GCSE Mathematics. Paper 1 Foundation Tier. Mark scheme November Version: 1.0 Final

Transcription

1 GCSE Mathematics Paper 1 Foundation Tier Mark scheme 8300 November 2017 Version: 1.0 Final

2 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers. This mark scheme includes any amendments made at the standardisation events which all associates participate in and is the scheme which was used by them in this examination. The standardisation process ensures that the mark scheme covers the students responses to questions and that every associate understands and applies it in the same crect way. As preparation f standardisation each associate analyses a number of students scripts. Alternative answers not already covered by the mark scheme are discussed and legislated f. If, after the standardisation process, associates encounter unusual answers which have not been raised they are required to refer these to the Lead Assessment Writer. It must be stressed that a mark scheme is a wking document, in many cases further developed and expanded on the basis of students reactions to a particular paper. Assumptions about future mark schemes on the basis of one year s document should be avoided; whilst the guiding principles of assessment remain constant, details will change, depending on the content of a particular examination paper. Further copies of this mark scheme are available from aqa.g.uk Copyright 2017 AQA and its licenss. All rights reserved. AQA retains the copyright on all its publications. However, registered schools/colleges f AQA are permitted to copy material from this booklet f their own internal use, with the following imptant exception: AQA cannot give permission to schools/colleges to photocopy any material that is acknowledged to a third party even f internal use within the centre.

3 Glossary f Mark Schemes GCSE examinations are marked in such a way as to award positive achievement wherever possible. Thus, f GCSE Mathematics papers, marks are awarded under various categies. If a student uses a method which is not explicitly covered by the mark scheme the same principles of marking should be applied. Credit should be given to any valid methods. Examiners should seek advice from their seni examiner if in any doubt. M A B ft SC M dep B dep Method marks are awarded f a crect method which could lead to a crect answer. Accuracy marks are awarded when following on from a crect method. It is not necessary to always see the method. This can be implied. Marks awarded independent of method. Follow through marks. Marks awarded f crect wking following a mistake in an earlier step. Special case. Marks awarded f a common misinterpretation which has some mathematical wth. A method mark dependent on a previous method mark being awarded. A mark that can only be awarded if a previous independent mark has been awarded. Or equivalent. Accept answers that are equivalent. eg accept 0.5 as well as 2 1 [a, b] Accept values between a and b inclusive. [a, b) Accept values a value < b 3.14 Accept answers which begin 3.14 eg 3.14, 3.142, Use of brackets It is not necessary to see the bracketed wk to award the marks. 3

4 Examiners should consistently apply the following principles Diagrams Diagrams that have wking on them should be treated like nmal responses. If a diagram has been written on but the crect response is within the answer space, the wk within the answer space should be marked. Wking on diagrams that contradicts wk within the answer space is not to be considered as choice but as wking, and is not, therefe, penalised. Responses which appear to come from increct methods Whenever there is doubt as to whether a student has used an increct method to obtain an answer, as a general principle, the benefit of doubt must be given to the student. In cases where there is no doubt that the answer has come from increct wking then the student should be penalised. Questions which ask students to show wking Instructions on marking will be given but usually marks are not awarded to students who show no wking. Questions which do not ask students to show wking As a general principle, a crect response is awarded full marks. Misread miscopy Students often copy values from a question increctly. If the examiner thinks that the student has made a genuine misread, then only the accuracy marks (A B marks), up to a maximum of 2 marks are penalised. The method marks can still be awarded. Further wk Once the crect answer has been seen, further wking may be igned unless it gs on to contradict the crect answer. Choice When a choice of answers and/ methods is given, mark each attempt. If both methods are valid then M marks can be awarded but any increct answer method would result in marks being lost. Wk not replaced Erased crossed out wk that is still legible should be marked. Wk replaced Erased crossed out wk that has been replaced is not awarded marks. Premature approximation Rounding off too early can lead to inaccuracy in the final answer. This should be penalised by 1 mark unless instructed otherwise. Continental notation Accept a comma used instead of a decimal point (f example, in measurements currency), provided that it is clear to the examiner that the student intended it to be a decimal point. 4

5 Rhombus a 9 5b 5

6 2 5c Division set up, with 8 and a remainder 3 seen in crect position answer < 840 but not a 834 A1 Build up method chunking method must lead to 830 answer < 840 to sce better 6

7 35 18 (+) A1 ft fractions with a crect common denominat and at least one crect numerat improper fraction mixed number ft f crect conversion of an improper fraction to a mixed number 6b F ft the mixed number must not be an integer Beware = 53 When attempts are made to cancel the fraction, full marks cannot be sced (attempt to cancel occurs befe conversion to mixed number) (attempt to cancel occurs after completely crect answer seen) M0 A1 A1 4 7a 7

8 at least four crect and intention to add their 28 4 dep 7b 7 A1 Totals other than 28 must be evidenced f M = 29, 29 4, answer = 7 A0 17 (days) may be implied their their eg build up must be fully crect method repeated addition can imply their number of days 1.36 A1ft ft their 17 accept 136p if sign deleted 8 16 (days) and (days) and 1.44 Answer only 1.28 Answer only 1.44 A1ft A1ft M0A0 M0A0 Beware digits arising from increct wk eg = 14.4(0) Condone 1.36p M0A0 A1 8

9 % fraction, decimal percentage Do not accept ratios Igne use of wds eg 3 out of 25 = a eg 3 in 25 (only) 12 Igne attempts to simplify 25 3 eg 25 3 = 8 1 (attempt to simplify) 3 = 0.03 (attempt to convert to a decimal) 25 3 = 3 : 25 (choice) 25 9

10 f 1 pair crect and 0 increct 9b E1, E3 and E3, E4 and C2, D2 B2 2 pairs crect and 0 increct 2 pairs crect and 1 increct 3 pairs crect and 1 increct E1, E3, (E3), E4, C2 and D2 listed, but not clearly in pairs and with no additional squares other than E2 listed Accept 1E f E1 etc Igne listing of E2 if included Igne any annotations on diagram If pairings seen in wking, allow list without pairings on answer line 10

11 Question Answer Mark Com8ments Fraction Percentage f each crect answer 30(%) B (%) Do not accept fractions with non-integer numerat denominat eg (unless it is an attempt to cancel after crect answer seen) 43 Igne attempts to cancel once crect fraction seen a eg multiples of 8 listed and 5 th one chosen with maximum one err 40 A1 SC b A0 40 out of 72 A1 11

12 8 12a 2 12b ( ) A1 eg 30% = 0.3 (recovered) = 1.5 (not recovered) = 0.3 (increct method) Embedded, crect answer, eg = unless 0.3 already seen 1 A1 M0A0 M0A0 A0 A0 12

13 points with integer values are x Identifies plots any two crect points y may be in a list 14 Crect straight ruled line from ( 3, 5) to (3, 1) A1 igne increct plots igne increct plots if crect line drawn Crect line, but not extending from ( 3, 5) to (3, 1) Two lines, one crect and one increct A0 A0 Alternative method 1 eg (0.5%) Method f finding a percentage beyond 5% 1% (6%) (10%) (2%) (15%) (3%) eg Fully crect method that would lead to the crect answer dep their 93 their 12.4 (their their 37.2) 2 their 62 + their A1 Alternative method 2 is on the next page 13

14 Alternative method may be implied From traditional method their their 620 their 26 + their cont digits 806 other than 80.6 dep at least one crect and placeholder of zero crect implied From grid method their their 20 + their their 6 at least three crect From Chinese / Napier s bones method at least three values crect from (0)/6, (0)/2, (0)/6 and 1/8 and then appropriate diagonal adding 80.6 A1 In all cases, accept repeated addition f multiplication eg accept f 2 31 Igne a % sign after

15 decimals B2 any two crect B3 1 in top centre cell 5 1 in centre cell 1 in bottom right cell 10 the product of the centre column and the diagonal from top left to bottom right are both 1 any one crect the product of the centre column the diagonal from top left to bottom right is 1 16 A response can be awarded B2 if it meets both ways of scing Eg one crect value and the product of the centre column is 1 (see example below right) Diagonal and centre column each have product Bottom right cell crect and centre column = 1 B2 Centre column has product 1 Diagonal has product

16 seen 8 their a 35 chosen A1 Subtract 11 and divide by 8 accept f wds subtract and divide but not / f divide 17b Do not accept use of algebra eg (x 11)/8 16

17 Alternative method 1 Angle DAB = 70 may be on diagram Angle ABC = 360 their Angle ABC = 80 Valid reason A1 may be on diagram eg = = 180 angles on a straight line add to 180 ( )/2 = Alternative method 2 wking backwards from x = 100 Angle ABC = Angle ABC = 80 may be on diagram Angle DAB = 360 their Angle DAB = 70 Valid reason dep A1 may be on diagram eg opposite angles are equal vertically opposite angles (are equal) = 110 and = 70 Increct angles seen anywhere around A B cannot sce the A1 17

18 eg Method f equating gallons to litres beyond 2 gallons = 9 litres eg Fully crect method that would lead to the crect answer dep 76.5 A1 their their their their 4.5 their 72 + their 4.5 their gallons = 9 litres 4 gallons = 18 litres 6 gallons = 36 litres (err with wking not shown) 8 gallons = 45 litres = 94.5 M0A0 2 gallons = 9 litres = 18 so 4 gallons = 18 litres = 36 so 6 gallons = 36 litres (method crect) so 8 gallons = 45 litres = 94.5 A0 18

19 n = an odd number and p = a prime number such that n + p is a square number eg n = 1 and p = 3 n = 9 and p = 7 20a Some of the early crect pairs are :- n p n = an odd number and p = a prime number such that np is a square number eg n = 3 and p = 3 n = 27 and p = 3 20b Some of the early crect pairs are :- n p

20 The arcs should be drawn from C from points the same distance from C The lines are different lengths, so you can t go from the ends 21a CB CD Not drawn an arc from C He put compass in wrong place. He should have started at C but he started at B and D Should be an arc on each line CB and CD Arcs in wrong place Arcs aren t equal His line isn t in the centre of B and D D has a longer line than B Arcs aren t the same radius Should be an arc from B to D Should be an arc from B to the line CD Should be an intersection on CB and CD 20

21 It should be a circle, (not a square) The cners are too far away eg accept circle constructed inside square, touching at midpoints of square to within 2mm A crect diagram takes precedence over statements, otherwise igne diagram Any distances if quoted, eg to cners, should be accurate to within 2mm Igne any reference to the top point P 21b The cners are me than 3 (km cm) away from the point Some points are me than 3 (km cm) away It isn t 3 (km cm) to the cners Each cner is [4.1, 4.5] (km cm) from P (values represent tolerance) Each cner is me than 4 away It should be a circle Each point is 4.2 km from P (not true) The cners of the square are 4 km (out of tolerance) The cners of the square are 4 km while the rest are 3 km Each cner will be me than 1 km away She s measured 3 cm from P without checking the cners It is not supposed to be this shape (but what should it be?) She has measured 4.3 km not 3 (not stated cners) She hasn t shown all the points that represent 3 km She hasn t plotted where all the 3 km points are It shouldn t be a square 21

22 One pair of equal, intersecting arcs from the vertices of one side of the rectangle Fully crect construction of line of symmetry with either two pairs of equal, intersecting arcs from the vertices of the same side of the rectangle one pair of equal, intersecting arcs from the vertices of one side of the rectangle and the diagonals drawn A1 tolerance ± 1 mm tolerance ± 1 mm line of symmetry may be solid dashed but must touch opposite sides of rectangle Crect line with no appropriately constructed arcs M0A0 21c 22

23 Alternative method 1 88 (7 + 4) = 88 their 8 7 and their 8 4 their 8 7 and 88 their value their 8 4 and 88 their value 56 and 32 their 8 (7 4) their 8 3 dep eg 8 7 = 63 and eg 8 4 = 30 and A1 Alternative method 2 22 One crectly evaluated trial f two numbers, other than 7 and 4, in the ratio 7 : 4 eg = and 32 dep eg = A1 Alternative method 3 using x : y = 7 : 4 (crect) 4x = 7y 4x = 7y and 4x + 4y = 352 and 7x + 7y = 616 fming equation from ratio and equating cfficients 11y = 352 y = 32 11x = 616 x = 56 dep equation in one variable 24 A1 Alternative method 4 is on the next page 23

24 Alternative method 4 using x : y = 4 : 7 (increct) 7x = 4y 7x = 4y and 4x + 4y = 352 and 7x + 7y = 616 fming equation from ratio and equating cfficients 11x = 352 x = 32 11y = 616 y = 56 dep equation in one variable their answer A0 Alternative method 5 using x : y = 7 : 4 (crect) x = 4 7 y y = 7 4 x x = 88 y y = 88 x making one variable the subject 22 cont 7y 11 + y = 88 y = x x = x = 88 dep equation in one variable 24 A1 Alternative method 6 using x : y = 4 : 7 (increct) y = 4 7 x x = 7 4 y x = 88 y y = 88 x 7 11 x + x = 88 x = y y = y = 88 dep making one variable the subject equation in one variable their answer A0 24, with no increct wking, implies 56 and 32 A0 x = 32 and y = 56 A0 24

25 Valid criticism referring to the line from (0, 0) to (10, 1) Valid criticism referring to the line from (15, 1) eg there shouldn t be a curve need to be specific about the line shape, it is not sufficient to simply say it is wrong eg he never gs 2 km from home Criticisms can be in either der A crect diagram takes precedence over statements, otherwise igne diagram F first : 23 The first part is curved The curve should be a straight line He has drawn a curve f constant speed The line is curved which shows his speed was not consistent/constant He s not going at a constant speed to the shop (crect referral to graph) All lines should be straight Constant speed should be a diagonal/straight line The line shouldn t curve The constant speed should be The curved line shows he decreased speed It should be a straight line from 0 to 10 It should be a straight line at the start A distance-time graph shouldn t have curves Continued on next page 25

26 It should be a straight line ( It seems to be referring to the whole graph) The curved line shows he increased and decreased speed He was walking at a range of speeds, so not consistent whole graph) (referral to The constant speed is drawn increctly (how?) The lines should be curved straight, not both The curve should be a line of best fit It should be a straight line from 0 to 15 (it should be to 10) The curve is wrong (how?) F 2nd : 23 cont The line should go down at the end He isn t walking home, he s walking further away He has walked away from home when he hasn t The line should go back to the bottom of the graph The graph should return to zero The last part should be decreasing (instead of increasing) The line f him walking home should have negative gradient The graph shows he didn t walk home The line f him walking home should have negative crelation The line f the journey home gs the wrong way The graph ds not show his journey home His house is 2 km away from the shop The line should be decreasing instead of increasing (which line?) His home is 1 km from the shop not 2 km 26

27 Alternative method 1 Three whole numbers that each are less than 80 and have units digit 4 States that each number must have units digit 4 82 A1 Alternative method 2 24 Crectly evaluated trial f three whole numbers, none of which are a multiple of 10, and that, when rounded, total 70 eg = A = 85 ( = 80) M0 Beware 82 from increct values, eg = 82 M0A0 Igne increctly evaluated trials that do not solely lead to the answer n

28 1 1 (b + 2b)h 3 bh bh 3 2 bh 3 bh 1 accept hb f bh 1 bh A (a) Crect expression with, brackets Condone units within expressions f only Condone the expression given within a fmula eg A = 1.5hb Condone crect expression stated and then equated to a value with values substituted A0 A1 A1 3b + 2s 3b = 2s 4s 26(b) 6b A1 eg b + b + b + b + b + b Condone the expression given within a fmula eg P = 6b A1 28

29 π π [113, ] 9 [3.14, 3.142] [28.26, 28.3] 9π 9 π π9 π 9 A1 accept [3.14, 3.142] f π 27 36π followed by an increct method eg 36π 2 = 18 π with answer 18 π Answer of 9π from π 3 2 9π and [28.26, 28.3] given on answer line πr 2 stated but followed by 36 9 A0 M0A0 A0 M0A accept a accept ( fraction) b If fraction given, igne attempts to cancel 29

30 (( 3 ) 2 =) 3 and (( 2 ) 2 =) 2 ( 6 ) A1 3 2 = 6 with answer eg M0A0 29 Condone 3 = 1.7, = 3 and 2 = 1.4, = 2, otherwise increctly evaluated ds not sce even if answer is 6 eg 3 = 1.5, = 3, answer 6 2 = 1, 1 2 = = 6, 6 = M0A0 M0A0 M0 M0A0 M0 30

31 Alternative method 1 x + 2x + 2x x + 10 x + 2x + 2x x x + 2x + 2x + 10 = x + 10 = 270 x + 2x + 2x = 360 5x = 360 5x = 260 dep (x =) 52 2x = 104 2x + 10 = (6..) (.6 )% 31.7% 32% A1 ft may be on diagram ft 2 their their angle f C 360 Alternative method 2 is on the next page 31

32 Alternative method x x 360 2x 10 5x x x P(C) = 1 + 2x x x x = 1 dep (x =) 52 2x = 104 2x + 10 = 114 A1 may be on diagram 30 cont (6..) (.6 )% 31.7% 32% ft ft 2 their their angle f C 360 Igne increct simplification conversion after 114 A x + 2x + 2x + 10 followed by 6x + 10 = 270 M0 Do not accept decimal within fraction f final answer if crect fraction not seen The follow through is not available if A1 awarded 32

33 (x 10)(x + 10) either der igne fw (x + 10)(x + 10) 31(a) Condone missing bracket at end only (x 10)(x + 10 (x 10(x + 10) (x 10)(x + 10) followed by attempt to solve, eg answer x = 10, x = 10 answer only x = 10, x = 10 7x 2x > 1 6 5x > > 2x 7x 5 > 5x 1 > x x > 1 1 < x A1 collecting terms SC1 increct sign eg x 1 x = 1 answer of 1 31(b) Answer x > Answer only x > 1 with -1 0, 1, 2,.. as the answer A0 SC0 A0 33