- Sections:
- 0, Chapter 1, Introduction,
### Introduction

After completing this unit you should be able to

- use and apply units of measure for length, mass and volume
- convert between metric measures
- understand how to calculate upper and lower bounds for measurements
- estimate areas
- approximate between SI units and basic Imperial units.

You have four sections to work through.

- Units and Measuring
- Upper and Lower Bounds
- Estimating Areas
- Conversion of Units

- 0, Chapter 2, Units and Measuring,
### Units and Measuring

Different units of measurement can be used to measure the same quantities. It is important to use sensible units. For example you would not think of the distance between London and Glasgow in terms of centimetres but rather kilometres.

Some important units are listed below.

Use the slider to explore worked examples.

- 1, Chapter 2, Task 1, Fitness Check,
### Exercises

Here are some questions to check your progress; there are more practice questions if needed.

**Exercise 1**Which units do you think would be the most suitable to use when measuring:

a. the distance between two towns

a. km or miles

b. the length of a sheet of paper

b. cm

c. the mass of a sheet of paper

c. mg or grams

d. the mass of a sack of cement

d. kg

e. the volume of water in a cup

e. ml

f. the volume of water in a large tank

f. m

^{3}or litres

**Exercise 2**What value does each arrow point to?

a.

a. 10.3

b.

b. 130

c.

c. 6.6

- 0, Chapter 3, Upper and Lower Bounds,
### Upper and Lower Bounds

When measurements are made, they can only be obtained to a limited degree of accuracy. For example if the length of a line is given as \(11\) mm, this means that it is \(11\) mm to the nearest mm. This means, for example, that the original measurement could have been \(11.3\), as \(11.3\) would round to \(11\); but the original measurement could also have had a value like \(10.6\), as \(10.6\) would also round to \(11\).

There are many values that the original measurement could have taken.

**In fact, if \(l\) is the length of \(11\) mm then it must lie in the range \(10.5\leq l\lt 11.5\),**

This just says that the length can be anything between \(10.5\) mm and \(11.5\) mmThe

**lower bound**is the smallest number which will round up to the given number,**upper bound**is the smallest number which will**not**round down to the given number.For example, \(9\) mm, measured to the nearest mm, means that

\(8.5\) mm \(\leq\) actual length \(\lt 9.5\) mm

Here \(8.5\) mm is the lower bound and \(9.5\) mm is the upper bound**Examples**For each length below we show the range of values within which the length must be.

- \( 18 \) cm ⇒ \(17.5\) cm \(\leq l \lt 18.5 \) cm
- \(12.7\) cm ⇒ \(12.65\) cm \(\leq l \lt 12.75\) cm
- \(11.06\) cm ⇒ \(11.055\) cm \(\leq l \lt 11.065 \) cm

- \( 18 \) cm ⇒ \(17.5\) cm \(\leq l \lt 18.5 \) cm
- 1, Chapter 3, Task 1, Worked Example,
### Worked Example

- 1, Chapter 3, Task 2, Fitness Check,
### Exercises

Here are some questions to check your progress; there are more practice questions if needed.

**Exercise 1**For each quantity below, state the range of values within which it must lie.

a. \(x = 4.7\)

a. \(4.65 \leq x \lt 4.75\)

b. \(r = 11.68\)

b. \(11.675 \leq r \lt 11.685\)

c. \(w = 218\)

c. \(217.5 \leq w \lt 218.5\)

d. \(v = 20.0\)

d. \(19.95 \leq v \lt 20.05\)

**Exercise 2**The quantities \(x\) and \(y\) are given to \(1\) significant figure as \(x = 30\) and \(y = 60\).

Find the maximum possible value of the expressions below:a. \(y − x\)

a. \(65 - 25 = 40\)

b. \(x + y\)

b. \(65 + 35 = 100\)

c. \(\frac{y}{x}\)

c. \(\frac{65}{25}\)\(\;=2.6\)

**Exercise 3**If the length of the sides of the rectangle are given to the nearest cm, find:

a. the maximum possible area

a. \(42.5 \times 32.5 = 1381.25\) cm

^{2}b. the range of possible perimeters.

b. \(146\) cm \(\leq\) perimeter \(\lt 152\) cm

- 0, Chapter 4, Areas Concept,
### Areas Concept

A square with sides of \(1\) cm has an area of \(1\) cm

^{2}The area of shapes can be either counted or estimated to see how many of these squares fit inside the shape.

Use the slider to explore worked examples.

- 1, Chapter 4, Task 2, Fitness Check,
### Exercises

**Exercise 1**Find the area of the shaded shape.

Answer: \(10\) cm

^{2}

**Exercise 2**Find the area of the shaded shape.

Answer: \(14\) cm

^{2}

**Exercise 3**Estimate the area of the shape shaded in the diagram.

Answer: You should get an estimate somewhere between

\(9\) cm^{2}and \(11\) cm^{2} - 0, Chapter 5, Conversion of Units,
### Conversion of Units

It is useful to be aware of both metric and imperial units and to be able to convert between them.

Imperial units are;

length (inch, foot, yard, mile)

mass (ounce, pound, stone)

volume (pint, gallon)SI (metric) units are;

length (mm, cm, m, km)

mass (g, kg)

volume (ml, litre)#### Conversion of units

The following list shows conversions within imperial units of measurement

\( 1\) foot \(= 12 \) inches

\( 1 \) yard \(= 3 \) feet

\( 1 \) mile \(= 1760 \) yards

\( 1 \) pound (lb) \(= 16 \) ounces

\( 1 \) stone \(= 14 \) pounds

\( 1 \) gallon \(= 8 \) pints

You can convert between metric and imperial units using the following facts.

**Conversion Facts**\( 1 \) kg is about \( 2.2 \) lbs.

\( 1 \) gallon is about \( 4.5 \) litres.

\( 1 \) litre is about \( 1.75 \) pints.

\( 5 \) miles is about \( 8 \) km.

\( 1 \) inch is about \( 2.5 \) cm.

\( 1 \) foot is about \( 30 \) cm.

- 1, Chapter 5, Task 1, Worked Example,
### Worked Example

Use the slider to explore worked examples.

- 1, Chapter 5, Task 2, Fitness Check,
### Exercises

Here are some questions to check your progress; there are more practice questions if needed.

**Exercise 1**Convert each quantity to the units given.

a. \(3\) inches to cm

a. \(7.5\) cm

b. \(18\) stone to pounds

b. \(252\) pounds

c. \(6\) lbs to ounces

c. \(96\) ounces

d. \(6\) feet \(3\) inches to inches

d. \(75\) inches

e. \(15\) kg to lbs

e. \(33\) lbs

f. \(3\) yards to inches

f. \(108\) inches

g. \(3\) feet to cm

g. \(90\) cm

h. \(5\) gallons to litres

h. \(22.5\) litres

i. \(45\) kg to lbs

i. \(99\) lbs

j. \(9\) litres to pints

j. \(15.75\) pints

k. \(45\) gallons to litres

k. \(202.5\) litres

l. \(8\) litres to pints

l. \(14\) pints

**Exercise 2**Jimar is \(6\) feet \(2\) inches tall and weighs \(11\) stone \(5\) pounds.

Sam is \(180\) cm tall and weighs \(68\) kg.

Who is the taller and who is the heavier?

Find your answer by converting Jimar’s measurements into metric measurements.Answer: Jimar is both taller and heavier than Sam (Jimar is \(185\) cm tall and weighs over \(70\) kg).

**Exercise 3**A car travels on average \(10\) km for every litre of fuel.

The car is driven from Chichester to Woking, a distance of \(41\) miles.a. How far does the car travel in km?

a. \(65.6\) km

b. How many litres of fuel are used?

b. \(6.56\) litres

c. How many gallons of fuel are used?

c. \(1.46\) gallons

- 0, Chapter 6, Summary,
### Summary

#### SI Units

length (mm, cm, m, km)

mass (g, kg)

volume (ml, litre)

#### Upper and Lower Bounds

For example, \(9\) mm, measured to the nearest mm, means that

\(8.5\) mm \(\leq\) actual length \(\lt9.5\) mm

Here \(8.5\) mm is the lower bound and \(9.5\) mm is the upper bound.#### Imperial Units

length (inch, foot, yard, mile)

mass (ounce, pound, stone)

volume (pint, gallon)

#### Conversion of Units

\(1\) km \(=1000\) m

\(1\) m \(=100\) cm \(=1000\) mm

\(1\) cm \(=10\) mm

\(1\) tonne \(=1000\) kg

\(1\) kg \(=1000\) g

\(1\) litre \(=1000\) ml

\(1\) m\(^3=1000\) litres \(=1000\) mm

\(1\) cm\(^3=1\) ml

\(5\) miles is approximately \(8\) km

\(1\) foot is approximately \(30\) cm

- 0, Chapter 1, Introduction,
- Interactive Exercises:
- Interactive Exercises - Units of Measurement, https://www.cimt.org.uk/sif/measurement/m3/interactive.htm
- Units and Measuring, https://www.cimt.org.uk/sif/measurement/m3/interactive/s1.html
- Upper and Lower Bounds, https://www.cimt.org.uk/sif/measurement/m3/interactive/s2.html
- Estimating Areas, https://www.cimt.org.uk/sif/measurement/m3/interactive/s3.html
- Conversion of Units, https://www.cimt.org.uk/sif/measurement/m3/interactive/s4.html

- YouTube URL: https://www.youtube.com/watch?v=jaexoIy3nXs&list=PL8ce1n6mRlCtS0rO_C7CNwtjhyqDVkr8g&index=18

- File Attachments: media/course-resources/Units-Of-Measurement_Presentation.pptxmedia/course-resources/Units-Of-Measurement_Text.pdfmedia/course-resources/Units-Of-Measurement_Answers.pdfmedia/course-resources/Units-Of-Measurement_Essential-Information.pdfmedia/course-resources/Units-Of-Measurement_Learning-Objectives.pdf