Transcript Slide 1

Mathematical
Similarity
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S4 Credit
Scales (Representative Fractions)
Scale Drawings
Working Out Scale Factor
Similar Triangles 1
Similar Triangles 2
Similar Triangles 3 with Algebra
Similar Figures
Scale Factor in 2D (Area)
Surface Area of similar Solids
Scale Factor 3D (Volume)
Starter Questions
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S4 Credit
Q1.
Factorise
Q2.
Find x and y when
Q3.
f(x) = x2 – 3x Find f(-2)
Q4.
Calculate
Sunday, 26 April 2020
x 2  5x  6
3x + y = 10
x - 3y = -10
3
1
1 2
7
6
Scales
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Learning Intention
1.
To explain the term scale
in the context of a map.
Success Criteria
1. To understand the term
scale.
2. Calculate the real life and
map distances.
26-Apr-20
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Scales
S4 Credit
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In order to make sense of a map or scale diagram
the representative fraction (scale factor) must be known.
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Scales
S4 Credit
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In order to make sense of a map or scale diagram
the scale factor must be known.
For this scale model
1 cm represents 1m
4 cm
This means for every 1 metre of the actual car,
1cm is drawn on the map.
26-Apr-20
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Scales
S4 Credit
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The scale of this drawing is
1cm = 5m
6cm
What is the actual length
of the tree ?
6 x 5 = 30m
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Scales
S4 Credit
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The scale of this drawing is
1cm = 90cm
What is the actual length
of the bus in metres ?
5cm
5 x 90 =
450cm
4.5m
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Scales
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S4 Credit
Now try Ex 2.1
Ch 3 (page 51)
Odd questions
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Starter Questions
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S4 Credit
Q1.
Find the roots for the quadratic to 1 decimal place
x  5x  3  0
2
Q2.
Solve the trig equation
Q3.
Find
Q4.
The sun is 92 million miles away.
Write his number in standard form
Sunday, 26 April 2020
3
3cos x – 2 = 0
64
Scaled Drawings using Bearings
S4 Credit
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Learning Intention
1.
26-Apr-20
Success Criteria
To explain how to
construct a scale drawing
using bearings.
1. Construct an accurate
scale drawing.
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Scaled Drawings using Bearings
S4 Credit
Make an accurate scale drawing of this sketch.
N
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N
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50o
15km
50o 15km
1cm represents 3km
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Scaled Drawings using Bearings
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S4 Credit
Make an accurate drawing of the plane journey
N
N
N
N
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120o
45o
8km
120o
12km
45o
8km
12km
1cm represents 2km
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Scale Drawings
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S4 Credit
Now try Ex 3.1
Ch 3 (page 54)
Odd questions
26-Apr-20
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Starter Questions
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S4 Credit
Q1.
Calculate
5.91 + 3.2 x 20
Q2.
Find the max point for f(x) = (x + 1)(x – 2)
Q3.
Rearrange to find the gradient and y intercept.
2y + 5x - 1 = 0
Sunday, 26 April 2020
Working out
Scale Factor
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S4 Credit
Learning Intention
Success Criteria
1. To work out a suitable
scale for a given set of
data.
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1. Calculate scales for given data.
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Working out
Scale Factor
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S4 Credit
Give one distance from the map and the corresponding
actual distance we can work out the scale of the map.
Example :
The map distance from Ben Nevis to Ben Doran is 2cm.
The real-life distance is 50km.
What is the scale of the map.
Map
2
1
Scale Factor 1
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

:
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Real Distance
50km
50 ÷ 2 = 25km
250 000
Working out
Scale Factor
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S4 Credit
Example :
The actual length of a Olympic size swimming pool is 50m.
On the architect’s plan it is 10cm.
What is the scale of the plan.
Plan
10
1
Scale Factor 1
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

:
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Real Distance
50m
5
500
Working out
Scale Factor
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S4 Credit
Now try Ex 4.1
Ch 3 (page 55)
Odd questions
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S4 Credit
Q1.
Solve
4sin x – 1 = 0
Q2.
Find the mini point for f(x) = (2 + x)(4 – x)
Q3.
Factorise
Q4.
Find the mean and standard deviation for the data
4d2 – 100k2
4
Sunday, 26 April 2020
10
2
3
6
Similar Triangles 1
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to similar
triangles.
1. Understand how the scale
factor applies to similar
triangles.
2. Solve problems using scale
factor.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Conditions for similarity
Two shapes are similar only when:
•Corresponding sides are in proportion and
•Corresponding angles are equal
All rectangles are not similar to one
another since only condition 2 is true.
If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B
8 cm
Not to scale!
5 cm
A
Scale factor = x2
Note that B to A
would be x ½
16 cm
10 cm
B
If two objects are similar then one is an enlargement of the other
The rectangles below are similar:
Find the scale factor of enlargement that maps A to B
8 cm
Not to scale!
5 cm
A
Scale factor = x1½
Note that B to A
would be x 2/3
12 cm
7½ cm
B
If we are told that two objects are similar and we can find the
scale factor of enlargement then we can calculate the value of an
unknown side.
13½ cm
The 3 rectangles are similar. Find the
unknown sides
Not to scale!
x cm
3 cm
y cm
A
8 cm
B
C
24 cm
Comparing corresponding sides in A and B: 24/8 = 3 so x = 3 x 3 = 9 cm
Comparing corresponding sides in A and C: 13½ /3 = 4½ so y = 4 ½ x 8 = 36 cm
If we are told that two objects are similar and we can find the
scale factor of enlargement then we can calculate the value of an
unknown side.
y cm
The 3 rectangles are similar. Find the
unknown sides
Not to scale!
7.14 cm
2.1 cm
A
26.88
cm
5.6 cm
B
C
x cm
Comparing corresponding sides in A and B: 7.14/2.1 = 3.4 so x = 3.4 x 5.6 = 19.04 cm
Comparing corresponding sides in A and C: 26.88/5.6 = 4.8 so y = 4.8 x 2.1 = 10.08 cm
Similar Triangles
Similar triangles are important in mathematics and their
application can be used to solve a wide variety of problems.
The two conditions for similarity between shapes as
we have seen earlier are:
•Corresponding sides are in proportion and
•Corresponding angles are equal
Triangles are the exception to this rule.
only the second condition is needed
Two triangles are similar if their
•Corresponding angles are equal
Similar Triangles
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S4 Credit
These two triangles are similar since
they are equiangular.
65o
70o
45o
70o
45o
50o
50o
55o
75o
These two triangles are similar since
they are equiangular.
If 2 triangles have 2 angles the
same then they must be equiangular
= 180 – 125 = 55
Scale factors
S4 Credit
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Enlargement Scale factor?
ESF =
8
8cm
3
=
2
Reduction Scale factor?
5cm
RSF =
12
12cm
2
=
3
Can you see the relationship
between the two scale factors?
7.5cm
Scale factors
S4 Credit
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Find a given ESF = 3
ESF = 3 =
9cm
b
5cm
a
9
27cm
a
By finding the RSF
Find the value of b.
1
b
5
RSF =
=
=
3
15 15
15cm
Finding Unknown sides (1)
20 cm
b
c
6 cm
12 cm
18 cm
Since the triangles are equiangular they are similar.
So comparing corresponding sides.
b
6

18
12
6x 18
b
 9 cm
12
c
12

20 18
c 
20x 12
 13.3 cm
18
Similar Triangles 1
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S4 Credit
Now try Ex 5.1
Ch 3 (page 58)
Q1 , Q3 and Q4 only
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S4 Credit
Q1.
Find the roots to 1 decimal place
1  7x  x 2  0
Q2.
Q3.
A freezer is reduced by 20% to £200 in a sale.
What was the original price.
Calculate
Sunday, 26 April 2020
3
1
3 1
4
3
Similar Triangles 2
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to similar
triangles with algebraic
terms.
1. Understand how the scale
factor applies to similar
triangles with algebraic terms.
2. Solve problems using scale
factor that contain algebraic
terms.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Determining similarity
A
B
Triangles ABC and DEC are
similar. Why?
C
Angle ACB = angle ECD (Vertically Opposite)
Angle ABC = angle DEC (Alt angles)
D Angle BAC = angle EDC (Alt angles)
E
Since
ABC
ABDE
is similar to
BCEC
DEC
we know that corresponding
sides are in proportion
ACDC
The order of the lettering is important in order to show which
pairs of sides correspond.
If BC is parallel to DE, explain why
triangles ABC and ADE are similar
A
B
Angle BAC = angle DAE (common to
both triangles)
C
Angle ABC = angle ADE (corresponding
angles between parallels)
Angle ACB = angle AED (corresponding
angles between parallels)
E
D
A line drawn parallel to any side of a triangle produces 2 similar triangles.
A
A
B
D
C
Triangles EBC and EAD are similar
B
E
D
C
E
Triangles DBC and DAE are similar
The two triangles below are similar:
Find the distance y.
C
B
20 cm
y
A
5 cm E
45 cm
D
AE
5
y
RSF =
=
=
AD
50
20
5x20
y=
= 2 cm
50
The two triangles below are similar:
Find the distance y.
Alternate
approach to
the same
problem
C
B
20 cm
y
A
5 cm E
45 cm
D
AD
50
20
ESF =
=
=
AE
5
y
5x20
y=
= 2 cm
50
In the diagram below BE is parallel to CD and all
measurements are as shown.
(a) Calculate the length CD
(b) Calculate the perimeter of the Trapezium EBCD
A
6m
4.5 m
B
3 cm
C
(a ) SF=
A
4.8 m
E
4m
D
8 cm
10 CD

6 4.8
4.8  10
CD 
 8 cm
6
10 m
7.5 cm
(b ) SF=
10 AC

6
4.5
C
4.5  10
AC 
 7.5 cm
6
8 cm
D
So perimeter
= 3 + 8 + 4 + 4.8
= 19.8 cm
Similar Triangles 2
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S4 Credit
Now try Ex 5.2
Ch 3 (page 60)
Q2, Q4, Q5 Q6
Last two questions in each
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Starter Questions
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S4 Credit
Q1.
Find the roots to 1 decimal place
2  9x  x 2  0
Q2.
Q3.
A 42” TV is reduced by 40% to £480 in a sale.
What was the original price.
Calculate
Sunday, 26 April 2020
3
3
4
1
1
3
Similar Triangles 3
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to similar
triangles with harder
algebraic terms.
1. Understand how the scale
factor applies to similar
triangles with harder algebraic
terms.
2. Solve problems using scale
factor that contain harder
algebraic terms.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
In a pair of similar triangles the ratio of the corresponding sides is
constant, always producing the same enlargement or reduction.
Find the values of x given that the triangles are similar.
A
Corresponding sides are in proportion
AB
AC
3m
E
D
4m
6m
B
1.5
x
C
3
3+x
=
EB
DC
=
4
6
3x 6
= 4(3 + x)
18
= 12 + 4x
4x
=
6
x
=
1.5
In a pair of similar triangles the ratio of the corresponding sides is
constant, always producing the same enlargement or reduction.
Find the values of x given that the triangles are similar.
Corresponding sides are in proportion
T
Q
3
PQ
ST
x
11.25
R
5
18
P
S
=
PR
PT
3
5
18 - x
x
=
3x
= 5(18 - x)
3x
= 90 - 5x
8x
=
90
x
=
11.25
Similar Triangles 3
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S4 Credit
Now try Ex 6.1
Ch 3 (page 62)
Q1 to Q5
26-Apr-20
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Starter Questions
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S4 Credit
Q1.
Expand (x + 2) (x2 + 3x + 2)
Q2.
A yacht has increase by 10% to £110 000 in a year.
Find the price before the increase.
Q3.
Find the gradient and where the line 2y - 6x – 3 = 0
cuts the x-axis.
Q4.
Calculate
Sunday, 26 April 2020
7 7

9 9
Similar Figures
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to other
similar figures.
1. Understand how the scale
factor applies to other similar
figures.
2. Solve problems using scale
factor.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Scale Factor applies to ANY SHAPES that are
mathematically similar.
2cm
z
1.4 cm
7 cm
y
3 cm
15 cm
6 cm
Given the shapes are similar, find the values y and z ?
15
Scale factor = ESF =
=5
3
y is 2 x 5 = 10
1
Scale factor = RSF =
= 0.2
5
z is 6 x 0.2 = 1.2
Scale Factor applies to ANY SHAPES that are
mathematically similar.
b
7.5 cm
a
4 cm
3 cm
8 cm
2cm
10 cm
Given the shapes are similar, find the values a and b ?
4
Scale factor = RSF =
= 0.4
10
a is 8 x 0.4 = 3.2
10
Scale factor = ESF =
= 2.5
4
b is 2 x 2.5 = 5
Similar Figures
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S4 Credit
Now try Ex 7.1
Ch 3 (page 66)
Q1 to Q4
26-Apr-20
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Starter Questions
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S4 Credit
Q1.
Find the mean and standard deviation for the data
6, 5, 3, 1, 5
Q2.
Solve the equation
2 cos xo  1  0
Sunday, 26 April 2020
Area of Similar Shape
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to area.
1. Understand how the scale
factor applies to area.
2. Solve area problems using
scale factor.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Area of Similar Shape
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S4 Credit
Draw an area with
sides 2 units long.
Draw an area with sides 4 units long.
2
y
2
x
Area = 2 x 2 = 4
4
y
x
4 Area = 4 x 4 = 16
It should be quite clear that second area is four times the first.
The scaling factor in 2D (AREA) is (SF)2.
For this example we have this case SF = 2
(2)2 = 4.
Another example of similar area ?
Work out the area of each shape
and try to link AREA and SCALE FACTOR
2cm
Connection ?
6cm
4cm
12cm
Small Area = 4 x 2 = 8cm2
Large Area = 12 x 6 = 72cm2
Scale factor = ESF =
12
=3
4
Large Area = (3)2 x 8 = 9 x 8 = 72cm2
Example
The following two shapes are said to be similar.
If the smaller shape has an area of 42cm2.
Calculate the area of the larger shape.
3cm
Working
4cm
4
ESF =
3
4
So area S.F =  
3
Area of
2nd
2
4
shape =  
3
2
X
16
42 =
9
X
42 = 74.67cm2
Questions
1.
2.
3.
4.
Area of Similar Shape
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S4 Credit
Now try Ex 8.1
Ch 3 (page 68)
Even Numbers Only
26-Apr-20
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Starter Questions
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S4 Credit
Q1.
Find the mean and standard deviation for the data
2, 3, 5, 1, 2
Q2.
Solve the equation
2 cos x  3  0
o
Sunday, 26 April 2020
Surface Area
of Similar Solids
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to surface
area.
1. Understand how the scale
factor applies to surface area.
2. Solve surface area problems
using scale factor.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
The same rule applies when dealing with Surface Area
Example : Work out the surface area of the larger cuboid.
3cm
4cm
2cm
12cm
Surface Area of small cuboid : 2(2x3) + 2(4x3) + 2(2x4) = 52cm2
Surface Area of large cuboid :
12
ESF =
=3
4
(3)2 x 52 = 468 cm2
Surface Area
of Similar Solids
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S4 Credit
Now try Ex 9.1
Ch 3 (page 70)
Odd Numbers Only
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Starter Questions
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S4 Credit
Q1.
Draw a box plot to represent the data.
2, 4, 3, 1, 2
Q2.
Find the
coordinates
where the line
and curve meet.
Sunday, 26 April 2020
Volumes of Similar Solids
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S4 Credit
Learning Intention
Success Criteria
1. To explain how the scale
factor applies to volume.
1. Understand how the scale
factor applies to 3D – volume.
2. Solve volume problems using
scale factor.
26-Apr-20
Created by Mr. Lafferty Maths Dept.
Volumes of Similar Solids
S4 Credit
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Draw a cube with sides 2 units long.
Draw a cube with sides 4 units long.
2
y z
x
2
2
4
Volume = 2 x 2 x 2 = 8
y
z
4
x
Scale factor = ESF =
4
2
4
Volume = 4 x 4 x 4 = 64
=2
Using our knowledge from AREA section, (SF)2.
For VOLUME the scale factor is (SF)3 = (2)3 = 8
Another example of similar volumes ?
Work out the volume of each shape
and try to link volume and scale factor
4cm
2cm Connection ?
3cm
2cm
V = 3 x 2 x 2 = 12cm3
Scale factor = ESF =
6cm
4cm
V = 6 x 4 x 4 = 96cm3
6
=2
3
Large Volume = (2)3 x 12 = 8 x 12 = 96cm3
Given that the two boxes are similar, calculate the
volume of the large box if the small box has a volume
of 15ml
2cm
6cm
6
ESF =
= 3
2
So volume of large box =
3  15
3
= = 405 ml
Example
ESF =
So volume of large jug =
 
30  3
20 2
3 3
2
 0.8 =
27
8
 0.8 = 2.7 litres
(a) SD : B = 4 : 3
3
(b) RSF =
4
B 5

(c) RSF =
M 6
SD : M = 8 : 5
Area B  

3 2
4
 1600  900cm2
VolumeB  

5 3
6
 2700  1562.5cm3
ESF =
So volume of large jug =
 
12
8
3 3
2

 40
3
2
=
27
8
 40 = £1.35
(SF )3 = 10
1
 10
SF = 2.1544
So surface area ratio = (SF)2 =
2.1544
2
= 4.64
Ratio of their surface area is 1 : 4.6 (to 1 d.p.)
Volumes of Similar Solids
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S4 Credit
Now try Ex 10.1
Ch 3 (page 72)
26-Apr-20
Created by Mr. Lafferty Maths Dept.