# Using arithmetic to solve engineering problems

These resources support the use of arithmetic to solve engineering problems with particular reference to perimeter and area of 2D shapes, volume of 3D shapes, and common measures. The resources support students to achieve the assessment outcomes of:

- calculating the perimeter and area of 2D shapes
- calculating the volume of 2D shapes
- solving practical problems requiring calculation with common measures.

In many cases the mathematical concepts are those found in the GCSE mathematics syllabus, but the application of these concepts in an engineering context 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.

### Estimating Length Using Standard Form

This resource is to raise awareness of the importance of estimating in engineering. **Whilst the main focus of the examples contained in the main body of the resource are at GCSE level, the principles can be applied to A level.**

Begin by asking the question "Why is estimation an important skill in engineering?” and follow this up by asking students to estimate a variety of lengths and distances. Card set A could be used for this activity. The table of distance estimations expressed as powers of ten found in the teachers’ notes may be useful here. This leads to a discussion about expressing numbers in standard form. Card sets B and C provide a useful matching exercise to practise expressing numbers in standard form. The blank cards enable students to suggest estimates of their own.

**The extension tasks on page 4.94 "What learners might do next" provides examples of how this resource can be applied to an electronics course or a mechanical engineering course.**

### Drinks Cans

This activity requires students to be able to calculate percentage waste and use mathematical thinking to reduce the percentage waste. Students are expected to use their understanding of tesselation which leads to more complex mathematical calculations using Pythagoras' theorem and trigonometry. **Whilst the mathematical topics are at GCSE level, the application of these skills is at level 3. This activity can also be extended to cover A level topics such as volumes of revolution and finding minimum surface area for a fixed volume using differentiation. Further extension can be made by investigating three dimensional packing.**

Drinks cans are made by stamping out circular discs from a sheet of tin. This activity could be given a practical element by asking students to cut circles from a rectangle of dough, with given dimensions, in such a way as to leave as little dough unused as possible. Students can weigh the original dough, the left over dough, and calculate the percentage waste. This value can be compared with the value calculated mathematically taking the area of the circles from the area of the rectangle and then calculating the percentage waste. The extension task can also be modelled, in which the waste material is used to create lids for the tins and the change in thickness of the material explored.

Students can then explore different patterns to cut out the circles from the sheet with the aim of creating less waste. Practical activity can then be modelled mathematically using Pythagoras’ theorem, and sine and cosine ratios, as seen in activities one and two.

Extension activities could consider how many tins can be made from each configuration if the waste material can be reused to make further tins, consider cutting shapes other than circles from the sheet and three dimensional packing problems such as how many tennis balls can be packed into a box.

### Coil Feed Line

Students will be familiar with rolls of kitchen roll, tin foil and cling film. The packaging on a roll of tin foil gives the width and the length of the tin foil. Challenge students to use this information to find the thickness of the tin foil. Students will be required to calculate the volume of the tin foil on the roll and understand that this volume will be the same when the roll is extended to form a very thin cuboid, hence the thickness of the tin foil can be calculated.

This task has a direct application to engineering featured in this resource in which a coil of tinplate is stored on a cylinder. The relevant information is provided including the density of the steel which entends the task to enable students to calculate the maximum weight of the coil of tinplate.

**Whilst this initial task uses GCSE level mathematics, the application uses mathematical skills beyond GCSE. The task can form an introduction to finding the volume of a shape by integrating to find the volume of revolution.**

The final activity in the resource requires students to find the percentage increase in the weight of the tinplate when other variables are changed. This can lead to students finding algebraic connections between the variables and graphing the connections.

### Reduction Mill

**The reduction mill activity requires students to use their understanding of geometric series, complex use of dimensions in an unfamiliar context and speed calcualtions. Extension work leads to the use of logarithms to solve equations.**

The reduction mill reduces the thickness of a strip of steel using a series of rollers, each roller making the steel slightly thinner.

Students could consider compound interest on savings in a bank account, circulating their savings value after one, two, three years etc. Students could then be asked, "what would happen to their savings if the bank, rather than added interest, charged for looking after your money?" Students could then be asked to calculate the percentage the bank charged if their £2,000 savings reduced to £1,920.80 in two years. The same mathematics can then be used to approach the reduction mill question.

The initial problem is then extended to look at the speed of the steel travelling through the mill. The volume of steel must leave the mill at the same rate as it entered. However, if the width of the steel stays the same throughout the process, but the thickness of the steel reduces, the speed at which the steel leaves the mill must increase. Given the speed the steel enters the mill, students are required to find the speed at which the steel leaves the mill. This increase in speed can be generalised for n rollers.

Further extension activity requires students to use logarithms to solve equations where the unknown is an index. The activity is further extended to model the process mathematically by introducing variables and finding algebraic relations between the given variables. This work can lead to graphing the relationships and analysing the graphs.

### Simple Gears and Transmission

Students are posed the question: "How are transmissions designed so that they provide the force, speed and direction required and how efficient is the design?"

Begin by considering a clock which contains a motor spinning at ten revolutions per minute. Ask students to explain why this motor could not be attached directly to the second hand of a clock. By what factor would the output from the motor have to be slowed down to make the clock work correctly?

Students then are required to understand what is meant by the pitch radius, the pitch and that gears must have the same pitch to mesh correctly. The interactive resources can be used to demonstrate this process. Students need to be confident finding the pitch of a gear using the pitch radius, circumference and number of teeth to be able to determine whether two gears will mesh correctly. Students should then return to the original clock task to determine

The accompanying resources provide visual stimulus. Further activities consider connecting three gears, torque transmission, transmission efficiency and gear boxes.

In completing this activity students evaluate expressions, work with fractions, solve problems involving ratio and proportion, understand and work with percentages, use scale drawings, simplify and evaluate expressions involving the use of indices, change the subject of the formula, solve problems involving angular motion, and convert between units of revolution speed.

### Arithmetic

**Mathcentre provide these resources which cover aspects of arithmetic, often used in the field of engineering.** They include:

- fractions and their associated arithmetic
- powers and roots
- scientific notation
- calculations involving surds
- using standard form
- understanding and drawing the graph of a function

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 arithmetic, will find them useful, as each topic includes a selection of questions to be completed, for which answers are provided.