Mark scheme for Paper 2

Transcription

1 259578_cover 2/4/04 10:06 AM Page FC1 Ma KEY STAGE 3 ALL TIERS 2004 Mathematics tests Mark scheme f Paper 2 Tiers 3 5, 4 6, 5 7 and

2 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Introduction Introduction The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to infm teachers. This booklet contains the mark scheme f paper 2 at all tiers. The paper 1 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking infmation f questions is set out in the fm of tables, which start on page 10 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available f that question part. The Crect response column usually includes two types of infmation: a statement of the requirements f the award of each mark, with an indication of whether credit can be given f crect wking, and whether the marks are independent cumulative; examples of some different types of crect response, including the most common. 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. The 2004 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 2

3 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 General guidance General guidance Using the mark schemes Answers that are numerically equivalent algebraically equivalent are acceptable unless the mark scheme states otherwise. In der to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed crect action. This is followed by further guidance, relating to marking of questions that involve money, time, codinates, algebra probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3

4 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 General guidance What if The pupil s response does not match closely any of the examples given. Markers should use their judgement in deciding whether the response cresponds with the statement of requirements given in the Crect response column. Refer also to the. The pupil has responded in a non-standard way. Calculations, fmulae and written responses do not have to be set out in any particular fmat. Pupils may provide evidence in any fm as long as its meaning can be understood. Diagrams, symbols wds are acceptable f explanations f indicating a response. Any crect method of setting out wking, however idiosyncratic, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma f a decimal point. The pupil has made a conceptual err. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, err. A computational err is a slip such as writing 4 t 6 e 18 in an otherwise crect long multiplication. A conceptual err is a me serious misunderstanding of the relevant mathematics; when such an err is seen no method marks may be awarded. Examples of conceptual errs are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 t 27; subtracting the smaller value from the larger in calculations such as to give the answer 21; increct signs when wking with native numbers. The pupil s accuracy is marginal accding to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, touches, the boundaries given, the mark(s) should be awarded. The pupil s answer crectly follows through from earlier increct wk. Follow through marks may be awarded only when specifically stated in the mark scheme, but should not be allowed if the difficulty level of the question has been lowered. Either the crect response an acceptable follow through response should be marked as crect. There appears to be a misreading affecting the wking. This is when the pupil misreads the infmation given in the question and uses different infmation. If the iginal intention difficulty level of the question is not reduced, deduct one mark only. If the iginal intention difficulty level is reduced, do not award any marks f the question part. The crect answer is in the wrong place. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a wd number response is expected, a pupil may meet the requirement by annotating a graph labelling a diagram elsewhere in the question. 4

5 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 General guidance What if The final answer is wrong but the crect answer is shown in the wking. Where appropriate, detailed guidance will be given in the mark scheme and must be adhered to. If no guidance is given, markers will need to examine each case to decide whether: the increct answer is due to a transcription err; If so, award the mark. in questions not testing accuracy, the crect answer has been given but then rounded truncated; If so, award the mark. the pupil has continued to give redundant extra wking which does not contradict wk already done; If so, award the mark. the pupil has continued, in the same part of the question, to give redundant extra wking which does contradict wk already done. If so, do not award the mark. Where a question part carries me than one mark, only the final mark should be withheld. The pupil s answer is crect but the wrong wking is seen. A crect response should always be marked as crect unless the mark scheme states otherwise. The crect response has been crossed rubbed out and not replaced. Mark, accding to the mark scheme, any lible crossed rubbed out wk that has not been replaced. Me than one answer is given. If all answers given are crect a range of answers is given, all of which are crect, the mark should be awarded unless prohibited by the mark scheme. If both crect and increct responses are given, no mark should be awarded. The answer is crect but, in a later part of the question, the pupil has contradicted this response. A mark given f one part should not be disallowed f wking answers given in a different part, unless the mark scheme specifically states otherwise. 5

6 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 General guidance Marking specific types of question Responses involving money F example: Accept Any unambiguous indication of the crect amount 3.20(p), 3 20, 3,20, 3 pounds 20, 3-20, 3 20 pence, 3:20, 7.00 The 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 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 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 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 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 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 General guidance Recding marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 a 0 entered in each marking space. Where 2m can be split into gained and lost, with no explicit der, then this will be recded by the marker as 1 0 The total marks awarded f a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recded on the front of the test paper. A total of 120 marks is available in 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 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 1 Crect response Spts a Shows a crect amount, with units b Shows a crect amount, with units 8.02! Value rounded In part (a), accept 182 but do not accept 181 unless a crect value is also seen In part (b), do not accept 8 unless a crect value is also seen! Units omitted Penalise only the first such occurrence c 3! Reference to money left over Accept the crect change shown 3 r ( )5.03 Do not accept reference to part of a racket 3.3(...) 2 a 24 Crect response Travelling by train b Completes the bar f girls crectly and in the crect position, ie! Bar not shaded lines not ruled accurate Accept provided the pupil s intention is clear and the top of the bar is not me than m from the line indicating 14 c 2m Gives all four crect entries, ie F 2m, zero omitted U2 Gives at least two crect entries 10

11 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 3 Crect response Maze a Identifies the crect square, ie Unambiguous indication Crect square marked A b Indicates the crect set of instructions, ie 6, south 3, east c 2m Indicates the crect set of instructions, ie 3, west 2, nth The only err is to der the instructions increctly, ie! F part (b), 6 south and 2 east given Condone Unambiguous indication, f part (b) 6.S 3.E s, 6 e, 3 Directions other than compass points used, f part (b) 6 down 3 right 2, nth 3, west One instruction is completely crect and crectly dered, even if the other instruction is increct omitted Both compass directions are crect and crectly dered 2 (err), W 3 (err), N 11

12 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 4 Crect response ABC Crect response Windmills a Completes the windmill pattern crectly, ie! Squares not shaded Accept provided the pupil s intention is clear b Completes the windmill pattern crectly, ie 12

13 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 6 Crect response Odd v even a Gives a crect counter example The most common crect counter examples:! Other trials shown Igne if at least one crect counter example is shown U1 Show an even number multiplied by three 2 t 3 = 6 which is even 3 t 10 = 30 Give an even number that is shown to be a multiple of 3 18 d 3 = 6 30 is in the 3 times table 3 goes into 12 b Gives a crect counter example U1 The most common crect counter examples: Show a multiple of four divided by two 8 d 2 = 4 which is even 1 of 12 is Give an even number that is multiplied by two to give another even number 2 t 10 = 20! Calculation not processed Accept if a crect comment is given, f part (a) 6 t 3 isn t odd 3 t 10 is even Even t 3 is even Otherwise, do not accept, f part (a) 6 t 3 Even t 3! Examples use addition subtraction rather than multiplication division F part (a), accept answers of the fm n p n p n where n is even, repeated addition of 3 where the number of 3s is even, accept 2 p 2 p 2 = 6 3 p 3 = 6 F part (b), accept answers of the fm 2n m n = n where n is even, n p n = 2n where n is even, accept 4 m 2 = 2 12 p 12 = 24! Crect counter example accompanied by an increct statement Igne increct statements, f part (a) accept 2 t 3 = 6, 6 isn t odd but most of the time the answer will be odd Increct notation, f part (a) 3 d 18 = 6 10 = 30 13

14 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 7 Crect response Triangular tiles a Shows how eight tiles join to make a square! Lines not ruled accurate Accept provided the pupil s intention is clear! Internal lines not shown Diagonal lines must be shown but pupils may use the given grid lines to represent hizontal vertical lines Internal lines increct! In both parts (a) and (b), tiles make an internal square even if there is no shading b Shows how four tiles join to make a square, ie Mark as 0, 1! In both parts (a) and (b), two tiles taken to be one larger tile Mark as 0, 1 14

15 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Recycling rubbish a a Gives a value between 6 and 16 inclusive Value qualified About 10 b b Indicates only Germany and Nway Unambiguous indication N, G 9 2 a a 18 Crect response Shaded shape b b Draws a rectangle of area 18cm 2 3 by 6 rectangle 2 by 9 rectangle 4 by 4.5 rectangle Follow through from part (a)! Lines not ruled accurate Accept provided the pupil s intention is clear 15

16 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, a a 6 Crect response Making b b Gives a crect explanation The most common crect explanations: Refer to the fact that an even number of 5p coins gives an even total, and that addition of 2p coins will keep the total even An even number of 5p coins gives an amount that is even, leaving an odd amount to make up 27p. You can t make an odd number with 2p coins An even number of 5s is even, adding 2s keeps it even, but 27 is odd An even number of 5s always ends in zero, leaving you to make an odd number with 2s which is not possible Minimally acceptable explanation An even number of 5s leaves an odd number and you can t make an odd number from 2s 27 is odd, so you have to have an odd number of 5ps the 2s would make it even Explanation refers only to 5s, only to 2s An even number of 5s is even but 27 is odd An even number of 5s always ends in zero You can t make an odd number with 2s Justification not given You can only make even totals You can only do it using an odd number of 5s Can t both be even 27 is an odd number Produce a set of possible solutions 0 t 5p = 0p leaving 27p, impossible 2 t 5p = 10p leaving 17p, impossible 4 t 5p = 20p leaving 7p, impossible 6 t 5p = 30p, which is too big You can t make 27, 17 7 using 2s! Only one case considered As this is a level 4 mark, condone, accept 2 t 5p = 10p leaving 17p, not possible 4 t 5p = 20p leaving 7p, can t You can t make 7 using 2s Two 5s make 10 and eight 2s that is as close as I can get Add 2ps to 10, you get 12, 14, 16, 18, 20, 22, 24, 26, U1 Justification not given 26 is as close as I can get You can make

17 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Patterns on a grid a a Gives the crect codinates, ie (2, 1) b b Gives both pairs of codinates in either der (3, 3) (4, 4) c c Gives both pairs of codinates in either der (16, 16) (17, 17) d d 2m Makes a crect decision and gives a crect explanation that shows implies 14 and justifies that 16 me are needed Yes, 1 2 p 2 2 p 3 2 p 4 2 = 30 There are enough because 1 p 4 p 9 = 14, 4 t 4 = 16 and 14 p 16 = 30 The next square is 16 tiles (4 by 4 square drawn) and you ve used up 14 of them, so there s just enough You have 16 tiles left and 4 t 4 = 16; all the tiles are used! 16 not justified Accept only if the response makes it clear that exactly 30 tiles are used, f 2m accept Used 14, got another 16 so you will use up all the 30 tiles 30 m 14 = 16, so yes you have exactly the crect amount, f 2m, do not accept 14 used, 16 left so yes you can 30 m 14 = 16, so yes you have enough U1 States implies that the next square uses 16 tiles You need 16 to make the next square Draws a 4 by 4 square with 16 cells 4 t 4 seen States implies that exactly 30 tiles will be used, but does not justify that 16 me are needed You need all 30 There would be no tiles left over It all adds up to 30 Identifies the pattern of differences p3, p5, p7! 4 by 4 square drawn crectly, but the number of squares increctly processed F, condone Their explanation could imply that 7 me squares are needed, ie a total of 21 so yes, there are enough 17

18 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Caribbean cdial 1 a a equivalent equivalent! Change of units Accept provided the new units are clearly shown, f the second mark accept 750ml 75cl! Increct units inserted in an otherwise crect response, f the first mark 0.5g Penalise only the first such occurrence b b

19 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Shape rotation a a Indicates the crect four faces Unambiguous indication Grey faces labelled G b b 2m Draws a crect view of the cuboid in either of the ientations below, using the isometric grid Increct no shading F 2m, internal lines omitted The only err is to draw the cuboid in the wrong ientation! Lines not ruled accurate Accept provided the pupil s intention is clear! Cuboid enlarged F 2m, accept provided a consistent scale fact has been used f all lengths Shape is not a cuboid The only err is to omit some external lines to show some hidden lines 19

20 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, a a 105 Crect response Multiples 108 b b Indicates Yes and gives a crect explanation interpreting the wd fact 140 will divide by 7 with no remainder 140 is a multiple of is in the 7 times table 7 goes into 140 exactly 7 t 20 = 140 Minimally acceptable explanation 140 will divide by 7 7 goes into t 2 = 140! Explanation refers to 14 rather than 140 Accept provided the relationship between 7 and 14 is shown implied, accept 7 goes into 14 7 t 2 = 14 7 times table goes 7, 14 and so on Otherwise do not accept 14 goes into 140! Use of repeated addition Condone, accept Keep going up in 7s and you get to 140! Use of it other ambiguous language Condone provided either is used, implying it is the other number, accept 7 goes into it 140 divides by it Otherwise do not accept It goes into it You can divide them! Response contains an increct statement Condone only if accompanying a crect response, accept Yes, 7 divides into 140 as it is a multiple of 140, do not accept 7 d 140 = 20 7 is a multiple of will go into 7 7 goes into 140 thirty times 20

21 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Nepal a a a 8 b b b 2m Draws a bar from m3 to 12, aligned with 5000 on the y-axis, and of the crect thickness! Lines not ruled accurate Accept provided the pupil s intention is clear Indicates that the maximum temperature is 12 m3 p 15 = 12 seen Draws a bar with a right-hand end at 12! F, bar increctly aligned with the 5000, bar of increct thickness Condone Indicates on the graph the crect positioning f m3 1 Draws a bar that is 15 units, ie 7 in length 2 squares, 21

22 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Angles a a a Indicates No and gives a crect explanation that shows the angle sum is increct 30 p 60 p 100 = 190 but it should sum to 180 They should add to 180 but these add to p 60 p 100 is 10 drees too big Minimally acceptable explanation Accept responses that state the angles should not add to 190, that the angles should add to 180 They add to 190 which is wrong Angles in a triangle add up to 180 The angles don t make 180 They should add to 180 Incomplete increct explanation The angles add to 190 When you add up the angles you get the wrong angle sum Angles add to 200 (err) not 180 U1! Increct units Igne, accept within a crect explanation 180ºC b b b 2m 130 Shows implies a crect method with not me than one computational err 360 m (70 p 70 p 90) 360 m t 70 p 90 = 200 (err), 360 m 200 = p 70 = 140, 140 p 90 = 330 (err), answer m 50 22

23 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Right angles a a a Draws any quadrilateral with exactly two right angles! Lines not ruled accurate Accept provided the pupil s intention is clear b b b Draws any quadrilateral with exactly one right angle 23

24 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Prime grid a a a Gives a crect explanation The most common crect explanations: State that 35 is a multiple of 5 and/ 7 35 is a multiple of 5 7 is a fact of 35 State that prime numbers have only two facts but that 35 has me than two facts A prime has 2 facts, 35 has 4 Minimally acceptable explanation 5 goes into it It s in the 7 times table 7 t 5 1, 5, 7, 35 It has me than two facts 35 divides by me than one and itself Incomplete explanation 35 is in some of the times tables 35 has facts Because it ends in 5 U1 State that the last digit of any prime number greater than 5 is 1, 3, 7 9 All prime numbers must end in 1, 3, 7 9 with the exception of 2 and 5! Crect explanation accompanied by a statement that uses mathematical language increctly Throughout the question, condone, f part (a) accept 35 has me than 2 facts, 35 goes into 5 5 goes into 35, so it has 2 facts b b b Gives a crect explanation The most common crect explanations: State imply the numbers in column Y will all be multiples of 6 ( 2, 3) They are all in the 6 times table, so they must be multiples of 6 They are all multiples of 3 State imply the numbers in column Y will all have a fact of 6 ( 2, 3) They all have a fact of 3 2 is the only prime that is even and all these numbers are even and greater than 2 Minimally acceptable explanation It s the 6 times table You can divide them by 3 They are all even The only even prime is 2 None of the numbers ends in 1, 3, 7 9 That column Y starts at 6 is not explicitly stated Condone, accept They are all even and even numbers are never prime Incomplete explanation They are all in times tables They all divide by something other than one and itself 6 d 3 = 2 It goes up 6 each time U1! Misunderstanding of prime A common misconception is to confuse prime with odd. Hence do not accept statements that refer only to odd, do not accept The numbers are not odd 24

25 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Prime grid (cont) c c c Gives a crect explanation U1 The most common crect explanations: State imply the numbers in column X will all be multiples of 3 They are all in the 3 times table, so they must be multiples of 3 State imply the numbers in column X will all have a fact of 3 They are all in the 3 times table, so they are all divisible by 3 Minimally acceptable explanation They are all in the 3 times table 3 goes into them Incomplete explanation They are all in times tables They will all divide by something other than one and itself All the other numbers have facts It goes up 3 each time! Misunderstanding of prime A common misconception is to confuse prime with odd. Hence do not accept statements that refer only to odd, do not accept The numbers are not odd Crect response Crisps 40! Increct units given Igne 25

26 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Shoe sizes 3m Indicates Yes and gives a crect explanation that shows implies both of the values and t 1.25 p 32 e 40.75, 7.5 t 1.25 p 32 e , so they both round to p 32 rounds to 41 and so does p gives 9 and gives 9 befe adding 32, so they will end up the same Minimally acceptable explanation, with Yes indicated They are both 41 They are and ! rounded truncated Accept 41, Do not accept 40! rounded truncated Accept 41, 41.4, 41.3, Do not accept 42 2m Shows implies both of the values and even if there is an increct no decision, increct further wking Tom wears 40.8 and Karl wears 41.4 so they don t wear the same size and so they both wear 40! from increct wking Note that pupils who add 1.25 rather than multiplying generate the shoe sizes and F 3m 2m, do not accept explanations based on such misconceptions They are both 41 as 7.5 p 1.25 p 32 e 41 7 p 1.25 p 32 e 41 Shows the value Shows the value with crect wking 7.5 t 1.25 p 32 e 41 The only err is to add 1.25 rather than multiplying Indicates No and shows the values and Indicates No and shows the values 41 and 40 26

27 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, 5 7, a a 8 Crect response Same area b b 2m 3, with no evidence of an increct method Shows the value 12 Fms a crect equation in w 1 4w e (6 t 4) 2 4 t w e 3 t 4 Shows a crect method with not me than one computational err 6 t 4 d 2 d 4 3 t t 4 d 2 e 20 (err), 20 d 4 e 5 6 d 2 Conceptual err 6 t 4 e 24, 24 d 4 e 6 27

28 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, 5 7, a a 2m Crect response Holiday! Value rounded Accept F 2m, do not accept 556 unless a crect method a me accurate value is seen Shows implies a complete crect method, even if there are rounding errs 17 t d 100 t % e 327.5(0) 5% e % e (0) p p 2 t % e 32.75, 33 (premature rounding) t 17 e 561 Shows the digits b b 2m 7.5(...)! Value rounded F 2m, do not accept 7 8 unless a crect method a me accurate value is seen Shows implies a complete crect method 1644 t Shows the digits 75(...) 7 Gives a value between 7 and 8 inclusive 28

29 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Straight lines a a a Completes the table with any three sets of crect codinates, indicating f each that x p y e 4 (x, y) (0, 4) (1, 3) (2, 2) x p y Incomplete processing, f (1, 3) 1 p 3! Values f (x, y) crect but some all of values f x p y omitted Accept provided a crect equation is given in part (b) b b b Gives a crect equation x p y e 4 y e 4 m x x emy p 4 c c c Draws the crect straight line through (0, 6) and (6, 0)! Line not ruled accurate Accept provided the pupil s intention is clear! Partial line drawn Do not accept lines that are less than 5cm in length! Points plotted Igne Points not joined 29

30 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Quiz a a a Gives both crect values, ie maximum of 40 and minimum of m20 Increct notation 20m b b b 14 c c c 2m Completes both rows crectly, in either der Completes one row crectly Crect response Cotton reel a a a 3π (...) 9.43 with no evidence of an increct method! Answer of 9 Accept provided a crect method a me accurate value is seen b b b 2m 970! Follow through from part (a) F 2m, accept 9100 d their (a), rounded crectly to the nearest ten, provided 9100 d their (a) is not a multiple of 10, from their (a) as 7.8, accept f 2m 1170, from their (a) as 7, do not accept f 2m 1300 Shows implies that the total length should be divided by the circumference, even if the units are increct there are rounding truncation errs 9100 d d 3π Digits 96(...) 97(...) seen F, follow through from part (a), even if their (a) is rounded truncated befe being used, from their (a) as 7.8, accept 9100 d 8 30

31 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Medicine a a 2m Indicates a crect value, with appropriate units, with a crect method shown 80 d 16, 5ml 20 t 4, litres 12 p 4 F 2m, increct incomplete method 20 d 4 e 5ml The only err is to omit units to give increct units Units of ml are given and the method shows implies crect substitution and understanding of algebraic notation f both multiplication and division 20 t 4 d 16, answer 50ml 20 t 4 = 100 (err), 12 p 4 e d 16 e 6.25ml 20 t 4 8 e (err in numerat) = 0.5ml 12 p 4 16 Answer of 10.6(...)ml 10.7ml 1l (only err is to omit necessary brackets when processing)! Units other than ml are given Accept provided the pupil shows such a change is intended and the change has been carried out crectly, accept 20 t 4 d 16 e 50, answer 0.05 litres An answer of 5ml, equivalent, is given with no wking b b 2m 12 (years) Shows a crect equation with the values 15 and 30 crectly substituted 30y 15 = 12 p y 15(12 p y) e 30 t y 1 = 2y 12 p y Shows the crect answer of 12 embedded, even if an increct value is chosen subsequently as the answer 30 t =, answer p 12! Use of? other symbol f y Accept if consistent, f accept 15 = 30 t? 12 p?! Units given within an equation Condone, f accept 30ml t y 15ml = 12 p y 31

32 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Recycling a a a 2m 8 Shows a crect angle f one me pupils, but not 5 pupils 60 d 5 e 12 f each one 3 pupils is 36 Shows a crect method with not me than one computational err 96 d (60 d 5) 96 d 60 e 1.6, 5 t 1.6 One pupil is 13 (err), and 96 d 13 e 7.38 so 7 pupils 96 Total pupils e 5 t 6 e 30, t e 0.083, 96 t b b b 2m 135 Shows a crect angle f one me pupils, but not 24 pupils 24 is 360, 1 is 15 3 pupils is 45 Shows a crect method with not me than one computational err 9 d 24 t d d 24 e 16 (err), 16 t 9 e Shows as a crect percentage %! 37.5 rounded truncated to an inter Do not accept unless a me accurate value is seen 37.5 without the percentage sign 32

33 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Russian dolls 1 a a Indicates both 6 and 10, in the crect der 2 Equivalent fractions decimals! 10.5 rounded truncated to an inter Do not accept unless a crect method a me accurate value is seen b b 2m Indicates both 5.1 and 7.7, in the crect der Indicates one crect value, even if not rounded, f the smallest doll (...), f the middle doll (...) Shows implies a crect method f both dolls, even if there is evidence of premature rounding 9 d 7 t 4, 9 d 7 t 6 9 = 1.3 (rounded), t 4 e 5.2, 1.3 t 6 e 7.8! 5.1(...) 7.7(...) rounded truncated to an inter Do not accept unless a crect method a me accurate value is seen! Answers are 5 and 8, round to 5 and 8 F to be awarded, 9 d (...) must be seen 33

34 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Sweets 2m 42, with sufficient wking to suppt a crect method! Method is trial and improvement Accept f 2m, but not f Increct method (39 p 40 p 41 p 42 p 43 p 44) d 6 e 42 Gives the answer 42 with no evidence of an increct method Shows the value 368 Shows the value 410 Shows a complete crect method with not me than one computational err (10 t 41) m (3 t 39 p 2 t 40 p 41 p 42 p 2 t 44) 117 p 80 p 41 p 42 p 84 (err) e m 364 e m (m2 t 3 p m1 t 2 p 1 p 3 t 2) m6 p m2 p 1 p 4 (err) em3 so there are 44 Shows the overall difference of the values given from the mean is m1 3(m2) p 2(m1) p 0 p 1 p 2(3) = m1 m6 p m2 p 1 p 6 e m1 34

35 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Marking overlay available Crect response Pentagonal pyramid a a Gives a crect explanation The most common crect explanations: Show state that the angles in a pentagon sum to 540, and that angle a is 540 d 5 The interi angle of a rular pentagon is 108, because 5 m 2 e 3, 3 t 180 e 540 and 540 d 5 Show state that the exteri angle of a rular pentagon is 72, and that angle a is 180 m d 5 e 72, 180 m 72 Show state that the angle at the centre of a rular pentagon is 72, and that angle a is 180 m d 5 e 72, (180 m 72) d 2 e 54, 54 t 2 Minimally acceptable explanation 540 d m 72 (with the exteri angle of 72 marked crectly on the diagram) The interi angle of a rular pentagon is m 72 (with the centre angle of 72 marked crectly on the diagram) Incomplete explanation The angles in a pentagon sum to t 5 e 540 (with no justification indication of the relevance of the 540) 180 m 72 e 108 (with no justification of the 72) The angle of a rular pentagon is 108 Angle of 108 marked on the diagram b b Indicates 36 and shows a crect method, using a large triangle (180 m 108) d 2, using a small triangle 180 m 2 t 72, using a kite 360 m (3 t 108) c c 2m Completes the perpendicular bisect, fulfilling four conditions below: 1. Ruled 2. Within the tolerance as shown on the overlay, including if their line were to be extended 3. At least 3cm in length 4. Evidence of crect construction arcs that are centred on C and D, the vertices next to C and D, are of equal radii, and show at least one intersection Minimally acceptable method 72 d 2 e 36 Spurious method 180 d 5 e 36! Use of construction arcs on the overlay Note that these are to give a visual guide as to whether the crect centres have been used, and do not indicate tolerance Side other than CD used Spurious construction arcs Do not accept arcs drawn without compasses arcs that do not show a distinct intersection, arcs that just touch Completes the perpendicular bisect with all of conditions 1 to 3 fulfilled Fulfils condition 4, even if the perpendicular bisect is increct omitted 35

36 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, a a 6 Crect response Running machine b b 20 c c 3 d 2m Draws a straight line on the graph joining the points (0935, 0) and (0959, 4)! Line not ruled Accept provided the pupil s intention is clear Shows implies the distance travelled is 4km 10 t 24 e 4 60 Their end point is on the line y = 4 The only err is to start at an increct time Shows a crect method f calculating the distance travelled, with not me than one computational err, then follows through crectly to draw their line 10 d 60 t 24 e 2.7 (err), then their line drawn from (0935, 0) to (0959, 2.7)! Line continued beyond (0959, 4) Accept a hizontal line, but f 2m do not accept the crect line continued! Their line is slightly inaccurate If their line starts at (0935, 0) and passes through (0941, 1) but continues to an increct value at 0959, then stops, continues hizontally, mark as 1, 0 36

37 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Squares a 2m Indicates only the values 0 and 1 Indicates one of the values 0 1, with no increct values Indicates both crect values with not me than one increct value! Use of infinity Igne, f 2m accept 1, 0, infinity! Answer(s) embedded in wking Accept provided there is no ambiguity and any statements made are crect, f 2m accept 2 1 e 1, 0 2 e 0 1, 1 2, 0, , 0 2 b 2m Indicates values between 0 and 1 not including the values 0 and 1 Numbers greater than nought but less than one 0 < x < 1 Minimally acceptable indication Between zero and one Numbers that bin 0.something Fractions that are positive and not improper Indicates values between 0 and 1 including either 0 1 both Indicates the crect upper limit, but without including 1 Numbers less than 1 All fractions that are not improper Gives at least one crect example of a number that is a member of this set and its square, with no increct examples = < and 0.01! Response ambiguous about the inclusion of 0 1 Numbers from zero to one Mark as 1, 0 F 2m, incomplete indication Fractions Decimals Increct statement Below 1 and must have 2 me decimal places 37

38 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Triangle calculations 2m Indicates No and gives a crect justification The most common crect justifications: Markers may find the following helpful: (134.56) Use Pythagas theem to show the sides are inconsistent p p e , but e (234.09) (75.69) Calculate what one side should be in der to make the triangle consistent The hypotenuse should be should be should be Use trigonometry to calculate two angles, which are then shown not to sum to 90, using cosine The angles are and p , using sine The angles are and should be 40.7! Values rounded truncated Accept values rounded truncated to 1 me decimal place(s). Otherwise, accept provided crect wking a me accurate value is seen F 2m, no indication of how values combine e e e Justification is from construction rather than calculation Shows sufficient wking to indicate crect application of Pythagas theem p m Shows sufficient wking to indicate a crect trigonometric ratio 8.7 sin e with the position of the relevant 15.3 angle indicated on the diagram No indication of which angle is being considered 38

39 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier Crect response Triangle calculations (cont) 2m Indicates No and gives a crect justification The most common crect justifications: Use trigonometry to show the sides are inconsistent, using sin 50 sin m1 (0.8) is not 50 sin sin 50 should be , = , using cos 40 cos t cos 40 12! No indication of which angle is being considered 12 sin e 15 Accept only if the trigonometric ratio is crect f the angle of 50 Calculate what one side should be in der to make the triangle consistent 15 sin 50 e not e not 15 sin50 (15 2 m 12 2 ) = 9 but 15 t cos 50 e Calculate what one angle should be in der to make the triangle consistent sin m1 (0.8) e not 50 The angle should be 53.1 The other angle is , but it should be 40 Shows implies a crect trigonometric ratio 12 sin 50 e t sin sin 50 39

40 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Algebraic expressions 1 a 2m 6 equivalent 2 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 2y m 8 e 5 5y e 3y p 13 2y e 13 2y em3 (terms in y simplified, err in simplification of numerical values) b 2m m18 Fms a crect equation 5y m 8 e 2(3y p 5) Fms the increct equation 2(5y m 8) e 3y p 5 and follows through crectly to give y e 3 10y m 16 e 3y p 5 7y e 21 y e 3! y = 3 without crect wking seen Accept provided at least the equation 2(5y m 8) e 3y p 5, equivalent, is seen. Note that trial and improvement alone, simply showing 5 t 3 m 8 e 7, 3 t 3 p 5 e 14, should not be considered as crect wking 40

41 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 15 Crect response What fraction? 2m Gives a crect expression n p 2 2n (n p 2) d 2n 1 1 p 2 n 2n m (n m 2) 2n Equivalent expressions F 2m, necessary brackets omitted n p 2 d 2n 2n m n m 2 2n Shows both the expressions n p 2 and 2n even if these are subsequently combined increctly n p 2 d 2n Gives an algebraic fraction in which the numerat is n p 2 Gives an algebraic fraction in which the denominat is 2n n p 2 seen but not in a fraction 41

42 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only Crect response Eating 17 2m 15 Crect response Equation solving Shows any two of the following three algebraic processes crectly: 1. Cross multiplication to remove the fraction 2. Multiplication division to remove brackets 3. Collecting like terms together 10y m 15 e 6y (err) 4y e 15 (Err in process 1) 5(2y m 3) e 9y 10y m 3 (err) e 9y, so y e 3 (Err in process 2) 5(2y m 3) e 9y 2y m 3 e 1.6y (err), so 0.4y e 3 (Err in process 2) 10y m 15 e 9y (Process 3 not shown) 42

43 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 18 2m (...) Crect response 3-D cut F 2m, length(s) found only through scale drawing Shows implies a crect method f the length of one side of the base (10 2 p 10 2 ) 14.14(...) 1.4(...) t sin cos 45! Length rounded Accept provided there is no evidence of an increct method 43

44 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 19 Crect response Tiles 3m Gives a complete crect justification that encompasses all four conditions below: 1. F the octagon, shows implies that the interi angle is 135, the exteri angle is F the square, shows implies that the interi exteri angle is F the hexagon, shows implies that the interi angle is 120, the exteri angle is Justifies why the hexagon will not fit! Explanation does not identify, on the diagram otherwise, whether interi exteri angles are being considered, to which shape the angles belong F 3m, accept only if there is no redundant infmation and the justification is unambiguous, accept 90 p 135 e 225, 360 m 225 e 135 but the angle in a hexagon is m (90 p 135) > p 120 p p 45 e 135 which is 15 too big 135 p 90 e 225 but it should be

45 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 19 Crect response Tiles (cont) 2m Shows at least one crect value from each of the following three sets of angles, even if it is not clear to which shape the angle belongs Shows implies the gap is p 45 e implied by a right angle symbol! Explanation confuses the terminology of interi and exteri angles F 2m, condone F 2m, increct angles marked further wking indicates confusion between interi and exteri angles Angle of 135 marked as 45 U1 Shows at least one crect value from two of the following three sets of angles, even if it is not clear to which shape the angle belongs Shows at least one crect value from each of the following three sets of angles, even if the angles are ascribed to increct shapes

46 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 20 Crect response Dissection 3m Gives a complete crect justification The most common crect justifications: Show the length of CD is 9, then use the similarity of triangles CDE and AEF to show through calculation that EF is Scale fact is, t 15 e The sides of triangle AEF are a third bigger than the cresponding sides of 1 triangle CDE, 15 t 1 = 20 3 Show the length of CD is 9, then use the similarity of triangles CDE and BDF to show through calculation that EF is Scale fact is 9 21 t 15 e 35, 35 m 15 e t 15 e 35, 35 e 20 p 15 3 x p 15 Let x e FE, then = 21 x p 15 e 35, x e Use trigonometry to calculate CDE as 53.1(...), DEC as 36.8(...), then use the similarity of triangles CDE and AEF ( CDE and BDF) to show through calculation that EF is 20 ( DF is 35) sin m1 12 e 53.1, 12 d cos 53.1 e EF taken as 20 then used to demonstrate the sides are in the crect ratio f similarity to hold, using triangles CDE and AEF e e FA e 20 2 m 12 2, so FA e 16, and e 16 12, using triangles CDE and BDF e e 15 9! Values rounded Accept values shown as rounded, but f 3m do not accept resultant increct values, f 3m accept 12 DEC e 37, e 20 sin 37, f 3m do not accept 15 EF e, 15 d 9 e 1.7, t 12 e 20.4 which rounds to 20 F 3m, justification uses only Pythagas and EF = 20 used within the argument Circular argument 20 2 m 12 2 e 16 2 so FA e p 12 2 e 400 so EF is 20 46

47 259578_KS3 Ma_MS P2 2/4/04 9:27 AM Page KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 20 Crect response Dissection (cont) 2m Shows implies a crect scale fact, even if rounded, f triangles CDE and AEF bigger 3, f triangles CDE and BDF Using a crect value f CDE DEC, even if rounded truncated, gives the cresponding angle within triangle AEF ( BDF) AEF ( BDF) is 53.1(...) EFA ( DFB) is 36.8(...) Shows implies the length of CD is 9 BD e 21 Shows CDE is 53.1(...), even if the value is rounded truncated Shows DEC is 36.8(...), even if the value is rounded truncated U1 Using their increct CD their increct CDE DEC, even if rounded truncated, shows their crect scale fact gives the cresponding angle within triangle AEF 47

48 259578_cover 2/4/04 10:06 AM Page BC1 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/