GCSE Mathematics Paper 3 Foundation Tier Mark scheme 8300 November 2017 Version: 1.0 Final
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.
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] [a, b) Accept values between a and b inclusive. Accept values a value < b 3.14 Accept answers which begin 3.14 eg 3.14, 3.142, 3.1416 Use of brackets It is not necessary to see the bracketed wk to award the marks. 3
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
1 1000 2 2 6 3 0.215 4 capacity 5
Alternative method 1 of 5 1.7(0) 2.5 0.68 170 2.5 68 their 0.68 3.25 their 68 3.25 221 dep 0.51 51 implies 2.21 A1 Alternative method 2 of 5 5 2.5 1.7(0) 1.47 2.5 170 0.0147 3.25 their 1.47 3.25 their 0.0147 221 dep 2.21 A1 Alternative method 3 of 5 3.25 2.5 1.3 their 1.3 1.7(0) 3.25 1.7(0) 2.5 dep 2.21 A1 Alternative method 4 continues on the next page 6
Alternative method 4 of 5 2.5 3.25 0.769 0.77 1.7(0) their 0.769 1.7(0) their 0.77 dep 2.21 A1 Alternative method 5 of 5 1.7(0) 10 0.17 and 3.25 0.25 13 5 cont their 0.17 their 13 dep 1.7(0) 10 their 13 2.21 A1 Condone 2.21p unless the sign has been crossed out A1 ( )0.51 51(p) is the cost of the extra 0.75 kg of carrots This implies the first on Alt 1 and achieves the second if added to 1.7(0) 170 Accept wk in grams rather than kilograms Do not allow a misread of 3.25 kg 7
BHS RHS f BHP BCS BCP RHP RCS RCP B2 four additional crect combinations with no errs repetitions five additional crect combinations with at most one err repetition 6a six seven additional crect combinations with at most two errs repetitions Do not allow repetition of BHS f B2 Ingredients may be written as full wds Accept letters wds in any der eg BPC f BCP Do not accept tree diagrams without combinations listed 2 1 8 4 ft ft their (a) with at least three additional combinations, at least one of which contains cheese and pickle igne further wking if attempting to simplify 6b 2 1 2 1 is, if not refer to (a) f possible ft 8 4 8 4 BHS, BHS, BHP, BCS, BCP, RHS, RHP, RCS and RCP in (a) with answer 9 2 ft Answer given only as decimal percentage 8
7a Right-angled triangle ABC drawn with A at ( 3, 2) and B at (1, 2) and C at ( 3, 4) (1, 4) B3 B2 f A, B and C crectly plotted with no triangle drawn A and B crectly plotted and a rightangled triangle drawn with A and B at two of the vertices C plotted on the line y = 4 and a rightangled triangle drawn with C at one of the vertices A and B crectly plotted with C plotted at (k, 4) with k 3 1 and triangle ABC drawn f A and B crectly plotted C plotted on the line y = 4 a right-angled triangle drawn Condone increct omitted labelling Alternative method 1 1 their base their height 2 7b 12 A1ft ft their triangle Alternative method 2 Evidence of counting squares seen 12 A1ft ft their triangle 9
Alternative method 1 7 in first box f any two crect and 2 in second box B2 and q in Output accept q = 7r 2 in Output Alternative method 2 8a 2 in first box 7 and 7 in second box B2 f any two crect and q in Output accept q = 7r 2 in Output Do not accept 7r 2 alone in Output Accept = q in Output Condone 7 in first box 3(x + 5) 3x + 15 Accept y = 3(x + 5) y = 3x + 15 8b Igne further wk if attempting to solve eg 3x + 15 = 0, x = 5 Do not igne further wk if attempting to simplify eg 3x + 15 = 18x (y =) x + 5 3 Do not accept (x + 5)3 3 (x + 5) (x + 5) 3 x3 + 15 10
Alternative method 1 10 20 200 and 15 12 180 and 25 6 150 9 10 20 + 15 12 + 25 6 their 200 + their 180 + their 150 530 dep 580 their 530 50 (eggs) dep 54 (10 + 15 + 25) 54 50 (boxes) 4 (me boxes) 1 (+) 2 (+) 1 11 boxes of 20 17 boxes of 12 26 boxes of 6 A1 Alternative method 2 continues on the next page 11
Alternative method 2 B4 f 11 boxes of 20 16 boxes of 12 28 boxes of 6 11 boxes of 20 9 cont 11 boxes of 20 17 boxes of 12 26 boxes of 6 B5 15 boxes of 12 30 boxes of 6 B3 f 580 eggs placed in boxes with two of these conditions satisfied at least 10 boxes of 20 eggs at least 15 boxes of 12 eggs at least 25 boxes of 6 eggs B2 f 580 eggs placed in boxes with one of the three conditions satisfied and at least one of each box f all three conditions satisfied with 54 boxes but a total number of eggs not equal to 580 Fourth mark may be awarded at any stage 10 + 15 + 25 = 50 is a total of boxes and ds not sce 1 (extra) boxes of 20 2 (extra) boxes of 12 1 (extra) boxes of 6 220, 204 and 156 (eggs) on answer line with 11, 17 and 26 (boxes) seen in wking Condone number of eggs on answer line if number of boxes seen in wking eg 220, 240 and 120 (eggs) on answer line with 11, 20 and 20 (boxes) seen in wking A1 B5 B3 12
10 Crect evaluation of the sum of three multiples of 10 where the sum is not a multiple of three and No eg 10 (+) 20 (+) 40 = 70 and No Crect evaluation of the sum of three multiples of 10 and she is only crect if the total is a multiple of 30 B2 f crect evaluation of the sum of three multiples of 10 eg 10 (+) 20 (+) 40 (=) 70 10 (+) 20 (+) 30 (=) 60 Igne increct evaluations alongside a crect evaluation The multiples do not have to be different eg 20 (+) 20 (+) 30 = 70 so she is not crect eg 10 (+) 10 (+) 10 = 30 3 10 = 30 B2 13
A in two sections B and C have equal number of sections P(B) = P(C) 0 and 12 sections labelled using only A, B, C D D in twice as many sections as A 11 2As, 3Bs, 3Cs, 4Ds 2As, 5Bs, 5Cs B and C have equal number of sections and 12 sections labelled using only A, B, C D 2As, 4Bs, 4Cs, 2Ds 2As, 2Bs, 4Cs, 4Ds 2As, 4Ds 2As, 4Bs, 4Cs only 10 sections labelled 2As, 3Bs, 4Cs, 3Ds 1A, 2Bs, 2Cs, 7Ds 1A, 2Bs, 2Cs, 3Ds only 8 sections labelled 12a 10 12b 35 12c 5 14
Alternative method 1 0.9² 0.81 4.86 A1 48 600 ft ft their 4.86 10 000 crectly evaluated their 4.86 cannot be 0.9 Alternative method 2 90 (cm) (their 90)² 8100 48 600 A1ft ft (their 90)² 6 crectly evaluated 13 In Alt 1, award the ft if their answer clearly comes from multiplying a value by 10 000, but not from 0.9 10 000 = 9000 0.9 m = 9 cm 9 9 = 81 (9 is their 90) 81 6 = 486 No conversion shown 9 9 = 81 (9 is their 90) 81 6 = 486 A1ft A1ft 0.9 0.9 = 0.81 and 0.81 0.9 = 0.729 M0 0.9 0.9 = 0.81 and 0.81 0.9 = 0.729 (0.729 10 000) = 7290 M0A0 ft 15
14 1700 0.04 68 1700 1.04 1768 4(%) 3 12(%) 1700 0.04 3 their 68 3 (their 1768 1700) 3 1700 (their 12 100) 1700 (1 + their 12 100) ( 1700) 1904 ( 1700) dep 204 A1 Answer of 1904 with without 204 seen in wking A0 1700 3 = 5100 and their 5100 0.04 Condone 1700 1.04 3 an answer of 212.26( ) 212.27 1912.26( ) 1912.27 f the first method mark M0A0 680 = 4% and 680 3 implies 4(%) 3 f the first mark only 680 is not their 68 f the second method mark [6.9, 7.1] (cm) 15a [345, 355] ft ft their [6.9, 7.1] 50 [345, 355] without sight of [6.9, 7.1] 16
f R marked [3.9, 4.1] cm from P 15b R marked [3.9, 4.1] cm due South of P B2 R marked due South of P 4 (cm) seen Alternative method 1 of 6 64 8 3 24 64 8 5 40 7 78 42 13 6 78 36 13 6 78 13 7 252 6 78 13 6 216 16 64 8 3 + 6 78 13 7 their 24 + their 252 dep 5 6 64 + 6 78 8 13 276 their 40 + their 216 256 64 + 6 78 64 + 468 532 their 532 2 266 dep dep on 3 rd method mark only 266 and 276 and Yes 266 and 256 and Yes A1 Alternative method 2 continues on the next page 17
Alternative method 2 of 6 64 8 3 24 64 8 5 40 7 78 42 13 6 78 36 13 6 78 13 7 252 16 cont 3 7 64 + 6 78 8 13 their 24 + their 252 dep 6 6 78 216 13 5 6 64 + 6 78 8 13 276 their 40 + their 216 256 64 + 6 78 64 + 468 532 their 532 their 276 dep dep on their 532 their 256 256 and 276 and Yes A1 Alternative method 3 continues on the next page 18
Alternative method 3 of 6 64 8 3 24 64 8 5 40 7 78 42 13 6 78 36 13 6 78 13 7 252 6 78 13 6 216 16 cont 3 7 64 + 6 78 8 13 their 24 + their 252 dep 5 6 64 + 6 78 8 13 276 their 40 + their 216 256 64 2 32 and (6 78) 2 468 2 234 their 32 + their 234 266 dep dep on 3 rd method mark only 266 and 276 and Yes 266 and 256 and Yes A1 Alternative method 4 continues on the next page 19
Alternative method 4 of 6 64 8 3 24 78 13 7 42 6 78 13 7 252 16 cont 64 8 3 + 6 78 13 7 their 24 + their 252 dep 276 64 + 6 78 64 + 468 532 their 276 their 532 0.51 0.52 their 532 their 276 1.9 1.93 532 and 276 and 0.51 0.52 and Yes 532 and 276 and 1.9 1.93 and Yes dep A1 dep on Alternative method 5 continues on the next page 20
Alternative method 5 of 6 64 8 3 24 64 8 5 40 7 78 42 13 6 78 36 13 6 78 13 7 252 6 78 13 6 216 16 cont 3 7 64 + 6 78 8 13 their 24 + their 252 dep 5 6 64 + 6 78 8 13 276 their 40 + their 216 256 their 276 2 552 dep their 256 2 512 64 + 6 78 64 + 468 532 532 and 552 and Yes 532 and 512 and Yes A1 Alternative method 6 continues on the next page 21
Alternative method 6 of 6 1 3 1 2 8 8 7 1 1 13 2 26 64 their 8 1 8 (under) dep 78 their 26 1 3 (over) 78 their 26 1 6 18 (over) dep 16 cont 64 their 8 1 8 (under) and 78 their 26 1 6 18 (over) dep May be subtracted 8 under (half) and 18 over (half) and Yes A1 10 over (half) and Yes 24 42 252 Condone f 24 f 42 f 252 f first method mark 64 468 468 276 and 10 over (266) and Yes implies 266 and 276 and Yes A1 In Alt 2 256 and 276 and Yes A1 In Alt 4 accept wking with unused seats leading to their 256 their 532 0.4 0.49 their 532 their 256 2.07 2.08 22
17 x 3 = 2 x 18 5 < x 9 Valid statement about proption eg there were me females than males Valid statement about average Valid statement about spread eg the average age of the females was higher eg the ages of the females were me spread out Condone increct values suppting statements Condone irrelevant statements with crect statements 19 Proption of the audience statements There were me women Are mostly female There were 66% me females than males The proption of women is high Females are a higher proption than males Less men than women The men were 17%, the women were 83% The males were 17% which is less than half The males were 17% The difference is 66% continues on the next page 23
Average age statements 19 cont The women had a higher mean Women were 5 years older Females were older than the males There were me females that were older than the males, this is why the mean age of the females is me Most males were younger than the females Me older women than men There are me younger males than females There are younger males than females Females have a high mean Average age 5.4 years difference The women s mean age range was higher Spread of ages statements The women had a higher range Me of an age gap in the females than the males Females have a higher spread Males ages are closer together than females Females have a wider age range The female age gap was high, the male age gap was low Ages were quite close together The female age gap was high Age range of males is younger than females 24
Alternative method 1 of 3 98 in the singles non-intersecting part and 34 in the doubles non-intersecting part 98 + x 34 + x 98 + x = 2(34 + x) dep 2 1 (98 + x) = 34 + x 98 + x = 68 + 2x dep 49 + 2 1 x = 34 + x 30 A1 20 Alternative method 2 of 3 98 in the singles non-intersecting part and 34 in the doubles non-intersecting part 34 2 68 98 2 49 98 34 64 98 their 68 2 (their 49 34) their 64 34 2 their 64 98 second implies third implies 30 A1 Alternative method 3 continues on the next page 25
Alternative method 3 of 3 One complete trial crectly evaluated eg 98 + 10 = 108 and 34 + 10 = 44 and 108 2 = 54 44 2 = 88 (and No) Second complete trial crectly evaluated eg 98 + 20 = 118 and 34 + 20 = 54 and 118 2 = 59 54 2 = 108 (and No) 108 2 = 54 44 2 = 88 is not required if a second trial is done 118 2 = 59 54 2 = 108 is not required if a third trial is done 20 cont Crect trial with both numbers and crectly evaluated 98 + 30 = 128 and 34 + 30 = 64 30 A1 Wking may be shown on Venn diagram 30 shown in intersection in Venn diagram unless contradicted by final answer A1 2 98 2 34 98 98 and 34 crectly positioned in Venn diagram may be replaced by wking have additional wking eg 34 in Venn diagram replaced by with 68 eg 98 in Venn diagram replaced by with 49 98 and 34 increctly positioned in Venn diagram may be recovered by wking 26
140 50 2.8 140 50 60 168 2 (hours) 48 (minutes) A1 258 (minutes) (after midday) implies A1 21a 4.18 (pm) A1ft ft their time in hours and minutes with awarded 140 50 2.8 = 2 hours 80 minutes = 3 hours 20 minutes, Answer 4.50 A0A1ft 140 50 2.8 = 2 hours 8 minutes, Answer 3.38 A0A1ft 140 50 2.8 = 2 hours 80 minutes = 3 hours 20 minutes, Answer 4.5 A0A0ft 140 50 2.8, Answer 4.10 A0A0ft 2 hours 8 minutes implies attempt at 140 50 27
Valid statement ft eg the arrival time will be later it will be later time will be me ft their time in (a) eg it will be after 4.18pm 21b It will be delayed The arrival time will be increased He will reach there late The time will go up It will go up The journey will take longer so the arrival time is later Take longer Longer Slower (restating question) You won t get there as quick Time will be longer Journey will be longer Longer is referring to a time period rather than an arrival time 28
Alternative method 1 of 2 PAB = 51 PAD = 51 APC = 180 51 APC = 129 ABP = 180 51 their 51 ABP = 180 102 ABP = 78 ADC = 180 their 51 their 51 ADC = 180 102 ADC = 78 dep PAB = 51 and PAD = 51 BAD = 102 22 BCD = 180 their 78 BCD = 360 their 129 their 51 their 78 BCD = 360 258 BCD = 102 eg BCD = (360 2 their 78) 2 4x = 180 their 78 4x = 360 their 129 their 51 their 78 4x = 360 258 4x = 102 dep 4x = (360 2 their 78) 2 102 4 25.5 A1 Alternative method 2 continues on the next page 29
Alternative method 2 of 2 22 cont ABC = 180 3x x ABC = 180 4x APC = 180 51 APC = 129 PAB = 2x APB = 2x 2x = 51 dep 51 2 dep 25.5 A1 Angles must be labelled shown on the diagram 30
Lists three from 3, 9, 27, 81, 243, 729 lists three from 1, 4, 9, 16,, 225, 256, 289 crectly evaluating a power of 3 + a square number crectly evaluating 268 a power of 3 crectly evaluating 268 a square number eg 27 + 25 = 52 3 3 + 5 2 = 52 eg 268 27 = 241 eg 268 49 = 219 23 243 + 25 3 5 + 5 2 A1 Addition sign must be seen in wking on answer line 3 5, 5 2 3 5 and 5 2 on answer line A0 268 243 = 25 A0 243, 25 243 and 25 on answer line A0 Beware of 5 3 + 5 2 24 y = x k 25 72 N 31
80 44 and 36 ft ft their 80 44 27 and 9 15 and 29 ft ft ft their 36 4 3 and ft their 36 4 ft 42 their 27 and ft 38 their 9 Total on ft must be 44 en 15 Yes 44 26a 80 omen 29 o en 27 36 omen 9 Mark diagram only, do not allow misread Values may be rounded up down to whole numbers provided the total is crect Penalise the use of relative frequencies on the first occurrence only If relative frequencies are shown the denominat must be 80 and not simplified eg 4 3 and 4 1 is continues on the next page 32
en 33 Yes 44 omen 11 80 ft o en 9 36 omen 27 26a cont en 30 Yes 44 80 omen 14 ft o en 12 36 omen 24 33
85% 0.85 26b 27.2 0.85 27.2 85 ( 100) 0.32 32(.00) dep A1 Crect money notation Allow 32.00p 32.0 A0 Alternative method 1 v u = at at = u v t = v u a u v t a A1 Alternative method 2 v u t a a v u t a a A1 27a t = (v u) a v u = at and t = v u a A1 A0 v u a u v a v u a a A0 a = v u t with without wking A0 t = v u a t = v u a M0A0 M0A0 34
(Speed) m/s ms 1 (Acceleration) m/s 2 ms 2 m/s/s B2 f one crect two mutually consistent units eg km/h and km/h 2 Accept mps f m/s and mps 2 f m/s 2 27b Allow units given in wds eg metres per second metres per second squared metres per second per second m/s 1 (speed) m/s 2 (acceleration) x 2 8x 8x + 64 x 2 16x + 64 A1 allow one err omission terms may be seen in a grid Igne fw eg if attempting to solve Do not igne fw if attempting to simplify x 2 16x (+ k) k 64 A0 28 x 2 8x + 64 x 2 16x + 64 = 15x 3 + 64 x 2 8x + 8x + 64 (one err) x 2 + 8x + 8x + 64 (one err) x 2 6x + 8x + 64 (two errs) x 2 + 64 (two errs) A0 A0 A0 A0 M0A0 M0A0 35