# Solving problems using the practical application of mathematics

Students are required to apply mathematical skills to resolve engineering problems by:

• correctly determining the solution to engineering problems

• use standard mathematical symbols, layouts and annotation

• selecting appropriate information from resources (such as data tables and formulae) to be able to evaluate engineering solutions

• selecting and applying standard mathematical techniques and methods to address real-world engineering problems

• use methods of communicating mathematical information, including formulas, tables and graphs

• analyse mathematical data

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

### Building Incubators

This set of teaching materials offer a cross-curricular approach to learning about bioengineering and the survival of premature babies. The context is designing a temperature-regulated environment to help premature babies to survive.

The activity requires students to:

• design and build a temperature controlled environment

• evaluate measuring instruments

• investigate body temperature

• investigate the effect of mass on heat loss

• use frequency graphs

• calculate surface area and volume

• interpret mortality statistics

• explore heat loss from babies

### Formula One Race Strategy

This resource, from Mathematics for Engineering Exemplars, shows the application of mathematics within F1 racing. Here students learn about the models which are used to develop race strategy, since every F1 team must decide how much fuel their cars will start each race with, and the laps on which they will stop to refuel and change tyres.

In completing the task students will:

- understand the idea of mathematical modelling
- understand the mathematical structure of a range of functions and be familiar with their graphs
- know how to use differentiation and integration in the context of engineering analysis and problem solving
- construct rigorous mathematical arguments and proofs in engineering context
- comprehend translations of common realistic engineering contexts into mathematics

Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.

### Underground sewage systems

This activity, from the Institution of Engineering and Technology (IET), challenges students to calculate the dimensions of an underground tunnel system. Students are encouraged to move beyond an ‘out of sight, out of mind’ approach to sewage as they use and develop their mathematical process skills within the real-world contexts presented.

Key ideas include:

• human impact

• environment

• water use

• waste water

• calculations

• volume of cylinder

• volume of cone

• approximation

• estimation

This resource is supported by the video Shifting Sewage.

### Shifting Sewage

This video considers the workings of the London sewerage system; a combined system in which dirty water from households and excess rainwater all ends up in the same place. When heavy rainfall fills the system it needs to be able to overflow into the river rather than the roads. Expelling sewerage into the River Thames is also a problem. Every year, 32 million cubic meters of untreated sewerage overflows into the Thames. The solution to this problem is the London Tideway Tunnels, which will connect 35 of the most polluting overflows, collect the discharge and transfer it to a treatment plant. This engineering project will be one of the largest ever carried out in London and will take eight years. This film explores the challenges involved.

### Analyse sewage tunnel design

This extension activity follows on from the How Much Waste? activity providing an engaging task to continue the learning focusing on the link between sewage and the underground tunnel system. Students are encouraged to think about the role of engineers in providing healthy sanitation and waste-water disposal systems.

Learning outcomes include:

• to develop an insight into the representation of large volumes

• to determine and select variables, then apply mathematical formulae to solve real-life problems

### Medical Emergencies

Medical emergencies is a set of teaching materials which offer a cross-curricular approach to learning about engineering. Students design and make a hanging storage device that could be folded up to make a rucksack or other carrying device. Students have the opportunity to communicate their findings in a variety of ways and look at how engineers and scientists need to be creative in their work using a range of skills and ideas.

In completing the activity students will:

• develop the design, write the specification and make the product

•consider the strength of thread and the transmittance of light

•consider the shape and position of the pocket and use scale diagrams

Whilst the activities are designed for post-16 learners, the ideas and principles are still applicable to an older audience.

### The Mathematics of Escalators on the London Underground

This resource shows the application of mathematics for the operation of escalators on the London underground. Students consider a variety of issues which include passenger numbers and flow, as well as carbon emissions, escalator speed and energy efficiency.

To complete the activity students will be required to:

- manipulate and use trigonometric functions (ratios in right-angled triangle)
- calculate percentages
- plot the graphs of simple functions using excel or other resources

### The mathematics of aircraft navigation

Aircraft navigation is the art and science of getting from a departure point to a destination in the least possible time without losing your way.

In this activity, students imagine that some people are stuck on a mountain in bad weather. Fortunately, with a mobile phone, they managed to contact the nearest Mountain Rescue base for help. The Mountain Rescue team needs to send a helicopter to save these people. With the signal received from the people on the mountain, they determine that the bearing from the helipad to the mountain and the approximate distance.

To solve the problem set students are required to use:

- vector methods (graphical and component representation of vectors, vector addition and subtraction)
- manipulation and use of trigonometric functions (sine and cosine rules, ratios in right-angled triangle)
- standard quadratic formula to find roots of quadratic equation

### Solar Detectives

Solar detectives is a set of teaching materials which offer a cross-curricular approach to learning about engineering. Students build and modify a model solar car and research the science and mathematics underlying the use of solar energy. Students have the opportunity to communicate their findings in a variety of ways and consider how engineers and scientists need to be creative in their work using a range of skill and ideas.

The Engineering a better world 15 minute video shows students exploring this case study.

In completing this activity students will:

• build and modify a model solar car

• consider how solar energy is developed

• compare the performance of different solar cars

### The Mathematics of Simple Beam Deflection

This resource, from Mathematics for Engineering Exemplars, shows the application of mathematics within a civil engineering environment. Here students apply standard deflection formulae to solve some typical beam deflection design problems. These formulae form the basis of the calculations that would be undertaken in real life for many routine design situations.

By completing the task students are required to:

- understand the idea of mathematical modelling
- construct rigorous mathematical arguments and proofs in engineering context
- comprehend translations of common realistic engineering contexts into mathematics

Detailed notes and examples are provided and there are extension activities for students to complete, together with learning outcomes and assessment criteria.