# Linking of mathematics to construction and engineering

There is a wide range of mathematical topics used in construction and engineering.

**Constructions** takes the mathematical description of the word construction, exploring mathematical constructions and scale diagrams in different contexts; skills which are vital when working with plans. The resource also stresses the importance of accuracy in calculations. This is a good lead in to the resource **Architect's Skills** in which students have to describe 3D models in two dimensions and decode plans. **Mathematics From Outdoors** and **Outdoor Trigonometry** continue the theme exploring more complex mathematical topics such as Pythagoras' theorem, trigonometry and circles. **Building a Town **and **Building Silos** are practical based activities in which students realise a three dimensional structure based upon two dimensional planning.

**Heat analysis** and **Heat Loss from Buildings** consider the mathematics used in other areas of construction such as the use of formulae, calculating perimeters, surface areas and volumes and being able to extract information from graphs. **Rigid Structures** is a practical activity designed to develop and use problem solving skills whilst **A Bridge Too Many** requires team work to build a model to demonstrate that their calculations work.

**Sewage Tunnels**, **How Much Waste** and **How Much Sewage** are part of a suite of resources exploring the mathematics used in some of the unseen areas of construction. Activities provides a context for work on volume of three dimensional shapes, approximation, estimation and a consideration of the wider implications of construction.

**The Mathematics of Escalators on the London Underground** provides a context for consideration of rates and speed. The list contains two short videos in which a Structural Engineer and a Civil Engineer talk about the jobs they do and the mathematics they use.

### Constructions

This resource looks at the military’s use of constructions and mathematical accuracy in warfare and peacetime defence.

Students are asked to draw constructions to represent how the military target hostile forces, stressing the importance of accuracy in calculations which consider projected paths. They will need to be able to construct scale drawings, an angle bisector, a perpendicular bisector, a triangle and a variety of loci.

### Architect's Skills

This activity **Architect's Skills** is designed to develop the process skills involved in understanding and making sense of mathematical information. Students have to give unambiguous descriptions to enable their partner to build an identical copy of a hidden shape, whilst the partner has to listen and interpret the descriptions accurately.

**Different views** is designed to engage pupils in visualising a three dimensional object from limited but sufficient information, given in code, for the front, side and plan views of shapes. Students then progress to **Decoding plans**, where they are invited to build 3-D models from the codes for each shape.

### Mathematics From Outdoors

This resource is designed to help teachers to develop classroom activities that will interest students and enable them to link the mathematics they learn with the world around them. The book covers many aspects of geometry from a practical perspective. Highlights include:

Fixing positions - an exploration of ways of identifying location. Mathematical topics covered are coordinates, estimating and measuring distances, constructing perpendiculars to find shortest distances and measuring angles

Uses of triangles - practical situations which explore the mathematical topics of angles of elevation, scale diagrams, trigonometry, area of triangles, angles in a triangle and angles in a polygon

Special triangles - looks at geometric construction of equilateral triangles and isosceles triangles plus the uses and ideas behind Pythagoras’ Theorem

Circles and other interesting curves - explores the need to find areas of circles, sectors of circles, tangents, radii and shows how to construct the parabola, hyperbola, cardioid and ellipse

### Motivating Maths at GCSE: Outdoor Trigonometry

In this video Andy Grice, teacher of mathematics at Brune Park Community College in Gosport, delivers a practical lesson on trigonometry to a class of GCSE students. Working in groups, they first estimate the height of four buildings on the school site. They then measure actual distances and angles which are used, back in the classroom, to calculate the heights of the buildings and compare these to their original estimates. Feedback from the students is positive and Andy and his colleagues analyse the success of the lesson and discuss some of the issues that emerged.

### Building a Town

In Building a town, students are encouraged to consider the mathematical shapes which form the net of a building. Some simple models are shown and students could use these to construct their nets, before building the models themselves.

This activity could be extended so that students produce models of buildings from their area. Buildings with sloping roofs will offer opportunities for experimenting, measuring, or applying Pythagoras’ theorem, in order to draw accurate nets, whilst conical roofs could provide opportunities to work with the formula for the circumference of a circle.

### Building Silos

This task offers pupils the opportunity to develop their personal learning and thinking skills. It involves using recycled materials to construct a scale model of a silo and is well suited to group work. The task can be extended for more able students by investigating the resizing of their model to accommodate an increased capacity.

Topics covered by this task include scale, ratio, measurement, volume of cylinders, unit conversions, graphs and substitution into formulae.

### Heat Analysis

This resource requires students to investigate heat loss from a building. The introduction looks at fabric heat loss (external walls, windows, roof etc.) and ventilation heat loss through cracks and openings in the external building fabric (around window frames, door, etc.).

Formulae for each type of heat loss are given and students apply this information to calculate the total heat loss from information about a specific building in winter conditions. A table to record results is given and questions regarding the reduction of carbon emissions, consideration of the major sources of heat loss as well as where and how it might be possible to reduce these values are posed.

This resource includes information which students could use, as an extension to the initial heat analysis investigation, to estimate the total heat loss from their own dwellings.

Thermal transmittance of a variety of elements is given for current and pre-2002 building regulations. The difference between the two sets of results will give an indication of how improvements to the fabric of their dwelling might reduce energy consumption and running costs.

### Heat Loss from Buildings

This resource asks the question: how can the most efficient design be determined, taking both building and running costs into account?

The introduction considers the 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

### Rigid Structures

This activity looks at the exciting mathematics connected with the regeneration going on in our cities. The three activities are based around the construction of rigid structures and provide opportunities for practical work and mathematical problem solving.

Finding rigidity introduces students to the principle of rigidity and provides the opportunity to study images of buildings and identify what makes them rigid, before investigating local buildings of their own choosing.

Testing rigidity and Bracing squares engage students in building structures using geostrips and challenges them to apply their experience to predict if diagrams of structures are rigid, before building them to test their conjectures.

### Sewage tunnels

This activity, from the Institution of Engineering and Technology (IET), challenges students to use their mathematical skills to calculate the critical dimensions of the tunnel that will be built. It could also be taught in design and technology or engineering, with the emphasis on how the development will affect the quality of life and how it could be realised.

The activity challenges students 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. This activity provides a context for work on volume a cylinder, volume of a cone, area of a circle and surface area of a cone.

This resource is supported by a full set of activities in the Shifting Sewage collection and by the video Shifting Sewage.

### How Much Waste?

This activity 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 a full set of activities in the Shifting Sewage collection and by the video Shifting Sewage.

### How Much Sewage?

This activity follows on from the How Much Waste? activity and provides an engaging task to continue the learning focusing on the link between sewage and the underground tunnel system. It encourages students to think about the role of engineers in providing us with 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

This resource is supported by a full set of activities in the Shifting Sewage collection and by the video Shifting Sewage.

### Structural Engineer

Katie Symons creates modern buildings and structures. This Department for Education video clip demonstrates the importance of STEM subjects in achieving a career in structural engineering.

Katie describes how her work involves designing buildings, which involves a great deal of science, engineering and mathematics skills. She explains how her role is to work with architects to help make their building designs come to life. This involves seeing the whole project from the early design stages to the final look of the building.

Katie says, "It's a really good way of using maths and science in designer buildings. I use maths and physics every day in my job when I'm designing building structures.''

The company she works for is currently involved in the re-development of Anfield Football Stadium, designing the biggest theme park in the world in Dubai and designing a skyscraper in St Petersburg.

### A Bridge Too Many

A Bridge too Many is a set of teaching materials which offer a cross-curricular approach to learning about engineering. The context for the activities is a challenge based on a large model bridge that has been pre-constructed using prefabricated components, with weak pins joining the structure together. The challenge is to remove the bridge from over a railway without dropping anything on the tracks.

Students, working in teams, have to use the information supplied (drawings, technical information on the weight, strength, etc., of the girders), information on how to calculate the loads in a pin-jointed framework, information on the weights of the various pipes, conveyor components, cladding sheets, technical data on a range of cranes and so on, to plan the dismantling and removal of the bridge. They should prepare a detailed report to the Railway Safety Officer (the Tutor), who approves or rejects it. Students have a set of components available if they wish to construct a model to confirm the results of their calculations experimentally.

Once the plan has been prepared, the team is required to remove the model bridge from over the railway, without accident. Successful completion of the task can result in a form of certification for the students.

### Liz Bland: Chartered Civil Engineer

In this video, Liz Bland talks about her work as a civil engineer. Liz manages the day-to-day running of construction sites, ordering materials, managing budgets and ensuring that jobs are completed in time. She enjoys problem-solving and finds the work very rewarding.

Liz talks about what made her consider a career in civil engineering and the qualities needed in her job. She described her ambition to work on major, prestigious construction programmes in the future.

### 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.

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