cambridge technicals in engineering - ocr · the suggested activities in this delivery guide listed...

TECHNICALSCambridge

CAMBRIDGE TECHNICALS IN ENGINEERINGLEVEL 3 UNIT 23 – APPLIED MATHEMATICS FOR ENGINEERING

DELIVERY GUIDEVersion 1

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CONTENTS

Introduction 3

Related Activities 4

Key Terms 5

General Delivery Guidance 7

Suggested Activities:

Learning Outcome (LO1) 8





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INTRODUCTIONThis Delivery Guide has been developed to provide practitioners with a variety of creative and practical ideas to support the delivery of this qualification. The Guide is a collection of lesson ideas with associated activities, which you may find helpful as you plan your lessons.

OCR has collaborated with current practitioners to ensure that the ideas put forward in this Delivery Guide are practical, realistic and dynamic. The Guide is structured by learning outcome so you can see how each activity helps you cover the requirements of this unit.

We appreciate that practitioners are knowledgeable in relation to what works for them and their learners. Therefore, the resources we have produced should not restrict or impact on practitioners’ creativity to deliver excellent learning opportunities.

Whether you are an experienced practitioner or new to the sector, we hope you find something in this guide which will help you to deliver excellent learning opportunities.

If you have any feedback on this Delivery Guide or suggestions for other resources you would like OCR to develop, please email [email protected].

Unit aimOnce the key mathematical techniques needed for engineering are learnt, they need to be applied to engineering problems. Understanding mathematics in an applied engineering context is what distinguishes the engineer from the pure mathematician.

The aim of this unit is to extend and apply the knowledge of the learner gained in Unit 1 Mathematics for engineering. It is therefore strongly recommended that learners have completed Unit 1 Mathematics for engineering prior to commencing the study of this unit.

By completing this unit learners will:

• be able to apply trigonometry and geometry to a range of engineering situations

• be able to apply knowledge of algebra, equations, functions and graphs to engineering problems

• be able to use calculus to analyse a range of problems

• understand applications of matrix and vector methods

• be able to apply mathematical modelling skills.

Unit 23 Applied mathematics for engineering

LO1Be able to apply trigonometry and geometry to a range of engineering situations

LO2Be able to apply knowledge of algebra, equations, functions and graphs to engineering problems

LO3 Be able to use calculus to analyse a range of problems

LO4 Understand applications of matrix and vector methods

LO5 Be able to apply mathematical modelling skills

Please note

The timings for the suggested activities in this Delivery Guide DO NOT relate to the Guided Learning Hours (GLHs) for each unit.

Assessment guidance can be found within the Unit document available from www.ocr.org.uk.

The latest version of this Delivery Guide can be downloaded from the OCR website.

mailto:resources.feedback%40ocr.org.uk?subject=

http://www.ocr.org.uk

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This unit (Unit 23) Title of suggested activity Other units/LOs

LO1 Formal lesson 1 Unit 1 Mathematics for engineering LO4 Be able to use trigonometry in the context of engineering problems

Practical exercises 1 Unit 1 Mathematics for engineering LO4 Be able to use trigonometry in the context of engineering problems

Formal lesson 2 Unit 1 Mathematics for engineering LO4 Be able to use trigonometry in the context of engineering problems


Formal lesson 3 Unit 1 Mathematics for engineering LO4 Be able to use trigonometry in the context of engineering problems


LO2 Tutorial 1 Unit 1 Mathematics for engineering LO1 Understand the application of algebra relevant to engineering problems

LO2 Be able to use geometry and graphs in the context of engineering problems

LO3 Understand exponentials and logarithms related to engineering problems

Practical exercises 4 Unit 1 Mathematics for engineering LO1 Understand the application of algebra relevant to engineering problems

Understand calculus relevant to engineering problems

Formal lesson 5 Unit 4 Principles of electrical and electronic engineering

LO2 Understand alternating voltage and current

LO3 Formal lesson 6 Unit 1 Mathematics for engineering Understand calculus relevant to engineering problems

Formal lesson 7 Unit 1 Mathematics for engineering Understand calculus relevant to engineering problems

Formal lesson 8 Unit 1 Mathematics for engineering Understand calculus relevant to engineering problems

Practical exercises 9 Unit 3 Principles of mechanical engineering

LO1 Understand systems of forces and types of loading on mechanical components

LO5 Understand principles of dynamic systems

LO5 Tutorial 3 All units, particularly Units 2, 3, 4, 12 and 15

The Suggested Activities in this Delivery Guide listed below have also been related to other Cambridge Technicals in Engineering units/Learning Outcomes (LOs). This could help with delivery planning and enable learners to cover multiple parts of units.

RELATED ACTIVITIES

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KEY TERMSUNIT 23 – APPLIED MATHEMATICS FOR ENGINEERING

Explanations of the key terms used within this unit, in the context of this unit

Key term Explanation

Trigonometry and Geometry

Learners should be familiar with common terms and concepts within this area of study including the following.

Basic formulae involving sine, cosine and tangent functions; inverses trigonometric functions; standard trigonometric identities; angles in degrees and radians; amplitude; frequency; period; phase angle; acute, obtuse, reflex and right angles; use of formulae to calculate angles and lengths; circle theorems; segment; cord; arc; area; volume; tangential; normal; equilateral, isosceles, scalene and right-angled triangle; pentagon, hexagon, octagon etc; prism, sphere, cone, cylinder, pyramid.

Algebra and Functions


Function; expression; equation; graphs; parabola; inequality; variable; constant; coefficient; product; quotient; quadratic; discriminate; powers; roots; surds; asymptote; pole; polynomial; partial fraction; logarithm; exponential; associative law; distributive law; communicative law; solution of an equation; simplification; factorisation; algebraic proof; least common denominator; highest common factor; real number; prime number; rational number; imaginary number; complex number; exponential form (of a complex number); polar form (of a complex number); argand diagram.

Calculus Learners should be familiar with common terms and concepts within this area of study including the following.

Differentiation; notation for differentiation; first derivative, second derivative; gradient; rate of change; local maximum; local minimum; turning point; point of inflection; derivative of common functions; derivative of a product; derivative of a quotient; derivative of a function of a function.

Indefinite integral; definite integral; constant of integration; limits of integration; integral of common functions; integration by parts; integration by substitution; area ‘under’ a curve; volume of revolution.

First order differential equation; second order differential equation; (solution of a differential equation by) direct integration (and) separation of variable; initial conditions.

Matrices and Vectors


Matrix notation; square matrix; row matrix; column matrix; unit matrix; (matrix operations) matrix addition, subtraction, multiplication, transpose, inverse; representation of simultaneous equations in matrix notation.

Vector notation; direction and magnitude (of a vector); unit vector; (vector operations) addition, subtraction, dot product, vector product; (vector applications) velocity, force.

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Explanations of the key terms used within this unit, in the context of this unit

Key term Explanation

General Modelling Learners should be aware of common terms and concepts which may be encountered when modelling and solving engineering problems including the following.

Electrical engineering: Common SI electrical units; voltage; current; power; charge; resistor; inductor; capacitor; reactance; DC; AC; frequency; period; amplitude; phase; circuit; open circuit; closed circuit; potential difference; Ohm’s law.

Mechanical engineering and dynamics: Common SI units; speed; linear velocity; angular velocity; mass; weight; acceleration; force; power; torque; inertia; moment; compression; tension; friction; Hooke’s law; Newton’s laws.

Miscellaneous: Pressure; turbulence; flow; vibration; heat; temperature; convection; radiation; exponential growth and decay.

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GENERAL DELIVERY GUIDANCEIt is recommended that delivery of this unit is conducted through a combination of formal lessons, tutorials and practical problem solving sessions. In addition to providing a solid grounding in the mathematical techniques specified in the Learning Outcomes (LOs), teachers should emphasise the practical application of all topics. This should be accomplished by discussing with learners practical, engineering problems similar to the examples given in LO5. Teachers should bear in mind that the nature of all questions in the examination will be set in an applied context. Examination questions will often involve more than one single LO and so it is important that all LOs are as treated equally important.

There is no formal requirement for coursework in this unit; however, learners would benefit from being provided regularly with example exercises. Teachers should encourage learners to explore alternative solutions to problems where possible and to check the feasibility and credibility of solutions. Learners should therefore review their

numerical results in the context of the situation to confirm that results obtained are practical and reasonable.

Although computers and programmable calculators are not allowed in the examination, learners should be encouraged to use available computer software to tabulate results, produce graphs and perform numerical calculations. This will allow learners to check their own work quickly and will encourage them to explore alternative formulae and solutions. The use of spreadsheets would be particularly appropriate here. For each of the three lesson elements given in this guide an example spreadsheet is provided.

Learners should be given access to Formulae Booklet for Level 3 Cambridge Technicals in Engineering (available from http://www.ocr.org.uk/qualifications/cambridge-technicals-engineering-level-3-certificate-extended-certificate-foundation-diploma-diploma-05822-05825/).

http://www.ocr.org.uk/qualifications/cambridge-technicals-engineering-level-3-certificate-extended-certificate-foundation-diploma-diploma-05822-05825/



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SUGGESTED ACTIVITIESLO No: 1

LO Title: Be able to apply trigonometry and geometry to a range of engineering situations

Title of suggested activity

Suggested activities Suggested timings Also related to

Formal lesson 1 Teachers could review basic trigonometry and apply this to shapes involving straight lines and angles.

http://www.mathwarehouse.com/trigonometry/

This site contains a number of lessons and interactive tools involving algebra, geometry and trigonometry.

It will also be useful for LO2 and all activities below.

1 hour Unit 1 LO4

Practical exercises 1 Learners could be asked to calculate perimeters and areas of 2-D shapes involving straight and curved sides.

Example Calculate the perimeter and area of the following metal component.

2-3 hours Unit 1 LO4

Formal lesson 2 Teachers could review basic formulae for lengths, areas and volumes of common 3-D shapes such as prisms, spheres, cones, cylinders and pyramids.

Reference should be made to the supplementary list of formulae available for this unit.

1 hour Unit 1 LO4

http://www.mathwarehouse.com/trigonometry/

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Title of suggested activity Suggested activities Suggested timings Also related to

Practical exercises 2

See Lesson Element Analysis of a cam

Learners should also complete the activities in LO2 and LO5.

http://auto.howstuffworks.com/camshaft.htm

This webpage provides a description of a cam, its use and its operation. An animation is also provided.

The spreadsheet ‘Cam_analysis’ is provided that will allow learners to enter various cam dimensions; the spreadsheet will then produce tabulated results and plot a graph of the cam lift against angle of rotation.


Formal lesson 3 Teachers could present basic trigonometric identities and show how these could be used to derive trigonometric relationships required in various problems.

Reference should be made to the standard list of formulae provided.

1 hour Unit 1 LO4

Practical exercises 3 Learners could be given a selection of practical situations involving electrical and mechanical engineering that require knowledge of amplitude, frequency and phase.

Examples

Determine the phase angle between the two voltages:v

1(t) = 12sin(1000t + 60o)

and v

2(t) = –6cos(1000t + 30o)

Show that the sum of two sinusoidal voltages:

15sin(ωt) + 12cos(ωt) can be expressed as βsin(ωt + α) and determine the values of α and β.

The period of a vibrating mechanism is 10ms and the amplitude varies between 5 mm and –5 mm.

Determine the frequency of oscillation in terms of cycles per second and express the vibration in the form βsin(ωt).


http://auto.howstuffworks.com/camshaft.htm

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LO No: 2

LO Title: Be able to apply knowledge of algebra, equations, functions and graphs to engineering problems


Formal lesson 4 Teachers could review techniques and principal areas specified for this LO. 1-2 hours

Tutorial 1 Learners could be presented with a range of exercises relating to this LO.

Examples

Solve for x in the equation

x + 2 =1 – 3x

x + 4

Find the equation of the straight line which is normal to the function y = –3x + 10 at the point where x = 2

Show on a graph the area enclosed by the inequalities:

–x + y ≤ 0x ÷ 2 + y ≥ 2x2 – 4y ≤ 0

Identify any poles, zeros and asymptotes of the function:

y =xe–x

+ x1 – x

Express the following as a sum of partial fractions:

x + 7

x2 – 7x + 10

3 hours Unit 1 LO1, LO2, LO3

SUGGESTED ACTIVITIES

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Rearrange the following to make v the subject:

–2In ( 4 + v )= t2

3 – v

Drawn a graph between x = 0 and x = 5 of the function:

y = xe–x + 1

also see: http://www.mathwarehouse.com/algebra/

This site contains number of lessons and interactive tools involving algebra, geometry and trigonometry.


See Lesson Element Roller Coaster

Learners should also complete the activities in LO3, LO4 and LO5.

The spreadsheet ‘Roller’ is provided that will produce a graph showing the profile of the roller coaster track based on relevant parameters.

2-3 hours Unit 1 LO1, LO5

Formal lesson 5 Teachers could present an introduction to complex numbers and their application in engineering. 2 hours Unit 4 LO2

Practical exercises 5 Learners could be provided with a range of problems requiring knowledge of complex numbers, the manipulation of functions involving complex numbers and the representation of complex numbers in different forms including argand diagrams.

Examples

Simplify the expressions:

(4 – j7)(2 + j3)

(2 – j3)(3 + j2)

(4 – j5)

if z1 = 2 + j, z

2 = –2 + j4 and

1 =

1 +

1

z3

z2

z1

2-3 hours

http://www.mathwarehouse.com/algebra/

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Evaluate z3 in the form a + jb

Given that z = 3 – j4 express Z in polar notation, exponential notation and in an argand diagram.

The voltage v(t) = 12sin(1000t + 30o) is applied to a 20 mH inductor. By using phasors, calculate the resultant current.

You are given that the voltage can be expressed as the phasor V = 12∠30 and that the phasor current is:

I =V

ωL∠90

Express your answer in the form i(t) = Icos (ωt + αo)

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LO No: 3

LO Title: Be able to use calculus to analyse a range of problems


Formal lesson 6 Teachers could present the fundamental concepts of differentiation and a range example problems involving differentiation.Applications should include maxima and minima problems and the derivation of straight line functions tangential to and normal to a given function at a given point.

http://www-math.mit.edu/~djk/calculus_beginners/

This site provides several chapters covering calculus with applications including differential equations. It will be useful in all activities below.

2 hours Unit 1 LO5

Practical exercises 6See Lesson Element Roller Coaster

Learners should also complete the activities in LO2, LO4 and LO5. See LO2

Formal lesson 7 Teachers could present further differentiation techniques including the differentiation of functions involving products, quotients and functions of a function.

2 hours Unit 1 LO5

Practical exercises 7 Learners could be presented with practical problems involving the differentiation of a range of common functions.Learners should be given a list of standard derivative formulae.

Examples

Differentiate with respect to x the following functions.

y = x2 cos(3x)

y =x4

(x + 1)2

y = cos(2x2 – e–2x)

Identify the coordinates of the stationary points of the function:

y = 2x3 + 3x2 – 36x + 12

2-3 hours


http://www-math.mit.edu/~djk/calculus_beginners/

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Determine the coordinates of any local maximum and minimum points of the function:

y = 2x3 + 3x2 – 36x + 12

Show that:

y =1

(2 cos4x – sin 4x)25

satisfies the equation:

d2y –5

dy + 6y = 2sin 4x

dx2 dx

Determine the equation of the straight line that is normal to the function

y =1

x where x = 2

Formal lesson 8 Teachers could present the fundamental concepts of integration and a range example problems involving integration. Reference should be made to the standard integrals given in the lists of formulae for this unit.

3 hours Unit 1 LO5

Practical exercises 8 Learners could be given a range of practical exercises including the calculation of areas between a curve and the x axis, volumes of revolution and other problems involving definite integrals.

Examples

Integrate the following functions:

y =x + 1

x2 – 3x + 2

y = sin3 xy = ecosx sin x

2-3 hours

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Calculate the absolute area between the function:y = x2 – 8x +15and the x axis between the ordinates x = 2 and x = 6

Find the volume generated when a plane figure bounded by the function y = 5cos2x, the x axis and

the ordinates x = 0 and

x =p4

rotates about the x axis through a complete revolution.

A car accelerates from rest and has a speed, v m s-1, given by

v = 40(1 – e–t/5) where t is time.

Calculate the distance travelled in the first 10s.

Calculate the RMS value of the voltage v = 324sin(100pt)

Formal lesson 9 Teachers could present an introduction to differential equations and their solutions. 2 hours

Tutorial 2 Learners could complete the problems given in the exemplification column of this LO. 2 hours


See Lesson Element Modelling the motion of a car

Learners should also complete the activities in LO2 and LO5.

The spreadsheet ‘Car_mation’ is provided which will allow various parameters to be inserted, speeds and distances to be calculated and graphs of speed v time and distance v time to be plotted.

3-4 hours Unit 3 LO1, LO5

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LO No: 4

LO Title: Understand applications of matrix and vector methods


Formal lesson 10 Teachers could introduce the fundamentals of matrices and matrix algebra.

http://stattrek.com/tutorials/matrix-algebra-tutorial.aspx

This site provides a tutorial in matrix algebra.

1-2 hours

Practical exercises 10 Learners could be presented with a range of problems similar to those provided in the exemplification column of this LO.

Other examples

Determine the inverses of the matrix A = [ –2 3

]4 –1

Use matrix methods to solve:

–2x + 3y =7

4x – y = –1

In an electrical circuit, voltages V1 and V

2 satisfy the following equations:

V1 ( 1

+ 1

) – V2( 1 ) = 1

12 5 5

V1( 1 ) + V

2 ( 1

+ 1

) = –35 5 7

Calculate the values of V1 and V

2

The tensions T1 and T

2 in two cables required to support a load of 300 kg in equilibrium satisfies the

following equations:

T1 cos50 + T

2 cos40 = 300

T1 sin50 – T

2 sin40 = 0

Calculate the values of T1 and T

2

1-2 hours


http://stattrek.com/tutorials/matrix-algebra-tutorial.aspx

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Practical exercises 11See Lesson Element Roller Coaster

Learners should also complete the activities in LO2 and LO3. See LO2

Formal lesson 11 Teachers could introduce the fundamentals of vectors and vector operations.

http://emweb.unl.edu/Math/mathweb/vectors/vectors.html

This site provides an introduction to vectors and vector algebra.

2 hour

Practical exercises 12 Learners could be presented with a range of problems relating vectors and their applications.

Example:

Draw a force diagram for the following situation in which a mass of 300 kg is being supported in equilibrium by two cables attached to masses M

1 and M

2.

Represent the forces in vector notation ai + bj and calculate the values of M1 and M

2

2-3 hours

http://emweb.unl.edu/Math/mathweb/vectors/vectors.html

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LO No: 5

LO Title: Be able to apply mathematical modelling skills


Tutorial 3 Learners could be presented with a range of practical situations which involve the formation of mathematical models and their solution. Some suggestions are shown in the exemplification column of the LO but others could be obtained from appropriate resources. The past examination papers, inserts and mark schemes for OCR’s H860 specification would be helpful here and are available at:

http://www.ocr.org.uk/qualifications/other-general-qualifications-mathematics-for-engineering-level-3-certificate-h860/

Where necessary, appropriate physical laws (e.g. Ohm’s law, Kirchhoff’s laws, Newton’s laws, Hooke’s law etc.) should be used. For examination purposes, learners are not required to remember such laws since these will be provided when and where necessary.

Example

A tank in the shape a 1 m cube is completely full of water when an outlet value at its base is opened. Water then flows out of the tank at a rate which is directly proportional to the height of water in the tank, h m. The height of water in the tank is modelled by the following equation.

dh = –

hdt 20

where t is time in seconds.

Calculate the time it will take for half of the water to be drained from the tank.

Calculate the height of water in the tank 30 s after the valve is opened.

At the time when the height of water in the tank becomes 0.5 m, calculate the rate at which water flows out of the valve in terms of litres per second.

Explain why, according to the model, the tank never becomes completely empty.

8-12 hours

(Much of this time will include work covered by other LOs)

All other engineering units in this qualification particularly Units 2, 3, 4, 12 and15.




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