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Introduction and guidance

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Succeed with maths – Part 2 is a free badged course which lasts 8 weeks, with approximately 3 hours' study time each week. You can work through the course at your own pace, so if you have more time one week there is no problem with pushing on to complete another week’s study.

You'll start with looking at different units of measurement for length and how to convert between these, before moving onto the same for volume and mass (known as weight in everyday language). In following weeks you’ll move on to look at scientific notation, basic geometry and ways of representing and analysing data. You’ll use plenty of real-life examples to help with this and get plenty of opportunities to practise your new understanding and skills.

Part of this practice will be the weekly interactive quizzes, of which Weeks 4 and 8 will provide you an opportunity to earn a badge to demonstrate your new skills. You can read more on how to study the course and about badges in the next sections.

After completing this course you will be able to:

understand the SI and imperial systems of measurement
understand and use scientific notation in different situations
describe and calculate the basic properties of circles, four sided shapes and triangles
understand some different ways of representing and analysing data.

Moving around the course

The easiest way to navigate around the course is through the 'My course progress' page. You can get back there at any time by clicking on 'Go to course progress' in the menu bar. From the quizzes click on 'Return to Succeed with maths − Part 2'.

It's also good practice, if you access a link from within a course page, including links to the quizzes, to open it in new window or tab. That way you can easily return to where you've come from without having to use the back button on your browser.

Viewing maths and equations

As you view the different text styles and symbols used to display maths throughout the course, you may notice that some of the text appears clickable or linked to something else. This is part of the equation processing software used by The Open University. You can ignore these clicks or links as they won't provide you with any useful information, but if you do click by accident you can just come back to where you were by using the back button.

What is a badged course?

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

While studying Succeed with maths – Part 2 you have the option to work towards gaining a digital badge.

Badged courses are a key part of The Open University’s mission to promote the educational well-being of the community. The courses also provide another way of helping you to progress from informal to formal learning.

To complete a course you need to be able to find about 24 hours of study time, over a period of about 8 weeks. However, it is possible to study them at any time, and at a pace to suit you.

Badged courses are all available on The Open University's OpenLearn website and do not cost anything to study. They differ from Open University courses because you do not receive support from a tutor. But you do get useful feedback from the interactive quizzes.What is a badge?

What is a badge?

Digital badges are a new way of demonstrating online that you have gained a skill. Schools, colleges and universities are working with employers and other organisations to develop open badges that help learners gain recognition for their skills, and support employers to identify the right candidate for a job.

Badges demonstrate your work and achievement on the course. You can share your achievement with friends, family and employers, and on social media. Badges are a great motivation, helping you to reach the end of the course. Gaining a badge often boosts confidence in the skills and abilities that underpin successful study. So, completing this course should encourage you to think about taking other courses.

How to get a badge

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Getting a badge is straightforward! Here's what you have to do:

read each week of the course
score 50% or more in the two badge quizzes in Week 4 and Week 8.

For all the quizzes, you can have three attempts at most of the questions (for true or false type questions you usually only get one attempt). If you get the answer right first time you will get more marks than for a correct answer the second or third time. Therefore, please be aware that for the two badge quizzes it is possible to get all the questions right but not score 50% and be eligible for the badge on that attempt. If one of your answers is incorrect you will often receive helpful feedback and suggestions about how to work out the correct answer.

For the badge quizzes, if you're not successful in getting 50% the first time, after 24 hours you can attempt the whole quiz, and come back as many times as you like.

We hope that as many people as possible will gain an Open University badge – so you should see getting a badge as an opportunity to reflect on what you have learned rather than as a test.

If you need more guidance on getting a badge and what you can do with it, take a look at the OpenLearn FAQs. When you gain your badge you will receive an email to notify you and you will be able to view and manage all your badges in My OpenLearn within 24 hours of completing the criteria to gain a badge..

Get started with Week 1.

Introduction

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Welcome to the first week of Succeed with maths – Part 2. The first two weeks of this course will focus on measurement and the units that are used to communicate this.

People use numbers to solve a wide variety of different problems, including when designing buildings, navigating and working out the fuel consumption of an aircraft. These examples all use measurements that involve distance and it is these types of situations this week will be concentrating on. For this, it is important that you have a working knowledge of various units of measurement and the two different measurement systems used in the UK – The Système Internationale (SI) or International System, on which the metric system is based, and imperial. Both of these will be covered over the next two weeks, and this week will look in particular at measuring length.

Watch the course author Maria Townsend introduce Week 1:

Download this video clip.Video player: swim2_week_1_1080.mp4 >

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Transcript

Maria Townsend:

This is my height measured in books, not a common unit of measurement but still an interesting one. This does give us a problem though. If I measure myself again using other books I’ll get a different answer as they’re not all the same size.

So to use the book as a unit of measurement I’d always have to use the same book and tell everybody what that book was. And something similar is true for all measurements as you’ll find out this week.

End transcript

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After this week’s study, you should be able to:

understand how the International System (SI) is formulated
understand the more common SI units for length and convert between them
recognise the more common imperial units for length and convert between these
understand how the SI and imperial units of length relate to each other.

If you haven’t seen Succeed with maths – Part 1 yet and would like to study this first please follow the link: Succeed with maths – Part 1.

The Open University would really appreciate a few minutes of your time to tell us about yourself and your expectations for the course before you begin, in our optional start-of-course survey. Participation will be completely confidential and we will not pass on your details to others.

1 Using units of measurement

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Everyday problems where you want to determine how much there is of a quantity involve measurement. Things are measured for a variety of different reasons, but in everyday life this is usually to find out how big, how long or how heavy something is, or its volume, length or mass.

For these numbers to have any meaning, you need to specify what units the measurement is in. If you were told that it was ‘three’ to the nearest hospital that wouldn’t be much good unless you knew the units as well. Is that three feet, three metres, three miles or three kilometres? Each is a very different distance from you. So, it is very important when dealing with measurements to always state the units being used.

The units will also tell you what system of measurement has been used. Most of the scientific world has adopted an internationally agreed system of measurement, using metric units such as metres and litres. The rules for their use form the Système Internationale (International System), simply known as the SI, whereas in everyday life you may still be using some imperial units, such miles and pints. The system you prefer will depend on what you are most comfortable with. This course looks at both systems of measurement because you probably encounter both in your daily life. This may well also be the case in any university level study. Although, most work will be carried out using the SI, it may be that work from other countries, such as the USA, and historical studies will employ imperial measures.

The next section starts with a look at how the SI is put together generally and then moves onto the SI and imperial units when working with length.

2 The International System (SI)

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

One major advantage to the SI is that everything you could ever want to measure can be measured using a few basic units, or combinations of them. As well as this, the different sizes of units to measure the same quantity, such as a distance, are all based upon the number ten. This makes calculations with SI units relatively straightforward compared to the imperial system, where there was no standard relationship between units, as you’ll see later.

Most of the scientific and engineering world agreed to use the SI system to standardise their work. However, there are some important exceptions to this, and the USA is one of these. This means that it is very important to check the units being used if an international team is working on a project, as the following costly example demonstrates:

In 1999, a NASA Mars Orbiter spacecraft was destroyed on arrival in the Martian atmosphere at a loss of $125 million. An inquiry established that the flight system software on board the Orbiter was written to calculate thruster performance in metric newtons (N), but mission control on earth was inputting course corrections using the imperial measure, pound-force (lbf). One newton is about 0.22 pounds-force, so there was a considerable difference between the two. An expensive mistake to make!

The SI system works with a combination of base units and prefixes. An example of a base unit is the metre – this is the unit of measurement on which the other length units are based. You may well recognise some of prefixes, such as ‘kilo-’, ‘centi-’ and ‘milli-’. Combined with the metre, these give kilometre, centimetre and millimetre respectively. All these prefixes have specific numerical meanings, as shown below:

‘kilo-’ a thousand or 1000
‘centi-’ a hundredth or 0.01
‘milli-’ a thousandth or 0.001.

When they are added to a base unit, such as the metres in our example, they alter the size of the unit by an amount defined by the prefix.

So, kilo combined with a base unit means a thousand times the size of the base unit, centi means a hundredth of the size and milli means a thousandth of the size of the base unit.

This idea extends to cover all seven SI base units, which are shown in Table 1:

Table _unit2.2.1 Table 1 The seven base SI units

Unit name	Unit abbreviation	Measurement
metre	m	distance
kilogram	kg	mass
second	s	time
ampere	A	electrical current
kelvin	K	temperature
mole	mol	number of particles
candela	cd	light intensity

In these two weeks of study you will be using the units for distance and mass from the SI and volume from the related metric system. The SI unit for volume is based upon the metre, but in everyday situations the litre is used, since that is a more appropriate size. If you move on to study science or engineering you will come across some, if not all, of the other SI base units.

For now though, it is time to look at measurements of length.

3 Length: SI units

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Length is one of the most common measurements that is used every day. This can tell you how far away the nearest town is, the width of a fridge or your height. In science it can be used on very different scales to measure the size of the universe, or at the other extreme, the diameter of an atom.

The base unit for length in the SI is the metre, abbreviated with a lower case m. An upper case M has a very different meaning, it is the prefix for a million times larger, so care needs to be taken with this! For those of you who are more familiar with imperial measurements, a metre is very roughly the same size as a yard.

Putting the metre together with the prefixes covered in the previous section gives:

kilometre (km) – one thousand times bigger than a metre
centimetre (cm) – a hundredth the size of a metre
millimetre (mm) – a thousandth of the size of a metre.

This not only gives an idea of the size of these units but also how they relate to each other; that is, how many centimetres and millimetres there are in a metre and how many metres there are in a kilometre. This is important knowledge for when you want to change between units – usually known as converting.

So, if one centimetre is a hundredth of a metre, that means one metre must contain 100 centimetres. Similarly, one metre contains 1000 millimetres and one kilometre is the same as 1000 metres.

This can be summarised as follows:

1 km = 1000 m
1 m = 100 cm
1 m = 1000 mm.

or in a diagram that shows how to change between units, as in Figure 1 below:

Figure _unit2.3.1 Figure 1 Relationships between SI units of length

This should make sense from what has been established so far about how the different units are connected.

For example, looking at kilometres, you know that there are 1000 m in 1 km, so in 2 km there will be 2000 m or 2 x 1000 m. So, to change from kilometres to metres you just need to multiply by 1000, as shown in Figure 1.

You’ll look at how to convert between different SI units of length in more detail in the next section.

3.1 Converting between SI units and length

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

You started to consider converting between different SI units of length in the last section. Take another look at Figure 1 below, showing how these relate to each other:

Figure _unit2.3.2 Figure 1 Relationships between SI units of length

Notice that to convert from a physically larger unit, such as kilometre (km), to a physically smaller unit, such as metre (m), you always multiply. This makes sense – you will need a lot more of the smaller units to make one of the larger units! On the other hand, to convert a measurement in a smaller unit to a larger unit, you always divide.

Another way to think about this is whether you expect your final answer, when converting between units, to be a bigger or smaller number than the original value you had. If it should be a larger number, then multiply, if smaller, then divide.

Let’s see how this might work in an example:

A girl’s height has been measured in metres, but for a school project this needs to be shown in centimetres. Her height was measured as 1.32 metres.

Looking at Figure 1, to convert from metres to centimetres you need to multiply by 100. That makes sense, as a centimetre is a physically smaller unit than a metre, so you would expect the final answer to be a bigger number than the original value that you started with. So the

Use these ideas to have a go at this first activity of the course.

Activity _unit2.3.1 Activity 1 SI units of length

Allow approximately 5 minutes

a.Write 6.78 metres in centimetres.

Comment

Do you expect a bigger or smaller number for your answer?

Answer

a.There are 100 cm in 1 m and you are converting from a larger unit to a smaller unit, so multiply by 100.

b.Write 327 centimetres in metres.

Answer

b.There are 100 cm in 1 m. This time you are converting from a smaller unit to a larger unit, so divide by 100 to change cm into m. Hence,

c.The distance to your friend’s house is about 2324 m. How far is this in kilometres?

Answer

c.There are 1000 m in 1 km. You are converting from a physically smaller unit to a physically larger unit, so divide by 1000 to change metres into kilometres. So,

Well done, you’ve just completed your first activity for the week and the course. Many people find converting between units a tricky skill to master, but using the ideas introduced here should help you to feel more confident with this. You’ll be getting lots of practice over this week and the next to build on this as well.

The next section introduces imperial units of measurement before turning your attention back to length.

4 Length: imperial measurements

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Before the 1970s, the UK used the imperial system of measurement, which had its basis in historical measurements and the need to have common weights and measures to enable the sale of goods and services to operate efficiently. For example, the foot, a measurement of length of around 30 cm (or the length of standard ruler), was first defined in law by Edward I in 1305, and is thought to be derived from the length of a man’s foot with shoes.

For those not brought up using the imperial system for measurement it may seem to be a very difficult way to measure things. It does not operate on a system of base units and standard prefixes, like the SI, so this means that there are lots of different relationships to remember for each set of measurements. These are also not based upon the number ten (as the SI is), so calculations and conversions between units isn’t quite so straightforward.

However, the same techniques can be used to help decide whether you need to divide or multiply when converting, as you’ll see.

The common units used for measuring length in the imperial system are inches (in), feet (ft), yards (yd) and miles (mi). These units are listed in increasing order of size.

If you haven’t worked in these units before you may not have a good idea of their actual sizes. Table 2 below shows approximate values for how the imperial units relate to the SI units to help with this.

Table _unit2.4.1 Table 2 Size of imperial units of length

Imperial unit	SI unit
1 inch	2.5 cm
1 foot (singular of feet)	30 cm
1 yard	0.9 m
1 mile	1.6 km

Now back to how these imperial units of length relate to each other.

There are:

12 inches in one foot
3 feet in one yard
1760 yards in one mile.

So as you can see, no regular relationship based on tens here!

This means that when you convert between the different units of length it becomes more important to think carefully about the answer that you will be expecting. Should it be bigger or smaller than the number you started with?

Let’s have a look at an example before you have a go yourself. If you have a photograph that measures eight inches by ten inches, as shown in Figure 2, what is the total distance around the photo in feet and inches?

Figure _unit2.4.1 Figure 2 What is the perimeter?

The top and bottom of the photo have the same measurement, eight inches. Similarly, the left- and right-hand sides share a measurement of ten inches.

Or as there are two sides measuring 8 inches and two sides measuring 10 inches:

Now looking at how many feet there are in 36 inches:

You already know that:

So if a measurement is in feet instead of inches, the converted number should be smaller than the original. This means the number of inches has to be divided by 12. So,

This means that 36 inches is the same as 3 feet. Thus, the length of the border of the picture is 3 feet.

Now it’s your turn. Remember to think about the size of the final answer you are expecting.

Activity _unit2.4.1 Activity 2 Length in imperial units

Allow approximately 10 minutes

For each of the following scenarios, use the appropriate operation, multiplication or division, and unit to determine the answer. Click on ‘reveal comment’ if you would like a hint to get going.

a.The height of a three-year-old girl is measured as 34 inches. Over her lifetime, she grows another 34 inches. When fully grown, how tall is she in feet and inches?

Discussion

If you have studied Succeed with Maths Part 1 you may remember that drawing pictures can help to visualise a situation.

Answer

a.The girl is 34 inches when she is three and then grows another 34 inches before reaching her full height.
There are two ways to calculate the final height, and either is perfectly fine!
Or
So for a measurement using feet instead of only inches, the number should be smaller than the original. This means the number of inches has to be divided by 12.
This is the same as 5 feet and 0.67 feet.
Now convert 0.67 feet into inches.
This time you are converting from a larger to a smaller unit, so you need to multiply.
This means 68 inches is the same as 5 feet and 8 inches.

b.A rectangular garden measures 12 feet by 18 feet. What is the length of the border of the garden in yards?

Answer

b.Here’s a quick sketch of the garden to help visualise the problem:
Figure _unit2.4.2 Figure 3 A rectangular garden measuring 18 feet by 12 feet
Just as with the photo frame problem, to calculate the length of the border you need to first add the length of all the sides together.
As you are converting from a smaller to a larger unit, you would therefore expect the converted answer to be smaller than the original value. Hence, you need to divide in this case.
So the
Thus, the length of the garden’s border is 20 yards

Now you’ve looked at units of length in both the SI and the imperial system, it’s time to look at converting between the systems of measurement in the next section.

5 Relating the SI and imperial systems of measurement

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This section of the week has two aims: to give a better idea of the size of unfamiliar units and to provide more practice at converting between units. You can’t get too much practice at this as it is one of those skills that just needs a bit of extra work to perfect.

In order to be able to do this, you need to know in a bit more detail about how units in the SI relate to units in the imperial system. So, listed below are some of the more common conversion factors:

1 metre = 39.4 inches
1 metre = 1.1 yards
1 yard = 0.9 metres
1 mile = 1.6 kilometres
1 inch = 2.54 cm.

A practical way that you can help yourself become more familiar with units that you don’t usually use is to use them in your everyday life. So, for example, if you measure materials for DIY projects in millimetres, see what this would be if you used inches instead. This should help you to get a much better feel for the units than just looking at a lot of numbers!

Now it is time to use this information.

5.1 Converting between different unit systems

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Many people find converting between units quite challenging at first – but the more you practise, the easier it will become. Hopefully, you are already finding this yourself. The process that you have been following has three steps:

Find out how the two units are related to each other – the ‘conversion factor’.
Ask yourself if the final answer should be bigger or smaller than the original value.
Divide or multiply by the conversion factor to get your answer.

If you are lucky, and with practice, you may be able to miss step two out altogether – but that may depend on what you are trying to convert.

The process to use when converting between metric and imperial, or imperial and metric, is just the same as outlined above.

Let’s look at an example to see this approach in action.

A man's height

A man measures 1.8 m and needs to know his height in feet and inches.

The first step is to find out how the units relate to each other. From the previous section, 1 m is the same as 39.4 inches.

This means that the answer in inches must be bigger than the original value, so multiply by 39.4. So, the

Now, this needs to be converted to feet and inches. There are 12 inches in 1 foot, so this time the conversion is from a physically smaller unit to a physically larger unit and the answer will be smaller. So now divide by 12 to convert from inches to feet.

Now you know that 1.8 m is the same as 5.91 feet, which is 5 feet and 0.91 of a foot. However, the final answer should be in feet and inches, so now find out what 0.91 of a foot is in inches.

This is a conversion from a larger to a smaller unit, so multiply by the conversion factor in this case.

Putting this altogether, a man who is 1.8 m is about 5 feet 11 inches.

You can apply the same technique for any units, so see how you get on with these next activities. Take your time and think carefully about whether you need to divide or multiply.

Activity _unit2.5.1 Activity 3 Converting length measurements between systems

Allow approximately 10 minutes

a.The distance between Fort William and Glasgow is about 108 miles. What is this distance in kilometres, to the nearest km? Click on ‘reveal comment’ if you feel you need more guidance.

Discussion

1 mile = 1.6 km

Answer

You know that 1 mile = 1.6 km

This means converting from a physically larger unit to a smaller one, so the expected converted answer will be larger. Hence, you need to multiply by the conversion factor.

So, the distance from Fort William to Glasgow is 173 km.

b.The North Downs Way is a footpath in the South of England. The total length of the route is about 250 km. What is this distance in miles?

Discussion

1 mile = 1.6 km

Answer

b.You know that 1 mile = 1.6 km
This means converting from a physically smaller unit to a bigger one, so the expected converted answer will be smaller. Hence, you need to divide by the conversion factor.
So, the North Downs Way is 156 miles long.

c.A reel of thread contains 182 yards of cotton. How long is the thread in metres, to the nearest metre?

Discussion

1 yard = 0.9 metres

Answer

c.You know that 1 yard = 0.9 metres
This means converting from a physically smaller unit to a larger one. However, the conversion factor is less than one so the expected converted answer will be larger. Hence, you need to multiply by the conversion factor.
So, the reel has 164 metres of cotton on it.

You’ve got one final activity to look at before finishing this week. This brings together your new conversion skills, as well as giving you a refresher on problem solving from Succeed with maths – Part 1.

6 Home improvement project

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This section looks at a real-world problem that uses measurement. The following instructions were given on a DIY store’s website for calculating how many rolls of wallpaper are needed to decorate a room:

Step 1: A standard roll of wallpaper is approximately 33 ft long and 1 ft 9 in wide. If you measure the height of the walls from the skirting board to the ceiling, you can determine how many strips of paper you can cut from a standard roll – four strips are about average.
Step 2: Measure around the room (ignoring doors and windows) to work out how many roll widths you need to cover the walls. Divide this figure by the number of strips you can cut from one roll to calculate how many rolls you need to buy. Make a small allowance for waste.

Now try this next activity.

Activity _unit2.6.1 Activity 4 Following the instructions

Allow approximately 5 minutes

Read through the directions above. Write down the important information you need to know to work out the number of rolls of wallpaper required. Remember, you can click on ‘reveal comment’ for additional help.

Discussion

Try drawing a sketch, and think through the steps you would take to apply new wallpaper in your kitchen.

Answer

This is how Rebecca, a student, tackled the problem. Her notes are given below.

There are three calculations:

working out how many strips of paper you can cut from a standard roll
working out how many roll widths there are round the room
calculating how many rolls are needed.

A roll of wallpaper measures 33 feet in length and 1 ft 9 in, in width.

You may have a slightly different answer here, but as long as the main points convey the same ideas then that is fine!

Now use your notes from this activity to complete this next one.

Activity _unit2.6.2 Activity 5 How many rolls?

Allow approximately 10 minutes

The room that you want to wallpaper measures 3.2 m by 4 m and the height of the walls is 2.34 m. Work out how many rolls of wallpaper you will need.

Click on ‘reveal comment’ if you would like a quick hint.

Discussion

As well as using your notes, drawing a diagram of the room from above and one wall, may help you to visualise what you need to do.

Answer

The first thing you need to do is to make all the units the same. The room has been measured in metric and the wallpaper is imperial. It doesn’t matter if you changed from imperial to metric or metric to imperial!

Measurements for wallpaper in metric units:

For these purposes this can be called 10 m. As the assumption is that each roll is slightly shorter rather than longer, buying too few rolls would not therefore cause a problem.

Number of strips from one roll?

The first calculation is to work out how many lots of the height (2.34 m) you can get from a roll of wallpaper 10 m long.

As 0.27 of a strip is not very useful, so it is best to say that you can get 4 strips from each roll.

How many strips are required to cover the room?

It is helpful to draw a quick sketch to show what the walls would look like if they were flattened out into one big wall:

Figure _unit2.6.1 Figure 4 A rough sketch representing how the walls would look as one big wall

Here again you could either calculate this using centimetres or metres.

To make sure there is enough paper, this has to be rounded-up, giving 28 strips altogether.

How many rolls of wallpaper?

These calculations show:

That 28 strips of wallpaper are needed
Each roll will give 4 strips.

So, the

This should be plenty, as doors and windows have not been taken into account.

Well done for completing this activity. It had lots of steps to get to the answer and some unit conversions as well. It was therefore much more involved than any other activity this week and required you to make use of your problem-solving skills as well as your measurement knowledge.

That completes the study for this week, except for this week’s quiz.

7 This week's quiz

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Box _unit2.7.1

Go to:

Week 1 practice quiz.

Open the quiz in a new tab or window (by holding ctrl [or cmd on a Mac] when you click the link).

8 Summary

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Congratulations for making it to the end of the first week in Succeed with maths – Part 2. You will use the skills that you have developed this week as you work through Week 2, which will introduce the measurement of volume and mass. So try and think back to what you have learnt as you continue your studies, particularly when converting between units.

You should now be able to:

understand how the International System (SI) is formulated
understand the more common SI units for length and convert between them
recognise the more common imperial units for length and convert between these
understand how the SI and imperial units of length relate to each other.

You can now go to Week 2.

Introduction

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This week follows on from Week 1, which looked at measurement of length in both the International System (SI) and imperial, to consider two other common measurements: mass and volume. As well as learning about the units for these measurements in both systems, you’ll get much more practice in converting between units. Conversions can often be a challenge, so the more practice you get with this the better!

Watch Maria introduce Week 2:

Download this video clip.Video player: swim2_week_2.mp4 >

Skip transcript

Transcript

Maria Townsend

This week you’re going to finish your study of measurement by covering mass and volume. You’ll also get more practice at converting between units to continue to build your confidence.

End transcript

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Interactive feature not available in single page view (see it in standard view).

After this week’s study, you should be able to:

understand the more common SI units for mass and volume, and convert between them
recognise the more common imperial units for mass and volume, and convert between these
understand how the SI and imperial units of mass and volume relate to each other.

1 Using units of measurement

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Before looking at mass, here is a quick reminder of how the SI works from last week.

The SI system uses a combination of base units and prefixes. The base unit that you used in Week 1 was the metre. This is then combined with a prefix, if relevant, to complete the unit. What unit you use will depend upon the size of the measurement that you are taking. For example, it wouldn’t make much sense to measure the length of a finger in kilometres! The most common everyday prefixes are:

‘kilo-’ a thousand
‘centi-’ a hundredth
‘milli-’ a thousandth.

This gives the following units of length:

kilometre = 1000 metres
centimetre = 0.01 metres
millimetre = 0.001 metres

This idea is extended to all the SI units, as you’ll see in the next sections on mass.

2 Mass

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

You’ll start by quickly looking at why this section is called mass rather than weight. Weight is the word that most people use in everyday language to describe how much ‘stuff’ is in an object. However, in science weight does not measure this quantity. In fact, the correct term for the amount of ‘stuff’ in an object is ‘mass’. The two quantities are related, but have different meanings.

Weight, on the other hand, is the force due to gravity on an object (or a mass). As mass is a measure of the amount of ‘stuff’ in an object, this doesn’t change unless you physically alter that object. However, the same object would not have the same weight on the moon as it does on the earth. Its weight would reduce because the moon has a lower gravity than the earth. They are also measured in different units! So, to be strictly accurate, the course uses the correct term: ‘mass’.

Now it’s time to look at the SI units for mass.

2.1 Mass: SI units

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Unlike all the other SI base units, the base unit for mass has a prefix. You may have noticed that in Week 1, when you looked at the summary of the seven SI base units.

The base unit for mass in the SI is the kilogram (abbreviated to ‘kg’), not the gram as you may have logically thought. However, the gram can be seen as the base unit from the point of view of adding prefixes to produce the other units.

Adding the prefixes gives the milligram (mg), as well as the kilogram (kg). Note that you could have a centigram as well, but this is not a unit that is commonly used.

Use your knowledge from the previous section on length in this next short activity.

Activity _unit3.2.1 Activity 1 SI units of mass

Allow approximately 5 minutes

a.What would you need to do to convert from kilograms (kg) to grams (g)?

Answer

Kilo means a thousand, so kilogram is the same as saying 1000 grams.

Hence to convert from kilograms to grams you should multiply by 1000.

b.What would you need to do to convert from grams (g) to milligrams (mg)?

Answer

Milli means a thousandth, so milligram is the same as saying a thousandth of a gram. This means that 1 gram = 1000 milligrams.

Hence, to convert from grams to milligrams you should multiply by 1000.

c.What would you need to do to convert from grams (g) to kilograms (kg)?

Answer

From part (a) to convert from kilograms to grams you needed to multiply by 1000, so the opposite must be true to convert from grams to kilograms.

Hence, to convert from grams to kilograms you should divide by 1000.

The findings from this activity can be summarised as:

kilograms (kg) to grams (g) – multiply by 1000
grams (g) to kilograms (kg) – divide by 1000
grams (g) to milligrams (mg) – multiply by 1000
milligrams (mg) to grams (g) – divide by 1000.

Use this information to help you in the next activity, as well as the steps from Week 1 to convert between different units. These are summarised again here for you:

Find out how the two units are related to each other – the conversion factor.
Ask yourself if the final answer should be bigger or smaller than the original value.
Divide or multiply by the conversion factor to get your answer.

You can click on ‘reveal comment’ if you would like some additional guidance.

Activity _unit3.2.2 Activity 2 Everyday mass

Allow approximately 10 minutes

a.During a shopping trip you buy 1.5 kg of apples, 450 g of bananas and 0.75 kg of pears. How much fruit in grams have you bought altogether?

Discussion

Remember that 1 kg = 1000 g.

Answer

a.As there is a mixture of different units, start by converting all the units into the same one. The final answer should be in grams, so it makes sense to convert all the masses into grams to start with.
So,

b.One batch of mortar mix required for building a brick wall requires 250 g of cement and 750 g of sand. If you need 5 batches to finish the wall, how much sand and cement will you need and what is the total mass in kilograms?

Answer

b.
This time you need to divide by 1000 to convert from g to kg
So,

c.The daily recommended intake of vitamin C in the UK for adults is 0.04 g. Convert this to milligrams, the more usual way to quote this value.

Answer

c.1 milligram (mg) is a thousandth of a gram.
This tells us that:
So, the

As with length, imperial units for mass are not quite as straightforward as the SI units. So, you may need to take your time over this next section.

2.2 Mass: imperial units

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

To measure mass in the imperial system, ounces (abbreviated to ‘oz’), pounds (lb) and stones (st) are used. These are listed in increasing order of size and you can see the relationship between these units below:

16 ounces in 1 pound
14 pounds in 1 stone

It's easy to confuse the two!

These relationships can in part be explained by the fact that an ounce was originally based on the mass of that very useful item, the barleycorn, with 480 barleycorns to the ounce (although it couldn't have been much fun counting 480 of them!). Having 16 ounces to the pound was useful because 16 can be divided into halves (8, 4, 2, 1) easily, which was very useful for shop keepers measuring out and selling their wares.

The same logic also applies to changing, or converting, between these different mass units as with any others. So, if you convert a value from ounces to pounds, you would expect the final number to be smaller, and hence need to divide by 16 (16 ounces in 1 pound). Using the knowledge you have already gained this week you may be able to work out this next example without looking at our workings. See how you get on!

Let’s suppose you were moving boxes of electronic equipment. The first box has a mass of 14 pounds 11 ounces and the second, 17 pounds 8 ounces. You want to find the total mass of both boxes in stones, pounds and ounces to ensure they can be moved safely.

First, add the pounds together:

Next add up the ounces:

Since there are 16 ounces in 1 pound, this means:

There are 14 pounds in 1 stone.

So,

So the total mass is 2 stones, 4 pounds and 3 ounces.

Now, have a go at this next activity yourself.

Activity _unit3.2.3 Activity 3 Mass in imperial

Allow approximately 10 minutes

For each of the following scenarios, use the appropriate operation and unit conversion factors to determine the answer.

a.At birth, Samuel’s mass is 7 pounds 4 ounces. After one week, he has gained 13 ounces. What is Samuel’s mass at the end of the week in pounds and ounces?

Answer

There are 16 ounces in 1 pound, so you can convert 17 ounces in pounds and ounces.

So,

The correct way to say this is that Samuel’s mass is now 8 pounds 1 ounce.

b.When he started dieting, Derek’s mass was 203 pounds. He lost 37 pounds. What is Derek’s mass now in stones and pounds?

Answer

b.
There are 14 pounds in a stone. So, this means that our final answer needs to be smaller, and that it is necessary to divide by 14 to achieve this.
So, now Derek’s mass is 11 stone and 12 pounds.

As with length, you need to be able to relate the two different systems of measurement to each other for mass – this is the subject of the next section and will help give you an idea of the size of the different units.

2.3 Relating SI and imperial: mass

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

For the units covered in the course, the two systems are related as shown below:

Box _unit3.2.1 Mass

1 kg = 2.2 lb
1 ounce = 28.3 g
1 stone = 6.4 kg

If you are more familiar with imperial measurements, this will hopefully show how small a gram actually is. To get a feel for this, a paperclip has a mass of around a gram – so not very much in everyday terms!

To give yourself a good feel for both systems of measurement, try weighing a few everyday objects around you. Note down the mass in both systems of measurement.

Hopefully, you will now feel confident enough to convert between the SI and imperial system for mass without an example. So, it’s straight into our next activity. Remember to click on ‘reveal comment’ if you need a hint.

Activity _unit3.2.4 Activity 4 Relating SI and imperial units of mass

Allow approximately 5 minutes

a.A baby weighs 3.7 kg when it is born. How much is this in pounds and ounces?

Discussion

This means converting from a physically larger unit to a smaller one. The final answer should therefore be larger.

Answer

a.
So the mass is 8 lb and 0.14 lb
So, the baby weighed 8 lb and 2 oz (to the nearest ounce) when it was born.

b.One litre of water has a mass of 1 kg. If a rainwater butt contains 120 litres of water, what mass of water does it contain?

Answer

Now, onto our next section, which is about how the capacity of something is measured – that is, its volume. This is another of those measurements that is very familiar in everyday life, whether it is the quantity of milk in a container or how much fuel is needed to fill a car.

3 Volume

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Here the units that are used in everyday life differ from the SI units for volume. In the SI, units of volume are based upon the metre, with a cubic metre being standard. However, this is a very large volume and not of much use to us when measuring everyday objects. To get an idea of how large 1 cubic metre is, imagine a box that is 1 metre high by 1 metre wide by 1 metre long – that is 1 cubic metre and would be a lot of milk!

The base unit for volume that is used in everyday situations is the litre (abbreviated as ‘l’ – lower case ‘L’, not upper case ‘i'). This is from the metric system of measurement on which the SI is based. Adding the prefixes as before gives us the related units of millilitre (ml) and centilitre (cl). Note this could also be continued to include a kilolitre, but larger volumes are usually measured in cubic metres, where 1 cubic metre is the same as 1000 litres (or a kilolitre).

Again, using the knowledge of prefixes it can be deduced that a millilitre is a thousandth of a litre, and a centilitre is a hundredth of a litre. From this the following can be stated:

1 litre = 1000 ml
1 litre = 100 cl.

These relationships can also be displayed in a diagram, showing how to convert between the different units, as shown below:

Figure _unit3.3.1 Figure 1 Converting between units of volume

Now you’ve had lots of practice with converting between units, see how you get on with this next activity. Think back to the start of Week 1 – do you feel that the activities are now becoming more straightforward?

Activity _unit3.3.1 Activity 5 Volume

Allow approximately 5 minutes

a.You fill your car with 35.6 litres of fuel one day and four days later, another 15.2 litres. How much fuel in total have you put in your car this week in centilitres? Remember, clicking on ‘reveal comment’ will give you additional hints and tips.

Answer

Converting from a physically larger to a physically smaller unit, means you need to multiply.

b.A bottle contains 14 cl of medicine. The dose is 5 ml. How many doses can be given from this bottle?
Click on reveal comment if you need a hint.

Discussion

You are dealing with different units here, so you need to start by converting 14 cl to ml or 5 ml to cl.

You may find it easier to convert 14 cl to ml.

Answer

To find the number of doses, the volume of the bottle and the dose need to be in the same units.

There are 10 ml in 1 cl.

Therefore,

You have probably guessed by now that the next topic will be the imperial units for volume. Some of these will no doubt prove more familiar than others, as they are still used in some everyday situations, such as buying drinks.

3.1 Volume: the imperial units

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

To measure volume, the imperial system uses, amongst other units, gallons and pints. You may also have heard of a gill and a quart. As with length and mass in the imperial system there are no constant relationships between all these units, although there is more similarity with these than with some!

There are:

4 gills in 1 pint
2 pints in 1 quart
4 quarts in 1 gallon.

For our purposes it is most useful to know that:

8 pints = 1 gallon.

As the methods used to convert between different units of volume are just the same as those already covered for both length and mass, you can move straight on to an activity to complete. Bear in mind what you have already learned over this week and the last. You can always look back at the previous sections or click on ‘reveal comment’ for some hints if you need them!

Activity _unit3.3.2 Activity 6 Imperial units of volume

Allow approximately 5 minutes

For each of the following scenarios, use the appropriate operation and unit to determine the answer.

a.At the beginning of the month a garage purchases 240 pints of motor oil. Each week it uses 44 pints. Assuming that there are four weeks in this month, how many pints of oil will be left over at the end of the month? What is this value in gallons?

Discussion

Write down all the information from the question in a list and think about the size of the answer that you expect.

Answer

a.There are 44 pints used each week.
The garage started with 240 pints.
There are 8 pints in a gallon.
So,
Therefore there are 8 gallons of oil remaining at the end of the month.

b.A large barrel holds 50 gallons of beer. If in one week 14.5 gallons of beer are sold from the barrel and the next week 15.5 gallons are sold, how many pints of beer will have been sold in total and what will be left in the barrel (in pints)?

Answer

There are 8 pints in a gallon.

So,

To double-check the answer add the pints sold to the pints left in the barrel and then convert back to gallons. This should be the same as the original capacity given:

You can now confidently state that 240 pints were sold and there were 160 pints remaining in the barrel.

Finally, you need to know how these two measurement systems relate to each other for volume. This is the subject of the next brief section.

3.2 Relating the two systems: volume

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

For volume you’re going to look at how just two units compare, that is pints and litres. This is because there is no real equivalent of the gallon in the metric system.

To see how the two are related, it is helpful to use an example.

Milk is still sold in containers that are exact sizes in pints, although the quantity in litres is displayed with equal prominence. On a 4-pint container of milk the label states that this is also 2.272 litres.

You can use this information to determine how pints and litres are related to each other. Try now in this next activity.

Activity _unit3.3.3 Activity 7 How do pints relate to litres?

Allow approximately 3 minutes

A milk bottle label states that it contains 4 pints or 2.272 litres. Use this information to work out how pints and litres are related by stating how many pints are in 1 litre. Remember you can click on ‘reveal comment’ for hints and tips.

Discussion

The information on the label means that

Answer

You know that:

So 1 litre is 2.272 times smaller than shown here. This means to determine how many pints are in 1 litre you must divide by 2.272.

4 Less familiar units

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

You’ve nearly reached the end of this week but there is one final activity to check your newly honed skills with converting between units. This involves some units that you may not be familiar with. However, all the techniques that you have been building over the last two weeks are still relevant, so bear these in mind as you work your way through Activity 8. Don’t forget to click on ‘reveal comment’ for an additional hint.

Activity _unit3.4.1 Activity 8 Converting between less familiar units

Allow approximately 5 minutes

a.Given the fact that:
Now convert 322 kWh to J.

Discussion

Remember to consider whether the final answer should be bigger or smaller than the original value given. This will tell you whether to multiply or divide by the conversion factor.

Answer

This means that for every 1 kWh there are 3 600 000 J. That is a lot of joules for each kilowatt-hour! So the final answer will be much larger than the original value. That means multiplying to convert between the two.

b.Given that 1 chain = 66 feet, convert 4345 feet to chains.

Answer

b.This time the final answer will be smaller than the original value, so you need to divide.

c.Given that:
convert 3.5 carats to mg.

Answer

c.The final answer needs to be larger than the original value. This means you must multiply.

5 This week's quiz

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Hopefully, the last activity has given you a feel for how the basic ideas that you have been building up and using in the last two weeks can be applied in any situation where you need to convert between different units. Always keep in mind the three steps that are needed:

Find out how the two units are related to each other – the ‘conversion factor’.
Ask yourself if the final answer should be bigger or smaller than the original value.
Divide or multiply by the conversion factor to get your answer.

Box _unit3.5.1

Go to:

Week 2 practice quiz.

Open the quiz in a new tab or window (by holding ctrl [or cmd on a Mac] when you click the link).

6 Summary

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Congratulations for making it to the end of another week! Many people find converting between different units a challenge, even at higher levels of study, but you now have a set of tools that will help in the future. This means that if you go on to study, in many different subject areas, you will already be another step on your way to achieving your goals. And even if this is not your ultimate aim, you will still have gained some very valuable skills for your everyday and work life.

You should now be able to:

understand the more common SI units for mass and volume, and convert between them
recognise the more common imperial units for mass and volume, and convert between these
understand how the SI and imperial units of mass and volume relate to each other.

You can now go to Week 3.

Introduction

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Patterns occur everywhere in art, nature, science and especially mathematics. Being able to recognise, describe and use these patterns is an important skill to have when tackling a variety of different problems and in understanding the world, for example, predicting the effect on fish numbers for a given amount harvested from the sea. This course explores some of these patterns, ranging from ancient number patterns to the latest mathematical research. It also looks at useful practical applications that are relevant to everyday life. Your study of this subject is spread over two weeks, and includes plenty of activities for you to do to gain confidence with these ideas.

In the following video Maria introduces you to Week 3:

Download this video clip.Video player: swim2_week_3.mp4 >

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Transcript

Maria Townsend:

I can’t see any clear patterns in this piece of artwork at the Open University but if I look at the brick wall below it’s a very different situation. This is made from rows of rectangles with each row overlapping the next by half a brick. And spotting the pattern means I can describe it concisely to other people.

In maths we’re always looking out for patterns and this week you’ll find out how useful they can be.

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After studying this unit, you should be able to:

recognise patterns in a variety of different situations
use word formulas to help solve a problem
write your own word formulas to help solve a problem.

1 Exploring patterns and processes

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Patterns occur in many different ways in everyday life. They include how petals are arranged on a flower and the repeating notes of bird song. These patterns give us a powerful tool in understanding the world and universe. Many centuries ago ancient civilisations recognised the yearly pattern of changing sunset and sunrise and used this to build places of worship, such as Stonehenge in the UK. Today, patterns in the progress of the sun and moon are used to work out accurate heights and times of tides. This course does not look at anything so complicated, but this example does show that patterns are important in many different ways.

The first section starts by looking at a do it yourself (DIY) problem using tiles. See how you get on with describing the pattern in this first activity.

Activity _unit4.1.1 Activity 1 Tile pattern

Allow approximately 5 minutes

Suppose that somebody is tiling a bathroom, and the last row of square tiles is going to be a decorative border made up of blank tiles and patterned tiles, as shown in Figure 1 below:

Figure _unit4.1.1 Figure 1 Tiles

A friend has offered to help with the job. How would you describe the pattern so that they laid the tiles correctly? Click on ‘reveal comment’ if you would like a hint.

Discussion

There is no one correct answer to how to tackle this description, so just get stuck in! Think about what might be the best way to describe this to somebody who didn’t have the pattern in front of them.

Answer

There are lots of ways you could have tackled this. For example, you might say that you will need some blank tiles and some patterned tiles with the ‘bridges’ on. Start with a bridge tile, and then put a blank tile next to it. Take another bridge tile, but turn it around so that the bridge is upside down, like a smile, and put it next to the blank tile in the same line. Then put another blank tile next to the smile tile. This is the pattern: bridge, blank, smile, blank, bridge, blank, smile …

Or maybe you numbered the tiles, something like this:

bridge
blank
smile

The pattern would then be:

1,2,3,2,1,2,3,2 etc.

The decorative border is an example of a type of geometric pattern that has many applications in art, crafts and design. You may have something similar in your own home. If you have, see if you can describe this as some extra practice.

The next section looks at a number pattern that has fascinated people for many centuries.

1.1 Pascal’s triangle

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Numbering the tiles in Activity 1 gave a number pattern that described a geometric pattern. This next example is a number pattern that appeared in China and Persia more than 700 years ago, but is still used by students in mathematical and statistical problems today – it even appears in chemistry. It is known as ‘Pascal’s triangle’ after the French mathematician Blaise Pascal who studied the properties of this triangle. The first part of Pascal’s triangle is shown in Figure 2:

Figure _unit4.1.2 Figure 2 Pascal's triangle

You can continue the triangle indefinitely by following the pattern. So how is it created?

If you look at the triangle you can see that each row of numbers starts and ends with the number 1. Now look at the numbers in the second to last line (1 4 6 4 1) and the last line of the triangle. Look carefully at them both - can you see a way that you can make the numbers in the last line from the line above? If you add each pair of numbers in the second to last line, this gives you numbers in the last line.

Starting from the left-hand side the pairs of numbers are 1 and 4, 4 and 6, 6 and 4, and 4 and 1. So:

This process, plus adding a 1 at end of the line, generates the next row of the triangle. Have a go at creating the next line for yourself in this next activity.

Activity _unit4.1.2 Activity 2 Pascal’s triangle – next line

Allow approximately 5 minutes

Create the next line of Pascal’s triangle by adding the pairs of numbers in the last line. Remember to add the 1s to the end of the rows when you are done.

Answer

Starting from the left-hand side the pairs of numbers are 1 and 5, 5 and 10, 10 and 10, 10 and 5, and lastly 5 and 1.

Adding these pairs gives us:

So the next line in the triangle should look like the bottom one in Figure 3:

Figure _unit4.1.3 Figure 3 How to generate row 7 of Pascal’s triangle from row 6

You can watch a larger version of Pascal’s triangle being built in this video:

Interactive feature not available in single page view (see it in standard view).

Now you’ve seen how to build Pascal’s triangle by adding pairs of numbers it’s time to see if there are more patterns hiding in this number triangle in the next section.

1.2 Pascal’s triangle – a closer look

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

As well as the patterns within the triangle being interesting in their own right, they are used in maths and science. If you continue your study at university level in these areas you may well come across Pascal’s triangle and will already be prepared for its delights. Also, having a go at spotting the patterns now will give you a chance to practise your pattern-spotting skills and you may well be surprised by how many there are!

Activity _unit4.1.3 Activity 3 Identifying patterns

Allow approximately 10 minutes

Study this six-row version of Pascal’s triangle and note down any patterns that you can spot.

Click on ‘reveal comment’ for tips on how to get started.

Figure _unit4.1.4 Figure 4 A six-row version of Pascal’s triangle

Discussion

Play around with the numbers to find patterns. Try looking down the diagonals and adding the numbers in each row.

Answer

Each row starts and ends with a 1.
The counting numbers, 1, 2, 3, 4, 5, are in the diagonal rows next to the sides.
In the next diagonal row, there is the sequence 1, 3, 6, 10. These numbers are known as the ‘triangular numbers’ because they create triangular patterns as shown in Figure 5:

Figure _unit4.1.5 Figure 5 Triangular numbers

Pascal’s triangle is symmetrical, too. If you draw a vertical line down through 1, 2, 6, one side of the triangle is a mirror image of the other.
The sums of the first six rows are 1, 2, 4, 8, 16, 32. The total for the next row is double the total for the current row.

These numbers can also be written as 1, 2¹, 2², 2³, 2⁴ and 2⁵.

The pattern in the powers of 2 shown in Figure 6 suggests that the first total, 1, might be written as 2⁰. Check on your calculator to see if this is correct. So it can also be said that the sums of each row are square numbers and these can be generated by raising 2 to the row number minus 1. Figure 6 may help you to see this more clearly than a written description:

Figure _unit4.1.6 Figure 6 Powers in Pascal's triangle

Well done if you spotted some or all of those patterns. There are many more than those looked at here, so this is just a flavour of what can be found in Pascal’s triangle.

What do the two examples have to do with mathematics? Well, recognising patterns in shapes, sets of numbers, processes or problems, and noticing what is the same and what is different about situations, often makes a task easier to solve. You saw how recognising the tiling pattern in Activity 1 made it easier to remember and describe, and by using the number patterns in Pascal’s triangle, you could work out the sum of each row without adding the individual numbers.

If you can spot a pattern and then describe what happens in general, this can lead to a rule or formula. If you can prove that this rule will always work, it can be used elsewhere. For example, if you can work out the general process for calculating a quarterly electricity bill and then give these instructions to a computer, many electricity bills can be generated, printed and sent out in just a few minutes. It may also help you to understand your bill!

In the next section you will start to look at how this process of looking for relationships can be used to write rules.

2 Looking for relationships

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This section considers the relationship between quantities in two practical situations and shows you how to describe these relationships by writing down a general rule or a word formula. You can think of these as a set of mathematical instructions. A good way to start is to use an everyday example.

Suppose you are planning a visit abroad. Your map marks all the distances between places in kilometres rather than miles, which you are used to dealing with. How can you work out what these distances are in miles? The first type of information you will need is how long one kilometre is, measured in miles. If you want to, you could use the internet to verify that 1 km is equivalent to approximately 0.6214 miles.

The next step is to work out how, using maths operations, you would convert kilometres into miles. You may be able to see how to convert from kilometres into miles immediately from your study of weeks 1 and 2 of this course – but, if not, try to visualise a few simple examples. The diagram in Figure 7 should help with this, as it shows how kilometres relate to miles:

Figure _unit4.2.1 Figure 7 Converting miles to kilometres

So if the distance is 3 km, you will have 0.6214 miles three times; if the distance is 10 km, you will have 0.6214 miles 10 times; if the distance is 250 km, you will have 0.6214 miles 250 times, and so on.

You can write this last example down mathematically as:

(to the nearest whole number)

Notice that in each of the examples above, the process for calculating the number of miles was the same: multiply the number of kilometres by 0.6214 (the conversion factor). This technique will work for any distance, and so the following word formula can be written:

Note that what the word formula works out is stated first – this is the standard way of presenting any formula.

You can use this formula to convert any distance in kilometres into miles. For example, suppose you wanted to convert 500 km, using the formula and replacing ‘distance in kilometres’ with 500 gives:

So 500 km, rounded to the nearest whole number, is equivalent to 311 miles.

In this example, you used the formula by replacing the phrase ‘distance in kilometres’ on the right side of the equation by the corresponding value, 500. This is known mathematically as substituting the value into the formula.

Have a go at this yourself in our next activity.

Activity _unit4.2.1 Activity 4 How many miles?

Allow approximately 5 minutes

Using the word formula just worked out, calculate how far 350 km is in miles. Show your answer to the nearest mile.

Answer

Substituting 350 for ‘distance in km’ gives:

So, 350 km is approximately 217 miles to the nearest mile.

Now that you have used a word formula, you can move on to finding your own word formula in an activity with a similar idea behind it. Remember that drawing diagrams can help you with spotting relationships and patterns!

2.1 Finding a word formula

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Instead of looking for a way to show the relationship between miles and kilometres in a word formula, the next activity asks you to find a word formula to help work out the value of one currency in another. This is very similar to the previous example, as the focus is again on how much of one quantity is represented by another. The activity takes you step-by-step through the process of writing your own word formula.

(Again, remember that drawing diagrams can help you to spot relationships and patterns)

Activity _unit4.2.2 Activity 5 Exchanging currencies

Allow approximately 10 minutes

Suppose somebody is visiting Europe, and they want to exchange some money from pounds (£) into euros (€). One agency offered an exchange rate of £1.00 to €1.18 and did not make any additional charges.

a.How many euros would you get for £5? How many euros would you get for £10?

Click on ‘reveal comment’ if you would like a hint to get going.

Discussion

Try drawing yourself a diagram to help visualise what you have been asked to find.

Answer

a.For each pound, you would get €1.18.
So for two pounds, you get two lots of €1.18, and for three pounds, three lots of €1.18 and so on.
Similarly, if £10 were exchanged, the person would get 10 lots of €1.18.

b.Now, write down a word formula that can be used to convert pounds into euros.

Answer

b.To change pounds into euros, multiply the number of pounds by the exchange rate of €1.18. The word formula to represent this is:
You may also have thought of the more general word formula that will work with any exchange rate of:

c.Check that your formula works by using it to convert £5 into euros.

Answer

c.Substituting 5 for ‘number of pounds’ gives:
This agrees with the answer in part (a) as expected, giving confidence that the word formula is correct.

These examples illustrate how a word formula can be used to summarise a mathematical process such as converting units of length or currencies. Once a formula has been derived, it can then be used in other situations, both for calculations by hand or by computer – for example, for currency transactions in a bank. You will be able to practise writing your own formulas next week.

Now let's look at using formulas in some more real-life examples.

3 Real-life examples of using formulas

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The previous section considered how a formula could be built and then how it could be used. This section considers some more complicated formulas that have already been developed and are used in a variety of different situations – such as cooking, health care, business and archaeology. These examples illustrate some of the broad applications of maths and how mathematical relationships can be used in making decisions. As you work through these examples, you might consider where else maths could be used in each of these topics.

When you are trying to solve a real-life problem mathematically, you often use formulas that have already been developed, so it is important to understand how to apply them. This section will help you do that.

3.1 Formulas in cooking

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The time to cook a fresh chicken depends on its mass (commonly called weight in everyday language), as given by the following formula:

Roughly how long will a chicken with a mass of 2.2 kg take to cook?

To use the formula, you need to substitute the mass of the chicken into the right side of the formula and then work out the resulting calculation. However, the formula asks for the mass in grams, so the first step is to convert 2.2 kilograms (kg) into grams (g).

Since there are 1000 g in 1 kg,

Substituting 2200 for the ‘mass in grams’ in the formula gives:

Following the BEDMAS rules, the multiplication and division must be carried out first, which gives:

(If you are unsure about what BEDMAS is, you can find out more in Succeed with maths – part 1.)

So the cooking time would be about 125 minutes, or 2 hours and 5 minutes.

Now use the same formula for a different sized chicken in the next activity. Remember to check carefully what units of mass are required.

Activity _unit4.3.1 Activity 6 A smaller bird

Allow approximately 5 minutes

How long would it take to cook a chicken with a mass of 1.6 kg?

Answer

First, convert 1.6 kg into grams, since the formula requires the mass in grams.

So,

Substituting this value into the formula gives:

The chicken would take about 95 minutes, or 1 hour and 35 minutes to cook.

The next example is an application from health care.

3.2 Formulas in health care

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The body mass index (BMI) is sometimes used to help determine whether an adult is underweight or overweight. It is calculated as follows:

Although care needs to be taken in interpreting the results – for example, the formula isn’t appropriate for children, old people or those with a very muscular physique – a BMI of less than 20 suggests the person is underweight and a BMI of over 25 suggests the person is overweight.

In this formula, the units have been included in the expression on the right-hand side of the equals sign. It is important to change any measurements into these units before you substitute the values into the formula. (No units have been included for the BMI on the left-hand side, in line with current health care practice.) This is true for any formula that you will come across, so it is very important to remember.

This time, instead of giving you an example first it is straight into an activity to apply the formula.

Activity _unit4.3.2 Activity 7 Body mass index

Allow approximately 5 minutes

If an adult man is 178 cm tall and weighs 84 kg, calculate his BMI and decide whether he is overweight.

Answer

The formula needs the mass in kg and the height in metres. You have the man’s height in centimetres, so this must first be converted into metres.

So, his

Substituting the mass and height into the formula for the BMI gives:

(to the nearest whole number)

Since his body mass index is over 25, the man is probably overweight.

Hopefully, you can see that this relatively simple formula gives any health care professional a way to check the health status of any patient.

The next section example introduces a bit more complexity in a formula, but all the same principles apply that you have already used. These are, converting to required units before substituting into a formula and following BEDMAS as necessary.

3.3 Formulas in business

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

One of the advantages of identifying the general features of a calculation and then describing it mathematically is that the formula can then be used in a computer to work out different calculations quickly and efficiently. Many utility suppliers (gas, water, electricity, telephone) have rates based on a fixed daily charge and a further charge based on how much you have used during the billing period.

For example, a mobile phone network charged £20 per month for 300 minutes (or less) of phone calls. Extra calls above the 300 minutes were charged at 40p per minute.

Assuming that more than 300 minutes were used per month, the formula for the total monthly cost can be expressed by the following word formula:

For simplicity data usage or texting are not included.

In this next activity you’ll unpick how this formula relates to the information given about the way in which bill is worked out. This insight will help when moving onto building your own formulas.

Activity _unit4.3.3 Activity 8 Understanding the formula

Allow approximately 10 minutes

Suppose that in one month, 375 minutes of phone calls were made. Explain how you would calculate the cost for the extra minutes (above the 300 minute allowance) and then how to calculate the total cost for that month. Can you then explain how the formula has been put together?

Answer

Because 300 minutes are included in the £20 charge, the number of minutes that are charged separately is:

Each extra minute costs 40p. This is the same as saying £0.40 per minute, as . So 75 extra minutes will cost

So the total charge for that month is

The formula says that the fixed charge is £20, and then each minute in excess of the 300 minutes allowed costs 40p. How this information relates to our formula can be shown by including some extra notes with it, as shown in Figure 8:

Figure _unit4.3.1 Figure 8 Annotated word formula

Now you know how the formula works, and has been put together (or derived), try applying it in this next activity.

Activity _unit4.3.4 Activity 9 Mobile Phone bills

Allow approximately 5 minutes

Use the formula to work out the total monthly cost for the following total number of minutes used:

a.348 minutes

Answer

a.Substituting 348 for the total number of minutes gives the total cost in pounds as:
So, the bill for the month is £39.20.
Note how this calculation is set out with all the working and a concluding sentence that answers the question precisely. This follows the ideas for correct mathematical communication that were covered at the end of Week 8 in Succeed with Maths Part 1.

b.250 minutes

Answer

b.The formula only applies if more than 300 minutes of calls are made, so it cannot be used in this case. This is another important step in using formulas – check that they apply in your situation before using them. Here, the charge is £20 for up to 300 minutes of calls, so the charge for this month is £20.

If you have a mobile phone with a similar deal, you could work out a formula that fits yours and use it to check your bills – and get some more practice at the same time. It might also help you to decide if a different deal may suit you better.

Moving on now from everyday formula to those used in academic study, the next section looks at how one set of formulas are used in archaeology to work out the approximate height of some of our ancestors.

3.4 Formulas in archaeology

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Footprints from prehistoric human civilisations around the world have been found preserved in either sand or volcanic ash. From these tracks, it is possible to measure the foot length and the length of the stride. These measurements can be used to estimate both the height of the person who made the footprint and also whether the person was walking or running. This can be done by using three formulas:

Note that no units have been included in these formulas – so it is important to make sure that you use the same units, such as centimetres, throughout the calculation.

The formula for relative stride length is used to establish if the person was walking or running. If the value of the relative stride length is less than 2, the person was probably walking; if the value is greater than 2.9, the person was probably running. Between these two values, it is difficult to be sure if the person was running or walking. The expected value for relative stride length is usually between 0 and 5.

Notice that the relative stride lengths quoted have no units. The stride length and the hip height are both measured in cm, and when one is divided by the other, these units cancel each other out, just like cancelling numbers when dealing with fractions. If you go on to study physics or other sciences, you will find many examples like this.

Before trying the next activity, if you would like to see one of these formulas being used in practice, have a look at this short video.

Activity _unit4.3.5 Activity 10 Footprints in the sand

Allow approximately 10 minutes

From one set of footprints, the length of the foot is measured as 21.8 cm and the stride length as 104.6 cm. What does the data suggest about the height and the motion of the person who made these footprints? Remember you can click on ‘reveal comment’ if you get stuck.

Discussion

Break the problem down into simpler parts. Start by asking yourself what you need to know to tell if the person was running or walking.

Answer

So,

(to the nearest cm).

To work out if the person was running or walking, it is important to know the relative stride length. This is given by:

The question gives the information about the stride length but not the hip height. To calculate the hip height use this formula.

Fortunately, this can be done, as the foot length is also given in the question.

So,

This has given all the information needed to calculate the relative stride length. So, substituting into the formula:

(rounded to 1 decimal place).

As the relative stride length is less than 2, the person was probably walking.

This was a slightly more complicated problem than our previous examples as it required a step in the middle to calculate some extra information needed. So, well done for having a go!

3.5 Combining formulas

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

There is a quicker way to the answer which requires two of the formula to be combined to give a new formula for relative stride length.

The formula for relative stride length was:

It is known that:

This means that it is possible to substitute the formula for hip height in the formula for relative stride length. This then gives:

This new formula could be used directly with the information that was given in the question. This version would also make more sense in the field, as it contains the two pieces of information that would be known on site.

In the final activity for this week you’ll consider why it is important to always think about whether the answer you have obtained looks sensible. This is something that you should always ask yourself. It can act as a prompt to check over your working or logic in any problem.

Activity _unit4.3.6 Activity 11 Relative stride length: what’s wrong?

A student used the formula to calculate the relative stride length in the footprints activity.

The student entered the key sequence on their calculator for the calculation as: 104.6 ÷ 4 × 21.8. This gave an answer of approximately 570, so the student concluded that the person was probably running very fast. Can you explain where the student made the mistake and why they should have been suspicious of their answer?

Answer

The calculator will perform this calculation from left to right, using the BEDMAS rules (order of operations), which treat multiplication and division as equally important. So, it will first divide 104.6 by 4 to get 26.15, and then multiply by 21.8 to get approximately 570. However, this is not the correct calculation from the formula.

The stride length (104.6) should be divided by (4 × foot length), so the calculation should be 104.6 ÷ (4 × 21.8). When explaining the formula at the start of this section you were told that the expected value for relative stride length lies between 0 and 5. So, an answer of 570 should have immediately set alarm bells ringing for this student that something had gone wrong somewhere. As well as checking your answer using known information, carrying out a quick estimate of what size of answer you are expecting can also be very useful. So, if you are expecting a value in the order of hundreds, say around 200, and your answer is in the millions, then those alarm bells should be ringing again!

Well done, you’ve just completed the last activity for Week 3! You’ve just got one final section to look at before moving on to the Week 3 quiz. This summarises the top tips for using any formula that you've covered here.

4 Tips for using a formula

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The following tips for using any formula you come across will be useful, both in next week’s study and any further study at university level you may undertake, particularly if you are interested in maths, science or technology. You might find it useful to make a note of these now, or bookmark them to refer to later:

Check that the formula can be applied to your particular problem.
Check what values you need to substitute and also what units these should be measured in. Convert the measurements if necessary before substituting.
Make a rough estimate for the answer.Substitute the values into the formula carefully.
Use BEDMAS to work out the resulting calculation, step by step. Explain your steps carefully using words like ‘substituting’ and ‘converting’.
Check that your answer seems reasonable, both practically and from the estimate. Round your answer appropriately.

Remember your concluding sentence should both answer the question and include units.

5 This week's quiz

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Use the tips as you are completing the questions in this week’s quiz.

Box _unit4.5.1

Go to:

Week 3 practice quiz.

Open the quiz in a new tab or window (by holding ctrl [or cmd on a Mac] when you click the link).

6 Summary

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Congratulations for making it to the end of another week. You may not have realised it, but you have now started on groundwork that will lead you to taking your first steps with algebra. This is a fundamental tool of maths and is used in many different subject areas and other, possibly unexpected subjects. In fact anywhere you want to work out an unknown value from numbers that you already know will probably need some algebra. This is just the start of your journey though, and this course takes you through it step-by-step to help you build your confidence. What you should now feel more confident in doing is:

recognising patterns in a variety of different situations
using word formulas to help solve a problem
writing your own word formulas to help solve a problem.

You can now go to Week 4.

Introduction

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

You finished last week by using some word formulas from different aspects of everyday life. Some of these examples will not have been relevant for some people (not everyone is an archaeologist!) but they should still have all given you some great practice at using different formulas. This week the course turns to writing your own word formula. You’ll start by looking at spreadsheets and then do some number tricks to amaze your friends and family with. Finally, you’ll use doing and undoing diagrams to change what is being calculated by a formula. This is another aspect of mathematical study that is directly related to building skills for algebra. The final section concentrates on inequalities. These are used in the place of the usual equals sign, to show that something is greater or less than something else.

Watch Maria introduce Week 4:

Download this video clip.Video player: swim2_week_4.mp4 >

Skip transcript

Transcript

Maria Townsend:

Once you’ve completed your study of patterns and formulas this week you’ll be able to do your first badge quiz. This is in a very similar format to the ones from Weeks 1 to 3. So if you’ve already completed those you’ll have no trouble doing this one.

Take your time to read over the questions very carefully as that will give you the best chance of showing your skills.

Good luck and see you next week.

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After this week's study, you should be able to:

understand how to read a spreadsheet
understand how a spreadsheet is created
use doing and undoing diagrams to change formulas
interpret and use notation for inequalities.

1 Your formulas: using a spreadsheet

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Last week, the course looked briefly at how to construct a word formula but concentrated on how to use formulas you were given successfully. This week moves onto building your own formulas, starting with a practical application using spreadsheets. As well as having relevance to future university level studies, if you need to work with data that you or somebody else has collected, being able to work with spreadsheets may well come in useful at home or at work, now and in the future. It is also a useful way to learn more about constructing your own formulas.

Before getting going though, it is worth pausing to think about what a spreadsheet is, how they may be useful and the way they are set up.

At the basic level, spreadsheets can be used to display data, such as fuel prices over a number of years, in tables. Once, the data are in a spreadsheet then the real strengths of spreadsheets can be used. You can perform calculations on the data to work out other information from the data and even use a spreadsheet to create charts for you to display the data or results. Spreadsheets come into their own when you have a mountain of data and calculations to perform on it.

Figure 1 shows part of a spreadsheet that has been constructed to record monthly income and outgoings, or expenditure. It is similar to a bank balance sheet that you might draw up by hand to check your monthly income and expenses, the totals, and the overall balance. However, the spreadsheet has been created on a computer and has formulas inserted into it to enable automatic calculations without using a calculator.

Figure _unit5.1.1 Figure 1 A spreadsheet used for a monthly budget

As with any other spreadsheet, this example is made up of rows and columns containing boxes. These boxes are called ‘cells’. The columns are identified by letters and the rows by numbers. This enables you to identify each cell in the spreadsheet. For example in this spreadsheet, the number 221.12 is in column B, row 11, so is in cell B11.

Cells can thus be found, from their reference, by looking down the column and across the row. So cell A3 can be found by looking down column A and across row 3. This cell contains the word ‘Salary’. Notice that cells can contain either text or numbers.

Use this knowledge when completing your first activity of the week.

Activity _unit5.1.1 Activity 1 Spreadsheet cells

Allow approximately 5 minutes

Use the spreadsheet in Figure 1 to answer the following questions.

a.What word or number is contained in the following cells?
- i.A5
- ii.A12
- iii.B15
- iv.B1

Answer

a.
- i.The cell that is in both column A and row 5 contains the words ‘Total Income’.
- ii.Down column A and across row 12, cell A12 contains the word ‘Other’.
- iii.The cell that is in column B, row 15 contains the number 166.72.
- iv.The cell that is in column B, row 1 contains the heading ‘Amount (£)’.

b.What is the reference for the cells that contain the following?
- i.The number 1585.18.
- ii.The word ‘Food’.

Answer

b.
- i.1585.18 is in column B and row 13, so its reference is B13.
- ii.‘Food’ is in column A and row 9, so its reference is A9.

So, remember when you are referring to a particular cell, you always state the column letter, followed by the row number.

Looking again at Figure 1 you can see what other information it shows. For example, if you look at row 3, this shows that the monthly salary is £1700.56. Although cell B3 only contains the number 1700.56, you know that this is measured in pounds from the heading in cell B1 ‘Amount (£)’. Overall, the spreadsheet shows the items that make up the monthly income and where money has been spent over the month. If you were keeping these records by hand, you would then need to calculate the total income, the total expenses and the balance.

For example, to find the total income for the month, you would need to add the salary of £1700.56 to the other income of £51.34. This gives the total income of £1751.90. In other words, to calculate the value in cell B5, you need to add the values in cells B3 and B4. This can be written as the following formula:

1.1 Finding the balance

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

A similar approach can be used for working out other values in the spreadsheet. Have a go yourself in this next activity.

Activity _unit5.1.2 Activity 2 Working out the balance

Allow approximately 5 minutes

Formulas have also been used to calculate the total monthly expenses and the balance. If you were working these calculations out by hand, what would you do? Write this using the appropriate cell numbers and maths operators as shown in the previous example. Finally, check these comparing your answers using the formulas with the values in cell B13 and B15.

Answer

To calculate the total monthly expenses, you need to add the individual expenses of ‘Rent’, ‘Food’, ‘Transportation’, ‘Regular Bills’, and ‘Other’. The formula will therefore be:

To find the balance, you need to take the expenses away from the total income. So, the formula to calculate the balance will be:

Checking these now:

Comparing these with the values in the spreadsheet, it is clear that they agree. This should give us confidence that the formulas built are correct!

How are these formulas actually put into the spreadsheet though? To do this you type the formula directly into the relevant cell, starting with an equals sign to tell the software that you are entering a formula rather than a word. This is shown in Figure 2:

Figure _unit5.1.2 Figure 2 Finding the balance

Notice that in cell B13, a shorthand form for the sum (total) has been entered. The formula ‘= B8 + B9 + B10 + B11 + B12’ could have also been entered, but it is much quicker to use the shorthand form ‘= SUM(B8:B12)’, which is shown here. This instruction tells the computer to add the values in all the cells from B8 to B12.

If you have access to spreadsheet software, try and recreate this example now and see how you get on with entering formulas yourself.

Another advantage of using a spreadsheet is that if you change some of the numbers, all the calculations that use that particular number will be automatically updated to reflect the change. For example, in this budget, the amounts for the salary, rent and the regular bills are likely to remain the same from one month to the next, and may only be updated once or twice a year. However, the amounts for food, transportation, other bills and other income will probably change from month to month. These values can be changed easily on the spreadsheet and the revised balance produced immediately, taking some of the work from you and reducing the risk of calculation errors.

1.2 Spreadsheet formulas

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

There is one final activity in this section on spreadsheets, bringing all these ideas together.

Activity _unit5.1.3 Activity 3 Spreadsheet formulas

Allow approximately 10 minutes

The formulas used in a spreadsheet are displayed as shown in Figure 3. Note that the asterisk (*) is the notation for multiplication used in spreadsheets and that all entries have been rounded to the closest penny. This example shows UK VAT, which was 20 per cent in 2014 (20 per cent as a decimal is 0.2).

Figure _unit5.1.3 Figure 3 Spreadsheet formulas

a.The values in columns C and D will be displayed to two decimal places because they represent an amount of money. What values will be displayed in cells C3 and D3?

Answer

a.The formula in C3 is: (to 2 decimal places)
So,
The value in D3 is obtained by adding together the values in B3 and C3.
So,

b.What do you think is being calculated in the cells in column D? Can you suggest a suitable heading for this column to be entered into D1?

Answer

b.Column D represents the total cost of the item plus VAT. A suitable heading might be ‘Total (£)’. You can think of other correct titles such as ‘Final Price (£)’.

c.Cell D5 calculates the sum of the values in D2, D3 and D4. Write down a formula that could be entered in cell D5. What does this value represent?

Answer

c.The sum can be found by adding together the values in cells D2, D3 and D4. The formula ‘=D2+D3+D4’ could therefore be entered into cell D5. (Note that you do not use the formulas that are present in each of the cells you are adding. Just the cell title is sufficient.) Alternatively, you could use ‘SUM(D2:D4)’, which also adds together the cells from D2 to D4. The resulting entry represents the total cost (including VAT) of the radio, kettle and fan together.
With these changes, and the title ‘Total (£)’ typed into cell C5, the spreadsheet will look like the example shown here:
Figure _unit5.1.4 Figure 4 Spreadsheet

This section has served a few purposes. It has introduced spreadsheets and shown you how you can use these to carry out calculations as well as helping you make a start on writing your own formula. The next section leaves spreadsheets behind to continue with this latter skill, to build your confidence with formulas further.

2 Your own formulas

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Let’s start this section by looking at a number trick and seeing if a formula can derived, or built, for it. The aim of the trick is to end up with the name of an animal.

To do this follow the instructions listed below. The final number represents a letter of the alphabet, numbered in ascending order from A to Z. So, A is 1, B is 2, C is 3 etc. Then write down the name of an animal beginning with that letter.

The instructions for the trick are:

Think of a number.
Add 5.
Double this.
Subtract 8.
Divide by 2.
Take away the number you first thought of.
Add 4.
The final number represents a letter of the alphabet, numbered in ascending order from A to Z. So, A is 1, B is 2, C is 3 etc.
Now write down the name of an animal beginning with that letter.

Activity _unit5.2.1 Activity 4 What animal?

Allow approximately 5 minutes

Carry out the number trick, starting with the number 3 and then any other number you would like to start with. What do you notice about the result in both cases?

Answer

Starting with 3, the instructions work out as follows

Table _unit5.2.1

Instruction	Expression
Add 5
Double it
Subtract 8
Divide by 2
Take away the number you first thought of
Add 4

You will find that you always get 5, whatever number you start with! This gives the letter E. Most people will then think of elephant, although you cannot always guarantee that. The odd eel or emperor penguin may sneak in!

What is really interesting here though, to a mathematician, is how the trick works. The activity in the next section will help you to work this out.

2.1 Understanding the number trick

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

One way to understand how the previous number trick works is to use a visual representation of each stage of the instructions, and that is how you are asked to approach this next activity. Take your time, and read over the instructions and hints carefully to start you off.

Activity _unit5.2.2 Activity 5 How does this trick work?

Activity 5 How does this trick work?

To see how this trick works, read through the instructions from before. Because you could have thought of any number, replace this unknown number with a thought bubble like this:

Keep the thought bubble as part of the new number that you get at each stage. So after the first step you will have something like this:

+ IIIII (to represent 5).

Discussion

When doubling, draw the cloud and associated number twice.

Answer

This is what our visualised representation of the number trick looks like:

Figure _unit5.2.1 Figure 5 The number trick explained

This shows that the numerical answer is always going to be 5. It does not depend on which number was chosen first since there are no clouds, representing the initial number, involved in the final answer. So the letter of the alphabet chosen is always E. There are not many animals with names beginning with E and most people do think of an elephant first, but we’ve already thought about the pitfalls of that!

Rather than using a cloud to represent the number and explaining the trick visually, you could write this in a more mathematical way by using either a word or letter to represent the initial number. In maths unknown numbers are usually represented using a letter. Using ‘n’ to stand for this number the trick can be written out again without the clouds like this:

Table _unit5.2.2 Table 1 Showing an algebraic representation of the number trick

Instruction	Expression
Think of a number
Add 5
Double it	This can be re-written as is the same as 2 lots of , so can be written in shorthand as So
Subtract 8
Divide by 2	Dividing each part by 2 separately is the same as dividing the whole of , by 2. Taking this former approach helps us show the next step more clearly. 2 lots of n divided by 2 gives 1 lot of n, and clearly 2 divided by 2 gives 1. So, finally giving:
Take away the number you first thought of
Add 4

This gives the same result, where n is not involved in the final answer. This time a more conventional mathematical representation was used rather than a cloud!

At first this may have appeared to be quite a daunting example, using both numbers and letters but hopefully you found that by working your way carefully through each stage you were able to understand how the final answer was obtained. Working step by step carefully through problems will really pay dividends as problems and concepts start to look more complicated. Remember this when completing the next activity with another number trick.

2.2 Another number trick

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Here is another number trick which uses the same techniques you learned in the last section to see how it works.

Activity _unit5.2.3 Activity 6 Another number trick

Activity 6 Another number trick

Try the following trick several times; make a note of the number you started with and your final answer each time:

Think of a number between 1 and 10 (this will work with numbers greater than 10, but the restriction is to keep the arithmetic manageable).
Multiply by 4.
Add 6.
Divide by 2.
Subtract 3.
Divide by 2.

What do you notice about the answer? See if you can explain why this happens, either by using a diagram or by writing down the expressions for the answer at each stage, replacing your initial number with n again.

Answer

You should find that this time the answer is always the number you chose at the start. Both ways of showing why the number trick works are shown below.

Figure _unit5.2.2 Figure 6 Graphic representation of how the number trick works

Table _unit5.2.3 Table 2 Table format showing how the number trick works

Instruction	Expression
Think of a number between 1 and 10
Multiply by 4
Add 6
Divide by 2
Subtract 3
Divide by 2

These two number tricks are sets of instructions that can be used to build, or derive, a formula. They are in essence not much different from those you looked at for spreadsheets. They just involved a few more steps to reach the answer.

Sometimes the answer and the set of instructions are available, while, for some reason, the starting value is not known. For example, imagine something had happened to your bank balance spreadsheet and the starting balance was no longer displayed but you still knew the final balance and all the formulas. Could you then rebuild the spreadsheet to work out initial balance? The next section shows you how.

3 Doing and undoing

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The next example helps you to start to think about how to work out an initial number if you know the answer and instructions using a more straightforward example than our spreadsheet. This time there are only two instructions:

Think of a number.
Add 4.

If my answer is 11, can you work out what number I was thinking of? You might have said ‘What number do I have to add to 4 to get 11?’ or perhaps ‘If I take away 4 from 11, what number do I get?’. In both cases, you should have arrived at the answer 7.

In the second method, ‘subtracting 4’ undoes the ‘adding 4’ in the original instructions, and this can be illustrated with a ‘doing-undoing diagram’ (see Figure 7).

Figure _unit5.3.1 Figure 7 A doing-undoing diagram

In the ‘doing’ part of the diagram, start with the number and write down the operations applied in turn until you get the answer. Here there is just one operation: ‘add 4’.

For the ‘undoing’ part of the diagram, start on the right with the answer, in this case 11. Then work back towards the left, undoing each operation in turn until you find the starting number. In this case, ‘subtract 4’ undoes ‘add 4’ and

You can then construct an undoing diagram that starts from the left, as shown in Figure 8. Notice how the arrows indicate the direction to read the diagram.

Figure _unit5.3.2 Figure 8 Doing-undoing diagram

So this also shows that the number first thought of was 7.

3.1 Doing some doing and undoing

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Now you’ve had a look at an example, see how you get on with the next activity.

Activity _unit5.3.1 Activity 7 Working out the number

Allow approximately 13 minutes

Try to work out what number was thought of in the following problems. You may find it helpful to use some doing-undoing diagrams to visualise, and keep track of these.

a.Think of a number, subtract 3, and the answer is 2.

Answer

Figure _unit5.3.3 Figure 9 Doing-undoing diagram for part (a)

So, the initial number was 5.

Remember, you can check your answer by using it with the instruction to check you obtain the correct answer given. So, 5 − 3 = 2!

b.Think of a number, multiply by 5, and the answer is 35.

Answer

Figure _unit5.3.4 Figure 10 Doing-undoing diagram for part (b)

So, the initial number was 7. (Check 7 × 5 = 35).

c.Think of a number, add 4, double it, and the answer is 14.
Click on ‘reveal comment’ if you would like a hint

Discussion

When you have to deal with more than one operation, just take each step in turn. Write down the doing diagram and draw the undoing one underneath, working backwards to undo each operation.

Answer

Figure _unit5.3.5 Figure 11 Doing-undoing diagram for part (c)

So, working from the left in the undoing diagram, 14 ÷ 2 = 7 and 7 − 4 = 3.

If you like, you can write the output from each box on the arrow as shown in Figure 13 above.

So, the initial number was 3. (Check: 3 + 4 = 7, then 7 × 2 = 14).

d.Think of a number, add 5, multiply by 3 and subtract 4. The answer is 17.

Answer

Figure _unit5.3.6 Figure 12 Doing and undoing diagrams to solve a multistep problem

So, if the answer is 17, undoing the steps by working from right to left gives:

Table _unit5.3.1

Instruction	Expression
Add 4
Divide by 3
Subtract 5

So, the initial number was 2 this time. (Check: 2 + 5 = 7, then 7 × 3 = 21 and 21 − 4 = 17).

You may be wondering what is the point of all this doing/undoing business! Sometimes there is a need to change a formula so that you can work out something different from it. This is called rearranging the formula and does occur quite frequently in any subject area that uses formulas. The same technique that you have been looking at in this section on ‘undoing’ instructions can be used when rearranging formulas, as you’ll see in the next section.

4 Rearranging formulas

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Rearranging a formula makes it possible to show the formula in a different way, whilst making sure the relationships between the various elements that it describes are not changed. If these relationships are changed, the formula will no longer work as expected. Thinking about rearranging using the ‘doing’ and ‘undoing’ technique from the last section will help to ensure that the relationships are maintained.

Instead of a using a set of instructions, the next example is from Week 3. Hopefully, you recognise this word formula:

How could a visitor from Europe use this word formula to convert euros into pounds, say, while shopping on holiday? They would need a formula for the number of pounds based on the number of euros. This can tackled by drawing the ‘doing and undoing’ diagrams for this situation.

In this case, the starting point for the doing diagram will be number of pounds on the left, the operation in the box will be multiply by 1.18 and output on the right will be number of euros (our answer). Therefore, to undo the operation in the box divide by 1.18. So, the diagrams will look like those shown in Figure 13.

Figure _unit5.4.1 Figure 13 Doing and undoing diagrams for converting pounds to euros and euros to pounds

From the undoing diagram the new word formula can be built. This gives:

This just needs to be swapped around, so that the formula is shown in the conventional way with what you are calculating stated first. This results in:

Now use what you have learned here in this next activity.

Activity _unit5.4.1 Activity 8 Miles and kilometres

Allow approximately 5 minutes

To change kilometres into miles, you can use the formula:

Starting with ‘distance in kilometres’, draw a doing diagram to show how to calculate the distance in miles. Then draw the undoing diagram and write down the formula for changing miles into kilometres.

Your formula should start ‘distance in kilometres = ...’.

Answer

The doing and undoing diagrams are shown in Figure 14.

Figure _unit5.4.2 Figure 14 Doing and undoing diagrams for converting kilometres to miles and miles to kilometres

So the formula for converting miles into kilometres is:

4.1 Time formula

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

There is just one last activity for you to complete now before moving on to a different subject area. This will give you another opportunity to rearrange a word formula but with a few more elements to consider.

Activity _unit5.4.2 Activity 9 How many minutes for chatting?

Allow approximately 5 minutes

This was the formula for the monthly cost in £ of a mobile phone that was used in week 3:

The owner wishes to stick to a monthly budget of £45. To do this they need to know the maximum number of minutes they can use, so that they don’t exceed the budget. Use doing and undoing diagrams to work the maximum number of minutes. Substitute ‘Total monthly cost in £’ for £45.

See if you can then write a word formula that will allow the number of minutes to be calculated for any budget.

Answer

Here is the doing diagram:

Figure _unit5.4.3 Figure 15 Doing diagram for £45 monthly budget

Undoing each step in turn gives the following undoing diagram.

Figure _unit5.4.4 Figure 16 Undoing diagram for number of minutes

Applying these operations in turn gives:

Subtracting 20 gives:
Dividing by 0.4 gives:
Adding 300 gives:

So, 362 minutes can be used for a monthly budget of £45.

Now all you need to do to provide a word formula for any monthly budget is to change from £45 back to ‘Total monthly cost in £’.

So the word formula will be:

Mathematicians prefer the second way of showing this formula just because it looks much neater – otherwise there is no difference between them.

The last part of the activity was more challenging than our other examples, so well done for having a go at it. What you may not realise is that during these last few sections you have been taking your first steps into the wonderful world of algebra. This is one of the fundamental tools of most maths, science and technology, so if you continue into university level study in any of these areas you will certainly find these skills useful. You won’t be using doing and undoing diagrams but techniques used for rearranging formulas (and equations) will be just the same – looking for what ‘undoes’ each operation.

In the next section you are going to look at inequalities. This is the name mathematicians use for expressions that use notation meaning, for example ‘greater than’ and ‘less than’. You have actually come across three examples of these in this week’s and last week’s study, as you will discover when you move on.

5 Inequalities: greater than or less than?

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

There have been three occasions in these last two weeks when checks have been made to see whether a result is greater than or less than some other value. These were when:

calculating the BMI and determining whether the person was overweight or underweight
determining from footprints whether a person was walking or running
checking whether a phone had been used for more than 300 minutes.

In all these situations the statements could have also been written mathematically using inequality symbols.

This checking of whether values are greater than, or less than, some limit happens frequently – occasions when you may see this could be safety issues or age limits. For example, medicines may have to be stored at a temperature of 25 °C or less; child train tickets can be bought for children who are over 5 but under 16 years old.

Rather than writing out ‘greater than’ or ‘less than’, shorthand notation is often used – as shown below.

> greater than
≥ greater than or equal to
< less than
≤ less than or equal to

If you have difficulty remembering these symbols, you can think of them as arrows that point to the smaller number or note that ‘<’ looks like an ‘L’, which stands for ‘less than’.

The symbols are read from left to right. For example, ‘11 > 9’ is read as ‘11 is greater than 9’; the cost of a holiday in pounds < 1000 is read as ‘the cost of a holiday is less than £1000’.

To use the symbols in your own writing, decide what you want to say first, then use the symbol. For example, since 10 is greater than 5, this would be written as ‘10 > 5’. On the number line, because –4 lies to the left of –3, –4 is less than –3; this would be written as ‘–4 < –3’.

Similarly, the instructions for the medicine that has to be stored at a temperature of 25 °C or less could be written as ‘Medicine storage temperature in °C ≤ 25’.

Sometimes, it can be helpful to draw a number line to visualise this kind of information. For example, the ages that children are eligible for the child train fare are from their fifth birthday up to, but not including, their sixteenth birthday. This means that the age has to be greater than or equal to 5 and less than 16. This range is shown on the number line below. Does this help you to visualise the information?

Figure _unit5.5.1 Figure 17 Ages that children are eligible for the child train fare

The empty circle means that this number (16) is not included and the filled-in circle means that this number (5) is included in the interval. This is then written as ‘5 ≤ age for child train fare < 16’.

Note carefully the format in which this last inequality is written. The variable that is being described – that is, the age for a child train fare – is always in the centre when defining a range of values with an upper and lower limit. This is the maths convention that is followed with ranges of values. This can be a little confusing, as you have to turn your logic around for the lower limit. You are in effect saying that 5 is less than, or equal to, a child train fare, rather than a child train fare is greater than, or equal to, 5. This amounts to exactly the same thing, but how you may usually think about the lower limit is turned on its head!

Now put these ideas into practice in this final activity for the week.

Activity _unit5.5.1 Activity 10 Inequalities

Allow approximately 10 minutes

a.Which symbol (< or >) should go in the blank spaces below? Click on ‘reveal comment’ if you would like a hint to get going.
- i.4__7
- ii.18__10
- iii.3__–2

Discussion

Write the inequality using words first and then convert this to the correct symbol.

Answer

a.
- i.4 < 7
- ii.18 > 10
- iii.3 > –2

b.Work out what the following mathematical statements mean. Write your answers as full sentences and be as precise as you can.
- i.Balance in account > 0.
- ii.Speed (in mph) on motorway ≤ 70.
- iii.18 ≤ age (in years) ≤ 50.

Answer

b.Your answers may be worded slightly differently, but they should still have the same meaning:
- i.The balance in the account is greater than zero.
- ii.The speed on the motorway is less than or equal to 70 mph.
- iii.The age in years is between 18 and 50, inclusively.

c.Rewrite the following sentences as statements using the inequality symbols:
- i.The refrigerator temperature should be less than 4 °C.
- ii.There must be at least five people on the committee.
- iii.For an ideal weight, a person’s BMI should be greater than 20 and less than 25.

Answer

c.Your answers may be worded slightly differently again, but they should still have the same meaning:
- i.Refrigerator temperature (in ºC) < 4.
- ii.Number of people on committee ≥ 5.
- iii.20 < BMI < 25

As well as in the examples given here, inequalities are also an important part of many computer programs. So you can see that maths pops up in all sorts of areas of study and everyday lives.

This is the end of this section on inequalities and the week as a whole. You may have found some ideas that were new to you here and others that were more familiar. Whatever the case for you, hopefully you feel even more confident with your maths skills now.

6 This week's quiz

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

Well done, you've just completed the last of the activities in this week's study before the weekly quiz.

Box _unit5.6.1

Go to:

Week 4 compulsory badge quiz.

Remember, this quiz counts towards your badge. If you're not successful the first time you can attempt the quiz again in 24 hours.

Open the quiz in a new tab or window (by holding ctrl [or cmd on a Mac] when you click the link).

7 Summary

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This week completes your study of patterns and formulas. Starting in Week 3 with describing a visual pattern and looking for number patterns in the famous Pascal’s triangle, you moved on to using word formulas and writing your own. Finally, you learned how to rearrange a word formula so that you could find out something new from it before taking a quick look at inequalities. These have been a busy two weeks as well as an important two weeks, as you’ve been developing skills that will stand you in good stead for any future studies that involve algebra. So, congratulations on making it to the end of another week. Next week will be all about using powers, and in particular how they can help to write very large and very small numbers concisely. See you there!

You should now be able to:

understand how to read a spreadsheet
understand how a spreadsheet is created
use doing and undoing diagrams to change formulas
interpret and use notation for inequalities.

You are now half way through the course. The Open University would really appreciate your feedback and suggestions for future improvement in our optional end-of-course survey, which you will also have an opportunity to complete at the end of Week 8. Participation will be completely confidential and we will not pass on your details to others.

You can now go to Week 5.

Introduction

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

This week extends your study of exponents, or powers (Succeed with maths – Part 1 provides an introduction to this). You’ll consider how to use these to write large and small numbers to make them easier to understand, write and work with. To do this, a system called scientific notation is used, based on different powers of ten, which allows small and large numbers to be written using fewer digits. Once such numbers have been rewritten this system enables calculations to be carried out with the numbers efficiently. Finally, this week will cover the operation that ‘undoes’ taking the power of a number – roots. This will all give you a good basis to continue your study in maths and many other subject areas, including technology and science.

Watch Maria introduce what this week involves:

Download this video clip.Video player: swim2_week_5.mp4 >

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Transcript

Maria Townsend:

This radio telescope can see over a thousand million million kilometres in to space. That’s a 1 followed by 15 zeroes. A very large number that is both inconvenient to read and write.

So instead of writing it like this a thousand million million can be written like this. One times 10 to the power of 15.

This is all based on powers of 10 as you’ll see this week.

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After this week's study, you should be able to:

understand and use scientific notation
perform calculations with numbers written in scientific notation
understand and calculate square roots of a number.

1 Big and small

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

The numbers that you encounter in everyday situations are usually of a reasonably manageable size, that is, they are generally not very big or very small. A person’s height may be 1.8 metres or the balance in a current account around a few hundred pounds. Both of these are numbers that are easy to understand and read. But what about the UK’s national debt? This is often quoted in billions of pounds, a number with at least ten digits! That is not quite so easy to write or understand when written down. Large numbers like this are often encountered in science and technology, as well as very small numbers. This is where scientific notation comes into its own.

One example of a very large distance is the width of the observable universe. This is about 92 billion light years, where a light year is about 9.5 trillion kilometres. A trillion is one thousand billion or a million million – which means a number followed by 12 zeroes. So to find the width of the universe in kilometres, you would need to multiply 92 billion by 9.5 trillion, since each light year is 9.5 trillion kilometres. How would you do that?

Well, you may want to use a calculator, but there’s a problem: 92 billion is 92 000 000 000, and that is too big a number for many calculators.

So how can you work this out? You could have worked the calculation out on paper, or multiplied 92 by 9.5 to get 874 and deduced that the distance must therefore be 874 billion trillion kilometres – or you may just have been bewildered by the enormity of the numbers! This is an example where scientific notation, which is based upon different powers of the number ten, could usefully be employed.

The next section will be a quick refresher on powers of ten, before moving onto how to use these in scientific notation. If you would like to refresh your knowledge of powers, take a look at Succeed with maths – Part 1.

1.1 Powers of 10

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

100 can be calculated by multiplying 10 by itself, that is 100 = 10 × 10 and, any number multiplied by itself can also be written using power notation. The value of the power being the number of times the number is multiplied by itself. Written in power notation, 10 × 10 is 10², said as ten to the power of two or ten squared.

Similarly, 1000 is the same as 10 × 10 × 10, or 10³. This can be extended indefinitely, to give larger and larger numbers and their corresponding powers of 10.

Our first activity will give you the chance to practise writing numbers using powers of 10, before moving on to how to use these in scientific notation. If you need a hint to get going, click on Reveal comment.

Activity _unit6.1.1 Activity 1 Powers of 10

Allow approximately 5 minutes

Write each of the following numbers as a power of ten.

a.10 000

Discussion

a.Start by working out how many tens you need to multiply together, to give you 10 000, then use the knowledge that .

b.1 000 000

Answer

c.1 000 000 000

Answer

d.What do you notice about the number of zeroes in the original number and the power of 10 in parts (a) to (c)?

Answer

d.The number of zeroes in the original number is equal to the power of 10.

Use the answer to part d) to write each of the following as a number using zeroes and then as a power of 10.

e.There are about one hundred thousand hairs on an average human head.

Answer

e.One hundred thousand is 100 000, or 10⁵.

f.By 2050, the population of earth may be about 10 billion people.

Answer

f.Ten billion is 10 000 000 000, or 10¹⁰.

g.In 1961, the French poet Raymond Queneau wrote a book called A Hundred Thousand Billion Poems.

Answer

g.One hundred thousand billion is 100 000 000 000 000, or 10¹⁴.

If you would like to know more about Queneau’s book, click on ‘reveal comment’.

Discussion

Queneau’s book contained ten sonnets, each with 14 lines. Each page, containing one sonnet, was cut into 14 strips with one line on each strip, so it was possible to combine lines from different sonnets to form a new sonnet. There are 10¹⁴ different ways of making a sonnet in this way.

A digital version of A Hundred Thousand Billion Poems allows you to change lines in one sonnet. The number of sonnets created by visitors to website already is displayed at the bottom of the page. When I first visited the site, fewer than one million had been created.

Now let’s look at how to use powers of ten to write large numbers using scientific notation. You’ll learn about its use with small numbers later in the week.

2 Scientific notation and large numbers

The Open University — Fri, 06 Feb 2015 13:29:49 GMT

So how can you use powers of ten to write numbers in scientific notation? Let’s look at the example of six million to start with. Written in full this number is six followed by six zeroes: 6 000 000. This can also be written as: . Now 6 million is shown like this it can be written using a power of ten by noting that:

10 multiplied by itself 6 times is the same as 10⁶

So,

So, this now means that 6 million can be written as:

You would therefore write 6 million using scientific notation as: .

Similarly, if the example had been six and a half million (6 500 000), this can be written in scientific notation as:

or in scientific notation.

So, a number written in scientific notation takes the form of: a number between 1 and 10 multiplied by a whole number power of 10. This can be shown mathematically as:

Thus, there are two steps to writing a number using scientific notation, as follows:

Work out what the number between 1 and 10 will be.
From this, decide on the power of 10 required.

So, taking the example of 130 000, the number between 1 and 10 must be 1.3, as it cannot be 0.13 or 13. 0.13 is less than 1, and 13 is greater than 1.

So, 130 000 written in scientific notation is . Now, it’s your turn to try some examples.

Activity _unit6.2.1 Activity 2 Understanding and writing numbers in scientific notation

Allow approximately 10 minutes

Write the following numbers without using powers of 10.

Answer

Write the following numbers in scientific notation.

d.92 billion

Discussion

1 billion is 1 followed by 9 zeroes.

Answer

e.400 trillion

Discussion

1 trillion is a million million.

Answer

f.9 500 000 000 000

Answer

g.Which of these numbers is the biggest?

Discussion

Compare the powers of ten.

Answer

g.The highest power of the three numbers in this activity is 14, so 400 trillion is the biggest number here.

Now you’ve found out how to write large numbers using scientific notation, in the next section you’ll turn your attention back to the problem posed at the beginning of this week: how to work out the width of the observable universe in kilometres.