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Ma Mathematics tests KEY STAGE 3 for Paper 2 Tiers 3–5, 4–6, 5–7 and 6–8 2007 ALL TIERS Mark scheme National curriculum assessments Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 1 http://www.SATs-Papers.co.uk 19/12/06 2:21:14 pm 2007 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 2007 key stage 3 mathematics tests and mark schemes were developed by the Test Development team at Edexcel. 2 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 2 http://www.SATs-Papers.co.uk 19/12/06 2:21:15 pm 2007 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. Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 3 http://www.SATs-Papers.co.uk 3 19/12/06 2:21:15 pm 2007 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. 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 275669_MS_P2.09.indd 4 http://www.SATs-Papers.co.uk 19/12/06 2:21:15 pm 2007 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. Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 5 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. http://www.SATs-Papers.co.uk 5 19/12/06 2:21:16 pm 2007 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: –2 Accept 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 275669_MS_P2.09.indd 6 http://www.SATs-Papers.co.uk 19/12/06 2:21:16 pm 2007 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 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 7 ! 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 http://www.SATs-Papers.co.uk 7 19/12/06 2:21:16 pm 2007 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 ) 8 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 8 y x ( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x – 5, y – 7 ) http://www.SATs-Papers.co.uk 19/12/06 2:21:17 pm 2007 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 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 9 http://www.SATs-Papers.co.uk 9 19/12/06 2:21:17 pm 2007 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 25 June 2007. 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. 10 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 10 http://www.SATs-Papers.co.uk 19/12/06 2:21:18 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Rules 3-5 4-6 5-7 6-8 1 Correct response 1m 11, 14 1m 23, 47 1m 41, 122 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 11 Additional guidance ! First new term for each sequence correct with second terms all incorrect or omitted Mark as 0, 0, 1 http://www.SATs-Papers.co.uk 11 19/12/06 2:21:18 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 3–5 only Homework 3-5 4-6 5-7 6-8 2 Correct response a 1m Monday and Wednesday, in either order b 1m Maths, English and Technology, in any order c 1m Additional guidance 3 Tier & Question ! Names of days or subjects abbreviated Accept provided unambiguous eg, for part (a) accept M and W eg, for part (b) do not accept M, E and T Odd one out 3-5 4-6 5-7 6-8 3 Correct response a 1m D 1m b E 1m Completes the sentence correctly with a correct property eg … equal sides … lines of symmetry … as the order of rotation symmetry Additional guidance Minimally acceptable response eg … sides the same … line symmetry … rotation symmetry … identical lines ! Incorrect or irrelevant statement Ignore alongside a correct response eg, accept … equal sides and right angles eg, do not accept … right angles Incomplete or incorrect response eg … sides … equal angles … squares for the area U1 12 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 12 http://www.SATs-Papers.co.uk 19/12/06 2:21:18 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Hibernation 3-5 4-6 5-7 6-8 4 Correct response a 1m 5 Additional guidance Value qualiﬁed eg About 5 ! Value inaccurate Accept values between 4.9 and 5.1 inclusive, or between 4 months 27 days and 5 months 3 days inclusive b 1m Indicates Yes and gives a correct explanation The most common correct explanations: State or imply that they sleep for more than 6 months eg 1 They sleep for 6 months, which is more 2 than half of 12 Half a year is 6 months but they sleep for just over 6 months Minimally acceptable explanation eg 1 6 months 2 Just over 6 months More than 6 boxes are shaded November to May is six months and then half of October Half a month more ! Exact value given Accept values between 6.4 and 6.6 months inclusive, or between 6 months 12 days and 6 months 18 days inclusive eg, accept 6 months and 2 weeks Incomplete or incorrect explanation eg They sleep for more than half the year They sleep from halfway through October to the end of April Half a year is 6 months but they sleep for 7 months 1 6 2 Refer to the area shaded or unshaded and its relation to the whole circle eg More than halfway round the circle is shaded The white bit for dormice doesn’t reach round half the circle Minimally acceptable explanation eg More than halfway round More than half the chart is shaded More is shaded than unshaded Incomplete explanation eg It shows more shaded months U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 13 http://www.SATs-Papers.co.uk 13 19/12/06 2:21:19 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Concert 3-5 4-6 5-7 6-8 5 Correct response a 1m £ 98.35 b 2m Additional guidance 5 or 1m Shows the digits 8225 or Shows or implies a complete correct method with not more than one error, even if their ﬁnal answer is not an integer or is rounded or truncated eg (155.75 – 3 × 24.50) ÷ 16.45 73.5 + 16.45 + 16.45 + 16.45 + 16.45 + 16.45 = 155.75 U1 Tier & Question Cake 3-5 4-6 5-7 6-8 6 Correct response a 1m 450 b 1m Indicates the correct position on the scale, ie Additional guidance 200 c 1m 2pm or 14:00 d 1m 300 Unambiguous indication ! Inaccurate indication Accept indications that are closer to 275 than either 250 or 300 Time ambiguous or incorrect eg 2 o’clock 14:00am Indicates Cylinder, ie 14 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 14 http://www.SATs-Papers.co.uk 19/12/06 2:21:19 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Bar chart 3-5 4-6 5-7 6-8 7 Correct response 2m Additional guidance Completes all labels for both axes correctly, ie Unambiguous indication of item names eg, for Glue G 24 20 16 12 8 4 0 Glue Pens Rulers or 1m Completes at least two labels on the vertical axis correctly U1 Tier & Question Coordinates 3-5 4-6 5-7 6-8 8 1 Correct response a a Additional guidance 1m 1m Gives C as (4, 3) ! Answers for A and C transposed but otherwise completely correct If this is the only error, ie gives A as (4, 3) and gives C as (0, 6), mark as 0, 1 1m b b Gives A as (0, 6) Indicates point D on the graph at (2, 7) ! Point inaccurate, not labelled or marked only with the letter D Condone any unambiguous indication within 2mm of the correct intersection of the grid Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 15 http://www.SATs-Papers.co.uk 15 19/12/06 2:21:20 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Fitting tiles 3-5 4-6 5-7 6-8 9 2 Correct response a a 1m Indicates correctly two congruent F-tiles on the diagram eg Additional guidance ! Tile not shaded or inaccurately indicated Accept provided the pupil’s intention is clear and there is no ambiguity Tiles overlapping 1m Indicates two congruent tiles on the diagram eg 1m b b Indicates two congruent tiles on the diagram, different from any previously credited U1 16 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 16 http://www.SATs-Papers.co.uk 19/12/06 2:21:20 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Names 3-5 4-6 5-7 6-8 10 3 Correct response Additional guidance Unambiguous indication of name eg, for Claire C a a 1m Claire b b 1m Gives the names Claire then Tom U1 Tier & Question Leaves 3-5 4-6 5-7 6-8 11 4 Correct response a a 1m Additional guidance Willow Oak Beech 1m smallest area Writes the leaves in the correct order for perimeter, ie Willow smallest perimeter Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 17 W largest area smallest area b b Unambiguous indication eg, for part (a) Writes the leaves in the correct order for area, ie Beech O B largest area ! Order given as largest to smallest for both parts (a) and (b) Mark as 0, 1 Oak largest perimeter ! Responses for parts (a) and (b) transposed but otherwise correct Mark as 0, 1 http://www.SATs-Papers.co.uk 17 19/12/06 2:21:21 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Marbles 3-5 4-6 5-7 6-8 12 5 Correct response 2m Additional guidance Matches all three questions correctly, ie 10 ! Question matched with more than one calculation For 2m or 1m, do not accept as a correct match 10 × 7 10 × 12 12 × 7 10 × 12 × 7 10 + 12 + 7 or 1m Matches any two of the questions correctly Tier & Question a and b 3-5 4-6 5-7 6-8 13 6 Correct response 1m Gives a pair of numbers for a and b, such that b=a+4 eg a=5 b=9 a = 1.5 b = 5.5 1m Additional guidance Values embedded eg 4+5=9 a=4+5 b=9 Gives a pair of numbers for a and b, such that b = a + 4, different from any previously credited U1 18 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 18 http://www.SATs-Papers.co.uk 19/12/06 2:21:22 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Turning 3-5 4-6 5-7 6-8 14 7 Correct response 1m Additional guidance Indicates the correct shape, ie Tier & Question Party 3-5 4-6 5-7 6-8 15 8 Correct response 2m Additional guidance Completes all four entries in the table correctly, ie 4.95 5 24.75 3.20 13 41.60 1.95 10 19.50 Total: 85.85 or 1m Completes at least three entries in the table correctly or ! For 1m, follow through Where the only error is in the total cost of balloons, for the overall total accept their total cost of balloons + 61.10 Completes all four entries correctly with some or all amounts of money given in pence U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 19 http://www.SATs-Papers.co.uk 19 19/12/06 2:21:23 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6 Tier & Question Survey 3-5 4-6 5-7 6-8 16 9 Correct response Additional guidance a a 1m 10 10% b b 2m Completes the percentage bar chart correctly, ie 50% labelled No 40% labelled Don’t know 10% labelled Yes, with bars in any order eg Unambiguous labelling eg No 0% 20% Don’t know Yes 40% 60% 80% 100% ? 0% 20% 40% ? 60% ? ? 80% 100% ! Lines not ruled or accurate Accept provided the pupil’s intention is clear or 1m Indicates sections corresponding to 50%, 40% and 10% but fails to label, labels incorrectly or bars are not continuous eg ? ? 0% 20% 40% ? 60% ? 80% 100% or Shows or implies the values 50, 40 and 10 eg 0% 20% 40% 60% 80% 100% or Indicates a correct bar for either Don’t know or Yes, and labels correctly 20 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 20 http://www.SATs-Papers.co.uk 19/12/06 2:21:23 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Frog spawn 3-5 4-6 5-7 6-8 17 10 1 Correct response Additional guidance a a a 1m 15th February (1997) Unambiguous or commonly used date notation eg 15/2 2/15 [US notation] b b b 1m Gives a possible description of the weather eg In 1991 it was colder than the other years It must have been less warm than usual Minimally acceptable response eg Cold Not warm It got warmer later ! Response implies a preference Condone provided the pupil’s intention is clear eg, accept It must have been nasty weather It was rainy and not sunny Bad Incomplete or incorrect response eg They were seen later than in other years Very cold so the eggs were seen quicker U1 Tier & Question Simplifying 3-5 4-6 5-7 6-8 18 11 2 Correct response a a a 1m Indicates 4a + 3, ie b b b 1m Additional guidance 8b + 3 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 21 http://www.SATs-Papers.co.uk 21 19/12/06 2:21:24 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Containers 3-5 4-6 5-7 6-8 19 12 3 Correct response 1m Additional guidance Indicates A and gives the value 250 Tier & Question Triangles 3-5 4-6 5-7 6-8 20 13 4 Correct response a a a 1m Gives the values 60, 60 and 60 b b b 1m Additional guidance Single answer of 60 given Gives the values 90, 45 and 45, in any order Tier & Question Spinners 3-5 4-6 5-7 6-8 21 14 5 Correct response a a a 1m Indicates B b b b 1m Additional guidance Indicates A and D, in either order 22 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 22 http://www.SATs-Papers.co.uk 19/12/06 2:21:25 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Faces 3-5 4-6 5-7 6-8 22 15 6 Correct response a a a 1m 8 b b b 1m Draws a solid with 6 faces in any orientation, using the isometric grid correctly eg Additional guidance Some or all internal lines shown eg ! Lines not ruled Accept provided the pupil’s intention is clear ! Drawing not accurate Accept vertices within 2mm of the dots of the grid ! Some or all hidden lines shown Do not accept unless the lines are clearly identiﬁed as hidden lines eg, accept eg, do not accept Isometric grid not used correctly eg Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 23 http://www.SATs-Papers.co.uk 23 19/12/06 2:21:25 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Fir trees 3-5 4-6 5-7 6-8 23 16 7 Correct response a a a 1m £ 30(.00) b b b 1m 4 and 5, in either order Additional guidance ! Upper bound taken to be just under 5 For the upper bound, accept values between 4.9 and 5 inclusive U1 Tier & Question Rectangles and squares 3-5 4-6 5-7 6-8 24 17 8 Correct response a a a 1m 4 b b b 1m 5 Additional guidance ! Value repeated Accept provided there is no ambiguity eg, for part (a) accept 4 by 4 eg, for part (a) do not accept 4×4 ! For parts (a) and (b), response of 16 then 20 Mark as 0, 1 U1 24 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 24 http://www.SATs-Papers.co.uk 19/12/06 2:21:26 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6, 5–7 Lemonade 3-5 4-6 5-7 6-8 25 18 9 Correct response 2m Additional guidance 80 p or 1m Shows the value 0.8(0) or Shows or implies a complete correct method with not more than one computational error eg 6 × 1.20 – 4 × 1.60 (120 ÷ 4 – 160 ÷ 6) × 24 7.40 (error) – 6.40 = 1.00 or Shows the value 720 or 7.2(0) and 640 or 6.4(0) U1 Tier & Question Three angles 3-5 4-6 5-7 6-8 26 19 10 Correct response 1m Indicates No and gives a correct explanation eg 24 + 93 + 61 = 178 but it should equal 180 for a straight line 24 + 93 + 61 is 2 degrees too small for a straight line 4 + 3 + 1 = 8, so they couldn’t add to 180 Additional guidance Minimally acceptable explanation that states or implies the angles should add to 180 or that they add to less than 180 eg The angles don’t make 180 They should add to 180 Too small by 2 The total ends in 8, but this should be 0 It totals 178°, so it would be an obtuse angle Incomplete or incorrect explanation eg 24 + 93 + 61 = 178 which is not straight The angles add to 188 not 180 The angles add to 178° so it will look straight U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 25 http://www.SATs-Papers.co.uk 25 19/12/06 2:21:27 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7, 6–8 Tier & Question Solving 3-5 4-6 5-7 6-8 27 20 11 Correct response 1m 14 1m 13 Additional guidance ! Incorrect notation eg, as an answer for the ﬁrst mark x = × 14 Penalise only the ﬁrst occurrence ! Incomplete processing eg, as an answer for the ﬁrst mark x = 448 32 Penalise only the ﬁrst occurrence Tier & Question Marking overlay available Newspaper 3-5 4-6 5-7 6-8 21 12 1 Correct response 2m Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels correctly Additional guidance Unambiguous abbreviation eg E for Evening newspaper, N for No newspaper or 1m Draws the sectors for Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels correctly or Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay but fails to label or labels incorrectly or Shows or implies that 5 people are represented by 30° or that 1 person is represented by 6° eg 5 people = 30° 150 ÷ 5 = 30 360 ÷ 60 = 6 60, 90 seen 26 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 26 http://www.SATs-Papers.co.uk 19/12/06 2:21:27 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Completing rules 3-5 4-6 5-7 6-8 22 13 2 Correct response 1m Gives two correct values in the correct order, and a correct expression in x eg 3, 1, 3x + 1 1, 9, x + 9 –2, 21, –2x + 21 1m ! Unconventional notation eg, for x + 9 1×x+9 Condone Gives two correct values in the correct order, and a correct expression in x eg x 2, 11, + 11 2 x 0.5, 5, 2x + 5 (or + 5) 0.5 1, 9, x + 9 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 27 For the ﬁrst mark, given example repeated Gives two correct values in the correct order, and a correct expression in x eg 4, 3, 4x – 3 –2, –21, –2x – –21 x, 3, x2 – 3 1m Additional guidance http://www.SATs-Papers.co.uk 27 19/12/06 2:21:27 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Parallelogram 3-5 4-6 5-7 6-8 23 14 3 Correct response 2m Additional guidance Gives the correct value with a correct unit eg 35cm2 or 1m Shows the value 35 or Shows a complete correct method with not more than one computational error and with a correct unit for area shown at least once eg 7 × 5 and cm2 seen (10 – 3) × 5 and cm2 seen 10 × 5 – 3 × 5 and cm2 seen 50 – 7.5 – 7.5 and cm2 seen 4 × 5 + 2 × 1.5 × 5 and cm2 seen 50 – 2 × 6.5 (error) = 37 and cm2 seen Tier & Question For 1m, necessary brackets omitted eg 10 – 3 × 5 Relationships 3-5 4-6 5-7 6-8 24 15 4 Correct response 1m 1m 28 9 100 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 28 Additional guidance ! Incomplete processing eg, for the ﬁrst mark 10 – 1 eg, for the second mark 102 Penalise only the ﬁrst occurrence http://www.SATs-Papers.co.uk 19/12/06 2:21:28 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Pi 3-5 4-6 5-7 6-8 25 16 5 Correct response a a a 1m Indicates Equivalent fractions or decimals 3.1416 b b b 1m Tier & Question Additional guidance 355 , ie 113 Enlarging Marking overlay available 3-5 4-6 5-7 6-8 26 17 6 Correct response 2m Shows a correct enlarged shape with all ﬁve vertices within the tolerances as shown on the overlay ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Construction lines drawn Ignore, even if incorrect or 1m Additional guidance Shows at least three vertices within the tolerances as shown on the overlay or Shows a correct enlarged shape with all ﬁve vertices within the tolerances as shown on the overlay, but in an incorrect position and/or orientation Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 29 http://www.SATs-Papers.co.uk 29 19/12/06 2:21:28 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Values 3-5 4-6 5-7 6-8 27 18 7 Correct response a a a 1m 15 b b b 1m 5 Additional guidance c c 1m 1 or equivalent 2 Indicates that e > 5 eg It has to be higher than 5 Any number over 5 Minimally acceptable indication eg >5 Above 5 More than half of 10 ! Range includes 5 eg 5 or over Condone Negative values of f excluded eg 5 < e 10 Between 5 and 10 Incorrect indication eg e can be 6, 7, 8 and so on e must be 5.1 or more Incomplete indication eg e = 10 – f f e 30 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 30 http://www.SATs-Papers.co.uk 19/12/06 2:21:29 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Travelling by car 3-5 4-6 5-7 6-8 28 19 8 Correct response a a a 1m 72 b b b 1m Additional guidance 4 c c 2m 1.8 or equivalent ! Answer of 2 For 2m, do not accept unless a correct method or a more accurate value is seen ! For 2m or 1m, follow through from part (b) Accept follow through as 18 ÷ (their (b) + 6) or as (their (b) + 14) ÷ (their (b) + 6), rounded or truncated to at least 2 s.f. or 1m Shows or implies a correct method eg 18 ÷ (4 + 4 + 2) 18 10 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 31 For 1m, necessary brackets omitted eg 18 ÷ 4 + 4 + 2 http://www.SATs-Papers.co.uk 31 19/12/06 2:21:29 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 4–6, 5–7, 6–8 Brackets 3-5 4-6 5-7 6-8 29 20 9 a a 1m Correct response Additional guidance Gives a correct explanation The most common correct explanations: Give the correct expansion of the expression eg 3(2a + 1) = 6a + 3, not 6a + 1 It should be 2 greater, ie 6a + 3 Minimally acceptable explanation eg 6a + 3 She needs to add 2 Incomplete or incorrect explanation eg 3(2a + 1) 6a + 1 3(2a + 1) = 6a + 2 3(2a + 1) = 6a + 3 = 9a Address the misconception eg Both things in the brackets should be multiplied by 3, but she has forgotten the 1 Minimally acceptable explanation eg 3×1 All bits need to be multiplied by 3 You have to multiply everything in the brackets She hasn’t multiplied the 1 Incomplete explanation eg She hasn’t multiplied out the brackets correctly The 1 is incorrect Give a correct counter example eg When a = 1 then 3(2a + 1) = 9, but 6a + 1 = 7 If a is 2, 3(2 × 2 + 1) 6 × 2 + 1 Minimally acceptable explanation eg When a = 1 you get 9 and 7 Incomplete explanation eg When a = 1 you get different answers for each side, so it can’t be right 32 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 32 http://www.SATs-Papers.co.uk 19/12/06 2:21:30 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Brackets (cont) 3-5 4-6 5-7 6-8 29 20 9 b b 1m Correct response Additional guidance Gives a correct explanation The most common correct explanations: Give the correct expansion of the expression eg (k + 4)(k + 7) = k2 + 11k + 28, not k2 + 28 He should get k2 + 4k + 7k + 28 He has missed out 4k + 7k so it should be k2 + 11k + 28 Minimally acceptable explanation eg k2 + 11k + 28 k2 + 4k + 7k + 28 11k is missing There should be 4k and 7k as well ! Correct expression equated to zero eg k2 + 11k + 28 = 0 Condone Incomplete or incorrect explanation eg (k + 4)(k + 7) k2 + 28 k2 + 11k + 28 = k2 + 39 It’s 11k Address the misconception eg Both things in the ﬁrst brackets should be multiplied by both things in the second brackets, but he has done k × k and 4 × 7 Minimally acceptable explanation eg He hasn’t multiplied the 4 or the 7 by k There should be a k term It should have been like this: ( k + 4)(k + 7) Incomplete explanation eg There should be 3 terms in the answer The ks should be added You have to multiply everything in the second brackets by everything in the ﬁrst brackets He hasn’t multiplied the ﬁrst set of brackets by the second set properly Give a correct counter example eg When k = 1 then (k + 4)(k + 7) = 40, but k2 + 28 = 29 If k is 2, (2 + 4)(2 + 7) 22 + 28 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 33 Minimally acceptable explanation eg When k = 1 you get 40 and 29 Incomplete explanation eg When k = 1 you get different answers for each side, so it can’t be right http://www.SATs-Papers.co.uk 33 19/12/06 2:21:30 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Vowels 3-5 4-6 5-7 6-8 30 21 10 Correct response a 2m Additional guidance 0.61 or equivalent probability or 1m Shows the digits 61 or Shows the value 0.39 or equivalent probability or Shows or implies a complete correct method with not more than one computational error eg 1 – (0.08 + 0.13 + 0.07 + 0.08 + 0.03) 0.08 + 0.13 + 0.07 + 0.08 + 0.03 = 0.38 (error) 1 – 0.38 = 0.62 b 2m 0.000936 or 9.36 × 10–4, or equivalent probability For 2m, 9.36 –04 or 1m Shows the digits 936 or Shows or implies a complete correct method with not more than one computational error eg 0.13 × 0.08 × 0.09 9.4 × 10–4 34 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 34 http://www.SATs-Papers.co.uk 19/12/06 2:21:31 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Beams 3-5 4-6 5-7 6-8 22 11 Correct response 3m Additional guidance Indicates the 1st way, and gives the correct difference of 1320 or 2m Shows the digits 132(0) or Shows the digits 484(0) and 352(0) or Shows or implies correct substitution of all values into the formula and the intention to subtract eg 5 × 112 × 8 – 5 × 82 × 11 5 × 11 × 8(11 – 8) 440 × 3 5 × (968 – 704) 5 × 264 or Shows a complete correct method with not more than one computational error, and gives a correct decision for their values eg 5 × 112 × 8 = 4440 (error) 4440 – 3520 = 920 so 1st way, difference 920 or 1m Shows the digits 484(0) or 352(0) or Indicates the 1st way and gives an answer of 264 [the only error is to omit to multiply the substituted values by 5] or Indicates the 1st way and gives an answer of 6600 [the only error is to process 5 × 112 × 8 as (5 × 11)2 × 8 and 5 × 82 × 11 as (5 × 8)2 × 11] Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 35 http://www.SATs-Papers.co.uk 35 19/12/06 2:21:31 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Car park 3-5 4-6 5-7 6-8 23 12 Correct response 3m Additional guidance 15 or 2m Shows the values 24 and 160 or Shows a correct method with not more than one computational or rounding error eg (208 – 136) ÷ 3 ÷ (240 ÷ 1.50) 208 – 136 = 72, 72 ÷ 3 = 26 (error), 26 + 136 = 162 26 ÷ 162 × 100 = 16.25 or 1m Shows the value 24 or 160 or Shows a correct method with not more than two computational or rounding errors eg 208 – 136 = 62 (error), 62 ÷ 3 = 21 (premature rounding), 21 ÷ 160 × 100 = 13.125 U1 Tier & Question Volume of prisms 3-5 4-6 5-7 6-8 24 13 Correct response a a 1m 36 120 b b 1m Additional guidance 450 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 36 http://www.SATs-Papers.co.uk 19/12/06 2:21:32 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Marking overlay available Tiers 5–7, 6–8 Straight lines 3-5 4-6 5-7 6-8 25 14 Correct response a a 1m Draws a different straight line with gradient 1, within the tolerance as shown on the overlay when the y-axes are aligned b b 1m 20 c Gives a correct equation eg y = 5x + 10 5x – y = –10 Additional guidance c 1m Tier & Question ! Line short As the line could be positioned anywhere on the grid, accept lines of at least one diagonal unit in length provided they are within the tolerance as shown on the overlay Responses consisting of longer lines must be entirely within tolerance ! Unconventional notation eg y1 = 5 × x + 10 Condone Two semicircles 3-5 4-6 5-7 6-8 26 15 Correct response 2m 25 + 10, 88.6, 88.5(…) or 89 Additional guidance ! Value of 88 For 2m, do not accept unless a correct method or a more accurate value is seen or 1m Shows one entry from the following list: 25 10 15 20 50 50 (or 78.6, 78.5(…), 79) (or 31.(…)) (or 47.(…)) (or 62.8(…), 63) and 30 (or 94.(…)) (or 157.(…)) + 10 (or 167.(…)) or Shows or implies a complete correct method with not more than one computational or rounding error eg 20 30 + + 30 – 20 2 2 25 × 3.14 + 10 Value of 88, with no correct method or more accurate value seen U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 37 http://www.SATs-Papers.co.uk 37 19/12/06 2:21:32 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Which pupil? 3-5 4-6 5-7 6-8 27 16 Correct response 2m Additional guidance Indicates Class 9A and gives a correct justiﬁcation The most common correct justiﬁcations: Use the proportions of boys in each class, in a form that enables comparison eg 12 168 13 169 but = = 26 364 28 364 336 338 and You get 728 728 13 12 = 0.464(…), = 0.461(…) (or 0.462) 28 26 A gives 46.4% and B gives 46.2% 28 ÷ 13 = 2.15(…) 26 ÷ 12 = 2.16(…) (or 2.17) 13 12.07(…) 12.1 = (or ) 28 26 26 12 12.9(…) = 28 26 13 × (12 + 14) = 338, 12 × (15 + 13) = 336 For 2m, minimally acceptable justiﬁcation eg 169 168 , 364 364 0.464(…), 0.461(…) (or 0.462) 46.4, 46.2 13 × 26 > 12 28 For 2m, incomplete or incorrect justiﬁcation eg 13 12 > 28 26 13 > 12 Use the ratios of boys to girls or girls to boys in each class, in a form that enables comparison eg 9A is 0.86(…) boys for every girl, 9B is 0.85(…) 9A is 0.87 boys for every girl, 9B is 0.86 13 : 15 = 1 : 1.15(…) 12 : 14 = 1 : 1.16(…) (or 1 : 1.17) 13 12.1(…) = 14 15 12 13 = 15 13.8(…) 182 180 , 210 210 Reason generally about the differences between the numbers of boys and girls eg A difference of 2 out of the bigger total in 9A is less than out of the smaller total in 9B 2 2 < 28 26 38 For 2m, minimally acceptable justiﬁcation eg 0.86(…), 0.85(…) 0.87, 0.86 1.15(…), 1.16(…) (or 1.17) For 2m, minimally acceptable justiﬁcation eg There are two fewer boys than girls in both, but 9A is bigger Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 38 http://www.SATs-Papers.co.uk 19/12/06 2:21:33 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Which pupil? (cont) 3-5 4-6 5-7 6-8 27 16 Correct response Additional guidance or 1m Shows a correct justiﬁcation but makes an incorrect or no decision eg 12 13 = 0.46, = 0.46 so equal 26 28 or Shows a correct justiﬁcation with not more than one computational error then makes the correct decision for their values eg 338 346 , (error), 9B indicated 728 728 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 39 http://www.SATs-Papers.co.uk 39 19/12/06 2:21:33 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Pythagoras 3-5 4-6 5-7 6-8 28 17 a 1m Correct response Gives a correct explanation Additional guidance Explanation uses only accurate or scale drawing The most common correct explanations: Show that the values 6, 8 and 10 work using Pythagoras’ theorem eg 62 + 82 = 36 + 64 = 100 = 102 102 – 82 = 100 – 64 = 36 = 62 State or imply that the triangle is an enlargement of a 3, 4, 5 right-angled triangle eg A 3, 4, 5 triangle is right-angled and 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 It’s just a 3, 4, 5 triangle with the lengths of the sides doubled Because 6, 8 and 10 make a Pythagorean triple b 1m Gives a correct justiﬁcation eg 6.9 × 8 = 9.2 6 8 × 1.15 = 9.2 9.2 ÷ 1.15 = 8 3 6.9 ÷ 9.2 = 4 3 6÷8= 4 6 6.9 is a 15% increase 8 × 0.15 = 1.2 8 + 1.2 = 9.2 8 = 53.1… 6 6.9 × tan 53.1… = 9.2 tan–1 40 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 40 Minimally acceptable explanation eg 62 + 82 = 102 36 + 64 = 100 The square of the longest side is equal to the sum of the squares of the other two sides Incomplete explanation eg 62 + 82 36 + 64 Minimally acceptable explanation eg It’s an enlarged 3, 4, 5 triangle 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 Incomplete explanation eg It’s like a 3, 4, 5 triangle Minimally acceptable explanation eg 6.9 ×8 6 8 × 1.15 6.9 6 = 9.2 8 Incomplete explanation eg 9.2 ÷ 1.15 Explanation attempts to use Pythagoras’ theorem eg 6.92 + 9.22 = 11.52 http://www.SATs-Papers.co.uk 19/12/06 2:21:33 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Pythagoras (cont) 3-5 4-6 5-7 6-8 28 17 Correct response c 1m 1m Shows the digits 115 eg 1.15 × 108 115 000 000 11.5 Additional guidance ! Zero(s) given after the last decimal place within standard form notation Condone eg, for both marks in part (c) accept 1.150 × 108 Shows the correct value in standard form, ie 1.15 × 108 Tier & Question Expressions 3-5 4-6 5-7 6-8 18 Correct response 2m Gives all three correct expressions, ie y + 15 2y y + 3a Additional guidance ! Expressions unsimpliﬁed or use unconventional notation eg, for the third expression y+a+a+a 1y + 3 × a Condone or 1m Gives two correct expressions U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 41 http://www.SATs-Papers.co.uk 41 19/12/06 2:21:34 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Gorillas 3-5 4-6 5-7 6-8 19 Correct response 2m Gives an integer value between 16 500 and 17 000 inclusive eg 17 000 16 700 16 667 Additional guidance ! Gives a non-integer value within the correct range eg 16 666.(…) Condone or 1m Shows the digits 166(…) or 167 or Shows a complete correct method with not more than one computational or rounding error eg 5000 ÷ 0.3 5000 ÷ 3 × 10 100 × 5000 30 5000 ÷ 30 = 200 (premature rounding), 200 × 100 = 20 000 Tier & Question Houses 3-5 4-6 5-7 6-8 20 Correct response 2m 2.9 or equivalent Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen or 1m Shows the value 29 or 290 or Shows a complete correct method with not more than one computational or rounding error eg 2.5 × 60 + 3.3 × 30 + 4.1 × 10 100 (2.5 × 6 + 3.3 × 3 + 4.1) ÷ 10 150 + 99 + 41 = 300 (error), 300 ÷ 100 = 3 42 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 42 For 1m, necessary brackets omitted eg 2.5 × 6 + 3.3 × 3 + 4.1 ÷ 10 http://www.SATs-Papers.co.uk 19/12/06 2:21:35 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Subtracting and squaring 3-5 4-6 5-7 6-8 21 Correct response 2m Gives the number as 13 and shows a complete correct method for solving algebraically eg (x – 25)2 = x2 – 25 2 x – 50x + 625 = x2 – 25 50x = 650 x = 13 Additional guidance Method used is trial and improvement or 1m Shows a correct expression without brackets that is equivalent to (unknown – 25)2 eg x2 – 50x + 625 n2 – 25n – 25n + 625 a × a – 50 × a + 25 × 25 or Shows a correct equation eg (x – 25)2 = x2 – 25 U1 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 43 http://www.SATs-Papers.co.uk 43 19/12/06 2:21:35 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only Light years 3-5 4-6 5-7 6-8 22 Correct response Additional guidance a 1m 9.43 × 1012 ! Zero(s) given after the last decimal place within standard form notation eg, for part (a) 9.430 × 1012 Condone b 1m 7.35(54) × 1013 or 7.36 × 1013 or 7.4 × 1013 ! For part (b), follow through Accept 7.8 × their (a) provided this is written correctly in standard form to at least 2 s.f. Tier & Question Octagon 3-5 4-6 5-7 6-8 23 Correct response 2m 2 2, 8 or 2.8(…) Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen or 1m Shows or implies a correct equation in y eg y2 = 8 y2 + y2 = 42 2y2 = 16 y×y+y×y=4×4 2y = 4 4sin 45 (or 4cos 45) U1 44 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 44 http://www.SATs-Papers.co.uk 19/12/06 2:21:36 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier & Question Tier 6–8 only x, y, a and b 3-5 4-6 5-7 6-8 24 Correct response a 1m a–b b 2m 2b – a Additional guidance ! For part (a), unsimpliﬁed expression or unconventional notation Condone For 2m or 1m, follow through from part (a) or 1m Shows a correct expression for x, even if it is unsimpliﬁed, uses unconventional notation or there is subsequent incorrect working eg 2×b–a b – (a – b) a – 2(a – b) or Shows a complete correct method with not more than one error eg x + 2y = a 2x + 2y = b (error) x=b–a or Forms two correct equations that would allow elimination of y eg x + 2y = a 2x + 2y = 2b For 1m, second equation doubled without the ﬁrst equation restated eg 2x + 2y = 2b seen or Attempts to solve by substitution and forms a correct equation in x eg x+a–b=b x + 2(a – b) = a x + 2(b – x) = a Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 45 http://www.SATs-Papers.co.uk 45 19/12/06 2:21:36 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Index Index to mark schemes Tier 3–5 4–6 Question 5–7 Page 6–8 1 Rules 11 2 Homework 12 3 Odd one out 12 4 Hibernation 13 5 Concert 14 6 Cake 14 7 Bar chart 15 8 Coordinates 15 9 2 Fitting tiles 16 10 3 Names 17 11 4 Leaves 17 12 5 Marbles 18 13 6 a and b 18 14 7 Turning 19 15 8 Party 19 16 9 Survey 20 17 10 1 Frog spawn 21 18 11 2 Simplifying 21 19 12 3 Containers 22 20 13 4 Triangles 22 21 14 5 Spinners 22 22 15 6 Faces 23 23 16 7 Fir trees 24 24 17 8 Rectangles and squares 24 25 18 9 Lemonade 25 26 19 10 Three angles 25 27 46 1 20 11 Solving 26 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 46 http://www.SATs-Papers.co.uk 19/12/06 2:21:37 pm 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3–5 Index Question Page 4–6 5–7 6–8 21 12 1 Newspaper 26 22 13 2 Completing rules 27 23 14 3 Parallelogram 28 24 15 4 Relationships 28 25 16 5 Pi 29 26 17 6 Enlarging 29 27 18 7 Values 30 28 19 8 Travelling by car 31 29 20 9 Brackets 32 30 21 10 Vowels 34 22 11 Beams 35 23 12 Car park 36 24 13 Volume of prisms 36 25 14 Straight lines 37 26 15 Two semicircles 37 27 16 Which pupil? 38 28 17 Pythagoras 40 18 Expressions 41 19 Gorillas 42 20 Houses 42 21 Subtracting and squaring 43 22 Light years 44 23 Octagon 44 24 x, y, a and b 45 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 47 http://www.SATs-Papers.co.uk 47 19/12/06 2:21:37 pm QCA wishes to make its publications widely accessible. Please contact us if you have any specific accessibility requirements. 29 Bolton Street London W1J 8BT Telephone: 08700 60 60 40 Minicom: 020 7509 6546 Fax: 020 7509 5908 Email: tests@naa.org.uk Website: www.naa.org.uk/tests First published 2007 © Qualifications and Curriculum Authority 2007 ISBN 1-85838-875-9 Reproduction, storage, adaptation or translation, in any form or by any means, of 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, providing full acknowledgement is given. Printed 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 Sourced from SATs-Papers.co.uk 275669_MS_P2.09.indd 48 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/06/2778 http://www.SATs-Papers.co.uk 275669 19/12/06 2:21:37 pm