Mark scheme for Test 1

Transcription

1 Mathematics Mark scheme f Test 1 Tiers 3 5, 4 6, 5 7 and 6 8

2 Introduction Introduction The markers will follow the mark scheme in this booklet, which is provided here to infm teachers. This booklet contains the mark scheme f paper 1 at all tiers. The paper 2 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking infmation f questions is set out in the fm of tables, which start on page 12 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available f that question part. The Crect response column usually includes two types of infmation: a statement of the requirements f the award of each mark, with an indication of whether credit can be given f crect wking, and whether the marks are independent cumulative; examples of some different types of crect response, including the most common. The Additional guidance 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. 2

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

4 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 Additional guidance. 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 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 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 x x Increct ambiguous use of pounds pence 320, 320p 700p, p not in the answer space. x 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 Any unambiguous indication 2.5 (hours), 2h 30 Digital electronic time ie 2:30 Take care! Do not accept x x 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 x x Increct time 8.4am, 8.40pm x Increct placement of separats, spaces, etc increct use omission of 0 840, 8:4:0, 084, 84 6

7 General guidance Responses involving the use of algebra F example: 2 p n n p 2 2n n n 2 2 Accept Take care! Do not accept x Unambiguous use of a different case variable N used f n x used f n! Unconventional notation n t 2 2 t n n2 n p n f 2n n t n f n 2 n d 2 f n 1 n p 1n f 2 p n 2 p 0n f 2 Within a question that demands simplification, do not accept as part of a final answer involving algebra. Accept within a method when awarding partial credit, within an explanation general wking. x Embedded values given when solving equations in solving 3x p 2 = 32, 3 t 10 p 2 = 32 f x = 10 Wds used to precede follow equations expressions t = n p 2 tiles tiles = t = n p 2 f t = n p 2 Unambiguous letters used to indicate expressions t = n p 2 f n p 2 To avoid penalising the two types of err below me than once within each question, do not award the mark f the first occurrence of each type within each question. Where a question part carries me than one mark, only the final mark should be withheld.! Wds units used within equations expressions n tiles p 2 n cm p 2 Do not accept on their own. Igne if accompanying an acceptable response. x Ambiguous letters used to indicate expressions n = n p 2 f n p 2 7

8 General guidance Responses involving codinates F example: ( 5, 7 ) Accept Unconventional notation ( 05, 07 ) ( five, seven ) x y ( 5, 7) ( x e 5, y e 7 ) Do not accept x x Increct ambiguous notation ( 7, 5 ) y x ( 7, 5) ( 5x, 7y ) ( 5 x, 7 y ) ( x m 5, y m 7 ) Responses involving native numbers F example: 2 Accept Do not accept x To avoid penalising the err below me than once within each question, do not award the mark f the first occurrence of the err within each question. Where a question part carries me than one mark, only the final mark should be withheld. x Increct notation 2m 8

9 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only. F example: Accept % Take care! Do not accept x Equivalent decimals, fractions and percentages 0.700, 70, 35, 70.0% A probability crectly expressed in one acceptable fm which is then increctly converted, but is still less than 1 and greater than 0 70 e The first four caties of err below should be igned if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of err below me than once within each question, do not award the mark f the first occurrence of each type of err unaccompanied by an acceptable response. Where a question part carries me than one mark, only the final mark should be withheld.! A probability that is increctly expressed 7 in 10 7 over 10 7 out of 10 7 from 10! A probability expressed as a percentage without a percentage sign.! A fraction with other than inters in the numerat and/ denominat.! A probability expressed as a ratio 7 : 10, 7 : 3, 7 to 10 x A probability greater than 1 less than 0 9

10 General guidance Recding marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 a 0 entered in each marking space. Where 2m can be split into gained and lost, with no explicit der, then this will be recded by the marker as 1 0 The total marks awarded f a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recded on the front of the test paper. A total of 120 marks is available in each of tiers 3 5 and 4 6. A total of 121 marks is available in each of tiers 5 7 and 6 8. 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 20 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. 10

11 Tier 3 5 only BLANK PAGE 11

12 Average heights 1 Crect response Additional guidance a 133 b 7 2 Crect response Additional guidance Making 24 2m Gives three different crect pairs of numbers 2 t 12 3 t 8 4 t 6 24 t 1 12 t 2 6 t 4 Fractions, decimals native numbers! F 2m, crect pair of numbers repeated, but in reverse der Do not accept as a different crect pair Gives two different crect pairs of numbers 12

13 Tier 3 5 only Write a number 3 Crect response Additional guidance a Gives a value that is greater than 1000, but less than Fractions decimals x F part (a), number given in wds b Gives a decimal that is greater than 0, but less than Point two x F part (b), number given as a fraction 4 Crect response Additional guidance a Indicates C! Unambiguous indication Accept, f part (b) accept Cube and cuboid b Indicates A and E in either der, f part (b) do not accept Square and rectangle 3-D shapes c 7 13

14 Tier 3 5 only 5 Crect response Additional guidance Digits a Gives all four crect numbers, ie in any der b Identifies the smallest and the biggest numbers from their list (including the two given numbers), provided their list has at least four numbers F both marks, follow through Crectly adds any numbers they identify, even if they are not from their list, provided their numbers each have at least three digits and the addition requires at least one carry 357 p 753 = p 753 = p 777 = p 375 p 537 p 573 p 735 p 753 = 3330 Markers may find the following sums using the values from a crect list useful: Gives the value 1110, without identifying their smallest and biggest numbers 14

15 Tier 3 5 only Different shapes 6 Crect response Additional guidance 3m Gives all four different crect shapes in any ientations with none increct duplicated! Lines not ruled accurate, shapes not shaded internal lines omitted Accept provided the pupil s intention is clear! F 3m, crect shapes duplicated even if ientation is different Condone duplication of the given shape, ie a 1 by 4 rectangle F 3m, do not accept other duplicates x Squares not joined crectly side-to-side Do not accept as a crect shape 2m Gives at least three different crect shapes, even if there are other increct duplicated shapes U1 Gives two different crect shapes, even if there are other increct duplicated shapes 15

16 Tiers 3 5, 4 6 Food and drink 7 Crect response Additional guidance a 1.55 b 2m Indicates the crect item of food and the crect drink, ie Pizza and juice, in either der Unambiguous indication P, J Shows the digits 24(0) U1 8 1 Crect response Additional guidance Number lines m Gives both the values 9 and (p)3 in the crect positions Gives one crect value in the crect position Gives both the values (p)3 and 9 but with the positions reversed 16

17 Tiers 3 5, 4 6 Shapes 9 2 Crect response Additional guidance 2m Gives all three crect areas, ie Gives any two crect areas! F, follow through Provided their 2 nd < their 3 rd < their 1 st, accept the following: F their 2 nd, accept follow through as their 1 st d 4 F their 3 rd, accept follow through as their 1 st d 2 their 2 nd t 2, f accept 20 (err), 5, 10 1 (err), 1, , 2 (err), 4, f do not accept 16, 8 (err), 16 Computation 10 3 Crect response Additional guidance a a 10.2 equivalent b b 9.5 equivalent c c 1270 d d 57 17

18 Tiers 3 5, Crect response Additional guidance Scales a a 900 U1 200! Follow through Accept follow through as 1100 their value f the first mark, provided this gives a positive value b b Indicates 1000, ie

19 Tiers 3 5, 4 6 Range of ages 12 5 Crect response Additional guidance a a Gives two ages with a difference of 7 years 1 and 8 7 and 14 7 and 0 20 and 13! Ages given using part-years Accept provided the difference is 7 years, accept 6 months and b b 0! Response given in wds Accept provided there is no ambiguity, accept Zero Nothing, do not accept No range! Units amended Accept responses giving a sht time interval, accept A few minutes A couple of hours 19

20 Tiers 3 5, 4 6 Placing fractions 13 6 Crect response Additional guidance 2m Gives all four fractions in the crect positions, ie Unambiguous indication of fractions F 1 as a decimal, accept 0.33 better 3 F 1 as a decimal, accept 0.13 better 8, f 2m accept Gives at least two fractions in the crect positions Converts at least three of the four crect values into a fm enabling comparison, even if the positions are increct and there are other errs At least three of: 90, 15, 40, At least three of: 0.75, 0.125, 0.33, , 5, , 3, , 20,

21 Tiers 3 5, 4 6 Survey results 14 7 Crect response Additional guidance a a Draws a crect bar f Don t know that indicates 9 people! Bar not ruled, accurate shaded Accept provided the pupil s intention is clear, and the height of the bar is closer to 9 than to either 8 10! Bar increctly positioned of an increct width Condone b b Indicates 3 circles f Don t know! Circles not shaded inaccurate in size Accept provided the pupil s intention is clear! Follow through from part (a) Accept the number of circles drawn as the height of their bar f Don t know d 3 If this results in a part circle, condone any inaccuracy in their part circle U1 21

22 Tiers 3 5, 4 6, Crect response Additional guidance Percentages a a a 7! F the first mark, out of 10 repeated 7 10 Condone 50 b b b Completes the sentence crectly with two values that are in the ratio 1 : 20 1 out of 20 5 out of out of out of out of 50 Completes the sentence crectly, in a different way from one previously credited! Follow through Accept as two values in the same ratio as their two values f the first mark, provided their first value < their second value, from their first mark as 1 out of 5 accept 2 out of 10 U1 22

23 Tiers 3 5, 4 6, 5 7 Marking overlay available Crect response Additional guidance Rotating 2m Gives crect triangles f both grids with vertices within the tolerances as shown on the overlay, ie! Lines not ruled accurate Accept provided the pupil s intention is clear Gives a crect triangle f either grid with vertices within the tolerances as shown on the overlay, even if the other is increct omitted (err) Completes two rotations of 90 clockwise that do not use the given centre of rotation Fails to complete the first rotation crectly but draws a shape that is a triangle, then follows through to rotate their triangle crectly through 90 clockwise about the given centre of rotation 23

24 Tiers 3 5, 4 6, 5 7 What is my number? Crect response Additional guidance 2m 21 Shows implies that 2 t my number is 42 2 t my number e 357 m 315 e 42 2x e d 2 U1 Shows a complete crect method with not me than one computational err, even if their choice between alternative answers is increct omitted 15 t 10 e 150, 150 p 150 p 15 e 315, so it s 10 p 10 p m 170 m 170 m 17 m 17 (err) e 0, so it s 10 p 10 p 1 p 1 e 22 1 (err) (err) Crect response Additional guidance Completing 32 12! F the first and second marks, incomplete processing Penalise only the first occurrence, f the first and second marks 4 t 8 48 d 4 Mark as 0, 1 Gives a crect expression in x with a value of 48 when x is 8 6x x p 40 3x p 24! F the third mark, unconventional notation Condone, f the third mark accept 6 t x x6 24

25 Tiers 3 5, 4 6, 5 7 Mean and median Crect response Additional guidance a a a Shows that the mean is 10 9 p 11 p 10 e 30, 30 d 3 (9 p 11 p 10) d 3 10 is already 10, then 9 is 1 below and 11 is 1 above Minimally acceptable explanation 30 d 3 30 d 10 e 3 9 p 11 e 20, 20 d 2 Add one to 9 and take one off is halfway between 9 and 11 Method described You add them up then divide by how many there are x Increct statement 9 p 10 p 11 d 3 e 10 3 d 30 e 10 Gives a crect explanation of why the median is is the middle number when the numbers are in der The median is the middle number when the numbers go from smallest to largest Minimally acceptable explanation It is the middle number It s the middle largest It s the second smallest It is in between x Incomplete increct explanation is halfway between 9 and 11 b b b Gives four values that total 40 and whose middle two numbers, when dered, add to 20, with none of the values being Fractions, decimals and natives U1 25

26 Tiers 3 5, 4 6, Crect response Additional guidance Angles Shows angle a as 50 Shows angle b as 130! F the second mark, follow through Accept follow through as 180 m their a, provided their a < 90 and is not 54 to 56 inclusive Shows angle c as 20! F the third mark, follow through Accept follow through as 150 m their b their a m 30, provided this gives a positive value Crect response Additional guidance Equations 5! Increct notation, f the first mark t5 Penalise only the first occurrence 3! Incomplete processing, f the first mark 15 3 Penalise only the first occurrence 26

27 Tiers 4 6, 5 7, 6 8 Long multiplication Crect response Additional guidance 2m 8602 Shows a complete crect method with not me than one computational err 3740 p 3740 p 374 t 3 = 7480 p so 6000 p 1400 p 80 p 900 p 210 p (err) x Conceptual err Crect response Additional guidance a a a (60, 60) Midpoint b b b Gives M as (0, 100) Gives N as (60, 0)! Answers f M and N transposed but otherwise completely crect If this is the only err, ie gives M as (60, 0) and gives N as (0, 100), mark as 0, 1 U1! x- and y-codinates transposed but otherwise crect f both M and N If this is the only err, ie gives M as (100, 0) and gives N as (0, 60), mark as 0, 1 27

28 Tiers 3 5, 4 6, 5 7, Crect response Additional guidance Square cut 2m 42, with no evidence of an increct method x Increct method 12 p 2 e 14, 14 t 3 e 42 Shows implies that the square is a 9(cm) by 9(cm) square 7 t 6 seen 6 7 Area of square = 81 Shows implies a crect method in which the only err is to use an increct value f the shter hizontal side of rectangle A 12 d 2 = 8 (err), 8 p 3 = m 2 = 9, 8 t 9 = 72 4 (err) U1 5 Answer: 20 28

29 Tiers 4 6, 5 7, Crect response Additional guidance Making zero a a a Indicates only the second statement, ie b b b Indicates that the other number is zero 0 Zero Minimally acceptable indication 0 p 0 Same! Use of native sign 0 Condone U1 Gives a crect pair of non-zero values that add to make zero 1 and 1 45 and p45 x and x x Operation changed 1 m 1 [ and crossed out] 29

30 Tiers 4 6, 5 7, Crect response Additional guidance Cuboid 2m Draws a 1 by 3 by 4 cuboid in any ientation, using the isometric grid Some all internal lines omitted! Lines not ruled Accept provided the pupil s intention is clear Draws a crect view, using the isometric grid and maintaining three dimensions, but either omits one me external lines shows some hidden lines! Drawing not accurate F 2m, accept vertices within 2mm of the dots of the grid F, accept a less accurate drawing provided the pupil s intention is clear! Cuboid enlarged F 2m, accept provided a consistent scale fact has been used f all lengths, and any internal lines divide the cuboid into only 12 smaller cubes Draws a view of a cuboid, using the isometric grid and with all external lines and no hidden lines shown, but with only one dimension increct, by not me than one unit! F 2m, hidden lines shown Do not accept unless the lines are clearly identified as hidden lines, f 2m, accept, f 2m, do not accept x F 2m, external lines omitted Shows a 1 by 3 by 4 cuboid in any ientation, but does not use the isometric grid crectly 30

31 Tiers 4 6, 5 7, 6 8 Dividing fractions Crect response Additional guidance 3m 2m Gives the first value as 2 and the second value as 6 Gives an increct omitted first value but crectly gives the second value as 6 Gives an increct first value but follows through crectly f the second value as their first value t 3, provided their first value is a positive inter first value: 4 second value: 12 Gives the crect first value and shows implies a crect method f the second value with not me than one computational err 2 t 3 3 d t d Answer of 6 equivalent, with no 8 evidence of an increct method F the second value 6 1! Eighths repeated Accept as the final answer f the first value, f the value 2 accept 2 8 Do not accept as the final answer f the second value, f the value 6 do not accept 6 8 x F 2m, conceptual err d e Gives an increct omitted first value but shows implies a crect method f the second value with not me than one computational err! F, follow through F, accept follow through as the intention to multiply their first value by 3 shown implied, accept first value: 4 8 second value: 1 1 equivalent 2 first value: 4 8 then 4 t 3 seen 8 31

32 Tiers 4 6, 5 7, 6 8 Refer to the new algebra general guidance Solving an equation Crect response Additional guidance 2m 25 equivalent 4 x F 2m, 25 seen but with increct further 4 wking 25 = Shows implies a crect first step of algebraic manipulation that either reduces the number of terms collects variables on one side of the equation and numbers on the other 2t e 25 m 2t 25 p 2t e m2t 2t p 2t e 100 m p 4t e 100 4t e d 4 seen! Method used is trial and improvement Note that no partial credit can be given Crect response Additional guidance Angle p 2m 140 Shows the value Shows implies a complete crect method with not me than one computational err 360 m 2 t (180 m 35 t 2) 360 m (360 m 4 t 35) 70 t p p 35 e 80 (err), 180 m 80 e m 100 t 2 e

33 Tiers 4 6, 5 7, Crect response Additional guidance Speed bumps a a 2m Completes both sentences crectly, with all four values in the crect positions, ie 12 46! Throughout the question, key not interpreted, f the value Penalise only the first occurrence 3 35 Gives at least two values in the crect positions Shows the values 46, 12, 35 and 3, even if their positions are increct b b Gives a crect justification 38 m 28 e 10 It falls from 38 to 28 Minimally acceptable justification 38 and 28 identified, with no evidence of an increct method! Ambiguous notation 28 m 38 Condone x Incomplete increct justification The difference between the middle numbers befe and after is 10 Indicates both values of 8 cresponding to the units of 38 and 28 on the diagram, but with no interpretation of the key Befe the median was 39, after the median was 29, so it fell by 10 33

34 Tiers 4 6, 5 7, 6 8 Refer to the new algebra general guidance Straight line graph Crect response Additional guidance a a a Indicates that the y-codinate is 146 Indication is within a pair of crect codinates, f part (a) (50, 146), f part (b) (18, 50) b b b Indicates that the x-codinate is 18! Answers to parts (a) and (b) transposed but otherwise crect Mark as 0, 1 c c Indicates Yes and gives a crect explanation with no evidence of increct wking When x e m10, y e 3 t m10 m 4 e m30 m 4 e m34 3x m 4 e m34 3x e m30 x e m10 Minimally acceptable explanation m30 m 4 e m34 m30 d 3 e m10 When x e m10, 3x m 4 e m34 The second number is equal to the first number multiplied by 3, minus 4 x Incomplete increct explanation When x e m10, y e m34 3x m 4 e m34 3x e m34 m 4 3x e m30 x e m10 34

35 Tiers 5 7, Crect response Additional guidance 64 3m Gives four different crect pairs of values f x and y x e 64 y e 1 x e 8 y e 2 x e 4 y e 3 x e 2 y e 6 x e 1 64 y e m1 x e 4096 y e 1 2 x e 8 y e 4 x e 8 y e 2 2m Gives three different crect pairs of values f x and y, even if there are errs, omissions repeats Gives two different crect pairs of values f x and y, even if there are errs, omissions repeats 35

36 Tiers 5 7, Crect response Additional guidance Sixths Gives a crect justification The most common crect justifications:! Response contains an increct statement Igne alongside a crect response, accept 1 is 33 and 100 d 6 = 16 3 State imply that 2 e 1 and use the known 6 3 fact that 1 rounds to 33% e which is 33 to the nearest per cent 6 3 Minimally acceptable justification 1 is 33 3 x Incomplete justification It s 33% not 34% Show imply that the percentage should be 33 by showing a me accurate percentage, a crect method It s 33 1 % so it rounds to 33 not d 6 t 2 e 33.33, so 33 Double 16.7 is t 3 e 102, but 33 t 3 e 99 which is closer to 100 Minimally acceptable justification d 6 t 2 gives t 3 e 102 but 33 t 3 e 99! Me accurate percentage rounded truncated F 1, accept 33.3% better 3 F 1, accept 16.7% 16.66% better 6 x Incomplete justification 100 d 6 t 2 34 t 3 e 102 ( 33 t 3 e 99) Refer to the effect of the premature rounding, f example by giving a possible value f 1 in 6 the range 16.5 to inclusive, to 17.5 inclusive, and shows implies the percentage f 2 could be If 1 were 16.6%, it would be 17% to the 6 nearest per cent, but double 16.6 is could be 17.4, but 17.4 t 2 e Minimally acceptable justification 17% is rounded not exact, so when you double it, you double the err 17 is not exact, so it could be x Incomplete justification 1 2 rounds to 17, so could round to Keep adding 17 and you don t get to t 6 e 102 x Increct justification that implies hypothetical values are the crect values 1 2 = 16.5% so = 33%

37 Tiers 5 7, Crect response Additional guidance Tyres a a 5 b b Gives a value between 3500 and 5500 inclusive! Increct units inserted 5000 miles Igne 37

38 Tiers 5 7, 6 8 Refer to the new algebra general guidance Which triangles? Crect response Additional guidance a a Indicates the crect triangle, ie and gives a crect equation linking a, b and c f the other triangle a 2 p c 2 e b 2 b 2 m a 2 e c 2 b 2 m c 2 e a 2 b e a 2 p c 2 b b Indicates the crect triangle, ie and gives a crect explanation f the other triangle The most common crect explanations: State imply that the third angle in the triangle on the right is not m 75 m 25 e 80 not 90, so you can t use Pythagas Theem Angle C is not 90, so it s not a right-angled triangle Minimally acceptable explanation Not 90 Not a right angle It only wks when it s right-angled x Incomplete explanation that does not refer explicitly to m 75 m 25 e 80 You can t use Pythagas Theem The angles are wrong Show that if the third angle in the triangle on the right were 90, the triangle would not be possible If the missing angle is 90, the angles add up to 190 not 180 Minimally acceptable explanation If you put 90 in you don t get 180 A right-angled triangle is impossible with those angles, they should make 180 The angles add up to 190 [right angle marked on right hand triangle] They add up to 100 not 90 x Incomplete explanation that does not refer explicitly to 90 The angles would add up to 190, not

39 Tier 5 7, Crect response Additional guidance Sweet peas a a equivalent probability equivalent probability! Unconventional notation, but equivalent value, f the first mark Condone! Estimates transposed but otherwise crect Mark as 0, 1 b b Indicates Ravi and gives a crect explanation that states implies that he used me seeds The me trials you have the me accurate your estimate of probability is likely to be The number of seeds in each packet was the same but Ravi had me packets than M so he had a greater number of trials There were me seeds to consider 200 seeds is me than 100 seeds Minimally acceptable explanation Me seeds Me packets He tested me He had 200, not 100 Ravi had 10, M had 5! Irrelevant statement Ravi s results were me accurate He had me chance of a bigger number germinating Igne alongside a crect response, otherwise do not accept U1 x Incomplete, ambiguous increct explanation Me A bigger number Ravi s = 170 which is me than Me of his seeds germinated He had 5 me seeds M s numbers were me complicated and harder to wk out 39

40 Tiers 5 7, 6 8 How many digits? Crect response Additional guidance 2m Gives a crect response that satisfies the following four conditions: 1. Indicates the minimum is 4 2. Shows a crect justification f the minimum, f condition t 10 e 1000 Minimally acceptable justification f the minimum [condition 2] t 10 ( 10 t 100) 3. Indicates the maximum is 5 4. Shows a crect justification f the maximum, f condition t 99 e t 100 e , a 5-digit number and subtracting 999 does not change it from being a 5-digit number is just over the biggest possible so this must have the same number of digits 100 t 1000 e , but this is the smallest possible 6-digit number, so 99 t 999 must have 5 digits Minimally acceptable justification f the maximum [condition 4] m m ( ) is just over t 100 ( 100 t 1000) x Incomplete increct justification f the maximum [condition 4] 999 t m m 999 Gives a response that satisfies at least condition 4, even if condition 3 is not satisfied U2 Gives a response that satisfies condition 1, satisfies condition 4 with not me than one computational err, then follows through crectly to give their maximum x Conceptual err 999 t t 999 = m 99 =

41 Tiers 5 7, 6 8 Refer to the new algebra general guidance Simultaneous Crect response Additional guidance 3m 2m Gives both x = 3 equivalent and y = 5 2 and shows a complete crect method f solving algebraically 4x p 3y e 21 4x p 2y e 16 so y e 5 2x p 5 e 8 so x e x p 3y e 21 6x p 3y e 24 so 2x e 3 therefe x e 1.5 and y e 5 2x p y e 8 2x p 2y e 13 so y e 5 and x e 3 2 4x p 3(8 m 2x) e m 2x e 21 x e 1.5, so y e 5 Shows a complete crect method f solving algebraically with not me than one err 4x p 3y e 21 4x p 2y e 16 so y e 4 (err) 2x p 4 e 8 so x e 2 4x p 3(8 m 2x) e 21 4x p 24 m 2x (err) e 21 2x e m3 x e m1.5 and y e 11 ( 9) Fms two crect equations that would allow elimination of either x y 4x p 3y e 21 4x p 2y e 16 4x p 3y e 21 6x p 3y e 24 Attempts to solve by substitution and fms a crect equation in either x y 4x p 3(8 m 2x) e m 4x 8 m 2x e 3 8 m y e 10.5 m 1.5y x Method used is trial and improvement! Only err is to use the wrong operation, spuriously eliminating either x y 4x p 3y e 21 4x p 2y e 16 5y e 37, so y e 7.4 2x p 7.4 e 8 so x e 0.3 Mark as 1, 1, 0! F, equations subtracted without the second equation restated Accept, f accept 2x p 2y e 13 seen 41

42 Tiers 5 7, 6 8 Marking overlay available Angle bisect Crect response Additional guidance 2m Completes a crect angle bisect that fulfils all four of the following conditions: 1. Ruled 2. Within the tolerance as shown on the overlay, even if their line were to be extended 3. At least 3cm in length from A through the acute angle BAC 4. Evidence of crect construction arcs that are centred on two points on lines AB and AC equidistant from A, are of equal radii and have one point of intersection! Use of construction arcs on the overlay Note that these are to give a visual guide as to whether a crect pair of centres has been used, and do not indicate tolerance! Section of angle bisect extending from A through reflex angle BAC Accept if needed as part of the 3cm required, provided the section is within the tolerance as shown by the dashed lines on the overlay. Otherwise, igne! Extra arcs drawn Igne x Spurious construction arcs F 2m, do not accept arcs drawn without compasses, arcs centred on points on the lines that are not equidistant from A Gives a response that fulfils condition 4, even if the angle bisect is incomplete, increct omitted 42

43 Tier 6 8 only 18 Crect response Additional guidance a 8! Units given Condone responses of 8cm only Star shapes b Gives a different pair of dimensions in the ratio 5 : 2 2 : 5 2 and 5 (either der) 10 and 25 (either der) 1 and 2.5 (either der) 12 and 30 (either der) x Dimensions of either given diagram Do not accept value 6 and 15 (either der) 8 and 20 (either der) 19 Crect response Additional guidance Straight lines a Gives A as (0, m8) Gives B as (2, 0)! Answers f A and B transposed but otherwise completely crect If this is the only err, ie gives A as (2, 0) and gives B as (0, m8), mark as 0, 1 b Gives a crect equation f the straight line y e 2x y m 2x e 0 x e y 2! Unconventional notation y e 2 t x y e 2x p 0 Condone 43

44 Tier 6 8 only 20 Crect response Additional guidance Acns a Gives a crect explanation The most common crect explanations: Show imply that the median f group A is 26, and f group B is 29 Median A m median B e 29 m 26 e 3 26 p 3 e 29 and A is 26, B is 29 Indicate, in wds on the diagram, the locations of the medians f A and B The vertical lines on the shaded part of the box plots represent the medians and they are 3mm apart on the graph! Median line referred to as the middle centre Condone, accept The lines in the middle are at 26 and 29 The centre points of the boxes are 3mm apart Minimally acceptable explanation 26, 29 A is 29 m 3 B is 26 p 3 x Incomplete explanation 29 m 3 26 p 3 Minimally acceptable explanation The lines in the shaded bit are 3 apart The lines in the boxes are the medians Arrows indicating both medians on the diagram x Incomplete explanation The vertical lines are 3mm apart on the graph The lines f the medians are 3mm apart on the graph! Throughout the question, increct units Condone, f part (a) accept The lines in the boxes are 3cm apart! Throughout the question, ambiguous notation, f part (a) 26 m 29, f part (b) 24 m 29 > 27 m 31 Condone 44

45 Tier 6 8 only 20 Crect response Additional guidance Acns (cont) b Indicates A and gives a crect explanation The most common crect explanations: Show imply that the inter-quartile range f A is 5 and f B is 4 F A the IQ range is 29 m 24 e 5, f B the IQ range is 31 m 27 e 4 The distance between 24 and 29 is greater than that between 27 and 31 The IQR is m bigger f group A Indicates, in wds on the diagram, the sizes of the inter-quartile ranges f A and B The shaded box in A is longer than in B, so A has a bigger inter-quartile range The box f group A covers 6 whole numbers, but f B only 5! Inter-quartile range referred to as range Condone, accept Range f A e 5, range f B e 4 The boxes show the range and A s is longer Minimally acceptable explanation 5, 4 29 m 24 > 31 m 27 1 me x Incomplete increct explanation 5 is the larger inter-quartile range 31 m 27 is less The inter-quartile range f A is 4cm and f B is 3.2cm [scale igned] Minimally acceptable explanation The box is bigger Distances between lower and upper quartiles f both A and B indicated It covers 6 numbers, the other covers 5 c U1 Gives a crect reason The most common crect reasons: Refer to possible differences in the conditions of the two samples The two groups could have collected the samples at different times of year Group A could have picked from one side of the tree and group B from the other side One group could have picked from the tree, the other from the ground Group B may have collected first and taken most of the larger ones Refer to possible differences in the sizes of the two samples One group could have collected a much larger number of acns than the other One sample may be less representative as they didn t collect enough Minimally acceptable reason Different times Different areas of the tree B s acns may have had me sunlight x Incomplete increct reason Different areas They used different trees Minimally acceptable reason Different numbers of acns You don t know how many acns x Incomplete reason You don t know how many One group could have spent longer There could have been me people to collect acns in one of the groups 45

46 Tier 6 8 only Standard fm 21 Crect response Additional guidance a Gives a crect justification (4 t 10 8 ) t (8 t 10 4 ) e (4 t 8) t (10 8 t 10 4 ) = 32 t = 3.2 t t 8 e 32, 8 p 4 e 12, so you get 32 t e 3.2 t t e = 3.2 t Minimally acceptable justification 32 t t 8 t t = [12 zeros shown] x Incomplete justification = 3.2 t t = 3.2 t (4 t 8) t (10 8 t 10 4 ) = 3.2 t b 2m 5 t 10 3 Shows a value equivalent to 5 t t Shows implies a crect method that demonstrates understanding of how to process the indices and places the multiplication symbol crectly, with not me than one err (8 4) 4 d 8 t 10 4 t 10 8 d 8 t 10 4 e 2 (err) t 10 4! Zero(s) given after the decimal point within standard fm notation Condone, f 2m accept t

47 Tier 6 8 only 22 Crect response Additional guidance Data sets 2m Gives both crect values, ie median = 90 mean = 97! Incomplete processing Condone, f 2m accept median e 90 mean e 95 p 2 U1 Gives one crect value Shows the value 9700 Marking overlay available Drawing a rhombus 23 Crect response Additional guidance 2m Draws a crect rhombus that fulfils all three of the following conditions: 1. Ruled 2. Crect intersecting construction arcs f at least one vertex, using compasses at either 8cm and 10cm 8cm and 8cm, within the tolerances as shown on the overlay 3. Vertices within the tolerances as shown on the overlay Gives a response that fulfils either condition 2 condition 3! Different ientations Markers should rotate and/ turn over the overlay as appropriate in der to check tolerances f construction arcs and/ vertices! Arcs extended extra arcs Igne inaccuracies in sections of arcs extending beyond the tolerances as shown on the overlay, arcs not indicated on the overlay, even if increct! Spurious arcs Do not accept as crect arcs drawn without compasses 47

48 Tier 6 8 only Refer to the new algebra general guidance 24 Crect response Additional guidance a and b 3m 2m Gives a crect justification b p b + 2 = b(b p 2) 2b p 2 e b 2 p 2b 2 e b 2 b e 2 which is not an inter, so a cannot be an inter either 2a m 2 e a 2 m 2a a 2 m 4a p 2 e 0 which doesn t factise, so a is not an inter Shows crect expressions f the sum and product of a and b using only one of the two variables b p b p 2, b(b p 2) 2a m 2, a 2 m 2a Minimally acceptable justification 2b p 2 e b 2 p 2b 2 e b 2! Variables a and b transposed but justification otherwise completely crect a p a p 2 e a(a p 2) 2a p 2 = a 2 p 2a 2 e a 2 a e 2 Mark as 1, 1, 0! Numerical examples given Igne U3 Shows implies the use of expressions f a and b involving only one of the two variables b, b p 2 a, a m 2 2b p 2 a 2 m 2a Shows a different crect equation involving both the variables a and b a p b e ab 48

49 Tier 6 8 only Refer to the new algebra general guidance 25 Crect response Additional guidance Temperature 2m Gives the value 10 and shows implies a crect method f solving algebraically 9C p 32 e 2C p C e 2C m 2 5 9C e 10C m = C 2C m 9C e 32 m C m 9C e 2 5 C e 2 5 x Method used is trial and improvement Shows implies a crect first step of algebraic manipulation using a crect equation in terms of C, that either reduces the number of terms collects unknowns on one side of the equation and numbers on the other 9C p 2 e 2C 5 0.2C p 30 e 32 2C m 9C e 32 m 30 5 C = t 5 49

50 Index Index to mark schemes Tier Question Page Average heights 12 2 Making Write a number D shapes 13 5 Digits 14 6 Different shapes 15 7 Food and drink Number lines Shapes Computation Scales Range of ages Placing fractions Survey results Percentages Rotating What is my number? Completing Mean and median Angles Equations Long multiplication Midpoint Square cut Making zero Cuboid Dividing fractions Solving an equation Angle p Speed bumps Straight line graph Sixths Tyres Which triangles? 38 50

51 Index Tier Question Page Sweet peas How many digits? Simultaneous Angle bisect Star shapes Straight lines Acns Standard fm Data sets Drawing a rhombus a and b Temperature 49 51