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254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 2 Mathematics tests Ma KEY STAGE 3 ALL TIERS 2003 2003 Mark scheme for Paper 2 Tiers 3–5, 4–6, 5–7 and 6–8 3 3 K E Y S TA G E K E Y S TA G E 3 K E Y S TA G E Sourced from SATs-Papers.co.uk K E Y S TA E G http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 2 2003 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 10 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. The 2003 key stage 3 mathematics tests and mark schemes were developed by the Mathematics Test Development Team at QCA. 2 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 3 2003 KS3 mathematics test mark scheme: Paper 2 General guidance General guidance Using the mark schemes Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise. In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance, relating to marking of questions that involve money, time, coordinates, algebra or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 4 2003 KS3 mathematics test mark scheme: Paper 2 General guidance What if … The pupil’s response does not match closely any of the examples given. The pupil has responded in a non-standard way. The pupil has made a conceptual error. 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. 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. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, error. A computational error is a slip such as writing 4 t 6 e 18 in an otherwise correct long multiplication. A conceptual error is a more serious misunderstanding of the relevant mathematics; when such an error is seen no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 t 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers. The pupil’s accuracy is marginal according to the overlay provided. Overlays can never be 100% accurate. However, provided the answer is within, or touches, the boundaries given, the mark(s) should be awarded. The pupil’s answer correctly follows through from earlier incorrect work. Follow through marks may be awarded only when specifically stated in the mark scheme, but should not be allowed if the difficulty level of the question has been lowered. Either the correct response or an acceptable follow through response should be marked as correct. There appears to be a misreading affecting the working. 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 http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 5 2003 KS3 mathematics test mark scheme: Paper 2 General guidance What if … The final answer is wrong but the correct answer is shown in the working. Where appropriate, detailed guidance will be given in the mark scheme and must be adhered to. If no guidance is given, markers will need to examine each case to decide whether: the incorrect answer is due to a transcription error; in questions not testing accuracy, the correct answer has been given but then rounded or truncated; 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 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. The correct response has been crossed or rubbed out and not replaced. If so, award the mark. the pupil has continued to give redundant extra working which does not contradict work already done; The pupil’s answer is correct but the wrong working is seen. If so, award the mark. If so, do not award the mark. Where a question part carries more than one mark, only the final mark should be withheld. A correct response should always be marked as correct unless the mark scheme states otherwise. Mark, according to the mark scheme, any legible crossed or rubbed out work that has not been replaced. 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 http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 6 2003 KS3 mathematics test mark scheme: Paper 2 General guidance Marking specific types of question Responses involving money For example: £3.20 £7 Accept ✓ Do not accept ✓ Any unambiguous indication of the correct amount eg £3.20(p), £3 20, £3,20, 3 pounds 20, £3-20, £3 20 pence, £3:20, £7.00 Incorrect or ambiguous use of pounds or pence eg £320, £320p or £700p, or 3.20 or 3.20p not in the answer space. Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 ✓ The £ sign is usually already printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the £ sign, accept an answer with correct units in pounds and/or pence eg 320p, 700p eg £3.2, £3 200, £32 0, £3-2-0, £7.0 Responses involving time A time interval For example: 2 hours 30 mins Accept ✓ Take care ! Do not accept ✓ Any unambiguous indication eg 2.5 (hours), 2h 30 ✓ Digital electronic time ie 2:30 Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min ! The time unit, hours or minutes, is usually printed in the answer space. Where the pupil writes an answer other than in the answer space, or crosses out the given unit, accept an answer with correct units in hours or minutes, unless the question has asked for a specific unit to be used. A specific time For example: 8.40am, 17:20 Accept ✓ ✓ Any unambiguous, correct indication eg 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 ✓ Unambiguous change to 12 or 24 hour clock eg 17:20 as 5:20pm, 17:20pm Do not accept Incorrect time eg 8.4am, 8.40pm Incorrect placement of separators, spaces, etc or incorrect use or omission of 0 eg 840, 8:4:0, 084, 84 6 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 7 2003 KS3 mathematics test mark scheme: Paper 2 General guidance Responses involving coordinates For example: ( 5, 7 ) Accept ✓ Do not accept ✓ Unambiguous but unconventional notation eg ( 05, 07 ) ( five, seven ) x y ( 5, 7 ) ( x e5, y e7 ) Incorrect or ambiguous notation eg ( 7, 5 ) ( 5x, 7y ) ( x5, y7 ) ( 5x, 7y ) Responses involving the use of algebra For example: 2 p n n p 2 2n Accept ✓ ✓ The unambiguous use of a different case eg N used for n ✓ Unconventional notation for multiplication eg n t 2 or 2 t n or n2 or n p n for 2n n t n for n2 ✓ Multiplication by 1 or 0 eg 2 p 1n for 2 p n 2 p 0n for 2 ✓ Words used to precede or follow equations or expressions eg t e n p 2 tiles or tiles e t e n p 2 for t e n p 2 ✓ Unambiguous letters used to indicate expressions eg t e n p 2 for n p 2 ✓ Embedded values given when solving equations eg 3 t 10 p 2 e 32 for 3x p 2 e 32 Take care ! Do not accept ! Words or units used within equations or expressions should be ignored if accompanied by an acceptable response, but should not be accepted on their own eg do not accept n tiles p 2 n cm p 2 Change of variable eg x used for n Ambiguous letters used to indicate expressions eg n e n p 2 However, to avoid penalising any of the three types of error above more than once within each question, do not award the mark for the first occurrence of each type within each question. Where a question part carries more than one mark, only the final mark should be withheld. Embedded values that are then contradicted eg for 3x p 2 e 32, 3 t 10 p 2 e 32, x e 5 7 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 8 2003 KS3 mathematics test mark scheme: Paper 2 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction or percentage only. For example: 0.7 Accept ✓ Take care ! Do not accept ✓ A correct probability that is correctly expressed as a decimal, fraction or percentage. ✓ Equivalent decimals, fractions or percentages eg 0.700, 70 35 , , 70.0% 100 50 ✓ A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 eg 70 18 e 100 25 The following four categories of error should be ignored if accompanied by an acceptable response, but should not be accepted on their own. ! A probability that is incorrectly expressed eg 7 in 10, 7 out of 10, 7 from 10 ! A probability expressed as a percentage without a percentage sign. ! A fraction with other than integers in the numerator and/or denominator. However, each of the three types of error above should not be penalised more than once within each question. Do not award the mark for the first occurrence of each type of error unaccompanied by an acceptable response. Where a question part carries more than one mark, only the final mark should be withheld. ! A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 or less than 0 8 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 9 2003 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 tiers 3–5, 4 – 6 and 6–8. A total of 122 marks is available in tier 5–7. 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 QCA website www.qca.org.uk from Monday 23 June 2003. 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. 9 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 10 2003 KS3 mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Hexagons 3-5 4-6 5-7 6-8 1 Correct response a 1m Gives a correct explanation eg ■ Each shape has six sides ■ They all have six corners ■ 6 sides Additional guidance ✓ Minimally acceptable explanation eg ◆ ◆ ◆ 6 edges 6 lines 6 points ! Incorrect statement alongside a correct explanation Condone eg, accept ◆ 6 equal sides b 1m Draws a regular hexagon of any size with vertices on the dots of the grid ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Internal lines shown Ignore provided the outer shape is a regular hexagon eg, accept ◆ Tier & Question Cities 3-5 4-6 5-7 6-8 2 Correct response a 1m 172 b 1m Indicates York and London, in either order c 1m Indicates London and gives the value 13 d 2m 332 or 1m Additional guidance Shows the three correct values 120, 91 and 121 or Shows three values, two of which are correct, then adds them correctly eg ■ 120 p 91 p 134 (error) = 345 10 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 11 2003 KS3 mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Number cards 3-5 4-6 5-7 6-8 3 a Correct response 1m Four hundred and nine Additional guidance ✓ Correct words even if cards not completed, or completed incorrectly or ! Digits used Nine hundred and four 1m Four hundred and ninety or Nine hundred and forty Accept provided the place value is interpreted eg, for the first mark, accept ◆ 4 hundred and 9 ◆ 400 and 9 ◆ 400 and 9 units ! Omission of the word ‘and’ Accept if unambiguous eg, for the first mark, accept ◆ Four hundred nine ◆ 4 hundred – nine ◆ 4 hundreds 9 units ◆ 4 hundred p nine ◆ 400 p 9 eg, for the first mark, do not accept ◆ 400 9 ! Within their number in words, digits other than 4 and 9 used Provided both their digits are non-zero, and the number shown by the cards and the number in words are the same, penalise only the first occurrence Otherwise, do not accept Place value not interpreted eg, for the first mark ◆ Four, zero and nine b 1m 853 1m 538 U1 11 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 12 2003 KS3 mathematics test mark scheme: Paper 2 Tier 3–5 only Tier & Question Late 3-5 4-6 5-7 6-8 4 Correct response a 1m 10 b 1m 19 c 1m Indicates Wednesday and gives a correct explanation Additional guidance ✓ Unambiguous abbreviation eg ◆ The most common correct explanations: Refer to the number of pupils or to the number of lates eg ■ 24 were late that day, more than on any other day ■ More pupils were late than on any other day ■ Biggest number of lates Refer to the heights of the bars for the three year groups eg ■ That day is always the tallest ■ It’s the one that was highest most often ◆ Wed W Markers may find the following totals useful: Monday 10 Tuesday 12 Wednesday 24 Thursday 8 Friday 8 ✓ Minimally acceptable explanation eg, for the first category ◆ 24 ◆ 8 in Y7, 7 in Y8, 9 in Y9 ◆ It has many lates eg, for the second category ◆ The charts show more were late then ◆ Taller bars ! Explanation could be referring to all three year groups/charts or to just one year group/ chart Do not accept explanations that refer explicitly to one year group/chart eg ◆ The tallest bar is 9 and that is a Wednesday ◆ More pupils were late in yr 9 on that day ◆ The chart shows more were late then ◆ Tallest bar Otherwise accept eg ◆ Wednesday is the highest ◆ Most pupils ! Ambiguous statement eg That was the day everyone was late Year 7, 8 and 9 all came late Ignore if accompanied by a correct response, but do not accept on its own ◆ ◆ U1 12 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 13 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Slicing cubes 3-5 4-6 5-7 6-8 5 Correct response a Tier 3–5 only 1m Additional guidance ! Number of faces of both pieces given 6 Accept if unambiguous eg, for part (a) accept ◆ 6 and 6 ◆ 6 p 6 ! Total number of faces given b 1m 6 c 1m Penalise only the first occurrence, provided answers of 12 are given for both parts (a) and (b). Otherwise do not accept eg ◆ 12, 12, 10 Mark as 0; 1; 1 ◆ 12, 12, 9 Mark as 0; 1; 0 ◆ 12, 14, 10 Mark as 0; 0; 0 5 ! For part (c), follow through Accept follow through as their (b) m 1, provided their (a) is equal to their (b) eg ◆ 5 (part (a)) 5 (part (b)) 4 (part (c)) Tier & Question Buying a bicycle 3-5 4-6 5-7 6-8 6 Correct response 2m £ 26.89 Additional guidance ! Answer rounded to 27 Accept for 2m only if a correct method or a more accurate value is seen For 2m, negative value eg ◆ m26.89 or 1m Shows the digits 20688 or Shows or implies a correct method eg ■ 8.62 t 24 m 179.99 ■ 27 with no evidence of an incorrect method ■ m26.89 ■ Digits 2689 seen ! Incorrect order for subtraction Condone eg, accept ◆ 179.99 m 8.62 t 24 U1 13 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 14 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Kings and queens 3-5 4-6 5-7 6-8 8 1 Correct response a a 1m 50 b b 1m Elizabeth (I) c 1m Indicates (81, 63) on the chart Additional guidance c Point identified but not interpreted eg ◆ f ! Point not accurately indicated Accept provided the point is nearer to (81, 63) than to any other point with integer coordinates 14 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 15 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 School uniform 3-5 4-6 5-7 6-8 7 2 Correct response a a 1m 54 b b 1m 16 c 2m Additional guidance Gives all three correct values correctly positioned, ie 10, 20, 10 c or 1m Gives any two correct values correctly positioned or Gives three values that sum to 40, one of which is correct and correctly positioned or Gives the correct value for No, ie 20, and gives values for Yes and Don’t know that are the same eg ■ 5, 20, 5 d d 1m Gives two labels in the two boxes of either the first row or the first column specifying sex and gives two labels in the two boxes of either the first row or the first column specifying yes or no, or other mutually exclusive labels that address the question eg, for sex ■ Boys, girls ■ Female, male ■ G, B eg, for yes or no ■ Yes, no ■ ✓, ■ N, Y ■ Have a pet, Do not have a pet U1 15 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 16 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Admission 3-5 4-6 5-7 6-8 9 3 Correct response 3m or 2m Additional guidance £ 5.65 Shows the digits 565 eg ■ 56.50 or Shows the values 11.15 and 16.8(0) or ! Values or differences shown in working in pence, without units given Accept for 2m, provided both values or all differences are in pence Shows one of the values 11.15 or 16.8(0), then follows through using their incorrect value to give their correct saving eg ■ 11.15 before, 4.90 p 3.50 p 3.50 p 4.90 e 14.80 (error) after, 14.80 m 11.15 e 3.65 or Shows the correct difference for each category eg ■ 1.7, 1, 1.95 ■ 1.7 p 2 t 1 p 1.95 or Shows the correct difference for two of the categories, then follows through using their incorrect difference to give their correct saving eg ■ 1.7 p 1 p 1 p 1.85 (error) = 5.55 or Shows the value £ 4.65 (from calculating using one child, rather than two) 16 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 17 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Admission (cont) 3-5 4-6 5-7 6-8 9 3 Correct response or 1m Additional guidance Shows any of the following: Digits 1115 or Digits 168(0) or Any two of the correct differences 1.7(0), 1, 1.95 ! Values or differences shown in working in pence, without units given Accept for 1m or Digits 465 (from calculating using one child, rather than two) or The values 8.65 and 13.3(0) (from calculating using one child, rather than two) U1 17 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 18 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 3–5, 4–6 Cubes in bags 3-5 4-6 5-7 6-8 12 4 Correct response a a 1m 27 b b 2m Additional guidance Both correct, ie 24 and 28, either order or 1m At least one correct and not more than one incorrect eg ■ 20, 24, 28 ■ 24, 27 or Gives the values 6 and 7 Tier & Question Temperature 3-5 4-6 5-7 6-8 10 5 Correct response a a 1m 1.5 or equivalent b b 1m 37.9 or equivalent c 2m Additional guidance 46.5 or equivalent c or 1m Shows or implies a complete correct method with not more than one error eg ■ (115.7 m 32) t 5 d 9 ■ ■ ■ ■ ■ ! For 1m, necessary brackets omitted As this is a level 4 mark, condone eg, accept ◆ 115.7 m 32 t 5 d 9 115.7 m 32 t5 9 115.7 m 32 e 82.7 (error), 82.7 t 5 d 9 e 45.9(…) 46 47 Digits 465 seen 18 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 19 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Drawing 3-5 4-6 5-7 6-8 11 6 1 a a a Correct response 1m Draws a rectangle of area 12 eg ■ 1 by 12 ■ 2 by 6 ■ 3 by 4 ■ 1.5 by 8 b b 1m 1m ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ✓ Edge of grid used as edge of shape Draws a rectangle of area 12, with different dimensions from one credited in part (a) c Additional guidance Draws a triangle of area 6 eg ■ Base 6, perpendicular height 2 ■ Base 4, perpendicular height 3 ■ Base 5, perpendicular height 2.4 c b 19 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 20 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Ages 3-5 4-6 5-7 6-8 14 7 2 a a a Correct response 2m Gives complete correct interpretations for both Barry and Carol, by referring to both the following aspects: The given context of age The meaning of the given numbers and operations eg, for Barry ■ One year younger (than Tina) ■ Aged one less (than T) eg, for Carol ■ Twice as old (as T) ■ Double her age ■ 2 t Tina years old or 1m Gives a complete correct interpretation for either Barry or Carol by referring to both aspects Additional guidance ! Incomplete interpretation Do not accept as complete an interpretation that lacks reference to one of the two aspects eg, for Barry ◆ Tina minus 1 [no reference to the given context] ◆ Younger [no reference to the m1] ◆ One year different [ambiguous reference to subtraction] eg, for Carol ◆ Twice Tina [no reference to the given context] ◆ Much older than Tina [no reference to the t 2] ◆ 2 Tina’s age [no reference to the multiplication] ! Interpretation using comparison with age of person other than Tina Accept provided the interpretation is unambiguous eg, accept as complete and correct for Barry ◆ Four years younger than Ann or Gives interpretations for both Barry and Carol that give the meaning of the given numbers and operations but contain no reference to the given context of age eg ■ For Barry, Tina minus 1 For Carol, Twice Tina 20 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 21 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Ages (cont) 3-5 4-6 5-7 6-8 14 7 2 b b b Correct response 2m Gives all three correct expressions in their simplest forms eg ■ n p 4, n, 2n p 1 Additional guidance ✓ 1n or n1 for n in a fully simplified expression n ±0 as a fully simplified expression for n ! Use of multiplication sign If a multiplication sign is used, an expression cannot be accepted as fully simplified eg, for Carol, do not accept as fully simplified ◆ 2 t n + 1 or 1m Gives any two correct expressions in their simplest forms or Gives all three correct expressions, even if not simplified c c c 1m 61 1m 62 Incomplete processing eg, for the first mark ◆ 2 t 30 p 1 eg, for the second mark ◆ 2 t 31 Incorrect notation eg, for the first mark ◆ 61n 21 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 22 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Grid percentages 3-5 4-6 5-7 6-8 13 8 3 a a a Correct response Additional guidance 60 1m b b b 1m 60 ! Percentage of diagram not shaded given Provided correct percentage unshaded is given consistently, ie 40 given for both, mark as 0, 1 1m Gives a correct explanation in which both ✓ Minimally acceptable explanation 1 and the link to 100% are shown or implied 8 eg 1 1 1 and of 100 is 12 8 8 2 ■ It’s ■ 1 out of 8 is equivalent to 12.5 out of 100 ■ 8 t 12 ■ 100 d 8 = 12.5 ■ It’s , and 1 d 8 = 0.125 1 = 100 2 1 8 eg 1 2 ◆ 8 squares is 100 so 1 is 12 ◆ 100 d 8 100 divided by the number of squares ◆ ◆ 1 = 0.125 8 ! The link is to a different percentage Accept provided the relevant fraction is shown or implied eg, accept ◆ 2 squares is 25%, 1 square is half of that ◆ 4 squares is 50%, 50 d 4 Incomplete explanation eg ◆ 8 squares is 100% ◆ 1 square out of 8 shaded ◆ 1 2 12 % = 1 8 Incorrect order of division eg 1 ◆ 8 d 100 = 12 2 c c c 1m Indicates a total of three squares on the diagram 22 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 23 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Data collection 3-5 4-6 5-7 6-8 15 9 4 a a a Correct response 1m Indicates 1 or 2 and gives a correct explanation Additional guidance ✓ Minimally acceptable explanation for 1 or 2 eg eg, for 1 ■ It will take a lot of time to write the name every time ■ You won’t have time to put the whole name ■ It will not tell you straightaway how many of each type there are ■ It will just give a long list of words ■ It would take ages to count up all the trees at the end ■ You could easily miscount the totals ■ It’s hard to draw a graph from it ■ It will take up a lot of paper U1 b b b 1m eg, for 2 ■ It will not tell you straightaway how many of each type there are ■ It will just give a long list of letters ■ It would take ages to count up all the trees at the end ■ You could easily miscount the totals ■ It’s hard to draw a graph from it ■ It will take up a lot of paper ■ Some names of trees might start with the same letter ■ You might not have a code for the type of tree you see Indicates 3 and gives a correct explanation eg ■ It is quick to do a tally chart ■ Tally marks are easy to write ■ It’s easy to see the number of each type ■ It shows clearly which types are most common ■ It’s easy to see the mode ■ You can count up the totals quickly ■ It is less likely you will miscount ■ It’s more likely to be accurate ■ It’s easy to draw a graph from a tally chart ■ It does not take up much space U1 ◆ ◆ ◆ Too long Not efficient It does not tell you how many there are ! Explanation for 1 or 2 that refers to an improvement to the design Accept provided the improvement relates to one of the correct explanations eg, for 1, accept ◆ It’s quicker to write only the first letter eg, for 1 or 2, accept ◆ Using a tally chart tells you how many there are eg, for 1 or 2, do not accept ◆ Using a tally chart is better Explanation for 1 or 2 that refers to pupils not knowing what type the trees are eg ◆ They might not know the trees’ names Explanation for 2 that refers to use of codes eg ◆ They might find the codes confusing ◆ They could forget the key ◆ It does not list the actual names ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ It’s quick It’s efficient You just put a line It collects the data together It’s easy to understand It’s simple to use It’s organised It tells you how many there are Incomplete explanation eg ◆ It’s easy ◆ It’s simple ◆ It’s effective ◆ It’s clear ◆ It can be understood ◆ It’s not confusing ! Reference to disadvantages of the design eg ◆ There might be lots of ‘Other’ and they will not know what type they were ◆ They have to decide in advance which sorts to include Ignore alongside a correct explanation 23 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 24 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Coins 3-5 4-6 5-7 6-8 16 10 5 a a a Correct response 1m Gives a correct explanation eg ■ ■ ■ ■ 2 1 = 4 2 Additional guidance ✓ Minimally acceptable explanation eg ◆ Two of the four coins are 10p so half of them are 10p 1 4 1 1 1 p p =1 4 4 2 1 1 1 1 Each coin has chance and p = 4 4 4 2 20p is , so is 1p, and ◆ ◆ ◆ 1m Identifies the values of the four coins as 1 20, 10, 2 and 1 and gives the probability , or 4 equivalent probability Two out of four Two is half of four Two are tens, two not Incomplete explanation eg ◆ It’s 50/50 ◆ There are two tens, a twenty and a 1p ◆ There are two 10ps ◆ Half the coins are 10ps ◆ b b b 2 4 20p is 1 , so is 1p 4 ! Values of coins identified but doubt expressed as to whether this is the only possible combination Condone Probability stated without values of coins identified U1 24 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 25 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Explaining why 3-5 4-6 5-7 6-8 17 11 6 Correct response 1m Indicates AD and CD are both 12, and justifies that triangle ACD is equilateral eg ■ The sides are the same length ■ All sides are 12 ■ AC = AD = CD Additional guidance ✓ Minimally acceptable justification eg ◆ ◆ Sides are the same They are equal Incorrect justification eg ◆ The sides are even ! Reference to angles Ignore, ie do not accept a justification based on angles alone and do not penalise incorrect information about angles given alongside a correct response U1 1m Indicates angle y is 60 and gives a correct justification either as a calculation or as a known fact eg ■ 180 d 3 ■ 60 t 3 = 180 ■ That’s how many degrees there are in one angle in an equilateral triangle ✓ Minimally acceptable justification eg ◆ ◆ ◆ 60 t 3 60 p 60 p 60 All the angles are the same Incomplete justification eg ◆ Angles in a triangle add up to 180 ! Incorrect notation Ignore for both this mark and the next eg, for angle y as 60, accept ◆ 60°C U1 1m Indicates angle x is 30 and gives a correct justification eg ■ Triangle ADB is a reflection of triangle ABC so x is half y ■ All angles in an equilateral triangle are 60° The reflection shows half so it must be 30° ■ Angles in ABC add up to 180, and 180 m 90 m 60 = 30 ✓ Minimally acceptable justification eg ◆ ◆ ◆ ◆ ◆ x is half y 2x = y 60 d 2 It is half 180 m 90 m 60 ! Follow through Accept for angle x as their y d 2 provided it is accompanied by a correct justification that either does not use a value for y or uses their value for y, and provided their y is not 0, 90 or greater than or equal to 180 U1 25 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 26 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 3–5, 4–6, 5–7 Tier & Question Water 3-5 4-6 5-7 6-8 18 12 7 a a a Correct response 2m 8 Additional guidance ✓ Value qualified eg ◆ or 1m About 8 Shows a complete correct method eg 1.8 t 1000 ■ 225 1.8 d 0.225 ■ ■ 225 t 2 = 450 450 t 2 = 900 900 t 2 = 1800 2t2t2 or Shows the value 1800 or 0.225 b b b 1m 48 26 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 27 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Ratio of ages 3-5 4-6 5-7 6-8 13 8 1 Correct response a a a 1m 7:5 b b b 1m 7:6 c Indicates No and gives a correct explanation eg ■ That would make their ages equal which is not possible as the sister is 6 years younger ■ They will never be the same age as he is always 6 years older ■ To make them the same age, Paul would have to stop getting older for a number of years c c 1m Additional guidance ! Ratio correct but not written as simply as possible Provided there is no incorrect simplification, penalise only the first occurrence ! Incorrect order If the only error is to write each ratio in the incorrect order, ie 5 : 7 and 6 : 7, do not award the mark for part (a) but award the mark for part (b) ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ ◆ ◆ ◆ They’d be the same They are not the same age His sister is 6 years younger Paul is older They were born in different years That would mean Paul had stopped getting older for a number of years That means they would’ve had to be the same age in the first place Incorrect statement eg ◆ She will always be 8 years younger U1 No or incomplete interpretation eg ◆ 7 : 7 is the same as 1 : 1 ◆ It wouldn’t be equal 27 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 28 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Sizing 3-5 4-6 5-7 6-8 14 9 2 Correct response 2m Gives the correct order of A, C, B accompanied by one of the following explanations, whether stated or implied: Additional guidance ! Correct order given in unconventional way Accept provided it is unambiguous eg, accept ◆ Area, perimeter, side length Side lengths of A and C are 6 (or √36) and 9 (or 36 d 4) respectively ✓ For 2m or 1m, side length of A implied by Area of C is 81 (or 9 t 9) ! For 2m or 1m, incorrect working or incorrect units alongside a correct response Ignore Perimeter of A is 24 (or 6 t 4) or 1m Gives one of the correct explanations as above, but does not order or orders incorrectly or Gives the correct ordering but justifies only with reference to the side length of A as 6 (or √36) or Gives the correct ordering but justifies only with reference to the side length of C as 9 (or 36 d 4) 6 t 6 seen ! Their explanation does not explicitly state which property and/or square is being considered Accept provided the explanation links the relevant values to 36 and accompanies the correct ordering eg, for side lengths of A and C for 2m accept ◆ Area 36 so 6 perimeter 36 so 9, A, C, B ◆ 36 = 6 t 6 36 = 9 p 9 p 9 p 9 A, C, B for 1m accept ◆ 36 = 6 t 6 A, C, B However, as many of the relevant values can be obtained from incorrect reasoning, do not accept only values 6 and 9 seen or Shows both that the area of B is 1296 (or 36 t 36) and the perimeter of B is 144 (or 36 t 4) U1 28 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 29 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Nets 3-5 4-6 5-7 6-8 15 10 3 Correct response a a a 1m Additional guidance Shows that the surface areas are different The most common correct explanations: Calculate A as 38, B as 32 eg ■ A is 4 t 8 p 6 = 38, B is 3 t 8 p 8 = 32 ✓ Minimally acceptable explanation eg ◆ ◆ ◆ State that the difference is 6 eg ■ A has 6 more squares than B Manipulate the nets to a form where comparison may be drawn without further computation eg ■ A is 6 t 8 m 10 but B would be 6 t 8 m 16 b b b 2m Shows that the volume of A is equal to that of B eg ■ length width height volume A: 4 3 1 12 B: 3 2 2 12 ■ ■ ■ or 1m A is 3 t 4 t 1 = 12, B is 2 t 3 t 2 = 12 3t4t1=2t3t2 A is one layer of 12 cubes and B is two layers of 6 cubes Shows the value 12, with no evidence of an incorrect method for this value 38, 32 4 t 8 p 6 isn’t the same as 3 t 8 p 8 6 more Incomplete explanation eg ◆ I counted the squares ◆ There are more squares in A than in B ! Units given Ignore eg, accept ◆ 382 cm, 322 ✓ Minimally acceptable explanation eg ◆ ◆ Both 12 12, 12 Incomplete explanation eg ◆ Both the same ! Units given Ignore ! Responses to parts (a) and (b) transposed but otherwise correct Mark part (a) as 0 but mark part (b) as 1, 0 29 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 30 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Beaches 3-5 4-6 5-7 6-8 16 11 4 Correct response a a a 2m Additional guidance Gives three correct integer values that sum to 1620, ie 535 1085 534 or 0 530 1086 1090 or 0 ✓ For 2m, entry for flies left blank ✓ For 2m, proportion taken as 1 and 2 3 3 eg ◆ 540 0 1080 0 or 1m Gives a value for sandhoppers that is either between 534 and 535 inclusive or is 540, or a value for beetles that is either between 1085 and 1086 inclusive or is 1080 or Gives all three correct integer values but in the wrong order b b b 2m Gives three correct integer values, ie 2 8 5 or 1m Gives any two correct integer values or Gives all three correct integer values but in the wrong order or Gives two correct values not rounded to the nearest integer, with the third either correct but not rounded to the nearest integer, or such that the total is 15, ie 1.95 7.95 2.1 or 4.95 7.95 4.95 1.95 or 8.1 4.95 ! Correct values are truncated to 1dp For 1m, accept provided all three correct values are truncated, or two correct values are truncated and the total is 15, ie 1.95 or 1.9 7.95 7.9 5.1 4.9 2.2 or 7.9 4.9 1.9 or 8.2 4.9 1.9 or 7.9 5.2 30 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 31 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Beaches (cont) 3-5 4-6 5-7 6-8 16 11 4 c c c Tiers 4–6, 5–7, 6–8 Correct response 1m Explains that there are fewer animals altogether on the cleaned beach, and also refers to at least one of the following aspects: The presence or increase of flies The smaller proportion of sandhoppers or beetles The relative proportions of sandhoppers or beetles The increase in the number of species Additional guidance ! Follow through from parts (a) and (b) Do not accept if their incorrect values lead to different conclusions about the changes. However condone use of their incorrect values within an otherwise correct statement Values stated without interpretation eg ◆ There were 1620 animals on the uncleaned beach and 15 on the cleaned, with 33% sandhoppers on the uncleaned and 13% on the cleaned eg ■ ■ ■ ■ ■ Attracts some flies, but fewer animals overall There are much less of them, and they’re not all sandhoppers and beetles There aren’t as many animals, and a smaller percentage of them are sandhoppers The numbers have gone down, but there are always more beetles than anything else Without cleaning you get more animals but fewer types of animals No reference to the change in the total number of animals eg ◆ It attracts some flies but kills sandhoppers and beetles 31 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 32 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Equations 3-5 4-6 5-7 6-8 17 12 5 Correct response 1m Additional guidance 12 3 4 Answer of t = 1m 1 2 7 or equivalent 4 6 ✓ 15 2 Answer of r = 5 7.5 32 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 33 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Star design 3-5 4-6 5-7 6-8 18 13 6 Correct response a a 2m or 1m Additional guidance 34 Shows a complete correct method with not more than one computational error eg ■ 26 t 6 e 156, 360 m 156 e 204, 204 d 6 ■ (360 d 6) m 26 ■ (180 m 3 t 26) d 3 ■ w p 26 e 60, so 60 m 26 ■ 360 m 156 e 214 (error) ! Decimal values rounded or truncated Condone 214 e 35.6(…) 6 b b 2m or 1m 52 Shows a complete correct method with not more than one computational error eg ■ 180 m (13 p 13) e 154 360 m 2 t 154 ■ 26 t 2 ■ 180 m 2 t 13 e 157 (error) 180 m 157 e 23 23 t 2 e 46 33 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 34 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Ks and ms 3-5 4-6 5-7 6-8 19 14 7 Correct response a a 1m Draws any shape with perimeter 4k p m eg Additional guidance ! Lines not ruled or drawn accurately Accept provided the pupil’s intention is clear ■ ! Multiple attempts with some incorrect Accept provided it is clear which is the pupil’s final answer 1m Draws any shape with perimeter 2(2k p m) eg ■ ■ b b 1m c c 1m Gives a correct expression eg 2 ■ 2m 2 ■ 2 t m ■ 2m t 2m d 2 ! Units given Ignore Correct expression that is incorrectly simplified eg, for part (b) ◆ 2m t m e 3m Gives a correct expression eg 2 ■ 4k 2 ■ 4 t k ■ 4k t 2k d 2 ■ (4k t 4k) d 4 34 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 35 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 4–6, 5–7, 6–8 Tier & Question Ks and ms (cont) 3-5 4-6 5-7 6-8 19 14 7 Correct response d d 1m Gives a correct explanation eg 2 2 ■ They are the same triangle so 2m = 4k 2 = 2k2 hence m ■ The areas of the triangles are equal, and if you divide 2m2 by 2 you get m2, divide 4k2 by 2 you get 2k2 ■ If you multiply both sides by 2, you get the areas of the two triangles. Since they are the same, they must be equal Additional guidance ✓ Explanation uses Pythagoras’ theorem eg ◆ Using Pythagoras, m2 e k2 p k2 ✓ Minimally acceptable explanation eg ◆ 2m2 e 4k2 ! Explanation equates expressions for parts (b) and (c) that are correct but not simplified Accept as minimal provided neither expression is a quotient or includes a division sign eg, accept ◆ 2 t m t m e 2 t k t 2 t k eg, do not accept ◆ 2m t 2m d 2 e 4k t 2k d 2 ! Follow through Accept only if both parts (b) and (c) are incorrect, but not m2 and 2k2, and neither expression is a quotient or contains a division sign, and they lead to the relationship shown in part (d) eg, from (b) as 4m2 and (c) as 8k2, accept ◆ 4m2 e 8k2 Incomplete explanation eg ◆ The areas of the triangles are equal ◆ Divide them by 2 U1 35 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 36 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Giraffe 3-5 4-6 5-7 6-8 15 8 Correct response 3m Gives a correct answer in days, ie 14 or 14.1 or 14.2 or 14.16(…) or Gives a correct answer in days and hours, ie 14 days and 4 hours Additional guidance ! Answer of 15 For 3m, do not accept unless a correct method or a more accurate value is seen eg, accept ◆ 31.2 t 10 e 312, 31.2 t 5 e 156 so 15 days e 626 cm ! Range of values given Accept range between 14 and 15 inclusive eg, accept ◆ 14 to 15 ! Method used is trial and improvement or or 2m Shows a complete correct method, with not more than one computational error eg ■ 1.3 t 24 e 31.2cm per day, 600 m 158 e 442cm to grow, 442 d 31.2 ■ 600 m 158 e 442 442 d 1.3 e 340 340 d 24 ■ 6 m 1.58 e 4.42 4.42 d 0.013 d 24 ■ 600 158 m 31.2 31.2 or Gives an answer of 15 counting on For 3m, do not accept a correct value shown in working unless it has been identified as the answer eg, accept ◆ 158 p 14 t 31.2 e 594.8 so 14 days For 2m, accept trial and improvement and counting on as types of correct method eg, accept ◆ 158 p 14 t 31.2 e 594.8 ◆ 0 days e 1.58 1 day e 1.892 : : 14 days e 5.948 ✓ For 2m, values approximated to 2 significant figures or better eg ◆ 1.3 t 24 is about 31cm per day Needs to grow about 440 cm 440 d 31 ◆ 0 days e 1.58 1 day e 1.89 : : 14 days e 5.92 For 2m, inconsistent units eg ◆ 4.42 d 1.3 d 24 36 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 37 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Giraffe (cont) 3-5 4-6 5-7 6-8 15 8 Correct response or 1m Additional guidance Shows the digits 142 or 141(…) eg ■ 0.141 or Shows the digits 312 and 442 or Shows the digits 34 eg ■ 340 or Shows or implies a complete correct method in which the only error is to omit the subtraction of the height at birth eg ■ Answer of 19.2(…) ■ Digits 192(…) seen ■ 600 d 1.3 d 24 ✓ Trial and improvement or counting on as types of correct method U1 37 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 38 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Long-eared owls 3-5 4-6 5-7 6-8 16 9 Correct response a a 1m Shows a complete correct method eg ■ 1 t 9 p 2 t 17 p 3 t 24 p 4 t 6 p 5 t 5 p 6t1 ■ 9 p 34 p 72 p 24 p 25 p 6 Additional guidance ✓ Correct description of method eg ◆ Multiply the number of mammals found by the frequency, then add them Incomplete method eg ◆ 9, 34, 72, 24, 25, 6 ◆ Multiply the number of mammals found by the frequency b b 2m or 1m c c 2m or 1m 2.7 Shows or implies a correct method eg ■ 2.74(…) ■ 170 d 62 ■ 170 d (9 p 17 p 24 p 6 p 5 p 1) 38 000 ✓ For 2m, follow through from part (b) as their (b) t 14 000, rounded to the nearest thousand, provided their (b) is such that rounding is required Shows or implies a complete correct method eg ■ 2.7(…) t 1.4 t 10 000 ■ 37 800 ■ 38 387 ■ Their (b) t 14 000 or Shows the digits 38 38 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 39 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Pictures 3-5 4-6 5-7 6-8 17 10 Correct response a a 1m Gives a correct explanation eg ■ 6d4 ■ 4 t 1.5 e 6 ■ ■ ■ ■ b b 1m c 1m d 1m A is 2 3 of B, so B is of A 3 2 One and a half lots of 4 is 6 Half of 4 is 2, then add it on Using areas, 33.6 d 22.4 e 1.5 and the heights are the same Additional guidance ✓ Minimally acceptable explanation eg ◆ ◆ ◆ ◆ Every 1cm wide on A is 1.5cm on B Width A p half again It’s A enlarged by half its original size A fits into B one and a half times Incomplete explanation eg ◆ 4 t 1.5 ◆ Using areas, 33.6 d 22.4 = 1.5 7 Gives a correct explanation by referring to both dimensions eg ■ Horizontal and vertical stretch factors are different ■ 1.25 is not the same as 1.5 ■ The enlargements would have to be the same ■ 6 d 4 ≠ 7 d 5.6 ■ 4 d 6 e 0.67, but 5.6 d 7 e 0.8 ■ 4 d 5.6 ≠ 6 d 7 ■ 5.6 t 1.5 e 8.4, but h e 7 ! Follow through Accept follow through from their (b), provided their (b) is not 8.4 Restatement of question with no interpretation eg ◆ Horizontal is 1.5 Vertical is 1.25 4.2 or equivalent 39 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 40 2003 KS3 mathematics test mark scheme: Paper 2 Tiers 5–7, 6–8 Tier & Question Coffee 3-5 4-6 5-7 6-8 18 11 Correct response a 2m Gives three correct values that sum to 100 eg ■ Retailers 25 Growers 3 Others 72 ■ or 1m Retailers Growers Others 25.14 2.86 72 Additional guidance Note to markers: Accept as correct the following values Retailers: 25, 25.1, 25.14(…) or 25 Growers: 3, 2.9, 2.86, 2.85(…) or 2 Others: 72 Gives any two correct values or Gives three correct values that sum to 100 but in an incorrect order or Shows or implies a complete correct method with not more than one computational or rounding error eg ■ 44 d 175 t 100 e 25 5 d 175 t 100 e 2 (rounding error) 100 m 25 m 2 e 73 or Shows or implies a complete correct method with not more than two rounding errors provided the three values sum to 100 eg ■ Retailers 25.2 (rounding error) Growers 2.8 (rounding error) Others 72 40 Sourced from SATs-Papers.co.uk 1 7 http://www.SATs-Papers.co.uk 6 7 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 41 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Coffee (cont) 3-5 4-6 5-7 6-8 18 11 Correct response b 2m or 1m Gives the value £ 1.91 or £ 1.92 Additional guidance ! Method used is trial and improvement Accept correct values from this method for 2m, but do not accept as a complete correct method for 1m Shows or implies a complete correct method with not more than one computational or rounding error eg ■ ■ ■ 0.44 t 100 23 44 23 23% 10% 67% So £ = 44 = 19 = 127 (rounding error) 1.90 or Shows the digits 191(...) or 192 eg ■ 191 ■ 19.13 41 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 42 2003 KS3 mathematics test mark scheme: Paper 2 Tier & Question Tiers 5–7, 6–8 Cissoid of Diocles 3-5 4-6 5-7 6-8 19 12 Correct response 1m 3.227486(...) Additional guidance Negative value or fewer than six digits shown after the decimal point Correct value shown in working but final answer incorrect 2m Gives both correct values, ie 3.23 and m3.23, in either order ✓ For 2m, answer given as ± 3.23 ! For 2m, follow through from the first mark Accept, even from a negative value, provided their incorrect value has at least 4sf and both the positive and negative values are consistent eg, from 1.5117 for the first mark, accept ◆ 1.51 and m1.51 or 1m Gives one correct value, ie 3.23 or m3.23 or Gives consistent positive and negative values even if there is no, or incorrect, rounding eg ■ 3.22748 and m3.22748 ■ 3.22 and m3.22 ■ ±3.2 ! For 1m, their incorrect value for the first mark correctly rounded to 3sf Accept, even from a negative value, provided their incorrect value has at least 4sf, even if the other value is incorrect or omitted eg, from 3.4882 for the first mark, accept ◆ 3.49 ! For 1m, both the positive and negative of their incorrect value for the first mark given Accept provided both the positive and negative values are consistent, even if there is no, or incorrect, rounding eg, from 3.4882 for the first mark, accept ◆ 3.488 and m3.488 42 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 43 2003 KS3 mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Populations 3-5 4-6 5-7 6-8 13 Correct response a 1m India Additional guidance ✓ Unambiguous indication eg ◆ 1m b 3m or 2m 1.0 t 109 for India Gambia Gives a correct value with no evidence of an incorrect method eg ■ 220 Shows correct values for both countries eg ■ UK 250, US 30.(…) ! Limits used or answer not given to 2sf Accept values Difference UK US in the following ranges: 212 to 229 inclusive 242 to 258 inclusive 29 to 31 inclusive For 3m or 2m, incorrect method eg ◆ 2.8 t 108 m 6 t 107 ! Incorrect order of division or Do not treat as a misread, ie do not accept Shows or implies a complete correct method with not more than one computational error eg 7 5 ■ 6.0 t 10 d 2.4 t 10 then subtract 8 2.8 t 10 d 9.3 t 106 ■ 250 m 2.8 t 108 d 9.3 t 106 ■ Answer of m220 or 1m Shows a correct method or value for one of the countries eg ■ 6.0 t 107 d 2.4 t 105 ■ 2.8 t 108 d 9.3 t 106 43 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 44 2003 KS3 mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Joining 3-5 4-6 5-7 6-8 14 Correct response a 2m or 1m b 2m Gives a complete correct justification eg ■ AC = √(28.82 p 122) = 31.2 CD = √(52 p 122) = 13 31.2 p 13 p 28.8 p 5 Additional guidance ✓ Minimally acceptable justification eg ◆ ◆ ◆ 31.2 p 13 p 28.8 p 5 31.2 p 13 p 33.8 78 m 28.8 m 5 m 13 m 31.2 = 0 Shows at least one of the values 31.2, 13 or 44.2 Gives a correct justification eg ■ 33.82 = 1142.44 132 p 31.22 = 169 p 973.44 = 1142.44 Pythagoras’ theorem works so ACD is right-angled ■ 5 t 2.4 = 12 and 12 t 2.4 = 28.8 so ABC and BCD are similar right-angled triangles, and Angle ACD = angle ACB p angle BCD = angle CDB p angle BCD = 90° ■ Area of ACD 1 1 AD t BC = t 33.8 t 12 = 202.8 2 2 1 1 and CD t AC = t 13 t 31.2 = 202.8 2 2 = so ∠ACD is 90° ✓ Minimally acceptable justification eg ◆ 132 p 31.22 = (28.8 + 5)2 ! For 2m or 1m, use of trigonometry to show angle ACD is a right angle Accept for 2m provided both angle ACB and angle BCD are correctly evaluated and understanding is shown that they should sum to 90 Accept as correct for angle ACB values 67.4 or 67.38(…) Accept as correct for angle BCD any value in the range 22.58 to 22.64 inclusive eg ◆ tan –1 28.8 5` p tan –1 = 67.4 p 22.6 = 90 12 12 67.38 and 22.58 together make 90 Accept for 1m either angle ACB or angle BCD correctly evaluated ◆ or 1m Indicates understanding of the way Pythagoras’ theorem can be used to prove that triangle ACD is right-angled eg ■ If right-angled, AC2 p CD2 = AD2 ■ 1142.44 ✓ For 1m, follow through using their values for AC and CD from part (a) or Shows that triangles ABC and BCD are similar eg ■ 5 t 2.4 = 12 and 12 t 2.4 = 28.8 so ABC and BCD are similar triangles U2 44 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 45 2003 KS3 mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Squares X 3-5 4-6 5-7 6-8 15 Correct response a 1m Additional guidance Gives a correct explanation The most common correct explanations: Give at least one counter-example for 9 p y 2 eg 2 ■ When y = 5, 9 p y = 34 which is not a square number ■ Let y = 1, 9 p 1 = 10, and √10 is not an integer State that 9 p y 2 only produces a square number for particular cases eg ■ It is only true when y = m4, 0 or 4 ■ It will only give the square numbers 9 or 25 ✓ Minimally acceptable explanation eg ◆ When y = 5, 9 p y 2 = 34 Incomplete explanation eg ◆ y 2 is a square number, but adding 9 doesn’t make it still a square number Incorrect explanation eg ◆ It is only square when y is 0 U1 b 1m Gives a correct explanation eg ■ 16y 2 = (4y)2 ✓ Minimally acceptable explanation eg ◆ ◆ ■ 4y ◆ ◆ 4y ■ ■ Two square numbers multiplied make a square number, ie a2b2 = (ab)2, and 16 is 42 √(16y 2) = 4y It’s 4y t 4y You multiply by 4 before you square it a2b2 = (ab)2 Two square numbers multiplied make a square number Incomplete explanation eg ◆ y 2 is a square number and 16 is a square number ◆ √16 = 4, √y 2 = y Explanation lacks generality eg ◆ y = 3, 16 t 32 = 144 which is a square number U1 45 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 46 2003 KS3 mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Cylinder 3-5 4-6 5-7 6-8 16 Correct response 3m or 2m or 1m 0.72 or 18 25 Shows or implies a correct method, even if values are rounded or truncated prematurely eg ■ 4.5 d 2.52 ■ 4.5π d 6.25π ■ 14.1 d 19.6 Additional guidance ! Answer given as 0.7 or 0.71(...) Accept for 3m only if a correct method, even if partial, or a more accurate value is seen For 2m, incorrect method eg ◆ 5 d 4.5π t 2 = 0.71 Forms a correct equation relating the height and the volume of 4.5π eg ■ 2.52 t π t h = 4.5 t π or Shows or implies a correct method for calculating the area of the cross-section eg ■ 2.52π ■ 19.6(…) 46 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 47 2003 KS3 mathematics test mark scheme: Paper 2 Tier 6–8 only Tier & Question Triangle 3-5 4-6 5-7 6-8 17 Correct response 1m Additional guidance Forms a correct equation for the equal sides, and shows a correct first step of algebraic manipulation eg ■ a e 4b ! Correct equation for the equal sides implied a 4 ■ be ■ 1m 8b e 2a ◆ 6 1 t a = 91 2 Award both the first and second marks Forms a correct equation for the perimeter of the triangle, and simplifies eg ■ 3a p 14b e 91 ■ 5a p 6b e 91 ■ 22b p a e 91 ■ 26b e 91 ■ 1m by equation for the perimeter but not stated explicitly eg ◆ 26b = 91 6 1 t a e 91 2 Gives both correct values, ie a = 14 and b = 7 or equivalent, even if these 2 do not follow from a correct algebraic method Index to mark schemes Tier Question Page 3–5 4–6 5–7 6–8 Tier Question Page 3–5 4–6 5–7 6–8 1 Hexagons 10 13 8 1 Ratio of ages 27 2 Cities 10 14 9 2 Sizing 28 3 Number cards 11 15 10 3 Nets 29 4 Late 12 16 11 4 Beaches 30 5 Slicing cubes 13 17 12 5 Equations 32 6 Buying a bicycle 13 18 13 6 Star design 33 19 14 7 Ks and ms 34 8 1 Kings and queens 14 7 2 School uniform 15 15 8 Giraffe 36 9 3 Admission 16 16 9 Long-eared owls 38 12 4 Cubes in bags 18 17 10 Pictures 39 10 5 Temperature 18 18 11 Coffee 40 11 6 1 Drawing 19 19 12 Cissoid of Diocles 42 14 7 2 Ages 20 13 Populations 43 13 8 3 Grid percentages 22 14 Joining 44 15 9 4 Data collection 23 15 Squares X 45 16 10 5 Coins 24 16 Cylinder 46 17 11 6 Explaining why 25 17 Triangle 47 18 12 7 Water 26 47 Sourced from SATs-Papers.co.uk http://www.SATs-Papers.co.uk 254656 KS3 Ma_MS P2 5/16/03 11:42 AM Page 1 EARLY YEARS NATIONAL CURRICULUM 5–16 GCSE GNVQ GCE A LEVEL First published in 2003 NVQ © Qualifications and Curriculum Authority 2003 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. OTHER VOCATIONAL QUALIFICATIONS Excerpts may be reproduced for the purpose of research, private study, criticism or review, or by educational institutions solely for educational purposes, without permission, provided full acknowledgement is given. Produced in Great Britain by the Qualifications and Curriculum Authority under the authority and superintendence of the Controller of Her Majesty’s Stationery Office and Queen’s Printer of Acts of Parliament. The Qualifications and Curriculum Authority is an exempt charity under Schedule 2 of the Charities Act 1993. Qualifications and Curriculum Authority 83 Piccadilly London W1J 8QA www.qca.org.uk/ Further teacher packs may be purchased (for any purpose other than statutory assessment) by contacting: QCA Publications, PO Box 99, Sudbury, Suffolk CO10 2SN (tel: 01787 884444; fax: 01787 312950) Order ref: QCA/03/963 Sourced from SATs-Papers.co.uk 254656 http://www.SATs-Papers.co.uk