BTEC National Level 3, HNC and HND (Y201 onwards) - Engineering Maths - UNIT02

General Engineering Mathematics, also known as Analytical Methods for Engineers / or Technicians
Covers the following unit codes;
C&G 401/439 / MCA LvI&II / H7K0 33 / H7K1 34 / 2288U / 21717P / 2314U / 2288U / D/101/9312 / F/101/1008 / F/101/1039 / A/600/0253 / A/601/1401

Mandatory Sections include;

0- Pre-requisites and Units

1.1- Mental and basic arithmetic in common use

1.2- Algebraic methods

2- Trigonometric methods/functions

3- Calculus

4- Statistics and probability

There are no optional units for this module.

Pre-requisites for this unit:
English- essay writing ability and speech, literacy, correct use of punctuation, verbs and adjectives, presentation skills, practical demonstrations, verbal reasoning, use of the telephone, library reference system, Harvard referencing system, organisational skills, self discipline and attendance, communication
Mathematics; geometry, algebra and trigonometry, core skills- application of number (see below).

Skills You Should Have for Engineering Science:

Below is the list of the skills you should be confident with before starting the FE Engineering Science course:

Mathematics skills;

See the value in maths: Why numeracy matters (video);

basic mensuration (units and measuring)
basic algebra (non-calculator), inc. transposition
simplifying algebraic expressions
laws of indices
expanding and factorising expressions (one term outside)
laws of indices for all rational exponents (positive, negative, fractions)
quadratic functions (non-calculator)
plotting graphs of quadratic functions
expanding and factorising quadratics (two brackets)
solving quadratic equations by factorising
solving quadratic equations using the formula
equations and inequalities (non-calculator)
solving simultaneous linear equations by elimination
solving simultaneous linear equations by substitution
solving linear inequalities
trigonometric functions (SOHCAHTOA)
sine rule (SIN)
cosine rule (COS)
use of tangent function (TAN)
using the sine rule to find missing sides and angles (SOH: Oranges Have Segments)
using the cosine rule to find missing sides and angles (CAH: Apples Have Cores)
using the tangent rule to find missing sides and angles (TOA: O.A.T. ... think Quaker Oats!)
using sine rule, cosine rule, trig ratios and pythagoras in problems

mathcentre, UK site (useful mathematics resources, common formulae sheets etc.)

Maths for ALL site - Maths from Elementry Level all the way through to pHD!

Science skills:
knowledge of practical physics
law of gravity
mass, acceleration/gravity and force relationships

BTEC Level 3



Higher National Engineering by Mike Tooley and Lloyd Dingle provides full coverage of the following core units of the Edexcel Foundation's new Higher National Certificate and Higher National Diploma schemes for Engineering.

Web-links; The sites listed in the table below will provide you with additional resources and support material:

Unit Code / Year
Section / Sub-Sec.

Learning Outcome









Technician Introduction and Syllabus Contents (inc. Health and Safety)

This unit includes, but is not limited to;
units, arithmetic, geometry, mensuration, algebra, power series, trigonometry, trigonometric and hyperbolic functions, partial fractions, calculus, differentiation, integration, statistics, probability, mathematical engineering problems and analytical methods.

Rachel Riley - How to improve your maths skills at home (video);

A student is expected, where a topic is only covered in part in class, to self study a minimum of 1-2 hours per unit, per week, on the remaining sub-topic areas (contained herein) to ensure full coverage of the units expectations, engineering council minimum specification requirements at level 3-5 and engineering institutions guidance on minimum subject requirements for EngTech level.
Students are reminded, in line with UKSPEC, to uphold the following four fundamental Ethical principles;
1. Honesty and integrity;
2. Respect for life, law, the environment and public good;
3. Accuracy and rigour;
4. Leadership and communication.
In addition, I add the following;
Question everything;
Don't always assume everything is 'correct';
Apply safe working practices- have a "Safety First" mindset;
Measure twice, Cut once!
Innovate, Conceptualize, Design, Develop, and Create;
Learn empathy and perseverance;
Avoid corrosive personalities- these people never "learn" or "change" (i've met a few in my time!).
Initiate Change- be an active change promotor/agent.
"We must learn by our mistakes, not make new ones";
Be prepared to argue somthing if YOU believe that it is the safe way and not the old way or those with the"we have always done it that way" attitude.
Prevention always overrules corrective action if it can be avoided in the first place (or prevent an accident).
Have a CAN DO approach.
Work as a Team, and help each other, within your peer group/ department at work- remember we have to get along with everyone.
... and above all;
Five Mantras; (1) Equality, (2) Opportunity, (3) Wisdom, (4) Freedom and (5) Love.

Health and Safety Law: The main principles of health and safety law in Great Britain are that:
1) Employers have to look after the health, safety and welfare of all their employees
2) Employees and the self employed have to look after their own health and safety;
3) Everyone has to take care of the health and safety of others, for example members of the public who may be affected by their work.
These principles are set out in the Health and Safety at Work etc Act 1974 (the HSW Act).



















Mental and basic arithmetic in common use (1.1) and Algebraic methods (1.2) (inc. Units and Units of Measure)

Mental arithmetic (no calculators) -ability to carry out mental calculations using: (1) addition (+), subtraction (-), multiplication (x) and division (÷), (2) time (t), (3) fractions (x/y), decimals and percentages (0.00x, %), (4) proportions involving fractions, decimals and percentages, (5) measurement, including money (£,p), distance (L, m), area (A, m^2) and volume (V, m^3) and (6) conversions (mph -> kph etc.) - ability to use conversion tables and unity factor method is important for Engineering Mathematics and Science
Conversion example: Degrees to radian measure.

Written arithmetic and written data- ability to interpret and use written data to: (1) identify trends correctly, (2) make comparisons in order to draw conclusions and (3) interpret numerical information accurately, sign (integer) rules, notation and precedence rules, BODMAS (orders of operations; Brackets, Order, Division, Multiplication, Addition, Subtraction), vulgar and decimal fractions, lowest common multiple (LCM) and highest common factor (HCF), ratios and constant of proportionality, significant figures and estimation techniques

Sign (integer) rules:
Addition and Subtraction;
(+) + (+) = + (same signs, add and keep the same sign)
(-) + (-) = -
(+) + (-) = + (different signs, subtract and keep larger sign)
(-) + (+) = -
Multiplication and Division;
(+) x (+) = + (same signs, opposite sign)
(-) x (-) = +
(+) x (-) = - (different signs, negative sign)
(-) x (+) = -

A vulgar fraction is a another name for a common fraction. A common fraction is one where the numerator and the denominator are both integers (not fractions) placed above and below a fraction bar. The use of fractions in common use is decreasing in favour of decimalisation, however their use on mathematics is still important and when dealing with imperial units of measure.

Arithmetic problems, which include: (1) time, (2) money, (3) fractions, decimals and percentages, (4) proportion and ratio, (5) measurements (eg distance, area), (6) conversions, (7) averages (including mean, median and mode), (8) range, and (9) using simple formulae.

Basic calculus concepts; familiarity with the concept of the differential and integral calculus (from A-level or level 2 or 3 study, National level) i.e. 'the change in y over x', differentiate simple polynomial and trigonometric functions using basic rules, integrate simple polynomial and trigonometric functions using standard rules

A&P, General Mathematics, King Schools, refresher;

NB: students NOT meeting the above standard need to be enrolled onto an appropriate bridging study or A-level mathematics top-up course.

Algebraic methods- Basic arithmetic; define numerator and denominator of a fraction, define vulgar fractions, mixed number and improper fraction, use common denominator in addition and subtraction of fractions, multiplies and divides fractions, use cancellation to reduce fractions, apply cancellation to manipulation of fractions, simplifies fractions involving the use of brackets, solve problems involving fractions, define decimal fractions, convert vulgar fractions to decimal fractions and visa-versa, add and subtract algebraic fractions using common denominator, add and subtract decimal fractions, multiply and divide decimal fractions, multiply binomial factors, identify a recurring decimal fraction, divide quadratic function by linear factor, round off a decimal fraction to a specified number of places (max. 4 s.f.), state a decimal number correct to a specified number of significant figures (s.f.).

Linear equations and graphs- state that the equation of a straight line is of the form y = ax + b, states that, for a straight line graph, the constant b in the equation of the graph y = ax + b is the intercept of the graph on the y axis ie where x = 0 and that the constant "a" is the slope (gradient) of the graph, identify a statement of equality as an equation, solve simple algebraic equations, state the axioms: (a) If equal quantities be added to two quantities that are already equal, the result will be equal, (b) If equal quantities be subtracted from two quantities that are already equal, the remainders will be equal, or (c) Equal quantities when multiplied or divided by the same quantity, will give results that are equal

Algebraic methods; state that arithmetic operations can be carried out by generalising and employing letters to represent quantities, add and subtract algebraic quantities, both positive and negative, identify the effect of plus or minus signs in front of a bracketed quantity or quantities, multiplication and division of quantities, laws of algebra, evaluation and transposition of formulae, solve problems involving transposition of formulae, algebraic operations, factorisation, linear, simultaneous and quadratic equations

Indices and Logs- laws of indices and logarithms, state the laws of indices for multiplication and division of numbers, define the "power" and the "index" of a quantity, define a fractional index of a quantity power, define a negative index of a quantity power, define the zero index of a quantity power, state the "Law of Distribution", common and Naperian logarithms, define logarithm, uses the log definition to put numbers in power form (ie 10X if given log10X), define the parts of a log viz; characteristic and mantissa, and their properties ie positive and/or negative, express numbers in standard form and so finds the characteristic, obtain logarithm of a number from tables or calculator, obtain antilogarithm from tables or calculator indicating the position of the decimal point from the characteristic, apply logarithms to multiplication and to division of numbers, evaluate a number raised to a positive integral power using logarithms, square or cube root of a number using logarithms, a string of figures raised to different powers or roots, indicial equations, direct and inverse proportion, inequalities

Algebraic functions and polynomial division- functional notation and manipulation of algebraic functions, polynomial division, quotients and remainders, use of factor and remainder theorem, rules of order for partial fractions including- linear, repeated and quadratic factors, reduction of algebraic fractions to partial fractions

Exponential, trigonometric and hyperbolic fractions; the nature of algebraic functions, relationship between exponential and logarithmic functions, reduction of exponential laws to linear form, solution of equations involving exponential and logarithmic expressions, relationship between trigonometric and hyperbolic identities, solution of equations involving hyperbolic functions, derive expressions and equations for engineering situations that exponential, trigonometric and hyperbolic fractions and find solutions to such equations

Arithmetic and geometric; notation for sequences, arithmetic and geometric progressions, the limit of a sequence, Sigma notation, the sum of a series, arithmetic and geometric series, Pascal's triangle and the binomial theorem, solve scientific problems that involve arithmetic and geometric series

Power series; variables expressed as power series functions, standard series, Maclaurin's series, binomial series, Pascal's Triangle, approximate values, L'Hopital's rule, use power series methods to determine estimates of engineering variables expressed in power series form

















Trigonometric methods/functions

Trigonometric methods- Geometry and Mensuration; state that a triangle has three sides and three angles and that the latter add up to 180 degrees, state that a right angle triangle has one angle of 90 degrees, state that an equilateral triangle has all angles equal and all sides equal, state that an isosceles triangle has two sides equal and two angles equal, state that a scalene triangle has no equal sides or angles

Two dimensional construction (2 planes, x & y)- constructs a triangle to scale given: (a) all sides; (b) two sides and an included angle; and (c) one side and two angles, using trigonometry where appropriate, bisect a line and an angle, and erects a perpendicular to a line, state that similar triangles have equal angles, state that congruent triangles have equal angles and equal sides, define radius, diameter, circumference, arc, sector, chord, segment of a circle, state that a circle contains 360 degrees, state that a tangent to the circle at a given point is perpendicular to the radius at that point, state the area of a triangle and deduces the area given: (a) the base and vertical height; and (b) two sides and the included angle, deduce the area of a parallelogram, determine the mean height of a figure from measurement of area and length, state the area of a circle in terms of radius/diameter and deduces the area of an annulus, define the Mid-Ordinate Rule and determines the area of a figure using the Mid-Ordinate Rule

Three dimensional construction (3 planes; x, y, z)- deduce the surface area of a cylinder, pyramid and cone, state the surface area of a sphere, deduce the volume of a cylinder, states the volume of a pyramid, cone and sphere, solves problems in mensuration relating to volumes of solid shapes, solves total internal volume of a pressure vessel with spherical ends and cylindrical section of more than 3xd, relationship between volume and material thickness of sheet metal parts, difference between internal and external surface area of formed sheet metal surfaces, determine the mass of a solid using volume and density, determine the ratios of masses and volumes of similar solids, material removal and total mass (rivet holes)

Trigonometry; express acute angles in degrees and radians and states the relationship between degrees and radians, define acute, right, obtuse and reflex angles, defines complementary angles and supplementary angles, defines sine, cosine, tangent, and the relationship between them, obtains numerical values of sine, cosine, tangent for any angle between 0 and 90 degrees from tables or calculator, obtains angle, given numerical value of its sine (Sin), cosine (Cos) or tangent (Tan) from tables or calculator, state Pythagoras's Theorem (c = sqrt(a^2 + b^2)), solve problems using Pythagoras's Theorem, solve right-angled triangles given two facts about the triangle

Sinusoidal functions; review of the basic trigonometric ratios (sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent) and their inverses, trigonometric ratios for the four quadrants (CAST rule etc.), solution of triangles (Sin, Cos and Tan relations), calculation of areas and volumes of solids, Cartesian and polar co-ordinate systems, properties of the circle, pi (Π), radian measure, sinusoidal functions, use sinusoidal functions and radian measure to solve engineering problems, angular velocity (ω, rad/s), angular acceleration (α, rad/s^2), centripetal and centrifugal force, relationship between angular velocity and frequency (f), amplitude (A) and phase (Ф), production of complex waveforms by sinusoidal geographical synthesis, AC waveforms and phase shift, use trigonometric functions to solve engineering problems involving static forces, relative motion, frameworks, metrology, friction, electrical motor torque and electrical-mechanical energy problems

Trigonometric identities; relationship between trigonometric and hyperbolic identities, double angle and compound angle formulae and the conversion of products to sums and differences, solve trigonometric equations using identities, simplify complex trigonometric expressions using identities

















Introduction to the Calculus; the concept of the limit and continuity, definition of the derivative, derivatives of standard functions (basic rules), notation of the derivative and rates of change, differentiation of simple functions using the (1) product, (2) quotient and (3) function of a function rules, introduction of integral calculus as the calculation of area and the inverse of differentiation, the indefinite integral and the constant of integration (+c term), standard integrals (basic rules) and the application of algebraic and trigonometric functions, the definite integral and area under curves

Further differentiation; determine second order and higher derivatives, logarithmic differentiation, implicit functions, differentiation of inverse trigonometric functions, differential coefficients of inverse hyperbolic functions, partial derivatives and partial differentiation, perform implicit and partial differentiation

Further integration; (1) integration by parts, (2) integration by substitution, (3) integration using partial fractions and reduction formulae, analyse engineering situations and solve engineering problems using calculus

Applications of the calculus; maxima and minima, points of inflexion, rates of change of temperature, distance and time, electrical capacitance, RMS values, electrical circuit analysis, AC theory, electromagnetic fields, velocity and acceleration problems, complex stress and strain, engineering structures, simple harmonic motion (SHM), centroids, volumes of solids of revolution, second moments of area, moments of inertia, rules of Pappus, radius of gyration, thermodynamic work and heat energy

Engineering problems; stress and strain, torsion, motion, dynamic systems, oscillating systems, force systems, heat energy and thermodynamic systems, fluid flow, AC theory, electrical signals, information systems, transmission systems, electrical machines, electronics













Statistics and probability

Statistics and probability- Tabular and graphical form; data collection methods, histograms, bar charts, line diagrams, define co-ordinate, axes and origin in graph making, state on which axes the dependent and independent variable are plotted, plots ordered pairs on graph paper having been given or having calculated x and y values, joins the points with a straight line, or smooth curve, depending upon the graph or position of the plotting points, cumulative frequency diagrams, scatter plots, representation of engineering data in tabular and graphical forms, convert vulgar fractions and decimal fractions to percentages, express "gain", "decrease" and "error" as a percentage, state that a ratio is a comparison of quantity magnitudes, state that proportion is an equation of ratios, solve problems on ratio and proportion

Microsoft Excel (use and formulas);

MEAN - Excel formula; =SUM('data range')/('count of data range') or =AVERAGE('data range')
MODE - Excel formula; =MODE('data range')
MEDIAN - Excel formula; =MEDIAN('data range')
RANGE - Excel formula; =MAX('data range')-MIN('data range')
MAX - Excel formula; =MAX('data range')
MIN - Excel formula; =MIN('data range')
S.DEV - Excel formula; =STDEV('data range')
CofV - Excel formula; =S.DEV/MEAN
CV / % - Excel formula; =(S.DEV/MEAN)*100

...where 'data range' above, is the selected data set.

Central tendency and dispersion; introduction to the concept of central tendency and variance measurement, mean, median, mode, standard deviation (S.D.), variance (CofV), and inter-quartile range, identifies direct, inverse and the constant of variation, application to engineering production, form linear equations consistent with data provided in a question

Regression, linear correlation; product moment formula for determining linear correlation coefficient, least squares regression lines, application to experimental work, batch production and quality control, sample the quality of engineering operations

Probability; interpretation of probability, probabilistic models, empirical variability, events and sets, mutually exclusive events, independent events, conditional probability, sample, space and probability, addition law, product law, Baye's theorem, estimate reliability and quality of engineering components and systems

Probability distributions; discrete and continuous distributions, binomial, Poisson and normal distributions, linear regression and confidence intervals, application to sampling, component and system reliability, batch production sampling and quality control

Key2 websites- Mike Tooley's Links to the my key2 websites are available below:

If at first you don't succeed,
Try, try, try again.

- proverb by William Edward Hickson, ref. Oxford Dictionary of Quotations (3rd edition). Oxford University Press. 1979. p. 251.

Note: if you find any errors with any of the notes or contents on this page, suspect the information contained within a resource, description is inaccurate or significantly out of date, then please email me at;

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