Mark scheme for Paper 1

Transcription

1 Ma KEY STAGE 3 ALL TIERS 2004 Mathematics tests Mark scheme f Paper 1 Tiers 3 5, 4 6, 5 7 and

2 2004 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 10 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available f that question part. The 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 UAM 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 2004 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 2

3 2004 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, time, codinates, algebra probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3

4 2004 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 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 2004 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 2004 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 sign is usually already printed in the answer space. Where the pupil writes an answer other than in the answer space, crosses out the sign, accept an answer with crect units in pounds and/ pence 320p, 700p Do not accept Increct ambiguous use of pounds pence 320, 320p 700p, p not in the answer space. Increct placement of decimal points, spaces, etc increct use omission of 0 3.2, 3 200, 32 0, 3-2-0, 7.0 Responses involving time A time interval F example: 2 hours 30 mins 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 time unit, hours minutes, is usually printed in the answer space. Where the pupil writes an answer other than in the answer space, crosses out the given unit, accept an answer with crect units in hours minutes, unless the question has asked f a specific unit to be used. A specific time F example: 8.40am, 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 6

7 2004 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving codinates F example: ( 5, 7 ) Accept Unambiguous but unconventional notation ( 05, 07 ) ( five, seven ) x y (5,7) ( xe5, ye7 ) Do not accept Increct ambiguous notation (7,5) (5x, 7y ) ( x5, y7 ) (5 x,7 y ) Responses involving the use of algebra F example: 2 p n n p 2 2n Accept Take care! Do not accept The unambiguous use of a different case N used f n Unconventional notation f multiplication n t 2 2 t n n2 n p n f 2n n t n f n 2 Multiplication by p 1n f 2 p n 2 p 0n f 2 Wds used to precede follow equations expressions t e n p 2 tiles tiles e t e n p 2 f t e n p 2 Unambiguous letters used to indicate expressions t e n p 2 f n p 2 Embedded values given when solving equations 3 t 10 p 2 e 32 f 3x p 2 e 32! Wds units used within equations expressions should be igned if accompanied by an acceptable response, but should not be accepted on their own do not accept n tiles p 2 n cm p 2 Change of variable x used f n Ambiguous letters used to indicate expressions n e n p 2 However, to avoid penalising any of the three types of err above me than once within each question, do not award the mark f the first occurrence of each type within each question. Where a question part carries me than one mark, only the final mark should be withheld. Embedded values that are then contradicted f 3x p 2 e 32, 3 t 10 p 2 e 32, x e 5 7

8 2004 KS3 Mathematics test mark scheme: Paper 1 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only. F example: 0.7 Accept Take care! Do not accept A crect probability that is crectly expressed as a decimal, fraction percentage. Equivalent decimals, fractions percentages ,,, 70.0% A probability crectly expressed in one acceptable fm which is then increctly converted, but is still less than 1 and greater than 0 70 e The following four caties of err should be igned if accompanied by an acceptable response, but should not be accepted on their own.! A probability that is increctly expressed 7 in 10, 7 out of 10, 7 from 10! A probability expressed as a percentage without a percentage sign.! A fraction with other than inters in the numerat and/ denominat. However, each of the three types of err above should not be penalised me than once within each question. Do not award the mark f the first occurrence of each type of err unaccompanied by an acceptable response. Where a question part carries me than one mark, only the final mark should be withheld.! A probability expressed as a ratio 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 less than 0 8

9 2004 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 tiers 3 5 and 6 8. A total of 121 marks is available in tiers 4 6 and 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 QCA website from Monday, 21 June QCA 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. 9

10 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 1 Answer of Pupils a 3 Pupils identified A, M, S Mike and two others b Drama c Paul Pupil not identified 6 d Sule 10

11 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 3 Number pyramids a Completes the pyramid crectly, ie b Completes the first pyramid crectly Numbers used are decimals, fractions, natives zero 20 Zeros omitted Completes the second pyramid crectly, in a different way from one credited f the first pyramid! Numbers credited f the first pyramid but shown in a different der Accept if the centre numbers of the bottom rows are different, accept, do not accept U1 11

12 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 3 5 only 4 Stacking a Gives all three crect and in the crect der ie 9, 18 and 27 b 30! In both parts (a) and (b), bottom layer not included ie 0, 9 and 18 [f part (a)] 24 [f part (b)] Mark as 0; 1 c 6 5 Calculations a 523 b 182 c 147 d 40 12

13 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Coins 3m 2m Shows all five crect ways, with none increct duplicated Shows at least four crect ways, with not me than one increct duplicated Zeros omitted! Values of coins given Provided this is the only err, mark as 1, 0, 0 Shows at least three crect ways, with not me than two increct duplicated 7 2 a a 10.6 Matchboxes Equivalent fractions decimals 7.2 3(.0) b b 8! Answer of 4 Accept only if it is clearly stated that another 4 boxes are needed, accept 4 me, do not accept 4 13

14 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Folding shapes a a Indicates the crect diagram, ie b b Completes the diagram crectly, ie! Lines not ruled accurate Accept provided the pupil s intention is clear Completes the diagram crectly, ie Completes the diagram crectly, ie 14

15 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Television 2m 130 Shows implies both 900 and 3, and carries out at least one of these calculations crectly = 330 (err) = Digits 13(0) seen 10 5 Measuring U1 Gives a crect explanation that shows the relationship between the volume of the jug and one litre It s 2 jugs Fill the jug once, pour it in the bucket and fill it again He uses A jug is half a litre Empty into the bucket twice Minimally acceptable explanation Fill it twice 500ml t 2 Jug assumed to be calibrated Put 200ml in the jug, then repeat to give a total of 5 times 11 6 Grid shapes a a B and E in either der Shape A given alongside a crect response b b D and E in either der! Responses f parts (a) and (b) transposed but otherwise crect Mark as 0; 1 c c 30 The given shape C excluded 29 me 29 15

16 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Club a a Indicates False and gives a crect explanation U1 The most common crect explanations: Identify the statement is increct f week 2 True f the first and last weeks only Identify the statement is increct f one of the Wednesdays The most popular day was a Wednesday The highest ever bar was Wednesday One Wednesday there were 27 Minimally acceptable explanation Not true f one of the weeks Wed was higher! Explanation unclear as to whether it refers to one week all three weeks Condone, accept Wed was the most popular day Do not accept increct explanations Each week Wed was most popular! Number of pupils identified Where the value is a multiple of 5, do not accept increct values. Otherwise, within a crect response, accept inter values between the relevant multiples of 5, f Monday of week 3 accept 26, 27, Incomplete explanation Not always true b b Indicates True and gives a crect explanation The most common crect explanations: Minimally acceptable explanation 20 The bars are the same U1 Identify that f each week 20 pupils attended 20 pupils went each Friday Identify the relevant feature of the charts The bars are all the same height Increct explanation, incomplete explanation that simply restates the infmation given They are all 25 (err) Same amount went It s the same number each week 16

17 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, Club (cont) c c Indicates Not enough infmation and gives a crect explanation The most common crect explanations: State that names are not shown It doesn t give their names so we don t know who went each week Minimally acceptable explanation No names State that the people could be different Same amount went each week but it could be different people Different pupils might have gone on different Fridays Minimally acceptable explanation It doesn t tell you which pupils Could be different each week State that only the total is shown It doesn t say the same pupils went. It just says 20 pupils went on Friday It doesn t tell you about each pupil, it tells you about the total Minimally acceptable explanation It only gives the total It just says 20 All it says is how many U1 Incomplete explanation You don t know The charts don t show it It doesn t give that much detail 17

18 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Points of intersection a a a Draws three straight lines intersecting at one point! Ruler not used Condone, provided the pupil s intention is clear Lines meet rather than intersect, f part (a) b b Draws three straight lines intersecting at three different points, f part (b) in tiers 3 5 and 4 6! Diagrams f parts (a) and (b) in tiers 3 5 and 4 6 transposed but otherwise crect Mark as 0; 1! Other diagrams shown Igne, as these may be wking f the last part of the question Diagram is ambiguous The drawing must clearly show the crect number of points of intersection, f part (b) in tiers 3 5 and 4 6 do not accept 18

19 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Points of intersection (cont) c c b Parallel! Wds used to describe parallel Accept if applicable to all sets of parallel lines Never meeting At the same angle In the same direction Not touching each other Do not accept if applicable to only some Vertical Hizontal U1 Incomplete response describing parallel Like railway tracks Apart 19

20 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Daylight hours 3m 2m Gives a complete crect response with both months identified crectly and crect values given within the ranges as shown below, ie June 18.5 to 19.5 inclusive December 5 to 6 inclusive Makes not me than one err, but if the err is in identifying a month the pupil must follow through from that increct month Jun 20 (err) Dec 6 June 19 February (err) 10 Makes not me than two errs omissions, but if the err is in identifying month(s) the pupil must follow through from that increct month(s) June 12 (err) Dec 7 (err) July (err) 18 Oct (err) 9 June 12 (err) Jan (err) 7! Months not written in full Accept unambiguous indications, f December D Do not accept ambiguous indication that could refer to other months, f June J! Dates given Igne, f June accept June 15th! Follow through Note that follow through must be applied from increct months. Ranges f crect values are shown below Jan Feb Mar Apr May (Jun Jul Aug Sep Oct Nov (Dec 6.5 to 7.5 inclusive 9.5 to 10 inclusive 12 to 12.5 inclusive 15 to 16 inclusive to inclusive 18.5 to 19.5 inclusive) 17.5 to 18 inclusive 15 to 15.5 inclusive 12 to 12.5 inclusive 9 to 9.5 inclusive 6.5 to 7.5 inclusive 5 to 6 inclusive)! Months omitted months identified ambiguously Treat each omission ambiguous response as one err, f 2m accept J (ambiguous) 19 Dec 5.8, f accept (omits) 19 (omits)

21 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Plasters a a a b b b ! Answer given as a decimal a percentage without a crect fraction shown Accept decimals within the following ranges, their percentage equivalents: part (a) to 0.03 inclusive part (b) 0.45 to 0.46 inclusive part (c) 0.54 to 0.55 inclusive! Wds given alongside a crect probability Igne, f part (a) accept Unlikely, 1 35 c c c

22 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Calculats 2m Shows the digits Shows a complete crect method f how to multiply 1.25 by 22, with not me than one computational err, but with the decimal point crectly positioned p p 1.25 p t 2.50 e 10 t 2.50 p (err) so Conceptual err 125 t so 5.00! Method is repeated addition F, at least some multiplication must be shown implied, f do not accept 1.25 p 1.25 p.. 22

23 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Delivery charges a a a Completes the table crectly, ie 8 7.(00) F 9 books, a value between 7.55 and 7.65 inclusive! 7.60 shown as 7.6 Condone b b b 60 p! Follow through from part (a) Accept provided their 7.60 > their 7.00 c c c Draws the crect straight line y = x, at least of length 6cm, including the point of intersection with the given line, with no errs! Line not dashed Condone! Line not ruled accurate Accept provided the pupil s intention is clear Series of points that are not joined d d d 6 U1! Follow through from an increct line in part (c) Provided there is only one point of intersection, follow through as the closest inter value above their x-value, from their intersection as (7.2, 6.5), accept 8, from their intersection as (4, 4.6), accept 5! Maximum of 10 books assumed Condone, accept 6 to 10 books 23

24 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 3 5, 4 6, Magic square a a a 2m Gives all six crect values, ie Incomplete processing Gives at least three crect values b b b 2m Gives all three crect values, ie a e 16, b e 4, c e 9 Gives the crect value f b the crect value f c Fractions equivalent fraction equivalent fraction! Decimals used 1 F, accept 0.33 better 3 7 F, accept 0.58, 0.583(...) 12 1 F, accept 0.17, 0.16, 0.166(...) equivalent fraction 24

25 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Functions a a a Gives both crect values, ie Incomplete processing, f part (a) b b b Gives both crect values, ie, f part (b) Increct notation, f part (a) c c c 2m Gives two different crect functions Examples of crect functions are shown below n 5 n n m 20 n m 10 3! Unconventional notation f n n Condone! n 5 Accept as a crect function, provided nothing that could be an increct operation is shown, do not accept n p 5 U1 Gives one crect function F 2m, same functions written with different symbols same functions but unsimplified n and n d 5 5 n and n t n m 20 and n m 10 p 30 25

26 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Cuboids a a a Indicates Cuboid A and gives a crect explanation The most common crect explanations: Show the crect surface area f both A and D The surface area of A is 66, but D is 40 Consider the number of cube faces that are not visible Each cube in D has 3 4 faces that cannot be seen but each cube in A has only 1 2 Fewer faces of the cubes are touching each other in A Consider the number of cube faces that are visible In A the cubes show 4 5 faces, but in D it s 2 3 There are me cube faces facing out on A than on D! Units inserted Igne Minimally acceptable explanation, f the crect surface areas 66 and 40 seen 4 t 16 p 2 is bigger than 4 t 8 p 8, f cube faces that are not visible There are fewer hidden faces in A D is me compact, f cube faces that are visible Cubes in A show 4 me faces, D shows less than 4 A has me faces showing A is me spread out! Use of sides f cube faces Condone, accept Me sides face out on A! Descripts of cube faces Note that pupils use a wide range of terms to describe the cube faces, f cube faces that are not visible Hidden faces Faces pointing inside Faces touching, f cube faces that are visible Faces facing out Faces showing Faces you can see Condone provided the pupil does not explicitly refer to the area of only one of the faces of each cuboid, do not accept You can see 8 faces on D and 16 faces on A Use of square f cube cuboid You can see me of each square s surface in A than in D Explanation is simply a description of one both of the cuboids In A all 16 are in a line and not on top of each other D is two cubes high U1 Increct statement Each cube in A shows 4 faces; D is 3 26

27 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Cuboids (cont) b b b Indicates All the same c c c 4 d d d 3m Shows, in any der, all four crect sets of dimensions ! Repeated sets of dimensions (repeated) (repeated) Igne the repeats and mark as 1, 0, 0 2m Shows three crect sets of dimensions Native non-inter dimensions used Shows two crect sets of dimensions 27

28 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Shading a a a Indicates No and gives a crect explanation The most common crect explanations: State imply that the sides are not all the same length The sides are not all the same length Only 2 sides are the same Minimally acceptable explanation The lengths are different An equilateral triangle has equal sides It is isosceles One side is 4, the others are 4.5 The angles are different It has rotation symmetry of der 1 It doesn t have rotation symmetry There is only one line of symmetry State imply that the angles are not all the same The angles are not all equal The angles aren t 60 Increct explanation No sides are equal No equal angles U1 State imply that the der of rotation symmetry is not 3, that the shape does not have 3 lines of symmetry b b b Indicates Yes and gives a crect explanation, even if the fact that the shape is a quadrilateral is not stated explicitly U1 The most common crect explanations: State imply there are two pairs of adjacent equal length sides The long sides are next to each other and they are the same length. So are the sht Two isosceles triangles on either side of the same base Two pairs of equal length sides, but opposite sides are not parallel State imply that the quadrilateral has exactly one line of symmetry through opposite vertices The only line of symmetry is a diagonal State imply that one diagonal bisects the other at right angles One diagonal is the perpendicular bisect of the other! Minimally acceptable explanation (sides) Note the explanation must make it explicit that the sides are both equal and adjacent, accept The top two sides are the same and the bottom two sides are the same Two joining sides equal, other two also equal It s two isosceles triangles, do not accept Two pairs of equal length sides It has a big triangle and a little triangle Opposite sides are equal in length Incomplete explanation There are two equal opposite angles Minimally acceptable explanation (symmetry) Relevant line of symmetry identified on diagram Incomplete explanation (symmetry) It has one line of symmetry [no line increct line of symmetry shown on diagram] 28

29 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Shading (cont) c c c Indicates Yes and gives a crect explanation The most common crect explanations: Minimally acceptable explanation Same sides, same angles State imply both that the sides are equal and the angles are equal 4 equal sides and 4 right angles It has 4 sides the same length and a right angle Incomplete explanation 4 sides that are the same length 4 right angles Sides are the same length and if you rotate it it s a square Same sides and it has rotation symmetry State imply that the der of rotation symmetry is 4 U1 State imply that the shape has 4 lines of symmetry Sums and products Both crect, ie 2 15 Both crect, ie 3 24! Second and third columns completely crect, fourth column increct omitted Mark as 0, 1 29

30 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, Thinking fractions a a 2m 1 2 F 2m, decimal fraction of Shows the fraction other unsimplified 30 but crect fraction Shows crect cancelling to t, even if 2 1 there are subsequent conceptual errs t e Shows implies a crect method using fractions with not me than one computational err, and with their fraction given in its simplest fm t = (err) = Shows implies a crect method using decimals recurring t 0.6 Conceptual err t = = (numerats added) t = (denominats added) Decimal rounded 0.83 t

31 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Thinking fractions (cont) 3 b b 2m equivalent fraction decimal 5 U1 Shows implies that the fractions should be multiplied, even if there are subsequent conceptual computational errs 3 4 t of is, then times t t % Shows a complete crect method involving finding fractions of an arbitrary amount, with not me than one computational err 4 3 of 100 e 80, of 80 e 60, so it is out of t 20 e 15, t 15 e 3 (err) so it s The use of of to imply multiplication 3 4 of 4 5 As the phrase is suggested by the question, do not accept as the only evidence Incomplete method To be complete, their final answer must show the connection between the arbitrary amount and the calculated value, do not accept 4 3 of 100 e 80, of 80 = 60 without 5 4 subsequent expression of 60 out of 100 equivalent 31

32 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 4 6, 5 7, a a a m 4 Rearrange c 4 c d 4 4k p 3 b b 2m Rearranges crectly w m 2 5 w m 10 5 F 2m, native denominat 10 m w m5! F 2m, use of division sign Accept provided there is no ambiguity, accept w d 5 m 2 (w m 10) d 5, do not accept w m 10 d 5 Shows implies a crect first step of algebraic manipulation w 2 p t e 5 10 p 5t e w 5t e w m 10 w m 10 d 5 32

33 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Journey 2m 24 1 Shows the journey time is 2 (hours) 2 Shows a complete crect method 60 d d (100 d 40) 60 t 2 d 5 40 t e of 100, so of The only err is to misread A f B, giving an answer of ! Answer given as a decimal Accept (...) Do not accept 67 unless a crect method a me accurate value is seen 15 8 Facts again a a Indicates (y p 2)(y p 6), ie b b 2m Gives a crect simplified expression y 2 p 11y p 18 11y p 18 p y 2! Use of multiplication sign in simplified expression Accept either y t y 11 t y, but not both Multiplies out the brackets crectly, even if there is increct no further simplification y 2 p 9y p 2y p 18 The only err is in the constant term but the pupil simplifies crectly to give an expression of the fm ay 2 p by p c y 2 p 9y p 2y p 11 (err) e y 2 p 11y p 11 a, b c as zero 33

34 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Marking overlay available Rodents a a Indicates the crelation is positive! Positive qualified Igne, accept Strong positive Direct positive Sign of crelation not indicated High Strong! Relationship quantified Igne if alongside a crect response Otherwise, do not accept Relationship described without reference to crelation The longer the body, the longer the feet b b Draws a line of best fit within the tolerance, and at least of the length, as shown on the overlay! Line not ruled accurate Accept provided the pupil s intention is clear! Line of best fit is increct beyond the dashed lines on the overlay Condone, accept A crect line of best fit that is then joined to the igin c c Indicates 7 34

35 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Rodents (cont) d d Indicates No and gives a crect explanation The most common crect explanations: Refer to the point being too far removed from the others It would be an outlier It would be a long way from the other points It would be too far from the line of best fit Minimally acceptable explanation It s on its own on the graph It doesn t fit the crelation The body to foot ratio doesn t fit with the others Incomplete explanation It s on its own It doesn t fit with the others Conceptual misunderstanding The point isn t on the line of best fit Refer to the general relationship between foot length and body length f these species A rodent as long as this would have much longer feet F such a small foot, the body would be smaller Minimally acceptable explanation Foot too small Body too long This one is long but it has small feet Rodents with small feet are small in length too U1 Use, implicitly explicitly, the values 228 and 22 Ratio of foot to body is too different from the others 228 is over ten times 22, which is too much Incomplete explanation It doesn t fit the graph! Their line of best fit used to estimate the foot length the body length if the animal were one of these species of rodents Accept provided the value is crect accding to their line (even if their line is increct) within the following ranges: Foot length 2, a range of 5 that includes their value Body length 15! Relationship quantified Accept provided the approximate nature of the relationship is recognised, and body length is shown as between 4 and 6 (inclusive) times foot length, foot length as between 16% and 25% (inclusive) of body length, accept 22 is not about a fifth of is a fifth of 228 and 22 is not close to this 228 is much me than about 6 times 22 35

36 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, m Gives the value equivalent probability, 2 and gives a crect justification The most common crect justifications: Two dice Use a systematic approach to illustrate all outcomes, either numerically as even odd e e e e 3 o o o o 4 e e e e 5 o o o o Minimally acceptable justification p 2, 3 p 4, 3 p 6, 3 p 8 5 p 2, 5 p 4, 5 p 6, 5 p 8 (odd outcomes only) 2, 2 2, 4 2, 6 2, 8 4, 2 4, 4 4, 6 4, 8 (even outcomes only implied)! Reversals included to give 32 outcomes Accept as a crect method Use separate probabilities f each dice which are then multiplied 4 2 t st dice all even so probability is 1, 2 nd dice two even so probability is 0.5, 1 t 0.5 Reason generally You are always adding an even from one dice. Half the time you add to another even which gives an even, half the time you add to an odd which gives an odd Minimally acceptable justification Using the 3 gives 4 odd numbers Using the 5 gives 4 odd numbers, and the other 8 must be even Even p even e even, even p odd e odd same amount of each Gives a crect probability without sufficient justification with a non-systematic approach Uses a systematic approach to show at least 12 crect outcomes with not me than one increct, even if an increct no probability is given Increct, spurious no justification 2 p 2 e 4, 4 p 3 e 7, 6 p 4 e 10, 8 p 5 e 13 1 so answer e with no further wking

37 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Juice 2m Indicates all three crect values, ie Equivalent fractions decimals Orange Cranberry Grape Gives a crect value f cranberry grape, with no evidence, seen implied, of an increct method f this value Gives the crect value f ange and shows wking indicating that one of the other amounts should be multiplied by t t d 2 e (err), p t F each type of juice, shows the crect amount to be added 1 1 1,, Increct method shown implied Answer of,, Answer of,, ( added to each) Answer of,, ( added to each) ! Unconventional notation F, condone 1, f accept ! Decimals rounded within wking 1 F, accept rounded to 0.33 better 3 1 and rounded to better 6 37

38 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Triangles 3m Gives a complete justification that identifies the four possible triangles as 4, 4, 7 5, 5, 5 6, 6, 3 7, 7, 1 and makes a crect deduction that allows them to reject other possibilities The most common crect deductions: State that the length of the two equal sides must sum to me than the length of the third Call the sides a, a and b, then 2a > b, so 1, 1, 13 1 p 1 < 13 2, 2, 11 2 p 2 < 11 3, 3, 9 3 p 3 < 9 4, 4, 7 5, 5, 5 6, 6, 3 7, 7, 1 It is not possible to make the base 9 me as each side must be less than the sum of the other two Minimally acceptable deduction There are no me because the combined total of the equal sides must be me than the other side it wouldn t meet [with four possible triangles identified] All sides must be < 8 the other two sides would not reach, only possible solutions are 5 p 5 p 5 7 p 7 p 1 6 p 6 p 3 4 p 4 p 7! Deduction is that the sides won t meet F 3m, pupils must consider explicitly the 3, 3, 9 triangle, f 3m accept 7, 7, 1 6, 6, 3 5, 5, 5 4, 4, 7 3, 3, 9 is not possible because the sides won t touch State that the length of the non-equal side must be less than 8 ( 7.5) 2x p y e 15, 2x > y so 0 < y < 7.5 when y e 7, x e 4 when y e 5, x e 5 when y e 3, x e 6 when y e 1, x e 7 38

39 2004 KS3 Mathematics test mark scheme: Paper 1 Tiers 5 7, Triangles (cont) 2m Makes a crect deduction that 2a > b that b < 8, even if the four possible triangles are not identified Identifies the four possible triangles and states that the 3, 3, 9 triangle will not wk, but gives an incomplete no explanation as to why Triangles identified only through unlabelled scale drawings Identifies the four possible triangles and gives an explanation that the sides on others won t meet, without explicitly considering the 3, 3, 9 triangle There are no me as the sides wouldn t meet [with four possible triangles identified] 2, 2, 11 and 1, 1, 13 won t wk as the sides are too sht to reach to make a triangle [with four possible triangles identified] Identifies the four possible triangles, with no impossible triangles identified as possible U3 Makes a crect statement about the sides of the triangles The sum of the sides that are equal must be even One side must be odd 39

40 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 13 Births a 1920 Unambiguous indication 1.13 t 10 6 b 2m 4.5 t 10 4 Shows implies the value t t 10 5 Increct value 45 t Facts a a e 4 and b e 3 b 7! F parts (a) and (b), values embedded Accept embedded values but do not accept increct statements, f part (a) accept 2 4 and 2 3 seen, f part (a) do not accept a e 2 4 b e 2 3 F part (b), follow through from part (a) as the sum of their values f a and b 40

41 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 15 Population a Indicates False and gives a crect explanation Although the number of under 20s is constant, the population size has changed It s a smaller proption of the whole population The overall number of people has increased so the percentage will drop U1 It s out of me people Minimally acceptable explanation There are me people (in 2050)! Values evaluated approximated Accept within the following inclusive ranges: 1998 No. of people < 20: to 2.4 (billion) Total no. of people: 5.9 to 6.1 (billion) Proption of people < 20 33% to 45% 2050 No. of people < 20: to 2.4 (billion) Total no. of people: 8.8 to 9.2 (billion) Proption of people < 20 20% to 30% 1998 to 2050 Proptional increase needed 45% to 55%, accept To keep the number of under 20s about the same it would need to be about 50% me b Gives a value between 45 and 55 inclusive c Gives a value between 250 and 350 inclusive d Makes a crect statement that refers both to the increase in the population as a whole and to the increase in the proption of the population who are aged 60 over,, minimally, 40 over By 2050 the wld s population is expected to have risen by 50%. Much of this increase will be from people aged 60 over The whole population will be bigger but the proption of young people will be less! Use of old young Accept old f people over 60,, minimally, over 40 Accept young f people under 20,, minimally, under 40 Implicit reference to the increase in the population as a whole Number of young people stays the same but old people increases! Follow through Accept provided this does not invalidate the crect conclusion Incomplete interpretation Me people in 2050, me over 60 The wld population will be bigger and people are expected to live longer Proption of young people will be less U1 No interpretation The wld population will increase by 50% and the number of people over 60 will increase by 300% 41

42 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 16 Box plots a 2m Draws a crect box plot, in which shtest e 136 tallest > 156 IQR < 10! Value f median shown, other labels given Igne, even if increct! All four points of location shown crectly but box plot not drawn Mark as 1, 0 Their box plot has shtest e 136, and tallest > 156 Their box plot has IQR < 10 b Up to 3m are available from the caties shown on the opposite page, all of which compare year 9 with year 7 girls Note that a maximum of 2m can be awarded from the minimally acceptable interpretations f the caties, ie f all 3m at least some valid comparison must be made! Year group(s) not specified Accept provided the statement is crect f year 9, accept The range of heights is greater! Increct statement interpretation Within each caty, do not accept contradicty statements increct data, do not accept In year 9 the IQR was 8, that s higher (err) than f year 7 The year 9 range was bigger and it was 40 (err) Markers may find the following helpful: year 9 year 7 shtest tallest range LQ median UQ IQR

43 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 16 Box plots (cont) b Uses the range, both the shtest and tallest, to make a valid comparison with year 7 girls The range of heights f older girls is greater The shtest girl in year 9 is (about) the same as the shtest girl in year 7, but the tallest girl in year 9 is (much) taller than the tallest girl in year 7 The shtest girls are the same, but the tallest girl in year 9 is about 16cm taller Shtest girl in year 9 is 136, same as year 7 Tallest in year 9 is 172, in year 7 it is 156 Minimally acceptable interpretation Identifies the year 9 range as 36 ( 2) Identifies f year 9 both the shtest as 136 ( 1) and the tallest as 172 ( 1) Values not interpreted 136, 172 (with no further indication of the meaning of these values) Ambiguous interpretation Year 9 girls are generally taller (with no further detail given) Uses the median to make a valid comparison with year 7 girls The median is higher in year 9 The median f year 9 is nearly the same as the tallest girl in year 7 Year 9 median is 153 but year 7 is 144 Minimally acceptable interpretation Identifies the median f year 9 as 153 ( 1)! Average used in place of median Condone Uses the IQR, both the LQ and the UQ, to make a valid comparison with year 7 girls The IQR f older girls is smaller Both quartiles are lower f year 7 pupils The IQR is 9 f year 9 but 10 f year 7 Year 7 has a lower quartile of 140 and an upper quartile of 150 but year 9 has a lower quartile of 150 and an upper quartile of 157 Minimally acceptable interpretation Identifies f year 9 both the LQ as ( 1) and the UQ as 157 ( 1) Identifies the year 9 IQR as 7.5 ( 2) Marks the median, the LQ and UQ on the x-axis of the cumulative frequency diagram, even if specific values labels are not given U2 F year 9, interprets the IQR to make a valid comparison with year 7 girls There is less variability within the middle 50% of girls in year 9 In year 9 the middle 50% are me bunched up A lot of girls in year 9 are just a bit bigger just a bit smaller than average! Incomplete interpretation There is less variability in year 9 Results me consistent f year 9 As these could be increctly referring to the range rather than the IQR, do not accept 43

44 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 17 Graphs 2m Gives all five crect letters in the crect der, ie D C B A E Gives at least three crect letters 18 Proving 3m Gives a crect proof The most common crect proofs: Use algebra to manipulate expressions representing two consecutive numbers, interpreting the results n and n p 1 are consecutive numbers n 2, (n p 1) 2 = n 2 p 2n p 1 n 2 p n 2 p 2n p 1 e 2n 2 p 2n p 1 e 2(n 2 p n) p 1, which is odd (2x) 2 e 4x 2 (2x m 1) 2 e 4x 2 m 2x m 2x p 1, 4x 2 p 4x 2 m 2x m 2x p 1 is even p even m even m even e even, then p 1 makes it odd Reason generally about odd and even numbers, showing explicitly the following four steps 1. Of the two numbers, one must be odd ( one must be even) 2. Odd 2 is odd 3. Even 2 is even 4. Odd p even is odd Out of the two you pick, one will be even and so have an even square. One will be odd and so have an odd square. An odd number added to an even number gives you an odd number! Numbers used Igne if used to illustrate but do not accept explanations that lack generality, do not accept 2 3 e 9, 4 2 e 16 9 p 16 e 25, which is odd Minimally acceptable proof, using algebra 2 n p (n p 1) 2 e 2n 2 p 2n p 1 = 2(n 2 p n) p 1 2 n p (n p 1) 2 e 2n 2 p 2n p 1 e even p even p 1 2 (2x) p (2x m 1) 2 e 4(2x 2 m x) p 1, reasoning generally One is odd, odd t odd e odd even t even e even odd p even e odd Consecutive numbers are odd and even, and consecutive square numbers alternate between being odd and even. Odd p even e odd F 3m, incomplete mathematical communication One is odd, one is even Square them both and you have one odd number, and odd p even is odd 44

45 2004 KS3 Mathematics test mark scheme: Paper 1 Tier 6 8 only 18 Proving (cont) 2m Uses algebraic expressions to represent the squares of any two consecutive numbers, then expands the brackets crectly, even if expressions are not simplified n 2, (n p 1) 2 n 2, n 2 p 2n p 1 (2x) 2 = 4x 2 (2x m 1) 2 e 4x 2 m 2x m 2x p 1 Reasons generally about odd and even numbers but omits one of the four steps shown above Odd 2 = odd, even 2 = even, Odd p even e odd [step 1 not explicit] Consecutive square numbers alternate between being odd and even, odd 2 e odd, an odd number added to an even number is always odd [step 3 not explicit] If the inters are consecutive, one of them will be even, the square of an odd number is always odd, and the square of an even number is always even [step 4 not explicit] F 2m, minimally acceptable response One is odd, one is even. Square them both and you have one odd number. Odd p even is odd Uses algebraic expressions to represent any two consecutive numbers n, n p 1 2x m 1, 2x U1 Attempts to reason generally, showing at least one of the four steps Of two consecutive numbers, one is odd and one is even Odd 2 = odd Even t even e even One in every two consecutive squares is odd Odd p even e odd F, minimally acceptable response One is odd 45

46 EARLY YEARS NATIONAL CURRICULUM 5 16 GCSE GNVQ GCE A LEVEL First published in 2004 NVQ Qualifications and Curriculum Authity 2004 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 Publications, PO Box 99, Sudbury, Suffolk CO10 2SN (tel: ; fax: ) Order ref: QCA/04/