GCSE MARKING SCHEME MATHEMATICS - TWO TIER LEGACY

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1 GCSE MARKING SCHEME MATHEMATICS - TWO TIER LEGACY NOVEMBER 2011

2 INTRODUCTION The marking schemes which follow were those used by WJEC for the November 2011 examination in GCSE MATHEMATICS - TWO TIER LEGACY. They were finalised after detailed discussion at examiners' conferences by all the examiners involved in the assessment. The conferences were held shortly after the papers were taken so that reference could be made to the full range of candidates' responses, with photocopied scripts forming the basis of discussion. The aim of the conferences was to ensure that the marking schemes were interpreted and applied in the same way by all examiners. It is hoped that this information will be of assistance to centres but it is recognised at the same time that, without the benefit of participation in the examiners' conferences, teachers may have different views on certain matters of detail or interpretation. WJEC regrets that it cannot enter into any discussion or correspondence about these marking schemes. Page Mathematics Paper 1 - Foundation Tier 1 Mathematics Paper 2 - Foundation Tier Mathematics Paper 1 - Higher Tier Mathematics Paper 2 - Higher Tier 9

3 PAPER 1 FOUNDATION TIER 2011 Autumn Paper 1 (Non calculator) Foundation Tier 1. (a) (i) 2302 (ii) eighty five thousand (one) hundred (and) five (b) (i) 23, (ii) 88, (iii) 9 (c) (i) 30 (ii) 00 (d) 1, 3,, 1 (e) (i) 9832 (ii) 2398 Marks FINAL POST CONFERENCE MARK SCHEME Comments (1/11/2011) (Page 1) Accept OR 2 for any 2 or 3 factors and no incorrect numbers. OR the correct factors and 1 incorrect number. 2. (a) (b) E (Total =) 0 (c) A, E, N, S, T along one axis Uniform scale for the frequency axis starting at 0 (No numbers interpreted as 1-12 in ones) Five bars at correct heights 11 8 for any three correct (tallies and) frequencies. F.T. their frequencies. If frequencies get 0 F.T. their table of frequencies B0 for 12, but E and 12 is OR indicated on the bars themselves F.T. their table of frequencies. Use of any other scale must be clearly indicated on graph for at least 3 correct bars on F.T. Bars can be of varying width for correct bars with frequency polygon Frequencies take precedence over tallies If freq. score 0 then all correct tallies score 3. (a) cuboid trapezium pentagon equilateral triangle Line to parallelogram gets B0 (b) parallel line (c) Lines Curve. (a) (i) 0 (ii) 8 (b) 3 2. (a) 11 (b) 32 (cm) % = cm 2 U1 Clear intent to be parallel. F.T. the ends of their lines Or equivalent. Correct answer OR F.T. their values. Candidates who get then multiply by 2 etc get M0, A0. Independent of other marks 1

4 . (a) 2011 Autumn Paper 1 (Non calculator) Foundation Tier Marks FINAL POST CONFERENCE MARK SCHEME Comments (1/11/2011) (Page 2) (b) (c) (i) 90 (ii) 100. All 3 quadrants correct. B3 for each correct quadrant. O 8. (a) = 128 (bottles) 3 Any correct complete method for the multiplication of 3 by 8 For either 288 or 10 Place value errors get M0, A0, A0 9. (, 1) (, ) ( 3, 1) ( 3, ) 10. (b) e.g. 1% = 3 AND 8% = Procedure for marking Q9 1. Mark any writen coords. up to 8 independent ones as if correct, 1 if incorrect down to If the mark is or 3 then STOP at that point. 3. If the mark is 2 award an extra if all correct points are plotted unambiguously.. If the mark is 1 or 0 then use the SCs in the right hand column of this Mark Scheme. (a) x 8 (p) I.S.W. (b) 0b (g) I.S.W. (c) 1 (d) c 3d (e) 2x + I.S.W Any valid method for 300 OR SC1 for (un)supported 2% OR (0).2, A0 if they go on to 300±2 2 gets, Reversed coordinates get 0 each time. Penalise -1 once only for incorrect coordinate notation e.g. (x,y1). SC2 for ONLY all the correct points on the diagram OR SC1 for any 2 correct points (out of a maximum of points) on diagram. Ignore x= OR =x OR y= etc. Ignore use of capital letter but use of a different letter is penalised 1 once only. Allow 0 b OR b 0 OR b0. 0+b=0b gets B0 for either OR 20 for either c OR 3d in an expression of the form ac±bd for c + 3d 2

5 Autumn Paper 1 (Non calculator) Foundation Tier (a) (180 2) / 2 = (º) (x=) 11 (º) (b) Internal angle = 9 (y = ) = 9 (º) 12. (a) AB = 11 cm AB (in km) = 11 = (km) (b) Bearing Distance 13. (a) False (indicated or implied) AND a counterexample, e.g. halving 1 ends up as 8 (b) False (indicated or implied) AND a counterexample, e.g. 33 is not prime 1.(a) Strategy, e.g. knowing that the probabilities add to 1, or that RED with BLUE is 0., or Green the same as Red RED 0.18 BLUE 0.32 (b) (0) or 30% H1 1.(a) 90 (kg), 12 (cm) (b) Positive (c) Suitable line, with some points above and below (d) Answers in the range >0 (kg) but < 0 (kg) (e) No, with a suitable reason e.g. No as there is no data around 210cm, or graph doesn t go that high H2 1.(a) 0/ (%) (b) 300/12 OR 300/12 ( )12 ( ) 1 (c) / OR ¾ 2.80 OR 281.(0) + 10.(0) ( )22.1(0) H 1.(a) n + 2 (b) 3x + x = 8 - x = /10 (or equivalent) ISW. H 18.(a) E.g not even powers OR No WHOLE number multiplied by itself gives 2 OR 2 is not a whole number. (b) Method that produces at least 2 correct prime factors Sight of correct factors (2, 2, 2, 2, ) 2 H Marks E1 FINAL POST CONFERENCE MARK SCHEME Comments (1/11/2011) (Page 3) Look at their diagram also F.T. 180 their, but F.T. 2 only if it is shown to be the right hand base angle in the triangle. F.T. their 9 Watch for which is correct for the M0A0 for y = 9 only seen Allow ±2mm. That is, allow 11 3 to 11. F.T. their AB Use overlay Allow ±2º Allow ±2mm for an explanation that is on the correct lines, but has an error, e,g. 3 divides into 2 to make 18 or 3 2 = 1 for an explanation that is on the correct lines, but has an error. Also for their P(Red) + their P(Blue) = 0. Either correct response implies also. for either or if reversed Do not accept a description No requirement to pass through the means OR Suitable answer from their line of best fit No maybe implied in their statement Allow descriptions of the scale to go up to 200. Allow SC1 for /100 Allow for 300/ their + OR OR equivalent for n +n = n gets B0 FT until 2 nd error Accept x = -/-10 Accept no number times itself gives 2 only with and given. Accept =1 and =2. Do not accept 1, 2 only. Do not accept not even powers without Before 2 nd error Ignore 1s seen FT their factors (with at least one index >1 used). Do not ignore 1s. 3

6 PAPER 2 FOUNDATION TIER 2011 Autumn Paper 2 (Calculator allowed) Foundation Tier 1. (a) ( ) (0) ( ) 10.(0) I.S.W. (b) e.g. 10% = 1.0 % = 1.0/2 = ( ) km kg l m (g) 190 (g) (30 190)/0 = (g). (a) Wage = = ( ) 2 ISW (b) Number of hours = (20 180) / 1 =. (a) Counting squares (b) d c b a. (a) B C A 0 1 (b) No number of pupils is odd, (so not equal number of girls and boys.) OR depends on how many tickets the girls and boys bought.. Units used = 2 OR Cost of units = ( ) 29.2 OR 292(p) Total cost = ( ) 1. OR 1(p) 8. (a) angle at A = º AC = 12 Complete triangle (b) Angle 9. (a) Sum = 11 Mean = 11/8 = 1 (cm) (b) (c) 3 (cm) 1 Marks B E1 m1 FINAL POST CONFERENCE MARK SCHEME Comments (21/11/2011) (Page 1) F.T. their figures for one error F.T. their 10. Award, for ( )12. OR their ( )12. on F.T. Accept cm 3 OR cc OR ml F.T. their 30 their 190 Correctly substitution including the division Accept embedded answers such as 20 = F.T. their 2 8 correctly calculated for each A marked at 1. B should be almost at 0 ( 0 to under the p in probabilty) C marked at centre of the line. Along these lines No may be implied in their explanation Reason overides the 'Yes' or 'No'' in the comments column F.T. their units. for 292 F.T. for their cost of units ± 2º ± 2mm Only if at least one already awarded F.T. if completed triangle joining BC. (Allow 0 in a good triangle) Complete correct triangle but reflected gets. For attempt to add the numbers ( will imply ) For dividing a number by 8 (dependent on the ) Arranging the numbers in order (ascending or descending) (Award the for of the numbers in correct order)

7 2011 Autumn Paper 2 (Calculator allowed) Foundation Tier 10. Door to 8 ft OR 1 8 to 2 metres Door 2 cm Building = 10cm Multiplying factor = Height = door s estimate their SF (2-) = SC1 for answers which: (a) give only door s height as 2.cm and building height as 10cm OR (b) a proper attempt at dividing the building s height into equal door heights 11. (a) D (b) C (c) Ran a little then turned back to the start 12. (a) Marks E1 3 feet metres FINAL POST CONFERENCE MARK SCHEME Comments (21/11/2011) (Page 2) Unsupported answers marked as follows: SC1, (inc) SC F.T. their door s height estimate AND scale factors 2 inc Correct units for their numerical answer must be shown somewhere in their working for this Along these lines for any (b) 3/1 ISW 3 (c) 80 1 F.T. their table in (b) and (c) for a numerator of 3 in a fraction less than 1. for the 1 in a fraction <1. Do not penalise incorrect reduction of fractions. F.T. their 3/1 if a fraction <1. but not ½. NOTES Penalise 1 for use of words such as 3 out of 1, 3 in 1 OR 3:1 OR 3 over 1. When fraction and wrong notation seen, DO NOT penalise wrong notation. 13. (a) 2p (b) (i) (x = ) (ii) 3y = (y = ) /3 ISW (=2) (c) 2 (d) 3 = 2 + 3M 3M = 2 M = 2/3 ISW (=9) = 1 1. Cost of all adult tickets = 88.8(0) 2 ( ).(0) = ( ) 298.8(0) Cost per ticket = 298.8(0)/2 = ( ) (a) 11/0 ISW (b) (0 10) = 8 If they have incorrectly reduced their answer in part (b) and use it in part (c), then it is, A0 in part (c). 1/80 gets, A0 Accept embedded answers such as / = 1 Accept embedded answers such as = 1 F.T. until 2 nd error. Final B0 for 3 F.T. until 2 nd error. F.T. am=b (a 1) Accept embedded answers such as 3 = F.T. their but NOT ( )88.80 Note: Pupils who interchange the 2 and 2 should be marked as if correct then MR-1. for 11/m in a fraction < 1, for n/0 (if < 1) Penalise 1 for incorrect notation such as 11:0 OR 11 out of 0 etc Allow one term to be incorrect or missing. Allow, A0 for /0

8 2011 Autumn Paper 2 (Calculator allowed) Foundation Tier 1 1. (a) ( ) = ( ).3 Internet price = ( ) OR ( ) Internet price = ( )223.2 (b) 1/3 2 2 = 20/3 tins 1. (a)(i) Their readings at 1:30 & 13:00 and intention to subtract = 2 (km) (ii) 2/ 1. 3.(km / h) (b) Explanation, e.g. first half journey in just over an hour, steeper to start H1 18.(a) 0.2 (b) Any correct 8% of a value used in workings 100 8% of 100 (= ) 12 8% of 12 (= ) ( ) 13.2 H3ac = ($)118 (Canadian dollars) Conclusion, has to buy 1100 (Canadian dollars) H = 0.3(1..) 20.(a) ½ ( ). 9.8 (cm 2 ) 0.3 (b) ½ π = 8.( ) to 88.2(9) Hbc Marks E1 m1 M2 FINAL POST CONFERENCE MARK SCHEME Comments (21/11/2011) (Page 3) For a correct method of finding 1% and subtracting from 29 F.T. their.3 if M awarded OR for a correct method of finding 83% for sight of 83 F.T. their 20/3 rounded up. 11. Allow for their readings with intention to subtract FT their (i) Watch for 1/ (= 3) which gets M0, A0 Accept rounded or truncated. However, do not accept an answer of 3 without working or from incorrect working SC1 for an answer of 0 Accept, e.g. more vertical, line increases means it is quicker for 0.(19.) Could be ( )128 OR ( )2 OR their 11. OR M2 for ( for ) FT their 128. Penalise extra working -1 Appreciate: Possible and SC1 for ( )18.2 Simple depreciate: Possible and (13) FT their rounding down to nearest 0 provided FT their amount of Canadian dollars if changed to a $0 amount. Do NOT F.T. 118 OR their 118 must be given. Accept 0.3p For example, 118 changed to 110 (incorrect) gets,, A0, but on F.T. 110/1. gets, 01.21(9) gets and finally gets giving marks in total. Accept an answer of 29.2 in credit or similar, but an answer of 29.2 would not get the final A mark, but do award the previous as alternative method Accept 80 from working, but unsupported 80 gets M0, A0. Allow π to 1. SC1 for answer 311 to

9 PAPER 1 HIGHER TIER Higher Tier November 2011 Paper 1 Mark Comments (Final Post-conference version ) 1.(a) Strategy, e.g. knowing that the probabilities add to 1, or that RED with BLUE is 0., or Green the same as Red RED 0.18 BLUE 0.32 (b) (0) or 30% or equivalent Also for their P(Red) + their P(Blue) = 0. Either correct response implies also 2.(a) 90 (kg), 12 (cm) (b) Positive (c) Suitable line, with some points above and below (d) Answers in the range >0 (kg) but < 0 (kg) (e) No, with a suitable reason e.g. No as there is no data around 210cm, or graph doesn t go that high 3.(a) a = 0, b = 110, c = 0 (b) 30/ /10 1 (0) (c)(i) Bearing (ii) from C from A D indicated or implied by point. (a) Correct reflection in the line x = -1 (b) Correct translation (c) Enlargement ½ Correct position (d) Bisector of angle CAB Arc radius cm centre A Correct region shaded.(a) 0/ (%) (b) 300/12 OR 300/12 ( )12 ( ) 1 (c) / OR ¾ 2.80 OR 281.(0) + 10.(0) ( )22.1(0).(a) n + 2 (b) 3x + x = 8 - x = /10 (or equivalent).(a) E.g not even powers, (b) Method that produces at least 2 correct prime factors Sight of correct factors (2, 2, 2, 2, ) 2 (c) 3/ or 0. (d) /3 2/ = 8 8.(a) 20 and 20 (b) Sight of least width 103 (mm) (mm) B E1 for either or if reversed Do not accept a description No requirement to pass through the means OR Suitable answer from their line of best fit No maybe implied in their statement for each. FT b = 180 a and c = a or c = b OR alternatively: 8 triangles at 180 or 10 (8 180) 10 1 Depends on at least 1 M mark for a reflection in any line indicated or either axis, OR for drawing x =-1 Note: Scale factor 2 is NOT MR, M0 A mm FT for intention of bisector & arc Allow SC1 for /100 Intention 300/(+) then or OR equivalent e.g. attempting to find lots of 10% and adding % for n. B0 for n + = n FT until 2 nd error ISW. Accept x = -/-10 Accept no number times itself gives 2 only with and given. Accept = 1 and = 2. Do not accept 1, 2 Do not accept not even powers without Accept 2 not a whole number Before 2 nd error Ignore 1s seen FT their factors (with at least on index >1 used). Do not ignore 1s. Unsimplified answer award only for each. Accept ( ) not 20.9 FT their least length, not 200 AND their least width, not 100

10 Higher Tier November 2011 Paper 1 Mark Comments (Final Post-conference version ) 9. (a) ( x ) ( x + 2 ) for ( x ) ( x 2 ) with no or incorrect signs (b) 2x 2 9x - (c) 21 2x = 20 x 3x = -1 x = -1/3 (= ) ISW for 2x 2 - or -9x as part of a trinomial. ISW solving FT until second error Do not accept -0.3 unless -1/3 seen. FT: 21 2x = 20 x B0, -x = -1, x = 1 FT: 21 2x = x B0, 3x = -1, x = -1/ x = x is 2 errors so no FT (d) 2 x 10 y (e) a 3 10.(a) 0, 0,, 0 (b) At least plots correct horizontally At least plots correct vertically All points plotted correctly and joined (c) (i) Median from their cum. freq. diagram Difference of heart rate reading for & 1 Interquartile range 11.(a) (3x+2)(2x-) x = -2/3 and x = /2 (b) (2y+9)(2y-9) 12.(a) 0.3, 0.2, 0.8, 0.2, 0.8 (b) = (a) Method for either (i) or (ii) (i) 3a +2b (ii) 9a + b (b) KM = 12a+8b seen or implied Showing p = (c) Collinear (or parallel)and times length, OR Collinear with ratio KL:LM as 1:3, OR equivalent 1.(a) x = 3, 1, (b) Tangent at x = Gradient = change y / change x 11 from a tangent or ft reasonable tangent (c) Line y = 10 stated or shown Solution ~ -2., ~0., ~.3 (d) Using trapezium rule or evidence of summation of areas. At least 2 correct non zero y values. Correct expression for total area. Answer 0 1. (a) (x=) & (100x=).222 with attempt to find the difference / 990 (b) 10 E2 11 for any two factors number, x & y correct, or correct but with times left in expression FT to (b) only if cumulative in (a). B0 for bars, for vertical lines. Accept plots, e.g 89 to 90 for <90 for bars or vertical lines Joined with a curve or a straight line FT their cumulative frequency or other cumulative diagram in (c) Allow consistent misread of the scale. Correct for their cumulative freq. diagram for (3x 2)(2x- ) FT their pair of brackets for (2y 9)(2y 9) 0.3 with one other correct, or 0.2 & 0.8 as a pair FT from their tree, not 0.s and must be <1 (Accept missing brackets if no other marks in (a)) Simplifying -(2a + b) + a + 3b correctly Simplifying -(a + 3b) + 1a + 9b correctly FT (i) + (ii) E1 for parallel OR collinear OR times length Accept all on straight line for collinear All three required Independent of tangent drawn or not, no values required Maybe implied FT from incorrect line for 2 solutions, or 3 solutions from consistent MR Must be for required area Equal strips gives (-3,0),(-2,21),(-1,2),(0,1),(1,0) Allow 1 error in y value An answer of 20 gets M0,, A0, A0 but then SC1 Or equivalent A final answer of./99 is only for 32= 13 or 32 = 13 or partial simplification or shown by division 8

11 PAPER 2 HIGHER TIER Higher Tier November 2011 Paper 2 Mark Comments Final 28/11/11 1.(a)(i) Their readings at 1:30 & 13:00 and intention to subtract = 2 (km) 11. Allow for their readings with intention to subtract (ii) 2/ 1. FT their (i) 3.(km / h) Accept rounded or truncated. However, do not accept an answer of 3 without working or from incorrect working SC1 for an answer of 0 (b) Explanation, e.g. first half journey in just over an hour, E1 Accept, e.g. more vertical, line increases means it is quicker steeper to start 2.(a) -, -2, 3 (b) x(x - ) (c) 120(2y -3) (d) 12x = 0 OR 3x-11 = 0/ 12x = 0+ OR 3x = x = 8/12 (ISW) OR x = 21/3 (ISW) OR x = (e) 9 3.(a) 0.2 (b) 18/100 (= 8.1(0)) + their tax ( )3.1(0) (c) Any correct 8% of a value used in workings 100 8% of 100 (= ) 12 8% of 12 (= ) ( ) = ($)118 (Canadian dollars) Conclusion, has to buy 1100 (Canadian dollars) = 0.3(1..) 0.3.(a) Mid points, 12 and 20 ( ) (OR = 122) 100 = 12.2() (b)polygon with at least 3 vertices correctly plotted (vertical & horizontal) All vertices of the polygon correct.(a) 2 π.2 =.2(1 ) to.3 (cm) Degree of accuracy, whole or 1 d.p. (b) ½ π = 8.( ) to 88.2( cm 2 ) (c) ½ ( ). 9.8 (cm 2 ). (a) 18k q = dk + 18k dk = q + k ( 18 d) = q + k = (q+)/ (18-d) (b)(i). 10 (ii) m1 m1 9 m1 M2 8 for any two terms in correct position. Award for -, -, -2 for correct partially factorised, or 120(2y ) or 120( -3) B0 for 20(y 1.) FT until 2 nd error in (d) Accept embedded answer for 0.(19.) Alternatively allow M2 for 1.18 OR M2 for ( for ) FT their 128. Penalise extra working -1 Appreciate: Possible and SC1 for ( )18.2 Simple depreciate: Possible and (13) FT their rounding down to nearest 0 provided FT their amount of Canadian dollars but not their 118, for M and 1 st A only, however FT multiples of 0 for all marks must be given. Accept 0.3p Accept an answer of 29.2 in credit or similar, but an answer of 29.2 would not get the final A mark, but would do award the previous as alternative method Two shown is sufficient if no error Attempt fx for their mid-points that fall within the intervals including bounds Attempt their fx divided by 100. Accept 12 only if all working shown No polygon M0. Ignore bars. Mid points - allow intention (e.g. from 10 to 12 inclusion) SC1 for a correct polygon translated horizontally or all correct plots with no polygon (or curved polygon!). Ignore joining to axis or to form a complete shape FT rounding to whole or 1d.p. provided awarded Allow π FT 1... to 1.3 SC1 for answer 311 to 312. Accept 80 from working Expand FT each stage for equivalent level of difficulty Collect until 2 nd error Factorise Divide Penalise incorrect notation once only 9

12 Higher Tier November 2011 Paper 2 Mark Comments Final 28/11/11 x 2x 3 + x (a) One correct evaluation, 3 x 2 correct evaluations, 3. <x <3., one either side of 0 2 correct evaluations, 3. < x <3., one either side of 0 OR correct evaluation of 3. if previous awarded 3. No calculations shown: accept too high, >, etc (b) Correctly setting up 2 equations for eliminating 1 variable First variable s value Correctly substituting their first variable Second variable s value 9.(a)./3 2. (b) 3. /./3 =. (cm) = 2 (cm) 10.(a) (AD 2 =) (AD 2 =) AD = 10.(23 cm) (b) Strategy, idea to find BC and CD sin1 = BC/12.3 OR cos1 = CD/12.3 BC = 12.3 sin1 OR CD = cos BC = 8.09 OR CD = Use of correct method to find the other side Other side correct CD or BC Area BCD = ½ BC CD Answers between 3. and 3. (cm 2 ) 11. (a) 30 (seconds) (b) Histogram drawn with at least frequency densities correct Correct histogram drawn 12.(a) y α x 2 OR y = kx 2 = k0. 2 y = 1x 2 (b) x y (a) 2(x+)(x+) +2(x+) +2(x+) 2x 2 + 2x = 0 (b) {-2 + ( )}/ ( )/ 1.28 and (c) (,).28,.28 1.(a) Mean. Σx 2 = 32 or Σ(x x ) 2 = 33.1 SD = 20.8(233 ) (b) New mean. SD = 20.8(233 ) Explanation e.g. spread unchanged 8 S M2 E1 Or alternate substitution method, allow one slip in multiplication in non-eliminate Either x = 2 or y = - FT their first variable FT their first variable Or equivalent calculation that could lead to correct answer Or equivalent calculation that could lead to correct answer If no marks in (a) or (b) then award SC1 for sight of scale factor 1.8 Or idea to find DC and use ½ cbsind Implies previous Correct Pythagoras substitution, or trig Allow FT from rounding errors FT their CD and BC provided at least S1 FT Frequency density 1, 1., 2.,.2 and 0. FT non linear only Maybe implied in part (b) for each value. FT their non linear expression for area of any 2 of the faces, or 1 of the 3 terms. Must follow from working, convincing Allow 1 slip. Incorrect formula is M0 OR M2 ½ absind correctly substituted ( if 1 slip, not the wrong angle) A2 Correct answer FT x + and x +. (Accept.3 and.3). Allow FT from +ve only. B0 if +ve and ve given FT mean + 2 FT their SD unchanged Depends on in (b). Understanding, not calculated. If no calculations accept simple statement all marks went up 10

13 Higher Tier November 2011 Paper 2 Mark Comments Final 28/11/11 S1 Accept 21. FT candidate s AC 1.Overall strategy, cosine rule followed by sine rule AC 2 = cos19 AC 2 = 3. AC = 20.8(81...) sinb = sin AC 28 sinb = sin AC/ (9. ) to 33(.2 ) Implies previous GCSE Mathematics - 2 Tier Legacy MS/November

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