Using algebra to solve engineering problems
These resources support the use of algebra to solve engineering problems with particular reference to the:
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use of equations to solve engineering problems
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manipulation of equations to change the subject
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simplification of equations and functions
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indices
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quadratic equations
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simultaneous linear equations
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partial fractions
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interpret changes in engineering systems from graphs
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expressing equations of a straight line, trigonometrical and exponential functions using graphs
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rules of indices and laws of logarithms, including changing the base
In some cases the mathematical concepts are those found in the GCSE mathematics syllabus, but the application of these concepts in an engineering concepts requires skills beyond GCSE level. It is vital that students have the ability to apply the mathematics they know in unfamiliar and more challenging contexts. This will thoroughly test their mathematical understanding in preparation for tackling extension tasks at level 3 in other areas of the mathematics curriculum
Air Tracking of a Ground Object
Students are asked the question: "How can a camera, mounted on a helicopter, be used to track the location of a ground-based object?"
The discussion of ideas will touch on many different aspects of mathematics including 2D and 3D co-ordinates, the calculation of distance and vectors, but the main focus of this activity is the algebraic skills of being able to:
- understand how 2D and 3D co-ordinate geometry is used to describe lines, planes and conic sections within engineering design and algebra
- understand the methods of linear algebra
- know how to use algebraic processes
- comprehend translations of common realistic engineering contexts into mathematics
There are two interactive files to accompany the tasks. The first practises using the scalar product, the second graphs the distance from a fixed point to a helicopter moving in a straight line.
Calculating Power of JCB Dieselmax Engines
This resource shows the application of mathematics in mechanical engineering and construction machinery. Students encounter the formulae used to calculate the power of the engine which was used to power the JCB Dieselmax LSR car to a world land speed record of 350mph in August 2006.
The activities cover:
- the calculation of power
- the plotting of engine power vs engine speed
- calculating the equation of a straight line
Extension activities require students to:
- create a spreadsheet to plot automatically the power curve of an engine
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compare the power curves of a turbocharged engine and a non-turbo charged engine
Detailed notes and examples are provided, together with learning outcomes and assessment criteria.
Functions and Graphs
These resources cover aspects of functions and graphs often used in the field of engineering. They include descriptions of the hyperbolic function and identities, the logarithm function and its graph as well as the graphs of the trigonometric functions.
Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of functions and graphs will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.
Power Demand
One important area of civil engineering is electrical power production. In order to plan for future building, which may take many years to prepare, design and construct, demand forecasts are often used to indicate the quantity and size of new power stations required.
Students are asked: "How can you predict future power requirements? " To solve the problem, students are required to complete a table by substituting values into a formula and plot a graph. The interactive file can be used to demonstrate some of the important aspects of growth and decline. The activity offers good opportunities to consolidate work on geometric progression.
The mathematics covered in this activity are:
- be able to write the rule for a sequence in symbolic form
- change the subject of a formula
- be able to plot data
- be able to draw a graph by constructing a table of values
- solve problems using the laws of logarithms
- solve problems involving exponential growth and decay
F1 Challenge
In this set of teaching materials students are required to build and modify a model F1 car. The challenge involves the construction and testing of a scale model F1 car powered by compressed gas.
Teachers' notes and the starter activity are included. Note that the web address for the F1 challenge is www.f1inschools.co.uk which provides the latest information about the challenge and the resources available. Video clips that might be used in the starter activity are available from this website.
In the starter activity students have to decide whether their design is built for speed or acceleration.
Students are required to consider friction, mass, and streamlining to help in the construction of an effective vehicle. Skills of measurement, analysis, and application are required in order to simulate acceleration.
Students then calculate the energy of moving objects to show how important it is to understand how the mass of the model and the energy of the gas engine will influence speed. Students use distance-time graphs, use formulae connecting force, mass and acceleration and analyse graphical data in order to refine and improve the design for the model car.
Formula One Race Strategy
This resource shows the application of mathematics within F1 racing. Students are required to use mathematical models to develop race strategy, deciding how much fuel cars will start the race with and the laps on which the car will stop to refuel and change tyres.
Students deal with formulae, rates such as fuel consumption, the effects of weight on these rates, and lap times. Extension activities require students to use the solution to an integral formula to calculate different stint times and to analyse different strategies.
Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.
Heat Loss from Buildings
In this activity, students are asked the question: "How can the most efficient design be determined, taking both building and running costs into account?"
Students consider thermal conductivity of different materials graphically to help decide which material should be used. There follows an explanation of the concept of kilowatt hours.
A video accompanies the resource explaining thermal conductivity.
The mathematics covered in this activity is:
- solve problems involving area, perimeter and volume
- use scale drawings
- work with formulae for the areas and perimeters of plane shapes
- work with formulae for surface areas and volumes of regular solids
- be able to draw graphs by constructing a table of values
- be able to extract information from a graph
Understanding the Motion of the Wheels
This activity features the application of mathematics within the field of mechanical engineering by exploring the design and construction of wheels, to be used to challenge a land speed record. These principles also apply in engines and gear-boxes, which typically consist of rotating machinery and understanding the loads imposed on components is of great importance for safety considerations.
The mathematics used by students in this activity includes:
- using frames of reference
- parametric equations of a circle and cycloid
- the calculation of velocity and acceleration to explore acceleration
- the equations relating to forces and stresses on a wheel
Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.
Algebra
These resources cover a wide range of algebraic topics, many of which are used in the field of engineering. The topics covered are:
- solving linear equations
- solving simultaneous equations
- solving quadratic equations
- dealing with inequalities
- the modulus symbol
- graphical solutions of inequalities
- the laws of logarithms, the exponential function and solving equations
- Sigma notation
- partial fractions
- rearranging formulas
- factorials and
- the laws of indices
Comprehensive notes, with clear descriptions, for each resource are provided, together with relevant diagrams and examples. Students wishing to review, and consolidate, their knowledge and understanding of algebraic principles will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.
Rearranging Equations
These materials cover two sessions designed to take students through the basic principles of manipulating and solving equations. The students work through building an equation, checking the equation, and solving the equation. They are then asked to create their own equation and swap it with a partner to be ‘undone’ step. Finally, students move to an activity which uses cards to further reinforce how equations are rearranged.
The resource includes detailed lesson plans and teacher notes for the sessions which are designed that students are able to:
- develop confidence with the notation used in equations develop the use of brackets by creating and solving equations
develop the skills needed to change the subject of a range of different equations
- develop an understanding of the nature of an equation and the principles that are applied when rearranging them
- learn from each other
