Mark scheme for Paper 2 Ma - Emaths

MaKEY STAGE3ALL TIERS2004Mathematics testsMark scheme forPaper 2Tiers 3–5, 4–6, 5–7 and 6–82004

2004 KS3 Mathematics test mark scheme: Paper 2 IntroductionIntroductionThe test papers will be marked by external markers. The markers will followthe mark scheme in this booklet, which is provided here to inform teachers.This booklet contains the mark scheme for paper 2 at all tiers. The paper 1mark scheme is printed in a separate booklet. Questions have been givennames so that each one has a unique identifier irrespective of tier.The structure of the mark schemesThe marking information for questions is set out in the form of tables, whichstart on page 10 of this booklet. The columns on the left-hand side of eachtable provide a quick reference to the tier, question number, question part, andthe total number of marks available for that question part.The Correct response column usually includes two types of information:■■a statement of the requirements for the award of each mark,with an indication of whether credit can be given for correct working,and whether the marks are independent or cumulative;examples of some different types of correct response,including the most common.The Additional guidance column indicates alternative acceptable responses,and provides details of specific types of response that are unacceptable. Otherguidance, such as when ‘follow through’ is allowed, is provided as necessary.Questions with a UAM element are identified in the mark scheme by anencircled U with a number that indicates the significance of using andapplying mathematics in answering the question. The U number can beany whole number from 1 to the number of marks in the question.The 2004 key stage 3 mathematics tests and mark schemes were developedby the Mathematics Test Development Team at QCA.2

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceGeneral guidanceUsing the mark schemesAnswers that are numerically equivalent or algebraically equivalent areacceptable unless the mark scheme states otherwise.In order to ensure consistency of marking, the most frequent proceduralqueries are listed on the following two pages with the prescribed correctaction. This is followed by further guidance, relating to marking ofquestions that involve money, time, coordinates, algebra or probability.Unless otherwise specified in the mark scheme, markers should apply thefollowing guidelines in all cases.3

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceWhat if …The pupil’s responsedoes not matchclosely any of theexamples given.Markers should use their judgement in deciding whether the responsecorresponds with the statement of requirements given in the Correct responsecolumn. Refer also to the Additional guidance.The pupil hasresponded in anon-standard way.Calculations, formulae and written responses do not have to be set out in anyparticular format. Pupils may provide evidence in any form as long as itsmeaning can be understood. Diagrams, symbols or words are acceptable forexplanations or for indicating a response. Any correct method of setting outworking, however idiosyncratic, is acceptable. Provided there is no ambiguity,condone the continental practice of using a comma for a decimal point.The pupil has made aconceptual error.In some questions, a method mark is available provided the pupil has madea computational, rather than conceptual, error. A computational error isa slip such as writing 4 t 6 e 18 in an otherwise correct long multiplication.A conceptual error is a more serious misunderstanding of the relevantmathematics; when such an error is seen no method marks may be awarded.Examples of conceptual errors are: misunderstanding of place value, such asmultiplying by 2 rather than 20 when calculating 35 t 27; subtracting thesmaller value from the larger in calculations such as 45 – 26 to give theanswer 21; incorrect signs when working with negative numbers.The pupil’s accuracyis marginalaccording to theoverlay provided.Overlays can never be 100% accurate. However, provided the answer iswithin, or touches, the boundaries given, the mark(s) should be awarded.The pupil’s answercorrectly followsthrough from earlierincorrect work.Follow through marks may be awarded only when specifically stated in themark scheme, but should not be allowed if the difficulty level of the questionhas been lowered. Either the correct response or an acceptable follow throughresponse should be marked as correct.There appears to be amisreading affectingthe working.This is when the pupil misreads the information given in the question anduses different information. If the original intention or difficulty level of thequestion is not reduced, deduct one mark only. If the original intention ordifficulty level is reduced, do not award any marks for the question part.The correct answer isin the wrong place.Where a pupil has shown understanding of the question, the mark(s) shouldbe given. In particular, where a word or number response is expected, a pupilmay meet the requirement by annotating a graph or labelling a diagramelsewhere in the question.4

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceWhat if …The final answer iswrong but the correctanswer is shown inthe working.Where appropriate, detailed guidance will be givenin the mark scheme and must be adhered to. If noguidance is given, markers will need to examine eachcase to decide whether:the incorrect answer is due to a transcription error;If so, award the mark.in questions not testing accuracy, the correct answerhas been given but then rounded or truncated;If so, award the mark.the pupil has continued to give redundant extraworking which does not contradict work alreadydone;If so, award the mark.the pupil has continued, in the same part of thequestion, to give redundant extra working whichdoes contradict work already done.If so, do not award themark. Where aquestion part carriesmore than one mark,only the final markshould be withheld.The pupil’s answer iscorrect but the wrongworking is seen.A correct response should always be marked as correct unless the mark schemestates otherwise.The correct responsehas been crossedor rubbed outand not replaced.Mark, according to the mark scheme, any legible crossed or rubbed out workthat has not been replaced.More than oneanswer is given.If all answers given are correct or a range of answers is given, all of which arecorrect, the mark should be awarded unless prohibited by the mark scheme.If both correct and incorrect responses are given, no mark should be awarded.The answer is correctbut, in a later partof the question,the pupil hascontradicted thisresponse.A mark given for one part should not be disallowed for working or answersgiven in a different part, unless the mark scheme specifically states otherwise.5

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceMarking specific types of questionResponses involving moneyFor example: £3.20 £7Accept ✓✓ Any unambiguous indication of thecorrect amounteg £3.20(p), £3 20, £3,20,3 pounds 20, £3-20,£3 20 pence, £3:20,£7.00✓ The £ sign is usually already printedin the answer space. Where the pupilwrites an answer other than in theanswer space, or crosses out the £sign, accept an answer with correctunits in pounds and/or penceeg 320p,700pDo not accept Incorrect or ambiguous use of poundsor penceeg £320, £320p or £700p,or 3.20 or 3.20p not inthe answer space. Incorrect placement of decimalpoints, spaces, etc or incorrect use oromission of 0eg £3.2, £3 200, £32 0,£3-2-0,£7.0Responses involving timeA time interval For example: 2 hours 30 minsAccept ✓ Take care ! Do not accept ✓ Any unambiguous indicationeg 2.5 (hours), 2h 30✓ Digital electronic timeie 2:30 Incorrect or ambiguous time intervaleg 2.3(h), 2.30, 2-30, 2h 3,2.30min! The time unit, hours or minutes, isusually printed in the answer space.Where the pupil writes an answerother than in the answer space, orcrosses out the given unit, accept ananswer with correct units in hours orminutes, unless the question hasasked for a specific unit to be used.A specific time For example: 8.40am, 17:20Accept ✓✓ Any unambiguous, correct indicationeg 08.40, 8.40, 8:40, 0840, 8 40,8-40, twenty to nine,8,40✓ Unambiguous change to 12 or 24 hourclockeg 17:20 as 5:20pm, 17:20pmDo not accept Incorrect timeeg 8.4am, 8.40pm Incorrect placement of separators,spaces, etc or incorrect use oromission of 0eg 840, 8:4:0, 084, 846

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceResponses involving coordinatesFor example: ( 5, 7 )Accept ✓✓ Unambiguous but unconventionalnotationeg ( 05, 07 )( five, seven )x y(5,7)( xe5, ye7 )Do not accept Incorrect or ambiguousnotationeg (7,5)(5x, 7y )( x5, y7 )(5 x ,7 y )Responses involving the use of algebraFor example: 2 p n n p 2 2nAccept ✓ Take care ! Do not accept ✓ The unambiguous use of a differentcaseeg N used for n✓ Unconventional notation formultiplicationeg n t 2 or 2 t n or n2or n p n for 2nn t n for n 2✓ Multiplication by 1 or 0eg 2 p 1n for 2 p n2 p 0n for 2✓ Words used to precede or followequations or expressionseg t e n p 2 tiles ortiles e t e n p 2for t e n p 2✓ Unambiguous letters used to indicateexpressionseg t e n p 2 for n p 2✓ Embedded values given when solvingequationseg 3 t 10 p 2 e 32for 3x p 2 e 32! Words or units used within equationsor expressions should be ignored ifaccompanied by an acceptableresponse, but should not be acceptedon their owneg do not acceptn tiles p 2n cm p 2 Change of variableeg x used for n Ambiguous letters used to indicateexpressionseg n e n p 2However, to avoid penalising any ofthe three types of error above morethan once within each question, donot award the mark for the firstoccurrence of each type within eachquestion. Where a question partcarries more than one mark, only thefinal mark should be withheld. Embedded values that are thencontradictedeg for 3x p 2 e 32,3 t 10 p 2 e 32, x e 57

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceResponses involving probabilityA numerical probability should be expressed as a decimal, fraction orpercentage only.For example: 0.7Accept ✓ Take care ! Do not accept ✓ A correct probability that is correctlyexpressed as a decimal, fraction orpercentage.✓ Equivalent decimals, fractions orpercentages70 35eg 0.700, , , 70.0%100 50✓ A probability correctly expressed inone acceptable form which is thenincorrectly converted, but is still lessthan 1 and greater than 070eg e 18100 25The following four categories of errorshould be ignored if accompanied byan acceptable response, but shouldnot be accepted on their own.! A probability that is incorrectlyexpressedeg 7 in 10,7 out of 10,7 from 10! A probability expressed as apercentage without a percentagesign.! A fraction with other than integers inthe numerator and/or denominator.However, each of the three types oferror above should not be penalisedmore than once within each question.Do not award the mark for the firstoccurrence of each type of errorunaccompanied by an acceptableresponse. Where a question partcarries more than one mark, only thefinal mark should be withheld.! A probability expressed as a ratioeg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 orless than 08

2004 KS3 Mathematics test mark scheme: Paper 2 General guidanceRecording marks awarded on the test paperAll questions, even those not attempted by the pupil, will be marked, with a1 or a 0 entered in each marking space. Where 2m can be split into 1m gainedand 1m lost, with no explicit order, then this will be recorded by the marker as 10The total marks awarded for a double page will be written in the box at thebottom of the right-hand page, and the total number of marks obtained on thepaper will be recorded on the front of the test paper.A total of 120 marks is available in tiers 3–5 and 6–8.A total of 121 marks is available in tiers 4–6 and 5–7.Awarding levelsThe sum of the marks gained on paper 1, paper 2 and the mental mathematicspaper determines the level awarded. Level threshold tables, which show themark ranges for the award of different levels, will be available on the QCAwebsite www.qca.org.uk from Monday, 21 June 2004. QCA will also senda copy to each school in July.Schools will be notified of pupils’ results by means of a marksheet, which willbe returned to schools by the external marking agency with the pupils’ markedscripts. The marksheet will include pupils’ scores on the test papers and thelevels awarded.9

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 onlyTier & Question3-5 4-6 5-7 6-81Correct responseAdditional guidanceSportsa 1m Shows a correct amount, with unitseg■ £181.99b 1m Shows a correct amount, with unitseg■ £8.02! Value roundedIn part (a), accept £182 but do not accept£181 unless a correct value is also seenIn part (b), do not accept £8 unless a correctvalue is also seen! Units omittedPenalise only the first such occurrencec 1m 3! Reference to money left overAccept the correct change showneg◆ 3 r (£)5.03Do not accept reference to part of a racketeg◆ 3.3(...)Tier & Question3-5 4-6 5-7 6-82a 1m 24Correct responseTravelling by trainAdditional guidanceb 1m Completes the bar for girls correctly and in thecorrect position, ie! Bar not shaded or lines not ruledor accurateAccept provided the pupil’s intention is clearand the top of the bar is not more than 1mmfrom the line indicating 14c 2m Gives all four correct entries, ie✓ For 2m, zero omitted0 184 4or1mU2Gives at least two correct entries10

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 onlyTier & Question3-5 4-6 5-7 6-83Correct responseAdditional guidanceMazea 1m Identifies the correct square, ie✓ Unambiguous indicationeg◆ Correct square marked Ab 1m Indicates the correct set of instructions, ie6, south3, eastc 2m Indicates the correct set of instructions, ieor1m3, west2, northThe only error is to order the instructionsincorrectly, ie! For part (b), 6 south and 2 east givenCondone✓ Unambiguous indicationeg, for part (b)◆ 6.S3.E◆ s, 6e, 3 Directions other than compass points usedeg, for part (b)◆ 6 down3 right2, north3, westorOne instruction is completely correct andcorrectly ordered, even if the other instruction isincorrect or omittedorBoth compass directions are correct andcorrectly orderedeg■ 2 (error), W3 (error), N11

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 onlyTier & Question3-5 4-6 5-7 6-84Correct responseAdditional guidanceABC1m 341m 81m 4Tier & Question3-5 4-6 5-7 6-85Correct responseAdditional guidanceWindmillsa 1m Completes the windmill pattern correctly, ie! Squares not shadedAccept provided the pupil’s intention is clearb 1m Completes the windmill pattern correctly, ie12

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 onlyTier & Question3-5 4-6 5-7 6-86Correct responseAdditional guidanceOdd v evena 1m Gives a correct counter exampleThe most common correct counter examples:! Other trials shownIgnore if at least one correct counter exampleis shownU1Show an even number multiplied by threeeg■ 2 t 3 = 6 which is even■ 3 t 10 = 30Give an even number that is shown to be amultiple of 3eg■ 18 d 3 = 6■ 30 is in the 3 times table■ 3 goes into 12b 1m Gives a correct counter exampleU1The most common correct counter examples:Show a multiple of four divided by twoeg■ 8 d 2 = 4 which is even1■ of 12 is 62■ 16 → 8Give an even number that is multiplied by twoto give another even numbereg■ 2 t 10 = 20! Calculation not processedAccept if a correct comment is giveneg, for part (a)◆ 6 t 3 isn’t odd◆ 3 t 10 is even◆ Even t 3 is evenOtherwise, do not accepteg, for part (a)◆ 6 t 3◆ Even t 3! Examples use addition or subtraction ratherthan multiplication or divisionFor part (a), accept answers of the formn p n p n where n is even, orrepeated addition of 3 where the number of3s is eveneg, accept◆ 2 p 2 p 2 = 6◆ 3 p 3 = 6For part (b), accept answers of the form2n m n = n where n is even, orn p n = 2n where n is eveneg, accept◆ 4 m 2 = 2◆ 12 p 12 = 24! Correct counter example accompanied byan incorrect statementIgnore incorrect statementseg, for part (a) accept◆ 2 t 3 = 6, 6 isn’t odd but most of thetime the answer will be odd Incorrect notationeg, for part (a)◆ 3 d 18 = 6◆ 10 = 3013

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 onlyTier & Question3-5 4-6 5-7 6-87Correct responseTriangular tilesAdditional guidancea 1m Shows how eight tiles join to make a squareeg■! Lines not ruled or accurateAccept provided the pupil’s intention is clear! Internal lines not shownDiagonal lines must be shown but pupilsmay use the given grid lines to representhorizontal or vertical lines Internal lines incorrect■! In both parts (a) and (b), tiles make aninternal square even if there is no shadingeg◆b 1m Shows how four tiles join to make a square, ieMark as 0, 1! In both parts (a) and (b), two tiles taken tobe one larger tileeg◆Mark as 0, 114

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6Tier & Question3-5 4-6 5-7 6-88 1Correct responseRecycling rubbishAdditional guidancea a 1m Gives a value between 6 and 16 inclusive✓ Value qualifiedeg◆ About 10b b 1m Indicates only Germany and Norway✓ Unambiguous indicationeg◆ N, GTier & Question3-5 4-6 5-7 6-89 2a a 1m 18Correct responseShaded shapeAdditional guidanceb b 1m Draws a rectangle of area 18cm 2eg■ 3 by 6 rectangle■ 2 by 9 rectangle■ 4 by 4.5 rectangle✓ Follow through from part (a)! Lines not ruled or accurateAccept provided the pupil’s intention is clear15

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6Tier & Question3-5 4-6 5-7 6-810 3a a 1m 6Correct responseAdditional guidanceMaking 271m 11b b 1m Gives a correct explanationThe most common correct explanations:Refer to the fact that an even number of 5pcoins gives an even total, and that addition of2p coins will keep the total eveneg■ An even number of 5p coins gives anamount that is even, leaving an odd amountto make up 27p. You can’t make an oddnumber with 2p coins■ An even number of 5s is even, adding 2skeeps it even, but 27 is odd■ An even number of 5s always ends in zero,leaving you to make an odd number with 2swhich is not possible✓ Minimally acceptable explanationeg◆ An even number of 5s leaves an oddnumber and you can’t make an oddnumber from 2s◆ 27 is odd, so you have to have an oddnumber of 5ps or the 2s would make iteven Explanation refers only to 5s, or only to 2seg◆ An even number of 5s is even but27 is odd◆ An even number of 5s always ends inzero◆ You can’t make an odd number with 2s Justification not giveneg◆ You can only make even totals◆ You can only do it using an odd numberof 5s◆ Can’t both be even◆ 27 is an odd numberProduce a set of possible solutionseg■ 0 t 5p = 0p leaving 27p, impossible2 t 5p = 10p leaving 17p, impossible4 t 5p = 20p leaving 7p, impossible6 t 5p = 30p, which is too big■ You can’t make 27, 17 or 7 using 2s! Only one case consideredAs this is a level 4 mark, condoneeg, accept◆ 2 t 5p = 10p leaving 17p, not possible◆ 4 t 5p = 20p leaving 7p, can’t◆◆You can’t make 7 using 2sTwo 5s make 10 and eight 2s that is asclose as I can get◆ Add 2ps to 10, you get 12, 14, 16, 18,20, 22, 24, 26, 28 .....U1 Justification not giveneg◆ 26 is as close as I can get◆ You can make 26 or 2816

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6Tier & Question3-5 4-6 5-7 6-811 4Correct responsePatterns on a gridAdditional guidancea a 1m Gives the correct coordinates, ie (2, 1)b b 1m Gives both pairs of coordinates in either ordereg■ (3, 3) (4, 4)c c 1m Gives both pairs of coordinates in either ordereg■ (16, 16) (17, 17)d d 2m Makes a correct decision and gives a correctexplanation that shows or implies 14 andjustifies that 16 more are neededeg■ Yes, 1 2 p 2 2 p 3 2 p 4 2 = 30■ There are enough because 1 p 4 p 9 = 14,4 t 4 = 16 and 14 p 16 = 30■ The next square is 16 tiles (4 by 4 squaredrawn) and you’ve used up 14 of them, sothere’s just enough■ You have 16 tiles left and 4 t 4 = 16; all thetiles are used! 16 not justifiedAccept only if the response makes it clearthat exactly 30 tiles are usedeg, for 2m accept◆ Used 14, got another 16 so you will useup all the 30 tiles◆ 30 m 14 = 16, so yes you have exactlythe correct amounteg, for 2m or 1m, do not accept◆ 14 used, 16 left so yes you can◆ 30 m 14 = 16, so yes you have enoughor1mU1States or implies that the next square uses16 tileseg■ You need 16 to make the next square■ Draws a 4 by 4 square with 16 cells■ 4 t 4 seenorStates or implies that exactly 30 tiles will beused, but does not justify that 16 more areneededeg■ You need all 30■ There would be no tiles left over■ It all adds up to 30orIdentifies the pattern of differenceseg■ p3, p5, p7! 4 by 4 square drawn correctly, but thenumber of squares incorrectly processedFor 1m, condone Their explanation could imply that 7 moresquares are needed, ie a total of 21eg◆so yes, thereare enough17

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6Tier & Question3-5 4-6 5-7 6-812 5Correct responseCaribbean cordialAdditional guidance1a a 1m or equivalent21m341m 450or equivalent! Change of unitsAccept provided the new units are clearlyshowneg, for the second mark accept◆ 750ml◆ 75cl! Incorrect units inserted in an otherwisecorrect responseeg, for the first mark◆ 0.5gPenalise only the first such occurrenceb b 1m 20018

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-813 6 1Correct responseShape rotationAdditional guidancea a 1m Indicates the correct four faceseg■✓ Unambiguous indicationeg◆ Grey faces labelled Gb b 2m Draws a correct view of the cuboid in either ofthe orientations below, using the isometric grid✓ Incorrect or no shading✓ For 2m, internal lines omittedeg◆or1mThe only error is to draw the cuboid in thewrong orientationeg■! Lines not ruled or accurateAccept provided the pupil’s intention is clear! Cuboid enlargedFor 2m or 1m, accept provided a consistentscale factor has been used for all lengths Shape is not a cuboidorThe only error is to omit some external lines orto show some hidden lineseg■■19

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6Tier & Question3-5 4-6 5-7 6-814 7a a 1m 105Correct responseAdditional guidanceMultiples1m 108b b 1m Indicates Yes and gives a correct explanationinterpreting the word factoreg■ 140 will divide by 7 with no remainder■ 140 is a multiple of 7■ 140 is in the 7 times table■ 7 goes into 140 exactly■ 7 t 20 = 140✓ Minimally acceptable explanationeg◆ 140 will divide by 7◆ 7 goes into 140◆ 70 t 2 = 140! Explanation refers to 14 rather than 140Accept provided the relationship between7 and 14 is shown or impliedeg, accept◆ 7 goes into 14◆ 7 t 2 = 14◆ 7 times table goes 7, 14 and so onOtherwise do not accepteg◆ 14 goes into 140! Use of repeated additionCondoneeg, accept◆ Keep going up in 7s and you get to 140! Use of ‘it’ or other ambiguous languageCondone provided either 7 or 140 is used,implying ‘it’ is the other numbereg, accept◆ 7 goes into it◆ 140 divides by itOtherwise do not accepteg◆ It goes into it◆ You can divide them! Response contains an incorrect statementCondone only if accompanying a correctresponseeg, accept◆Yes, 7 divides into 140 as it is a multipleof 140eg, do not accept◆ 7 d 140 = 20◆ 7 is a multiple of 140◆ 140 will go into 7◆ 7 goes into 140 thirty times20

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-815 8 2Correct responseAdditional guidanceNepala a a 1m 8b b b 2m Draws a bar from m3 to 12, aligned with 5000on the y-axis, and of the correct thickness! Lines not ruled or accurateAccept provided the pupil’s intention is clearor1m Indicates that the maximum temperature is 12eg■ m3 p 15 = 12 seen■ Draws a bar with a right-hand end at 12! For 1m, bar incorrectly aligned withthe 5000, or bar of incorrect thicknessCondoneorIndicates on the graph the correct positioningfor m3or1Draws a bar that is 15 units, ie 7in length2squares,21

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-816 9 3Correct responseAdditional guidanceAnglesa a a 1m Indicates No and gives a correct explanationthat shows the angle sum is incorrecteg■ 30 p 60 p 100 = 190 but it shouldsum to 180■ They should add to 180 but these addto 190■ 30 p 60 p 100 is 10 degrees too big✓ Minimally acceptable explanationAccept responses that state the anglesshould not add to 190, or that theangles should add to 180eg◆ They add to 190 which is wrong◆ Angles in a triangle add up to 180◆ The angles don’t make 180◆ They should add to 180 Incomplete or incorrect explanationeg◆ The angles add to 190◆ When you add up the angles you getthe wrong angle sum◆ Angles add to 200 (error) not 180U1! Incorrect unitsIgnoreeg, accept within a correct explanation◆ 180ºCb b b 2m 130or1mShows or implies a correct method with notmore than one computational erroreg■ 360 m (70 p 70 p 90)■ 360 m 230■ 2 t 70 p 90 = 200 (error), 360 m 200 = 160■ 70 p 70 = 140, 140 p 90 = 330 (error),answer 30■ 180 m 5022

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-817 10 6Correct responseAdditional guidanceRight anglesa a a 1m Draws any quadrilateral with exactly tworight angleseg! Lines not ruled or accurateAccept provided the pupil’s intention is clear■b b b 1m Draws any quadrilateral with exactly oneright angleeg■23

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-818 11 4Correct responseAdditional guidancePrime grida a a 1m Gives a correct explanationThe most common correct explanations:State that 35 is a multiple of 5 and/or 7eg■ 35 is a multiple of 5■ 7 is a factor of 35State that prime numbers have only two factorsbut that 35 has more than two factorseg■ A prime has 2 factors, 35 has 4✓ Minimally acceptable explanationeg◆ 5 goes into it◆ It’s in the 7 times table◆ 7 t 5◆ 1, 5, 7, 35◆ It has more than two factors◆ 35 divides by more than one and itself Incomplete explanationeg◆ 35 is in some of the times tables◆ 35 has factors◆ Because it ends in 5U1State that the last digit of any prime numbergreater than 5 is 1, 3, 7 or 9eg■ All prime numbers must end in 1, 3, 7or 9 with the exception of 2 and 5! Correct explanation accompanied by astatement that uses mathematical languageincorrectlyThroughout the question, condoneeg, for part (a) accept◆ 35 has more than 2 factors, eg 35 goesinto 5◆ 5 goes into 35, so it has 2 factorsb b b 1m Gives a correct explanationThe most common correct explanations:State or imply the numbers in column Y will allbe multiples of 6 (or 2, or 3)eg■ They are all in the 6 times table, so theymust be multiples of 6■ They are all multiples of 3State or imply the numbers in column Y will allhave a factor of 6 (or 2, or 3)eg■ They all have a factor of 3■ 2 is the only prime that is even and all thesenumbers are even and greater than 2✓ Minimally acceptable explanationeg◆ It’s the 6 times table◆ You can divide them by 3◆ They are all even◆ The only even prime is 2◆ None of the numbers ends in 1, 3, 7 or 9✓ That column Y starts at 6 is not explicitlystatedCondoneeg, accept◆ They are all even and even numbers arenever prime Incomplete explanationeg◆ They are all in times tables◆ They all divide by something other thanone and itself◆ 6 d 3 = 2◆ It goes up 6 each timeU1! Misunderstanding of primeA common misconception is to confuseprime with odd. Hence do not acceptstatements that refer only to oddeg, do not accept◆ The numbers are not odd24

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-818 11 4Correct responsePrime grid (cont)Additional guidancec c c 1m Gives a correct explanationU1The most common correct explanations:State or imply the numbers in column X will allbe multiples of 3eg■ They are all in the 3 times table, so theymust be multiples of 3State or imply the numbers in column X will allhave a factor of 3eg■ They are all in the 3 times table, so they areall divisible by 3✓ Minimally acceptable explanationeg◆ They are all in the 3 times table◆ 3 goes into them Incomplete explanationeg◆ They are all in times tables◆ They will all divide by something otherthan one and itself◆ All the other numbers have factors◆ It goes up 3 each time! Misunderstanding of primeA common misconception is to confuseprime with odd. Hence do not acceptstatements that refer only to oddeg, do not accept◆ The numbers are not oddTier & Question3-5 4-6 5-7 6-819 12 5Correct responseAdditional guidanceCrisps1m 40! Incorrect units givenIgnore25

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7Tier & Question3-5 4-6 5-7 6-820 13 7Correct responseAdditional guidanceShoe sizes3mIndicates Yes and gives a correct explanationthat shows or implies both of the values 40.75and 41.375eg■ 7 t 1.25 p 32 e 40.75,7.5 t 1.25 p 32 e 41.375,so they both round to 41■ 8.75 p 32 rounds to 41 and so does9.375 p 32■ 8.75 gives 9 and 9.375 gives 9 beforeadding 32, so they will end up the same✓Minimally acceptable explanationeg, with Yes indicated◆ They are both 41◆ They are 40.75 and 41.375! 40.75 rounded or truncatedAccept 41, 40.8 or 40.7Do not accept 40! 41.375 rounded or truncatedAccept 41, 41.4, 41.3, 41.38 or 41.37Do not accept 42or2mShows or implies both of the values 40.75 and41.375 even if there is an incorrect or nodecision, or incorrect further workingeg■ Tom wears 40.8 and Karl wears 41.4 sothey don’t wear the same size■ 40.75 and 41.375 so they both wear 40! 40.75 from incorrect workingNote that pupils who add 1.25 ratherthan multiplying generate the shoe sizes40.25 and 40.75For 3m or 2m, do not accept explanationsbased on such misconceptionseg◆ They are both 41 as7.5 p 1.25 p 32 e 417 p 1.25 p 32 e 41or1m Shows the value 41.375orShows the value 40.75 or 41with correct workingeg■ 7.5 t 1.25 p 32 e 41orThe only error is to add 1.25 rather thanmultiplyingeg■ Indicates No and shows the values 40.75and 40.25■ Indicates No and shows the values 41and 4026

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-821 14 8 1a a 1m 8Correct responseAdditional guidanceSame areab b 2m 3, with no evidence of an incorrect methodor1m Shows the value 12orForms a correct equation in weg1■ 4w e (6 t 4)2■ 4 t w e 3 t 4orShows a correct method with not more thanone computational erroreg■ 6 t 4 d 2 d 43 t 4■4■ 6 t 4 d 2 e 20 (error), 20 d 4 e 5■ 6 d 2 Conceptual erroreg◆ 6 t 4 e 24, 24 d 4 e 627

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-822 15 9 2a a 2m £ 556.75Correct responseAdditional guidanceHoliday! Value roundedAccept 557 or 560For 2m, do not accept 556 unless a correctmethod or a more accurate value is seenor1mShows or implies a complete correct method,even if there are rounding errorseg17■ t 3275100■ 3275 d 100 t 17■ 556■ 10% e 327.5(0)5% e 163.751% e 32.75327.5(0) p 163.75 p 2 t 32.75■ 1% e 32.75,33 (premature rounding) t 17 e 561orShows the digits 55675b b 2m 7.5(...)! Value roundedFor 2m, do not accept 7 or 8 unless a correctmethod or a more accurate value is seenor1mShows or implies a complete correct methodeg1644■ t 10021842■ Shows the digits 75(...)■ 7orGives a value between 7 and 8 inclusive28

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-816 10 3Correct responseStraight linesAdditional guidancea a a 1m Completes the table with any three sets ofcorrect coordinates, indicating for each thatx p y e 4eg■(x, y) (0, 4) (1, 3) (2, 2)x p y 4 4 4✓ Incomplete processingeg, for (1, 3)◆ 1 p 3! Values for (x, y) correct but some or all ofvalues for x p y omittedAccept provided a correct equation is givenin part (b)b b b 1m Gives a correct equationeg■ x p y e 4■ y e 4 m x■ x emy p 4c c c 1m Draws the correct straight line through(0, 6) and (6, 0)! Line not ruled or accurateAccept provided the pupil’s intention is clear! Partial line drawnDo not accept lines that are less than 5cmin length! Points plottedIgnore Points not joined29

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-817 11 4Correct responseAdditional guidanceQuiza a a 1m Gives both correct values, iemaximum of 40 and minimum of m20 Incorrect notationeg◆ 20mb b b 1m 14c c c 2m Completes both rows correctly, in either ordereg■13 2 514 4 2or1mCompletes one row correctlyTier & Question3-5 4-6 5-7 6-818 12 5Correct responseAdditional guidanceCotton reela a a 1m 3π or 9.4 or 9.42(...) or 9.43with no evidence of an incorrect method! Answer of 9Accept provided a correct method or a moreaccurate value is seenb b b 2m 970! Follow through from part (a)For 2m, accept 9100 d their (a), roundedcorrectly to the nearest ten, provided9100 d their (a) is not a multiple of 10eg, from their (a) as 7.8, accept for 2m◆ 1170eg, from their (a) as 7, do not accept for 2m◆ 1300or1mShows or implies that the total length shouldbe divided by the circumference, even if theunits are incorrect or there are rounding ortruncation errorseg■ 9100 d 9.42■ 91 d 3π■ Digits 96(...) or 97(...) seen✓ For 1m, follow through from part (a), evenif their (a) is rounded or truncated beforebeing usedeg, from their (a) as 7.8, accept◆ 9100 d 830

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-820 13 6Correct responseAdditional guidanceMedicinea a 2m Indicates a correct value, with appropriateunits, with a correct method showneg■ 80 d 16, 5ml■20 t 412 p 4, 0.005 litres For 2m, incorrect or incomplete methodeg◆ 20 d 4 e 5mlor1mThe only error is to omit units or to giveincorrect unitsorUnits of ml are given and the method shows orimplies correct substitution and understandingof algebraic notation for both multiplicationand divisioneg■ 20 t 4 d 16, answer 50ml■ 20 t 4 = 100 (error), 12 p 4 e 16100 d 16 e 6.25ml20 t 4 8■ e (error in numerator) = 0.5ml12 p 4 16■ Answer of 10.6(...)ml or 10.7ml or 11ml(only error is to omit necessary bracketswhen processing)! Units other than ml are givenAccept provided the pupil shows such achange is intended and the change has beencarried out correctlyeg, accept◆ 20 t 4 d 16 e 50, answer 0.05 litresorAn answer of 5ml, or equivalent, is given withno workingb b 2m 12 (years)or1m Shows a correct equation with the values 15and 30 correctly substitutedeg30y■ 15 =12 p y■ 15(12 p y) e 30 t yor■ 1 =2y12 p yShows the correct answer of 12 embedded, evenif an incorrect value is chosen subsequently asthe answereg30 t 12■ 15 = , answer 1512 p 12! Use of ? or other symbol for yAccept if consistenteg, for 1m accept◆ 15 =30 t ?12 p ?! Units given within an equationCondoneeg, for 1m accept30ml t y◆ 15ml =12 p y31

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8Tier & Question3-5 4-6 5-7 6-819 14 7Correct responseAdditional guidanceRecyclinga a a 2m 8or1mShows a correct angle for one or more pupils,but not 5 pupilseg■ 60 d 5 e 12° for each one■ 3 pupils is 36orShows a correct method with not more thanone computational erroreg■ 96 d (60 d 5)■ 96 d 60 e 1.6, 5 t 1.6■ One pupil is 13 (error), and96 d 13 e 7.38 so 7 pupils■ Total pupils e 5 t 6 e 30,96t 30360■5e 0.083, 96 t 0.08360b b b 2m 135or1mShows a correct angle for one or more pupils,but not 24 pupilseg■ 24 is 360°, 1 is 15°■ 3 pupils is 45orShows a correct method with not more thanone computational erroreg■ 9 d 24 t 36024■ 360 d9■ 360 d 24 e 16 (error), 16 t 9 e 144or9Shows as a correct percentage24eg■ 37.5%! 37.5 rounded or truncated to an integerDo not accept unless a more accurate valueis seen 37.5 without the percentage sign32

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-815 8Correct responseRussian dollsAdditional guidance1a a 1m Indicates both 6 and 10 , in the correct order2✓ Equivalent fractions or decimals! 10.5 rounded or truncated to an integerDo not accept unless a correct method or amore accurate value is seenb b 2m Indicates both 5.1 and 7.7, in the correct orderor1mIndicates one correct value, even if not roundedeg, for the smallest doll36■7■ 5.1(...)eg, for the middle doll54■7■ 7.7(...)orShows or implies a correct method for bothdolls, even if there is evidence of prematureroundingeg■ 9 d 7 t 4, 9 d 7 t 69■ = 1.3 (rounded),71.3 t 4 e 5.2, 1.3 t 6 e 7.8! 5.1(...) or 7.7(...) rounded or truncated toan integerDo not accept unless a correct method or amore accurate value is seen! Answers are 5 and 8, or round to 5 and 8For 1m to be awarded, 9 d 7 or 1.3or 1.28(...) must be seen33

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-816 9Correct responseAdditional guidanceSweets2m42, with sufficient working to support acorrect method! Method is trial and improvementAccept for 2m, but not for 1m Incorrect methodeg◆ (39 p 40 p 41 p 42 p 43 p 44) d 6 e 42or1mGives the answer 42 with no evidence of anincorrect methodorShows the value 368orShows the value 410orShows a complete correct method with notmore than one computational erroreg■ (10 t 41) m (3 t 39 p 2 t 40 p 41 p 42p 2 t 44)■ 117 p 80 p 41 p 42 p 84 (error) e 364410 m 364 e 46■ 41 m (m2 t 3 pm1 t 2 p 1 p 3 t 2)■ m6 pm2 p 1 p 4 (error) em3so there are 44orShows the overall difference of the valuesgiven from the mean is m1eg■ 3(m2) p 2(m1) p 0 p 1 p 2(3) = m1■ m6 pm2 p 1 p 6 em134

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-817 10Marking overlay availableCorrect responsePentagonal pyramidAdditional guidancea a 1m Gives a correct explanationThe most common correct explanations:Show or state that the angles in a pentagonsum to 540, and that angle a is 540 d 5eg■ The interior angle of a regular pentagon is108, because 5 m 2 e 3, 3 t 180 e 540 and540 d 5Show or state that the exterior angle of aregular pentagon is 72, and that angle a is180 m 72eg■ 360 d 5 e 72, 180 m 72Show or state that the angle at the centre of aregular pentagon is 72, and that angle a is180 m 72eg■ 360 d 5 e 72, (180 m 72) d 2 e 54, 54 t 2✓ Minimally acceptable explanationeg◆ 540 d 5◆ 180 m 72 (with the exterior angle of 72marked correctly on the diagram)◆The interior angle of a regular pentagonis 108◆ 180 m 72 (with the centre angle of 72marked correctly on the diagram) Incomplete explanationeg◆ The angles in a pentagon sum to 540◆ 108 t 5 e 540 (with no justification orindication of the relevance of the 540)◆ 180 m 72 e 108 (with no justification ofthe 72)◆ The angle of a regular pentagon is 108◆ Angle of 108 marked on the diagramb b 1m Indicates 36 and shows a correct methodeg, using a large triangle■ (180 m 108) d 2eg, using a small triangle■ 180 m 2 t 72eg, using a kite■ 360 m (3 t 108)c c 2m Completes the perpendicular bisector, fulfillingfour conditions below:1. Ruled2. Within the tolerance as shown on theoverlay, including if their line were to beextended3. At least 3cm in length4. Evidence of correct construction arcsthat are centred on C and D, or the verticesnext to C and D, are of equal radii, andshow at least one intersection✓ Minimally acceptable methodeg◆ 72 d 2 e 36 Spurious methodeg◆ 180 d 5 e 36! Use of construction arcs on the overlayNote that these are to give a visual guide asto whether the correct centres have beenused, and do not indicate tolerance✓ Side other than CD used Spurious construction arcsDo not accept arcs drawn without compassesor arcs that do not show a distinctintersection, eg arcs that just touchor1mCompletes the perpendicular bisector with allof conditions 1 to 3 fulfilledorFulfils condition 4, even if the perpendicularbisector is incorrect or omitted35

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-818 11a a 1m 6Correct responseRunning machineAdditional guidanceb b 1m 20c c 1m 3d 2m Draws a straight line on the graph joining thepoints (0935, 0) and (0959, 4)! Line not ruledAccept provided the pupil’s intention is clearor1mShows or implies the distance travelled is 4kmeg10■ t 24 e 460■ Their end point is on the line y = 4orThe only error is to start at an incorrect timeorShows a correct method for calculating thedistance travelled, with not more than onecomputational error, then follows throughcorrectly to draw their lineeg■ 10 d 60 t 24 e 2.7 (error), then their linedrawn from (0935, 0) to (0959, 2.7)! Line continued beyond (0959, 4)Accept a horizontal line, but for 2m do notaccept the correct line continued! Their line is slightly inaccurateIf their line starts at (0935, 0) and passesthrough (0941, 1) but continues to anincorrect value at 0959, then stops, orcontinues horizontally, mark as 1, 036

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-819 12Correct responseAdditional guidanceSquaresa 2m Indicates only the values 0 and 1or1mIndicates one of the values 0 or 1, withno incorrect valuesorIndicates both correct values withnot more than one incorrect value! Use of infinityIgnoreeg, for 2m accept◆ 1, 0, infinity! Answer(s) embedded in workingAccept provided there is no ambiguity andany statements made are correcteg, for 2m accept2◆ 1 e 1, 0 2 e 0◆ 1, 12, 0, 0 2◆ 12, 0 2b 2m Indicates values between 0 and 1 not includingthe values 0 and 1eg■ Numbers greater than nought butless than one■ 0 < x < 1✓ Minimally acceptable indicationeg◆ Between zero and one◆ Numbers that begin 0.something◆ Fractions that are positive and notimproperor1mIndicates values between 0 and 1 includingeither 0 or 1 or bothorIndicates the correct upper limit, but withoutincluding 1eg■ Numbers less than 1■ All fractions that are not improperorGives at least one correct example of a numberthat is a member of this set and its square, withno incorrect exampleseg■ 0.5 2 = 0.251 1■

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-820 13Correct responseTriangle calculationsAdditional guidance2mIndicates No and gives a correct justificationThe most common correct justifications:Markers may find the following helpful:11.6 2(134.56)Use Pythagoras’ theorem to show the sides areinconsistenteg■ 11.6 2 p 8.7 2 ≠ 15.3 2■ 134.56 p 75.69 e 210.25,but 15.3 2 e 234.0915.3 2(234.09)8.7 2(75.69)Calculate what one side should be in order tomake the triangle consistenteg■ The hypotenuse should be 14.5■ 8.7 should be 9.9764...■ 11.6 should be 12.5857...Use trigonometry to calculate two angles,which are then shown not to sum to 90eg, using cosine■ The angles are 55.3454... and 40.6968...55.3 p 40.7 ≠ 90eg, using sine■ The angles are 49.3031... and 34.6545...34.6 should be 40.7! Values rounded or truncatedAccept values rounded or truncated to 1 ormore decimal place(s). Otherwise, acceptprovided correct working or a more accuratevalue is seen For 2m or 1m, no indication of howvalues combineeg2◆ 11.6 e 134.568.7 2 e 75.6915.3 2 e 234.09 Justification is from construction ratherthan calculationor1mShows sufficient working to indicate correctapplication of Pythagoras’ theoremeg■ 11.6 2 p 8.7 2■ 210.25■ 15.3 2 m 11.6 2orShows sufficient working to indicate a correcttrigonometric ratioeg8.7■ sin e with the position of the relevant15.3angle indicated on the diagram No indication of which angle is beingconsidered38

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8Tier & Question3-5 4-6 5-7 6-820 13Correct responseTriangle calculations (cont)Additional guidance2mIndicates No and gives a correct justificationThe most common correct justifications:Use trigonometry to show the sides areinconsistenteg, using sin 50■ sin m1 (0.8) is not 50■ sin 50 ≠ 0.812■ sin 50 should be 0.7660..., = 0.815eg, using cos 40■ cos 40 ≠ 0.8■ 15 t cos 40 ≠ 12! No indication of which angle is beingconsideredeg12◆ sin e15Accept only if the trigonometric ratio iscorrect for the angle of 50°Calculate what one side should be in order tomake the triangle consistenteg■ 15 sin 50 e 11.4906... not 1212■ e 15.6648... not 15sin50■ √(15 2 m 12 2 ) = 9 but 15 t cos 50 e 9.6418...Calculate what one angle should be in order tomake the triangle consistenteg■ sin m1 (0.8) e 53.1301... not 50■ The angle should be 53.1■ The other angle is 36.8698...,but it should be 40or1mShows or implies a correct trigonometric ratioeg12■ sin 50 e15■ 15 t sin 50■12sin 5039

2004 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8Tier & Question3-5 4-6 5-7 6-821 14Correct responseAlgebraic expressionsAdditional guidance1a 2m 6 or equivalent2or1mShows or implies a correct first step ofalgebraic manipulation that either reduces thenumber of terms orcollects unknowns on one side of the equationand numbers on the othereg■ 2y m 8 e 5■ 5y e 3y p 13■ 2y e 13■ 2y em3 (terms in y simplified, error insimplification of numerical values)b 2m m18or1mForms a correct equationeg■ 5y m 8 e 2(3y p 5)orForms the incorrect equation 2(5y m 8) e 3y p 5and follows through correctly to give y e 3eg■ 10y m 16 e 3y p 57y e 21y e 3! y = 3 without correct working seenAccept provided at least the equation2(5y m 8) e 3y p 5, or equivalent, is seen.Note that trial and improvement alone, orsimply showing 5 t 3 m 8 e 7, 3 t 3 p 5 e 14,should not be considered as correct working40

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-815Correct responseWhat fraction?Additional guidance2mGives a correct expressionegn p 2■2n■ (n p 2) d 2n1 1■ p2 n2n m (n m 2)■2n✓ Equivalent expressions For 2m, necessary brackets omittedeg◆ n p 2 d 2n2n m n m 2◆2nor1mShows both the expressions n p 2 and 2neven if these are subsequently combinedincorrectlyeg■ n p 2 d 2norGives an algebraic fraction in which thenumerator is n p 2orGives an algebraic fraction in which thedenominator is 2n n p 2 seen but not in a fraction41

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-8161m 7 or 6.7 or 6.67Correct responseAdditional guidanceEatingTier & Question3-5 4-6 5-7 6-8172m 15Correct responseEquation solvingAdditional guidanceor1mShows any two of the following three algebraicprocesses correctly:1. Cross multiplication to remove the fraction2. Multiplication or division to removebrackets3. Collecting like terms togethereg■■■■10y m 15 e 6y (error)4y e 15(Error in process 1)5(2y m 3) e 9y10y m 3 (error) e 9y, so y e 3(Error in process 2)5(2y m 3) e 9y2y m 3 e 1.6y (error), so 0.4y e 3(Error in process 2)10y m 15 e 9y(Process 3 not shown)42

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-8182m 30√2 or 42 or 42.(...)Correct responseAdditional guidance3-D cut For 2m or 1m, length(s) found only throughscale drawingor1mShows or implies a correct method for thelength of one side of the baseeg■ 10√2■ √200■ √(10 2 p 10 2 )■ 14.14(...)■ 1.4(...) t 1010■sin 45■10cos 45! Length roundedAccept 14 or 14.1 provided there is noevidence of an incorrect method43

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-819Correct responseAdditional guidanceTiles3mGives a complete correct justification thatencompasses all four conditions below:1. For the octagon, shows or implies thatthe interior angle is 135°, orthe exterior angle is 45°2. For the square, shows or implies thatthe interior or exterior angle is 90°3. For the hexagon, shows or implies thatthe interior angle is 120°, orthe exterior angle is 60°4. Justifies why the hexagon will not fiteg■! Explanation does not identify, on thediagram or otherwise, whether interior orexterior angles are being considered, or towhich shape the angles belongFor 3m, accept only if there is no redundantinformation and the justification isunambiguouseg, accept◆ 90 p 135 e 225, 360 m 225 e 135but the angle in a hexagon is 120◆360 m (90 p 135) > 120135 p 120 p 90 ≠ 360■135 ≠ 120■90 p 45 e 135°which is 15° too big■135 p 90 e 225but it should be 24044

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-819Correct responseAdditional guidanceTiles (cont)or2mShows at least one correct value from each ofthe following three sets of angles, even if it isnot clear to which shape the angle belongsor135 or 4590120 or 60Shows or implies the ‘gap’ is 135°eg■ 90 p 45 e 135■✓ 90 implied by a right angle symbol! Explanation confuses the terminology ofinterior and exterior anglesFor 2m or 1m, condone For 2m, incorrect angles marked or furtherworking indicates confusion betweeninterior and exterior angleseg◆ Angle of 135 marked as 451mU1Shows at least one correct value from two ofthe following three sets of angles, even if it isnot clear to which shape the angle belongs135 or 4590120 or 60orShows at least one correct value from each ofthe following three sets of angles, even if theangles are ascribed to incorrect shapes135 or 4590120 or 6045

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-820Correct responseAdditional guidanceDissection3mGives a complete correct justificationThe most common correct justifications:Show the length of CD is 9, then use thesimilarity of triangles CDE and AEF to showthrough calculation that EF is 20eg12 12■ Scale factor is , t 15 e 209 9■ The sides of triangle AEF are a thirdbigger than the corresponding sides of1triangle CDE, 15 t 1 = 203Show the length of CD is 9, then use thesimilarity of triangles CDE and BDF to showthrough calculation that EF is 20eg21■ Scale factor is921t 15 e 35, 35 m 15 e 2091■ 2 t 15 e 35, 35 e 20 p 153x p 15■ Let x e FE, then =21x p 15 e 35, x e 20159Use trigonometry to calculate∠CDE as 53.1(...)°, or∠DEC as 36.8(...)°,then use the similarity of triangles CDE andAEF (or CDE and BDF) to show throughcalculation that EF is 20 (or DF is 35)eg■ sin m1 12 e 53.1, 12 d cos 53.1 e 2015✓ EF taken as 20 then used to demonstrate thesides are in the correct ratio for similarity toholdeg, using triangles CDE and AEF20 15◆e12 920 12◆e15 92◆ FA e 20 2 m 12 2 , so FA e 16, and20 15e16 12eg, using triangles CDE and BDF15 35◆e9 2135 21◆e15 9! Values roundedAccept values shown as rounded, but for 3mdo not accept resultant incorrect valueseg, for 3m accept12◆ ∠DEC e 37°, e 20sin 37eg, for 3m do not accept15 EF◆e , 15 d 9 e 1.7,9 121.7 t 12 e 20.4 which rounds to 20 For 3m, justification uses only Pythagorasand EF = 20 used within the argument Circular argumenteg◆ 202m 12 2 e 16 2 so FA e 1616 2 p 12 2 e 400 so EF is 2046

2004 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 onlyTier & Question3-5 4-6 5-7 6-820Correct responseDissection (cont)Additional guidanceor2mShows or implies a correct scale factor, even ifroundedeg, for triangles CDE and AEF12■■913biggereg, for triangles CDE and BDF21■9■ 2.33orUsing a correct value for ∠CDE or ∠DEC,even if rounded or truncated, givesthe corresponding angle withintriangle AEF (or BDF)eg■ ∠AEF (or ∠BDF) is 53.1(...)°■ ∠EFA (or ∠DFB) is 36.8(...)°or1m Shows or implies the length of CD is 9eg■ BD e 21orShows ∠CDE is 53.1(...)°,even if the value is rounded or truncatedorShows ∠DEC is 36.8(...)°,even if the value is rounded or truncatedorU1Using their incorrect CD or their incorrect∠CDE or ∠DEC, even if rounded or truncated,shows their correct scale factor or gives thecorresponding angle within triangle AEF47

EARLY YEARSNATIONALCURRICULUM5–16GCSEGNVQGCE A LEVELFirst published in 2004NVQ© Qualifications and Curriculum Authority 2004Reproduction, storage, adaptation or translation, in any form or by any means, ofthis publication is prohibited without prior written permission of the publisher,unless within the terms of licences issued by the Copyright Licensing Agency.Excerpts may be reproduced for the purpose of research, private study, criticism orreview, or by educational institutions solely for educational purposes, withoutpermission, provided full acknowledgement is given.OTHERVOCATIONALQUALIFICATIONSProduced in Great Britain by the Qualifications and Curriculum Authority under theauthority and superintendence of the Controller of Her Majesty’s Stationery Officeand Queen’s Printer of Acts of Parliament.The Qualifications and Curriculum Authority is an exempt charity under Schedule 2of the Charities Act 1993.Qualifications and Curriculum Authority83 PiccadillyLondonW1J 8QAwww.qca.org.uk/Further teacher packs may be purchased (for any purpose other than statutoryassessment) by contacting:QCA Publications, PO Box 99, Sudbury, Suffolk CO10 2SN(tel: 01787 884444; fax: 01787 312950)Order ref: QCA/04/1203 259578