 21, 36 U1 U1 Gives a crect statement that shows implies an appreciation that some have stayed nearby Me nests have got further away over the years, as well as some staying nearby At first they were only in the small circle, but then some went to the outer circle The range of distances has become bigger Over the years they gradually spread out over a wider area It got crowded in the centre so nests were built further away Markers may find the following useful: Year Eggs No gs Minimally acceptable statement Some moved a bigger distance away There were nests further away Me got further away They spread out They expanded They covered me area The average distance increased Incomplete statement They got further away over the years The distances increased They went up 19

20 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, a a 2022 Crect response Place value b b equivalent! Follow through from part (a) Accept as their (a) d 100! Answer of Do not accept unless is also seen, is from their follow through c c 2m 0.45 equivalent Shows the values 5.85 and 5.4 equivalent Shows the values 0.2 and 0.25 equivalent Shows implies a complete crect method with not me than one computational err (err) Answer: m 4.4 (err) e 1.45 Conceptual err within a crect method 3.5 p 2.35 e p 2.35 e is bigger than 2.1 by

21 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Crect response Completing quadrilaterals a a Completes the crect square, ie! Lines not ruled accurate Accept provided the pupil s intention is clear! F part (a), vertices of the square not on the intersections of the grid Accept vertices within 2mm of the intersections of the grid! Given line extended Do not accept in part (a), but condone in part (b) b b Completes a crect trapezium with exactly one pair of parallel sides! F part (b), vertices of the trapezium not on the intersections of the grid Accept provided the pupil s intention is clear and the conditions have been satisfied! F part (b), parallel lines wrongly labelled Igne U1 21

22 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response 28 times table a a a Gives a crect method to show that 9 t 28 e t [with evidence of the 7 tens] so 72 p t 28 e 280, 280 m 28 9 t 30 m 18 2 t 28 e 56, 4 t 28 e 112, 8 t 28 e 224, so 224 p [with evidence of the 7 tens] Minimally acceptable indication 72 p m m 18! Method uses repeated addition Accept provided there is evidence of how the addition has been carried out, accept : , 56, 84, 112, 140, 168, 196, 224, 252 Final answer increct 28 t b b b 2m 756 Shows implies a complete crect method with not me than one computational err The most common crect methods: Use the relationship between 27 t 28 and 9 t 28 3 t 252 ( their increct value from (a)) 252 p 252 p 252 F, method uses repeated addition Calculate 27 t 28 directly 10 t 27 e 270, 270 t 3 e 810, 810 m Conceptual err 28 t t 20 e 400, 7 t 8 e p 56 e t (err) so 400 p 160 p 140 p 56 22

23 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response Matches the second statement with 5(k p m) Matching expressions Statement matched with me than one expression Note to markers: The following shows the crect responses: 5k Matches the third statement with 5m m 5k 5m 5 5m 500 5m Matches the fourth statement with 500 m 5m 5k + m 5(k + m) 5m 5k 5k 5m 23

24 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response Paper a a a 3m Completes all six entries crectly, ie Area Perimeter Square A Rectangle B Square C m Completes at least four entries crectly Completes either column crectly Completes any one row crectly Gives an increct value f the area of square A, but not 8, then follows through crectly, by halving each time, to find the other two areas Area Square A 60 (err) Rectangle B 30 Square C 15 Gives the two columns transposed but otherwise crect b b b 32! Follow through from part (a) Accept as half their area of square A, provided this was not 8, their area of rectangle B, provided this was not 4 c c c Indicates that the perimeter is greater than 24cm and gives a crect explanation 8 p 8 p a number bigger than 8 is bigger than 24 The hypotenuse is longer than 8cm and the other two are 8cm The diagonal is the longest side so it is greater than 8cm 8 2 p 8 2 e 128, 128 > 8 Minimally acceptable explanation It doesn t have 3 equal sides The slope is the biggest side The fold is 11.( ) ( 128) > 8 The longer side is about 10 The perimeter would be me like 26! Increct units inserted Igne Increct statement The longer side is 10 The perimeter is 26 U1 Note to markers: The length of the hypotenuse is 11.3cm and the perimeter is 27.3cm, to 3 s.f. 24

25 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response CD player a a a 2m Gives all three crect values, ie 8.4! Units given Igne equivalent Gives two crect values! Follow through F, allow follow through from an increct value that is crectly divided by 2, 1 provided their values are not 10, 5, , 42, 21, f accept (err) (err) 2.15 b b b 2m Shows the digits 987! Follow through from part (a) F 2m, allow follow through as 84 p the sum of their three values from part (a), provided at least one of their values is not an inter, and the total is rounded truncated to a whole number of pence Shows implies the addition of the three 1 values cresponding to 10%, 5% and 2 % p 4.2 p seen The sum of their 3 values from part (a) seen [with without addition to 84] Shows implies a complete crect method with not me than one computational err t p t

26 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response Solving 4! Increct notation, as an answer f the first mark k et4 Penalise only the first occurrence m7! Incomplete processing, as an answer f the first mark k 8 e 2 Penalise only the first occurrence 26

27 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response Odd even? a a a Indicates that the number must be even and gives a crect explanation The most common crect explanations: State imply that 4 itself is an even number 4 is even, so all its multiples must be too 4 is even and even t even e even, and even t odd e even You add fours together to get the multiples and even numbers added give even answers Minimally acceptable explanation 4 is even ( a multiple of 2) All the 4 times table is even If you start with an even number, you end up with one too If the multiple is odd, the number itself would have to be odd 4 is a multiple of an even number Anything t an even number is even Any multiple of an even number is even Even p even e even U1 Show imply the link between multiples of 4 and even numbers It s 2 t 2 t something, which must be even To get the 4 times table, you double the 2 times table Multiples of 4 always end in the evens 0, 2, 4, 6 8, 4, 8, 12, 16, 20 Minimally acceptable explanation It s 2 t 2 t something They re every other even number It s the 2 times table doubled They end in 0, 2, 4, 6 8 Incomplete increct explanation All multiples of 4 are even Any odd number divided by 4 leaves a remainder Any even number t even e even 3 t 4 e 12 which is even 4, 8, 12, 16, 20 They all end in even numbers They end in 2, 4, 6 8 b b b Indicates that the number could be odd even and gives a crect explanation that shows implies at least one odd and one even fact Facts of 20 are 1, 2, 4, 5, 10 and 20, some are odd and some are even There are two odd facts and four even facts of 20 It could be 4 (even) 5 (odd) 4 t 5 = is even, but 1 is odd and goes into everything U1 Minimally acceptable explanation 1, 2, 4, 5, 10 and 20 It could be 4 5! Incomplete list of facts given Condone, provided none is increct and at least one odd and one even fact are shown, accept The facts of 20 are 1, 2, 4 and 5 Incomplete increct explanation Facts of 20 can be odd even It could be 5 It could be 2 (even) 3 (odd) 27

28 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Crect response Hexagon patterns 2m 61 F 2m, increct notation, f 2m 61n Shows the value 21 40, with no evidence of an increct method a method using counting on f the value Shows a crect method f both types of tile with not me than one computational err 20 p 1, 20 t 2 20 t 3 p 1 F, method shown uses counting on Shows a crect expression f the total number of hexagons, in which the terms in n have been collected together 3n p 1 n t 3 p 1 28

29 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Dice Gives all three numbers crectly f the first net, ie Gives all three numbers crectly f the second net, ie

30 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Sizing a a a 2m Gives the four values in the crect der smallest largest smallest largest Shows any three of the values 25, 9, 27, 16, with no evidence of an increct method f a crect value Gives the four values in der of size, largest to smallest b b b 2m Follow through using their value f 5 2 from part (a) Shows the value , even if there is subsequent increct wking Shows implies a complete crect method, with at least some crect processing, with not me than one computational err 3125 t 100 e , d t 5 e t t (err) Conceptual err 3125 t e 3125, 5 2 e 25, 3125 p 25 e e 10, 3125 t 10 e t 10 d 2 e (err) t 10 d 2 e

31 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Operations 2m Gives all four crect operations, ie m d p t Gives any two crect operations Crect response 1 2m 6 equivalent 2 Finding y Shows implies a crect first step of algebraic manipulation that either reduces the number of terms collects unknowns on one side of the equation and numbers on the other 14 e 2y p 1 3y p 13 e 5y 14 m 1 = 5y m 3y 13 = 2y 13 d 2 31

32 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Favourite spt a a a Indicates No and gives a crect explanation You can only find the mean of a set of numbers The data are in wds not in figures so the mean cannot be found You can t add wds up then divide by how many there are There are no numerical values Minimally acceptable explanation They are wds You need numbers There are no quantities ( figures) You need to add them together You can t divide them (by 10) Incomplete explanation You can t find the mean of spts You can t have fractions of a wd Not enough infmation Their explanation shows misconceptions about the mean You can t add them up and divide by 5 You can t divide a wd by a wd You can t find the mean of wds unless you use the frequencies It doesn t say whether Hanif asked them to give the spts marks out of ten You can t put them in der because they are wds not numbers U1 Numerical values assigned Yes, football and swimming are 8 letters, cricket and netball are 7 and hockey is 6 b b b Indicates Yes and gives a crect explanation The mode is the most common thing, so you can find it f numbers wds The mode is football as it was chosen most often, by four people You can see from the table what was the most popular spt Minimally acceptable explanation Most common Most popular Me like football Highest is football Football is favourite Mode identified but not explained The mode is football Four of the ten chose football so this is the mode Football appears me than once Incomplete increct explanation Most You can see how many picked each spt There s me than one of some results You can find the mode from both numbers and wds Football was chosen the most as five people said that 32

33 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Consideration a a a Gives a crect counter example, using a value that is less than equal to one m4 t 2 = m8 which is not greater than t 2 = 0.2, 0.2 < 2 2 t 1 = 2 which is not greater than 2 U1 Gives a crect general explanation Two times a native number is less than 2 Double a number between 0 and 1 is not greater than 2 b b b Gives a crect counter example, using a value that is less than equal to zero 2 m (m3) = 5, 5 > 2 2 m 0 = 2 which is not less than 2 U1 Gives a crect general explanation Two minus a native number is greater than 2! Throughout the question, the result of their counter example is not shown and/ the comparison is not explicit Condone provided only one of these aspects is omitted, f part (a) accept m4 t 2 = m8 m4 t 2 < 2 However, penalise only the first occurrence of both aspects omitted, f part (a) m4 t 2! Throughout the question, their general statement makes no explicit comparison Condone, f part (a) accept Multiply it by a native number Numbers less than 1, f part (b) accept Take away a native number Numbers less than 0, f part (c) accept Take a number from 0 to 1 and square it Positive numbers that are decimals starting with nought point! Throughout the question, other numerical examples general reasoning given alongside a crect response Igne other numerical examples, even if they are increct suppt the given statement If a crect counter example is given, igne any general explanation unless it contradicts the counter example given c c Indicates No and gives a crect counter example, using a value that is greater than equal to zero and less than equal to one 1 2 = 1 which is equal not bigger 0 t 0 = 0, so it stays the same 1 ( ) = but < t 0.1 = 0.01, not greater than 0.1 U1 Indicates No and gives a crect general explanation When you square a number between 0 and 1 the answer gets smaller not bigger Fractions bigger than zero that are not top heavy get smaller when squared F part (c), minimally acceptable counter example 1 2 = 1 0 t 0 is not greater than 0 1 ( ) 2 1 < 2 2 F part (c), increct response (m2) 2 = m4 which is less than m2 It s not true f native numbers It is only true f numbers that are bigger than 1 It is not true f numbers that are smaller than 1 It s not true f decimals fractions It s only false when the number is 1 33

34 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Test a a a N! N identified only on scatter graph Accept provided unambiguous! Highest total mark given Igne if given with N If N is not given, accept a value between 82 and 83 inclusive b b b Indicates True and gives a crect explanation The range f coursewk is 40, but the range f the test is 30 Coursewk goes from 10 to 50, test from 10 to 40 Both start at 10 but coursewk goes to 50 rather than to 40 Minimally acceptable explanation 30, 40 seen Highest to lowest is bigger f coursewk marks than f test marks Coursewk marks spread over 8 squares of the graph, test marks over 6 squares The points are me spread out along the x-axis than along the y-axis They had a wider span of marks There s me variation in the cwk marks They re me scattered ( spread out) C/w results start at the same mark as test results, but finish at a higher mark! Ambiguous notation Test marks 10 m 40 Coursewk 10 m 50 Condone! Increct use of % sign Igne Incomplete explanation Coursewk has a greater range than test marks Coursewk has lowest 10, highest 50 Coursewk went up to 50, test went up to 40 Coursewk goes from 10 to 50 but test goes from 10 to 30 except f 2 pupils Coursewk marks were varied, but test marks were mostly between 10 and 25 U1 Increct explanation The range f coursewk was 40, but the range f test was 20 The test marks are me scattered 34

35 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, c c c 70 Crect response Value on the line excluded Me than 70 Just over Test (cont)! Range of total marks given Accept provided all values win prizes, accept At least me, do not accept About 70! Increct use of % sign Igne d d 2m Indicates the crect rion, ie 50 Unambiguous indication of rion Crect rion labelled R Indicates both the lines x = 25 and y = 25, even if there are other errs Indicates the line x p y = 65, even if there are other errs! F 2m, lines dotted dashed Accept unless the intention is only to indicate specific points! Lines not ruled accurate Accept provided the pupil s intention is clear! Line(s) drawn below crect position in der to allow the rion to include points on the line(s) Condone provided their line is parallel to the crect line, and is closer to the crect mark than to the crect mark m5, f x p y = 65 accept Line parallel to x p y = 65 and closer to x p y = 65 than to x p y = 60! F, line(s) not full length Accept provided each line spans at least 10 marks 35

36 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Crect response Fractions 7 12 equivalent F either calculation shows, implies by a crect answer otherwise, a crect method that would enable addition subtraction of fractions The most common crect methods: Show imply crect fractions with common denominats, f the first calculation 3 4, seen =, = , f the second calculation the denominat of the fraction =, = ( = ) seen with no attempt to change F the first and third marks, increct notation increct further wking, f the first mark ! Throughout the question, decimal percentage values rounded truncated 7 F, accept better, percentage 12 equivalents 8 F, accept 0.53 better, percentage 15 equivalents 1 F, accept 0.33 better, percentage 3 equivalents 1 F, accept better, 15 percentage equivalents Convert crectly to decimals percentages, even if their value is rounded truncated, f the first calculation 0.25 and 0.33 seen 25 and 33.3 seen, f the second calculation 0.6 and seen 8 15 equivalent 36

37 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Triangle 3m 2m 15, with no evidence of an increct method Shows implies at least two crect deductions about a and b 2a p (180 m b) = 180, 3b p 90 = 180 2a = b, b = 30 b p 2b p 90 = 180, 2a p 2b p 90 = 180 b = 30, a p b = 45 2a = 180 m (180 m b), a p a p 2b = 90 2a = b, a p b = 45 (a = ) 30 d 2 Shows implies a complete crect method with not me than one computational err 3 t b p 90 = 180, b = 25 (err) 180 m 25 = 155, 180 m 155 = 2 t a, so a = 25 d 2 = 12.5 Increct method 180 m 90 = d 2 = d 3 = 15 Note to markers: From the three triangles, the following simplified deductions may be made about a and b 1. 2a = b 2. b = a p b = 45 U1 Shows implies at least one crect deduction about b about a and b b = 2 t a b = 30 (180 m 90) d 3 = 30 2b p 2a = 180 m 90 b = 2a implied by values shown a as 22.5 and b as 45 37

38 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Multiplication grids 3m Completes both multiplication grids crectly, ie t 8 m m45 m6 m48 30 t m Completes one of the grids crectly and makes not me than one err omission in the other grid! F 2m, follow through F the first grid, accept follow through only from their m5 but note that their m5 must be native t 8 m6 (err but native) 9 72 m54 follow through Completes one of the grids crectly Makes not me than one err omission in each grid m6 m48 30 F the second grid, accept follow through only from their 15 t (err) 2 6 follow through 38

39 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Building a a Gives a crect explanation The highest was 11.7, the smallest was m 6.5 = m 6.5 Count up from 6 to 11, that s 5, then count up from 5 to 7 f the 0.2 Minimally acceptable explanation 6.5 to ! Ambiguous notation 6.5 m 11.7 Condone Maximum and minimum values given, but with no indication of how the range is found 11.7, 6.5 Values not identified identified increctly It s the largest the smallest 11 7 m 6 5 = 5.2 b b 8.7 equivalent Unambiguous indication of crect value! Value identified increctly 8 7 Condone if this err was penalised in part (a), otherwise do not accept c c ! Value rounded Accept 33 better 39

40 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Quiz 2m Indicates B can win and gives a crect explanation that makes explicit implicit reference to the questions still remaining A has 54 points, B has 45 points. B could get 10 me right and A none, so B would win on 55 Person A Person B Points so far Maximum possible Minimum possible So B could win A is 54, B is 45, 45 p 10 = 55 A = 54, B = 45, 54 m 45 = 9, but there are still 10 questions left 10% of 90 is 9, so A is only 9 ahead with 10 remaining F 2m, minimally acceptable explanation 54, p 10 = 55 B can get 55, and A is on 54 A = 54, B = 45, but there are 10 questions left A has 54 crect and B has 45 crect B can win if she gets all the rest right 54 and 45, so B can only win if A gets all the rest wrong 10% of 90 is 9, but there are 10 remaining A is 9 ahead with 10 remaining F 2m, incomplete explanation A has 54 and B has 45 A is only 9 ahead of B B can win if they get all the rest right and A gets all the rest wrong Shows both the values 54 and 45 States implies that there is a difference of 9 between the sces of A and B A is only 9 ahead of B 10% of 90 = 9 Gives a complete crect explanation that refers to the questions still remaining, with not me than one err, and follows through to make their crect decision B is at 45 and 45 p 10 = 55 A is at 56 (err) so B cannot win draw A is 60 (err), B is 45, 60 m 45 = 15 but there are only 10 questions left so B cannot win draw U1 Gives a partially crect explanation that fails to show imply the actual number of questions still remaining, but with a crect decision B can win if they get all the rest right and A gets all the rest wrong 40

41 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response x and y 3m Gives both x = m1 and y = 3 and shows a complete crect method f solving algebraically 3x p 7y = 18 3x p 6y = 15 so y = 3 x p 6 = 5 so x = m1 3(5 m 2y) p 7y = m 6y p 7y = 18 y = 3, so x = m1 Method used is trial and improvement 2m Shows a complete crect method f solving algebraically with not me than one err 3x p 7y = 18 3x p 6y = 15 so y = 4 (err) x p 8 = 5 so x = m3 3(5 m 2y) p 7y = m 5y (err) p 7y = 18 2y = 3 y = 1.5, so x = 2 18 m 7y = 15 m 6y my = 3 (err) so y = m3 and x = 5 m (2 tm3) = 11 F 2m, the only err is to use the wrong operation, spuriously eliminating either x y 3x p 7y = 18 3x p 6y = y = 33, so y = x p 5 = 5 so x = m Shows two crect equations that would allow elimination of either x y 3x p 7y = 18 3x p 6y = 15 6x p 14y = 36 7x p 14y = 35 Attempts to solve by substitution and fms a crect equation in only one variable 18 m 7y = 15 m 6y 3(5 m 2y) p 7y = 18 41

42 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Line of best fit a a Indicates that the crelation is native! Native qualified Igne qualifiers that accompany native, accept Strong native A bit native Do not accept without native, do not accept Strong Inverse Relationship described without reference to crelation The me time spent studying, the less time spent watching television b b Gives a crect explanation The most common crect explanations: Use the gradient its meaning That would be positive crelation, not native That would mean the me studying you do the me TV you watch The gradient should be native, but y e x p 40 has a positive gradient The gradient is m1, not 1 It would slope up not down Minimally acceptable explanation It would not be native [with native given f part (a), implying crelation] The gradient is not native It would need to be mx It would slope the wrong way It would go up not down It would look me like this: [sketches any positive gradient] [ caty continued on next page] Incomplete explanation It would not be native [without native given f part (a)] The equation needs a minus sign The line is going in the wrong direction It s at the wrong angle 42

43 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Line of best fit (cont) b b Refer to the fact that the values of y do not go cont above 40 When x is positive, y would always be me than 40, but none of the points are like this Adding 40 to the x codinates of the points gives different y codinates from theirs Nobody watched TV f me than 40 hours, but this equation would give hours watching TV as hours studying plus 40 Use a point points on the line of best fit, the meaning of its codinates When x is 10, y e 10 p 40 e 50, but on the graph it is 30 When y is 0, 0 e x p 40 so x em40, but the line of best fit goes to (40, 0) Someone studying f 20 hours would watch TV f about 20 hours, not 60 y = x p 40 means that y > x, but this is not true f some of the points on the line Give a crect equation f the line of best fit The line of best fit goes to 40 on both axes so the equation should be x p y e 40 It should be y emx p 40 Minimally acceptable explanation The vertical scale only goes up to 40 All the ys are below 40 [condone that this is not true at (0, 40)] 10 p 40 e 50, the graph doesn t go up that far Nobody watched TV f me than 40 hours Incomplete explanation The scale only goes up to 40 Everything is below 40 Everything would be higher on the graph Nobody studied f me than 40 hours Minimally acceptable explanation When x e 10, y e 30 so y x p 40 Line goes through (40, 0) but y e 40 p 40 y e p 40 The equation shows that y > x, but this is not always true f the line Incomplete increct explanation 5 30 p 40 [indicating use of the cross at (30, 5) rather than a value on the line of best fit] Minimally acceptable explanation x p y e 40 y emx p 40 F part (b), increct statement alongside a crect explanation It should be x p y e 40, y e x p 40 would be a vertical line That would be positive crelation, the equation should be y = x m 40 43

44 2006 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Crect response Thinking diagonally a Gives a crect explanation that shows the crect application of Pythagas theem (5 2 p 5 2 ) e (25 p 25) 5 t 5 p 5 t 5 e 50, so y e 50 y 2 e 5 2 p 5 2 y 2 e 50 y e y so y 2 e 50 and y e 50 It s an enlargement of a 1, 1, 2 triangle, so it s 5 2 and 5 2 = 50 Minimally acceptable explanation (5 2 p 5 2 ) 5 2 p 5 2 (25 p 25) (2 t 25) 2 t 25 e 50 [with no evidence of a misconception, about area] 5 2 e (5 2 t 2)! Throughout the question, increct notation increct further wking alongside a crect explanation Condone, f part (a) accept y 2 e 5 2 p 5 2 y 2 e 50 y 2 (err) e 50 5 t 5 p 5 t 5 e 50, so length is 50 e 7.5 (err) Incomplete increct explanation y 2 e 50 Use Pythagas 25 p 25 It s 2 t 25 5 t 10 e 50 and y e 50 Area = 5 t 5 t 2 = 50 5 t 5 e 25 which is half the square, so 25 t 2 e 50 b Indicates 200 and gives a crect explanation 2 50 e 4 t 50 e 200 The sides would be 10cm (10 2 p 10 2 ) e e 100 t 2 e is 10, 10 d 2 e 5 but the length of the diagonal of the small square is > e 10, but 50 5 Minimally acceptable explanation 2 50 e 4 t 50 (10 2 p 10 2 ) 10 2 p t 10 e 100, 100 t 2 Incomplete increct explanation 200 e e t 2 then t 2 again Area = 10 t 10 t 2 =

45 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 16 Crect response Rounding a 2m 8.7 t 10 4 Shows the value , not expressed in any kind of index fm! Throughout the question, zero(s) given after the last decimal place within standard fm notation Condone, f 2m in part (a) accept t 10 4 Shows the digits 87 b 2m 1 t 10 3 Shows the value equivalent, not expressed in any kind of index fm Shows the value equivalent 0.1 t 10 2 Shows the value equivalent 9.0 t t

46 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 17 Crect response Factising a Gives x and 4 in either der! F part (a), terms in x increct, but constants crect 2x and 4, 2x then 10 Mark as 0, 1 Gives x then 10 m10 then mx! Throughout the question, unconventional notation, f the first mark of part (a) 1 t x and 4, f the first 2 marks of part (b) (x m 2) t (x p 9) Condone b 2m Factises the expression crectly and fully (x m 2)(x p 9)! Throughout the question, quadratic expressions equated to zero Igne, even if there are errs in a subsequent attempt to solve it, f the first 2 marks of part (b) accept (x m 2)(x p 9) e 0 so x e 2 x e 9 (err) Shows one crect fact Identifies the digits 2 and 9 9 m 2 e 7, 9 t m2 em18 The numbers must be 2 and 9 m2, m9 (x m 9)(x p 2) Factises the expression crectly and fully (x p 7)(x m 7) 46

47 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 18 Crect response Mean of zero 2m Makes all three crect decisions, ie Must Could Cannot be true be true be true Makes two crect decisions U1 47

48 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 19 Crect response Equation a (p)20 and m20, in either der Answer of ± 20 b Gives a crect explanation The denominat is zero, and fractions with denominats of zero are not defined 60 isn t defined 0 Minimally acceptable explanation The denominat would be zero You can t divide by 0 There s nothing to divide 60 by 60 0! Use of infinity Condone, accept The closer the denominat gets to 0, the me the fraction tends towards infinity Anything divided by 0 e infinity 60 e 0 Incomplete increct explanation 60 It s and that s impossible 0 Because 10 m 10 e 0 You cannot divide by zero and you cannot find the square root of zero The denominat would be zero but 60 e e 0 0 c Gives a value less than 10 Crect set of values described x < 10 Less than 10 48

49 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 20 Marking overlay available Crect response Journeys 3m Draws a complete crect curve within the tolerance as shown on the overlay F 3m, points joined with straight lines f a curve 2m Draws a curve within the tolerance as shown on the overlay between (2, 50) and (5, 20), even if the curve is increct omitted elsewhere Indicates at least 5 crect points on the graph, even if the points are not joined joined with straight lines Indicates at least 3 crect points on the graph Gives the codinates of at least 5 crect points with x values greater than 0 but less than equal to 10! F 2m, points inaccurately plotted Accept provided the pupil s intention is clear! F 2m, points not explicitly plotted Accept unambiguous indications of the locations of points on the graph, f example the tops of vertical lines Note to markers: The five points with inter codinates are (1, 100), (2, 50), (4, 25), (5, 20) and (10, 10) 49

50 2006 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 21 Crect response Tangent 2m Gives a crect proof that shows implies the following three facts: 1. AC is a diameter (of the large circle) 2. APC is PC is a radius (of the small circle) Because AC is a diameter of the larger circle, APC must be 90. PC is a radius of the smaller circle and since AP is at right angles to PC, AP must be a tangent of the smaller circle PC is a radius given APC = 90 angle in a semicircle [1 implied] AP is a tangent An angle subtended by a diameter is 90, [1 implied] so the line through A and P is at right angles [2 implied] to a radius of the smaller circle [3 implied] a 2a 180 m b 2a e so b e 90 m a 2 therefe a p b e 90 [1 p 2 implied] and PC is a radius of the small circle a b Minimally acceptable proof Since AC goes through B, [1 implied] angle P is a right angle [2 implied] and P is joined to the centre [3 implied] ABCD goes through the diameters [1 implied] so AP touches the small circle at right angles [2 implied] to the radius [3 implied] F 2m, incomplete proof APC is 90, so AP and radius PC are at right angles and AP must be a tangent to the smaller circle [1 omitted] AC is a diameter and AP touches the small circle at P where PC is a radius [2 omitted] APC = 90 because AP and PC are joined to either end of a diameter, so AP is a tangent as it s at right angles to PC [3 omitted] U2 States implies that AC is a diameter (of the large circle) States implies that APC is 90 States implies that PC is a radius (of the small circle) 50