Mathematics tests. Mark scheme KEY STAGE 3. for Paper 2 Tiers 3 5, 4 6, 5 7 and 6 8 ALL TIERS. National curriculum assessments

Transcription

1 Ma KEY STAGE 3 ALL TIERS Mathematics tests Mark scheme f Paper 2 Tiers 3 5, 4 6, 5 7 and National curriculum assessments

2 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 infm teachers. This booklet contains the mark scheme f 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 infmation f questions is set out in the fm 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 f that question part. The Crect response column usually includes two types of infmation: a statement of the requirements f the award of each mark, with an indication of whether credit can be given f crect wking, and whether the marks are independent cumulative examples of some different types of crect 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. F 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

3 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance General guidance Using the mark schemes Answers that are numerically equivalent algebraically equivalent are acceptable unless the mark scheme states otherwise. In der to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed crect action. This is followed by further guidance relating specifically to the marking of questions that involve money, native numbers, algebra, time, codinates probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases. 3

4 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. The pupil has responded in a non-standard way. The pupil has made a conceptual err. The pupil s accuracy is marginal accding to the overlay provided. The pupil s answer crectly follows through from earlier increct wk. There appears to be a misreading affecting the wking. The crect answer is in the wrong place. Markers should use their judgement in deciding whether the response cresponds with the statement of requirements given in the Crect response column. Refer also to the Additional guidance. Calculations, fmulae and written responses do not have to be set out in any particular fmat. Pupils may provide evidence in any fm as long as its meaning can be understood. Diagrams, symbols wds are acceptable f explanations f indicating a response. Any crect method of setting out wking, however idiosyncratic, is acceptable. Provided there is no ambiguity, condone the continental practice of using a comma f a decimal point. In some questions, a method mark is available provided the pupil has made a computational, rather than conceptual, err. A computational err is a slip such as writing 4 6 = 18 in an otherwise crect long multiplication. A conceptual err is a me serious misunderstanding of the relevant mathematics; when such an err is seen no method marks may be awarded. Examples of conceptual errs 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 to give the answer 21; increct signs when wking with native numbers. Overlays can never be 100% accurate. However, provided the answer is within, touches, the boundaries given, the mark(s) should be awarded. 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 crect response an acceptable follow through response should be marked as crect. This is when the pupil misreads the infmation given in the question and uses different infmation. If the iginal intention difficulty level of the question is not reduced, deduct one mark only. If the iginal intention difficulty level is reduced, do not award any marks f the question part. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a wd number response is expected, a pupil may meet the requirement by annotating a graph labelling a diagram elsewhere in the question. 4

5 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance What if The final answer is wrong but the crect answer is shown in the wking. Marking procedure 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 increct answer is due to a transcription err If so, award the mark. in questions not testing accuracy, the crect answer has been given but then rounded truncated the pupil has continued to give redundant extra wking which does not contradict wk already done the pupil has continued, in the same part of the question, to give redundant extra wking which does contradict wk already done. If so, award the mark. If so, award the mark. If so, do not award the mark. Where a question part carries me than one mark, only the final mark should be withheld. The pupil s answer is crect but the wrong wking is seen. The crect response has been crossed rubbed out and not replaced. Me than one answer is given. The answer is crect but, in a later part of the question, the pupil has contradicted this response. A crect response should always be marked as crect unless the mark scheme states otherwise. Mark, accding to the mark scheme, any lible crossed rubbed out wk that has not been replaced. If all answers given are crect a range of answers is given, all of which are crect, the mark should be awarded unless prohibited by the mark scheme. If both crect and increct responses are given, no mark should be awarded. A mark given f one part should not be disallowed f wking answers given in a different part, unless the mark scheme specifically states otherwise. 5

6 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance Marking specific types of question Responses involving money F example: Accept Do not accept Any unambiguous indication of the crect amount 3.20(p), 3 20, 3,20, 3 pounds 20, 3-20, 3 20 pence, 3:20, 7.00 The unit, 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 with given in the answer space, accept Given units amended with crossed out in the answer space, accept 320p 700p Increct ambiguous indication of the amount 320, 320p 700p Ambiguous use of units outside the answer space with given in the answer space, do not accept 3.20p outside the answer space Increct placement of decimal points, spaces, etc increct use omission of 0 3.2, 3 200, 32 0, Responses involving native numbers F example: 2 Accept Do not accept To avoid penalising the err below me than once within each question, do not award the mark f the first occurrence of the err within each question. Where a question part carries me than one mark, only the final mark should be withheld. Increct notation 2 6

7 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving the use of algebra F example: 2 + n n + 2 2n n n 2 2 Accept Take care! Do not accept Unambiguous use of a different case variable N used f n x used f n! Unconventional notation n 2 2 n n2 n + n f 2n n n f n 2 n 2 f n n 2 + 1n f 2 + n 2 + 0n f 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, within an explanation general wking. Embedded values given when solving equations in solving 3x + 2 = 32, = 32 f x = 10 To avoid penalising the two types of err below me than once within each question, do not award the mark f the first occurrence of each type within each question. Where a question part carries me than one mark, only the final mark should be withheld. Wds used to precede follow equations expressions t = n + 2 tiles tiles = t = n + 2 f t = n + 2 Unambiguous letters used to indicate expressions t = n + 2 f n + 2! Wds units used within equations expressions n tiles + 2 n cm + 2 Do not accept on their own. Igne if accompanying an acceptable response. Ambiguous letters used to indicate expressions n = n + 2 f n + 2 7

8 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving time A time interval F example: 2 hours 30 minutes Accept Take care! Do not accept Any unambiguous indication 2.5 (hours), 2h 30 Digital electronic time ie 2:30 Increct ambiguous time interval 2.3(h), 2.30, 2-30, 2h 3, 2.30min! The unit, hours and/ minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, crosses out the given unit, accept answers with crect units, unless the question has specifically asked f other units to be used. A specific time F example: 8:40am 17:20 Accept Do not accept Any unambiguous, crect indication 08.40, 8.40, 8:40, 0840, 8 40, 8-40, twenty to nine, 8,40 Unambiguous change to hour clock 17:20 as 5:20pm, 17:20pm Increct time 8.4am, 8.40pm Increct placement of separats, spaces, etc increct use omission of 0 840, 8:4:0, 084, 84 Responses involving codinates F example: ( 5, 7 ) Accept Do not accept Unconventional notation ( 05, 07 ) ( five, seven ) x y ( 5, 7 ) ( x = 5, y = 7 ) Increct ambiguous notation ( 7, 5 ) y x ( 7, 5 ) ( 5x, 7y ) ( 5 x, 7 y ) ( x 5, y 7 ) 8

9 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance Responses involving probability A numerical probability should be expressed as a decimal, fraction percentage only. F example: % 10 Accept Take care! Do not accept Equivalent decimals, fractions and percentages 0.700, , 35 50, 70.0% A probability crectly expressed in one acceptable fm which is then increctly converted, but is still less than 1 and greater than = The first four caties of err below should be igned if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of err below me than once within each question, do not award the mark f the first occurrence of each type of err unaccompanied by an acceptable response. Where a question part carries me than one mark, only the final mark should be withheld.! A probability that is increctly expressed 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 inters in the numerat and/ denominat.! A probability expressed as a ratio 7 : 10, 7 : 3, 7 to 10 A probability greater than 1 less than 0 9

10 2007 KS3 Mathematics test mark scheme: Paper 2 General guidance Recding marks awarded on the test paper All questions, even those not attempted by the pupil, will be marked, with a 1 a 0 entered in each marking space. Where 2m can be split into gained and lost, with no explicit der, then this will be recded by the marker as 1 0 The total marks awarded f 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 recded 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 f the award of different levels, will be available on the NAA website from Monday 25 June 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 sces on the test papers and the levels awarded. 10

11 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 1 Crect response Additional guidance Rules 11, 14 23, 47 41, 122! First new term f each sequence crect with second terms all increct omitted Mark as 0, 0, 1 11

12 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 2 Crect response Additional guidance Homewk a b Monday and Wednesday, in either der Maths, English and Technology, in any der! Names of days subjects abbreviated Accept provided unambiguous, f part (a) accept M and W, f part (b) do not accept M, E and T c 3 3 Crect response Additional guidance a E Odd one out D b Completes the sentence crectly with a crect property equal sides lines of symmetry as the der of rotation symmetry Minimally acceptable response sides the same line symmetry rotation symmetry identical lines! Increct irrelevant statement Igne alongside a crect response, accept equal sides and right angles, do not accept right angles Incomplete increct response sides equal angles squares f the area U1 12

13 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 4 Crect response Additional guidance a 5 Value qualified About 5 Hibernation! Value inaccurate Accept values between 4.9 and 5.1 inclusive, between 4 months 27 days and 5 months 3 days inclusive b Indicates Yes and gives a crect explanation The most common crect explanations: State imply that they sleep f me than 6 months They sleep f 6 1 months, which is me 2 than half of 12 Half a year is 6 months but they sleep f just over 6 months Minimally acceptable explanation months Just over 6 months Me than 6 boxes are shaded November to May is six months and then half of October Half a month me! Exact value given Accept values between 6.4 and 6.6 months inclusive, between 6 months 12 days and 6 months 18 days inclusive, accept 6 months and 2 weeks Incomplete increct explanation They sleep f me than half the year They sleep from halfway through October to the end of April Half a year is 6 months but they sleep f 7 months Refer to the area shaded unshaded and its relation to the whole circle Me than halfway round the circle is shaded The white bit f dmice doesn t reach round half the circle Minimally acceptable explanation Me than halfway round Me than half the chart is shaded Me is shaded than unshaded U1 Incomplete explanation It shows me shaded months 13

14 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 3 5 only 5 Crect response Additional guidance a Concert b 2m 5 U1 Shows the digits 8225 Shows implies a complete crect method with not me than one err, even if their final answer is not an inter is rounded truncated ( ) = Crect response Additional guidance Cake a 450 b Indicates the crect position on the scale, ie Unambiguous indication ! Inaccurate indication Accept indications that are closer to 275 than either c 2pm 14:00 Time ambiguous increct 2 o clock 14:00am d Indicates Cylinder, ie 14

15 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance Bar chart 2m Completes all labels f both axes crectly, ie Unambiguous indication of item names, f Glue G Glue Pens Rulers U1 Completes at least two labels on the vertical axis crectly 8 1 Crect response Additional guidance a a Gives A as (0, 6) Codinates Gives C as (4, 3)! Answers f A and C transposed but otherwise completely crect If this is the only err, ie gives A as (4, 3) and gives C as (0, 6), mark as 0, 1 b b Indicates point D on the graph at (2, 7)! Point inaccurate, not labelled marked only with the letter D Condone any unambiguous indication within 2mm of the crect intersection of the grid 15

16 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance Fitting tiles a a Indicates crectly two congruent F-tiles on the diagram! Tile not shaded inaccurately indicated Accept provided the pupil s intention is clear and there is no ambiguity Tiles overlapping b b Indicates two congruent tiles on the diagram U1 Indicates two congruent tiles on the diagram, different from any previously credited 16

17 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance a a Claire Unambiguous indication of name, f Claire C Names b b U1 Gives the names Claire then Tom 11 4 Crect response Additional guidance Leaves a a Writes the leaves in the crect der f area, ie Willow Oak Beech smallest area largest area Unambiguous indication, f part (a) W O B smallest area largest area b b Writes the leaves in the crect der f perimeter, ie Willow Beech Oak smallest perimeter largest perimeter! Order given as largest to smallest f both parts (a) and (b) Mark as 0, 1! Responses f parts (a) and (b) transposed but otherwise crect Mark as 0, 1 17

18 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance Marbles 2m Matches all three questions crectly, ie ! Question matched with me than one calculation F 2m, do not accept as a crect match Matches any two of the questions crectly 13 6 Crect response Additional guidance a and b Gives a pair of numbers f a and b, such that b = a + 4 a = 5 b = 9 a = 1.5 b = 5.5 Values embedded = 9 a = b = 9 U1 Gives a pair of numbers f a and b, such that b = a + 4, different from any previously credited 18

19 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance Turning Indicates the crect shape, ie 15 8 Crect response Additional guidance Party 2m Completes all four entries in the table crectly, ie Total: U1 Completes at least three entries in the table crectly Completes all four entries crectly with some all amounts of money given in pence! F, follow through Where the only err is in the total cost of balloons, f the overall total accept their total cost of balloons

20 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, Crect response Additional guidance a a 10 10% Survey b b 2m Completes the percentage bar chart crectly, ie 50% labelled No 40% labelled Don t know 10% labelled Yes, with bars in any der No Don t know Yes 0% 20% 40% 60% 80% 100% Unambiguous labelling???? 0% 20% 40% 60% 80% 100%! Lines not ruled accurate Accept provided the pupil s intention is clear Indicates sections cresponding to 50%, 40% and 10% but fails to label, labels increctly bars are not continuous???? 0% 20% 40% 60% 80% 100% Shows implies the values 50, 40 and 10 0% 20% 40% 60% 80% 100% Indicates a crect bar f either Don t know Yes, and labels crectly 20

21 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Additional guidance Frog spawn a a a 15th February (1997) Unambiguous commonly used date notation 15/2 2/15 [US notation] b b b Gives a possible description of the weather In 1991 it was colder than the other years It must have been less warm than usual Minimally acceptable response Cold Not warm It got warmer later! Response implies a preference Condone provided the pupil s intention is clear, accept It must have been nasty weather It was rainy and not sunny Bad U1 Incomplete increct response They were seen later than in other years Very cold so the gs were seen quicker Crect response Additional guidance a a a Indicates 4a + 3, ie Simplifying b b b 8b

22 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Additional guidance Indicates A and gives the value 250 Containers Crect response Additional guidance a a a Gives the values 60, 60 and 60 Single answer of 60 given Triangles b b b Gives the values 90, 45 and 45, in any der Crect response Additional guidance a a a Indicates B Spinners b b b Indicates A and D, in either der 22

23 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Additional guidance Faces a a a 8 b b b Draws a solid with 6 faces in any ientation, using the isometric grid crectly Some all internal lines shown! 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 all hidden lines shown Do not accept unless the lines are clearly identified as hidden lines, accept, do not accept Isometric grid not used crectly 23

24 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Additional guidance a a a 30(.00) Fir trees b b b U1 4 and 5, in either der! Upper bound taken to be just under 5 F the upper bound, accept values between 4.9 and 5 inclusive Rectangles and squares Crect response Additional guidance a b a b a b 4 5! Value repeated Accept provided there is no ambiguity, f part (a) accept 4 by 4, f part (a) do not accept 4 4 U1! F parts (a) and (b), response of 16 then 20 Mark as 0, 1 24

25 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, Crect response Additional guidance Lemonade 2m 80 p Shows the value 0.8(0) Shows implies a complete crect method with not me than one computational err ( ) (err) 6.40 = 1.00 U1 Shows the value (0) and (0) Crect response Additional guidance Three angles Indicates No and gives a crect explanation = 178 but it should equal 180 f a straight line is 2 drees too small f a straight line = 8, so they couldn t add to 180 Minimally acceptable explanation that states implies the angles should add to 180 that they add to less than 180 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 U1 Incomplete increct explanation = 178 which is not straight The angles add to 188 not 180 The angles add to 178 so it will look straight 25

26 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 3 5, 4 6, 5 7, Crect response Additional guidance 14! Increct notation, as an answer f the first mark x = 14 Penalise only the first occurrence 13! Incomplete processing, as an answer f the first mark x = Penalise only the first occurrence Solving Marking overlay available Crect response Additional guidance Newspaper 2m Draws the sects f Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels crectly Unambiguous abbreviation E f Evening newspaper, N f No newspaper Draws the sects f Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels crectly Draws the sects f Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay but fails to label labels increctly Shows implies that 5 people are represented by 30 that 1 person is represented by 6 5 people = = = 6 60, 90 seen 26

27 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, 6 8 Completing rules Crect response Additional guidance Gives two crect values in the crect der, and a crect expression in x 3, 1, 3x + 1 1, 9, x + 9 2, 21, 2x + 21 F the first mark, given example repeated! Unconventional notation, f x x + 9 Condone Gives two crect values in the crect der, and a crect expression in x 4, 3, 4x 3 2, 21, 2x 21 x, 3, x 2 3 Gives two crect values in the crect der, and a crect expression in x 2, 11, x , 5, 2x + 5 ( x ) 1, 9, x

28 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, 6 8 Parallelogram Crect response Additional guidance 2m Gives the crect value with a crect unit 35cm 2 Shows the value 35 Shows a complete crect method with not me than one computational err and with a crect unit f area shown at least once 7 5 and cm 2 seen (10 3) 5 and cm 2 seen and cm 2 seen and cm 2 seen and cm 2 seen (err) = 37 and cm 2 seen F, necessary brackets omitted Relationships Crect response Additional guidance 9 100! Incomplete processing, f the first mark 10 1, f the second mark 10 2 Penalise only the first occurrence 28

29 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Additional guidance Pi a a a Equivalent fractions decimals b b b Indicates , ie Marking overlay available Crect response Additional guidance Enlarging 2m Shows a crect enlarged shape with all five vertices within the tolerances as shown on the overlay Shows at least three vertices within the tolerances as shown on the overlay Shows a crect enlarged shape with all five vertices within the tolerances as shown on the overlay, but in an increct position and/ ientation! Lines not ruled accurate Accept provided the pupil s intention is clear! Construction lines drawn Igne, even if increct 29

30 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Additional guidance a a a 15 Values b b b 5 1 equivalent 2 c c Indicates that e > 5 It has to be higher than 5 Any number over 5 Minimally acceptable indication > 5 Above 5 Me than half of 10! Range includes 5 5 over Condone Native values of f excluded 5 < e 10 Between 5 and 10 Increct indication e can be 6, 7, 8 and so on e must be 5.1 me Incomplete indication e = 10 f f e 30

31 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, 6 8 Travelling by car Crect response Additional guidance a a a 72 b b b 4 c c 2m 1.8 equivalent! Answer of 2 F 2m, do not accept unless a crect method a me accurate value is seen! F 2m, follow through from part (b) Accept follow through as 18 (their (b) + 6) as (their (b) + 14) (their (b) + 6), rounded truncated to at least 2 s.f. Shows implies a crect method 18 ( ) F, necessary brackets omitted

32 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Additional guidance a a Gives a crect explanation The most common crect explanations: Brackets Give the crect expansion of the expression 3(2a + 1) = 6a + 3, not 6a + 1 It should be 2 greater, ie 6a + 3 Minimally acceptable explanation 6a + 3 She needs to add 2 Incomplete increct explanation 3(2a + 1) 6a + 1 3(2a + 1) = 6a + 2 3(2a + 1) = 6a + 3 = 9a Address the misconception Both things in the brackets should be multiplied by 3, but she has fgotten the 1 Minimally acceptable explanation 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 She hasn t multiplied out the brackets crectly The 1 is increct Give a crect counter example When a = 1 then 3(2a + 1) = 9, but 6a + 1 = 7 If a is 2, 3( ) Minimally acceptable explanation When a = 1 you get 9 and 7 Incomplete explanation When a = 1 you get different answers f each side, so it can t be right 32

33 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, 6 8 Brackets (cont) Crect response Additional guidance b b Gives a crect explanation The most common crect explanations: Give the crect expansion of the expression (k + 4)(k + 7) = k k + 28, not k He should get k 2 + 4k + 7k + 28 He has missed out 4k + 7k so it should be k k + 28 Minimally acceptable explanation k k + 28 k 2 + 4k + 7k k is missing There should be 4k and 7k as well! Crect expression equated to zero k k + 28 = 0 Condone Incomplete increct explanation (k + 4)(k + 7) k k k + 28 = k It s 11k Address the misconception 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 He hasn t multiplied the 4 the 7 by k There should be a k term It should have been like this: (k + 4)(k + 7) Incomplete explanation 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 crect counter example When k = 1 then (k + 4)(k + 7) = 40, but k = 29 If k is 2, (2 + 4)(2 + 7) Minimally acceptable explanation When k = 1 you get 40 and 29 Incomplete explanation When k = 1 you get different answers f each side, so it can t be right 33

34 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 4 6, 5 7, Crect response Additional guidance Vowels a 2m 0.61 equivalent probability Shows the digits 61 Shows the value 0.39 equivalent probability Shows implies a complete crect method with not me than one computational err 1 ( ) = 0.38 (err) = 0.62 b 2m , equivalent probability F 2m, Shows the digits 936 Shows implies a complete crect method with not me than one computational err

35 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Additional guidance Beams 3m Indicates the 1st way, and gives the crect difference of m Shows the digits 132(0) Shows the digits 484(0) and 352(0) Shows implies crect substitution of all values into the fmula and the intention to subtract (11 8) ( ) Shows a complete crect method with not me than one computational err, and gives a crect decision f their values = 4440 (err) = 920 so 1st way, difference 920 Shows the digits 484(0) 352(0) Indicates the 1st way and gives an answer of 264 [the only err is to omit to multiply the substituted values by 5] Indicates the 1st way and gives an answer of 6600 [the only err is to process as (5 11) 2 8 and as (5 8) 2 11] 35

36 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Additional guidance Car park 3m 15 2m Shows the values 24 and 160 Shows a crect method with not me than one computational rounding err ( ) 3 ( ) = 72, 72 3 = 26 (err), = = Shows the value U1 Shows a crect method with not me than two computational rounding errs = 62 (err), 62 3 = 21 (premature rounding), = Volume of prisms Crect response Additional guidance a a 120 b b

37 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, 6 8 Marking overlay available Crect response Additional guidance Straight lines a a Draws a different straight line with gradient 1, within the tolerance as shown on the overlay when the y-axes are aligned! Line sht 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 b b 20 c c Gives a crect equation y = 5x x y = 10! Unconventional notation y1 = 5 x + 10 Condone Two semicircles Crect response Additional guidance 2m , 88.6, 88.5( ) 89! Value of 88 F 2m, do not accept unless a crect method a me accurate value is seen Shows one entry from the following list: 25 ( 78.6, 78.5( ), 79) 10 ( 31.( )) 15 ( 47.( )) 20 ( 62.8( ), 63) and 30 ( 94.( )) 50 ( 157.( )) ( 167.( )) U1 Shows implies a complete crect method with not me than one computational rounding err Value of 88, with no crect method me accurate value seen 37

38 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Additional guidance Which pupil? 2m Indicates Class 9A and gives a crect justification The most common crect justifications: Use the proptions of boys in each class, in a fm that enables comparison = but = You get and = 0.464( ), = 0.461( ) ( 0.462) A gives 46.4% and B gives 46.2% = 2.15( ) = 2.16( ) ( 2.17) = 12.07( ) ( ) = 12.9( ) ( ) = 338, 12 ( ) = 336 F 2m, minimally acceptable justification , ( ), 0.461( ) ( 0.462) 46.4, > F 2m, incomplete increct justification > > 12 Use the ratios of boys to girls girls to boys in each class, in a fm that enables comparison 9A is 0.86( ) boys f every girl, 9B is 0.85( ) 9A is 0.87 boys f every girl, 9B is : 15 = 1 : 1.15( ) 12 : 14 = 1 : 1.16( ) ( 1 : 1.17) = 12.1( ) = ( ) , F 2m, minimally acceptable justification 0.86( ), 0.85( ) 0.87, ( ), 1.16( ) ( 1.17) Reason generally about the differences between the numbers of boys and girls A difference of 2 out of the bigger total in 9A is less than out of the smaller total in 9B 2 28 < 2 26 F 2m, minimally acceptable justification There are two fewer boys than girls in both, but 9A is bigger 38

39 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, 6 8 Which pupil? (cont) Crect response Additional guidance Shows a crect justification but makes an increct no decision = 0.46, = 0.46 so equal Shows a crect justification with not me than one computational err then makes the crect decision f their values , 346 (err), 9B indicated

40 2007 KS3 Mathematics test mark scheme: Paper 2 Tiers 5 7, Crect response Additional guidance Pythagas a Gives a crect explanation The most common crect explanations: Explanation uses only accurate scale drawing Show that the values 6, 8 and 10 wk using Pythagas theem = = 100 = = = 36 = 6 2 Minimally acceptable explanation = = 100 The square of the longest side is equal to the sum of the squares of the other two sides Incomplete explanation State imply that the triangle is an enlargement of a 3, 4, 5 right-angled triangle 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 Pythagean triple Minimally acceptable explanation It s an enlarged 3, 4, 5 triangle 3 2 = 6, 4 2 = 8 and 5 2 = 10 Incomplete explanation It s like a 3, 4, 5 triangle b Gives a crect justification = = = = = is a 15% increase = = 9.2 tan = tan 53.1 = 9.2 Minimally acceptable explanation = 6 8 Incomplete explanation Explanation attempts to use Pythagas theem =

41 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only Pythagas (cont) Crect response Additional guidance c Shows the digits ! Zero(s) given after the last decimal place within standard fm notation Condone, f both marks in part (c) accept Shows the crect value in standard fm, ie Crect response Additional guidance Expressions 2m Gives all three crect expressions, ie y y y + 3a! Expressions unsimplified use unconventional notation, f the third expression y + a + a + a 1y + 3 a Condone U1 Gives two crect expressions 41

42 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 19 Crect response Additional guidance Gillas 2m Gives an inter value between and inclusive ! Gives a non-inter value within the crect range ( ) Condone Shows the digits 166( ) 167 Shows a complete crect method with not me than one computational rounding err = 200 (premature rounding), = Crect response Additional guidance Houses 2m 2.9 equivalent! Value of 3 F 2m, do not accept unless a crect method a me accurate value is seen Shows the value Shows a complete crect method with not me than one computational rounding err ( ) = 300 (err), = 3 F, necessary brackets omitted

43 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only Subtracting and squaring 21 Crect response Additional guidance 2m Gives the number as 13 and shows a complete crect method f solving algebraically (x 25) 2 = x 2 25 x 2 50x = x x = 650 x = 13 Method used is trial and improvement U1 Shows a crect expression without brackets that is equivalent to (unknown 25) 2 x 2 50x n 2 25n 25n a a 50 a Shows a crect equation (x 25) 2 = x

44 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 22 Crect response Additional guidance Light years a ! Zero(s) given after the last decimal place within standard fm notation, f part (a) Condone b 7.35(54) ! F part (b), follow through Accept 7.8 their (a) provided this is written crectly in standard fm to at least 2 s.f. 23 Crect response Additional guidance Octagon 2m 2 2, 8 2.8( )! Value of 3 F 2m, do not accept unless a crect method a me accurate value is seen U1 Shows implies a crect equation in y y 2 = 8 y 2 + y 2 = 4 2 2y 2 = 16 y y + y y = 4 4 2y = 4 4sin 45 ( 4cos 45) 44

45 2007 KS3 Mathematics test mark scheme: Paper 2 Tier 6 8 only 24 Crect response Additional guidance x, y, a and b a a b! F part (a), unsimplified expression unconventional notation Condone b 2m 2b a F 2m, follow through from part (a) Shows a crect expression f x, even if it is unsimplified, uses unconventional notation there is subsequent increct wking 2 b a b (a b) a 2(a b) Shows a complete crect method with not me than one err x + 2y = a 2x + 2y = b (err) x = b a Fms two crect equations that would allow elimination of y x + 2y = a 2x + 2y = 2b F, second equation doubled without the first equation restated 2x + 2y = 2b seen Attempts to solve by substitution and fms a crect equation in x x + a b = b x + 2(a b) = a x + 2(b x) = a 45

46 2007 KS3 Mathematics test mark scheme: Paper 2 Index Index to mark schemes Tier Question Page Rules 11 2 Homewk 12 3 Odd one out 12 4 Hibernation 13 5 Concert 14 6 Cake 14 7 Bar chart Codinates Fitting tiles Names Leaves Marbles a and b Turning Party Survey Frog spawn Simplifying Containers Triangles Spinners Faces Fir trees Rectangles and squares Lemonade Three angles Solving 26 46

47 2007 KS3 Mathematics test mark scheme: Paper 2 Index Tier Question Page Newspaper Completing rules Parallelogram Relationships Pi Enlarging Values Travelling by car Brackets Vowels Beams Car park Volume of prisms Straight lines Two semicircles Which pupil? Pythagas Expressions Gillas Houses Subtracting and squaring Light years Octagon x, y, a and b 45 47

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