MS GENERAL CERTIFICATE OF SECONDARY EDUCATION TYSTYSGRIF GYFFREDINOL ADDYSG UWCHRADD MARKING SCHEME MATHEMATICS - 2 TIER

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1 MS.00 GENERAL CERTIFICATE OF SECONDARY EDUCATION TYSTYSGRIF GYFFREDINOL ADDYSG UWCHRADD MARKING SCHEME MATHEMATICS - 2 TIER SUMMER 2008

2 INTRODUCTION The marking schemes which follow were those used by WJEC for the Summer 2008 examination in GCSE MATHEMATICS - 2 TIER. 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.

3 FOUNDATION TIER - PAPER 1 GCSE Mathematics June 2008 Foundation paper 1 2 tier examination 1. (a) (i) 12 2 or 2 12 (ii) 2 or 2 (iii) 9 or 9 (iv) 0 or 0 (b) 2 (c) 8 / 1 ISW 2. Rectangle Trapezium Pentagon Labels on axes Uniform scale Bars correct. (a) ( )20 (b) 12, 1, 18 Mark. Correct image 2. (a) 2 / 8 / 28 (b) Rent ( )100, saving ( )12 ( )100 AND ( )12 seen 1 / = 20% 1 / = 20 / 100, 2% = 2 / 100 Correct conclusion drawn (c) 0(%) 2(%) ¼ (0). 0(%) or equivalent (d) Oranges cost ( )2.88 or 288(p) Change ( ).12 or 12(p). Estimate for height of car 9 FINAL VERSION for 1 / 0 EQUIVALENT At least one label given for uniform scale from their table Accept vertical line diagram for attempt to multiply AND add. ( )0 gets M0 A0 for any two correct when numbers shown. 18, 2,, gets -1 for each incorrect vertex. for 1 correct or (1 or 2 correct and 1 incorrect) for 100 or 12 AND an attempt at finding the second value for change to %age or fractions which can be compared. Allow 2% = ¼, ¼ > 1 / for for their percentages 1.2m to 2m ft to ft inches Height of house 10cm (9.cm) height of car 2cm (1.9cm) (equal) intervals marked on the diagram Height of house = their height of the car Height of house 8. (a) (i) Median (ii) Range (b) P at ½ Q at 1 R at 1 / 1 + 2mm Accept (or ) their height + a part of their height for their height Answer only which is greater than their height BUT less than or equal to their height gets Attempt at ordering the numbers SC2 correct probabilities given but points not marked on the diagram. >0 but < ¼

4 GCSE Mathematics June 2008 Foundation paper 1 2 tier examination 9. (a) Area = = 12 (m 2 ) (b) Cost of carpet = 1 their area ( )180 Total cost = ( ) = ( ) (a) x = ( ) = 1(º) (b) Base angle = (180 )/2 = (º) y = 11 (º) (c) 0 ( ) 0 z = 120 (d) Correct reflection 11. (a) Any correct method that will lead to 2 e.g = 128 (b) (i) (0) 0(00 ) (ii) 9.88 (c) 2 (d) 8/9 /9 = 2/9 (e) (i) 1 (ii) 2 (f) (i) (0).21 (ii) 1.18 Mark m1 9 A Correct enlargement 2 1. (a) or angles correct and correctly labelled. B or angles correct, labels not fully correct. 2 angles correct and correctly labelled. 2 angles correct, labels not fully correct. 1 angle correct and correctly labelled. If 0 1 for their diagram or no diagram, 0/90 Angles are 100, 10, 80, 0 (b) Probabilities must add up to 1. B E1 from (a) Award for FINAL VERSION Allow marks for y = 11 F.T 180 their M for correct place value For 188 and 1110 for 1 error One error. Answer only 0 gets A0 Stop at 2 nd error. for 8 9 seen For dealing with the process correctly equivalent fractions, but, A0 for 889 = 222 equivalent in % for 1 incorrect vertex. MR 1 for use of incorrect factor > 1. Use the overlay and allow + 2. Correct labels (Words NOT the frequency angle). correct labels is enough. If only is scored for the diagram, and all the angles given correctly, then cancel the and award, for 2 marks. If B0 scored for the diagram, check the angles and the method to see if the and the can be awarded. (1 is) gets the. SC1 for all correct percentages: 2 8, 8.9, 22.2, The probability should be 0.8 (not 0.9). 2

5 GCSE Mathematics June 2008 Foundation paper 1 2 tier examination 1. (a) x x = 0 x = 200 x = 0 (b) x = 0 y = 180 ( their ) = 1. (a) (i) x x = 1 x = 12 x = 12/ ISW () (ii) x + 2 = x x = x = / ISW (b) (i) r r 11r + 2 ISW (ii) p + 9 2p + 2 p Circle (m 2 ) Divide area into rectangles 2 rectangles Area = = 100 (m 2 ) Required area = 18 Correct units given m 2 1. Sight of 0. Sight of 00 and 20 Answer 0,000 Allow Sight of 0. Sight of 00 and 20 Answer 2,000 Sight of 0.2 or 0. Sight of 00 and 20 Answer 0,000 or 0,000 Mark 10 U1 FINAL VERSION for LHS or expression in x = 0 if LHS is of equivalent difficulty and RHS = 0 or 180 if of the form ax = b. No equation formed but correct answer given B0. In (a) and (b) stop at 2 nd error. Accept embedded answers in a(i), a(ii). 12 gets B0 gets B0 Do not ignore subsequent working for any other correct method if BOTH M marks given For this value accept (02 or 01) and 20. Answers 2,080 2,00 For either of these values accept (02 or 01) and 20. Answers 0,200 or 0,100 0,100 or 0,00 If any other value is used in the denominator allow SC1 for the sight of 00 AND 20 in the numerator.

6 FOUNDATION TIER - PAPER 2 GCSE Mathematics June 2008 Foundation paper 2 Pilot examination 1. (a) (0) (b) 10 their total /100 or equiv. ( ) (a) (b) (a) (i) 8 (ii) Pointer at (b) 0. (a) (i) (ii) (b) Shade any 2 sections (c) (d) (i) 0 (ii) 000. (-, -2) (1,). (a) A and D G and B (b) P and V 8. (a) 98 2 (b) A line of length 1cm or 9cm or 12cm A second line having a correct length Completed correct triangle 9. (a) (b) 2 correct lines Mark 2 FINAL VERSION for 1 error Accept any of the above giving from (a) for part (b) All four must be attempted for Unsupported ( )2.2 gets for attempting to count squares. Accept fractional answers in the given range. > but < 8 for 2 or factors. B0 if any incorrect factors are given. for 1 correct or (1 or 2 correct and 1 incorrect) + 2 If is given then 1 for incomplete triangle + 2 Allow + 2mm for each length for 1 correct line or (2 correct lines and 1 or 2 diagonals)

7 GCSE Mathematics June2008 Foundation paper 2 Pilot examination 10. (a) (i) Sausages ( ) 1.1 Steak ( ) (1.1+1.) their.88 /8 ( )(0).8 8p (ii) 20/ / (p) ( )(0).2 (b) / (a) Attempt to add the numbers their sum /9 9 (b) (i) (ii) /20 ISW 0% (0). 12. (a) 28(cm) ISW 20(cm) ISW (b) (c) (a) (i) 12/ ISW () (ii) 8 (b) x 21 (c) 18 (d) p 8q (e) x + 1 ISW I.S.W. 1.Accept the names of the shapes R Triangular prism P Cuboid Q Tetrahedron S Octagon 1. (a) 0( ). (b) Line from Llandudno Line from Blackpool Lines intersecting 1. 0 / 1.88 ( ) 20 2 / 1. ( ) 22 Colin ( )2 Mark m1 10 m1 8 B Allow if 2.92 seen M0 for FINAL VERSION for their cost of sausage and steak Unsupported 0.8p gets m1 A0 Accept 20/89 for the. Ignore incorrect units, even %. 1 1 for each error max 2 for denominator 20 for numerator in a fraction < 1 Incorrect notation used for probability 1. for sight of areas and 0 Do not penalise extra =x or x= or =n or n= in this question. Accept embedded answers in (a). Allow x for the x 12y gets B0. for p 8q (NOT 8q) If is given 1 for further incorrect work Accept 2x x ISW All correct for any 2 correct for 1 correct. If 2 letters are put in the same cell, they are both incorrect. + 2 Use overlay if at least one given 1. euros = $1.88 $1.88 = 1. euros 1 euro = $1.88/1. $1 = 1./1.88 euros 2 euros = /1. $0 = 1. 0/1.88 = $2 = 0 euros Colin $ Colin euros

8 GCSE Mathematics June2008 Foundation paper 2 Pilot examination 1. (a) Numbered uniform scales on both axes One correct point plotted Correct line drawn (b) Line y = 2 Point of intersection (0., 2) 18. Sams ( )8. Sams by ( ) (AC 2 ) = (902 02) = 000 AC =.(9 ) AC = (m) or. (m) , 201, 208 and 21 seen ( ) (8 ) One correct evaluation (1dp)1< x <2 2 correct evaluations 1.2 < x < 1. one either side of 0 2 correct evaluations 1.2 < x < 1. one either side of 0 correct evaluation of 1. if previous awarded. Correct conclusion 1. Mark FINAL VERSION Using standard convention When B0 is given: only if one scale is marked and the other unmarked but apparently the same NO SCALES MARKED but 2cm or 1cm is used as 1 unit on both axes when points are shown or lines are drawn. for their lines. for for the difference between their answer for Sams and 08 Give for 9 2 = AC for their mid points from within the groups or the use of bounds for correct sum of their fx terms for their Σfx/10 correctly evaluated. Unsupported 20.8(. ) or 20.9 or 20 given all marks. Use of bounds gives the following answers (lower200.8( ) upper (20.8( ) ) x x + 2x (9) (2) (0) 1. 0.() () 1. 1.() (9) 1..(1) 1.8.(2) 1.9.(9) 2 If calculations are not shown accept the use of statements such as greater than zero and less than zero or too high and too low. Unsupported 1. gets B0 B0 M0 A0

9 HIGHER TIER - PAPER 1 Paper 1 Two Tier 2008 Higher Tier 1. For example 90, 20 and 0 100, 200 and 0 Estimate 2. (a) 20/ = 0 (%) 2 For reasonable estimates that lead to a calculation that effectively only involves multiplication or division by single digit numbers. (100 not reasonable denominator) Must be a correct calculation using their figures, giving an estimate in the range C.A.O. (b) 110 0/12 = (large) eggs 00 (ml) (milk) AND 2 (g) (flour). ( 1, 1), (, 1), ( 2, 2), ( 2, ), ( 1, ). (a) 0 of 200 = 0 (b) Receipts = 10 Payouts = 10 Profit =( ) (p). (a) x+9+x+111 = 0 x = 200 x = 0 (b) x = 0 y = 180 ( their ) =. (a)(i) x x = 1 x = 12 x = 12/ ISW (ii) x + 2 = x x = x = / ISW (b) (i) r r 11r + 2 (ii) p + 9 2p + 2 p Circle.1 x (m 2 ) Divide area into rectangles 2 rectangles Area = 0x x0 0x10 = 100 (m 2 ) Required area = 18 m U1 Sight of any ingredient 0/12 2½ equivalent. C.A.O. on the first ingredient used. For the remaining 2 quantities Unsupported correct answers get the appropriate marks. For example, eggs gets,, then 00 and 2 would be needed for the rd mark. for a pentagon with at least 2 correct vertices, for a correct reflection about x = c (c 1). for a correct reflection in y = 1 Penalise 1 once only if there are extra drawings. If B0, then for sight of the line x = 1 with or without any number of diagrams. 0 out of 200 gets,. 0/200 gets, A0. Full method of p their perceived (a) 2 F.T. their 0. for LHS or expression in x = 0 their LHS, equivalent difficulty, and RHS=0 or 180 if of form ax = b No equation, but answer of 0 gets In (a) and (b) stop at 2 nd error. Accept embedded answers in (a) Do not ignore subsequent working in part (b) for any other correct method if both M marks given

10 8. Sight of 0. Sight of 00 and 20 Answer 0,000 Allow Sight of 0. Sight of 00 and 20 Answer 2,000 Sight of 0.2 or 0. Sight of 00 and 20 Answer 0,000 or 0,000 For this value accept (02 or 01) and 20. Answers 2,080 2,00 For either of these values accept (02 or 01) and 20. Answers 0,200 or 0,100 0,100 or 0, , 2,,, (a) n 1 or equivalent (b) n (n + 2) (a) Correct image (1, ) (, 1) ( 1, ) (b) Correct image ( 2, 0) ( 2, ) (, 2) 12. Correctly setting up two equations for eliminating one variable First variable s value. Correctly substituting their first variable Second variable s value 1. (a) 12p r 8 (b) a(2ab + ) 1. volume, length, none of these 1 1. (a) x 9-9 > -x x 9/ I.S.W. 2¼ ( NOT 9 ) (b) Smallest = 1. (a)equal angele are: <A=<E, <ACB =<DCE, < B =<D (b) DE/8 = 1/10 DE = 12 (cm) If any other value is used in the denominator allow SC1 for the sight of 00 AND 20 in the numerator. For a method that produces 2 prime factors from the set {2, 2,,, } before their second error. If their 2 nd prime and 2 nd error occurs at the same level then allow. C.A.O. for the five correct factors. (Ignore 1s). F.T. their answer if at least one index form used with at least a square. Ignore prime number requirement for this B mark. Use of brackets (2 2 )( ) dot 2 2. is. + is B0 The inclusion of any 1s in their index form gets B0. Do not penalise extra =x or x= or =n or n= in this question. Change of letter is penalised 1 once only. Allow n 1, n 1. for the n. Allow n n+2 without brackets. sight of 1n 2 for n expr. in n expr. in n n+2 If no brackets and no multiplication sign penalise 1 once. for any 2 correct vertices, but not reflection in x-axis Reflection in y=-x is MR-1 for anticlockwise rotation of 90º about ( 1,2). (0, ) (0, ) (2, ) :2 nd point will require extra grid lines. for clockwise rotation of 90º about (2, 1). (, 0) (, ) (2, 2) Allow one error in calculating the other coefficients. C.A.O. Either x = y = 2 F.T. F.T. B for both correct answers. Substitution method for correct substitution. Ignore any extra signs. for 12p r m 12p n r 8 kp r 8 for a(ab + 9) (2a 2 b + a) a(2ab + ) a( + ) Ignore =0, but, -1 for extra inappropriate algebra For all correct.. Use of = is no marks. Use of > is MR-1 Accept 2.2. until 2 nd error F.T. if their 2.2 is NOT a whole number AND a has been awarded. For any 1 of these AND an attempt at showing that the 2 triangles are equiangular. Complete explanation (may only involve 2 pairs of angles) Statement of scale factor is insufficient for. Calculation to be carried out needed for 8

11 1. (a) Interquartile range = 1 (2 to 2.) (b) (a) (p + ) (p - ) (b) (q - ) ( q + 2) 19. (a) 10h 20e = h k 10h h = 20e k h = (20e-k)/ = = Strategy, repeated use of Pythag. Thm. Any two correct Hypotenuses Perimeter Pattern = (a) ( =) 200 (b) 0 (seconds) 22. Volume x 2 (x+2) x 2 h Attempt to equate their volumes Πx 2 h = x 2 (x+2) h = x 2 (x+2) / Πx 2 h = ( x + 2 ) / Π 2. (a) E.g, 9, 1, 9/, (b) x=0.11..and 100x=.11.. /10+1/990 / P(no cherry) P(no cherry) =/12 x /11 x /10 (=120/120=1/11) 1200/120 (=120/12 = 10/11) 2. (a) x - y (b)(i) x + 2y (ii) -2x 2y (c) NO, CA multiple of MA, or equivalent B Award, for their graph showing that they are using and 1 AND (have read at least one value correctly or both values with a consistent misread of the scale) AND subtracted their 2 values. Misreads of the scale can be from the 20 or from the 2 etc Allow Unsupported answers in the range get,. C.A.O. for (p ) (p ) for (q ) ( q 2) Penalise further working -1 only until 2nd error Accept 10- as. Penalise further incorrect working -1 E.g. For Patterns 1 & 2 for any correct Hypotenuse ( 2 or or or etc). B marks are implied if awarded, i.e. B for correct answer with no working Allow for one correct area, or for any three correct 200/=0 gets B0 for their x 2 (x+2) and Πx 2 h from their equate isolating h, equivalent difficulty Not 1, ¼,.. Or alternative 10x with 1000x. ( for./99) Ignore incorrect cancel Read as 0.11 M0 A0 Or equivalent complete strategy, idea Seen alone not part of further probabilities. full alternative with correct values.. Ignore incorrect cancelling in final answer Penalise not simplifying only once. OM + x Allow clear with reasoning 9

12 HIGHER TIER - PAPER 2 Paper 2 Two Tier 2008 Higher Tier 1. Line from Blackpool Line from Llandudno Lines intersecting 2. 0/1.88 ( ) 20 2/ 1. ( ) 22 Colin 2.(a) Numbered uniform scale on both axes One correct point plotted Correct line drawn (b) Line y = -2 Point of intersection ( 0., -2 ). Sams 0 + 0x ( ) 8. Sams by ( ) AC 2 = = 000 AC =.(9 ) AC = (m) or. (m). 19, 201, 208 and 21 seen 2 x 19 + x x x21 their total (.18 ). One correct evaluation (1sf) 1<x<2 2 correct evaluations 1.2<x<1. one either side 0 2 correct evaluations 1.2<x<1. one either side 0 Correct evaluation of 1. if previous awarded Correct conclusion 1. If no values shown accept statements such as higher than zero or lower than zero 8. (a) 9 (b) = 2 (c) 2/ x = 1/ tins 9. (a) Vague, what do you mean by often? Use definite figures and/or a time period, such as 0 2,, + times per annum (b) (i) Overlapping ages (ii) What about the under 0s? if at least given 1. euros = $ euro = $ 1.88/1. 2 euros = 1.88x2/1. = $2 Colin $ $1.88 = 1. euros $1 = 1./1.88 euros $0 = 1.x0/1.88 = 0 euros Colin euros Using standard convention if only one scale marked, the other unmarked but apparently the same, or nothing marked using 2cm or 1cm as 1 unit on both their lines for 0x1.1 for the difference between their answer and 08 Give for 9 2 = AC their mid points from within the groups or use of bounds for correct sum of their fx terms for their fx/110 correctly evaluated Unsupported 20(.18.). give all marks. Use of bounds correct answers are 201(.18 ) or 20(.18 ) x x +2x for 88(1). 8 or. only gets. Partition methods should be worked out properly, e.g. 10%=, 1%=, 20%=0, %=1 SC1 for 0.2 or 2/100 Answer of (tins) gets all marks unless from obviously incorrect logic No box for never Along these lines 0 in 2 boxes and 0 in 2 boxes is, not two criticisms 10

13 (1.0) A ISW for compound interest If years used, mark up to years, then MR 1 provided the B or A mark has been awarded. 11. (a) Volume = π 2 2 = 10 (882) 11 cm (b) PR 2 = PR 2 = ( ) = 2 8 PR = 1 9(2) or (a) (i) (ii) (b) 9() 10 8 or x (a) x 2 12 x (b) m t = 2t + m = t + 2t + t + t + 2t + m = (c) (x 1)(x + 1) 1. x 2(x + ) = x = 20 x = 20/ ISW t + 1. (a) H = 1 tanº Height = (8) (m) (b) cos(angle) = 1/21 = 0 19(0) EBC = 1 () Total = 00 (Number in Country / 00) x 1.02, 10.81, 8.89,.2,.9 1, 11, 9,, 1.(a) y = k/x 2 or yα1/x 2 2 = k / 22 y = 0/x 2 (b). U1 M2 11 For a correct %. Alternatively for ( )120 ( )920. For the overall method ( stages of adding different %). The method for finding their three % must be a correct one otherwise it is M0. Arithmetical errors are allowed for the Candidates using depreciation: Allow SC1 for seeing an amount of ( )8.90 Accept ( )92 If 2 years used it is A0 and MR 1 provided the B mark has been awarded. Accept rounds to 10 upto 12 Correct substituted Pythagoras. Ignore incorrect writing C.A.O. Allow PR = 2 8. process if answers F.T. their PR 2 if awarded correct C.A.O. Penalise -1 once only for incorrect C.A.O. notation for or () only if written as a trinomial, e.g. x 2 x 12 (any order) Clearing bracket correctly F.T. until second error For isolating the term in m. F.T. equivalent difficulty, i.e. m = f(t)/a with a 1 for (x 2 1) or (2x-2)(2x+2) If B then penalise 1, once only, if any subsequent incorrect or inappropriate algebra such as using equations For correctly clear fractions by a valid method for any 2 terms. For correctly clear fractions by a valid method for all terms. NOTE: x 8x + 10 = would get,, A0 and the final can be gained for correct follow through (0) The two s are dependent on only one being awarded Not dealing with -2x+ correctly ok for M marks, lose A mark Collecting terms F.T. until 2 nd error starting after the M marks. x + If 0 marks awarded, SC1 for = Unsupported answer of x = gets all marks. Correct substituted tan ratio. Allow for tanº =Ht/1 C.A.O. Correct substituted cos ratio, or cos -1 (1/21) Unsupported 1 2 gets all marks. their total. Or alternative method Any correct non linear start only Maybe implied in (b)

14 18. Lines drawn: x + y = 8, y =x + and x = - Correct region indicated 19. (a) ( x + 1 ) ( 2x - ) x = -1/, x = /2 (b) {-+ ( 2 xx-11)}/2x = {-+ 18}/ 1.1, (a) BD 2 = x8.x12.1xcos80 BD 2 = 18. BD = 1.2.(cm) (b) Area CDB = ½.xBDxsin2 (=18.1 cm 2 ) Area ADB = ½ 8.x12.1xsin80 (=1.8 cm 2 ) Area quad..( cm 2 ) with no other values 22. n ( n + ) + ( n + ) as a numerator AND (n + ) ( n + ) as a denominator multiplying throughout by (n+) and (n+) n ( n + ) + ( n + ) = (n + ) ( n + ) n 2 + n + n + 21 = n 2 + n + n + 12 n = - 9/ 2 Award for any two correct lines, or for any one line drawn correctly for (x 1) ( 2x ) (x ) ( 2x 1) for middle term splitting method their pair of brackets. Ignore incorrect cancelling Use of formula gains no marks For substitution, allow one slip their BD 2 if awarded their BD if one of M marks awarded, and their calculations accurate 0 first answer, in range 0 to 0 Brackets required or implied later Award for either numerator or denominator, or multiplying throughout by (n+) or (n+) with an error for similar level of difficulty, with n 2 terms on both sides GCSE Mathematics - 2 Tier Marking Scheme (Summer 2008) 12

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GCSE MARKING SCHEME MATHEMATICS - TWO TIER LEGACY

GCSE MARKING SCHEME MATHEMATICS - TWO TIER LEGACY NOVEMBER 2011 INTRODUCTION The marking schemes which follow were those used by WJEC for the November 2011 examination in GCSE MATHEMATICS - TWO TIER LEGACY.