Ks3 Math Test Mark Scheme
Ma
KEY STAGE
Mathematics tests
3
ALL TIERS
Mark scheme for Paper 2 Tiers 3–5, 4–6, 5–7 and 6–8
2007
National curriculum assessments
2007 KS3 Mathematics test mark scheme: Paper 2
Introduction
Introduction
The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to inform teachers. This booklet contains the mark scheme for paper 2 at all tiers. The paper 1 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier.
The structure of the mark schemes
The marking information for questions is set out in the form of tables, which start on page 11 of this booklet. …show more content…
The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part and the total number of marks available for that question part. The Correct response column usually includes two types of information: a statement of the requirements for the award of each mark, with an indication of whether credit can be given for correct working, and whether the marks are independent or cumulative examples of some different types of correct response, including the most common. The Additional guidance column indicates alternative acceptable responses, and provides details of specific types of response that are unacceptable. Other guidance, such as when ‘follow through’ is allowed, is provided as necessary. Questions with a UAM element are identified in the mark scheme by an encircled U with a number that indicates the significance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet. The 2007 key stage 3 mathematics tests and mark schemes were developed by the Test Development team at Edexcel.
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2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
General guidance
Using the mark schemes
Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise. In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance relating specifically to the marking of questions that involve money, negative numbers, algebra, time, coordinates or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases.
3
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. 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 × 6 = 18 in an otherwise correct long multiplication. A conceptual error is a more serious misunderstanding of the relevant mathematics; when such an error is seen no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 × 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers. 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 has made a conceptual error.
The pupil’s accuracy is marginal according to the overlay provided. The pupil’s answer correctly follows through from earlier incorrect work. There appears to be a misreading affecting the working.
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. This is when the pupil misreads the information given in the question and uses different information. If the original intention or difficulty level of the question is not reduced, deduct one mark only. If the original intention or difficulty level is reduced, do not award any marks for the question part. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question.
The correct answer is in the wrong place.
4
2007 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.
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 incorrect answer is due to a transcription error in questions not testing accuracy, the correct answer has been given but then rounded or truncated the pupil has continued to give redundant extra working which does not contradict work already done the pupil has continued, in the same part of the question, to give redundant extra working which does contradict work already done. If so, award the mark. If so, award the mark.
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.
The pupil’s answer is correct but the wrong working is seen. The correct response has been crossed or rubbed out and not replaced. More than one answer is given.
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.
The answer is correct but, in a later part of the question, the pupil has contradicted this response.
5
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Marking specific types of question Responses involving money
For example: £ 3.20 £7
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 The unit, £ or p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units eg with £ given in the answer space, accept 3.20 7 or 7.00 Given units amended eg with £ crossed out in the answer space, accept 320p 700p
Do not accept
Incorrect or ambiguous indication of the amount eg £320, £320p or £700p
Ambiguous use of units outside the answer space eg with £ given in the answer space, do not accept 3.20p outside the answer space Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 eg £3.2, £3 200, £32 0, £3-2-0 £7.0
Responses involving negative numbers
For example: –2
Accept
Do not accept
To avoid penalising the error below more than once within each question, do not award the mark for the first occurrence of the error within each question. Where a question part carries more than one mark, only the final mark should be withheld. Incorrect notation eg 2 –
6
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving the use of algebra
For example: 2+n
n+2
2n
n
2
n2
Accept
Unambiguous use of a different case or variable eg N used for n x used for n
Take care ! Do not accept
!
Unconventional notation eg n × 2 or 2 × n or n2 or n + n for 2n n × n for n2 n ÷ 2 for n or 1 n 2 2 2 + 1n for 2 + n 2 + 0n for 2 Within a question that demands simplification, do not accept as part of a final answer involving algebra. Accept within a method when awarding partial credit, or within an explanation or general working. Embedded values given when solving equations eg in solving 3x + 2 = 32, 3 × 10 + 2 = 32 for x = 10
To avoid penalising the two types of error below more than once within each question, do not award the mark for the first occurrence of each type within each question. Where a question part carries more than one mark, only the final mark should be withheld. Words used to precede or follow equations or expressions eg t = n + 2 tiles or tiles = t = n + 2 for t = n + 2
!
Words or units used within equations or expressions eg n tiles + 2 n cm + 2 Do not accept on their own. Ignore if accompanying an acceptable response. Ambiguous letters used to indicate expressions eg n = n + 2 for n + 2
Unambiguous letters used to indicate expressions eg t = n + 2 for n + 2
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2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving time
A time interval For example: 2 hours 30 minutes
Accept
Any unambiguous indication eg 2.5 (hours), 2h 30 Digital electronic time ie 2:30
Take care ! Do not accept
Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min
!
The unit, hours and/or minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, or crosses out the given unit, accept answers with correct units, unless the question has specifically asked for other units to be used.
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 5:20 pm, 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
Responses involving coordinates
For example: ( 5, 7 )
Accept
Unconventional notation eg ( 05, 07 ) ( five, seven ) x y
Do not accept
Incorrect or ambiguous notation eg ( 7, 5 ) y x
( 5, 7 ) ( x = 5, y = 7 )
( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x – 5, y – 7 )
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2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving probability
A numerical probability should be expressed as a decimal, fraction or percentage only. 7 For example: 0.7 70% 10
Accept
Equivalent decimals, fractions and percentages 70 35 , 70.0% eg 0.700, , 100 50
Take care ! Do not accept
The first four categories of error below should be ignored if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of error below more than once within each question, do not award the mark for the first occurrence of each type of error unaccompanied by an acceptable response. Where a question part carries more than one mark, only the final mark should be withheld.
A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 70 = 18 eg 100 25
!
A probability that is incorrectly expressed eg 7 in 10 7 over 10 7 out of 10 7 from 10 A probability expressed as a percentage without a percentage sign. A fraction with other than integers in the numerator and/or denominator. A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10
! ! !
A probability greater than 1 or less than 0
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2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Recording marks awarded on the test paper
All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1 0 The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper. A total of 120 marks is available in each of tiers 3–5, 4–6, 5–7 and 6–8.
Awarding levels
The sum of the marks gained on paper 1, paper 2 and the mental mathematics paper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the NAA website www.naa.org.uk/tests from Monday 25 June 2007. QCA will also send a copy to each school in July. Schools will be notified of pupils’ results by means of a marksheet, which will be returned to schools by the external marking agency with the pupils’ marked scripts. The marksheet will include pupils’ scores on the test papers and the levels awarded.
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2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Rules
Correct response 1m 11, 14 Additional guidance
3-5 4-6 5-7 6-8
1
1m
23, 47
1m
41, 122
! First new term for each sequence correct with second terms all incorrect or omitted Mark as 0, 0, 1
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2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Homework
Correct response 1m Monday and Wednesday, in either order Additional guidance ! Names of days or subjects abbreviated Accept provided unambiguous eg, for part (a) accept M and W eg, for part (b) do not accept M, E and T
3-5 4-6 5-7 6-8
2 a
b
1m
Maths, English and Technology, in any order
c
1m
3
Tier & Question
Odd one out
Correct response 1m E Additional guidance
3-5 4-6 5-7 6-8
3 a
1m
D
b
1m
Completes the sentence correctly with a correct property eg … equal sides … lines of symmetry … as the order of rotation symmetry
Minimally acceptable response eg … sides the same … line symmetry … rotation symmetry … identical lines ! Incorrect or irrelevant statement Ignore alongside a correct response eg, accept … equal sides and right angles eg, do not accept … right angles Incomplete or incorrect response eg … sides … equal angles … squares for the area
U1
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2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Hibernation
Correct response 1m 5 Additional guidance Value qualified eg About 5 ! Value inaccurate Accept values between 4.9 and 5.1 inclusive, or between 4 months 27 days and 5 months 3 days inclusive
3-5 4-6 5-7 6-8
4 a
b
1m
Indicates Yes and gives a correct explanation The most common correct explanations:
State or imply that they sleep for more than 6 months eg 1 They sleep for 6 months, which is more 2 than half of 12 Half a year is 6 months but they sleep for just over 6 months
Minimally acceptable explanation eg 1 6 months 2 Just over 6 months More than 6 boxes are shaded November to May is six months and then half of October Half a month more ! Exact value given Accept values between 6.4 and 6.6 months inclusive, or between 6 months 12 days and 6 months 18 days inclusive eg, accept 6 months and 2 weeks Incomplete or incorrect explanation eg They sleep for more than half the year They sleep from halfway through October to the end of April Half a year is 6 months but they sleep for 7 months 1 6 2
Refer to the area shaded or unshaded and its relation to the whole circle eg More than halfway round the circle is shaded The white bit for dormice doesn’t reach round half the circle
Minimally acceptable explanation eg More than halfway round More than half the chart is shaded More is shaded than unshaded Incomplete explanation eg It shows more shaded months
U1
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2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Concert
Correct response 1m £ 98.35 Additional guidance
3-5 4-6 5-7 6-8
5 a
b
2m or 1m
5
Shows the digits 8225 or Shows or implies a complete correct method with not more than one error, even if their final answer is not an integer or is rounded or truncated eg (155.75 – 3 × 24.50) ÷ 16.45 73.5 + 16.45 + 16.45 + 16.45 + 16.45 + 16.45 = 155.75
U1
Tier & Question
Cake
Correct response 1m 450 Additional guidance
3-5 4-6 5-7 6-8
6 a
b
1m
Indicates the correct position on the scale, ie
Unambiguous indication ! Inaccurate indication Accept indications that are closer to 275 than either 250 or 300
200
300
c
1m
2pm or 14:00
Time ambiguous or incorrect eg 2 o’clock 14:00am
d
1m
Indicates Cylinder, ie
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Bar chart
Correct response 2m Completes all labels for both axes correctly, ie 24 20 16 12 8 4 0 Additional guidance Unambiguous indication of item names eg, for Glue G
3-5 4-6 5-7 6-8
7
Glue
Pens
Rulers
or 1m U1 Completes at least two labels on the vertical axis correctly
Tier & Question
Coordinates
Correct response 1m Gives A as (0, 6) Additional guidance
3-5 4-6 5-7 6-8
8 1 a a
1m
Gives C as (4, 3)
! Answers for A and C transposed but otherwise completely correct If this is the only error, ie gives A as (4, 3) and gives C as (0, 6), mark as 0, 1
b b
1m
Indicates point D on the graph at (2, 7)
!
Point inaccurate, not labelled or marked only with the letter D Condone any unambiguous indication within 2mm of the correct intersection of the grid
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Fitting tiles
Correct response 1m Indicates correctly two congruent F-tiles on the diagram eg Additional guidance ! Tile not shaded or inaccurately indicated Accept provided the pupil’s intention is clear and there is no ambiguity Tiles overlapping
3-5 4-6 5-7 6-8
9 2 a a
b b
1m
Indicates two congruent tiles on the diagram eg
1m U1
Indicates two congruent tiles on the diagram, different from any previously credited
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Names
Correct response 1m Claire Additional guidance Unambiguous indication of name eg, for Claire C
3-5 4-6 5-7 6-8
10 3 a a
b b
1m U1
Gives the names Claire then Tom
Tier & Question
Leaves
Correct response 1m Writes the leaves in the correct order for area, ie Willow smallest area Oak Beech largest area Additional guidance Unambiguous indication eg, for part (a) W smallest area O B largest
area
3-5 4-6 5-7 6-8
11 4 a a
b b
1m
Writes the leaves in the correct order for perimeter, ie Willow smallest perimeter Beech Oak largest perimeter
! Order given as largest to smallest for both parts (a) and (b) Mark as 0, 1 ! Responses for parts (a) and (b) transposed but otherwise correct Mark as 0, 1
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Marbles
Correct response 2m Matches all three questions correctly, ie
10 10 × 7 10 × 12 12 × 7 10 × 12 × 7 10 + 12 + 7
3-5 4-6 5-7 6-8
12 5
Additional guidance ! Question matched with more than one calculation For 2m or 1m, do not accept as a correct match
or 1m Matches any two of the questions correctly
Tier & Question
a and b
Correct response 1m Gives a pair of numbers for a and b, such that b=a+4 eg a=5 b=9 a = 1.5 b = 5.5 Additional guidance Values embedded eg 4+5=9 a=4+5 b=9
3-5 4-6 5-7 6-8
13 6
1m U1
Gives a pair of numbers for a and b, such that b = a + 4, different from any previously credited
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Turning
Correct response 1m Indicates the correct shape, ie Additional guidance
3-5 4-6 5-7 6-8
14 7
Tier & Question
Party
Correct response 2m Completes all four entries in the table correctly, ie 4.95 3.20 1.95 5 13 10 Total: 24.75 41.60 19.50 85.85 Additional guidance
3-5 4-6 5-7 6-8
15 8
or 1m Completes at least three entries in the table correctly or Completes all four entries correctly with some or all amounts of money given in pence U1 ! For 1m, follow through Where the only error is in the total cost of balloons, for the overall total accept their total cost of balloons + 61.10
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Survey
Correct response 1m 10 10% Additional guidance
3-5 4-6 5-7 6-8
16 9 a a
b b
2m
Completes the percentage bar chart correctly, ie 50% labelled No 40% labelled Don’t know 10% labelled Yes, with bars in any order eg
No 0% 20% 40% Don’t know Yes 60% 80% 100%
Unambiguous labelling eg
? 0% 20% 40% ? 60% ? ? 80% 100%
! Lines not ruled or accurate Accept provided the pupil’s intention is clear
or 1m Indicates sections corresponding to 50%, 40% and 10% but fails to label, labels incorrectly or bars are not continuous eg
? 0% or Shows or implies the values 50, 40 and 10 eg 20% ? 40% ? 60% ? 80% 100%
0% or
20%
40%
60%
80%
100%
Indicates a correct bar for either Don’t know or Yes, and labels correctly
20
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Frog spawn
Correct response 1m 15th February (1997) Additional guidance Unambiguous or commonly used date notation eg 15/2 2/15 [US notation]
3-5 4-6 5-7 6-8
17 10 1 a a a
b b b
1m
Gives a possible description of the weather eg In 1991 it was colder than the other years It must have been less warm than usual
Minimally acceptable response eg Cold Not warm It got warmer later ! Response implies a preference Condone provided the pupil’s intention is clear eg, accept It must have been nasty weather It was rainy and not sunny Bad Incomplete or incorrect response eg They were seen later than in other years Very cold so the eggs were seen quicker
U1
Tier & Question
Simplifying
Correct response 1m Indicates 4a + 3, ie Additional guidance
3-5 4-6 5-7 6-8
18 11 2 a a a
b b b
1m
8b + 3
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Containers
Correct response 1m Indicates A and gives the value 250 Additional guidance
3-5 4-6 5-7 6-8
19 12 3
Tier & Question
Triangles
Correct response 1m Gives the values 60, 60 and 60 Additional guidance Single answer of 60 given
3-5 4-6 5-7 6-8
20 13 4 a a a
b b b
1m
Gives the values 90, 45 and 45, in any order
Tier & Question
Spinners
Correct response 1m Indicates B Additional guidance
3-5 4-6 5-7 6-8
21 14 5 a a a
b b b
1m
Indicates A and D, in either order
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2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Faces
Correct response 1m 8 Additional guidance
3-5 4-6 5-7 6-8
22 15 6 a a a
b b b
1m
Draws a solid with 6 faces in any orientation, using the isometric grid correctly eg
Some or all internal lines shown eg
! Lines not ruled Accept provided the pupil’s intention is clear ! Drawing not accurate Accept vertices within 2mm of the dots of the grid ! Some or all hidden lines shown Do not accept unless the lines are clearly identified as hidden lines eg, accept
eg, do not accept
Isometric grid not used correctly eg
23
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Fir trees
Correct response 1m £ 30(.00) Additional guidance
3-5 4-6 5-7 6-8
23 16 7 a a a
b b b
1m
4 and 5, in either order
! Upper bound taken to be just under 5 For the upper bound, accept values between 4.9 and 5 inclusive
U1
Tier & Question
Rectangles and squares
Correct response 1m 4 Additional guidance ! Value repeated Accept provided there is no ambiguity eg, for part (a) accept 4 by 4 eg, for part (a) do not accept 4×4 ! For parts (a) and (b), response of 16 then 20 Mark as 0, 1 U1
3-5 4-6 5-7 6-8
24 17 8 a a a
b b b
1m
5
24
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Lemonade
Correct response 2m or 1m Shows the value 0.8(0) or Shows or implies a complete correct method with not more than one computational error eg 6 × 1.20 – 4 × 1.60 (120 ÷ 4 – 160 ÷ 6) × 24 7.40 (error) – 6.40 = 1.00 or Shows the value 720 or 7.2(0) and 640 or 6.4(0) U1 80 p Additional guidance
3-5 4-6 5-7 6-8
25 18 9
Tier & Question
Three angles
Correct response 1m Indicates No and gives a correct explanation eg 24 + 93 + 61 = 178 but it should equal 180 for a straight line 24 + 93 + 61 is 2 degrees too small for a straight line 4 + 3 + 1 = 8, so they couldn’t add to 180 Additional guidance Minimally acceptable explanation that states or implies the angles should add to 180 or that they add to less than 180 eg The angles don’t make 180 They should add to 180 Too small by 2 The total ends in 8, but this should be 0 It totals 178°, so it would be an obtuse angle Incomplete or incorrect explanation eg 24 + 93 + 61 = 178 which is not straight The angles add to 188 not 180 The angles add to 178° so it will look straight U1
3-5 4-6 5-7 6-8
26 19 10
25
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7, 6–8
Tier & Question
Solving
Correct response 1m 14 Additional guidance ! Incorrect notation eg, as an answer for the first mark x = × 14 Penalise only the first occurrence ! Incomplete processing eg, as an answer for the first mark x = 448 32 Penalise only the first occurrence
3-5 4-6 5-7 6-8
27 20 11
1m
13
Tier & Question
Marking overlay available Correct response 2m Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels correctly
Newspaper
Additional guidance Unambiguous abbreviation eg E for Evening newspaper, N for No newspaper
3-5 4-6 5-7 6-8
21 12 1
or 1m Draws the sectors for Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels correctly or Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay but fails to label or labels incorrectly or Shows or implies that 5 people are represented by 30° or that 1 person is represented by 6° eg 5 people = 30° 150 ÷ 5 = 30 360 ÷ 60 = 6 60, 90 seen
26
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Completing rules
Correct response 1m Gives two correct values in the correct order, and a correct expression in x eg 3, 1, 3x + 1 1, 9, x + 9 –2, 21, –2x + 21 Additional guidance For the first mark, given example repeated ! Unconventional notation eg, for x + 9 1×x+9 Condone
3-5 4-6 5-7 6-8
22 13 2
1m
Gives two correct values in the correct order, and a correct expression in x eg 4, 3, 4x – 3 –2, –21, –2x – –21 x, 3, x2 – 3
1m
Gives two correct values in the correct order, and a correct expression in x eg x 2, 11, + 11 2 x 0.5, 5, 2x + 5 (or + 5) 0.5 1, 9, x + 9
27
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Parallelogram
Correct response 2m Gives the correct value with a correct unit eg 35cm2 Additional guidance
3-5 4-6 5-7 6-8
23 14 3
or 1m Shows the value 35 or Shows a complete correct method with not more than one computational error and with a correct unit for area shown at least once eg 7 × 5 and cm2 seen (10 – 3) × 5 and cm2 seen 10 × 5 – 3 × 5 and cm2 seen 50 – 7.5 – 7.5 and cm2 seen 4 × 5 + 2 × 1.5 × 5 and cm2 seen 50 – 2 × 6.5 (error) = 37 and cm2 seen For 1m, necessary brackets omitted eg 10 – 3 × 5
Tier & Question
Relationships
Correct response 1m 9 Additional guidance ! Incomplete processing eg, for the first mark 10 – 1 eg, for the second mark 102 Penalise only the first occurrence
3-5 4-6 5-7 6-8
24 15 4
1m
100
28
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Pi
Correct response 3.1416 Additional guidance Equivalent fractions or decimals
3-5 4-6 5-7 6-8
25 16 5 a a a 1m
b b b 1m
Indicates
355 , ie 113
Tier & Question
Marking overlay available Correct response 2m Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay
Enlarging
Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Construction lines drawn Ignore, even if incorrect
3-5 4-6 5-7 6-8
26 17 6
or 1m Shows at least three vertices within the tolerances as shown on the overlay or Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay, but in an incorrect position and/or orientation
29
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Values
Correct response 15 Additional guidance
3-5 4-6 5-7 6-8
27 18 7 a a a 1m
b b b 1m
5
1 or equivalent 2
c
c 1m
Indicates that e > 5 eg It has to be higher than 5 Any number over 5
Minimally acceptable indication eg >5 Above 5 More than half of 10 ! Range includes 5 eg 5 or over Condone Negative values of f excluded eg 5 < e 10 Between 5 and 10 Incorrect indication eg e can be 6, 7, 8 and so on e must be 5.1 or more Incomplete indication eg e = 10 – f f e
30
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Travelling by car
Correct response 72 Additional guidance
3-5 4-6 5-7 6-8
28 19 8 a a a 1m
b b b 1m
4
c
c 2m
1.8 or equivalent
! Answer of 2 For 2m, do not accept unless a correct method or a more accurate value is seen ! For 2m or 1m, follow through from part (b) Accept follow through as 18 ÷ (their (b) + 6) or as (their (b) + 14) ÷ (their (b) + 6), rounded or truncated to at least 2 s.f.
or 1m Shows or implies a correct method eg 18 ÷ (4 + 4 + 2) 18 10 For 1m, necessary brackets omitted eg 18 ÷ 4 + 4 + 2
31
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Brackets
Correct response Gives a correct explanation The most common correct explanations: Additional guidance
3-5 4-6 5-7 6-8
29 20 9 a a 1m
Give the correct expansion of the expression eg 3(2a + 1) = 6a + 3, not 6a + 1 It should be 2 greater, ie 6a + 3
Minimally acceptable explanation eg 6a + 3 She needs to add 2 Incomplete or incorrect explanation eg 3(2a + 1) 6a + 1 3(2a + 1) = 6a + 2 3(2a + 1) = 6a + 3 = 9a
Address the misconception eg Both things in the brackets should be multiplied by 3, but she has forgotten the 1
Minimally acceptable explanation eg 3×1 All bits need to be multiplied by 3 You have to multiply everything in the brackets She hasn’t multiplied the 1 Incomplete explanation eg She hasn’t multiplied out the brackets correctly The 1 is incorrect
Give a correct counter example eg When a = 1 then 3(2a + 1) = 9, but 6a + 1 = 7 If a is 2, 3(2 × 2 + 1) 6 × 2 + 1
Minimally acceptable explanation eg When a = 1 you get 9 and 7 Incomplete explanation eg When a = 1 you get different answers for each side, so it can’t be right
32
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Brackets (cont)
Correct response Gives a correct explanation The most common correct explanations: Additional guidance
3-5 4-6 5-7 6-8
29 20 9 b b 1m
Give the correct expansion of the expression eg (k + 4)(k + 7) = k2 + 11k + 28, not k2 + 28 He should get k2 + 4k + 7k + 28 He has missed out 4k + 7k so it should be k2 + 11k + 28
Minimally acceptable explanation eg k2 + 11k + 28 k2 + 4k + 7k + 28 11k is missing There should be 4k and 7k as well ! Correct expression equated to zero eg k2 + 11k + 28 = 0 Condone Incomplete or incorrect explanation eg (k + 4)(k + 7) k2 + 28 k2 + 11k + 28 = k2 + 39 It’s 11k
Address the misconception eg Both things in the first brackets should be multiplied by both things in the second brackets, but he has done k × k and 4 × 7
Minimally acceptable explanation eg He hasn’t multiplied the 4 or the 7 by k There should be a k term It should have been like this:
( k + 4)( k + 7)
Incomplete explanation eg There should be 3 terms in the answer The ks should be added You have to multiply everything in the second brackets by everything in the first brackets He hasn’t multiplied the first set of brackets by the second set properly Give a correct counter example eg When k = 1 then (k + 4)(k + 7) = 40, but k2 + 28 = 29 If k is 2, (2 + 4)(2 + 7) 22 + 28 Minimally acceptable explanation eg When k = 1 you get 40 and 29 Incomplete explanation eg When k = 1 you get different answers for each side, so it can’t be right
33
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Vowels
Correct response 0.61 or equivalent probability Additional guidance
3-5 4-6 5-7 6-8
30 21 10 a 2m or 1m
Shows the digits 61 or Shows the value 0.39 or equivalent probability or Shows or implies a complete correct method with not more than one computational error eg 1 – (0.08 + 0.13 + 0.07 + 0.08 + 0.03) 0.08 + 0.13 + 0.07 + 0.08 + 0.03 = 0.38 (error) 1 – 0.38 = 0.62
b 2m
0.000936 or 9.36 × 10–4, or equivalent probability
For 2m, 9.36 –04
or 1m Shows the digits 936 or Shows or implies a complete correct method with not more than one computational error eg 0.13 × 0.08 × 0.09 9.4 × 10–4
34
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Beams
Correct response 3m Indicates the 1st way, and gives the correct difference of 1320 Additional guidance
3-5 4-6 5-7 6-8
22 11
or 2m Shows the digits 132(0) or Shows the digits 484(0) and 352(0) or Shows or implies correct substitution of all values into the formula and the intention to subtract eg 5 × 112 × 8 – 5 × 82 × 11 5 × 11 × 8(11 – 8) 440 × 3 5 × (968 – 704) 5 × 264 or Shows a complete correct method with not more than one computational error, and gives a correct decision for their values eg 5 × 112 × 8 = 4440 (error) 4440 – 3520 = 920 so 1st way, difference 920 or 1m Shows the digits 484(0) or 352(0) or Indicates the 1st way and gives an answer of 264 [the only error is to omit to multiply the substituted values by 5] or Indicates the 1st way and gives an answer of 6600 [the only error is to process 5 × 112 × 8 as (5 × 11)2 × 8 and 5 × 82 × 11 as (5 × 8)2 × 11]
35
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Car park
Correct response 3m or 2m Shows the values 24 and 160 or Shows a correct method with not more than one computational or rounding error eg (208 – 136) ÷ 3 ÷ (240 ÷ 1.50) 208 – 136 = 72, 72 ÷ 3 = 26 (error), 26 + 136 = 162 26 ÷ 162 × 100 = 16.25 or 1m Shows the value 24 or 160 or Shows a correct method with not more than two computational or rounding errors eg 208 – 136 = 62 (error), 62 ÷ 3 = 21 (premature rounding), 21 ÷ 160 × 100 = 13.125 U1 15 Additional guidance
3-5 4-6 5-7 6-8
23 12
Tier & Question
Volume of prisms
Correct response 120 Additional guidance
3-5 4-6 5-7 6-8
24 13 a a 1m
b b 1m
450
36
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Marking overlay available Correct response Draws a different straight line with gradient 1, within the tolerance as shown on the overlay when the y-axes are aligned
Straight lines
Additional guidance ! Line short As the line could be positioned anywhere on the grid, accept lines of at least one diagonal unit in length provided they are within the tolerance as shown on the overlay Responses consisting of longer lines must be entirely within tolerance
3-5 4-6 5-7 6-8
25 14 a a 1m
b b 1m
20
c
c 1m
Gives a correct equation eg y = 5x + 10 5x – y = –10
! Unconventional notation eg y1 = 5 × x + 10 Condone
Tier & Question
Two semicircles
Correct response 2m 25 + 10, 88.6, 88.5(…) or 89 Additional guidance ! Value of 88 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
26 15
or 1m Shows one entry from the following list: 25 10 15 20 50 50 or Shows or implies a complete correct method with not more than one computational or rounding error eg 20 30 + + 30 – 20 2 2 25 × 3.14 + 10 Value of 88, with no correct method or more accurate value seen U1 (or 78.6, 78.5(…), 79) (or 31.(…)) (or 47.(…)) (or 62.8(…), 63) and 30 (or 94.(…)) (or 157.(…)) + 10 (or 167.(…))
37
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Which pupil?
Correct response 2m Indicates Class 9A and gives a correct justification The most common correct justifications: Additional guidance
3-5 4-6 5-7 6-8
27 16
Use the proportions of boys in each class, in a form that enables comparison eg 12 168 13 169 but = = 26 364 28 364 336 338 You get and 728 728 13 12 = 0.464(…), = 0.461(…) (or 0.462) 28 26 A gives 46.4% and B gives 46.2% 28 ÷ 13 = 2.15(…) 26 ÷ 12 = 2.16(…) (or 2.17) 13 12.07(…) 12.1 = (or ) 28 26 26 12 12.9(…) = 26 28 13 × (12 + 14) = 338, 12 × (15 + 13) = 336
For 2m, minimally acceptable justification eg 169 168 , 364 364 0.464(…), 0.461(…) (or 0.462) 46.4, 46.2 13 × 26 > 12 28 For 2m, incomplete or incorrect justification eg 13 12 > 28 26 13 > 12
Use the ratios of boys to girls or girls to boys in each class, in a form that enables comparison eg 9A is 0.86(…) boys for every girl, 9B is 0.85(…) 9A is 0.87 boys for every girl, 9B is 0.86 13 : 15 = 1 : 1.15(…) 12 : 14 = 1 : 1.16(…) (or 1 : 1.17) 13 12.1(…) = 15 14 13 12 = 15 13.8(…) 182 180 , 210 210
For 2m, minimally acceptable justification eg 0.86(…), 0.85(…) 0.87, 0.86 1.15(…), 1.16(…) (or 1.17)
Reason generally about the differences between the numbers of boys and girls eg A difference of 2 out of the bigger total in 9A is less than out of the smaller total in 9B 2 2 < 28 26
For 2m, minimally acceptable justification eg There are two fewer boys than girls in both, but 9A is bigger
38
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Which pupil? (cont)
Correct response or 1m Shows a correct justification but makes an incorrect or no decision eg 12 13 = 0.46, = 0.46 so equal 26 28 or Shows a correct justification with not more than one computational error then makes the correct decision for their values eg 338 346 , (error), 9B indicated 728 728 Additional guidance
3-5 4-6 5-7 6-8
27 16
39
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Pythagoras
Correct response Gives a correct explanation The most common correct explanations: Additional guidance Explanation uses only accurate or scale drawing
3-5 4-6 5-7 6-8
28 17 a 1m
Show that the values 6, 8 and 10 work using Pythagoras’ theorem eg 62 + 82 = 36 + 64 = 100 = 102 2 2 10 – 8 = 100 – 64 = 36 = 62
Minimally acceptable explanation eg 62 + 82 = 102 36 + 64 = 100 The square of the longest side is equal to the sum of the squares of the other two sides Incomplete explanation eg 62 + 82 36 + 64 Minimally acceptable explanation eg It’s an enlarged 3, 4, 5 triangle 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 Incomplete explanation eg It’s like a 3, 4, 5 triangle
State or imply that the triangle is an enlargement of a 3, 4, 5 right-angled triangle eg A 3, 4, 5 triangle is right-angled and 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 It’s just a 3, 4, 5 triangle with the lengths of the sides doubled Because 6, 8 and 10 make a Pythagorean triple
b 1m
Gives a correct justification eg 6.9 × 8 = 9.2 6 8 × 1.15 = 9.2 9.2 ÷ 1.15 = 8 3 6.9 ÷ 9.2 = 4 3 6÷8= 4 6 6.9 is a 15% increase 8 × 0.15 = 1.2 8 + 1.2 = 9.2 tan–1 8 = 53.1… 6 6.9 × tan 53.1… = 9.2
Minimally acceptable explanation eg 6.9 ×8 6 8 × 1.15 6.9 6 = 9.2 8 Incomplete explanation eg 9.2 ÷ 1.15 Explanation attempts to use Pythagoras’ theorem eg 6.92 + 9.22 = 11.52
40
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Pythagoras (cont)
Correct response Shows the digits 115 eg 1.15 × 108 115 000 000 11.5 Additional guidance ! Zero(s) given after the last decimal place within standard form notation Condone eg, for both marks in part (c) accept 1.150 × 108
3-5 4-6 5-7 6-8
28 17 c 1m
1m
Shows the correct value in standard form, ie 1.15 × 108
Tier & Question
Expressions
Correct response 2m Gives all three correct expressions, ie y + 15 2y y + 3a Additional guidance ! Expressions unsimplified or use unconventional notation eg, for the third expression y+a+a+a 1y + 3 × a Condone
3-5 4-6 5-7 6-8
18
or 1m U1 Gives two correct expressions
41
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Gorillas
Correct response 2m Gives an integer value between 16 500 and 17 000 inclusive eg 17 000 16 700 16 667 Additional guidance ! Gives a non-integer value within the correct range eg 16 666.(…) Condone
3-5 4-6 5-7 6-8
19
or 1m Shows the digits 166(…) or 167 or Shows a complete correct method with not more than one computational or rounding error eg 5000 ÷ 0.3 5000 ÷ 3 × 10 100 × 5000 30 5000 ÷ 30 = 200 (premature rounding), 200 × 100 = 20 000
Tier & Question
Houses
Correct response 2m 2.9 or equivalent Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
20
or 1m Shows the value 29 or 290 or Shows a complete correct method with not more than one computational or rounding error eg 2.5 × 60 + 3.3 × 30 + 4.1 × 10 100 (2.5 × 6 + 3.3 × 3 + 4.1) ÷ 10 150 + 99 + 41 = 300 (error), 300 ÷ 100 = 3 For 1m, necessary brackets omitted eg 2.5 × 6 + 3.3 × 3 + 4.1 ÷ 10
42
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Subtracting and squaring
Correct response 2m Gives the number as 13 and shows a complete correct method for solving algebraically eg (x – 25)2 = x2 – 25 2 x – 50x + 625 = x2 – 25 50x = 650 x = 13 Additional guidance Method used is trial and improvement
3-5 4-6 5-7 6-8
21
or 1m Shows a correct expression without brackets that is equivalent to (unknown – 25)2 eg x2 – 50x + 625 n2 – 25n – 25n + 625 a × a – 50 × a + 25 × 25 or Shows a correct equation eg (x – 25)2 = x2 – 25 U1
43
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Light years
Correct response 9.43 × 1012 Additional guidance ! Zero(s) given after the last decimal place within standard form notation eg, for part (a) 9.430 × 1012 Condone
3-5 4-6 5-7 6-8
22 a 1m
b 1m
7.35(54) × 1013 or 7.36 × 1013 or 7.4 × 1013
! For part (b), follow through Accept 7.8 × their (a) provided this is written correctly in standard form to at least 2 s.f.
Tier & Question
Octagon
Correct response 2m 2 2, 8 or 2.8(…) Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
23
or 1m Shows or implies a correct equation in y eg y2 = 8 y2 + y2 = 42 2y2 = 16 y×y+y×y=4×4 2y = 4 4sin 45 (or 4cos 45)
U1
44
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
x, y, a and b
Correct response a–b Additional guidance ! For part (a), unsimplified expression or unconventional notation Condone
3-5 4-6 5-7 6-8
24 a 1m
b 2m or 1m
2b – a
For 2m or 1m, follow through from part (a)
Shows a correct expression for x, even if it is unsimplified, uses unconventional notation or there is subsequent incorrect working eg 2×b–a b – (a – b) a – 2(a – b) or Shows a complete correct method with not more than one error eg x + 2y = a 2x + 2y = b (error) x=b–a or Forms two correct equations that would allow elimination of y eg x + 2y = a 2x + 2y = 2b or Attempts to solve by substitution and forms a correct equation in x eg x+a–b=b x + 2(a – b) = a x + 2(b – x) = a For 1m, second equation doubled without the first equation restated eg 2x + 2y = 2b seen
45
2007 KS3 Mathematics test mark scheme: Paper 2
Index
Index to mark schemes
Tier 3–5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 4–6 5–7 6–8 Rules Homework Odd one out Hibernation Concert Cake Bar chart Coordinates 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 11 12 12 13 14 14 15 15 16 17 17 18 18 19 19 20 21 21 22 22 22 23 24 24 25 25 26 Question Page
46
2007 KS3 Mathematics test mark scheme: Paper 2
Index
Tier 3–5 4–6 21 22 23 24 25 26 27 28 29 30 5–7 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 6–8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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? Pythagoras Expressions Gorillas Houses Subtracting and squaring Light years Octagon x, y, a and b
26 27 28 28 29 29 30 31 32 34 35 36 36 37 37 38 40 41 42 42 43 44 44 45
47
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275669
KEY STAGE
Mathematics tests
3
ALL TIERS
Mark scheme for Paper 2 Tiers 3–5, 4–6, 5–7 and 6–8
2007
National curriculum assessments
2007 KS3 Mathematics test mark scheme: Paper 2
Introduction
Introduction
The test papers will be marked by external markers. The markers will follow the mark scheme in this booklet, which is provided here to inform teachers. This booklet contains the mark scheme for paper 2 at all tiers. The paper 1 mark scheme is printed in a separate booklet. Questions have been given names so that each one has a unique identifier irrespective of tier.
The structure of the mark schemes
The marking information for questions is set out in the form of tables, which start on page 11 of this booklet. …show more content…
The columns on the left-hand side of each table provide a quick reference to the tier, question number, question part and the total number of marks available for that question part. The Correct response column usually includes two types of information: a statement of the requirements for the award of each mark, with an indication of whether credit can be given for correct working, and whether the marks are independent or cumulative examples of some different types of correct response, including the most common. The Additional guidance column indicates alternative acceptable responses, and provides details of specific types of response that are unacceptable. Other guidance, such as when ‘follow through’ is allowed, is provided as necessary. Questions with a UAM element are identified in the mark scheme by an encircled U with a number that indicates the significance of using and applying mathematics in answering the question. The U number can be any whole number from 1 to the number of marks in the question. For graphical and diagrammatic responses, including those in which judgements on accuracy are required, marking overlays have been provided as the centre pages of this booklet. The 2007 key stage 3 mathematics tests and mark schemes were developed by the Test Development team at Edexcel.
2
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
General guidance
Using the mark schemes
Answers that are numerically equivalent or algebraically equivalent are acceptable unless the mark scheme states otherwise. In order to ensure consistency of marking, the most frequent procedural queries are listed on the following two pages with the prescribed correct action. This is followed by further guidance relating specifically to the marking of questions that involve money, negative numbers, algebra, time, coordinates or probability. Unless otherwise specified in the mark scheme, markers should apply the following guidelines in all cases.
3
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. 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 × 6 = 18 in an otherwise correct long multiplication. A conceptual error is a more serious misunderstanding of the relevant mathematics; when such an error is seen no method marks may be awarded. Examples of conceptual errors are: misunderstanding of place value, such as multiplying by 2 rather than 20 when calculating 35 × 27; subtracting the smaller value from the larger in calculations such as 45 – 26 to give the answer 21; incorrect signs when working with negative numbers. 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 has made a conceptual error.
The pupil’s accuracy is marginal according to the overlay provided. The pupil’s answer correctly follows through from earlier incorrect work. There appears to be a misreading affecting the working.
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. This is when the pupil misreads the information given in the question and uses different information. If the original intention or difficulty level of the question is not reduced, deduct one mark only. If the original intention or difficulty level is reduced, do not award any marks for the question part. Where a pupil has shown understanding of the question, the mark(s) should be given. In particular, where a word or number response is expected, a pupil may meet the requirement by annotating a graph or labelling a diagram elsewhere in the question.
The correct answer is in the wrong place.
4
2007 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.
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 incorrect answer is due to a transcription error in questions not testing accuracy, the correct answer has been given but then rounded or truncated the pupil has continued to give redundant extra working which does not contradict work already done the pupil has continued, in the same part of the question, to give redundant extra working which does contradict work already done. If so, award the mark. If so, award the mark.
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.
The pupil’s answer is correct but the wrong working is seen. The correct response has been crossed or rubbed out and not replaced. More than one answer is given.
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.
The answer is correct but, in a later part of the question, the pupil has contradicted this response.
5
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Marking specific types of question Responses involving money
For example: £ 3.20 £7
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 The unit, £ or p, is usually printed in the answer space. Where the pupil writes an answer outside the answer space with no units, accept responses that are unambiguous when considered alongside the given units eg with £ given in the answer space, accept 3.20 7 or 7.00 Given units amended eg with £ crossed out in the answer space, accept 320p 700p
Do not accept
Incorrect or ambiguous indication of the amount eg £320, £320p or £700p
Ambiguous use of units outside the answer space eg with £ given in the answer space, do not accept 3.20p outside the answer space Incorrect placement of decimal points, spaces, etc or incorrect use or omission of 0 eg £3.2, £3 200, £32 0, £3-2-0 £7.0
Responses involving negative numbers
For example: –2
Accept
Do not accept
To avoid penalising the error below more than once within each question, do not award the mark for the first occurrence of the error within each question. Where a question part carries more than one mark, only the final mark should be withheld. Incorrect notation eg 2 –
6
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving the use of algebra
For example: 2+n
n+2
2n
n
2
n2
Accept
Unambiguous use of a different case or variable eg N used for n x used for n
Take care ! Do not accept
!
Unconventional notation eg n × 2 or 2 × n or n2 or n + n for 2n n × n for n2 n ÷ 2 for n or 1 n 2 2 2 + 1n for 2 + n 2 + 0n for 2 Within a question that demands simplification, do not accept as part of a final answer involving algebra. Accept within a method when awarding partial credit, or within an explanation or general working. Embedded values given when solving equations eg in solving 3x + 2 = 32, 3 × 10 + 2 = 32 for x = 10
To avoid penalising the two types of error below more than once within each question, do not award the mark for the first occurrence of each type within each question. Where a question part carries more than one mark, only the final mark should be withheld. Words used to precede or follow equations or expressions eg t = n + 2 tiles or tiles = t = n + 2 for t = n + 2
!
Words or units used within equations or expressions eg n tiles + 2 n cm + 2 Do not accept on their own. Ignore if accompanying an acceptable response. Ambiguous letters used to indicate expressions eg n = n + 2 for n + 2
Unambiguous letters used to indicate expressions eg t = n + 2 for n + 2
7
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving time
A time interval For example: 2 hours 30 minutes
Accept
Any unambiguous indication eg 2.5 (hours), 2h 30 Digital electronic time ie 2:30
Take care ! Do not accept
Incorrect or ambiguous time interval eg 2.3(h), 2.30, 2-30, 2h 3, 2.30min
!
The unit, hours and/or minutes, is usually printed in the answer space. Where the pupil writes an answer outside the answer space, or crosses out the given unit, accept answers with correct units, unless the question has specifically asked for other units to be used.
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 5:20 pm, 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
Responses involving coordinates
For example: ( 5, 7 )
Accept
Unconventional notation eg ( 05, 07 ) ( five, seven ) x y
Do not accept
Incorrect or ambiguous notation eg ( 7, 5 ) y x
( 5, 7 ) ( x = 5, y = 7 )
( 7, 5 ) ( 5x, 7y ) ( 5x, 7y ) ( x – 5, y – 7 )
8
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Responses involving probability
A numerical probability should be expressed as a decimal, fraction or percentage only. 7 For example: 0.7 70% 10
Accept
Equivalent decimals, fractions and percentages 70 35 , 70.0% eg 0.700, , 100 50
Take care ! Do not accept
The first four categories of error below should be ignored if accompanied by an acceptable response, but should not be accepted on their own. However, to avoid penalising the first three types of error below more than once within each question, do not award the mark for the first occurrence of each type of error unaccompanied by an acceptable response. Where a question part carries more than one mark, only the final mark should be withheld.
A probability correctly expressed in one acceptable form which is then incorrectly converted, but is still less than 1 and greater than 0 70 = 18 eg 100 25
!
A probability that is incorrectly expressed eg 7 in 10 7 over 10 7 out of 10 7 from 10 A probability expressed as a percentage without a percentage sign. A fraction with other than integers in the numerator and/or denominator. A probability expressed as a ratio eg 7 : 10, 7 : 3, 7 to 10
! ! !
A probability greater than 1 or less than 0
9
2007 KS3 Mathematics test mark scheme: Paper 2
General guidance
Recording marks awarded on the test paper
All questions, even those not attempted by the pupil, will be marked, with a 1 or a 0 entered in each marking space. Where 2m can be split into 1m gained and 1m lost, with no explicit order, then this will be recorded by the marker as 1 0 The total marks awarded for a double page will be written in the box at the bottom of the right-hand page, and the total number of marks obtained on the paper will be recorded on the front of the test paper. A total of 120 marks is available in each of tiers 3–5, 4–6, 5–7 and 6–8.
Awarding levels
The sum of the marks gained on paper 1, paper 2 and the mental mathematics paper determines the level awarded. Level threshold tables, which show the mark ranges for the award of different levels, will be available on the NAA website www.naa.org.uk/tests from Monday 25 June 2007. QCA will also send a copy to each school in July. Schools will be notified of pupils’ results by means of a marksheet, which will be returned to schools by the external marking agency with the pupils’ marked scripts. The marksheet will include pupils’ scores on the test papers and the levels awarded.
10
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Rules
Correct response 1m 11, 14 Additional guidance
3-5 4-6 5-7 6-8
1
1m
23, 47
1m
41, 122
! First new term for each sequence correct with second terms all incorrect or omitted Mark as 0, 0, 1
11
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Homework
Correct response 1m Monday and Wednesday, in either order Additional guidance ! Names of days or subjects abbreviated Accept provided unambiguous eg, for part (a) accept M and W eg, for part (b) do not accept M, E and T
3-5 4-6 5-7 6-8
2 a
b
1m
Maths, English and Technology, in any order
c
1m
3
Tier & Question
Odd one out
Correct response 1m E Additional guidance
3-5 4-6 5-7 6-8
3 a
1m
D
b
1m
Completes the sentence correctly with a correct property eg … equal sides … lines of symmetry … as the order of rotation symmetry
Minimally acceptable response eg … sides the same … line symmetry … rotation symmetry … identical lines ! Incorrect or irrelevant statement Ignore alongside a correct response eg, accept … equal sides and right angles eg, do not accept … right angles Incomplete or incorrect response eg … sides … equal angles … squares for the area
U1
12
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Hibernation
Correct response 1m 5 Additional guidance Value qualified eg About 5 ! Value inaccurate Accept values between 4.9 and 5.1 inclusive, or between 4 months 27 days and 5 months 3 days inclusive
3-5 4-6 5-7 6-8
4 a
b
1m
Indicates Yes and gives a correct explanation The most common correct explanations:
State or imply that they sleep for more than 6 months eg 1 They sleep for 6 months, which is more 2 than half of 12 Half a year is 6 months but they sleep for just over 6 months
Minimally acceptable explanation eg 1 6 months 2 Just over 6 months More than 6 boxes are shaded November to May is six months and then half of October Half a month more ! Exact value given Accept values between 6.4 and 6.6 months inclusive, or between 6 months 12 days and 6 months 18 days inclusive eg, accept 6 months and 2 weeks Incomplete or incorrect explanation eg They sleep for more than half the year They sleep from halfway through October to the end of April Half a year is 6 months but they sleep for 7 months 1 6 2
Refer to the area shaded or unshaded and its relation to the whole circle eg More than halfway round the circle is shaded The white bit for dormice doesn’t reach round half the circle
Minimally acceptable explanation eg More than halfway round More than half the chart is shaded More is shaded than unshaded Incomplete explanation eg It shows more shaded months
U1
13
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 3–5 only
Tier & Question
Concert
Correct response 1m £ 98.35 Additional guidance
3-5 4-6 5-7 6-8
5 a
b
2m or 1m
5
Shows the digits 8225 or Shows or implies a complete correct method with not more than one error, even if their final answer is not an integer or is rounded or truncated eg (155.75 – 3 × 24.50) ÷ 16.45 73.5 + 16.45 + 16.45 + 16.45 + 16.45 + 16.45 = 155.75
U1
Tier & Question
Cake
Correct response 1m 450 Additional guidance
3-5 4-6 5-7 6-8
6 a
b
1m
Indicates the correct position on the scale, ie
Unambiguous indication ! Inaccurate indication Accept indications that are closer to 275 than either 250 or 300
200
300
c
1m
2pm or 14:00
Time ambiguous or incorrect eg 2 o’clock 14:00am
d
1m
Indicates Cylinder, ie
14
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Bar chart
Correct response 2m Completes all labels for both axes correctly, ie 24 20 16 12 8 4 0 Additional guidance Unambiguous indication of item names eg, for Glue G
3-5 4-6 5-7 6-8
7
Glue
Pens
Rulers
or 1m U1 Completes at least two labels on the vertical axis correctly
Tier & Question
Coordinates
Correct response 1m Gives A as (0, 6) Additional guidance
3-5 4-6 5-7 6-8
8 1 a a
1m
Gives C as (4, 3)
! Answers for A and C transposed but otherwise completely correct If this is the only error, ie gives A as (4, 3) and gives C as (0, 6), mark as 0, 1
b b
1m
Indicates point D on the graph at (2, 7)
!
Point inaccurate, not labelled or marked only with the letter D Condone any unambiguous indication within 2mm of the correct intersection of the grid
15
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Fitting tiles
Correct response 1m Indicates correctly two congruent F-tiles on the diagram eg Additional guidance ! Tile not shaded or inaccurately indicated Accept provided the pupil’s intention is clear and there is no ambiguity Tiles overlapping
3-5 4-6 5-7 6-8
9 2 a a
b b
1m
Indicates two congruent tiles on the diagram eg
1m U1
Indicates two congruent tiles on the diagram, different from any previously credited
16
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Names
Correct response 1m Claire Additional guidance Unambiguous indication of name eg, for Claire C
3-5 4-6 5-7 6-8
10 3 a a
b b
1m U1
Gives the names Claire then Tom
Tier & Question
Leaves
Correct response 1m Writes the leaves in the correct order for area, ie Willow smallest area Oak Beech largest area Additional guidance Unambiguous indication eg, for part (a) W smallest area O B largest
area
3-5 4-6 5-7 6-8
11 4 a a
b b
1m
Writes the leaves in the correct order for perimeter, ie Willow smallest perimeter Beech Oak largest perimeter
! Order given as largest to smallest for both parts (a) and (b) Mark as 0, 1 ! Responses for parts (a) and (b) transposed but otherwise correct Mark as 0, 1
17
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Marbles
Correct response 2m Matches all three questions correctly, ie
10 10 × 7 10 × 12 12 × 7 10 × 12 × 7 10 + 12 + 7
3-5 4-6 5-7 6-8
12 5
Additional guidance ! Question matched with more than one calculation For 2m or 1m, do not accept as a correct match
or 1m Matches any two of the questions correctly
Tier & Question
a and b
Correct response 1m Gives a pair of numbers for a and b, such that b=a+4 eg a=5 b=9 a = 1.5 b = 5.5 Additional guidance Values embedded eg 4+5=9 a=4+5 b=9
3-5 4-6 5-7 6-8
13 6
1m U1
Gives a pair of numbers for a and b, such that b = a + 4, different from any previously credited
18
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Turning
Correct response 1m Indicates the correct shape, ie Additional guidance
3-5 4-6 5-7 6-8
14 7
Tier & Question
Party
Correct response 2m Completes all four entries in the table correctly, ie 4.95 3.20 1.95 5 13 10 Total: 24.75 41.60 19.50 85.85 Additional guidance
3-5 4-6 5-7 6-8
15 8
or 1m Completes at least three entries in the table correctly or Completes all four entries correctly with some or all amounts of money given in pence U1 ! For 1m, follow through Where the only error is in the total cost of balloons, for the overall total accept their total cost of balloons + 61.10
19
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6
Tier & Question
Survey
Correct response 1m 10 10% Additional guidance
3-5 4-6 5-7 6-8
16 9 a a
b b
2m
Completes the percentage bar chart correctly, ie 50% labelled No 40% labelled Don’t know 10% labelled Yes, with bars in any order eg
No 0% 20% 40% Don’t know Yes 60% 80% 100%
Unambiguous labelling eg
? 0% 20% 40% ? 60% ? ? 80% 100%
! Lines not ruled or accurate Accept provided the pupil’s intention is clear
or 1m Indicates sections corresponding to 50%, 40% and 10% but fails to label, labels incorrectly or bars are not continuous eg
? 0% or Shows or implies the values 50, 40 and 10 eg 20% ? 40% ? 60% ? 80% 100%
0% or
20%
40%
60%
80%
100%
Indicates a correct bar for either Don’t know or Yes, and labels correctly
20
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Frog spawn
Correct response 1m 15th February (1997) Additional guidance Unambiguous or commonly used date notation eg 15/2 2/15 [US notation]
3-5 4-6 5-7 6-8
17 10 1 a a a
b b b
1m
Gives a possible description of the weather eg In 1991 it was colder than the other years It must have been less warm than usual
Minimally acceptable response eg Cold Not warm It got warmer later ! Response implies a preference Condone provided the pupil’s intention is clear eg, accept It must have been nasty weather It was rainy and not sunny Bad Incomplete or incorrect response eg They were seen later than in other years Very cold so the eggs were seen quicker
U1
Tier & Question
Simplifying
Correct response 1m Indicates 4a + 3, ie Additional guidance
3-5 4-6 5-7 6-8
18 11 2 a a a
b b b
1m
8b + 3
21
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Containers
Correct response 1m Indicates A and gives the value 250 Additional guidance
3-5 4-6 5-7 6-8
19 12 3
Tier & Question
Triangles
Correct response 1m Gives the values 60, 60 and 60 Additional guidance Single answer of 60 given
3-5 4-6 5-7 6-8
20 13 4 a a a
b b b
1m
Gives the values 90, 45 and 45, in any order
Tier & Question
Spinners
Correct response 1m Indicates B Additional guidance
3-5 4-6 5-7 6-8
21 14 5 a a a
b b b
1m
Indicates A and D, in either order
22
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Faces
Correct response 1m 8 Additional guidance
3-5 4-6 5-7 6-8
22 15 6 a a a
b b b
1m
Draws a solid with 6 faces in any orientation, using the isometric grid correctly eg
Some or all internal lines shown eg
! Lines not ruled Accept provided the pupil’s intention is clear ! Drawing not accurate Accept vertices within 2mm of the dots of the grid ! Some or all hidden lines shown Do not accept unless the lines are clearly identified as hidden lines eg, accept
eg, do not accept
Isometric grid not used correctly eg
23
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Fir trees
Correct response 1m £ 30(.00) Additional guidance
3-5 4-6 5-7 6-8
23 16 7 a a a
b b b
1m
4 and 5, in either order
! Upper bound taken to be just under 5 For the upper bound, accept values between 4.9 and 5 inclusive
U1
Tier & Question
Rectangles and squares
Correct response 1m 4 Additional guidance ! Value repeated Accept provided there is no ambiguity eg, for part (a) accept 4 by 4 eg, for part (a) do not accept 4×4 ! For parts (a) and (b), response of 16 then 20 Mark as 0, 1 U1
3-5 4-6 5-7 6-8
24 17 8 a a a
b b b
1m
5
24
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7
Tier & Question
Lemonade
Correct response 2m or 1m Shows the value 0.8(0) or Shows or implies a complete correct method with not more than one computational error eg 6 × 1.20 – 4 × 1.60 (120 ÷ 4 – 160 ÷ 6) × 24 7.40 (error) – 6.40 = 1.00 or Shows the value 720 or 7.2(0) and 640 or 6.4(0) U1 80 p Additional guidance
3-5 4-6 5-7 6-8
25 18 9
Tier & Question
Three angles
Correct response 1m Indicates No and gives a correct explanation eg 24 + 93 + 61 = 178 but it should equal 180 for a straight line 24 + 93 + 61 is 2 degrees too small for a straight line 4 + 3 + 1 = 8, so they couldn’t add to 180 Additional guidance Minimally acceptable explanation that states or implies the angles should add to 180 or that they add to less than 180 eg The angles don’t make 180 They should add to 180 Too small by 2 The total ends in 8, but this should be 0 It totals 178°, so it would be an obtuse angle Incomplete or incorrect explanation eg 24 + 93 + 61 = 178 which is not straight The angles add to 188 not 180 The angles add to 178° so it will look straight U1
3-5 4-6 5-7 6-8
26 19 10
25
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 3–5, 4–6, 5–7, 6–8
Tier & Question
Solving
Correct response 1m 14 Additional guidance ! Incorrect notation eg, as an answer for the first mark x = × 14 Penalise only the first occurrence ! Incomplete processing eg, as an answer for the first mark x = 448 32 Penalise only the first occurrence
3-5 4-6 5-7 6-8
27 20 11
1m
13
Tier & Question
Marking overlay available Correct response 2m Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels correctly
Newspaper
Additional guidance Unambiguous abbreviation eg E for Evening newspaper, N for No newspaper
3-5 4-6 5-7 6-8
21 12 1
or 1m Draws the sectors for Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels correctly or Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay but fails to label or labels incorrectly or Shows or implies that 5 people are represented by 30° or that 1 person is represented by 6° eg 5 people = 30° 150 ÷ 5 = 30 360 ÷ 60 = 6 60, 90 seen
26
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Completing rules
Correct response 1m Gives two correct values in the correct order, and a correct expression in x eg 3, 1, 3x + 1 1, 9, x + 9 –2, 21, –2x + 21 Additional guidance For the first mark, given example repeated ! Unconventional notation eg, for x + 9 1×x+9 Condone
3-5 4-6 5-7 6-8
22 13 2
1m
Gives two correct values in the correct order, and a correct expression in x eg 4, 3, 4x – 3 –2, –21, –2x – –21 x, 3, x2 – 3
1m
Gives two correct values in the correct order, and a correct expression in x eg x 2, 11, + 11 2 x 0.5, 5, 2x + 5 (or + 5) 0.5 1, 9, x + 9
27
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Parallelogram
Correct response 2m Gives the correct value with a correct unit eg 35cm2 Additional guidance
3-5 4-6 5-7 6-8
23 14 3
or 1m Shows the value 35 or Shows a complete correct method with not more than one computational error and with a correct unit for area shown at least once eg 7 × 5 and cm2 seen (10 – 3) × 5 and cm2 seen 10 × 5 – 3 × 5 and cm2 seen 50 – 7.5 – 7.5 and cm2 seen 4 × 5 + 2 × 1.5 × 5 and cm2 seen 50 – 2 × 6.5 (error) = 37 and cm2 seen For 1m, necessary brackets omitted eg 10 – 3 × 5
Tier & Question
Relationships
Correct response 1m 9 Additional guidance ! Incomplete processing eg, for the first mark 10 – 1 eg, for the second mark 102 Penalise only the first occurrence
3-5 4-6 5-7 6-8
24 15 4
1m
100
28
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Pi
Correct response 3.1416 Additional guidance Equivalent fractions or decimals
3-5 4-6 5-7 6-8
25 16 5 a a a 1m
b b b 1m
Indicates
355 , ie 113
Tier & Question
Marking overlay available Correct response 2m Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay
Enlarging
Additional guidance ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Construction lines drawn Ignore, even if incorrect
3-5 4-6 5-7 6-8
26 17 6
or 1m Shows at least three vertices within the tolerances as shown on the overlay or Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay, but in an incorrect position and/or orientation
29
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Values
Correct response 15 Additional guidance
3-5 4-6 5-7 6-8
27 18 7 a a a 1m
b b b 1m
5
1 or equivalent 2
c
c 1m
Indicates that e > 5 eg It has to be higher than 5 Any number over 5
Minimally acceptable indication eg >5 Above 5 More than half of 10 ! Range includes 5 eg 5 or over Condone Negative values of f excluded eg 5 < e 10 Between 5 and 10 Incorrect indication eg e can be 6, 7, 8 and so on e must be 5.1 or more Incomplete indication eg e = 10 – f f e
30
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Travelling by car
Correct response 72 Additional guidance
3-5 4-6 5-7 6-8
28 19 8 a a a 1m
b b b 1m
4
c
c 2m
1.8 or equivalent
! Answer of 2 For 2m, do not accept unless a correct method or a more accurate value is seen ! For 2m or 1m, follow through from part (b) Accept follow through as 18 ÷ (their (b) + 6) or as (their (b) + 14) ÷ (their (b) + 6), rounded or truncated to at least 2 s.f.
or 1m Shows or implies a correct method eg 18 ÷ (4 + 4 + 2) 18 10 For 1m, necessary brackets omitted eg 18 ÷ 4 + 4 + 2
31
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Brackets
Correct response Gives a correct explanation The most common correct explanations: Additional guidance
3-5 4-6 5-7 6-8
29 20 9 a a 1m
Give the correct expansion of the expression eg 3(2a + 1) = 6a + 3, not 6a + 1 It should be 2 greater, ie 6a + 3
Minimally acceptable explanation eg 6a + 3 She needs to add 2 Incomplete or incorrect explanation eg 3(2a + 1) 6a + 1 3(2a + 1) = 6a + 2 3(2a + 1) = 6a + 3 = 9a
Address the misconception eg Both things in the brackets should be multiplied by 3, but she has forgotten the 1
Minimally acceptable explanation eg 3×1 All bits need to be multiplied by 3 You have to multiply everything in the brackets She hasn’t multiplied the 1 Incomplete explanation eg She hasn’t multiplied out the brackets correctly The 1 is incorrect
Give a correct counter example eg When a = 1 then 3(2a + 1) = 9, but 6a + 1 = 7 If a is 2, 3(2 × 2 + 1) 6 × 2 + 1
Minimally acceptable explanation eg When a = 1 you get 9 and 7 Incomplete explanation eg When a = 1 you get different answers for each side, so it can’t be right
32
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Brackets (cont)
Correct response Gives a correct explanation The most common correct explanations: Additional guidance
3-5 4-6 5-7 6-8
29 20 9 b b 1m
Give the correct expansion of the expression eg (k + 4)(k + 7) = k2 + 11k + 28, not k2 + 28 He should get k2 + 4k + 7k + 28 He has missed out 4k + 7k so it should be k2 + 11k + 28
Minimally acceptable explanation eg k2 + 11k + 28 k2 + 4k + 7k + 28 11k is missing There should be 4k and 7k as well ! Correct expression equated to zero eg k2 + 11k + 28 = 0 Condone Incomplete or incorrect explanation eg (k + 4)(k + 7) k2 + 28 k2 + 11k + 28 = k2 + 39 It’s 11k
Address the misconception eg Both things in the first brackets should be multiplied by both things in the second brackets, but he has done k × k and 4 × 7
Minimally acceptable explanation eg He hasn’t multiplied the 4 or the 7 by k There should be a k term It should have been like this:
( k + 4)( k + 7)
Incomplete explanation eg There should be 3 terms in the answer The ks should be added You have to multiply everything in the second brackets by everything in the first brackets He hasn’t multiplied the first set of brackets by the second set properly Give a correct counter example eg When k = 1 then (k + 4)(k + 7) = 40, but k2 + 28 = 29 If k is 2, (2 + 4)(2 + 7) 22 + 28 Minimally acceptable explanation eg When k = 1 you get 40 and 29 Incomplete explanation eg When k = 1 you get different answers for each side, so it can’t be right
33
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 4–6, 5–7, 6–8
Tier & Question
Vowels
Correct response 0.61 or equivalent probability Additional guidance
3-5 4-6 5-7 6-8
30 21 10 a 2m or 1m
Shows the digits 61 or Shows the value 0.39 or equivalent probability or Shows or implies a complete correct method with not more than one computational error eg 1 – (0.08 + 0.13 + 0.07 + 0.08 + 0.03) 0.08 + 0.13 + 0.07 + 0.08 + 0.03 = 0.38 (error) 1 – 0.38 = 0.62
b 2m
0.000936 or 9.36 × 10–4, or equivalent probability
For 2m, 9.36 –04
or 1m Shows the digits 936 or Shows or implies a complete correct method with not more than one computational error eg 0.13 × 0.08 × 0.09 9.4 × 10–4
34
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Beams
Correct response 3m Indicates the 1st way, and gives the correct difference of 1320 Additional guidance
3-5 4-6 5-7 6-8
22 11
or 2m Shows the digits 132(0) or Shows the digits 484(0) and 352(0) or Shows or implies correct substitution of all values into the formula and the intention to subtract eg 5 × 112 × 8 – 5 × 82 × 11 5 × 11 × 8(11 – 8) 440 × 3 5 × (968 – 704) 5 × 264 or Shows a complete correct method with not more than one computational error, and gives a correct decision for their values eg 5 × 112 × 8 = 4440 (error) 4440 – 3520 = 920 so 1st way, difference 920 or 1m Shows the digits 484(0) or 352(0) or Indicates the 1st way and gives an answer of 264 [the only error is to omit to multiply the substituted values by 5] or Indicates the 1st way and gives an answer of 6600 [the only error is to process 5 × 112 × 8 as (5 × 11)2 × 8 and 5 × 82 × 11 as (5 × 8)2 × 11]
35
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Car park
Correct response 3m or 2m Shows the values 24 and 160 or Shows a correct method with not more than one computational or rounding error eg (208 – 136) ÷ 3 ÷ (240 ÷ 1.50) 208 – 136 = 72, 72 ÷ 3 = 26 (error), 26 + 136 = 162 26 ÷ 162 × 100 = 16.25 or 1m Shows the value 24 or 160 or Shows a correct method with not more than two computational or rounding errors eg 208 – 136 = 62 (error), 62 ÷ 3 = 21 (premature rounding), 21 ÷ 160 × 100 = 13.125 U1 15 Additional guidance
3-5 4-6 5-7 6-8
23 12
Tier & Question
Volume of prisms
Correct response 120 Additional guidance
3-5 4-6 5-7 6-8
24 13 a a 1m
b b 1m
450
36
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Marking overlay available Correct response Draws a different straight line with gradient 1, within the tolerance as shown on the overlay when the y-axes are aligned
Straight lines
Additional guidance ! Line short As the line could be positioned anywhere on the grid, accept lines of at least one diagonal unit in length provided they are within the tolerance as shown on the overlay Responses consisting of longer lines must be entirely within tolerance
3-5 4-6 5-7 6-8
25 14 a a 1m
b b 1m
20
c
c 1m
Gives a correct equation eg y = 5x + 10 5x – y = –10
! Unconventional notation eg y1 = 5 × x + 10 Condone
Tier & Question
Two semicircles
Correct response 2m 25 + 10, 88.6, 88.5(…) or 89 Additional guidance ! Value of 88 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
26 15
or 1m Shows one entry from the following list: 25 10 15 20 50 50 or Shows or implies a complete correct method with not more than one computational or rounding error eg 20 30 + + 30 – 20 2 2 25 × 3.14 + 10 Value of 88, with no correct method or more accurate value seen U1 (or 78.6, 78.5(…), 79) (or 31.(…)) (or 47.(…)) (or 62.8(…), 63) and 30 (or 94.(…)) (or 157.(…)) + 10 (or 167.(…))
37
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Which pupil?
Correct response 2m Indicates Class 9A and gives a correct justification The most common correct justifications: Additional guidance
3-5 4-6 5-7 6-8
27 16
Use the proportions of boys in each class, in a form that enables comparison eg 12 168 13 169 but = = 26 364 28 364 336 338 You get and 728 728 13 12 = 0.464(…), = 0.461(…) (or 0.462) 28 26 A gives 46.4% and B gives 46.2% 28 ÷ 13 = 2.15(…) 26 ÷ 12 = 2.16(…) (or 2.17) 13 12.07(…) 12.1 = (or ) 28 26 26 12 12.9(…) = 26 28 13 × (12 + 14) = 338, 12 × (15 + 13) = 336
For 2m, minimally acceptable justification eg 169 168 , 364 364 0.464(…), 0.461(…) (or 0.462) 46.4, 46.2 13 × 26 > 12 28 For 2m, incomplete or incorrect justification eg 13 12 > 28 26 13 > 12
Use the ratios of boys to girls or girls to boys in each class, in a form that enables comparison eg 9A is 0.86(…) boys for every girl, 9B is 0.85(…) 9A is 0.87 boys for every girl, 9B is 0.86 13 : 15 = 1 : 1.15(…) 12 : 14 = 1 : 1.16(…) (or 1 : 1.17) 13 12.1(…) = 15 14 13 12 = 15 13.8(…) 182 180 , 210 210
For 2m, minimally acceptable justification eg 0.86(…), 0.85(…) 0.87, 0.86 1.15(…), 1.16(…) (or 1.17)
Reason generally about the differences between the numbers of boys and girls eg A difference of 2 out of the bigger total in 9A is less than out of the smaller total in 9B 2 2 < 28 26
For 2m, minimally acceptable justification eg There are two fewer boys than girls in both, but 9A is bigger
38
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Which pupil? (cont)
Correct response or 1m Shows a correct justification but makes an incorrect or no decision eg 12 13 = 0.46, = 0.46 so equal 26 28 or Shows a correct justification with not more than one computational error then makes the correct decision for their values eg 338 346 , (error), 9B indicated 728 728 Additional guidance
3-5 4-6 5-7 6-8
27 16
39
2007 KS3 Mathematics test mark scheme: Paper 2
Tiers 5–7, 6–8
Tier & Question
Pythagoras
Correct response Gives a correct explanation The most common correct explanations: Additional guidance Explanation uses only accurate or scale drawing
3-5 4-6 5-7 6-8
28 17 a 1m
Show that the values 6, 8 and 10 work using Pythagoras’ theorem eg 62 + 82 = 36 + 64 = 100 = 102 2 2 10 – 8 = 100 – 64 = 36 = 62
Minimally acceptable explanation eg 62 + 82 = 102 36 + 64 = 100 The square of the longest side is equal to the sum of the squares of the other two sides Incomplete explanation eg 62 + 82 36 + 64 Minimally acceptable explanation eg It’s an enlarged 3, 4, 5 triangle 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 Incomplete explanation eg It’s like a 3, 4, 5 triangle
State or imply that the triangle is an enlargement of a 3, 4, 5 right-angled triangle eg A 3, 4, 5 triangle is right-angled and 3 × 2 = 6, 4 × 2 = 8 and 5 × 2 = 10 It’s just a 3, 4, 5 triangle with the lengths of the sides doubled Because 6, 8 and 10 make a Pythagorean triple
b 1m
Gives a correct justification eg 6.9 × 8 = 9.2 6 8 × 1.15 = 9.2 9.2 ÷ 1.15 = 8 3 6.9 ÷ 9.2 = 4 3 6÷8= 4 6 6.9 is a 15% increase 8 × 0.15 = 1.2 8 + 1.2 = 9.2 tan–1 8 = 53.1… 6 6.9 × tan 53.1… = 9.2
Minimally acceptable explanation eg 6.9 ×8 6 8 × 1.15 6.9 6 = 9.2 8 Incomplete explanation eg 9.2 ÷ 1.15 Explanation attempts to use Pythagoras’ theorem eg 6.92 + 9.22 = 11.52
40
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Pythagoras (cont)
Correct response Shows the digits 115 eg 1.15 × 108 115 000 000 11.5 Additional guidance ! Zero(s) given after the last decimal place within standard form notation Condone eg, for both marks in part (c) accept 1.150 × 108
3-5 4-6 5-7 6-8
28 17 c 1m
1m
Shows the correct value in standard form, ie 1.15 × 108
Tier & Question
Expressions
Correct response 2m Gives all three correct expressions, ie y + 15 2y y + 3a Additional guidance ! Expressions unsimplified or use unconventional notation eg, for the third expression y+a+a+a 1y + 3 × a Condone
3-5 4-6 5-7 6-8
18
or 1m U1 Gives two correct expressions
41
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Gorillas
Correct response 2m Gives an integer value between 16 500 and 17 000 inclusive eg 17 000 16 700 16 667 Additional guidance ! Gives a non-integer value within the correct range eg 16 666.(…) Condone
3-5 4-6 5-7 6-8
19
or 1m Shows the digits 166(…) or 167 or Shows a complete correct method with not more than one computational or rounding error eg 5000 ÷ 0.3 5000 ÷ 3 × 10 100 × 5000 30 5000 ÷ 30 = 200 (premature rounding), 200 × 100 = 20 000
Tier & Question
Houses
Correct response 2m 2.9 or equivalent Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
20
or 1m Shows the value 29 or 290 or Shows a complete correct method with not more than one computational or rounding error eg 2.5 × 60 + 3.3 × 30 + 4.1 × 10 100 (2.5 × 6 + 3.3 × 3 + 4.1) ÷ 10 150 + 99 + 41 = 300 (error), 300 ÷ 100 = 3 For 1m, necessary brackets omitted eg 2.5 × 6 + 3.3 × 3 + 4.1 ÷ 10
42
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Subtracting and squaring
Correct response 2m Gives the number as 13 and shows a complete correct method for solving algebraically eg (x – 25)2 = x2 – 25 2 x – 50x + 625 = x2 – 25 50x = 650 x = 13 Additional guidance Method used is trial and improvement
3-5 4-6 5-7 6-8
21
or 1m Shows a correct expression without brackets that is equivalent to (unknown – 25)2 eg x2 – 50x + 625 n2 – 25n – 25n + 625 a × a – 50 × a + 25 × 25 or Shows a correct equation eg (x – 25)2 = x2 – 25 U1
43
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
Light years
Correct response 9.43 × 1012 Additional guidance ! Zero(s) given after the last decimal place within standard form notation eg, for part (a) 9.430 × 1012 Condone
3-5 4-6 5-7 6-8
22 a 1m
b 1m
7.35(54) × 1013 or 7.36 × 1013 or 7.4 × 1013
! For part (b), follow through Accept 7.8 × their (a) provided this is written correctly in standard form to at least 2 s.f.
Tier & Question
Octagon
Correct response 2m 2 2, 8 or 2.8(…) Additional guidance ! Value of 3 For 2m, do not accept unless a correct method or a more accurate value is seen
3-5 4-6 5-7 6-8
23
or 1m Shows or implies a correct equation in y eg y2 = 8 y2 + y2 = 42 2y2 = 16 y×y+y×y=4×4 2y = 4 4sin 45 (or 4cos 45)
U1
44
2007 KS3 Mathematics test mark scheme: Paper 2
Tier 6–8 only
Tier & Question
x, y, a and b
Correct response a–b Additional guidance ! For part (a), unsimplified expression or unconventional notation Condone
3-5 4-6 5-7 6-8
24 a 1m
b 2m or 1m
2b – a
For 2m or 1m, follow through from part (a)
Shows a correct expression for x, even if it is unsimplified, uses unconventional notation or there is subsequent incorrect working eg 2×b–a b – (a – b) a – 2(a – b) or Shows a complete correct method with not more than one error eg x + 2y = a 2x + 2y = b (error) x=b–a or Forms two correct equations that would allow elimination of y eg x + 2y = a 2x + 2y = 2b or Attempts to solve by substitution and forms a correct equation in x eg x+a–b=b x + 2(a – b) = a x + 2(b – x) = a For 1m, second equation doubled without the first equation restated eg 2x + 2y = 2b seen
45
2007 KS3 Mathematics test mark scheme: Paper 2
Index
Index to mark schemes
Tier 3–5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 4–6 5–7 6–8 Rules Homework Odd one out Hibernation Concert Cake Bar chart Coordinates 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 11 12 12 13 14 14 15 15 16 17 17 18 18 19 19 20 21 21 22 22 22 23 24 24 25 25 26 Question Page
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2007 KS3 Mathematics test mark scheme: Paper 2
Index
Tier 3–5 4–6 21 22 23 24 25 26 27 28 29 30 5–7 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 6–8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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? Pythagoras Expressions Gorillas Houses Subtracting and squaring Light years Octagon x, y, a and b
26 27 28 28 29 29 30 31 32 34 35 36 36 37 37 38 40 41 42 42 43 44 44 45
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First published 2007 © Qualifications and Curriculum Authority 2007 ISBN 1-85838-875-9 Reproduction, storage, adaptation or translation, in any form or by any means, of this publication is prohibited without prior written permission of the publisher, unless within the terms of licences issued by the Copyright Licensing Agency. Excerpts may be reproduced for the purpose of research, private study, criticism or review, or by educational institutions solely for educational purposes, without permission, providing full acknowledgement is given. Printed in Great Britain by the Qualifications and Curriculum Authority under the authority and superintendence of the Controller of Her Majesty’s Stationery Office and Queen’s Printer of Acts of Parliament. The Qualifications and Curriculum Authority is an exempt charity under Schedule 2 of the Charities Act 1993. Qualifications and Curriculum Authority 83 Piccadilly London W1J 8QA www.qca.org.uk
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