Area and perimeter L6 non calc
27 minutes
25 marks
Page 1 of 21
Q1.
L-shape
What is the area of this L-shape?
Show your working.
........................ cm2
2 marks
Q2.
Perimeters
Jenny and Alan each have a rectangle made out of paper.
One side is 10cm.
The other side is n cm.
Page 2 of 21
(a)
They write expressions for the perimeter of the rectangle.
Jenny writes
2n + 20
Alan writes
2(n + 10)
Tick ( ) the true statement below.
Jenny is correct and Alan is wrong.
Jenny is wrong and Alan is correct.
Both Jenny and Alan are correct.
Both Jenny and Alan are wrong.
1 mark
(b)
Alan cuts his rectangle, then puts the two halves side by side.
What is the perimeter of Alan’s new rectangle?
Write your expression as simply as possible.
2 marks
(c)
Jenny cuts her rectangle a different way, and puts one half below the other.
Page 3 of 21
What is the perimeter of Jenny’s new rectangle?
Write your expression as simply as possible.
2 marks
(d)
What value of n would make the perimeter of Jenny’s new rectangle the same value as
the perimeter of Alan’s new rectangle?
1 mark
Q3.
Areas algebraically
(a)
The diagram shows a rectangle.
Its dimensions are 3a by 5b
Write simplified expressions for the area and the perimeter of this rectangle.
Area: ........................................
1 mark
Perimeter: ................................
1 mark
Page 4 of 21
(b)
A different rectangle has area 12a 2 and perimeter 14a .
What are the dimensions of this rectangle?
Dimensions: ...................... by ...........................
1 mark
Q4.
Cuboids
You can make only four different cuboids with 16 cubes.
Dimensions
(a)
Cuboid A
1
1
16
Cuboid B
1
2
8
Cuboid C
1
4
4
Cuboid D
2
2
4
Which of the cuboids A and D has the larger surface area?
Tick ( ) the correct answer below.
Cuboid A
Cuboid D
Both the same
1 mark
Page 5 of 21
Explain how you know.
1 mark
(b)
Which cuboid has the largest volume?
Tick ( ) the correct answer below.
Cuboid A
Cuboid B
Cuboid C
Cuboid D
All the same
1 mark
(c)
How many of cuboid D make a cube of dimensions 4 × 4 × 4?
...............................
1 mark
Page 6 of 21
(d)
You can make only six different cuboids with 24 cubes.
Complete the table to show the dimensions.
Two have been done for you.
Dimensions
Cuboid E
1
1
24
Cuboid F
1
2
12
Cuboid G
Cuboid H
Cuboid I
Cuboid J
3 marks
Q5.
Square cut
The diagram shows a square.
Two straight lines cut the square into four rectangles.
The area of one of the rectangles is shown.
Not drawn accurately
Page 7 of 21
Work out the area of the rectangle marked A.
......................... cm2
2 marks
Q6.
Circle working
Kevin is working out the area of a circle with radius 4
He writes:
Area = π × 8
Explain why Kevin’s working is wrong.
1 mark
Page 8 of 21
Q7.
Cube
Look at the cube.
The area of a face of the cube is 9x 2
Write an expression for the total surface area of the cube.
Write your answer as simply as possible.
....................
1 mark
Q8.
Square tiles
The diagram shows a square with a perimeter of 12cm.
Not drawn accurately
Six of these squares fit together to make a rectangle.
Not drawn accurately
Page 9 of 21
What is the area of the rectangle?
You must give the correct unit with your answer.
.............................
2 marks
Q9.
Missing lengths
Look at the rectangle.
Not drawn accurately
Page 10 of 21
The total area of the rectangle is 40cm2
Work out lengths x and y
x = ....................... cm
y = ....................... cm
2 marks
Page 11 of 21
M1.
30
2
or
Shows a complete correct method with no more than one computational error
The most common correct methods:
Calculate the area as 2 non-overlapping parts, A and B, as shown in the diagram
•
4×6+3×2
•
24 + 6
•
4 × 6 = 28 (error), 28 + 6 = 34
Calculate the area as 2 non-overlapping parts, C and D, as shown in the diagram
•
7×2+4×4
•
14 + 16
•
4 × 4 = 18 (error), 18 + 14 = 32
Calculate the area as 3 non-overlapping parts, E, F and G, as shown in the diagram
•
4 × 4 + 4 × 2 +3 × 2
•
16 + 8 + 6
Page 12 of 21
Calculate the area as two parts that need to be subtracted, eg
•
6×7–4×3
•
42 – 12
!
Do not accept conceptual error, eg
• 6 × 4 + 7 × 2 (overlapping parts)
• 6 × 4 × 7 × 2 (given lengths multiplied)
Method not shown fully
If there is no evidence of an incorrect method, accept the following:
• 24 + error but not 14 (for A and B)
• 14 + error but not 24 (for C and D)
• 16 + 8 error but not 14 (for E, F, G)
• 16 + error not 24 or 14 + 6 (for E, F, G)
• Error but not 24 + 8 + 6 (for E, F, G)
• 42 – error (subtraction method)
• Error – 12 (subtraction method)
1
[2]
M2.
(a)
Both correct, ie
1
(b)
or
or
Correct simplified expression, eg
•
4n + 10
•
4 (n + 2.5)
•
2(2n + 5)
•
(2n + 5) × 2
For only 1 Correct expression seen, even if terms are not collected together, eg
•
5+n+n+n+n+5
•
2n + 2n + 10
For only 1 An otherwise correct simplified expression, of 4n + k, (k ≠ 10), eg
•
4n + (10 ÷ 2)
•
4(n + 5)
•
2(2n + 2)
Page 13 of 21
!
!
!
Incorrect working follows correct response
eg, for part (a)
• 4n + 10 = 40n Mark as 1, 0
Correct answer preceded by incorrect working
If the intended answer is unambiguous, ignore preceding work
eg, for part (b), accept
• 5 + 2n × 2 = 10 + 4n
An otherwise correct simplified expression with the only error being
that the brackets are omitted
eg, for part (b)
• 2n + 5 × 2 Mark as 1, 0
2
(c)
or
or
Correct simplified expression, eg
•
40 + n
•
2(20 + ½ n)
For only 1 Correct expression seen, even if terms are not collected together, eg
•
10 × 4 + n
•
40 + ½ n + ½ n
•
2(n ÷ 2 + 20)
For only 1 An otherwise correct simplified expression, of n + k,
(k ≠ 40 but is a multiple of 10) eg
•
2(n ÷ 2 + 10)
!
Value for n substituted into an otherwise creditworthy response
Ignore.
2
(d)
10
Accept 10 cm
1
[6]
M3.
(a)
Gives a correct simplified expression for the area, eg
•
15ab
•
15 × a × b
1
Page 14 of 21
Gives a correct simplified expression for the perimeter, eg
•
6a + 10b
•
2(3a + 5b)
•
6 × a + 10 × b
•
2 × (3a + 5b)
!
!
Partially simplified or unsimplified expressions
eg, for the area
• 3a5b
eg, for the perimeter
• 2(3a) + 2(5b)
• 2 × (3 × a + 5 × b)
If both expressions are correct but are partially simplified
or unsimplified, mark as 0, 1, provided neither has subsequently
been incorrectly simplified
Expressions transposed but otherwise correct and simplified
Mark as 0, 1
1
(b)
Gives both correct dimensions in either order, ie 4a and 3a
!
Correct dimensions embedded
Accept provided both the area and perimeter have been
considered,
eg, accept
• 12a² = 3a × 4a
14a = 2(3a + 4a)
!
Dimensions labelled as length or width
Ignore
1
[3]
M4.
(a)
Indicates Cuboid A and gives a correct explanation
The most common correct explanations:
Show the correct surface area for both A and D
eg
•
The surface area of A is 66, but D is 40
Consider the number of cube faces that are not visible
eg
•
Each cube in D has 3 or 4 faces that cannot be
seen but each cube in A has only 1 or 2
•
Fewer faces of the cubes are touching each other in A
Page 15 of 21
Consider the number of cube faces that are visible
eg
•
In A the cubes show 4 or 5 faces, but in D it’s 2 or 3
•
There are more cube faces facing out on A than on D
!
Units inserted
Ignore
Accept minimally acceptable explanation
eg, for the correct surface areas
• 66 and 40 seen
• 4 × 16 + 2 is bigger than 4 × 8 + 8
eg, for cube faces that are not visible
• There are fewer hidden faces in A
• D is more compact
eg, for cube faces that are visible
• Cubes in A show 4 or more faces, D shows less than 4
• A has more faces showing
• A is more spread out
!
Use of ‘sides’ for cube faces
Condone
eg, accept
• More sides face out on A
!
Descriptors of cube faces
Note that pupils use a wide range of terms to describe the
cube faces
eg, for cube faces that are not visible
• Hidden faces
• Faces pointing inside
• Faces touching
eg, for cube faces that are visible
• Faces facing out
• Faces showing
• Faces you can see
Condone provided the pupil does not explicitly refer to the area of
only one of the faces of each cuboid
eg, do not accept
• You can see 8 faces on D and 16 faces on A
Do not accept use of ‘square’ for cube or cuboid
eg
• You can see more of each square’s surface in A than in D
Do not accept explanation is simply a description of one or
both of the cuboids
eg
• In A all 16 are in a line and not on top of each other
• D is two cubes high
Do not accept incorrect statement
eg
• Each cube in A shows 4 faces; D is 3
U1
Page 16 of 21
(b)
Indicates All the same
1
(c)
4
1
(d)
Shows, in any order, all four correct sets of dimensions
eg
•
1
1
2
2
3
4
2
3
8
6
6
4
3
or
Shows three correct sets of dimensions
2
or
Shows two correct sets of dimensions
1
!
Repeated sets of dimensions
eg
• 1
3
8
1
8
3(repeated)
2
2
6
6
2
2 (repeated)
Ignore the repeats and mark as 1, 0, 0
Do not accept negative or non-integer dimensions used
[6]
M5.
42, with no evidence of an incorrect method
Do not accept incorrect method
eg
• 12 + 2 = 14, 14 × 3 = 42
2
or
Shows or implies that the square is a 9(cm) by 9(cm) square
eg
•
7 × 6 seen
•
•
Area of square = 81
Page 17 of 21
or
Shows or implies a correct method in which the only error is to use an
incorrect value for the shorter horizontal side of rectangle A
eg
•
12 ÷ 2 = 8 (error), 8 + 3 = 11
11 – 2 = 9, 8 × 9 = 72
•
Answer: 20
1
(U1)
[2]
M6.
Gives a correct explanation
The most common correct explanations:
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
Page 18 of 21
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
U1
[1]
Page 19 of 21
54x 2
M7.
Do not accept: unsimplified expression or unconventional
notation
eg
•
9x2 × 6
•
•
9x2 + 9x2 + 9x2 + 9x2 + 9x2 + 9x2
54xx
[1]
M8.
Gives a correct value for the area of the rectangle
eg
• 54
• 5400
U1
Shows the correct unit for their area
eg
• cm² [with 54]
• mm² [with 5400]
!
Area incorrect or omitted, but units given
If the mark for their correct area has not been awarded, condone
cm² seen for the second mark
1
[2]
M9.
Gives both correct lengths, ie
x = 10 and y = 3.9 or equivalent
2
or
Gives y = 3.9 or equivalent
or
Gives the two values transposed, ie
x = 3.9 or equivalent and y = 10
or
Shows a complete correct method with not more than one computational error
eg
• x = 10, 10 – 6.1 = 4.9 (error)
• 4 × 6.1 = 24.4, 40 – 24.4 = 16.6 (error)
16.6 ÷ 4 = 4.15, 4.15 + 6.1 = 10.25
• 40 ÷ 4 = 20 (error)
20 – 6.1 = 13.9
1
[2]
Page 20 of 21
Page 21 of 21