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Week 1: Measurement of length

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Week 1: Measurement of length

Introduction

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:

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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

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)

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 1 The seven base SI units
Unit nameUnit abbreviationMeasurement
metremdistance
kilogramkgmass
secondstime
ampereAelectrical current
kelvinKtemperature
molemolnumber of particles
candelacdlight 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

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:

A figure showing how to convert from kilometres, to metres, to centimetres, to millimetres.
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

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:

A figure showing how to convert from kilometres, to metres, to centimetres, to millimetres.
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

multiline equation line 1 girl apostrophe s height equals 1.32 times prefix multiplication of 100 cm line 2 equals 132 cm

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

Activity 1 SI units of length

Timing: 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. multiline equation line 1 6.78 m equation left hand side equals right hand side 6.78 multiplication 100 cm line 2 equals 678 cm
  • 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,

    multiline equation line 1 327 cm equation left hand side equals right hand side open 327 division 100 close m line 2 equals 3.27 m

  • 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,

    multiline equation line 1 distance to the house equation left hand side equals right hand side open 2324 division 1000 close km line 2 equals 2.324 km

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

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 2 Size of imperial units of length
Imperial unitSI unit
1 inch2.5 cm
1 foot (singular of feet)30 cm
1 yard0.9 m
1 mile1.6 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?

A rectangle. The width is shown as 8 inches and the height 10 inches.
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.

multiline equation line 1 The length of the border equals eight inches plus 10 postfix times inches plus eight postfix times inches plus 10 postfix times inches line 2 equation left hand side equals right hand side 36 postfix times inches

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

multiline equation line 1 The length of the border equals eight inches prefix multiplication of two plus 10 postfix times inches multiplication two line 2 equation left hand side equals right hand side 36 postfix times inches

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

You already know that: one foot equals 12 inches

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,

multiline equation line 1 length in feet equation left hand side equals right hand side 36 division 12 line 2 equals three feet

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 2 Length in imperial units

Timing: 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?

Comment

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!

    multiline equation line 1 Girl apostrophe s height when fully grown equals 34 inches multiplication two line 2 equals 68 inches

    Or

    multiline equation line 1 Girl apostrophe s height when fully grown equals 34 inches plus 34 inches line 2 equals 68 inches

    one feet equals 12 inches

    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.

    multiline equation line 1 Girl apostrophe s height in feet equals 68 inches prefix division of 12 line 2 equals 5.67 feet

    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.

    multiline equation line 1 0.67 feet equation left hand side equals right hand side 0.67 multiplication 12 inches line 2 equals eight inches

    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:
    A rectangular garden measuring 18 feet by 12 feet
    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.

    multiline equation line 1 Length of border equals 18 feet plus 12 feet plus 18 feet plus 12 feet line 2 equals 60 feet

    one yard equals three feet

    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

    multiline equation line 1 length of the border in yards equals 60 division three yards line 2 equals 20 yards

    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

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

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:

  1. Find out how the two units are related to each other – the ‘conversion factor’.
  2. Ask yourself if the final answer should be bigger or smaller than the original value.
  3. 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

multiline equation line 1 man apostrophe s height in inches equals left parenthesis 1.8 multiplication 39.4 right parenthesis inches line 2 equals 70.92 inches

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.

multiline equation line 1 Man apostrophe s height in feet equals left parenthesis 70.92 division 12 right parenthesis feet line 2 equals 5.91 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.

multiline equation line 1 Number of inches equals open 0.91 multiplication 12 close inches line 2 equals 10.92 inches line 3 equals 11 inches left parenthesis to the nearest inch right parenthesis

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 3 Converting length measurements between systems

Timing: 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.
Comment

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.

multiline equation line 1 Distance from Fort William to Glasgow equation left hand side equals right hand side 108 multiplication 1.6 km line 2 equals 173 km left parenthesis to the nearest kilometre right parenthesis

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?
Comment

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.

    multiline equation line 1 Length of route equation left hand side equals right hand side 250 division 1.6 km line 2 equals 156 miles

    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?
Comment

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.

    multiline equation line 1 Length of thread equation left hand side equals right hand side 182 multiplication 0.9 metres line 2 equals 164 metres left parenthesis to the nearest m right parenthesis

    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

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 4 Following the instructions

Timing: 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.

Comment

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 5 How many rolls?

Timing: 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.

Comment

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:

multiline equation line 1 Length of roll equals 33 feet line 2 equation left hand side equals right hand side open 33 multiplication 12 close inches line 3 equals 396 inches

one inch equals 2.54 cm

multiline equation line 1 Length of roll in cm equation left hand side equals right hand side open 396 multiplication 2.54 close cm line 2 equals 1005.84 cm

one m equals 100 cm

multiline equation line 1 Length of roll in m equation left hand side equals right hand side open 1005.84 division 100 close m line 2 equals 10.06 m left parenthesis to two decimal places right parenthesis

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.

multiline equation line 1 Width of wallpaper equals one ft nine in line 2 equals 12 inches prefix plus of nine inches line 3 equals 21 inches

multiline equation line 1 Width of wallpaper in cm equation left hand side equals right hand side open 21 multiplication 2.54 close cm line 2 equals 53.34 cm

one m equals 100 cm

multiline equation line 1 Width of wallpaper in m equation left hand side equals right hand side open 53.34 division 100 close m line 2 equals 0.53 m left parenthesis to two decimal places right parenthesis

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.

multiline equation line 1 Number of strips of wallpaper equals 10 m prefix division of 2.34 m line 2 equals 4.27 strips

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?

multiline equation line 1 The total length around the room equals 3.2 m plus 3.2 m plus four m plus four m line 2 equals 14.4 m

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:

This shows a rectangle, with 3 equally separated vertical dotted lines on the left hand side.
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.

multiline equation line 1 Number of strips required equals 14.4 m prefix division of 0.53 m line 2 equals 27.1 m

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:

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

So, the

multiline equation line 1 number of rolls needed equation left hand side equals right hand side 28 division four line 2 equals seven

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

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

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.

Acknowledgements

This course was written by Hilary Holmes and Maria Townsend.

Except for third party materials and otherwise stated (see FAQs), this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence.

Every effort has been made to contact copyright owners. If any have been inadvertently overlooked, the publishers will be pleased to make the necessary arrangements at the first opportunity.

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Week 2.