Sample Schemes of Work and
Lesson Plans
Engineering
OCR Level 3 Principal Learning in Engineering: H811
Unit F563: Mathematical techniques and applications for engineers
This Support Material booklet is designed to accompany the OCR Engineering Diploma
specification for teaching from September 2008.
© OCR 2009
Contents
Contents
2
Sample Scheme of Work: OCR Level 3 Principal Learning in Engineering: H811 Unit
F563 Mathematical techniques and applications for engineers
3
Sample Lesson Plan: OCR Level 3 Principal Learning in Engineering: H811 Unit
F563 Mathematical techniques and applications for engineers
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23
Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
60 GLH
Topic
Mathematical techniques and applications for engineers
Unit Overview: This unit will be externally assessed by a two-hour written examination paper. In Section A there will be fifteen short answer questions and
in Section B eight long answer questions. The learner is expected to answer all Section A questions and choose three questions from Section B.
It is intended that learners will develop knowledge and understanding of:
 Algebra
 Geometry and Trigonometry
 Calculus
 Statistics
From this knowledge and understanding of the theory will come the development of the ability to solve problems in the context of engineering. This unit has
not only been written to ensure success in the Diploma in Engineering but as preparation for the exciting world of engineering which lies ahead of the
learner.
At first the content of this unit looks extensive but it needs a closer inspection to realise that everything that a presenter needs to know when teaching the
unit is absolutely provided in the assessment criteria.
It is intended that the basic facts are taught, ideally by a mathematics specialist presenter together with an abundance of worked examples. The learner
having been provided with sufficient mathematical tools should then, possibly with some assistance, be able to carry out theoretical calculations. It is
essential to present wherever possible mathematical techniques in the context of practical engineering examples.
Personal, Learning and Thinking Skills (PLTS). Although not mapped because this is an externally assessed unit there are ample opportunities for
learners to demonstrate and develop a number of their PLT’s during the learning experiences provided within this unit. It is the responsibility of a Centre to
ensure that a learner has sufficiently covered the requirements for the development of PLTS.
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3 of 26
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Algebra
Topic outline
Suggested teaching and homework
activities
Suggested resources
Points to note
Algebraic brackets


1. 4(5 + x) = 20 + 4x

Binomial expressions

Algebraic factorisation
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State and apply the rule that when
algebraic brackets are removed, every
term within the bracket is multiplied by the
quantity outside the bracket
State and apply the rule for a binomial
expression, which means that the contents
of one bracket are multiplied by the
contents of a second bracket
State and apply the rule that a factor of an
algebraic expression is a letter or number
that can be taken from the expression, the
remainder of the expression being placed
in brackets
State and apply the rule for factorisation
that gives a result with two brackets
State and apply the rule that the principle
of the lowest common multiple (LCM) is
applied





Pratley, J.B. (1983)
Mathematics Level 1
McGraw-Hill
Pratley, J.B. (1985)
Mathematics Level 2
McGraw-Hill
Yates, J.C. (1993)
National Engineering Mathematics
The Macmillan Press Ltd
Bird, B.O. (1995)
Early Engineering Mathematics
Newnes
Greer, A and Taylor, G.W.
(1989)
Mathematics for Technicians
Stanley Thornes (Publishers) Ltd
Elen, L.W.F. (1971)
Mathematics
Thomas Nelson and Sons
2. (x + 2)(x + 5)
= x2 + 7x + 10
3. abc + ade = a(bc + de)
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Algebra
Topic outline
Suggested teaching and homework
activities
Suggested resources
Points to note
Algebraic fractions


4. x2 + 5x + 6
= (x + 2)(x + 3)


Algebraic equations


Simultaneous equations


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Engineering Diploma
State and apply the rule that an equation is
a statement that two algebraic expressions
are equal and the process of finding the
unknown is called solving the equation
Solve simultaneous equations by the
elimination method and the substitution
method
State that transposition of formulae means
that we change the subject of a given
formula, using the same rules as for the
solution of equations
Transpose formulae that are used in
engineering
Transpose formulae used in engineering
that contain a root or a power
Transpose formulae used in engineering
that contain two like terms
Solve quadratic equations using the
factorisation, completing the square or
formula method
Bleau, B.L. (2003)
Forgotten Algebra: a self teaching refresher
course
Barron’s
 Sterling, M.J. (2009)
Linear Algebra for Dummies
John Wiley and Sons
 Young, C.Y. (2008)
College Algebra
John Wiley and Sons
 Ashton, C.H. (2009)
College Algebra
Bibliolife
Useful websites can be found at:
www.algebra.com
www.algebrahelp.com
www.sosmath.com
www.kutasoftware.com
Algebra mathematics games:
5. (x + 2)/ 5 + (x + 4)/3 gives a common multiple
of 15 leading to a solution of (8x + 26)/15
6. 5(x – 3) – 7(6 – x)
= 12 – 3(8 – x)
leading to a solution that x = 5
7. Solve practical engineering problems using
simultaneous equations in the area of:
mechanics using the distance travelled by an
object is s = ut + ½ at2
electrical engineering using kirchhoff’s laws
fluid mechanics using p1 = rg( d – d1) and p2 =
rg( d – d2)
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5 of 26
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Transposition of formulae
Topic
Algebra
Suggested teaching and homework
activities

State and apply the rule that will split up a
single fraction whose denominator has
factors into two partial fractions
Suggested resources
Points to note
www.onlinemathlearning.com
Mathematics worksheets:
www.EdPlace.co.uk
8. R = R1 + R2 find R2
9. F = ma find m
10. PV =RT find T
11. V = u + at find t
Video results for algebra:
www.youtube.com
12. E = mv2/2g find v
13. T = 2 π √(K2/gh) find K
14. Mv + mu = MV + mU
find M or m
Quadratic equations
Partial fractions
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15. (i) Bending moment (M) of beams M = 0.3x2
+ 0.35x – 2.6 (ii) displacement S of a particle S
= 1.9t + 4.3t2 (iii) fabrication of steel boxes
when the volume of the box is 2(x – 4)(x – 4)
and “x” is a required dimension
16. 5/(x2 + x – 6) resolves into
1/(x – 2) – 1/(x + 3)
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Geometry And Trigonometry
Topic outline
Suggested teaching and homework
activities
Degrees and radians


Length of arc of a circle



Area of sector of a circle
Solution of a right angled
triangle
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
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Engineering Diploma
Define the terms angle and radian
State and apply the formula that 1 radian =
3600/2π degrees and that 1 degree =
2π/360 radians
State and apply the formula that the length
of an arc of a circle S = xr radians and
S = πx0r/180 where r is the radius of the
circle and x0 is the subtended angle
State and apply the formula that the area of
a sector A = πr2 x0/180
Explain what is meant by the term “solution
of a triangle”
State and apply the formula that the sine of
an angle is the ratio of the side opposite to
the hypotenuse in a right angled triangle
State and apply the formula that the cosine
of an angle is the ratio of the side adjacent
to the hypotenuse in a right angled triangle
Suggested resources
Points to note







Pratley, J.B. (1983)
Mathematics Level 1
McGraw-Hill
Pratley, J.B. (1985)
Mathematics Level 2
McGraw-Hill
Yates, J.C. (1993)
National Engineering Mathematics
The Macmillan Press Ltd
Bird, B.O. (1995)
Early Engineering Mathematics
Newnes
Greer, A and Taylor, G.W.
(1989)
Mathematics for Technicians
Stanley Thornes (Publishers) Ltd
Elen, L.W.F. (1971)
Mathematics
Thomas Nelson and Sons




1. A wheel rotating at the rate of 54
revolutions per minute. Determine the
angular speed in radians per minute
2. A shaft rotating at 100 revolutions per
minute. Express this in radians per second
3. A water main is 500 mm diameter, and is
more than half full of water. The angle
subtended at the centre by the horizontal
surface of the water is 2/3 π radians.
Calculate (a) the length of the surface that is
wet (b) the depth of the water
4. The braking surface of a brake lining is in
the form of an arc of a circle of radius 120
mm, and the angle subtended by the arc is
1200. Calculate the length of the braking
surface
5. A triangular template is being made for
use in a workshop. The hypotenuse of the
right angled triangle is 150 mm. One angle
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Suggested teaching and homework
activities

Graphs of trigonometrical
functions


Values of sin x, cos x and tan
x for angles between 00 and
3600

Sine rule

Cosine rule
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Geometry And Trigonometry

State and apply the formula that the tangent
of an angle is the ratio of the opposite side
to the adjacent side in a right angled triangle
Plot a graph of y = sin x, y = cos x and y =
tan x for a range of angles from 0o to 3600
Determine the sine, cosine and tangent of
any angle between 00 and 3600 from a
graph or by using a calculator
State and apply the sine rule for a non-right
angled triangle that a/sin A = b/sin B = c/sin
C where A, B and C are angles within the
triangle and a, b and c are the lengths of the
three sides
State and apply the cosine rule for a nonright angled triangle that a2 = b2 + c2 –
2bcCos A ,
b2 = a2 + c2 – 2acCosB and c2 = a2 + b2 –
2abCosC where A, B and C are angles
within the triangle and a, b, and c are the
lengths of the three sides
Suggested resources

Sterling, M.J. (2005)
Trigonometry for Dummies
John Wiley and Sons
 Sterling, M.J. (2005)
Trigonometry workbook for Dummies
John Wiley and Sons
 Abbott, P et al (2003)
Teach yourself trigonometry
Hodder
 Moyer, R. and Ayres, F. (2008)
Schaum’s outline of trigonometry
McGraw Hill
 Ross, D.A. (2009)
Master Math – Trigonometry
Cengage Learning
Useful websites can be found at:
 www.clarku.edu/˜djoyce/trig
 www.univie.ac.at/future.
 media/moe/galerie/trig
Points to note
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is 230. Determine the value of the other
angle and the lengths of the other two sides
6. Surveying – From two points A and B,
100 m apart, in a straight line with a tower,
the angles of elevation of the top of the
tower are 20 and 350
respectively. Determine the height of the
tower
7. Two pulleys A and B of diameter 100 and
70 mm respectively, are connected by an
open belt. The centre distance between the
pulleys is 140 mm. Calculate (a) the length
of the belt assuming it does not sag (b) the
rotational speed of pulley B if pulley is
rotating at 210 rev/min
8. A single phase load takes a current of 5
amperes on a 240 volt 50 hertz supply at a
lagging power factor of 0.8. Calculate the
value of a capacitor to correct the power
factor to unity
9. An engineering factory is supplied with a
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Area of a triangle
Topic
Suggested teaching and homework
activities
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Complementary angles

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Ratios of 300, 450 and 600
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Engineering Diploma
Geometry And Trigonometry


For a non right-angled triangle state and
apply the formulae for the area of a triangle
(a) Area = ½bh where b is the length of the
base and h is the perpendicular height
(b) Area = ½bc sin A where b and c are the
lengths of two sides and A is the angle
opposite the third side
(c) Area = √ [s(s - a)(s - b)(s – c)] where a,
b, and c are the lengths of the sides of the
triangle and
s = ½(a + b + c)
State and apply the formula that (a) sin A =
cos (90 – A) (b) cos A = sin (90 – A)
State and apply that for a 450 right angle
triangle (a) tan 450 = 1 (b) sin 450 = 1/ √2
and
(c) cos 450 = 1/ √2
State and apply that for a 600 equilateral
triangle (a) sin 600 = √3/2 (b) cos 600 = ½
Suggested resources







www.emteachline.com
www.tutor.com/subject/
trigonometry
www.mathsrevision.net/
alevel/pages
www.mathisfun.com/sine-cosinetangent.html
www.a-levelmathstutor.com/sin-cosrules.php
Video results for trigonometry:
 www.mathstu.com
 www.math-e-matics.co.uk
Points to note




2000 volt single phase 50 hertz supply, and
takes a load current of 400 amperes at 0.6
power factor lagging. Calculate the value of
a capacitor that will alter the power factor to
(a) unity, (b) 0.9 lagging and (c) 0.9 leading
10. An alternating e.m.f. is represented by v
= 25 sin x. Determine the value of v when x
equals (a) 300 ,(b) 600, (c) 900 , (d) 1800 (e)
2100, and (f) 2700
11. The instantaneous value of an
alternating current is given by i = 5 sin
314.2t amperes. Determine (a) peak value
(b) frequency (c) periodic time (d) the
current after
12 milliseconds, and (e) sketch a sine wave
showing the position of the current
calculated in part (d)
13. An oscillating mechanism has a
maximum displacement of 3 m and a
frequency of 60 Hz. At time t = 0 the
displacement is 75 m. Express the
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Suggested teaching and homework
activities

Reciprocal of sine, cosine and
tangent



Trigonometrical identities
Solid trigonometry with three
dimensional problems
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Geometry And Trigonometry


(c) tan 600 = √3 (d) sin 300 = ½ (e) cos
300 = √3/2 (f) tan 300 = 1/√3
State and apply that (a) the reciprocal of
sine is the cosecant (cosec), ie 1/sin x =
cosecant x (b) the reciprocal of cosine is the
secant (sec), ie 1/ cos x = secant x (c) the
reciprocal of tangent is cotangent (cot), ie1/
tan x = cotangent x.
Prove from first principles that (a) tan A =
sin A / cos A (b) cot A = cos A / sin A
(c) sin2 A + cos2 A = 1 (d) 1 + cot2A =
cosec2 A (e) tan2A + 1 = sec2A
State that a plane is a flat surface, defined
as a surface containing all of the straight
lines passing through a fixed point and also
intersecting a straight line in space
Solve problems in three dimensions by
drawing suitable triangles in different planes
and then calculating dimensions as required
Suggested resources
Points to note
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


displacement in the general form A sin (wt ±
x) where A = maximum displacement, w =
angular velocity, x = lagging or leading
angle in radians
14. A workshop 10 m wide has a span roof
which slopes at 300 on one side and 450 on
the other. Calculate the length of the roof
slopes
15. An e.m.f. of 20 volts lags another e.m.f.
of 40 volts by 600 . Determine the magnitude
of the total e.m.f. and its phase angle
16. A jib crane consisting of a tie AB and a
strut BC is fixed to two points CA fastened
to a vertical post. Point C is at ground level.
AB =10 m, BC = 15 m and CA = 12 m.
Sketch the jib crane and calculate the height
of the point B above ground level and the
horizontal distance of point B from the wall
17. In a reciprocating engine the lengths of
the crank AC and the connecting rod AB are
1 m and 4.8 m respectively. Calculate the
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Geometry And Trigonometry
Suggested teaching and homework
activities
Suggested resources
Points to note




= Innovative teaching idea
Engineering Diploma
value of angle ABC when the angle ACB is
860
18. Two alternating quantities are
represented by vectors of lengths 25 and
50, which act at the same point. The angle
between the vectors is 600. Calculate the
resultant quantity and the angle it makes
with the larger of the original quantities
19. The plan of a building plot is a
quadrilateral ABCD in which AB = 50 m, BC
= 60 m, CD = 32 m, DA =42 m and the
diagonal BD = 66 m. Calculate the area of
the plot of land
20. An iron casting has a uniform triangular
cross section with the following dimensions:
length of base of triangle = 400 mm, height
of triangle = 75 mm. Determine the cross
sectional area of the casting (iii) A copper
bar with a uniform cross section is in the
form of a regular hexagon with sides 15 mm
long. Calculate the cross sectional area of
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Geometry And Trigonometry
Suggested teaching and homework
activities
Suggested resources
Points to note







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the bar
21. Given that tan x = 4/3, find the value for
sin x and cos x
22. Using complementary angle, find a
value for (a) sin 400/cos 500 (b) sec
200/cosec 700
23. Show that the identity sin2x + cos2x = 1
is true when x = 1250
24. Prove that tan x = sec x/cosec x
25. Prove that sin2 x (cosec2x + sec2x) =
1/cos2x
26. A solid block of material is 67.5 mm x
45.8 mm x 23.6 mm. Determine (a) its
longest dimension and (b) the angle the
longest dimension makes with the base of
the block
27. A solid pyramid has a square base of
side 45 mm with a perpendicular height of
60 mm. Calculate the length of the diagonal
of the base (b) the length of one of the
sloping sides, and (c) the angle that the
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Geometry And Trigonometry
Suggested teaching and homework
activities
Suggested resources
Points to note

= Innovative teaching idea
Engineering Diploma
sloping side edge makes with the base
28. A television mast is held vertical by a
number of cables fastened to its top and
pegged to the ground. Two of these cables
are inclined at 380 to the vertical and
pegged 50 m from the bottom of the mast.
Calculate (a) the length of each cable (b)
the height of the mast (c) the distance
between the two pegs if the angle between
the two cables is 800
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Calculus
Topic outline
Suggested teaching and homework
activities
Suggested resources
Points to note
The gradient of a curve



1. Solve problems in the area of mechanics
using the formula for distance (s) travelled
by a body in t seconds is given by s = t3 –
5t2 – 3t. Express the velocity in terms of
time t. velocity = ds/dt and acceleration =
d2s/dt2

2. Given that the surface area S of a
cylindrical water tank is given by S = 2π(r2 +
6750/r). Calculate the dimensions of the
tank so that its total surface area is a
minimum
3. Given that an alternating voltage is given
by v = 20 sin 50t where v is in volts and t in
seconds. Calculate the rate of change of
voltage for a given time
4. Laws (i) linear expansion l = loeab (ii)
tension in belts T1 = Toeua (iii) biological
Differentiation from first
principles

Differentiation of algebraic
functions


Maximum and minimum
turning points


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14 of 26
Determine gradients to a simple curve using
a graphical method eg y = a.x2 where “a” is
a constant and determine the gradient of a
curve using a numerical method
Derive dy/dx for the functions y = a.xn, n = 0,
n = 1, n = 2, n = 3 ….. from first principles
State and apply the rule to differentiate
simple algebraic functions
Given a graph of y = x3 – 3x – 1, identify the
maximum and minimum turning point and
then determine the co-ordinates of the
turning points by differentiating the equation
twice. If d2y/dx2 is positive for a value of x
the turning point is at a minimum value for y.
If d2y/dx2 is negative for a value of x the
turning point is at a maximum value for y
Draw a graph and then derive the
differentiation of sin.x and cos.x





Pratley, J.B. (1983)
Mathematics Level 1
McGraw-Hill
Pratley, J.B. (1985)
Mathematics Level 2
McGraw-Hill
Yates, J.C. (1993)
National Engineering Mathematics
The Macmillan Press Ltd
Bird, B.O. (1995)
Early Engineering Mathematics
Newnes
Greer, A and Taylor, G.W.
(1989)
Mathematics for Technicians
Stanley Thornes (Publishers) Ltd
Elen, L.W.F. (1971)
Mathematics
Thomas Nelson and Sons


= ICT opportunity
Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Differentiation of sine and
cosine
Differentiation of the
exponential function
Differentiation of the
logarithmic function
Topic
Suggested teaching and homework
activities
Suggested resources
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

Indefinite integrals



= Innovative teaching idea
Engineering Diploma
Calculus
Differentiate functions of the form (i) y =
sin.x (ii) y = a.sin.x (iii) y = a.sin.bx (iv) y =
cos.x
(v) y = a.cos.x (vi) y = a.cos.bx (vii) y =
a.cos.x + b.sin x, where “a” and “b” are
constants
Define the differential properties of
exponential and logarithmic functions
State and apply the rule for differentiating
an exponential function
State and apply the rule for differentiating a
logarithmic function
Define indefinite integration as the reverse
process to differentiation and state that an
indefinite integral does not reveal a
calculated value
Recognise the symbol ∫ for integration
State and apply the rule to integrate simple
algebraic functions
eg If y = a xn then ∫ axn dx = a (xn + 1/ n + 1)
+ constant C. The term dx indicates the






Bigg, C. et al (2005)
Maths AS & A2
Longman
Ryan, M. (2003)
Calculus for Dummies
John Wiley and Sons
Gibilisco, S. (2008)
Calculus Know-it-all
McGraw-Hill
Mendelson, E. (2008)
Schaum’s Outline of Beginning Calculus
McGraw-Hill
Silverman, R.A. (2000)
Essential Calculus with Applications
Dover Publications
Thompson, S.P. (1998)
Calculus made easy
St. Martin’s Press
Lang, S. (2002)
A First Course in Calculus
St. Martin’s Press
Points to note


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
growth y = yoekt (iv) discharge of a capacitor
q =Qe-t/RC
(v) radioactive decay N = Noe-wt (vi)
atmospheric pressure p = poe-h/c (vii) decay
of current in an inductive circuit i = Ie-Rt/L
(viii) growth of current in a capacitive circuit
i = I(1 – e-t/RC)
5. If y = a.ebx then dy/dx = ba.ebx
6. If y = a.e-bx then dy/dx = -ba.e-bx
7. If y =ln.x then dy/dx = 1/x
8. If y = ln.3x then dy/dx = 1/x
9. If y = 4.ln.2x then dy/dx = 4/x
10. Integrate x3 + 3x2 + x with respect to x
11. Integrate x1.4 + 1/x3 with respect to x
12. Integrate 6x4 + √x with respect to x
13. Calculate a value for the definite integral
4
2∫ 6x dx
4
2
4
2
4
2∫ 6x dx = [ 6x /2 + C] 2 = [ 3x + C] 2. The
numerical values of 2 and 4 mean that x = 2
and x = 4.
When x = 4, integral =3x2 + C = 48 + C
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Suggested teaching and homework
activities
Definite integrals



Area under a curve
Calculus

variable that is the subject of the integration
process and n ≠ -1
State and apply the rule for a definite
integral
State that in all calculations for definite
integrals the constant C will disappear when
an upper and lower limit are given
State and apply that the interpretation of a
definite integral is that it represents the area
between the function f(x) and the x axis
between the limits given
State and apply that
(a) ∫ sin x dx = – cos x + C
(b) ∫ cos x dx = sin x + C
Suggested resources

Larsen, R.E. (2005)
Calculus
Houghton Mifflin
(Includes text-specific videos)
Useful websites can be found at:
 www. Calculus-without limits.com
 www. Understandingcalculus.com
 www.mathworld.wolfram.com
Points to note





Video results for calculus:
 www.a-levelmathstutor.com


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16 of 26
When x = 2, integral = 3x2 + C = 12 + C
So 2∫4 6x dx = (48 + C) – (12 + C) = 48 + C
– 12 – C = 36 ie 2∫4 6x dx = 36
14. Integrate 0∫2 4x dx
15. Integrate 0∫3 2x2 dx
16. Integrate 1∫2 3x + 2 dx
17. Integrate 1∫4 2x3 + x2 dx
18. Find the area between the curve y = x
and the x axis between the values x = 0 and
x =10
Equation: y = x
Area under the curve = 0∫10 x dx = [ x2/2 ]010
= 102/2 – 0 = 50 units
A check on y = x can be made by plotting a
graph of x against y
19. Find the area under the curve y = (x –
3)(x – 2) from x = 2 and x = 3
20. The brakes are applied to a train and the
velocity (v) at any time (t) seconds after
applying the brakes is given by v = (18 –
3.5t) metres per second. Calculate the
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Calculus
Suggested teaching and homework
activities
Suggested resources
Points to note
Integrals of sin x and cos x



= Innovative teaching idea
Engineering Diploma
distance travelled in 6 seconds if distance
(d) = t1∫t2 v dt
21. The force (F) newtons acting on a body
at a distance x metres from a fixed point is
given by
F = 4x + 3x2 . Calculate the work done (W) =
x2 F dx when the body moves from the
x1∫
position where x = 2 metres to that where x
= 5 metres.
22. Integrate the following with respect to x
(a) sin 2x (b) cos 3x (c) cos (2x +Ø)
23. Evaluate the following integrals all
between the limits of 0 and π/2 (a) sin x (b)
cos x (c) sin 3x
(d) cos 3x (e) 3 cos 5x (f) sin x + cos x (g) 4
sin 5x + 3 cos 4x (h) cos 4x – 5 sin x
(i) 2x3 + cos 4x – 3 sin 4x (j) sin 4x + 2 cos
6x + √x
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Statistics
Topic outline
Suggested teaching and homework
activities
Suggested resources
Points to note
Data Handling

Explain what is meant by the term “data
handling”


Histograms

Explain what is meant by the term
“histogram” and construct a histogram from
given data

Frequency polygon

Explain what is meant by the term
“frequency polygon” and construct a
frequency polygon from a histogram


Cumulative frequency

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Explain what is meant by the term
“cumulative frequency” and construct a
cumulative frequency curve from given data
Construct a table showing a tally diagram
and then draw a (a) histogram (b) frequency
polygon and (c) cumulative frequency
diagram




Pratley, J.B. (1985)
Mathematics Level 2
McGraw-Hill
Yates, J.C. (1993)
National Engineering Mathematics
The Macmillan Press Ltd
Bird, B.O. (1995)
Early Engineering Mathematics
Newnes
Elen, L.W.F. (1971)
Mathematics
Thomas Nelson and Sons
Bigg, C. et al (2005)
Maths AS & A2
Longman
Gonick,L. and Smith, W. (1993)
The cartoon guide to statistics
Harper
Takahshi, S. (2009)


Collection and analysis of data involves
populations and samples
1. The number of castings per box in a
sample of 21 boxes was as follows:
Number in box
70 71
72
73
74
75
Number of boxes
2
6
3
1
4
5
Draw a histogram and a frequency polygon
from the information.
2. The diameters of 30 components were
measured in millimetres with a micrometer,
with the following results:
5.8 6.2 6.0 6.2 5.9 6.1 5.9 5.7 6.1 5.5
5.8 5.9 6.2 6.1 6.0 6.0 5.9 6.0 6.0 5.9
6.0 6.1 5.9 6.1 6.2 6.3 6.3 6.3 6.3 6.2
Construct a table showing a tally diagram
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Statistics
Suggested teaching and homework
activities
Suggested resources
The manga guide to statistics
William Pollock
 Morrison, S.J. (2009)
Statistics for Engineers
John Wiley and Sons
 Jaising, L.R. (200)
Statistics for the utterly confused
McGraw-Hill
Useful websites can be found at:
 Statistic Lesson Plans
 LessonPlanet.com
 Minitab for Education
www.Minitab.com
 Teaching Statistics
www.rsscse.org.uk/ts/
 Statistics/Probability worksheets
www.teach-nology.com/
 worksheets/math/stats
= Innovative teaching idea
Engineering Diploma
Points to note


and then draw a histogram, a frequency
polygon and a cumulative frequency
diagram
3. The tensile strength for 15 sample of tin
are:
34.16 34.75 34.04 34.36 34.15 34.94
34.16 34.25 34.55 34.85 34.35 34.44
34.84 34.04 34.28
Determine the mean, mode and median
4. The diameter of 100 spindles was
measured giving the frequency distribution
shown in the table
Diameter:
10 10.2 10.4 10.6 10.8 11.00 11.2
Frequency:
4 8 15 28 32 10
3
Draw a cumulative frequency diagram and
from it determine the median
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Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic outline
Topic
Statistics
Suggested teaching and homework
activities
Suggested resources
Points to note

5. The values of mass obtained by weighing
200 components is shown.
Mass kg 93 94 95 96 97 98 99
Frequency 7 30 42 46 40 25 10

Explain and apply the following term to a set
of data (a) arithmetic mean (b) mode and (c)
median
Arithmetic mean, mode and
median
Percentiles and quartiles
= Innovative teaching idea
20 of 26

Explain and apply the following terms to a
set of data (a) percentiles (b) quartiles
Plot a cumulative frequency curve and find
the median and the upper and lower
quartiles
6. In a study exercise components being
assembled by a group of technicians were
timed in seconds as shown:
56
76
69
88
67
61
89
78
54
75
68 59 74 69 73 76 68 80 79 57
96 74 97 66 74 86 63 77 98 53
70 74 87 66 83 78 77 78 72
90 63 75 94 84 86 65 80 73 60
52
Construct a histogram and a frequency
polygon to represent the data. Determine
the (a) median (b) lower quartile and (c) the
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Engineering Diploma
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Statistics
Topic outline
Suggested teaching and homework
activities
Distribution curves

Standard deviation

Explain what is mean by the terms (a)
distribution curve (b) positive skew (c)
negative skew
Explain and apply the following terms to a
set of data (a) variance (b) standard
deviation
Suggested resources
Points to note


upper quartile
7. Calculate the standard deviation for a set
of test scores:
80% 70% 65% 40% 55% 50%
8. A random sample of components were
taken from a production line, measured on
dimensions nominally 6 +/- 0.5 units and put
into categories as follows:
Variation x
5.6
5.8
Frequency
f 2
8
Probability
= Innovative teaching idea
Engineering Diploma

Explain and apply the following terms to
data (a) probability (b) expectation (c)
dependent event without replacement (d)
independent event with replacement

6.0
6.2
6.4
19
15
6
Calculate the arithmetic mean and standard
deviation
9. Determine the probability of selecting at
random (a) a drill for use on bricks (b) a drill
for use in wood, from a box of drills
containing 25 brick drills and 40 wood drills
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21 of 26
Sample Diploma Scheme of Work
OCR Level 3 Principal Learning in Engineering: H811 Unit F563
Mathematical techniques and applications for engineers
Suggested
teaching time
15 hours
Topic
Statistics
Topic outline
Suggested teaching and homework
activities
Addition law of probability
Multiplication law of probability

Suggested resources
Points to note

State and apply the addition law of
probability and the multiplication law of
probability

= Innovative teaching idea
22 of 26
10. The probability of a resistor failing in one
year due to excessive temperature is 1/25,
due to excessive vibration is 1/30 and due
to excessive humidity is 1/55. Determine the
probabilities that over one year a resistor
fails due to excessive (a) temperature and
vibration (b) vibration or humidity
11. A batch of 30 castings contains 5 which
are defective. If a casting is chosen at
random and inspected for quality and then a
second casting is chosen at random,
determine the probability of having one poor
quality casting, both with and without
replacement
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Engineering Diploma
Sample Diploma Lesson Plan
OCR Level 3 Principal Learning in
Engineering: H811 Unit F563 Mathematical
techniques and applications for engineers
Algebra
OCR recognises that the teaching of this qualification above will vary greatly from school to school
and from teacher to teacher. With that in mind this lesson plan is offered, as a possible approach
but will be subject to modifications by the individual teacher.
Lesson length is assumed to be one hour.
Learning Objectives for the Lesson
Objective 1
Learners will be able to state and apply the rule that when algebraic brackets are
removed, every term within the bracket is multiplied by the quantity outside the
bracket
Objective 2
Learners will be able to state and apply the rule for a binomial expression, which
means that the contents of one bracket are multiplied by the contents of a second
bracket
Objective 3
Learners will be able to state and apply the rule for factorisation that gives a result
with two brackets
Recap of Previous Experience and Prior Knowledge
Learners have been introduced to the concept of algebra and are aware of:

algebraic addition

algebraic subtraction

algebraic multiplication

algebraic division
Content
Time
Content
10 minutes
Algebraic brackets
Verbal exposition and questioning
Worked examples.
1. Remove the brackets in the following:
(a) 4(5 + x) = 20 + 4x
(b) 3(2 – a) = 6 - 3a
(c) -2(8 + b) = -16 - 2b
(d) -5(x – 6) = -5x + 30
Engineering Diploma
23 of 26
Sample Diploma Lesson Plan
Time
Content
(e) 2( - 3x + 5) = -6x - 10
(f) -4(-x – 3) = 4x + 12
Activity.
1. Remove the brackets from the following:
(a) 5(3 + x)
(b) 6(4 – a)
(c) -3(5 + b)
(d) -2(x – 4)
Further activities can be found at:
Taylor, G. (2004)
Mathematics for Engineers
Thornes
Bird, J. (2005)
Basic Engineering Mathematics
Newnes
15 minutes
Binomial expressions
Verbal exposition and Questioning
Worked examples.
1. Multiply out and expand the following:
(a) (x + 2)(x + 3) = x2 + 5x + 6
(b) (x – 4)(x – 5) = x2 – 9x + 20
(c) ( x – 3)(x + 4) = x2 + x - 12
(d) (x + 3)(x – 6) = x2 – 3x - 18
(e) (2x – 5)(x + 2) = 2x2 –x - 10
Activity.
1. Multiply out and expand the following:
(a) (x + 4)(x + 2)
(b) (x – 3)(x – 6)
(c) ( x – 2)(x + 4)
(d) (x +3)(x – 6)
(e) (2x – 3)(x + 4)
Further activities can be found at:
Taylor, G. (2004)
Mathematics for Engineers
Thornes
24 of 26
Engineering Diploma
Sample Diploma Lesson Plan
Time
Content
Bird, J. (2005)
Basic Engineering Mathematics
Newnes
25 minutes
Algebraic factorization
Verbal exposition and Questioning
Worked example.
1. Factorize x2 + 6x + 8
By inspection x2 = (x)(x) and 8 = (1)(8) or (2)(4)
After trying each combination we find x2 + 6x + 8 = (x + 2)(x + 4)
Activity.
Factorize the following expressions:
(a) x2 + 5x + 6
(b) x2 + 7x + 12
(c) x2 + 3x – 10
(d) x2 - 6x + 8
Further activities can be found at:
Taylor, G. (2004)
Mathematics for Engineers
Thornes
Bird, J. (2005)
Basic Engineering Mathematics
Newnes
Consolidation
Time
Content
5 minutes
Quick fire questions on algebraic brackets and Binomial expressions
Class discussion – Has learning taken place?
5 minutes
5 minutes
Homework
Work can be found in:
Taylor, G. (2004)
Mathematics for Engineers
Thornes
Bird, J. (2005)
Basic Engineering Mathematics
Newnes
Engineering Diploma
25 of 26
Sample Diploma Lesson Plan
Time
Content
Tooley, M. & Dingle, L. (2007)
BTEC National Engineering – Unit 4
Newnes
Bird, J. (2005)
Basic Engineering Mathematics
Newnes
26 of 26
Engineering Diploma