GCSE Mathematics. Paper F Mark scheme F November Version/Stage: 1.0 Final

Transcription

1 GCSE Mathematics Paper F Mark scheme 43651F November 2016 Version/Stage: 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 2016 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. 1 e.g. accept 0.5 as well as 2 [a, b] Accept values between a and b inclusive. [a, b) Accept values a value < b 3.14 Accept answers which begin 3.14 e.g. 3.14, 3.142, Q Marks awarded f quality of written communication Use of brackets It is not necessary to see the bracketed wk to award the marks.

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 candidate has used an increct method to obtain an answer, as a general principle, the benefit of doubt must be given to the candidate. In cases where there is no doubt that the answer has come from increct wking then the candidate should be penalised. Questions which ask candidates to show wking Instructions on marking will be given but usually marks are not awarded to candidates who show no wking. Questions which do not ask candidates to show wking As a general principle, a crect response is awarded full marks. Misread miscopy Candidates often copy values from a question increctly. If the examiner thinks that the candidate 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 candidate intended it to be a decimal point. 4

5 MARK SCHEME GCSE MATHEMATICS 43651F NOVEMBER 2016 Paper 1 Foundation Tier 1(a) 72 1(b) 36 1(c) 46 1(d) 2 5 2(a) 436 2(b) 168 2(c) 42 Allow 042 5of 26

6 Alternative method 1 Bar chart vertical line graph (could be hizontal) 3 Linear scale starting at 0 increasing in 1s 2s Vertical axis labelled as frequency ( clear reference such as f freq) Bars/ lines labelled (allow A, B, O, P) Equal width f bars/ lines Equal spacing between bars/ lines All heights crect Title (accept this as a label of hizontal axis) Alternative method 2 B3 Pictogram (vertical hizontal) B2 5 6 conditions met 3 4 conditions met Pictogram key Consistent symbols f at least 2 rows Labels f trees (allow A, B, O, P) Equal spacing of rows Equal alignment of columns Crect number of symbols eg 8, 6, 7, 2 if 1 symbol f 1 tree 4, 3, 3.5, 1 if 1 symbol f 2 trees Title (accept this as a label of side bottom axis ) B3 B2 5 6 conditions met 3 4 conditions met is on the next page 6of 26

7 Bar graph ( line graph) could be hizontal 7: Title (could be label of hizontal axis) 2: Label of frequency () 6: Heights crect (allow 8, 6, 7, 2 4, 3, 3.5, 1 if no axis scale) 4: Equal widths f bars/lines 1:Linear scale starting at 0 5: Equal spacing between bars/ lines 3: Bars/ lines labelled () allow in bars 3 cont Pictogram could be vertical 3: Labels () 4: Equal spacing of rows 5: Equal alignment of columns Points only can sce the marks f conditions 1, 2, 3, 6 and 7 7: Title (could be label of vertical axis) 6: Crect number of symbols (allow 8, 6, 7, 2 4, 3, 3.5, 1 if no key) 2: Symbols consistent with key f at least 2 rows (If no key, symbols consistent with each other) 1: Key B2 max Only check spacing between bars not befe the first bar All values not needed f axis scale eg 0 can be implied Frequency may be Number How many Title must include the wd Tree 7of 26

8 4(a) (p) 96.0 is B0 4(b) ds not divide (exactly) by 6 36 is 6 people and 42 is 7 people 37 is not in the 6 times table 37 is odd / is prime It is not ( it should be) even / multiples of 6 are even (It is) not in 6 times table / not a multiple of 6 / must be a multiple of 6 It ends in a = 36 and = 6 with remainder = 36, 7 6 = 42 4(c) 6, 12, 18, 24, 30, 36, = 6.1 (Allow r1) No matter how many times you add 6 it dsn t end in 7 Only allow 37 dsn t go into 6 if a crect reason is also given eg 37 dsn t go into 6, 37 6 = dsn t go into 6, so it is not in the 6 times table Do not allow if an increct calculation seen eg 37 6 = 6.5 so 37 is not a multiple of 6 B0 1 too many B0 6 6 = 36 B0 37 dsn t go into 6 B0 Not a whole number B0 6, 12, 18, 24, 30, 36 (no further) B0 8of 26

9 Alternative method their 96 from (a) their 78 + their 66 + their 96 their dep Must be three products 4(d) their dep their 240 their A1ft ft their 96 from (a) if used Alternative method (0) (0) their 40 their 5.4 their 40 (6 their 0.6) dep dep on M2 216 A1 Alternative methods continued on the next page 9of 26

10 Alternative method their their their 36 6 (their 40 their 4) 6 dep dep 216 A1 4(d) cont Alternative method their 96 from (a) their (0) their (0) their (0) their (0) their (0) their (0) their their their 86.4 their 78 + their 66 + their 96 their 7.8 their 6.6 their 9.6 dep dep 216 A1ft ft their 96 from (a) if used Alternative methods and continued on the next page 10 of 26

11 Alternative method their their their dep their their their (d) cont their 36 6 dep ( their 4) A1 Calculation f 10% seen as part of build-up to percentage other than 90 ds not sce the method mark f percentage Build-up f percentages must be crect show full method However allow rounding truncation eg (f Alt 1) = % = 23 Answer 212 dep dep A0 11 of 26

12 5(a) Any two from (3, A), (3, B), (3, E) (3, F) Accept codinates transposed B2 one crect Accept ( 3A, 3B), (3E, 3F) B2 No line of 4 (whites) possible 5(b) Accept row, path reference to Connect 4 to imply line Accept maximum of 3 to imply not 4 5(c) (4, E) Allow (E, 4) Any rotationally symmetrical pattern with 8 squares shaded and no line symmetry eg f any rotationally symmetrical pattern with 8 squares shaded and line symmetry eg 6(a) B2 f any rotationally symmetrical pattern with 6 10 squares shaded and no line symmetry eg If answer pattern blank, mark practice pattern 12 of 26

13 seen diagram divided into 16 squares 16 seen 6(b) 6 16 A1 fraction eg 3 8 Igne any increct cancelling (except 3 ) once crect fraction seen A0 7(a) (0) 7.6(0) 240p 2.40p (0) ( )2.40 Q1 Strand (i) If building up down must be crect show full method 2, 20p, 20p ft ft smallest number of coins f their (a) Allow coins notes used f 5 me 7(b) Units needed Crect coins in wking lines followed by answer 3 (a) 8.10 (b) ( 5) p p ft 8(a) 81 8(b) of 26

14 Alternative method 1 (Red) (Silver) (Black) 30 (their 10 + their 6) 14 dep dep on at least sced Alternative method 2 A1 (20% =) 1 5 fraction Crectly converts 1 3 and their 1 5 to 9 fractions with a common denominat eg 5 15 and (their their 3 15 ) dep 7 15 Alternative method 3 A1 ( 1 3 =) 0.33(3) 33.(3)% At least 2 sf their (3) 20% + their 33% 53.(3)% 1 their % their 53% 47% dep At least 2 sf dep on % A1 If exact value seen allow subsequent rounding truncation is on the next page 14 of 26

15 9 cont red = 10 silver = 0.2 (30 10) = 4 30 (10 + 4) = = Answer = 0.53 Answer 0.47 Igne any increct cancelling change of fm once crect answer seen Igne any probability wds once crect answer seen M0 dep A0 B0 M0dep A0 dep A0 Crect, labelled pie chart Sizes of angles do not need to be labelled 10 B3 Mark intention f angles B2 Crect, unlabelled pie chart labelled, 3-sect pie chart with one sect crect and Brown > Blue > Green Pie chart (any number of sects) with one crect sect (labelled unlabelled) 60, 120 and 180 seen 1, 1 and 1 seen Crect angles seen in table Labels must be wds eg Br, Bl, Gr not just numbers angles Allow sects to be split if the split sects remain adjacent 15 of 26

16 21 11(a) Embedded answer only of 21 3 = = 7 B (b) Embedded answer only of = 12 B0 ±2w ±18 5w 3w = Terms in w constant terms collected 2w = 18 2w = A1 11(c) 9 A1ft ft on 2w = a where a 3 15 bw = 18 where b 5 3 2w = w = A0 A1ft A0 A1ft 3w = w = 18 6 Embedded answer of 9 If only decimal answer given must be accurate to at least 2 dp M0 A0 A0ft A1 A0 16 of 26

17 2, 3, 4, 6, 7, 8 B3 B2 Six numbers, median 5, total 30 Allow one of decimals, numbers < 2 repeated numbers eg 2, 2, 2, 8, 8, 8 2, 2.5, 4, 6, 7.5, 8 1, 2, 4, 6, 7, 10 Any six numbers with median 5 Allow cards to be in any der Combinations f B2 that have repeats 12 2, 2, 2, 8, 8, 8 2, 2, 3, 7, 7, 9 2, 2, 3, 7, 8, 8 2, 2, 4, 6, 6, 10 2, 2, 4, 6, 7, 9 2, 2, 4, 6, 8, 8 2, 2, 5, 5, 5, 11 2, 2, 5, 5, 6, 10 2, 2, 5, 5, 7, 9 2, 3, 3, 7, 7, 8 2, 3, 4, 6, 6, 9 2, 3, 5, 5, 5, 10 2, 3, 5, 5, 6, 9 2, 3, 5, 5, 7, 8 2, 4, 4, 6, 6, 8 2, 4, 4, 6, 7, 7 2, 4, 5, 5, 5, 9 2, 4, 5, 5, 6, 8 2, 4, 5, 5, 7, 7 2, 5, 5, 5, 5, 8 2, 5, 5, 5, 6, 7 B2 2, 2, 5, 5, 8, 8 If answer line blank, mark wking and apply usual rules f choice 17 of 26

18 1 ( ) 0.65 eg 65% 0.35 A1 eg 35% = Answer follows through A = = 0.25 Method even though answer wrong A = No method seen and answer ds not follow through M0 A A0 Answer only of 0.65 A No addition seen M0 Embedded answer = 1 A = 1 Answer 0.8 A0 18 of 26

19 Side of square = = 25 May be on diagram A1 May be on diagram = 400 is A1 14 Yes and 5 and their 16 Q1ft Strand (iii) Conclusion must be based on length not volume ft their 16 if awarded and crect conclusion Igne any volume calculations Square = 5 cm = 400 No A0 Q1ft 15(a) Café [336, 340] B2 f [334, 342] but not [336, 340] which sces B2 f [156, 160] 15(b) 340 B Igne extra compass directions eg 338 NW B2 19 of 26

20 Point within tolerance (on bold line) B2 Point [4, 5] squares on the line Nth of the Toilets Point between rays 15(c) Mark intention (point should be [0.2, 0.6] of a square down from top grid line) Crect bearing drawn that stops at bold line Crect bearing drawn that stops inside park B2 20 of 26

21 Alternative method 1 BCD = 105 DCE = 180 their CDE = 180 (their ) 75 DCE = 75 and CDE = 75 and two angles equal dep Q1 Calculation must be shown crect angle marked on diagram Calculation must be shown crect angle marked on diagram Strand (ii) Must sce M2 and have no increct angles calculations seen 16 C = 105 C = = 65 D = 180 ( ) = 85 BCD = 75 DCE = = 105 CDE = 180 ( ) = 45 BCD = 105 DCE = 65 CDE = 85 (no method shown) Alternative methods continued on the next page dep Q0 B0 dep Q0 M0 M0dep Q0 21 of 26

22 Alternative method 2 ABC = ADC = DCE = their 75 CDE = 180 (their ) 75 dep dep Calculation must be shown crect angle marked on diagram their 75 must be the same as their ABC their ADC Calculation must be shown crect angle marked on diagram 16 cont DCE = 75 and CDE = 75 and two angles equal B = = 75 C = 105 D = 180 ( ) = 45 ABC ( ADC) = = 65 DCE = 65 CDE = 85 (no method shown) Q1 Strand (ii) Must sce M3 and have no increct angles calculations seen M0dep M0dep Q0 dep M0dep Q0 ABC ( ADC) = = 75 DCE = 75 CDE = 180 ( ) = 65 Alternative methods continued on the next page dep dep Q0 22 of 26

23 Alternative method 3 BCD = 105 CDE = their DCE = 180 (their ) 75 DCE = 75 and CDE = 75 and two angles equal dep Q1 Calculation must be shown crect angle marked on diagram Calculation must be shown crect angle marked on diagram Strand (ii) Must sce M2 and have no increct angles calculations seen 16 cont C = 105 D = = 65 C = 180 ( ) = 85 BCD = 75 CDE = = 45 DCE = 180 ( ) = 105 BCD = 105 CDE = 65 DCE = 85 (no method shown) Alternative methods continued on the next page dep Q0 B0 dep Q0 M0 M0dep Q0 23 of 26

24 Alternative method 4 DCE CDE = (180 30) 2 75 Calculation must be shown one crect angle marked on diagram CDE and DCE = their 75 dep DCB = 180 their ABC = their 75 ADC = their 75 dep Calculation must be shown crect angle marked on diagram 16 cont DCE = 75 and CDE = 75 and DCB = 105 and opposite angles of parallelogram equal DCE = 75 and CDE = 75 and ABC ADC = 75 and allied (co)interi angles of parallelogram Q1 Strand (ii) Must sce M3 and have no increct angles calculations seen (180 30) 2 = 65 C = 65 and D = 65 C = 115 (no method shown) (180 30) 2 = 75 DCE = 75 and CDE = 75 DCB = = 105 dep M0dep Q0 dep dep Q0 24 of 26

25 2 ( ) 200 their dep is M2 their 50 their their 50 must follow dep A1 Perimeter = 100 Side of square = Side of square = 100 = = = = 100 Side of square = = 2400 Answer = = 4200 M0 M0dep A0 M0 M0dep A0 dep A0 M0 M0dep A0 3rd M0 25 of 26

26 Any crect product, division with answer of 210 that involves a prime number eg 2 105, 5 42, = 70, , 3, 5, A1 Product may be implied f by a prime fact tree, a prime fact ladder values written as pairs eg (2, 105) A0 19 6n + 3 3(2n + 1) B2 f 6n Accept 6 n n 6 but not n6 f n6 + 3 Accept any letter (10 2) A1 Answer only of 144 A1 26 of 26