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Ma Mathematics tests KEY STAGE 3 for Paper 1 2 Tiers 3–5, 4–6, 5–7 and 6–8 2008 ALL TIERS Mark scheme National curriculum assessments Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 1 http://www.SATs-Papers.co.uk 30/11/07 22:28:57 2008 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 mark scheme is printed in a separate booklet. 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: 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. Other guidance, such as when ‘follow-through’ is allowed, is provided as necessary. Questions with a UAM element are identiﬁed in the mark scheme by an encircled U with a number that indicates the signiﬁcance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. 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. The 2008 key stage 3 mathematics tests and mark schemes were developed by the Test Development Team at Edexcel. 2 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 2 http://www.SATs-Papers.co.uk 30/11/07 22:28:58 2008 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 speciﬁcally to the marking of questions that involve money, negative numbers, algebra, time, coordinates or probability. Unless otherwise speciﬁed in the mark scheme, markers should apply the following guidelines in all cases. 3 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 3 http://www.SATs-Papers.co.uk 30/11/07 22:28:58 2008 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 × 6 = 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 × 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. 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. The correct answer is in the wrong place. 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. 4 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 4 http://www.SATs-Papers.co.uk 30/11/07 22:28:58 2008 KS3 Mathematics test mark scheme: Paper 2 General guidance What if … Marking procedure 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. The answer is correct but, in a later part of the question, the pupil has contradicted this response. 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. 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. 5 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 5 http://www.SATs-Papers.co.uk 30/11/07 22:28:58 2008 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 indication of the amount eg £320, £320p or £700p The unit, £ or p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units eg with £ given in the answer space, accept 3.20 7 or 7.00 Ambiguous use of units outside the answer space eg with £ given in the answer space, do not accept 3.20p outside the answer space Given units amended eg with £ crossed out in the answer space, accept 320p 700p 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 Responses involving negative numbers For example: Accept –2 Do not accept To avoid penalising the error below more than once within each question, do not award the mark for the ﬁrst occurrence of the error within each question. Where a question part carries more than one mark, only the ﬁnal mark should be withheld. Incorrect notation eg 2 – 6 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 6 http://www.SATs-Papers.co.uk 30/11/07 22:28:58 2008 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving the use of algebra For example: 2+n n+2 2n n n2 2 Take care ! Do not accept Accept Unambiguous use of a different case or variable eg N used for n x used for n ! Unconventional notation eg n × 2 or 2 × n or n2 or n + n for 2n n × n for n2 n ÷ 2 for n or 1 n 2 2 2 + 1n for 2 + n 2 + 0n for 2 Within a question that demands simpliﬁcation, do not accept as part of a ﬁnal answer involving algebra. Accept within a method when awarding partial credit, or within an explanation or general working. Embedded values given when solving equations eg in solving 3x + 2 = 32, 3 × 10 + 2 = 32 for x = 10 To avoid penalising the two types of error below 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. Words used to precede or follow equations or expressions eg t = n + 2 tiles or tiles = t = n + 2 for t = n + 2 Unambiguous letters used to indicate expressions eg t = n + 2 for n + 2 ! Words or units used within equations or expressions eg n tiles + 2 n cm + 2 Do not accept on their own. Ignore if accompanying an acceptable response. Ambiguous letters used to indicate expressions eg n = n + 2 for n + 2 7 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 7 http://www.SATs-Papers.co.uk 30/11/07 22:28:59 2008 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving time A time interval For example: 2 hours 30 minutes Accept Take care ! Do not accept Any unambiguous indication eg 2.5 (hours), 2h 30 Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min ! Digital electronic time ie 2:30 A speciﬁc time For example: 8:40am The unit, hours and/or minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, or crosses out the given unit, accept answers with correct units, unless the question has speciﬁcally asked for other units to be used. 17:20 Accept Do not accept Any unambiguous, correct indication eg 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Incorrect time eg 8.4am, 8.40pm Unambiguous change to 12 or 24 hour clock eg 17:20 as 5:20pm, 17:20pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0 eg 840, 8:4:0, 084, 84 Responses involving coordinates For example: ( 5, 7 ) Accept Do not accept Unconventional notation eg ( 05, 07 ) ( ﬁve, seven ) Incorrect or ambiguous notation eg ( 7, 5 ) x y ( 5, 7 ) ( x = 5, y = 7 ) y x ( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x – 5, y – 7 ) 8 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 8 http://www.SATs-Papers.co.uk 30/11/07 22:28:59 2008 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. 7 For example: 0.7 70% 10 Accept Take care ! Do not accept Equivalent decimals, fractions and percentages 70 35 , 70.0% eg 0.700, , 100 50 The ﬁrst four categories of error below should be ignored if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the ﬁrst three types of error below 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 that is incorrectly expressed eg 7 in 10 7 over 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. ! A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 70 = 18 eg 100 25 A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 9 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 9 http://www.SATs-Papers.co.uk 30/11/07 22:28:59 2008 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. Awarding levels The sum of the marks gained on paper 1, paper 2 and the mental mathematics paper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the NAA website www.naa.org.uk/tests from Monday 23 June 2008. 10 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 10 http://www.SATs-Papers.co.uk 30/11/07 22:29:00 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Rounding 3-5 4-6 5-7 6-8 1 Correct response a 2m Additional guidance Matches all four numbers correctly, ie 912 ! Number matched to more than one nearest hundred For 2m or 1m, do not accept as a correct match 800 990 955 900 849 1000 881 or 1m 1m b Matches at least two numbers correctly Gives a value greater than or equal to 45 but less than 55 Fractions or decimals Value of exactly 55 given 1m Gives a different value greater than or equal to 45 but less than 55 from any credited for the ﬁrst mark U1 11 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 11 http://www.SATs-Papers.co.uk 30/11/07 22:29:00 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Cuboid 3-5 4-6 5-7 6-8 2 Correct response a 1m 6 b 1m 2 1m Additional guidance 3 Tier & Question Placing 40 3-5 4-6 5-7 6-8 3 Correct response 1m ! Inaccurate indication Accept provided their indication is closer to the correct marker than any other Indicates 40 in the correct position, ie 0 1m Additional guidance 200 ! Follow-through For the second mark, accept responses in which the distance between the arrow and zero is half as big as for the ﬁrst mark Indicates 40 in the correct position, ie 0 400 12 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 12 http://www.SATs-Papers.co.uk 30/11/07 22:29:00 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Directions 3-5 4-6 5-7 6-8 4 Correct response Additional guidance a 1m Indicates right then left Unambiguous indication eg, for part (a) r then l b 2m Gives directions that state or imply the following four steps (or equivalent) in the correct order: 1. (Come out of house A and) turn right 2. (Take the) second road on the left 3. Turn right 4. (House C is on the) right For part (b), unambiguous description for step 2, ie ‘second road on the left’ eg Cross the junction then turn left At the next turning, go straight on, then turn left or 1m Gives directions that state or imply all four steps, with not more than one error eg Right Left [indication of ‘second’ omitted] Right Right Turn right out of the house Take the second right (error) Take the ﬁrst right The house is on the right or Gives directions that state or imply steps 2 and 3 above, even if steps 1 and/or 4 are incorrect or omitted or Gives correct directions for getting from house C to house A: 1. (Come out of house C and) turn left 2. (At the end of the road) turn left 3. Turn right 4. (House A is on the) left U1 13 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 13 http://www.SATs-Papers.co.uk 30/11/07 22:29:00 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 3–5 only Writing cheques 3-5 4-6 5-7 6-8 5 Correct response 1m £ 102.70 1m £ 120.07 Additional guidance ! Non-standard notation Condone any unambiguous notation eg, for the ﬁrst mark accept £ 102 = 70 Tier & Question Theme park 3-5 4-6 5-7 6-8 6 Correct response a 1m 8 b 1m 7 c 1m Additional guidance 5 14 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 14 http://www.SATs-Papers.co.uk 30/11/07 22:29:00 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 3–5 only Adding odd 3-5 4-6 5-7 6-8 7 Correct response 1m Gives a correct counter example showing the sum of two odd numbers eg 1 + 3 = 4, which is even 5 and 7 makes 12 Odd = even + 1, so odd + odd = even + even + 2 = even Additional guidance Minimally acceptable example eg 1+3=4 Odd numbers taken to be equal eg 2 × 5 = 10 ! Response uses negative numbers and/or zero Accept negative odd numbers and zero as an even number within a correct response eg, accept –1 + 1 = 0 ! Other calculations or general reasoning given alongside a correct response Ignore other calculations, even if they are incorrect or do not relate to the given statement If a correct counter example is given, ignore any general explanation unless it contradicts the counter example given Incomplete or incorrect example eg 1 + 3 = even Odd + odd = even Only odd + even = odd 15 + 17 = 42 U1 Tier & Question Calculating 3-5 4-6 5-7 6-8 8 Correct response a 1m 2134 b 1m Additional guidance 663768 15 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 15 http://www.SATs-Papers.co.uk 30/11/07 22:29:01 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Time machine 3-5 4-6 5-7 6-8 9 1 Correct response 2m Additional guidance 6 or 1m Shows the value 94 or the values 4 and 2 or For 1m, necessary brackets omitted eg 100 – 46 + 48 Shows a complete correct method with not more than one computational error eg 100 – 46 – 48 100 – (46 + 48) 100 – 46 = 53 (error) 53 – 48 = 5 Tier & Question Four cards 3-5 4-6 5-7 6-8 10 2 Correct response 2m Additional guidance Matches all four statements to their correct positions, ie … 4 times table Certain ! Statement matched to more than one position For 2m or 1m, do not accept as a correct match … even number … odd number Even chance … less than 7 … the number 2 Impossible or 1m Matches any two statements to their correct positions 16 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 16 http://www.SATs-Papers.co.uk 30/11/07 22:29:01 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Sleep 3-5 4-6 5-7 6-8 11 3 Correct response Additional guidance –11 a a 1m 11 b b 2m 7pm or 19:00 ! Incorrect notation for time Condone eg, for 2m accept 7 19 7am or 1m Shows or implies that 12 hours’ sleep are needed eg 12 seen (30 – 6) ÷ 2 30 – 6 = 24, 24 ÷ 2 ! For 1m, necessary brackets omitted Condone eg, for 1m accept 30 – 6 ÷ 2 ! For 1m, incorrect order of operations shown Condone provided evaluation using the correct order is seen eg, for 1m accept 6 – 30 = 24, 24 ÷ 2 eg, for 1m do not accept (6 – 30) ÷ 2 For 1m, –12 U1 Tier & Question Sorting shapes 3-5 4-6 5-7 6-8 12 4 Correct response 2m Additional guidance Gives the three letters B, C and D in the correct places in the table, ie A D B C or 1m Unambiguous indication Gives at least two of the letters in the correct places in the table, with not more than one error or omission Any letter repeated in an incorrect place in the table eg, for 1 mark A A (error) D B C A A (error) D C (error) B C eg, for 0 marks 17 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 17 http://www.SATs-Papers.co.uk 30/11/07 22:29:01 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Shopping 3-5 4-6 5-7 6-8 13 5 Correct response a a 1m £ 1.15 b b 1m 5 Additional guidance ! Reference to remainder Condone reference to the correct amount of money left over eg, accept 5 with 20p change 5 r 20 eg, do not accept 5.5(…) or 5.6 5 with 55p change Tier & Question Speedometer 3-5 4-6 5-7 6-8 14 6 a a Correct response 1m Indicates the correct value on the scale, ie 60 40 20 Additional guidance ! Inaccurate indication Accept provided their marker would touch the circumference of the dial within 2mm of the correct position, if extended 80 mph 0 b b 1m 100 40 18 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 18 http://www.SATs-Papers.co.uk 30/11/07 22:29:02 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Football survey 3-5 4-6 5-7 6-8 15 7 Correct response 2m Gives the value 3 in the key and completes 3 circles for each of the Yes and No rows Additional guidance ! Circles not shaded, or inaccurate in size Accept provided the pupil’s intention is clear or 1m Shows or implies the value 9 eg Completes 9 circles for one or both rows or Draws the same number of circles for each of the Yes and No rows, provided this number is not 4, even if the value in the key is incorrect or omitted U1 Tier & Question Jug 3-5 4-6 5-7 6-8 16 8 Correct response 1m 750 1m 100 1m Additional guidance 1 or equivalent fraction or decimal 5 19 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 19 http://www.SATs-Papers.co.uk 30/11/07 22:29:02 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Double shape 3-5 4-6 5-7 6-8 17 9 a a Correct response 1m Indicates Yes and gives a correct explanation The most common correct explanations: Show or imply the correct areas eg The area of the rectangle is 18, the area of the square is 9 and 9 × 2 = 18 A is 18 and B is 18 ÷ 2 = 9 Additional guidance ! Incorrect units Condone eg, accept 18cm, 9cm 182, 92 Minimally acceptable explanation eg 18, 9 2 × 9 (or double 9), 9 18, 18 ÷ 2 Incomplete explanation eg The area of the rectangle is 18 Refer to the space taken up by each shape eg Two of the squares can ﬁt inside the rectangle If you draw a line down the middle of the rectangle, you get two of the squares A holds twice as many squares as B Minimally acceptable explanation eg A holds two squares You cut A in half to get B Rectangle divided into two squares on the diagram I counted the squares inside the shapes Incomplete explanation eg The area of A is twice the area of B B is half of A He’s just added another shape on I counted the squares Refer to the ratio of lengths together with the equal widths eg They are the same width but the rectangle is twice as long as the square 6 × 3 is twice 3 × 3 Minimally acceptable explanation eg Equal width, but the length is doubled Same height, but width is twice as long 6 × 3, 3 × 3 Incomplete explanation eg The rectangle is twice as long as the square Because A is 6 squares long and B is 3 squares long 20 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 20 http://www.SATs-Papers.co.uk 30/11/07 22:29:02 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Double shape (cont) 3-5 4-6 5-7 6-8 17 9 b b Correct response 1m Indicates No and gives a correct explanation Additional guidance ! Incorrect units Condone eg, accept 18cm2, 12cm2 The most common correct explanations: Show or imply the correct perimeters eg The perimeter of the rectangle is 18, the perimeter of the square is 12 but 2 × 12 18 2 × 9 is not twice 2 × 6 Minimally acceptable explanation eg 18, 12 2 × 9, 2 × 6 12 + 6, 12 It’s 6cm more but that’s not double 12 Incomplete or incorrect explanation eg The perimeter of the rectangle is 18 Area A = 18, area B = 12 Refer to the distance around each shape eg The length around the edge of the square goes more than halfway round the edge of the rectangle Minimally acceptable explanation eg It’s less than double the perimeter of the square B’s perimeter is more than half A’s I counted the distance round the sides Incomplete explanation eg The perimeter of A is not double the perimeter of B Refer to the rectangle’s additional lengths eg You only add two of the square’s sides to get the rectangle, not all four It’s increased by 50%, not doubled You join two squares, but two of their sides will be touching Minimally acceptable explanation eg It has two extra lengths of 3, not four It’s half as long again These sides are hidden Incomplete explanation eg It has two extra sides U1 21 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 21 http://www.SATs-Papers.co.uk 30/11/07 22:29:02 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Cube edges 3-5 4-6 5-7 6-8 18 10 1 Correct response 2m Additional guidance Completes the table correctly to show the further 5 ways with no errors or duplicates, ie A A A A A A Ways of moving from A to H B C B G D C D E F E F G H H H H H H Unambiguous indication eg, for A B G H ABGH ! Correct vertices, but in an incorrect order eg, for A B G H A G B H Do not accept as a correct way [rows in any order] or 1m Gives at least 3 of the correct ways, even if there are other errors or duplicates Tier & Question Track 3-5 4-6 5-7 6-8 19 11 2 Correct response a a a 1m 5 b b b 1m 6 Additional guidance ! Response assumes the piece of track shown has already been counted For answers of 4 for part (a) followed by 5 for part (b), mark as 0, 1 22 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 22 http://www.SATs-Papers.co.uk 30/11/07 22:29:03 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Matching expressions 3-5 4-6 5-7 6-8 20 12 3 Correct response 2m Additional guidance Matches all four statements correctly, ie 2 ! Statement matched to more than one expression For 2m or 1m, do not accept as a correct match 2–a a+2 2a a–2 2 a a2 a 2 or 1m Matches three of the statements correctly Tier & Question Area 3-5 4-6 5-7 6-8 21 13 4 Correct response 1m Additional guidance Gives both correct areas, ie 9 then 3 Tier & Question Values 3-5 4-6 5-7 6-8 22 14 5 Correct response a a a 1m 6 b b b 1m –2 Additional guidance ! Incomplete processing Penalise only the ﬁrst occurrence eg, for parts (a) and (b) 9–3 4–6 Mark as 0, 1 23 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 23 http://www.SATs-Papers.co.uk 30/11/07 22:29:04 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Symmetry patterns 3-5 4-6 5-7 6-8 23 15 6 Correct response a a a 1m Indicates two squares so that the shape has rotation symmetry of order 4, ie b b b 1m Indicates four squares in total [that include the same two squares required in part (a)] so that the shape has rotation symmetry of order 2 eg Additional guidance Unambiguous indication ! For part (b), response uses part squares Accept provided the intended symmetry is clearly correct 24 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 24 http://www.SATs-Papers.co.uk 30/11/07 22:29:05 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Shop 3-5 4-6 5-7 6-8 24 17 7 Correct response 2m Additional guidance £ 196.25 or Digits 19625 seen Markers may ﬁnd the following useful: or 1m Mon Tues Wed or Thur (Fri Sat Shows or implies the correct subtotals of pay for the hours worked at 6.35, or pay for the hours worked at 7.5(0) eg 158.75 25 × 6.35 37.5(0) 5 × 7.5(0) 15 and 22.5(0) seen or Shows the values 44.45, 40.4(0) and 22.5(0) 7 × 6.35 = 44.45 7 × 6.35 = 44.45 4 × 6.35 = 25.4 and 2 × 7.5 = 15 4 × 6.35 + 2 × 7.5 = 40.4(0) 7 × 6.35 = 44.45 0) 3 × 7.5 = 22.5 no. of hours worked pay per hour 25 5 6.35 7.5(0) total 158.75 37.5(0) 2 3 7.5(0) 7.5(0) 15(.00) 22.5(0) or or Shows or implies a complete correct method with not more than one computational error eg 7 + 7 + 4 + 7 = 26 (error), 26 × 6.35 + 5 × 7.5(0) = 202.60 or Gives an answer of 193.95 or 200.85 [the only error is to assume 6.35 or 7.50 for all hours on Wednesday] U1 25 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 25 http://www.SATs-Papers.co.uk 30/11/07 22:29:05 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Using algebra 3-5 4-6 5-7 6-8 25 18 8 Correct response Additional guidance 1m n+2 ! Unsimpliﬁed expression or unconventional notation eg, for Jo’s age n+1+1 1n + 2 eg, for Kate’s age 2 × (n + 2) n×2+4 Condone 1m 2(n + 2) or 2n + 4 ! For the second mark, follow-through Accept follow-through as 2 × their algebraic expression for Jo’s age provided there are no other errors eg, from Jo’s age as 2n accept 4n n×4 For the second mark, incomplete processing eg 2×n+2×2 For the second mark, necessary brackets omitted eg 2×n+2 2(n + 2 26 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 26 http://www.SATs-Papers.co.uk 30/11/07 22:29:05 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Goldbach 3-5 4-6 5-7 6-8 26 16 9 a a a Correct response 1m Additional guidance Gives a pair of prime numbers that sum to 16, ie 3 and 13, in either order or 5 and 11, in either order 1m Gives a different pair of prime numbers that sum to 16 from any credited for the ﬁrst mark Values credited for the ﬁrst mark repeated but in reverse order Completes the sentence correctly, giving an even number greater than 16 and a correct pair of prime numbers that sum to their number eg … even number 20 … … prime numbers 7 and 13 … even number 22 … … prime numbers 11 and 11 … even number 50 … … prime numbers 3 and 47 Their even number is less than or equal to 16 U1 b b b 1m Markers may ﬁnd the following values useful: Prime numbers up to 100 2, 3, 5, 7 11, 13, 17, 19 23, 29 31, 37 41, 43, 47 53, 59 61, 67 71, 73, 79 83, 89 97 U1 27 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 27 http://www.SATs-Papers.co.uk 30/11/07 22:29:05 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Side length 3-5 4-6 5-7 6-8 27 19 10 Correct response 2m Additional guidance 6.3 or equivalent or 1m Shows the value 25.2 or equivalent or For 1m, necessary brackets omitted eg 8.4 + 8.4 + 8.4 ÷ 4 Shows a complete correct method with not more than one computational error eg 8.4 × 3 ÷ 4 (8.4 + 8.4 + 8.4) ÷ 4 8.4 + 8.4 + 8.4 = 25.6 (error), 25.6 ÷ 4 = 6.4 U1 28 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 28 http://www.SATs-Papers.co.uk 30/11/07 22:29:06 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Value of x 3-5 4-6 5-7 6-8 20 11 1 Correct response a a a 1m Additional guidance Indicates … one particular number, ie U1 b b b 1m Indicates … any number at all, ie U1 Tier & Question Darts 3-5 4-6 5-7 6-8 21 12 2 Correct response 1m Additional guidance Gives all three correct numbers, ie 10, 15 and 20 [any order] 29 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 29 http://www.SATs-Papers.co.uk 30/11/07 22:29:06 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Conversions 3-5 4-6 5-7 6-8 22 13 3 Correct response 1m Gives a correct explanation The most common correct explanations: Additional guidance Explanation does not use the values in the given table eg 1 ounce is more like 28g They only use 25g as roughly equal, so those values are not accurate ! Explanation states or implies what values ‘should be’ or that the table is ‘incorrect’ Condone Show the values in grams do not consistently go up/down in steps of 25 per ounce eg It goes up in 25s until the step from 3 to 4 ounces when it suddenly goes up 35 It should go from 150g down to 125g, but it’s 110g instead Minimally acceptable explanation eg It goes up in 25s at ﬁrst but then changes It goes up 25, 25, 35, 40 and so it is not a steady pattern It should go 25, 50, 75, 100 The numbers should go up by the same amount each time Incomplete explanation eg 25, 25, 35, 40 4 ounces should be 100g and 10 ounces should be 250g They don’t go up in proportion Show that the relationship between two values in grams is not what other values would predict eg If 1 ounce is 25g, then 4 ounces should be 25 × 4 = 100g not 110g If 5 ounces is 150g, then 10 ounces should be 150 × 2 = 300g not 275g 10 ounces in grams should be 25 × 10 = 250, but it is 275 in the table 50 ÷ 2 = 25, but 150 ÷ 5 = 30 Minimally acceptable explanation eg 25 × 4 110 4 should be 25 × 4 = 100 150 × 2 275 If 5 is 150, then 10 should be 300 50 ÷ 2 150 ÷ 5 10oz should equal double 5oz but it doesn’t Incomplete explanation eg 1 ounce is 25g so 4 ounces shouldn’t be 110g 5 ounces = 150g, but 10 ounces = 275g U1 30 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 30 http://www.SATs-Papers.co.uk 30/11/07 22:29:06 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Concorde 3-5 4-6 5-7 6-8 23 14 4 Correct response 2m Additional guidance 1200 or 1m Shows or implies a correct rate, other than 1 mile every 3 seconds, even if it doesn’t use single units of time eg 20 (miles) per minute 1 (mile) in a sec 3 10 miles in 30 seconds 60 miles every 3 mins ! For 1m, unit(s) abbreviated Condone provided unambiguous within the context of the question eg, for 1m accept 20m per min 1 m/s [miles implied by given context] 3 eg, for 1m do not accept 20m per m [ambiguity between miles and minutes] or Shows or implies a complete correct method with not more than one computational or rounding error eg 20 × 60 60 × 60 3 1 × 3600 3 1 ÷ 3 = 0.33 (premature rounding) 0.33 × 602 = 1188 31 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 31 http://www.SATs-Papers.co.uk 30/11/07 22:29:06 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Counters in a bag 3-5 4-6 5-7 6-8 24 15 5 Correct response 2m Additional guidance Completes the sentence correctly with three positive integers r, w then y, such that w = 2r and y < r eg 2, 4 then 1 3, 6 then 1 or 2 4, 8 then 1, 2 or 3 or 1m Completes the sentence with three integers r, w then y, such that w = 2r and y = 0 eg 2, 4 then 0 3, 6 then 0 For 1m, values for r or w negative or zero eg –1, –2 then 0 0, 0 then 0 or Completes the sentence with three values r, w then y between zero and one, such that 1 r > , w = 2r and r + w + y = 1 4 eg 1 2 4 , then 7 7 7 0.3, 0.6 then 0.1 32 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 32 http://www.SATs-Papers.co.uk 30/11/07 22:29:07 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Perimeters 3-5 4-6 5-7 6-8 25 16 6 Correct response a a a 1m Additional guidance ! Unsimpliﬁed expression or unconventional notation eg 42a + 18 6 (42 × a + 18) ÷ 6 Condone 7a + 3 Necessary brackets omitted eg 42a + 18 ÷ 6 b b b 1m 5 c 24 c c 1m ! Units given Ignore, even if incorrect for a perimeter eg, accept 24cm 24cm2 Incomplete processing eg 4×6 Tier & Question Yoghurt 3-5 4-6 5-7 6-8 26 17 7 Correct response 2m Additional guidance 125 or 1m Shows or implies recognition of the need to divide by 7 eg 5 × 175 7 175 ÷ 7 25 seen or Shows the value 50 [mass of fruit] 33 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 33 http://www.SATs-Papers.co.uk 30/11/07 22:29:07 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Lawn 3-5 4-6 5-7 6-8 27 18 8 Correct response 2m Additional guidance 28.(…) or 9π or 1m Shows or implies a complete correct method for ﬁnding the area of the lawn, with no evidence of conceptual error and not more than one computational or rounding error eg Shows the digits 282(…) or 283 32 × π π = 3 (rounding error), 9 × 3 = 27 Tier & Question For 1m, conceptual error eg 32 × π = 19 or 18.8(…) or 6π π32 = 89 Area = 2 × 3 × π Triangular numbers 3-5 4-6 5-7 6-8 28 19 9 Correct response a a a 1m 55 b b b 1m Additional guidance 5050 34 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 34 http://www.SATs-Papers.co.uk 30/11/07 22:29:07 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Isosceles triangle 3-5 4-6 5-7 6-8 29 21 10 Correct response 2m Additional guidance Gives x = 74, y = 32 and z = 46 and gives a correct reason for each angle The most common correct reasons: For angle x, refer to the isosceles triangle eg It is an isosceles triangle, so it is equal to angle ADB The triangle is isosceles so it is the same as the 74° angle marked For angle y, refer to angles in a triangle eg Angles in a triangle, so 180 – 74 – 74 74 + 74 = 148 and 180 – 148 because they add up to 180 in a triangle Minimally acceptable reason eg Isosceles Incomplete reason without the correct geometrical property identiﬁed eg It is equal to angle ADB It is the same as the 74° angle marked Minimally acceptable reason eg Angles in a triangle ! Follow-through from their x For angle y, accept 106 – their x accompanied by a correct reason Incomplete reason without the correct geometrical property identiﬁed eg 180 – 74 – 74 74 + 74 = 148 and 180 – 148 For angle z, refer to angles in a triangle and angles on a straight line or just angles in a triangle or exterior angle of a triangle eg Angles in a triangle, 180 – 28 – 74 – 32 Angles on a straight line, 180 – 74 = 106, angles in a triangle, 180 – 106 – 28 Exterior angle of a triangle, 74 – 28 Minimally acceptable reason eg Angles in a triangle Angles on a straight line and angles in a triangle Exterior angle of a triangle ! Follow-through from their x and their y For angle z, accept 152 – their x – their y accompanied by a correct reason Incomplete reason without the correct geometrical property identiﬁed eg 180 – 28 – 74 – 32 180 – 74 = 106, 180 – 106 – 28 or 1m Gives two correct angles with a correct reason for each or U1 For 1m, follow-through Accept follow-through for each angle as detailed above Gives all three correct angles, even if reasons are incorrect or omitted 35 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 35 http://www.SATs-Papers.co.uk 30/11/07 22:29:08 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Journeys 3-5 4-6 5-7 6-8 30 20 11 Correct response a a 1m Additional guidance Gives all four names in the correct order, ie Chris Dee Ann Ben b b 2m Joins the points (0, 0), (15, 1), (30, 1.5) and (60, 4) with straight lines, ie Unambiguous indication eg C D A B ! Lines not ruled or accurate Accept provided the pupil’s intention is clear 4 3 2 1 0 0 10 20 30 40 50 60 or 1m Indicates at least two of the points (15, 1), (30, 1.5) and (60, 4) on the graph, even if they are not joined or are joined incorrectly or Shows or implies all three sets of coordinates (15, 1), (30, 1.5) and (60, 4) in working, even if the graph is incorrect or omitted ! For 1m, follow-through from their (15, 1) with an incorrect y-value For an incorrect y-value between 0.5 and 3 inclusive, accept their (30, 1.5) as (30, their incorrect y-value + 0.5) eg, for 1m accept 4 3 2 1 0 0 c c 1m 5 10 20 30 40 50 60 ! Follow-through from their graph in part (b) Provided their line for the ﬁnal section of the graph has a positive gradient and passes through (60, 4), accept follow-through as 2 × (4 – their y-coordinate for (30, 1.5)) 36 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 36 http://www.SATs-Papers.co.uk 30/11/07 22:29:08 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Special offer 3-5 4-6 5-7 6-8 22 12 Correct response 2m Indicates Both paid the same and gives a correct justiﬁcation eg Marie paid 96 – 9.60 = 86.40 Richard paid 108 – 21.60 = 86.40 0.9 × 96 = 86.4 0.8 × 108 = 86.4 Additional guidance For 2m, minimally acceptable justiﬁcation eg 96 – 9.6(0), 108 – 21.6(0) 0.9 × 96, 0.8 × 108 86.4(0) For 2m or 1m, incomplete justiﬁcation eg 10% off 96 is the same as 20% off 108 It works out to be the same or 1m Gives a correct justiﬁcation but makes an incorrect or no decision or For 1m, conceptual error eg 20% off 108 = 108 – (108 ÷ 20) = 108 – 5.40 = 102.60 Gives a correct justiﬁcation with not more than one computational or rounding error, but follows through to make their correct decision eg Marie paid 96 – 9.60 = 87.4(0) (error) Richard paid 108 – 21.60 = 86.4(0) [indicates Marie] U1 37 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 37 http://www.SATs-Papers.co.uk 30/11/07 22:29:08 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Planes Marking overlay available 3-5 4-6 5-7 6-8 23 13 Correct response a a 1m Indicates the correlation is positive Additional guidance ! Positive qualiﬁed Ignore eg, accept Strong positive Direct positive Sign of correlation not indicated eg High Strong ! Relationship quantiﬁed Ignore alongside a correct response Relationship described without reference to correlation eg The greater the wingspan, the more passengers it can hold b b 1m Draws a line of best ﬁt within the tolerance, and at least of the length, as shown on the overlay ! Line not ruled or accurate Accept provided the line is within tolerance, and at least of the length required ! Line of best ﬁt is incorrect beyond the dashed lines on the overlay Condone eg, accept A correct line of best ﬁt that is then joined to the origin c c 2m 3600 to 5200 inclusive or 1m Shows a value between 180 and 260 inclusive or Shows a value that follows through from their line of best ﬁt eg Their line passes through the point (40, 280), ﬁnal answer: 5600 ! For 1m, range for follow-through value If their line goes through (40, y) accept follow-through as 20 × (y ± 10) provided their line always has a positive gradient U1 38 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 38 http://www.SATs-Papers.co.uk 30/11/07 22:29:08 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Cubes 3-5 4-6 5-7 6-8 24 14 Correct response 2m Additional guidance 27 or 1m Shows the values 216 (or 63 or 6 × 6 × 6) and 8 (or 23 or 2 × 2 × 2), even if there are errors or Shows or implies that 3 of the smaller cubes will ﬁt along each edge of the larger cube eg 33 or 3 × 3 × 3 3 by 3 by 3 39 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 39 http://www.SATs-Papers.co.uk 30/11/07 22:29:09 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Best buy 3-5 4-6 5-7 6-8 25 15 Correct response 2m Indicates A and gives a correct justiﬁcation, based on correctly calculating a pair of comparable values The most common justiﬁcations: Compare pence (or pounds) per gram eg 159 ÷ 454 = 0.35(…) 125 ÷ 340 = 0.36(…) (or 0.37) Compare grams per penny (or per pound) eg 454 ÷ 159 = 2.8(…) (or 2.9) 340 ÷ 125 = 2.7(2) 454 ÷ 1.59 = 285.(…) (or 286) 340 ÷ 1.25 = 272 Reason proportionally using the prices eg 125 ÷ 340 × 454 = 166.(…) (or 167) That’s more than 159 159 ÷ 454 × 340 = 119.(…), which is < 125 1.59 × 340 = 540(.6) (or 541) 1.25 × 454 = 567(.5) (or 568) 2 × 340 = 680g, which is £2.50 1.5 × 454 = 681g, which is only £2.39 4 × 340g = 1360g for £5 3 × 454g = 1362g for £4.77 If A were decreased by 114g its price should go down by 40p (or 39.(..)p), but the difference is 34p so it’s a worse reduction 454 – 340 = 114g, £1.59 – £1.25 = 34p 114 but × 1.25 = 42p (or 41.(…)p) 340 or 1m Shows a correct pair of comparable values but makes an incorrect or no decision or Shows correct calculations for a pair of comparable values, with not more than one error if evaluation is attempted, then follows through to make their correct decision eg 159 ÷ 454 and 125 ÷ 340, so A 454 ÷ 159 = 2.8(…) 340 ÷ 125 = 27.2 (error), so B Additional guidance For 2m, correct decision and any pair of comparable values shown Note that common pairs are: 0.35(…) and 0.36(…) or 0.37 (p per g) 0.0035(…) and 0.0036(…) or 0.0037 (£ per g) 2.8(…) or 2.9 and 2.7(2) (g per p) 285.(…) or 286 and 272 (g per £) 159 and 166.(…) or 167 (p per 454g) 119.(…) and 125 (p per 340g) 540(.6) or 541 and 567(.5) or 568 (£ per 154 360g) 34 and 39.(…) or 40 (p for 114g extra compared to A) 34 and 41.(…) or 42 (p for 114g extra compared to B) ! Correct decision and comparison is per 454g or per 340g but given price is not restated Condone eg, for 2m accept 125 ÷ 340 × 454 = 167 ! Correct decision but units omitted, incorrect or inconsistent Condone provided any values used to make a decision are comparable eg, for 2m accept 1.59 ÷ 454 = 0.35 1.25 ÷ 340 = 0.37 ! Additional incorrect working Ignore For 2m or 1m, incomplete justiﬁcation eg 454 – 340 = 114g £1.59 – £1.25 = 34p Therefore jar A because you get 114g more for only 34p extra For 2m or 1m, comparable values, or the method to calculate them, not shown eg The big jar is 8p cheaper U1 40 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 40 http://www.SATs-Papers.co.uk 30/11/07 22:29:09 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Shadows 3-5 4-6 5-7 6-8 26 16 Correct response 2m Additional guidance 4.2 or equivalent or 1m Shows the value 2 3 or or equivalents 3 2 or ! For 1m, value rounded 2 For , accept 0.66(…) or 0.67 3 Shows or implies a complete correct method with not more than one computational or rounding error eg 1.8 ÷ 2.7 × 6.3 1.8 ÷ 2.7 = 0.6 (rounding error) 0.6 × 6.3 = 3.78 6.3 ÷ 2.7 = 2.3 (rounding error) 1.8 × 2.3 = 4.14 41 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 41 http://www.SATs-Papers.co.uk 30/11/07 22:29:09 2008 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question 1, 2, 4 3-5 4-6 5-7 6-8 27 17 Correct response 3m Additional guidance Gives a complete correct response that satisﬁes all four of the following conditions: 1. Indicates that A is 8 2. Indicates that B is 7 3. Indicates that C is 8 4. Shows or implies correct substitution at least for value C eg 4(42 – 3 × 4 + 8) 6 4 × 12 6 48 ÷ 6 or 2m Gives a response that satisﬁes three of the four conditions or 1m Gives a response that satisﬁes two of the four conditions 42 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 42 http://www.SATs-Papers.co.uk 30/11/07 22:29:10 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Triangles 3-5 4-6 5-7 6-8 28 18 Correct response a 2m 14.4(…), or 4 13, or 208 Additional guidance ! Value of 14 Do not accept unless a correct method or a more accurate value is seen For 2m or 1m, method uses accurate or scale drawing or 1m b 2m Shows a correct method that indicates at least the intention to square and subtract the two given lengths eg 172 – 92 289 – 81 208 seen 7.8 or 7.79(…) ! Value of 8 Do not accept unless a correct method or a more accurate value is seen For 2m or 1m, method uses accurate or scale drawing or 1m Shows or implies a correct trigonometric ratio involving not more than one unknown eg Answer of 7.7 12 tan 33 DF tan 33 = 12 tan 33 = 0.6 (premature rounding), 12 × 0.6 = 7.2 12 tan 57 = x ! For 1m, no indication of which angle is being considered eg DF tan = 12 For 1m, accept only if the trigonometric ratio is correct for the given angle DEF 43 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 43 http://www.SATs-Papers.co.uk 30/11/07 22:29:10 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Box plots 3-5 4-6 5-7 6-8 19 Correct response Additional guidance a 1m 6 b 1m Gives a correct justiﬁcation eg Median for year 10 = 56, Median for year 11 = 65, 65 – 56 = 9 The medians are the vertical lines inside the 1 grey boxes, they are 4 divisions apart and 2 this is 9 marks since 1 division = 2 marks Minimally acceptable justiﬁcation eg 56, 65 The medians are the vertical lines inside the boxes and they are 9 marks apart There is a gap of 9 [with both medians indicated on the graph] ! Ambiguous notation eg 56 – 65 Condone Incomplete justiﬁcation eg The difference between the medians is 9 marks on the graph c 1m Indicates Yes and gives a correct explanation, referring to the inter-quartile range eg Inter-quartile range for year 10 = 33, Inter-quartile range for year 11 = 18, so year 11 was more consistent The middle half of the year group was less spread out for year 11 than for year 10 The grey box shows the inter-quartile range and it is shorter for year 11 Minimally acceptable explanation eg 33, 18 Its inter-quartile range is 15 less The IQ range is smaller The IQ range is bigger for year 10 The box is shorter (or smaller) For Y10: 43 to 76, for Y11: 51 to 69 It is shorter [distance between upper and lower quartiles indicated on both box plots] ! ‘Inter-quartile range’ referred to as ‘range’ within an otherwise correct explanation Accept only if it is clear the response actually refers to the inter-quartile range eg, accept For year 10, range = 33 For year 11, range = 18 eg, do not accept The range is bigger for year 10 Incomplete explanation eg Year 11 is shorter than year 10 The results for year 10 are more spread out than in year 11 U1 44 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 44 http://www.SATs-Papers.co.uk 30/11/07 22:29:10 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Circle graph 3-5 4-6 5-7 6-8 20 Correct response a 2m Additional guidance Completes both pairs of coordinates correctly, ie (3, 4) and (3, –4), in either order or 1m Completes either pair of coordinates correctly or Shows the value 16 or Shows or implies a correct method for ﬁnding the value of y eg y2 = 25 – 32 b 1m 5 c 2m Gives P as (3.5, 3.5) –5 or ± 5 ! For 2m, gives P as (–3.5, –3.5) Condone For 2m, equivalent fractions or decimals or 1m Shows the value 3.5(…) or 12.5 or equivalent or Shows or implies a correct method for ﬁnding the value of x or y eg 2y2 = 25 x2 = 25 ÷ 2 45 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 45 http://www.SATs-Papers.co.uk 30/11/07 22:29:10 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Giant pandas 3-5 4-6 5-7 6-8 21 Correct response 2m 1100 Additional guidance ! For 2m upper bound used Since pupils could assume 1600 is given to the nearest 100 in the context of the question, accept use of upper bound provided a correct method is seen eg, for 2m accept 1650 ÷ 140 × 100, answer: 1200 or 1m Shows the digits 11(…) or Shows or implies a complete correct method eg 1600 ÷ 140 × 100 1600 1.4 160 000 140 For 1m, lower and/or upper bound used within a correct method eg, for 1m accept 1650 ÷ 140 × 100 1550 ÷ 1.4 46 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 46 http://www.SATs-Papers.co.uk 30/11/07 22:29:11 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Prism 3-5 4-6 5-7 6-8 22 Correct response 3m 6.9(…), or 4√3, or √48 or 2m Shows or implies a correct method with not more than one computational or rounding error eg √32 + 16 √32 = 5.6 (rounding error) AC2 = 5.62 + 42 AC = 6.8(…) √32 = 6 (premature rounding) √36 + 16 = 7.2 Additional guidance ! Value of 7 Do not accept unless a correct method or a more accurate value is seen For 3m, 2m or 1m, method uses accurate or scale drawing or 1m Shows sufﬁcient working to indicate correct application of Pythagoras’ theorem for at least one triangle eg 42 + 42 2 × 16 5.6(…) or 5.7 seen (Their BC)2 + 42 U1 47 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 47 http://www.SATs-Papers.co.uk 30/11/07 22:29:11 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Number cards 3-5 4-6 5-7 6-8 23 Correct response 2m Additional guidance Gives all three correct values, ie 15 20 25 in any order or 1m Gives any two correct values, with not more than one error or omission or States or implies that n is a multiple of 5 and that n there are square numbers 5 eg There must be 1 out of 5, 2 out of 10, 3 out of 15 etc for the fraction to be right 1 2 3 4 5, but should be only one 6 7 8 9 10, but should be only two 11 12 13 14 15, correct ! For 1m, minimally acceptable implication For 1m, accept responses in which there are at least three examples using multiples of 5, (with no examples not using multiples of 5) and some square numbers identiﬁed, even if there are errors or omissions eg 1, 2, 3, 4, 5, so n could be 5 6, 7, 8, 9, 10, so n could be 10 11, 12, 13, 14, 15 U1 48 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 48 http://www.SATs-Papers.co.uk 30/11/07 22:29:11 2008 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Window 3-5 4-6 5-7 6-8 24 Correct response 3m Additional guidance Gives an integer value between 3925 and 3928 inclusive or 2m Shows a non-integer value between 3925 and 3927.5 inclusive [rounding to the nearest whole number omitted] or Shows an integer value between 7850 and 7855 inclusive [division of whole circle area by 2 omitted] or Shows or implies a complete correct method with not more than one error, and follows through to give their value correct to the nearest whole number eg 1m ÷ 2 = 50cm, π × 502 integer response outside = correct range 2 π × 0.5 × 0.5 = 0.79 (premature rounding), 0.79 ÷ 2 = 0.395, 0.395 × 10000 = 3950 π × 0.52 × 100 (error) = 39 2 For 2m or 1m, conceptual error eg π × 100 ÷ 2 = 157 For 2m uses a radius of 25 or 0.25 or 1m Shows a non-integer value between 7850 and 7855 inclusive or Shows the value 0.39(…) or equivalent [ie, the correct area in m²] or Shows or implies a complete correct method with not more than one error but fails to follow through to give their value correct to the nearest whole number eg 1m ÷ 2 = 50cm, π × 502 non-integer response outside = correct range 2 π × 252 (error) ÷ 2 = 981.75 U1 49 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 49 http://www.SATs-Papers.co.uk 30/11/07 22:29:11 2008 KS3 Mathematics test mark scheme: Paper 2 Tier Index Question Page 1 Rounding 11 2 Cuboid 12 3 Placing 40 12 4 Directions 13 5 Writing cheques 14 6 Theme park 14 7 Adding odd 15 8 Calculating 15 3–5 4–6 5–7 6–8 9 1 Time machine 16 10 2 Four cards 16 11 3 Sleep 17 12 4 Sorting shapes 17 13 5 Shopping 18 14 6 Speedometer 18 15 7 Football survey 19 16 8 Jug 19 17 9 Double shape 20 18 10 1 Cube edges 22 19 11 2 Track 22 20 12 3 Matching expressions 23 21 13 4 Area 23 22 14 5 Values 23 23 15 6 Symmetry patterns 24 24 17 7 Shop 25 25 18 8 Using algebra 26 26 16 9 Goldbach 27 27 19 10 Side length 28 50 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 50 http://www.SATs-Papers.co.uk 30/11/07 22:29:12 2008 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 Index Question Page 4–6 5–7 6–8 20 11 1 Value of x 29 21 12 2 Darts 29 22 13 3 Conversions 30 23 14 4 Concorde 31 24 15 5 Counters in a bag 32 25 16 6 Perimeters 33 26 17 7 Yoghurt 33 27 18 8 Lawn 34 28 19 9 Triangular numbers 34 29 21 10 Isosceles triangle 35 30 20 11 Journeys 36 22 12 Special offer 37 23 13 Planes 38 24 14 Cubes 39 25 15 Best buy 40 26 16 Shadows 41 27 17 1, 2, 4 42 28 18 Triangles 43 19 Box plots 44 20 Circle graph 45 21 Giant pandas 46 22 Prism 47 23 Number cards 48 24 Window 49 51 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 51 http://www.SATs-Papers.co.uk 30/11/07 22:29:12 QCA wishes to make its publications widely accessible. 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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 Sourced from SATs-Papers.co.uk 282676_MS_P2.06.indd 52 For more copies: QCA Orderline, PO Box 29, Norwich NR3 1GN www.qca.org.uk/orderline email: orderline@qca.org.uk Tel: 08700 60 60 15 Fax: 08700 60 60 17 QCA/08/3286 282676 http://www.SATs-Papers.co.uk 30/11/07 22:29:12