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Mathematics tests Ma KEY STAGE 3 ALL TIERS 2001 Mark scheme for Paper 2 Tiers 3–5, 4–6, 5–7 and 6–8 KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST KEY STAGE 3 KEY STAGE 3 KEY STAG 3 KEY STAGE 3 KEY STAGE 3 KEY ST Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Introduction Introduction The test papers will be marked by external markers. The markers will follow the 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 1 and the extension paper mark schemes are printed in separate booklets. Questions have been given names so that each one has a unique identifier irrespective of tier. The structure of the mark schemes The marking information for questions is set out in the form of tables, which start on page 11 of this booklet. The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part, and the total number of marks available for that question part. The ‘Correct response’ column usually includes two types of information: I 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; I examples of some different types of correct response, including the most common and the minimum acceptable. The ‘Additional guidance’ column indicates alternative acceptable responses, and provides details of specific types of response that are unacceptable. Other guidance, such as when ‘follow through’ is allowed, is provided as necessary. For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet. 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance General guidance Using the mark schemes Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise. In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance, relating to marking of questions that involve money, time, coordinates, algebra or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 3 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance What if … The pupil’s response does not match closely any of the examples given. Markers should use their judgement in deciding whether the response corresponds with the statement of requirements given in the ‘Correct response’ column. Refer also to the additional guidance. The pupil has responded in a non-standard way. Calculations, formulae and written responses do not have to be set out in any particular format. Pupils may provide evidence in any form as long as its meaning can be understood. Diagrams, symbols or words are acceptable for explanations or for indicating a response. Any correct method of setting out working, 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 a conceptual error. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, error. A computational error is a ‘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 relevant mathematics; when such an error is seen no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 t 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers. The pupil’s accuracy is marginal according to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, or touches, the boundaries given, the mark(s) should be awarded. The pupil’s answer correctly follows through from earlier incorrect work. ‘Follow through’ marks may be awarded only when specifically stated in the mark scheme, but should not be allowed if the difficulty level of the question has been lowered. Either the correct response or an acceptable ‘follow through’ response should be marked as correct. There appears to be a misreading affecting the working. The correct answer is in the wrong place. 4 This is when the pupil misreads the information given in the question and uses different information. If the original intention or difficulty level of the question is not reduced, deduct one mark only. If the original intention or difficulty level is reduced, do not award any marks for the question part. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance What if … The final answer is wrong but the correct answer is shown in the working. Where appropriate, detailed guidance will be given in the mark scheme, and must be adhered to. If no guidance is given, markers will need to examine each case to decide whether: the incorrect answer is due to a transcription error; If so, award the mark. in questions not testing accuracy, the correct answer has been given but then rounded or truncated; If so, award the mark. the pupil has continued to give redundant extra working which does not contradict work already done; If so, award the mark. the pupil has continued, in the same part of the question, to give redundant extra working which does contradict work already done. If so, do not award the mark. Where a question part carries more than one mark, only the final mark should be withheld. The pupil’s answer is correct but the wrong working is seen. A correct response should always be marked as correct unless the mark scheme states otherwise. The correct response has been crossed (or rubbed) out and not replaced. Mark, according to the mark scheme, any legible crossed (or rubbed) out work that has not been replaced. More than one answer is given. If all answers given are correct (or a range of answers is given, all of which are correct), 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 correct but, in a later part of the question, the pupil has contradicted this response. A mark given for one part should not be disallowed for working or answers given in a different part, unless the mark scheme specifically states otherwise. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 5 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance Marking specific types of question Responses involving money For example: £3.20 £7 Accept Do not accept Any unambiguous indication of the correct amount eg £3.20(p), £3 20, £3,20, 3 pounds 20, £3-20, £3 20 pence, £3:20, £7.00 Incorrect or ambiguous use of pounds or pence eg £320, £320p or £700p, or 3.20 or 3.20p not in the answer space. Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 eg £3.2, £3 200, £32 0, £3-2-0, £7.0 The £ sign is usually already printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the £ sign, accept an answer with correct units in pounds and/or pence eg 320p, 700p Responses involving time A time interval For example: 2 hours 30 mins Accept Take care ! Do not accept Any unambiguous indication eg 2.5 (hours), 2h 30 Digital electronic time ie 2:30 Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min ! The time unit, hours or minutes, is usually printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the given unit, accept an answer with correct units in hours or minutes, unless the question has asked for a specific unit to be used. A specific time For example: 8.40am, 17:20 Accept Any unambiguous, correct indication eg 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Unambiguous change to 12 or 24 hour clock eg 17:20 as 5:20pm, 17:20pm 6 Sourced from SATs-Papers.co.uk Do not accept Incorrect time eg 8.4am, 8.40pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0 eg 840, 8:4:0, 084, 84 http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance Responses involving coordinates For example: ( 5, 7 ) Accept Do not accept Unambiguous but unconventional notation eg ( 05, 07 ) ( five, seven ) x y ( 5, 7 ) ( x = 5, y=7) Incorrect or ambiguous notation eg ( 7, 5 ) ( 5x, 7y ) ( x5, y7 ) ( 5x, 7y ) Responses involving the use of algebra For example: 2 + n n + 2 2n Accept The unambiguous use of a different case eg N used for n Unconventional notation for multiplication eg n t 2 or 2 t n or n2 or n + n for 2n n t n for n2 Multiplication by 1 or 0 eg 2 + 1n for 2 + n 2 + 0n for 2 Words used to precede or follow equations or expressions eg t = n + 2 tiles or tiles = t = n + 2 for t = n + 2 Take care ! Do not accept ! Words or units used within equations or expressions should be ignored if accompanied by an acceptable response, but should not be accepted on their own eg do not accept n tiles + 2 n cm + 2 Change of variable eg x used for n Ambiguous letters used to indicate expressions eg n = n + 2 Unambiguous letters used to indicate expressions eg t = n + 2 for n + 2 However, to avoid penalising any of the three types of error above more than once within each question, do not award the mark for the first occurrence of each type within each question. Where a question part carries more than one mark, only the final mark should be withheld. Embedded values given when solving equations eg 3 t 10 + 2 = 32 for 3x p 2 e 32 Embedded values that are then contradicted eg for 3x + 2 = 32, 3 t 10 + 2 = 32, x = 5 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 7 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction or percentage only. For example: 0.7 Accept A correct probability that is correctly expressed as a decimal, fraction or percentage. Equivalent decimals, fractions or percentages eg 70 35 0.700, , , 70.0% 100 50 A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 eg 70 18 e 100 25 Take care ! Do not accept The following four categories of error should be ignored if accompanied by an acceptable response, but should not be accepted on their own. ! A probability that is incorrectly expressed eg 7 in 10, 7 out of 10, 7 from 10 ! A probability expressed as a percentage without a percentage sign. ! A fraction with other than integers in the numerator and/or denominator. However, each of the three types of error above should not be penalised more than once within each question. Do not award the mark for the first occurrence of each type of error unaccompanied by an acceptable response. Where a question part carries more than one mark, only the final mark should be withheld. ! A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 General guidance Recording marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1 0 The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper. A total of 120 marks is available in each of tiers 3–5, 4 – 6, 5–7 and 6–8. The extension paper carries 42 marks. Awarding levels The sum of the marks gained on paper 1, paper 2 and the mental arithmetic paper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the QCA website (www.qca.org.uk) from Friday 22 June 2001. QCA will also send a copy to each school in July. Schools will be notified of pupils’ results by means of a marksheet, which will be returned to schools by the External Marking Agency with the pupils’ marked scripts. The marksheet will include pupils’ scores on the test papers and the levels awarded. The 2001 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 9 2001 KS3 Mathematics Test Mark Scheme: Paper 2 BLANK PAGE 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3–5 only Tier & Question Cards 3-5 4-6 5-7 6-8 1 Correct response a 1m £ 3.20 b 1m £ 102(.00) c 1m Additional guidance 14 Tier & Question No. 1 Singles 3-5 4-6 5-7 6-8 2 Correct response a 1m 7 b 1m Madonna c 1m 6 d 1m Abba and Spice Girls, either order Additional guidance Sourced from SATs-Papers.co.uk ! Reference to fourth place Ignore http://www.SATs-Papers.co.uk 11 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 3–5 only Tier & Question Using Number Lines 3-5 4-6 5-7 6-8 3 Correct response a 1m 50 and 75; correctly placed b 1m 20, 40, 60, 80; correctly placed c 2m Additional guidance 40, 80, 120, 160; correctly placed or 1m Any three correct, with follow through of steps of 40 from not more than one incorrect value eg I 40, 80, 120, 170 (error) I 40, 90 (error), 130, 170 I 50 (error), 90, 130, 170 ! Follow through as double their values from part (b) Accept provided their values form an increasing sequence eg, from part (b) as 20, 40, 50, 70 accept for 1m N 40, 80, 100, 140 Follow through values greater than 200 d 1m 4 Tier & Question Map 3-5 4-6 5-7 6-8 4 Correct response a 1m 5 b 1m West Additional guidance Abbreviations eg N N 1m North-east W NE Bearings eg, for W N 270 eg, for NE N 045 N 45 Unconventional but unambiguous notation eg, for North-east N East North c 12 1m 4 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Ruler 3-5 4-6 5-7 6-8 5 Correct response a 1m Additional guidance Equivalent fractions or decimals, or use of 1.5 words Distance in mm without units specified 1m 2m Indicates 4.5 and 11.5 or 1m b 5 One correct Accuracy within ± 2mm or Scale misread but arrows placed symmetrically about point E Tier & Question Getting There 3-5 4-6 5-7 6-8 6 1 a a Correct response 64 and 864 1m 675 1m 2520 1m b b 1m Additional guidance 15 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 13 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Squares 3-5 4-6 5-7 6-8 7 2 a a Correct response 1m Additional guidance ! Units given 9 Ignore b b 1m 4 c 1m 4 c ! Answers for part (c) reversed Mark as 0, 1 1m 14 Tier & Question Disco Costs 3-5 4-6 5-7 6-8 8 3 Correct response a a 1m £ 4.(00) b b 1m Additional guidance Correct explanation. The most common correct explanations: Minimally acceptable explanation eg Interpret the spreadsheet to explain why there is one charge eg I The hire of the hall is a fixed charge. I You only hire the hall once. I You only hire one hall. N Implication that only one hall is available eg N N N Explain the hire is independent of the number of people attending eg I You pay for the hall however many people come. I It is not affected by the other columns. c c 1m 19 It always costs the same to hire the hall. You use the same hall no matter how many people there are. The hall is always the same size. It’s the same hall. Incomplete explanation that does not interpret the spreadsheet eg N It’s the hire of the hall. N It’s always the same. ! Money quantified Ignore d d 1m 27 e 1m £ 28.50 14 e Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Cooking 3-5 4-6 5-7 6-8 9 4 a a Correct response 1m Additional guidance Correct answer in hours and minutes 51 eg, for part (b) N 4 hours 5 minutes b b 1m ! Incorrect conversion to hours and minutes 245 If the correct number of minutes is shown, ignore any further working. c c 2m 56 or 1m Shows either 39 or 95 Tier & Question Pieces 3-5 4-6 5-7 6-8 10 5 a a Correct response 1m Indicates Yes, and gives a correct explanation. Additional guidance Minimally acceptable explanation eg The most common correct explanation focuses on the complete area eg I They’re both 8 I Both have 7 wholes and 2 halves. I 8 is half of 16 N N N Same number of squares. I counted the squares and it was the same. The one square jutting out fills the two half squares missing on the right hand piece. Restatement of the question eg N Both have same space inside. Incorrect or incomplete explanation eg N Each one has 7 squares. N The area of both is 9 N When you work out area you don’t count the halves. ! Units incorrect Ignore b b 1m Correct piece, ie _ _ _ _ Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 15 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Areas 3-5 4-6 5-7 6-8 11 6 1 a a a Correct response 1m All correct, ie b b b Additional guidance – 2m 40 or 1m Shows the value 10 or Follows through from an incorrect side length to find the perimeter, provided the side length is not 25 eg I Side is 8, so perimeter is 32 Marking overlay available Tier & Question Ferry 3-5 4-6 5-7 6-8 12 7 2 a a a Correct response 2m The line representing the ferry crossing, within the tolerances shown by the overlay. Additional guidance Line(s) not ruled but within tolerance ! Pupil draws their own base line or 1m b b b c c c 1m 2m or 1m One angle drawn within the tolerance shown by the overlay, and at least of length as shown by the overlay, even if their angle does not start at the end of the given line. Their length ± 2mm (Note that the calculated value is 5.59) Accept for 2m provided the base line is the correct length within the tolerance shown. If the base line length is incorrect but the angles are correct, mark as 1, 0 ! Rounded to the nearest integer Accept if their measurement is within 2mm of an integer length, otherwise do not accept. Correct response using their (b) or their length eg I Their (b) t 20 and metres given. I Their (b) t 2000 and cm given. Their part (b), or their length, multiplied by either 20 or 2000, even if the units are incorrect or omitted. or Shows a correct method with consistent units eg I t 20 seen, and metres given. I t 2000 seen, and centimetres given. 16 Sourced from SATs-Papers.co.uk Correct units with no length http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Swimming 3-5 4-6 5-7 6-8 13 8 3 a a a Correct response 1m 48 and 72 Additional guidance No values within the table but correct points plotted on the graph b b b 2m or 1m 3 or 4 points plotted correctly ± 1mm, and joined with the correct ruled straight line. ! Line ruled but does not pass exactly through the correct points Accept provided the pupil’s intention is clear. ! Bar chart drawn 3 or 4 points plotted correctly ± 1mm, but not joined. Ignore bars. For 1m, follow through from part (a) or 3 or 4 points plotted correctly ± 1mm, but joined incorrectly or line not ruled. c c c 1m 50 and 64 No values within the table but correct points plotted on the graph d d d 2m or 1m 3 or 4 points plotted correctly ± 1mm, and joined with the correct ruled straight line. ! Line not ruled Accept if this error has already been penalised in part (b). ! Line does not pass exactly through the 3 or 4 points plotted correctly ± 1mm, but not joined. or correct points Accept provided the pupil’s intention is clear. ! Bar chart drawn Ignore bars. 3 or 4 points plotted correctly ± 1mm, but joined incorrectly or line not ruled. e e e 1m For 1m, follow through from part (c) 22 Follow through their graph, including non-integer values, even if rounded to the nearest integer ! Their graph shows more than one intersection All such values must be listed. ! Cost shown Ignore. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 17 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Mints 3-5 4-6 5-7 6-8 14 9 4 a a a Correct response 2m or 1m b b b 18 1m Additional guidance 5y + 6 and 6 + 5y, in either order Only one of the correct expressions given; the other incorrect or omitted. Indicates Yes, and gives a correct explanation eg I If you take away the 6, then it is divisible by 5 I Could be 10 in a packet. I 5 t 10 + 6 Sourced from SATs-Papers.co.uk Definitive statement eg N There must be 10 mints in a packet. http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Drinks Machine 3-5 4-6 5-7 6-8 10 5 1 Correct response 3m or 2m Additional guidance 49 Shows a complete correct method, with not more than one computational error. The most common correct methods are: Finding the total and dividing by 55 eg I 2695 d 55 I 15.50 + 4.40 + 4.10 + 2.95, then d 0.55 I (50 t 31 + 20 t 22 + 10 t 41 + 5 t 59) d 55 I 15.5 + 4.40 + 4.10 + 29.50 (error) e 53.5 53.5 d 0.55 Grouping the money for specific amounts of cans eg I 31 cans uses 31 t 50p + 31 t 5p; 11 uses 22 t 20p + 11 t 10p + 11 t 5p; 6 cans uses 30 t 10p + 6 t 5p; 1 can uses the remaining 11 t 5p I 31 t 50p + 31 t 5p is 31 cans; 22 t 20p + 11 t 10p + 11 t 5p is 22 (error) cans; 30 t 10p + 17 t 5p is another 7 cans. Dividing each sub-total by 55 eg I 31 t 50 = 1550, that’s 28 cans and 10p left. 22 t 20 = 440, that’s 8 cans. 41 t 10 = 410, that’s 7 cans and 25p left. 59 t 5 = 295, that’s 5 cans and 20p left. The money left is enough for one more can. or 1m Shows the digits 2695 or Shows a correct method for finding the total, but with more than one computational error. or d 55 is seen or implied. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 19 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Advert 3-5 4-6 5-7 6-8 11 6 2 Correct response a a a 2m or 1m b b b 2m or 1m Additional guidance £ 345 Correct method shown eg I 15 t 18 + 75 I Digits 345 seen, other than for the correct response. 36 Correct method shown eg I 615 – 75, then d 15 I Digits 36 seen, other than for the correct response. Tier & Question Speed 3-5 4-6 5-7 6-8 12 7 3 Correct response a a a 1m Correct response eg I 7.5 hours. I 7 hours 30 minutes. I b b b 1m 20 ! Answer of 8 hours Accept only if a more accurate value is seen. 1 hours. 2 465 c 7 Additional guidance 60 c c 1m Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7, 6–8 Trundle Wheel 3-5 4-6 5-7 6-8 13 8 4 Correct response a a a 2m or 1m b b b 1m Additional guidance 157.(...) or 50π Correct method eg I 50 t π I 3.14 t 2 t 25 Follow through as 87 t their (a) d 100, 137 rounded to the nearest metre Tier & Question Algebra Pairs 3-5 4-6 5-7 6-8 15 14 9 5 a Correct response 2m or 1m b 3m or 2m or 1m Additional guidance Both pairs correct, and no incorrect, ie At least one correct pair identified, with not more than one incorrect pair. All three pairs correct, and no incorrect, ie At least two correct pairs, and not more than one incorrect pair. At least one correct pair, and not more than two incorrect pairs. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 21 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Marking overlay available Tier & Question Books 3-5 4-6 5-7 6-8 15 10 6 Correct response a a a 2m or 1m Additional guidance Pie chart completed within the smaller tolerance as indicated by the overlay, and at least one of their sectors labelled correctly. Pie chart completed within the greater tolerance as indicated by the overlay, and at least one sector labelled correctly. or Pie chart completed within the smaller tolerance as indicated by the overlay, but sectors not labelled or labelled incorrectly. or A correct method for finding an angle or percentage is shown or implied eg I 13 d 20 t 360 (or t 100) I 4 d 20 t 360 (or t 100) I 54 d 3 t 4 = angle for Fantasy b b b 2m or 1m 24 Shows a correct method using angles eg I 360 d (165 d 11) I 360 t 11 165 I 360 d 15 or Gives a correct angle for 1 pupil eg I 15° or Correct number of pupils for other than 165° seen eg I 180° is 12 22 Sourced from SATs-Papers.co.uk Angle of 54 measured as 54 ± 2° Markers may find the following values helpful: Non-fiction 11 165° Romance 5 75° Crime 3 45° Fantasy 5 75° ! Correct method using percentages Accept correct methods eg 11 11 N 46% is 11; 1% is ; t 100 46 46 Accept percentages within the following inclusive ranges: Non-fiction 45 to 46 Romance 20 to 21 Crime 12 to 13 Fantasy 20 to 21 Number of pupils not given as an integer http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Yoghurt 3-5 4-6 5-7 6-8 16 11 7 Correct response a a 2m 3.6 Additional guidance ! Answer rounded Do not accept unless a correct method, or a more accurate value, is seen. or 1m b b 2m Shows a correct method eg I 4.5 d 125 t 100 I 4.5 d 5 t 4 I 25g = 0.9, 0.9 t 4 Indicates A, and gives a correct justification. The most common correct justification compares the same amount of grams eg I A has 2.22 for 25g but B has 2.183(...) I If B were 125g, it would be 10.916(...)g. I If A were 150g, it would be 13.32g. I 11.1 d 125 = 0.088(8), 13.1 d 150 = 0.087(...) or 1m Correct method but no, or incorrect, conclusion drawn, even if the values have been truncated or rounded eg I 6 t 11.1 = 66.6, 5 t 13.1 = 65.5 so B or Correct method with not more than one computational error, with a correct conclusion drawn for their figures. Sourced from SATs-Papers.co.uk Markers may find the following helpful: Grams A B 1 0.0888 0.0873(...) 25 2.22 2.183(...) 100 8.88 8.733(...) 125 11.1 (given) 10.916(...) 150 13.32 13.1 (given) 750 66.6 65.5 ! Values rounded or truncated Accept provided the comparison can be drawn eg N A has 2.2 for 25g and B has 2.2 Mark as 1, 0 ! Correct calculations for yoghurt per gram of carbohydrate Accept for 2m if correctly interpreted, otherwise mark as 1, 0 eg, for 2m N A: 125 d 11.1 = 11.26(...) B: 150 d 13.1 = 11.45(...) so A provides more carbohydrate. eg, for 1m N A: 125 d 11.1 = 11.26(...) B: 150 d 13.1 = 11.45(...) so B provides more carbohydrate. http://www.SATs-Papers.co.uk 23 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Missing Side 3-5 4-6 5-7 6-8 12 8 Correct response a a 2m 20.8(...) or √433 Additional guidance ! Answer 21 Do not accept unless a correct method, or a more accurate value, is seen. or 1m b b 2m Shows both squaring and adding eg 2 2 I 17 + 12 I 433 seen I 289 + 144 9.8(0) or 9.79(...) or √96 ! Answer truncated to 9.7 Accept if a correct method or more accurate value is seen. Otherwise mark as 1, 0 ! Answer 10 or 1m 24 Shows both squaring and subtracting eg 2 2 I 11 – 5 I 96 seen I 121 – 25 Sourced from SATs-Papers.co.uk Do not accept unless a complete correct method, including the need to square root, or a more accurate value is seen eg, mark the following as 1, 0 N 121 – 25 = 96, 9.6 so 10 http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Goldcrests 3-5 4-6 5-7 6-8 13 9 Correct response a a 1m 4.83 to 4.87 inclusive Additional guidance Incorrect notation eg 1 N 4.8 2 b b 1m 0.09 to 0.11 inclusive c Indicates (12.5, 4.5) and gives a justification based on the distance from the line of best fit eg I It’s an outlier. I It’s the furthest away. c 1m Minimally acceptable explanation eg N It’s the one that is most different. or Indicates (12.5, 4.5) and gives a justification based on the low mass given the time of day eg I It’s very small and getting late in the day. I The mass goes up by about 0.1g every hour so by 3pm the mass would only be about 4.7g which is very low. I Because at 12.30 it just weighs the same as it should have done much earlier in the day. or Minimally acceptable explanation eg N N Because it is the lightest around that time of the day. It’s the lightest and it is 12.30 No reference to the time of day eg N It’s very small and will freeze to death. N It’s the lightest. No reference to the mass eg N It’s very late in the day. Indicates (1.5, 4.8) or (2.5, 5.0) and gives a justification based on the lack of time to catch up eg I It’s late in the day and that one hasn’t eaten much. Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 25 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Triangles 3-5 4-6 5-7 6-8 17 14 10 a a a 1m Correct response A right-angled triangle of height 4 eg Additional guidance Lines not ruled Accept provided the pupil’s intention is clear. I b b b 1m An isosceles triangle of height 4 eg I ! AB used as one of the pairs of equal sides Accept if the height is clearly intended to be 4, and the apex is between 1 and 2cm to the right of the point above A (or to the left of the point above B) eg, accept N Do not accept if the apex is clearly intended to be at an intersection eg N 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6–8 only Tier & Question Triangles (cont) 3-5 4-6 5-7 6-8 10 c Correct response 2m or 1m Correct explanation eg 2 2 2 I AC = 4 + 3 = 25, so AC = 5 = AB I It’s a 3, 4, 5 triangle (correct triangle identified on the diagram), so AC = 5 and AB = 5 Partial explanation eg I It’s a 3, 4, 5 triangle (no identification) or Additional guidance Correct use of trigonometry eg N ∠B = tan 3 = 71.56... –1 ∠A = tan –1 3 = 36.86... so 4 ∠C = 180 – (71.57 + 36.87) = ∠B Length of sides stated with no reference to the 3, 4, 5 triangle eg N One side is 3cm, one is 4cm, the other side is 5cm. Shows a complete correct method using trigonometry with not more than one computational error, even if there are rounding errors. d 2m 71.6 or 71.57 or 71.56(...) ! Answer 71.5 or 72 As this could be obtained through measuring, accept only if a correct method or a more accurate value is seen. or 1m Any correct trigonometric ratio seen, even if in part (c) eg I tan ABC = I tan A = 3 1 3 4 ! Angle not identified Accept if referring to the angle at B eg –1 N tan 3 N tan = 3 Otherwise, do not accept eg N or tan = 3 4 Bisects the triangle through CB, then creates a correct trigonometric ratio using their measured half BC eg I cos ABC = 16 d 50 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 27 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5–7, 6–8 Marking overlay available Tier & Question Tree 3-5 4-6 5-7 6-8 15 11 Correct response 1m Draws the straight line parallel to the greenhouse, and both straight lines parallel to the edges of the vegetable plot, within the tolerance, and at least of length, as shown on the overlay. 1m Draws the correct arc within the tolerance, and at least of length, as shown on the overlay. 1m Indicates the complete correct region. Additional guidance ! Follow through from either or both of the previous marks Accept from their boundary provided there is no ambiguity. 28 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Earnings 3-5 4-6 5-7 6-8 16 12 Correct response a a 2m or 1m Additional guidance 51.8 Shows 6.16 d 11.89 or Shows the digits 51(...) or 52 Correct justification. The most common justifications are: Values not rounded to the nearest penny For 1998, calculating women’s earnings as a % of men’s eg I 72.(...)% b b 2m ! Further working Using their value from (a) to calculate what the earnings would have been eg I 51.8% of 420.30 = £217.72 Using ratio in a form that enables comparison eg I 1956 male : female earnings was 1.93 : 1; 1998 it was 1.38 : 1, so men got less. Comparing the rate of increase eg I 420.3 d 11.89 is about 35; 303.7 d 6.16 is about 49 so women’s salaries went up more than men’s. or 1m Ignore eg, accept for 2m N 72.3 – 51.8 = 20.5% increase in women’s wages. ! Values approximated If values are correctly approximated, accept provided the response makes it clear they are approximations and not exact eg, accept N 6.16 out of 11.89 is about 50% but 303.70 out of 420.30 is about 75% eg, accept (minimally acceptable) N 1998 is about 75% eg, do not accept N 1998 is 75% Also accept follow through from part (a), provided it is less than 67% Any complete correct method with not more than one computational error. or Gives a partial justification eg I 303.7 d 420.3 > 51.8% or The only error is to assume that there are equal numbers of male and female employees eg I 6.16 d (11.89 p 6.16) is 34% but 303.7 d (420.3 p 303.7) is 42% Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 29 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6–8 only Tier & Question Sale 3-5 4-6 5-7 6-8 13 Correct response 2m or 1m Additional guidance 45 Shows 38.25 d 85 eg I 38.25 d 85 t 100 I 0.45 seen Tier & Question Parabolas 3-5 4-6 5-7 6-8 14 Correct response a 2m or 1m Additional guidance All three correct, ie (0, 16), (4, 0), (– 4, 0) Any two correct. or All three correct but in an incorrect order. b 1m (4, 24) Follow through from their incorrect coordinates for B eg, for their B as (16, 0) N (16, 24) c 1m 2 y e x p 8, or equivalent expression eg 2 I y e 24 – (16 – x ) ! Follow through from their incorrect coordinates for A Accept provided the y ordinate > 12 eg, for their A as (0, 14) 2 N y e x p 10 Incomplete equation eg 2 N x p8 30 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier 6–8 only Tier & Question Which is Bigger? 3-5 4-6 5-7 6-8 15 Correct response a 2m or 1m Indicates B, and gives a correct justification eg I 3.2π > 3.125π I A is 9.8(...), B is over 10 I A is 125π d 40 but B is 128π d 40 Additional guidance π omitted eg N 3.2 > 3.125 ! Rounding Accept area of A as 3.13π or 3.12π or 3.1π but do not accept as 3π Shows a correct area for either A or B eg, for A I 9.8(...) I 3.125π eg, for B I 10.(0...) I 10.1 I 3.2π or Shows correct working for both A and B eg 25tπ 16tπ I , 8 5 Incomplete working eg 2 N πt5 8 2 N b 2m or 1m Indicates A, and gives a correct justification eg I 13.92699... > 13.02654... πt4 5 evaluated as 246.7.. or 30.8(...) 8 evaluated as 157.9.. or 31.5(...) 5 ! Values rounded or truncated Accept values rounded to 2 or more s.f. Accept values rounded or truncated to 1 or more d.p. Correct total perimeter seen for A or B eg I A, 13.9(...) I B, 13.0(...) or Correct arc length seen for both A and B I A is 3.9(...), B is 5.0(...) I A is 1.25π, B is 1.6π c 2m or 1m r 2.8 or 2.83 or 2.82(...) or 2r2 Correct method shown eg 2 I π t 16 d 2 e π t r 2 I r e 8 I r e r8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 31 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Tier & Question Tier 6–8 only Music Concert 3-5 4-6 5-7 6-8 16 Correct response 1m 1m Forms correct equations eg I 3x p 9y e 120, 5x p 5y e 90 I x p 3y e 40, x p y e 18 Additional guidance ! Change of variable from x and y Accept if unambiguous. ! Correct values for x and y and/or an answer of 112 from trial and improvement or other non-algebraic method Award the last mark only. Arranges their equations in a form that allows for the elimination of one variable eg I 15x p 45y e 600, 15x p 15y e 270 I 15x p 45y e 600, 45x p 45y e 810 or Rearranges their equation(s) to express one variable in terms of the other eg I x e 18 – y I x e 40 – 3y I y e 18 – x 120 – 9y I x e 3 1m Solves their equations algebraically for either x or y eg, from correct equations I x e 7 I y e 11 1m 112 minutes or Shows correct values for x and y but with no supporting correct algebraic method. 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 2001 KS3 Mathematics Test Mark Scheme: Paper 2 Index Index to mark schemes Tier 3-5 4-6 5-7 6-8 Question Page 1 Cards 11 2 No. 1 Singles 11 3 Using Number Lines 12 4 Map 12 5 Ruler 13 6 1 Getting There 13 7 2 Squares 14 8 3 Disco Costs 14 9 4 Cooking 15 10 5 Pieces 15 11 6 1 Areas 16 12 7 2 Ferry 16 13 8 3 Swimming 17 14 9 4 Mints 18 10 5 1 Drinks Machine 19 11 6 2 Advert 20 12 7 3 Speed 20 13 8 4 Trundle Wheel 21 14 9 5 Algebra Pairs 21 15 10 6 Books 22 16 11 7 Yoghurt 23 12 8 Missing Side 24 13 9 Goldcrests 25 14 10 Triangles 26 15 11 Tree 28 16 12 Earnings 29 13 Sale 30 14 Parabolas 30 15 Which is Bigger? 31 16 Music Concert 32 15 17 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 33 EARLY YEARS NATIONAL CURRICULUM 5 –16 GCSE GNVQ GCE A LEVEL NVQ First published in 2001 © Qualifications and Curriculum Authority 2001 Reproduction, storage, adaptation or translation, in any form or by any means, of OTHER VOCATIONAL QUALIFICATIONS this 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 or review, or by educational institutions solely for educational purposes, without permission, provided full acknowledgement is given. Produced in Great Britain by the Qualifications and Curriculum Authority under the authority and superintendence of the Controller of Her Majesty’s Stationery Office and Queen’s Printer of Acts of Parliament. The Qualifications and Curriculum Authority is an exempt charity under Schedule 2 of the Charities Act 1993. Qualifications and Curriculum Authority 83 Piccadilly London W1J 8QA www.qca.org.uk/ Further teacher packs may be purchased (for any purpose other than statutory assessment) by contacting: QCA Publications, PO Box 99, Sudbury, Suffolk CO10 2SN (tel: 01787 884444; fax: 01787 312950) SourcedOrderSATs-Papers.co.uk from ref: QCA/01/660 http://www.SATs-Papers.co.uk 00-6651/12