Cambridge TECHNICALS CAMBRIDGE TECHNICALS IN ENGINEERING LEVEL 3 UNIT 1 MATHEMATICS FOR ENGINEERING. DELIVERY GUIDE Version 1

Transcription

1 Cambridge TECHNICALS CAMBRIDGE TECHNICALS IN ENGINEERING LEVEL 3 UNIT 1 MATHEMATICS FOR ENGINEERING DELIVERY GUIDE Version 1

2 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING CONTENTS Introduction 3 Key Terms 4 Misconceptions 7 Suggested Activities: Learning Outcome (LO1) 8 Learning Outcome (LO2) 10 Learning Outcome (LO3) 11 Learning Outcome (LO4) 12 Learning Outcome (LO5) 14 Learning Outcome (LO6) 15 2

3 INTRODUCTION This 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 resources.feedback@ocr.org.uk. Unit aim Mathematics is one of the fundamental tools of the engineer. It underpins every branch of engineering and the calculations involved are needed to apply almost every engineering skill. This unit will develop learners knowledge and understanding of the mathematical techniques commonly used to solve a range of engineering problems. By completing this unit learners will develop an understanding of: algebra relevant to engineering problems the use of geometry and graphs in the context of engineering problems exponentials and logarithms related to engineering problems the use of trigonometry in the context of engineering problems calculus relevant to engineering problems how statistics and probability are applied in the context of engineering problems 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 The latest version of this Delivery Guide can be downloaded from the OCR website. Foreword: Resources There are a wide range of resources that may be useful in supporting the delivery of this unit, some of which have been listed in the separate Resources Link resource available from These include reference text books, web-based tutorials, web-based video tutorials and worked and practice questions. Many resources will singularly cover the entire content of this unit, and for this reason specific resources have not been listed against each topic area. Although there are many resources that cover mathematical concepts generically, some are dedicated to the application of mathematics in the context of engineering. Teachers and learners are encouraged to solve problems related to engineering where possible. The following two books (which also include further web-based resources and which are available as paperback or e-book) are highly recommended: Bird, John (2014) Basic Engineering Mathematics Routledge, UK Bird, John (2014) Engineering Mathematics Routledge, UK Unit 1 Mathematics for engineering 3 3 LO1 LO2 LO3 LO4 LO5 LO6 Understand the application of algebra relevant to engineering problems Be able to use geometry and graphs in the context of engineering problems Understand exponentials and logarithms related to engineering problems Be able to use trigonometry in the context of engineering problems Understand calculus relevant to engineering problems Be able to apply statistics and probability in the context of engineering problems Opportunities for Maths skills development This unit provides a range of activities entirely focussed on Mathematics for Engineering. The activities are not designed to replace you own subject knowledge and expertise in deciding what is most appropriate for your learners. Maths OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

4 4 KEY TERMS UNIT 1 MATHEMATICS FOR ENGINEERING Explanations of the key terms used within this unit, in the context of this unit Key term Algebra Angles and radians Arcs Explanation Algebra is an area of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations. Converting from angle to radians: 360 = 2 π radians i.e. x radians = 180 x/π degrees; x degrees = πx/180 radians Arc length s = rθ where θ is in radians Binomial expression A binomial is a mathematical expression (a polynomial) with two terms eg 3x 2 +2 or 5x -1 Circles - areas The area of a circle: area = πr 2 Circles co-ordinate equation Co-ordinate geometry Curve sketching Definite integrals Differentiation exponentials and logarithms Differentiation simple functions Differentiation trigonometric function Exponential function The co-ordinate equation of a circle: (x a) 2 + (y b) 2 = r 2 Co-ordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Curve sketching is the sketching of mathematical functions in order to visualise their shape and is often used to determine solution(s). A definite integral is an integral expressed as the difference between the values of the integral at specified upper and lower limits of the independent variable. Differentiation of expressions containing e ax and ln ax The derivative is the instantaneous rate of change of a function with respect to one of its variables. Simple functions are taken to contain terms of the formy = ax n Differentiation of expressions containing the terms sine (sin) and cosine (cos). An exponential function is a function whose value is a constant raised to the power of the argument. In this unit, it is taken to be of the form y = ex and y = e-x Factorisation Factorisation is the reverse of expanding brackets. For example putting 2x² + x - 3 into the form (2x + 3)(x - 1) OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

5 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Explanations of the key terms used within this unit, in the context of this unit Key term Graphical transformations Indefinite integrals Integration simple functions Integration trigonometric function Linear simultaneous equation Log laws Logarithmic function Explanation Graphical transformations are movements of a graph representing a function in the co-ordinate plane. In this unit, translation is taken to be by addition or multiplication. An indefinite integral is an integral expressed without limits, and so containing an arbitrary constant (C). Indefinite integrals do not reveal a definite value. Integration is the reverse process of differentiation. It is typically used to determine the area of a region of a function. Simple functions are taken to contain terms of the form y = ax n Integration of expressions containing the terms sine (sin) and cosine (cos). Linear simultaneous equations are a set of linear equations; the values of the variables in which are solved simultaneously eg Solve for x and y: 5x 2y = 4; x + 2y There are three log laws: log (A x B) = log A + log B; log (A/B) = log A log B; log A n = n log A In this unit, the logarithm is taken to be log to base 10 for problems involving the log laws. It is taken to be log to base e (the natural or Naperian logarithm) for problems involving solutions using the inverse of the exponential function. Polynomial A polynomial is an expression of more than two algebraic terms eg 4x 2 + 3x - 7 Probability Probability addition and multiplication laws Probability expectation, independent and dependent events, replacement Pythagoras Theorem In mathematical terms probability is the extent to which an event is likely to occur. The addition law recognised by the term OR joining the probabilities. If P A is the probability of event A happening and P B event B, the probability of event A OR event B happening is P A + P B The multiplication law recognised by the term AND joining the probabilities. If P A is probability of event A happening and P B event B, the probability of event A AND event B happening is P A x P B Expectation the expectation of an event happening is defined as the product of the probability of an event happening and the number of attempts made. Independent event two events, A and B, are independent if the fact that A occurs does not affect the probability of B occurring. Dependent event two events are dependent if the outcome or occurrence of the first affects the outcome or occurrence of the second, so that the probability of the second occurrence is changed. With Replacement: the events are independent - the chances don t change. Without Replacement: the events are dependent - the chances change. A theorem attributed to Pythagoras that the square on the hypotenuse of a right-angled triangle is equal in area to the sum of the squares on the other two sides. Quadratic equation A quadratic equation is an equation where the highest exponent of the variable (usually x ) is a square. It is usually written ax 2 +bx+c = 0 eg 2x 2 +5x-3 = 0 5

6 6 Explanations of the key terms used within this unit, in the context of this unit Key term Right-angled triangle Sectors - area Sine and cosine rule (non-right angled triangle) Sine, cosine and tangent operations Sine, cosine and tangent rule (right angled triangle) Statistics Statistics cumulative frequency graph Statistics frequency polygon Statistics histograms Statistics mean, mode and median Statistics variance and standard deviation Explanation A right-angled triangle is a triangle in which one of the angles is 90 degrees. Area of a sector = ½r 2 θ where θ is in radians Sine rule: a/sin A = b/sin B = c/sin C Cosine rule: a 2 = b 2 + c 2 2bc Cos A b 2 = a 2 + c 2 2ac Cos B c 2 = a 2 + b 2 2ab Cos C where A, B and C are angles within the triangle and a, b, and c are the lengths of the three sides. Are taken to be graphs of the functions y = sin x, y = cos x and y = tan x for a range of angles for 0 to 360 Rules relating to the lengths of the sides in a right-angled triangle. sine θ = opposite side/hypotenuse; cosine θ = adjacent side/hypotenuse; tangent θ = opposite side/adjacent side Statistics is a branch of mathematics concerned with the collection, organisation and analysis of data. A cumulative frequency graph is a visual representation of ranked categorical data. An example is grouping people by height categories: under five feet, five feet to six feet, and above six feet. The number of people in each category is the frequency. Frequency polygons are a graphical device for understanding the shapes of distributions of data. A histogram is a diagram consisting of rectangles whose area is proportional to the frequency of a variable and whose width is equal to the class interval. Mean is the average value of a set of data Mode is the value that is repeated most often in a set of data Median is the middle value in an ordered set of data Variance the average of the squared differences from the mean. Standard deviation the standard deviation is a measure of how spreads out numbers are in a set of data. Its symbol is σ where σ = variance (for population data). For sample data the standard deviation is termed s with formula OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

7 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Explanations of the key terms used within this unit, in the context of this unit Key term Statistics distribution curve, percentiles, quartiles and skew Transposition Triangles - area Turning points maxima and minima Explanation Distribution curve is taken to mean a theoretical frequency distribution for a set of variable data, usually represented by a bell-shaped curve symmetrical about the mean. This is often referred to as a normal distribution. Percentile a percentile (or a centile) is a measure used in statistics indicating the value below which a given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20 percent of the observations may be found. Quartile the quartiles of a ranked set of data values are the three points that divide the data set into four equal groups, each group comprising a quarter of the data. Skew skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. The skewness value can be positive or negative, or even undefined. Transposition of formulae is the process of changing the subject of the formula e.g. given E = mv 2 /2g, transpose the formula to make v the subject. Area for any triangle: ½ base x perpendicular height ½ bc sin A where b and c are the lengths of two sides and A is the angle opposite the third side [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) Turning points are points at which the rate of change of a function (the gradient) are zero. They can represent maximum and minimum points of the function. They can be determined by differentiating the function twice. 7

8 8 MISCONCEPTIONS Some common misconceptions and guidance on how they could be overcome What is the misconception? How can this be overcome? Resources which could help Algebra - mathematical order BODMAS/BIDMAS Application of units Linear (straight line) equations Inverting and reordering exponentials Use of radians Differentiation determining maxima and minima Integration indefinite integrals and use of constant C Differentiation and integration sine and cosine Statistics percentiles, quartiles and skew Learners may already be familiar with the importance of mathematical order i.e. performing mathematical operations in the correct order. BODMAS and BIDMAS are common acronyms representing the order in which mathematical operations should be performed (Brackets, Order, Division, Multiplication, Addition, and Subtraction). Teachers may wish to remind learners when solving problems manually, using a calculator or using ICT. Mathematical operations related to solving engineering problems will invariably result in an answer with units (e.g. newtons, volts, amperes, pascals etc). Learners should be encouraged to include the correct units when solving engineering problems. Learners may already be familiar with the equation of a straight line. Teachers might wish to recap on straight line mathematics before learners undertake more complex problems. Many engineering phenomena involving growth and decay are described using exponentials. It is often necessary to transpose exponential equations to change the subject of the equation. This can be done using natural logarithms. Teachers might develop suitable examples for learners to practice rearranging exponential formulae. Some formulae require the value of the angle to be in radians and not degrees. Teachers might wish to reinforce this with practice activities. Learners should also be aware that this may affect operation of a scientific calculator which often have degree and radian modes. To determine maxima and minima using differentiation requires differentiation of the function twice (i.e. where the rate of change or slope is zero). Problems with maxima and minima that learners can visualise graphically and solve numerically may prove useful to reinforce this. Learners often forget with indefinite integration to include the constant term C. Teachers could reinforce the importance of including C with different integration problems that result in the same integral. A useful way in which to remember the integral and derivative of sine and cosine is to write the terms in circular form ie clockwise: sin, cos, -sin, -cos. Differentiation is performed clockwise and integration anticlockwise. Practice problems might prove a useful way in developing knowledge of percentiles, quartiles and skew. Applied problems using engineering data would also prove useful. operation-order-bodmas.html maths/number/order_operation/ revision/2/ Practice engineering problems. equation_of_line.html algebra/exponential-growth.html geometry/radians.html calculus/maxima-minima.html calculus/integration-introduction. html Sine/cosine written in a circular form percentiles.html skewness.html OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

9 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING MISCONCEPTIONS Some common misconceptions and guidance on how they could be overcome What is the misconception? How can this be overcome? Resources which could help Statistics - variance, standard deviation (population and sample data) Probability dependence and independence Learners may already be confident in calculating mean, mode and median. Variance and standard deviation are an extension of this. The formula for standard deviation differs for population and sample data, and teachers may wish to reinforce this difference. Probability may involve independent or dependent events. Teachers may wish to reinforce the difference between both types of event using suitable examples. standard-deviation.html probability-events-types.html 9

10 10 SUGGESTED ACTIVITIES LO No: 1 LO Title: Understand the application of algebra relevant to engineering problems Title of suggested activity Suggested activities Suggested timings Application of algebra Simplification of polynomials Transposition of formulae How to simplify and solve equations The confident application of algebra is fundamental to solving engineering problems. This might include transposition and solution of engineering formulae in the areas of mechanical engineering, electrical and electronic engineering or materials science. Examples could include volumes and geometry of shapes, solution of laws and interpretation of data. Teachers might begin the delivery of algebra with common techniques including: multiplication by a constant, binomial expressions, removing a common factor, factorisation and using a lowest common multiple. Where possible learners could undertake problems relevant to engineering. Throughout this unit, teachers might adopt a style of showing worked engineering examples and tasking learners to solve many practice questions. Text and web-based resources may prove useful with the delivery of areas of this unit, and with the supply of worked and practice activities. Polynomials are often used in the solution of engineering problems that represent some form of relationship between quantities (which are sometimes presented and analysed graphically). They are also commonly used to define control system problems and those involving motion (e.g. a trajectory). Teachers might develop worked examples and learner tasks for the simplification of polynomials including: factorising a cubic, algebraic division and the remainder and factor theorems. Learners might be taught how to transpose formulae containing like terms, roots and powers. Where possible engineering examples should be used (e.g. transposition of fundamental engineering formulae or equations describing engineering laws and relationships). Practice with many examples may be useful to developing a thorough understanding. The simplification and ultimate solution of equations is an important aspect of solving engineering problems. Teachers might develop further worked and practice examples in order to consolidate learner s knowledge of the application of algebra, polynomials and transposition. OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

11 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Title of suggested activity Suggested activities Suggested timings Linear simultaneous equations Quadratic equations Linear simultaneous equations describe linear systems where there are multiple unknowns. Typical engineering examples include electrical circuit theory (e.g. current loops in Kirchhoff s Laws), systems of forces in mechanical engineering, and system modelling (eg force and pressure). Learners might be taught how to solve linear simultaneous equations using graphical and algebraic methods. These should be limited to two unknowns. Teachers might again develop suitable worked and practice examples. See Lesson Element Linear simultaneous equations. Quadratic equations describe problems where there is a non-proportional (or non-linear) relationship. They are sometimes termed second-term or order. Examples in engineering could include distance/rate/time, quantity/rate/time or trajectory-type problems. Learners might be taught how to solve quadratic equations using graph sketching, factorising method, completing the squares and using the classical formula: x = [- b ± (b 2 4ac)]/ 2a. Again teachers might develop suitable worked and practice examples. 11

12 12 SUGGESTED ACTIVITIES LO No: 2 LO Title: Be able to use geometry and graphs in the context of engineering problems Title of suggested activity Suggested activities Suggested timings Co-ordinate geometry straight lines Curve sketching Graphical transformations Engineering problems that relate to straight-line co-ordinate geometry include force vs displacement graphs eg linear spring, tensile testing force-elongation graphs and linear relationships such as Ohm s Law. There are many other examples of how engineering systems can be described. Learners should be taught how to solve problems to determine the equation of a straight line between two points, gradient of a line, mid-point and distance between points. Text and web-based resources may prove useful with the delivery of this topic area, and with the supply of worked and practice activities. Curve sketching is a useful method to visualise and solve more complex mathematical formulae. Learners might be taught how to sketch and solve graphs of the form y = kx n and also cubic functions. In addition to hand-sketching this might present an opportunity for learners to produce graphs using a spreadsheet thereby developing ICT skills. Learners could plot graphs from functions or raw data, add trend lines, and determine solutions. Graphical transformations include: translation by addition, transitions by multiplication (ie stretches, reflections and rotations). Engineering examples might include manipulation of an electrical waveform (eg amplification, attenuation or phase shift) or manipulation of a graph representing a mechanical quantity (eg trajectory or position). Teachers could use suitable worked examples to demonstrate graphical transformations which might relate to engineering problems, with learners solving practice problems. OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

13 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING SUGGESTED ACTIVITIES LO No: 3 LO Title: Understand exponentials and logarithms related to engineering problems Title of suggested activity Suggested activities Suggested timings Exponentials and logarithms Inverse function and log laws Exponential functions are commonly used in engineering to express quantities that grow or decay exponentially such as voltage in a capacitor, stress-strain relationships or radioactive decay. Logarithms are also used to represent and manipulate growth and decay such as decibels in signal attenuation and the Richter scale in earthquake intensity. Learners should be taught how to manipulate and solve problems of the form y=e ax. y=e ax and In y=ax Teachers might develop worked examples and learner tasks to develop understanding of the application of exponentials and logarithms. Leaners could sketch and interpret graphs showing exponential growth and decay. Graphs could be hand produced or produced using ICT. Teachers might continue the topic area by showing learners how to solve and manipulate exponential functions by rearranging using logarithms. Examples could include rearranging and solving RC circuit problems (eg of the form V c = V s E -t/(r/c) ) by inverting the function. Learners could also solve problems using the log laws (eg engineering functions representing growth and decay). Again, similar worked examples and practice activities may prove an approach to developing understanding. See Lesson Element Inverse function and log laws 13

14 14 SUGGESTED ACTIVITIES LO No: 4 LO Title: Be able to use trigonometry in the context of engineering problems Title of suggested activity Suggested activities Suggested timings Angles and radians Arcs, circles and sectors Right-angled triangles Non-right angled triangles There are an enormous number of uses for trigonometry and trigonometric functions. In mechanical engineering this might include locating position, length and dimensions or resolving forces in systems of pulleys, belts and levers. In electrical engineering, trigonometry might be used in AC theory to analyse waveforms or solve an impedance triangle. Teachers might begin the area of trigonometry by introducing learners to angles and radians. Learners could solve problems involving converting between angles and radians. Engineering examples might include problems involving rotating shafts and pulleys, or alternating electrical waveforms. As common throughout this unit, worked engineering examples and practice activities may prove a useful approach to developing understanding. Learners might be taught how to solve engineering problems involving arcs, circles and sectors. This could include: formula for length of an arc, formula for sector of a circle and use of the co-ordinate equation: (x a) 2 + (y b) 2 = r 2. Learners might be encouraged to solve problems both numerically and to prove their solutions graphically using scale sketches. Engineering examples might include those that require the solution of component geometry (e.g. mating of components, layout of belts, pulleys and levers etc). Undertaking many practice problems might again prove useful to developing understanding. The right-angled triangle has many applications in representing engineering systems (e.g. vector diagrams of mechanical forces, or phasor diagrams for voltage and current). Learners might be taught what is meant by the term solution of a triangle, Pythagoras Theorem, and use of sine, cosine and tangent rule for right-angled triangles. They could be introduced to the formulae for the area of a right-angled triangle. Learners could solve problems both graphically (using scale sketches) and numerically for right-angled triangles. Teachers might be able to find many suitable and relevant engineering examples. Non-right angled triangles can also represent engineering systems such as the geometric layout of components showing position and lengths. Teachers should show learners how to apply the sine rule, cosine rule and how to determine the area of non-right angled triangles. Practice with many examples may once again prove useful to developing understanding. 1 hour 3 hours 3 hours OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

15 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Title of suggested activity Suggested activities Suggested timings Common trigonometric identities Sine, cosine and tangent operations There are many common trigonometric identities that are useful for solving problems involving triangles. Teachers could develop problems where learners apply a range of common identities ie sin 60 = ( 3)/2 cos 60 = ½ tan 60 = 3 tan 45 = 1 sin 45 = 1/ 2 cos 45 = 1/ 2 sin 30 = ½ cos 30 = ( 3)/2 tan 30 = 1/ 3 sin A = cos (90 A) cos A = sin (90 A) Again, solution could be determined numerically with scale sketches used to prove their correct solution. Sine and cosine waveforms are fundamental to many areas of engineering such as AC waveforms in electrical engineering or excitation and vibration waveforms in mechanical engineering. Teachers might show learners to interpret and produce graphs from sine, cosine and tangent functions (y = sin x, y = cos x and y = tan x for a range of angles from 0 to 360 ). Learners could also determine the sine, cosine and tangent of any angle between 0 and 360. Learners could be encouraged to sketch graphs manually, or could use ICT to produce graphs and solve problems. 15

16 16 SUGGESTED ACTIVITIES LO No: 5 LO Title: Understand calculus relevant to engineering problems Title of suggested activity Suggested activities Suggested timings Differentiation graphical methods and simple algebraic expressions Differentiation of trigonometric functions Differentiation of exponentials and logs There are many practical situations in which an engineer has to analyse quantities that are changing such as the stresses in a beam, speed or velocity of an object or voltage and current in an electrical circuit. Differentiation is a technique for analysing changing quantities. Teachers might break-down the delivery of differentiation into a number of areas. Teachers might begin by introducing learners to graphical methods to solve differential problems, and to simple algebraic expressions (containing terms of the form ) Learners could practice solving problems to develop understanding. Trigonometric functions describe the behaviour of quantities in many engineering systems and their differentiation is useful in determining rate of change. Teachers might continue differentiation by introducing trigonometric functions ie y = sin. x y = a.sin.x y = a.sin.bx y = cos.x y = a.cos.x y = a.cos.bx y = a.cos.x + b.sin x, where a and b are constants Worked examples and practice problems may once again prove a useful way in which to develop understanding. Graphical explanation might be another way in which learners can embed understanding of the differentiation of sine and cosine functions. Exponential and logarithmic functions are often used to describe growth and decay in engineering systems and their differentiation (to determine rate of change) is equally as important. Learners could solve problems of the form y=e ax and y=ln ax Problems could be solved numerically and their solution proven using a sketch of the function to demonstrate understanding. OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

17 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Title of suggested activity Suggested activities Suggested timings Differentiation maxima and minima Indefinite integration simple algebraic functions Indefinite integration sine and cosine Definite integrals Turning points are useful for determining the points and values at which a function is at its maximum of minimum and find a ready application in engineering. Determination of turning points (maxima and minima) requires that the function is differentiated twice. Teachers could develop worked examples showing how to determine maxima and minima. Learners could again undertake solving practice problems with both numerical and graphical solution being used as appropriate. Integral calculus (integration) also plays a key part in analysing engineering problems. Typical examples include determining areas and volumes of objects, mean and r.m.s. values of waveforms and distances in velocity-time diagrams. There are many other useful applications. Teachers might introduce learners to integration through simple indefinite integrals of the form: y = a xn... axn dx = (xn + 1)/ n constant C The use of graphical methods to illustrate integration might prove a useful starting point to explain the purpose of integration. Teachers could show learners worked examples, with learners solving many practice problems. Teachers might continue integration by introducing the integration of common trigonometric identities i.e.: sin x dx = cos x + C cos x dx = sin x + C sin ax dx = -cos ax/a + C cos ax dx = sin ax/a + C Practice with many examples might one again prove a useful approach to developing a thorough understanding. Teachers might conclude integration by introducing problems involving definite integrals (i.e. integration between defined limits). Learners might be taught the form and rules for a definite integral including notation: a b fx = [F(x)] b a = F(b) F(a) Teachers could develop worked and practice problems for learners to solve using definite integrals. 3 hours 17

18 18 SUGGESTED ACTIVITIES LO No: 6 LO Title: Be able to apply statistics and probability in the context of engineering problems Title of suggested activity Suggested activities Suggested timings Histograms, frequency polygons and cumulative frequency curves Problem solving for a set of data mean, mode and median Problem solving for a set of data distribution, percentiles, quartiles and skew Problem solving for a set of data variance and standard deviation Statistics is about the collection, organisation, analysis and interpretation of data. It is often used when analysing quality, manufacturing and maintenance operations in engineering. Specific examples include calculating mean time to failure (MTTF) and mean time between failures (MTBF) of engineering components. Teachers might begin the study of statistics with histograms, frequency polygons and cumulative frequency curves. Learners could use engineering examples to solve problems. There are many ways in which a set of data (or values) can be analysed including determining its mean, mode and median. Teachers could develop suitable worked examples using sets of data or learners could gather real data relevant to engineering operations. Engineering examples might include quality control of machined parts, or reliability of electrical and electronic components. Problems might be solved numerically and with the aid of ICT in order to develop graphical solutions. Teachers might continue the analysis of data sets by introducing the concept of distribution of data including percentiles, quartiles and skew. Again, suitable engineering examples might be used, such as the reliability of measurement data or analysis of manufacturing production rates. Learners may also be able to gather and analyse real data, for example, for measurement of component dimensions. Practice at solving many and varied problems will undoubtedly prove useful in developing understanding. Teachers might conclude the analysis of data by introducing variance and standard deviation. Similar engineering examples might be used for example mechanical or electrical component tolerances. Learners might be tasked to research and explain the significance of variance and standard deviation in the context of quality control. See Lesson Element Problem solving for a set of data variance and standard deviation. OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING

19 OCR LEVEL 3 CAMBRIDGE TECHNICALS IN ENGINEERING MATHEMATICS FOR ENGINEERING Title of suggested activity Suggested activities Suggested timings Problem solving using probability Addition and multiplication laws of probability Probability is commonly used in prediction and analysis and is often used to model engineering systems. It is used to predict reliability of systems and their components, and in reliability-centred maintenance regimes for predicting breakdown and failures. Teachers should develop worked and practice problems relevant to engineering that include expectation, dependent events without replacement and independent events with replacement. Suitable engineering examples might include mechanical component and system reliability in terms of backup capacity, or likelihood of critical component failure in an electromechanical system. Teachers might conclude the topic area of probability by showing learners how to solve problems using the addition law of probability and the multiplication law of probability. Again, learners may be able to apply these to solving engineering examples, for example, systems in which there are multiple dependent components each with their own reliability figure. 1 hour 19

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