GCSE Mathematics. Paper F Mark scheme F June Version 1: Final Mark Scheme

Transcription

1 GCSE Mathematics Paper F Mark scheme 43651F June 2015 Version 1: Final Mark Scheme

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 2015 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 Use of brackets Marks awarded f quality of written communication It is not necessary to see the bracketed wk to award the marks.

4 AQA GCSE Mathematics (Linear) 43651F June 2015 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. 4

5 MARK SCHEME GENERAL CERTIFICATE OF SECONDARY EDUCATION MATHEMATICS 43651F JUNE 2015 Paper 1 Foundation Tier Tallies crect f coffee and ange ie Must use five bar gate f ange Milk tally must be as given 1(a) Frequencies crect f milk, coffee and ange ie 5, 3 and 8 ft ft their tallies f milk, coffee and ange Frequencies f coffee and ange may be crect follow through from their tallies 3 bars labelled M(ilk), C(offee), O(range) of equal width (1 square), with equal gaps and crect heights Q2ft Strand (i) Q1ft Fully crect apart from one err ie unequal widths, unequal missing gaps, increct missing labels one height increct ft their table (tally frequency) 1(b) Heights may be crect follow through from their table Labelled bars may be in any der. Assume unlabelled bars are in the given der Mark intention ie condone unruled bars Bars do not need to be shaded to have matching shading Only check the gaps between the four bars so spacing of 1, 2, 2, 2 squares is crect Equal widths may be eg 2 squares if Tea bar is changed to 2 squares Accept a key in place of labels Allow the bar chart to be extended to show a height of 9, f example 5of 21

6 seen 36 and 45 and 54 chosen dep 135 A1ft ft M0 crect total of their three numbers SC1 Crect total of three numbers between 31 and 59 inclusive 2(a) 39, 48 and 57 chosen and answer 144 (must be three numbers added) SC1 36, 45 and 54 chosen in wking and answer 3 M2A0 36, 45 and 56 chosen and answer 137 M0A1ft 135 must not come from increct wking A0 2(b) At least one of 9, 19, 89 and at least one of 90, 91, 97 seen 20 A1 SC1 Answer 19 3(a) Plot at (1, 2) Allow f vertex of rectangle at (1, 2) 3(b) 20 cm 2 4(a) triangle 4(b) pentagon 4(c) square less than 5 greater than B2 f 2 crect equal to 6of 21

7 Alternative method 1 (Price per tin f 2 tins) and and and and 90 Box of 4 tins identified eg 1.80 A1 Q1ft Strand (iii) ft their values if awarded SC2 45 and 50 seen SC seen Alternative method 2 (Scaling to an equal number of tins eg multiple of 12) eg 12 tins 2 3 and 3 1.8(0) 6 and 5.4(0) A1 6 Box of 4 tins identified eg 1.80 Q1ft Strand (iii) ft their values if awarded SC2 45 and 50 seen SC seen Alternative method 3 (Using small box to wk out cost of 6 tins) and their 45 2 their their 90 their 90 3 (6 tins = ) 270 A1 Box of 4 tins identified eg 1.80 Q1ft Strand (iii) ft their values if awarded SC2 45 and 50 seen SC seen Alternative method 4 (Using large box to wk out cost of 4 tins ) and 300 their 50 2 their their 100 their (4 tins =) 200 A1 Box of 4 tins identified eg 1.80 Q1ft Strand (iii) ft their values if awarded SC2 45 and 50 seen SC seen on next page 7of 21

8 Allow wking in and/ pence with without units throughout Check the diagram f values All schemes wk in the same way: f full method to wk out two directly comparable values (may only need one if they are matching to one given in the question). The M mark can be implied by the crect A value(s). A1 f evaluating their value(s) Q1ft, if A0 awarded and crect decision made f their evaluation(s) Some students wk out 45 and 50 crectly but then do an increct calculation which still means that 4 tins is the best offer. Do not award full marks if a calculation err is seen that is part of their chosen wking. However SC2 will always be available f these students. 6 cont Useful values: Box of 6 50p ( 3) Box of 4 45p 90p 1.35 ( 1.80) They must have two values to compare (one may have been given) otherwise they will not sce (apart from possible SC marks) eg Using box of 4, 8 tins cost 3.60 You get two tins me f only 60p ( each extra tin only costs 30p) Box of 4 Wking out tins per is impractical on a non-calculat paper. If seen, the divisions must be crect so use this scheme: 6 3 and 4 1.8(0) 2 and tins Box of 4 with no crect wking M0 A0 Q0 M0 at this stage A1 A1Q1 M0A0Q0 8of 21

9 Any crect shading eg Condone part triangles shaded if unambiguous eg 7(a) Igne any lines of symmetry drawn Any crect shading eg a rotation of any of these Condone part triangles shaded if unambiguous 7(b) Igne any lines of symmetry drawn If there is wking seen f Q8 below the centre line of the diagram, do not escalate the clip as this will be seen in Q8. Any wking seen above must be escalated May be on a diagram 25 A1 9of 21

10 (Area A =) 8 (Area B =) 8 A has the same area as B indicated ft May be on diagram Accept 8 16 May be on diagram Accept 8 16 crect ft decision f their two areas if awarded 9 They don t need to explicitly state both areas if same area box is indicated eg Area B = 8 and crect box indicated implies the first Both are 8 and crect box indicated Only indicating the same area box (with no increct values f areas) Area A = 4, Area B = 8 and A < B box indicated Perimeter Where possible award benefit of doubt eg student appears to be counting around the edge of shape A when they may just be counting squares/ unshaded squares If student mentions both area and perimeter, just mark the area part Even if student mentions only perimeter, you can still award the third f ticking A has the same area as B, however there would be no ft available f ticking one of the other boxes However, stating A = (ie 8) and B = is clearly perimeter so do not give credit Condone no units increct units B3 B3 B0B0 B0ft B0B0 B0B0B0 Two areas seen that are not labelled and not linked to the diagram eg 2 2 = 4 and = 8, then A < B B0ft 10 of 21

11 Q1 If using build-up, must see completely crect, full method Strand (i) p is Q0 SC1 59.5(0) 10(a) Build-up must be crect have full crect method 10% = 7, 5% = 2.5 Answer 9.50 M0Q0 10% = 7, 7 2 = 2.5 Answer 9.50 Q0 10.5(0) seen in wking but then added to deducted from 70 Q0 Alternative method ( 100) A1 SC (b) Alternative method 2 20 = 10% 10 = 5% 2 = 1% Crect % f any fact/multiple of A1 SC = 20% seen, then answer 40 A0 Assume 20 on the answer line is 20% not 20 A1 11 of 21

12 11(a) Scalene 11(b) Fully crect ie Angle at either end of line of [68, 72] Angle at other end of line of [58,62] Triangle drawn with ruled straight lines B2 one angle drawn within tolerance Igne angle labels If student uses own base line must be 9 cm ± 2 mm Angles must be at the ends of the lines, allow tolerance of ± 2 mm (but judge by eye) 12(a) A1 May be on diagram 30 f LHS total 16 f two hearts 12(b) = 36 May be on diagram 26 f two stars 13 A1 13(a) 33 Igne units eg ºC 13(b) (+)1 Igne units 13(c) K = 9 5 (F 32) One crect value in table ( 3, 11) (3, 7) May be implied from graph if table blank 14 Accurately plots at least two points ± ½ square ft their table Crect table and ruled straight line from ( 3, 11) to (3, 7) A1 Must be completed No increct plots ± ½ square 12 of 21

13 15(a) 10xy 15(b) 6 B2 18 = (1 ) 2 ( ) 3 ( ) 3 30 = (1 ) 2 ( ) 3 ( ) 5 one crect prime fact tree one crect prime fact ladder one complete set of facts (may be in product pairs) 1, 2, 3, 6, 9, 18 1, 2, 3, 5, 6, 10, 15, 30 SC1 Answer 2 and/ Alternative method their their dep 270 A1 SC1 digits 27 Alternative method 2 16 ( ) (ml) their 90 3 dep 270 A1 SC1 digits 27 Alternative method : and their dep 270 A1 SC1 digits 27 on next page 13 of 21

14 Students may convert wrongly to millilitres using a fact of 10 (ie 450) then convert back using the same wrong fact to get the crect answer. Allow this, as the method is valid. However, partial marks cannot be awarded if a wrong conversion fact is used but if digits 27 seen allow SC1 (1 gallon = ) 45 millilitres (3 gallons = ) 135 millilitres 135 millilitres 10 = 13.5 litres = 270, dep, A1 16 cont (1 gallon = ) 45 millilitres (3 gallons = ) 135 millilitres = 2700 If a build up method is used to get millilitres equivalent to 13.5 litres then it must be fully crect to get the dep SC = 200, 1 = 20, 3 20 = 80, 0.5 = = = 200, 1 = 20, 3 = 60, 0.5 = = 290 Gallons Litres Lawn feed (ml) dep A0 M0 A0 dep A1 14 of 21

15 Alternative method ( x) 6 = 18 their 108 ( ) dep eg complete repeated subtraction Look f total written under by table 19 A1 SC1 89 seen Alternative method 2 18 each value in table, eg 1.8, 0.1, +2.1, +0.2, 3 Allow one err 17 Totals their subtractions their ( ) 1 and adds to 18 dep 19 A = 34.2, = = 77.9, = = = 30.9 ( x) 6 = 18 x = x = = Allow increct solution of equation if full method dep A0 dep A0, dep, A0 15 of 21

16 18(a) 5x 18(b) w = z 3 w = 3 + z z 3 = w 3 + z = w Must have w = = w Many students write z like the number 2. Allow f this 2y(2y + 3) B2 f 2(2y 2 + 3y) y(4y + 6) Allow signs between numbers, brackets and letters, eg 2y (2y + 3) 2(2 y y) 18(c) Factising may be done in two steps, ie y(4y + 6) followed by 2y(2y + 3). If the second attempt is done wrongly, can still be awarded. y(4y + 6) 2y(2y + 6) 2(2y 2 + 3y) 2y(y + 3) B0 B0 16 of 21

17 Two of the four crect approximations , , 34 30, denominat must not come from an increct method eg Three crect approximations used in the crect calculation All four crect approximations , , dep eg , and 20 may be implied by 50 but 50 must not come from an increct method 40 and all approximations crect A1 Must show crect approximations Answer only of and answer = 53, = 2000, = 40 (50 from wrong method) = , answer 40 M0M0A0 A0 M0A0 M0A = 40 M0A0 17 of 21

18 Alternative method 1 Crectly lists first three bus times to X Y ie 7 25, 7 50, 8 15, Accept any notation f time eg 7.20, 7: , 0720, 7-20, 20 past 7, , 7 40, 8 00, Continues both lists at least as far as a common time ie 7 25, 7 50, 8 15, 8 40,... dep Allow one err up to and including their common time, igne errs after and 7 20, 7 40, 8 00, 8 20, 8 40, (am) after/in 100 minutes after/in 1h 40 minutes A1 SC2 No other wking and any time that is 7 am + 100n minutes eg 10 20, 12 00, etc Alternative method 2 Crectly lists first three multiples of ie 25, 50, 75, 20, 40, 60, Stops both lists at 100 identifies hour 40 minutes 25 4 and 20 5 dep 8.40 (am) after/in 100 minutes after/in 1h 40 minutes A1 SC2 No other wking and any time that is 7 am + 100n minutes eg 10 20, 12 00, etc on next page 18 of 21

19 7 25, 7 50, 8 15, 8 40, 9 05, 7 20, 7 40, 8 00, 8 20, 8 40, 9 00,.. dep (Answer =) 8 40 pm pm is wrong A0 (No wking) (Answer =) 8 40 pm Method by implication M2 7 25, 7 50, 8 05, 8 30, 8 55, 9 20 Second list crect f 3 values. 7 20, 7 40, 8 00, 8 20, 8 40, One err in first list. dep (Answer =) 9 20 Both lists taken to a common value A0 7 25, 7 50, 8 10, 8 30, 9 00, , 7 40, 8 00, 8 20, 8 40, 9 00 (Answer =) 9 00 Second list crect f 3 values. Both lists taken to a common value but me than one err in first list. M0dep A0 20 cont 25, 50, 75, 80, 20, 40, 60, 80, At least one list crect f 3 values. Ds not get to 100 M0 (Answer = ) 8 10 A0 7 00, 25, 50, 8 15, 40, 9 05, 7 00, 20, 40, 8 00, 20, 40, 9 00 Intention to list times clear dep 8 40 A1 As question asks f When.. rather than What time.. then the students do not have to say 8.40 but could qualify it as a length of time after 7am. If so then the wding must be clear. 7 25, 7 50, 8 15, 8 40, 9 05, 7 20, 7 40, 8 00, 8 20, 8 40, 9 00,.. (Answer =) 1 h 40 after , 7 50, 8 15, 8 40, 9 05, 7 20, 7 40, 8 00, 8 20, 8 40, 9 00,.. (Answer =) 1 h 40 Must make it clear that the time is after 7 (am) Not clear that the time is after 7 am dep A1 dep A0 19 of 21

20 7, 8, 9, 11, 11, 11 7, 7, 9, 11, 11, 11 7, 9, 9, 11, 11, 11 B3 B2 All three conditions met but not all whole numbers B2 two conditions met with six numbers (need not be integers) one condition met with six numbers (need not be integers) Numbers do not have to be in der 21 Mark answer line unless blank, then look f an obvious set of 6 numbers. Must be 6 numbers. 7, 9, 9 2 1, , 11, 11 Mode, range and median but not all whole numbers B Mode and range B Mode and range B Median and range B Mode and range (undered ok) B2 7.5, 8, 10, 11, 11, 11.5 Mode and range B Median 20 of 21

21 22(a) 0.4 (relative frequency of carp) 1 (bream) their roach frequency 10 (must be less than 1) 1 their carp relative frequency Fully crect table ie (4) (0.1) 0.5 A1 accept equivalent fractions percentages f relative frequencies throughout If table fully crect award 3 marks. If not check f Either sces. Then check last column/bottom row. If the roach relative frequency = roach frequency 10 if the total of the relative frequencies is 1 then award. Increase sample size Repeat it Check some me Catch me fish Count it again, catch me fish Last bit sces 22(b) Fish on me days Me implies increased sample Fish f longer Longer implies increased sample Fish on different days Different ds not imply increased sample B0 Do the estimate twice Not implying increasing sample B0 Catch them all Not a sample B0 Experiment at different times of day Not implying increasing sample B0 21 of 21