level 6 ans
Revision work answers Level 6 1.
6
1
12
1 [2]
2.
(a)
3.1416
1 Do not accept: equivalent fractions or decimals
(b)
Indicates
355 , ie 113
1
[2]
3.
54x2
1 Do not accept: unsimplified expression or unconventional notation eg •
9x2 × 6
• •
9x2 + 9x2 + 9x2 + 9x2 + 9x2 + 9x2 54xx [1]
4.
Walk Indicates ‘steady speed’, ie
1
[1]
5.
(a)
Gives a value between 7.2 and 7.5 inclusive, or equivalent
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1
1
(b)
Indicates A and gives a correct explanation
1
The most common correct explanations: Use the trend line for type A eg It is closest to the line for type A (3.2, 5.8) is close to (3, 6) which is on line A Type A have smaller diameters with bigger heights than the other types For A, the height is about double the diameter, and that’s roughly true for this one Refer to the diameters of type B being consistently larger than 3.2cm, or to the heights of type A differing from their diameters eg It’s between the lines for A and B, but all the type Bs have diameters between 6 and 7 It’s too far from the type C line so it’s A or B, and the A ones don’t have similar diameters and heights Accept minimally acceptable explanation eg It’s closest to that line The line goes through (3, 6) which is very close It is closest to type A [with point correctly plotted on graph] Type A have small diameters with big heights For A, height is bigger than diameter A tomatoes are thin but tall Do not accept incomplete or incorrect explanation eg It is closest to type A It’s in the A section For A, the height is double the diameter The graph shows it It is on A’s line Type A tomatoes have small diameters
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2
Accept minimally acceptable explanation eg B tomatoes have bigger diameters A tomatoes have diameters that are not roughly equal to their heights Do not accept incomplete explanation eg It could be A or B but it’s more like A (c)
Indicates B and gives a correct explanation The most common correct explanations:
1 U1
Refer to the position of its line on the graph B’s graph is closest to y = x (or h = d) The line for B is closest to the line drawn [line h = d correctly indicated on graph] Refer to the dimensions of the tomatoes eg The height and the diameter of a sphere are equal and that’s also roughly true for B The height and diameter of B are both around 6 A tomatoes are too tall for their diameter, but C tomatoes are too fat for their height Accept minimally acceptable explanation eg B’s line is about 45° through the middle It goes through (0, 0) then when d goes up by 1, so does h The x and y (or h and d) coordinates are nearly equal Do not accept incomplete or incorrect explanation B’s line is at about 45° B’s line is a diagonal through the middle
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The graph shows it B has h = 2 and d = 2
3
Accept minimally acceptable explanation eg Same height and diameter h and d are closest The two values are nearly equal
The others are either too tall and thin or too short and wide [3]
6.
Same area 3, with no evidence of an incorrect method
2
or Shows the value 12
1
or Forms a correct equation in w eg 4w = 1 (6 4) 2 4w=34 or Shows a correct method with not more than one computational error eg 6424
3 4 4 6 4 2 = 20 (error), 20 4 = 5 62 Do not accept conceptual error eg 6 4 = 24, 24 4 = 6
[2]
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4
7.
Overlay
Evening newspaper
No newspaper
Morning newspaper
No newspaper
Morning newspaper Evening newspaper
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5
or
Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels correctly.
2
Draws the sectors for Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels correctly
1
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 Accept: unambiguous abbreviation eg • E for Evening newspaper, N for No newspaper [2]
8.
0.61 or equivalent probability
2
or
Shows the digits 61
1
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 [2]
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6
9.
(a)
72
1
(b)
4
1 [2]
10.
Points and rules (a)
All correct for y = – 3, ie
1
All correct for x = y, ie
1
Accept unambiguous indication eg for y = – 3 D, E ticked, rest left blank eg for x = y
(b)
A, D ticked, rest left blank
x + y = 5 or equivalent algebraic equation Accept equation expressed in words using x and y eg x and y add up to 5 x and y = 5 Do not accept rule does not use x and y eg The coordinates add up to 5
1
[3]
11.
(a)
For 2m draws the correct shape in the correct orientation
2
For 2m either all lines showing small cubes should be shown individually, or the complete outline only should be given. Hence, Inconsistent use of lines eg:
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7
For only 1m draws the correct outline shape in the correct orientation but adds or omits lines eg:
For 2m or 1m the drawing need not be correctly aligned with the drawings in the question. or
Makes an isometric drawing to show the L-shapes from different views do not accept L-shape from a different view from any inconsistent use of lines shown in the question eg:
For L-shapes from different views do not accept inconsistent use of lines eg:
or
Draws an otherwise correct shape that is 2 or 4 cubes high eg:
For otherwise correct shapes that are 2 or 4 cubes high do not accept inconsistent use of lines. [2]
12.
Gives all three numbers correctly for the first net, ie
1
6 3
2
4
5
1
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8
Gives all three numbers correctly for the second net, ie
4 1 2
1
6
5
3 [2]
13.
Isosceles A completed triangle, sides ruled, with each of the three vertices in the region indicated on the marking overlay
or
2
All three vertices within tolerance, but ruler not used 1 or The ‘top’ vertex within tolerance, but lines not shown or The ‘top’ vertex within tolerance, but one or both of the other vertices outside the tolerance, lines ruled or The ‘top’ vertex is outside the tolerance, but construction arcs show understanding of the correct method ! Given base of 5cm not used For 2m or 1m, allow if the 5 cm line is drawn within the tolerance as shown on the overlay. If the triangle is rotated to make the base 7 cm, allow a maximum of 1 mark only. [2]
14.
Same volume (a)
Correct volume, ie 60 The value of 60 is shown to the power of 3 eg 60³
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1
603 cm Assume the power refers to units, ie mark as 1, 0
9
Correct units
1
eg
(b)
cm3
Centimetres cubed Accept informal but unambiguous language eg Cube centimetres
6
1 ! !
Follow through as their part (a) ÷ 10 provided the value is exact and not rounded Incorrect units inserted Ignore [3]
15.
Gives a = 50 and gives a correct reason eg
1 U1
▪
Angle a is on a straight line with 130, so a = 180 – 130
▪
a is supplementary with 130, so a + 130 = 180
▪
The angle vertically opposite 130 is 130, 360 – (130 + 130) = 100, (angles at a point) 100 a is = 50 (also vertically opposite) 2 Accept: minimally acceptable reason eg • On a straight line • Supplementary • Opposite angles and angles at a point Do not accept: informal reason without the correct geometrical property identified eg • 180 – 130 • 360 – 260 2 Do not accept: incomplete reason eg • It is adjacent to the 130° angle
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10
Gives b = 60 and gives a correct reason eg ▪
Angle b is vertically opposite the 60° angle, so it is also 60°
▪
The angle on a straight line with b is 120, so b is 360 – 120 – 120 – 60 (angles at a point) Accept: minimally acceptable reason eg • Opposite • Angles on a straight line and angles at a point Do not accept: informal reason without the correct geometrical property identified eg • b is equal to the 60° angle next to it Do not accept: incomplete reason eg • It is the same as the 60° angle
Gives c = 70 and gives a correct reason eg ▪
There are 180° in a triangle, so c = 180 – 50 – 60
▪
The exterior angle of a triangle is equal to the sum of the two opposite interior angles, so c = 130 – 60 Accept: minimally acceptable reason eg • Angles in a triangle • Exterior angle = sum of two opposite interior angles • We’ve already got 50 and 60 in the triangle !
1 U1
1 U1
Follow through Accept as 180 – (their a + b), alongside a correct reason referring to angles in a triangle, or as 130 – their b alongside a correct reason referring to an exterior angle of a triangle Do not accept: informal reason without the correct geometrical property identified eg • 180 – (a + b) • 130 – b Do not accept: incomplete reason eg • It is in a triangle • All the inside angles add up to 180° [3]
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11
16.
(a)
or
Gives the four values in the correct order eg
32 smallest
9 smallest
24 16
2
52 33 largest 25
27 largest
Shows any three of the values 25, 9, 27, 16, with no evidence of an incorrect method for a correct value
1
or Gives the four values in order of size, largest to smallest (b)
78 125
2 Do not accept follow through using their value for 52 from part (a)
or
Shows the value 78 125, even if there is subsequent incorrect working
1
or Shows or implies a complete correct method, with at least some correct processing, with not more than one computational error eg 3125 × 100 = 312 500, 312 500 ÷ 4 3125 × 5 = 15 625 15 625 × 5
3125 × 25 15525 (error) 62500 78025
3125 × 10 ÷ 2 = 15 125 (error) 15 125 × 10 ÷ 2 = 75 625 Do not accept conceptual error eg
3125 × 25 15525 6250 21875
55 = 3125, 52 = 25, 3125 + 25 = 3150
52 = 10, 3125 × 10 = 31 250 [4]
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12
17.
(a)
Indicates a correct probability eg:
35 36
0.97
1
Do not accept incomplete processing eg: ‘1 – 1/36’
(b)
Ticks the ‘
1 ‘ box 36
1
and correctly justifies why the probability is
1 eg: 36
1 1 × 6 6
Its
Its a 1 in 36 chance. Accept any indication provided it is clear the probability is still 1/36
or Explains it is the same as for the previous throw eg:
What you throw doesnt affect what you throw the next time.
It doesnt change.
A double is as likely to appear again.
Its still 1 in 36
Because it is fair.
The odds are the same.
The throws are independent.
Theyre still the same dice. Do not accept the word ‘even’ to mean equal eg:
‘There are even chances of getting double 6 on each throw.’ (c)
Indicates a correct probability eg:
1
1 36
Accept decimals or percentages rounded or truncated to 2 or more s.f. eg: ‘0.02777’ Accept a reference to earlier in the question eg: ‘Same as double 6’ (d)
Indicates a correct probability eg:
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1 13
6 36
1 6
Accept decimals or percentages rounded or truncated to 2 or more s.f. eg:
‘0.166’ Do not accept incomplete processing eg:
‘
1 × 6’ 36
[4]
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14
18.
Overlay
P
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15
or
Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay
2
Shows at least three vertices within the tolerances as shown on the overlay
1
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 ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Construction lines drawn Ignore, even if incorrect [2]
19.
Gives a correct explanation The most common correct explanations:
1 U1
Show the correct working eg ▪
It should be π × 16 not π × 8
▪
Needs to be π × radius2, not π × diameter Accept: minimally acceptable explanation eg • 16 • •
42 4×4
•
r2
• πr2 Do not accept: incomplete explanation eg • The 8 is wrong
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16
Address the misconception eg ▪
He is finding the circumference not the area
▪
He is using 2πr, not πr2
▪
He has done 4 × 2 instead of 42 Accept: minimally acceptable explanation eg • Circumference • It’s not 2πr [or πd] • He didn’t square the 4 • He didn’t square the radius ! Use of ‘perimeter’ for ‘circumference’ Condone Do not accept: incomplete explanation eg • He used the wrong formula • He used the diameter • He hasn’t used the radius • He doubled the radius
Show that his working gives an incorrect answer eg ▪
He gets 25.(…), but it should be 50.(…)
▪
His answer is half as big as it should be Accept: minimally acceptable explanation eg • 50, not 25 • It should be his answer × 2 Do not accept: incomplete explanation eg • 50 • His answer is too small [1]
20.
(a)
15
(b)
5
1
1 or equivalent 2
1 [2]
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17
21.
Overlay
(a)
Draws a straight line on the graph joining the points (0, 0) and (60, 30) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 30 km/hour
1
(b)
Draws a straight line on the graph joining the points (0, 0) and (30, 60) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 120 km/hour
1
Accept: unambiguous labelling eg, for 30 km/hour • 30 ! Labels omitted or incorrect For two correct lines of full length with labels omitted, mark as 0, 1 Do not accept incorrect labels ! Lines not of full length For two correct lines at least 5cm long but not of full length, mark as 0, 1 Do not accept lines less than 5cm long [2]
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18
22.
(a)
For 2m gives all 4 correct values eg:
2
5
6
10
16
17
32
For only 1m gives any 2 or 3 correct values. or Gives all 4 values but transposes the colours eg: 5
10
6
16
32
17
Ignore any reference to colour eg. accept 6g, 10w (b)
For 2m gives 2 correct expressions eg:
2
n
grey
white
n+1
2n
n
grey
white
2n – (n – 1)
n+n
For only 1m gives 1 correct expression. or Gives 2 correct expressions using words or multiple letters eg:
Pattern number + 1 for grey, double the pattern number for white.
1 + pn for grey, 2pn for white.
or Gives 2 correct expressions in the wrong order, even though no evidence of incorrect ordering has been shown in (a). For 2m or 1m accept correct expressions for (b) given in (c) eg: for 2m: + 1, ×2 in (b), n + 1, 2n in (c) for 2m: no response in (b), n + 1, 2n in (c) for 2m g, w in (b), g = n + 1, w = 2n in (c)
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19
For 2m or 1m allow follow through from part (a) for each column independently, provided the difficulty level has not been decreased. One of the columns must at least be of the form n + k (k 0). The other must be different and at least of the form an (a 1 or 0) eg: for 2m 5 4 12 16
25
34
n
n–1
2n + 2
5
3
15
16
14
48
n
n–2
3n
eg, for 1m 5 7
8
16
18
19
n
n+2
n+3
eg, do not accept 5 5 12
(c)
16
16
32
n
n
2n + 2
Indicates a correct, simplified expression eg:
3n + 1
1+n×3 Allow follow through from two algebraic expressions given in part (b), provided they are able to be simplified eg:
1
2n + 3 from n + 1 and n + 2 but not n2 + 1 from n + 1 and n2. Throughout the question ignore redundant brackets eg:
4 + (5n) but Incorrect brackets eg: (4 + 5)n [5]
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20
23.
120
1
84
1 !
Incorrect use of % sign Ignore [2]
24. No mark scheme available
25.
Straight lines (a)
Completes the table with any three sets of correct coordinates, indicating for each that x+y=4 eg
1
(x, y)
(0, 4)
(1, 3)
(2, 2)
x+y
4
4
4
!
(b)
Accept incomplete processing eg, for (1, 3) 1+3 Values for (x, y) correct but some or all of values for x + y omitted Accept provided a correct equation is given in part (b)
Gives a correct equation eg x+y=4 y=4x x = –y + 4
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1
21
(c)
Draws the correct straight line through (0, 6) and (6, 0) ! Line not ruled or accurate Accept provided the pupil’s intention is clear ! Partial line drawn Do not accept lines that are less than 5cm in length ! Points plotted Ignore Do not accept points not joined
1
[3]
26.
(a)
Gives a correct counter example, using a value that is less than or equal to one eg
1 U1
–4 × 2 = –8 which is not greater than 2 0.1 × 2 = 0.2, 0.2 < 2 2 × 1 = 2 which is not greater than 2 or Gives a correct general explanation eg Two times a negative number is less than 2 Double a number between 0 and 1 is not greater than 2 (b)
Gives a correct counter example, using a value that is less than or equal to zero eg,
1 U1
2 – (–3) = 5, 5 > 2 2 – 0 = 2 which is not less than 2 or Gives a correct general explanation eg Two minus a negative number is greater than 2
Hertfordshire La Schools
22
!
Throughout the question, the result of their counter example is not shown and/or the comparison is not explicit Condone provided only one of these aspects is omitted eg, for part (a) accept –4 × 2 = –8
–4 × 2 < 2 However, penalise only the first occurrence of both aspects omitted eg, for part (a) –4 × 2 !
Throughout the question, their general statement makes no explicit comparison Condone eg, for part (a) accept
Multiply it by a negative number Numbers less than 1 eg, for part (b) accept Take away a negative number Numbers less than 0 !
Throughout the question, other numerical examples or general reasoning given alongside a correct response Ignore other numerical examples, even if they are incorrect or support the given statement If a correct counter example is given, ignore any general explanation unless it contradicts the counter example given [2]
27.
Gives two correct values in the correct order, and a correct expression in x
1
eg •
3, 1, 3x + 1
•
1, 9, x + 9
•
–2, 21, –2x + 21
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23
Gives two correct values in the correct order, and a correct expression in x
1
eg •
4, 3, 4x – 3
•
–2, –21, –2x – –21
•
x, 3, x2 – 3
Gives two correct values in the correct order, and a correct expression in x
1
eg
x + 11 2
•
2, 11,
•
0.5, 5, 2x + 5 (or
•
1, 9, x + 9 !
x + 5) 0.5
Do not accept: for the first mark, given example repeated Unconventional notation eg, for x + 9 • 1×x+9 Condone [3]
28.
9
1
100
1 !
Incomplete processing eg, for the first mark • 10 – 1 eg, for the second mark • 102 Penalise only the first occurrence [2]
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24
29. or
1 or equivalent 2
2
Shows or implies a correct first step of algebraic manipulation that removes the brackets eg ▪
2 × 2n + 2 × 5 = 12
▪
4n + 10 = 12
▪
2n + 5 = 6
▪
4n = 2
▪
2n = 1
▪
2÷4
▪
1÷2 [2]
30.
Tickets For 2m indicates only the correct three combinations, eg:
BEDA, BEDC, BACD
ABCD, ABDE, BCDE
BD
|CA
|CE
|EA
2
Ben always goes, the other three tickets could be for Anna, Carl and Donna, or ADE or EDC. For 2m do not accept repeated or incorrect combinations, or permutations.
For only 1m lists only 2 combinations, both of which are correct. or
or
Lists only 4 combinations, the correct 3 and 1 incorrect, eg:
BEDA, BEDC, BACD, BACE
ABCD, ABDE, BCDE, ACDE
Lists the correct 3 combinations with no incorrect combinations, but repeats combinations and/or gives permutations, eg:
or
BEDA, BEDC, BACD, BEDA, BADE Correct combinations/permutations must include B and D. Note that BACE is incorrect.
Lists only 3 combinations which are otherwise correct but omit Ben, eg:
EDA, EDC, ACD [2]
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25
Hertfordshire La Schools
26
6
1
12
1 [2]
2.
(a)
3.1416
1 Do not accept: equivalent fractions or decimals
(b)
Indicates
355 , ie 113
1
[2]
3.
54x2
1 Do not accept: unsimplified expression or unconventional notation eg •
9x2 × 6
• •
9x2 + 9x2 + 9x2 + 9x2 + 9x2 + 9x2 54xx [1]
4.
Walk Indicates ‘steady speed’, ie
1
[1]
5.
(a)
Gives a value between 7.2 and 7.5 inclusive, or equivalent
Hertfordshire La Schools
1
1
(b)
Indicates A and gives a correct explanation
1
The most common correct explanations: Use the trend line for type A eg It is closest to the line for type A (3.2, 5.8) is close to (3, 6) which is on line A Type A have smaller diameters with bigger heights than the other types For A, the height is about double the diameter, and that’s roughly true for this one Refer to the diameters of type B being consistently larger than 3.2cm, or to the heights of type A differing from their diameters eg It’s between the lines for A and B, but all the type Bs have diameters between 6 and 7 It’s too far from the type C line so it’s A or B, and the A ones don’t have similar diameters and heights Accept minimally acceptable explanation eg It’s closest to that line The line goes through (3, 6) which is very close It is closest to type A [with point correctly plotted on graph] Type A have small diameters with big heights For A, height is bigger than diameter A tomatoes are thin but tall Do not accept incomplete or incorrect explanation eg It is closest to type A It’s in the A section For A, the height is double the diameter The graph shows it It is on A’s line Type A tomatoes have small diameters
Hertfordshire La Schools
2
Accept minimally acceptable explanation eg B tomatoes have bigger diameters A tomatoes have diameters that are not roughly equal to their heights Do not accept incomplete explanation eg It could be A or B but it’s more like A (c)
Indicates B and gives a correct explanation The most common correct explanations:
1 U1
Refer to the position of its line on the graph B’s graph is closest to y = x (or h = d) The line for B is closest to the line drawn [line h = d correctly indicated on graph] Refer to the dimensions of the tomatoes eg The height and the diameter of a sphere are equal and that’s also roughly true for B The height and diameter of B are both around 6 A tomatoes are too tall for their diameter, but C tomatoes are too fat for their height Accept minimally acceptable explanation eg B’s line is about 45° through the middle It goes through (0, 0) then when d goes up by 1, so does h The x and y (or h and d) coordinates are nearly equal Do not accept incomplete or incorrect explanation B’s line is at about 45° B’s line is a diagonal through the middle
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The graph shows it B has h = 2 and d = 2
3
Accept minimally acceptable explanation eg Same height and diameter h and d are closest The two values are nearly equal
The others are either too tall and thin or too short and wide [3]
6.
Same area 3, with no evidence of an incorrect method
2
or Shows the value 12
1
or Forms a correct equation in w eg 4w = 1 (6 4) 2 4w=34 or Shows a correct method with not more than one computational error eg 6424
3 4 4 6 4 2 = 20 (error), 20 4 = 5 62 Do not accept conceptual error eg 6 4 = 24, 24 4 = 6
[2]
Hertfordshire La Schools
4
7.
Overlay
Evening newspaper
No newspaper
Morning newspaper
No newspaper
Morning newspaper Evening newspaper
Hertfordshire La Schools
5
or
Draws the sectors for Evening newspaper and No newspaper within the smaller tolerance as shown on the overlay and labels correctly.
2
Draws the sectors for Evening newspaper and No newspaper within the larger tolerance as shown on the overlay and labels correctly
1
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 Accept: unambiguous abbreviation eg • E for Evening newspaper, N for No newspaper [2]
8.
0.61 or equivalent probability
2
or
Shows the digits 61
1
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 [2]
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6
9.
(a)
72
1
(b)
4
1 [2]
10.
Points and rules (a)
All correct for y = – 3, ie
1
All correct for x = y, ie
1
Accept unambiguous indication eg for y = – 3 D, E ticked, rest left blank eg for x = y
(b)
A, D ticked, rest left blank
x + y = 5 or equivalent algebraic equation Accept equation expressed in words using x and y eg x and y add up to 5 x and y = 5 Do not accept rule does not use x and y eg The coordinates add up to 5
1
[3]
11.
(a)
For 2m draws the correct shape in the correct orientation
2
For 2m either all lines showing small cubes should be shown individually, or the complete outline only should be given. Hence, Inconsistent use of lines eg:
Hertfordshire La Schools
7
For only 1m draws the correct outline shape in the correct orientation but adds or omits lines eg:
For 2m or 1m the drawing need not be correctly aligned with the drawings in the question. or
Makes an isometric drawing to show the L-shapes from different views do not accept L-shape from a different view from any inconsistent use of lines shown in the question eg:
For L-shapes from different views do not accept inconsistent use of lines eg:
or
Draws an otherwise correct shape that is 2 or 4 cubes high eg:
For otherwise correct shapes that are 2 or 4 cubes high do not accept inconsistent use of lines. [2]
12.
Gives all three numbers correctly for the first net, ie
1
6 3
2
4
5
1
Hertfordshire La Schools
8
Gives all three numbers correctly for the second net, ie
4 1 2
1
6
5
3 [2]
13.
Isosceles A completed triangle, sides ruled, with each of the three vertices in the region indicated on the marking overlay
or
2
All three vertices within tolerance, but ruler not used 1 or The ‘top’ vertex within tolerance, but lines not shown or The ‘top’ vertex within tolerance, but one or both of the other vertices outside the tolerance, lines ruled or The ‘top’ vertex is outside the tolerance, but construction arcs show understanding of the correct method ! Given base of 5cm not used For 2m or 1m, allow if the 5 cm line is drawn within the tolerance as shown on the overlay. If the triangle is rotated to make the base 7 cm, allow a maximum of 1 mark only. [2]
14.
Same volume (a)
Correct volume, ie 60 The value of 60 is shown to the power of 3 eg 60³
Hertfordshire La Schools
1
603 cm Assume the power refers to units, ie mark as 1, 0
9
Correct units
1
eg
(b)
cm3
Centimetres cubed Accept informal but unambiguous language eg Cube centimetres
6
1 ! !
Follow through as their part (a) ÷ 10 provided the value is exact and not rounded Incorrect units inserted Ignore [3]
15.
Gives a = 50 and gives a correct reason eg
1 U1
▪
Angle a is on a straight line with 130, so a = 180 – 130
▪
a is supplementary with 130, so a + 130 = 180
▪
The angle vertically opposite 130 is 130, 360 – (130 + 130) = 100, (angles at a point) 100 a is = 50 (also vertically opposite) 2 Accept: minimally acceptable reason eg • On a straight line • Supplementary • Opposite angles and angles at a point Do not accept: informal reason without the correct geometrical property identified eg • 180 – 130 • 360 – 260 2 Do not accept: incomplete reason eg • It is adjacent to the 130° angle
Hertfordshire La Schools
10
Gives b = 60 and gives a correct reason eg ▪
Angle b is vertically opposite the 60° angle, so it is also 60°
▪
The angle on a straight line with b is 120, so b is 360 – 120 – 120 – 60 (angles at a point) Accept: minimally acceptable reason eg • Opposite • Angles on a straight line and angles at a point Do not accept: informal reason without the correct geometrical property identified eg • b is equal to the 60° angle next to it Do not accept: incomplete reason eg • It is the same as the 60° angle
Gives c = 70 and gives a correct reason eg ▪
There are 180° in a triangle, so c = 180 – 50 – 60
▪
The exterior angle of a triangle is equal to the sum of the two opposite interior angles, so c = 130 – 60 Accept: minimally acceptable reason eg • Angles in a triangle • Exterior angle = sum of two opposite interior angles • We’ve already got 50 and 60 in the triangle !
1 U1
1 U1
Follow through Accept as 180 – (their a + b), alongside a correct reason referring to angles in a triangle, or as 130 – their b alongside a correct reason referring to an exterior angle of a triangle Do not accept: informal reason without the correct geometrical property identified eg • 180 – (a + b) • 130 – b Do not accept: incomplete reason eg • It is in a triangle • All the inside angles add up to 180° [3]
Hertfordshire La Schools
11
16.
(a)
or
Gives the four values in the correct order eg
32 smallest
9 smallest
24 16
2
52 33 largest 25
27 largest
Shows any three of the values 25, 9, 27, 16, with no evidence of an incorrect method for a correct value
1
or Gives the four values in order of size, largest to smallest (b)
78 125
2 Do not accept follow through using their value for 52 from part (a)
or
Shows the value 78 125, even if there is subsequent incorrect working
1
or Shows or implies a complete correct method, with at least some correct processing, with not more than one computational error eg 3125 × 100 = 312 500, 312 500 ÷ 4 3125 × 5 = 15 625 15 625 × 5
3125 × 25 15525 (error) 62500 78025
3125 × 10 ÷ 2 = 15 125 (error) 15 125 × 10 ÷ 2 = 75 625 Do not accept conceptual error eg
3125 × 25 15525 6250 21875
55 = 3125, 52 = 25, 3125 + 25 = 3150
52 = 10, 3125 × 10 = 31 250 [4]
Hertfordshire La Schools
12
17.
(a)
Indicates a correct probability eg:
35 36
0.97
1
Do not accept incomplete processing eg: ‘1 – 1/36’
(b)
Ticks the ‘
1 ‘ box 36
1
and correctly justifies why the probability is
1 eg: 36
1 1 × 6 6
Its
Its a 1 in 36 chance. Accept any indication provided it is clear the probability is still 1/36
or Explains it is the same as for the previous throw eg:
What you throw doesnt affect what you throw the next time.
It doesnt change.
A double is as likely to appear again.
Its still 1 in 36
Because it is fair.
The odds are the same.
The throws are independent.
Theyre still the same dice. Do not accept the word ‘even’ to mean equal eg:
‘There are even chances of getting double 6 on each throw.’ (c)
Indicates a correct probability eg:
1
1 36
Accept decimals or percentages rounded or truncated to 2 or more s.f. eg: ‘0.02777’ Accept a reference to earlier in the question eg: ‘Same as double 6’ (d)
Indicates a correct probability eg:
Hertfordshire La Schools
1 13
6 36
1 6
Accept decimals or percentages rounded or truncated to 2 or more s.f. eg:
‘0.166’ Do not accept incomplete processing eg:
‘
1 × 6’ 36
[4]
Hertfordshire La Schools
14
18.
Overlay
P
Hertfordshire La Schools
15
or
Shows a correct enlarged shape with all five vertices within the tolerances as shown on the overlay
2
Shows at least three vertices within the tolerances as shown on the overlay
1
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 ! Lines not ruled or accurate Accept provided the pupil’s intention is clear ! Construction lines drawn Ignore, even if incorrect [2]
19.
Gives a correct explanation The most common correct explanations:
1 U1
Show the correct working eg ▪
It should be π × 16 not π × 8
▪
Needs to be π × radius2, not π × diameter Accept: minimally acceptable explanation eg • 16 • •
42 4×4
•
r2
• πr2 Do not accept: incomplete explanation eg • The 8 is wrong
Hertfordshire La Schools
16
Address the misconception eg ▪
He is finding the circumference not the area
▪
He is using 2πr, not πr2
▪
He has done 4 × 2 instead of 42 Accept: minimally acceptable explanation eg • Circumference • It’s not 2πr [or πd] • He didn’t square the 4 • He didn’t square the radius ! Use of ‘perimeter’ for ‘circumference’ Condone Do not accept: incomplete explanation eg • He used the wrong formula • He used the diameter • He hasn’t used the radius • He doubled the radius
Show that his working gives an incorrect answer eg ▪
He gets 25.(…), but it should be 50.(…)
▪
His answer is half as big as it should be Accept: minimally acceptable explanation eg • 50, not 25 • It should be his answer × 2 Do not accept: incomplete explanation eg • 50 • His answer is too small [1]
20.
(a)
15
(b)
5
1
1 or equivalent 2
1 [2]
Hertfordshire La Schools
17
21.
Overlay
(a)
Draws a straight line on the graph joining the points (0, 0) and (60, 30) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 30 km/hour
1
(b)
Draws a straight line on the graph joining the points (0, 0) and (30, 60) within the tolerance as shown on the overlay (ie within 2mm), and labels the line 120 km/hour
1
Accept: unambiguous labelling eg, for 30 km/hour • 30 ! Labels omitted or incorrect For two correct lines of full length with labels omitted, mark as 0, 1 Do not accept incorrect labels ! Lines not of full length For two correct lines at least 5cm long but not of full length, mark as 0, 1 Do not accept lines less than 5cm long [2]
Hertfordshire La Schools
18
22.
(a)
For 2m gives all 4 correct values eg:
2
5
6
10
16
17
32
For only 1m gives any 2 or 3 correct values. or Gives all 4 values but transposes the colours eg: 5
10
6
16
32
17
Ignore any reference to colour eg. accept 6g, 10w (b)
For 2m gives 2 correct expressions eg:
2
n
grey
white
n+1
2n
n
grey
white
2n – (n – 1)
n+n
For only 1m gives 1 correct expression. or Gives 2 correct expressions using words or multiple letters eg:
Pattern number + 1 for grey, double the pattern number for white.
1 + pn for grey, 2pn for white.
or Gives 2 correct expressions in the wrong order, even though no evidence of incorrect ordering has been shown in (a). For 2m or 1m accept correct expressions for (b) given in (c) eg: for 2m: + 1, ×2 in (b), n + 1, 2n in (c) for 2m: no response in (b), n + 1, 2n in (c) for 2m g, w in (b), g = n + 1, w = 2n in (c)
Hertfordshire La Schools
19
For 2m or 1m allow follow through from part (a) for each column independently, provided the difficulty level has not been decreased. One of the columns must at least be of the form n + k (k 0). The other must be different and at least of the form an (a 1 or 0) eg: for 2m 5 4 12 16
25
34
n
n–1
2n + 2
5
3
15
16
14
48
n
n–2
3n
eg, for 1m 5 7
8
16
18
19
n
n+2
n+3
eg, do not accept 5 5 12
(c)
16
16
32
n
n
2n + 2
Indicates a correct, simplified expression eg:
3n + 1
1+n×3 Allow follow through from two algebraic expressions given in part (b), provided they are able to be simplified eg:
1
2n + 3 from n + 1 and n + 2 but not n2 + 1 from n + 1 and n2. Throughout the question ignore redundant brackets eg:
4 + (5n) but Incorrect brackets eg: (4 + 5)n [5]
Hertfordshire La Schools
20
23.
120
1
84
1 !
Incorrect use of % sign Ignore [2]
24. No mark scheme available
25.
Straight lines (a)
Completes the table with any three sets of correct coordinates, indicating for each that x+y=4 eg
1
(x, y)
(0, 4)
(1, 3)
(2, 2)
x+y
4
4
4
!
(b)
Accept incomplete processing eg, for (1, 3) 1+3 Values for (x, y) correct but some or all of values for x + y omitted Accept provided a correct equation is given in part (b)
Gives a correct equation eg x+y=4 y=4x x = –y + 4
Hertfordshire La Schools
1
21
(c)
Draws the correct straight line through (0, 6) and (6, 0) ! Line not ruled or accurate Accept provided the pupil’s intention is clear ! Partial line drawn Do not accept lines that are less than 5cm in length ! Points plotted Ignore Do not accept points not joined
1
[3]
26.
(a)
Gives a correct counter example, using a value that is less than or equal to one eg
1 U1
–4 × 2 = –8 which is not greater than 2 0.1 × 2 = 0.2, 0.2 < 2 2 × 1 = 2 which is not greater than 2 or Gives a correct general explanation eg Two times a negative number is less than 2 Double a number between 0 and 1 is not greater than 2 (b)
Gives a correct counter example, using a value that is less than or equal to zero eg,
1 U1
2 – (–3) = 5, 5 > 2 2 – 0 = 2 which is not less than 2 or Gives a correct general explanation eg Two minus a negative number is greater than 2
Hertfordshire La Schools
22
!
Throughout the question, the result of their counter example is not shown and/or the comparison is not explicit Condone provided only one of these aspects is omitted eg, for part (a) accept –4 × 2 = –8
–4 × 2 < 2 However, penalise only the first occurrence of both aspects omitted eg, for part (a) –4 × 2 !
Throughout the question, their general statement makes no explicit comparison Condone eg, for part (a) accept
Multiply it by a negative number Numbers less than 1 eg, for part (b) accept Take away a negative number Numbers less than 0 !
Throughout the question, other numerical examples or general reasoning given alongside a correct response Ignore other numerical examples, even if they are incorrect or support the given statement If a correct counter example is given, ignore any general explanation unless it contradicts the counter example given [2]
27.
Gives two correct values in the correct order, and a correct expression in x
1
eg •
3, 1, 3x + 1
•
1, 9, x + 9
•
–2, 21, –2x + 21
Hertfordshire La Schools
23
Gives two correct values in the correct order, and a correct expression in x
1
eg •
4, 3, 4x – 3
•
–2, –21, –2x – –21
•
x, 3, x2 – 3
Gives two correct values in the correct order, and a correct expression in x
1
eg
x + 11 2
•
2, 11,
•
0.5, 5, 2x + 5 (or
•
1, 9, x + 9 !
x + 5) 0.5
Do not accept: for the first mark, given example repeated Unconventional notation eg, for x + 9 • 1×x+9 Condone [3]
28.
9
1
100
1 !
Incomplete processing eg, for the first mark • 10 – 1 eg, for the second mark • 102 Penalise only the first occurrence [2]
Hertfordshire La Schools
24
29. or
1 or equivalent 2
2
Shows or implies a correct first step of algebraic manipulation that removes the brackets eg ▪
2 × 2n + 2 × 5 = 12
▪
4n + 10 = 12
▪
2n + 5 = 6
▪
4n = 2
▪
2n = 1
▪
2÷4
▪
1÷2 [2]
30.
Tickets For 2m indicates only the correct three combinations, eg:
BEDA, BEDC, BACD
ABCD, ABDE, BCDE
BD
|CA
|CE
|EA
2
Ben always goes, the other three tickets could be for Anna, Carl and Donna, or ADE or EDC. For 2m do not accept repeated or incorrect combinations, or permutations.
For only 1m lists only 2 combinations, both of which are correct. or
or
Lists only 4 combinations, the correct 3 and 1 incorrect, eg:
BEDA, BEDC, BACD, BACE
ABCD, ABDE, BCDE, ACDE
Lists the correct 3 combinations with no incorrect combinations, but repeats combinations and/or gives permutations, eg:
or
BEDA, BEDC, BACD, BEDA, BADE Correct combinations/permutations must include B and D. Note that BACE is incorrect.
Lists only 3 combinations which are otherwise correct but omit Ben, eg:
EDA, EDC, ACD [2]
Hertfordshire La Schools
25
Hertfordshire La Schools
26
