Mathematics (Specification A) | Measures

Welcome to Measures! Navigating the World of Units

Hey there! Ready to tackle the chapter on Measures? This might seem like a simple topic, but it is absolutely fundamental—not just for Geometry, but for all of Mathematics and Science!

In this chapter, we will learn how to accurately describe the size, weight, distance, and speed of objects and events. Measures allow us to communicate quantity clearly, and mastering unit conversion is a key skill for success in your exams!

Don't worry if conversion feels tricky at first; we have simple tricks and memory aids to help you master it. Let's dive in!

Key Terminology Quick Review

• Measure: A value that describes a quantity (like length or mass).
• Unit: The standard quantity used to express a measure (like metres or kilograms).
• Conversion: Changing a measure from one unit to another (e.g., centimetres to metres).

Section 1: The Metric System (SI Units)

The metric system is the standard international system (SI) used for most scientific and mathematical measurements. It is incredibly logical because it relies entirely on powers of ten (10, 100, 1000, etc.).

The Three Core Metric Base Units

We use different base units depending on what we are measuring:

1. Length: Measures distance or size. The base unit is the metre (m).
   Example: The height of a door is usually around 2 metres.
2. Mass (Weight): Measures the amount of matter in an object. The base unit is the gram (g).
   Example: A paper clip weighs about 1 gram.
3. Capacity (Volume): Measures how much a container can hold. The base unit is the litre (l or L).
   Example: A large bottle of soda might be 2 litres.

Section 2: Metric Unit Conversion - The Magic of 10

The beauty of the metric system is its prefixes, which indicate how many times bigger or smaller the unit is compared to the base unit (metre, gram, or litre).

The Metric Prefix Ladder

This ladder shows the most common prefixes used in IGCSE Maths. Moving one step changes the value by a factor of 10.

Memory Aid: King Henry Died By Drinking Chocolate Milk

Kilo- (k)
Hecto- (h)
Deca- (da)
Base Unit (B) (m, g, or L)
Deci- (d)
Centi- (c)
Milli- (m)

Key Conversions (The "Big Jumps")

The most important steps (jumps of 1000):

• 1 Kilometre (km) = 1,000 metres (m)
• 1 Kilogram (kg) = 1,000 grams (g)
• 1 Metre (m) = 100 centimetres (cm)
• 1 Metre (m) = 1,000 millimetres (mm)
• 1 Litre (L) = 1,000 millilitres (mL)

Step-by-Step Conversion Process

When converting, always ask yourself: Am I converting to a bigger unit or a smaller unit?

1. To convert from a BIG unit to a SMALL unit: You must MULTIPLY.
   (Example: To change km into m, you need many more metres, so you multiply by 1000.)
2. To convert from a SMALL unit to a BIG unit: You must DIVIDE.
   (Example: To change mm into cm, you need fewer centimetres, so you divide by 10.)

Example: Convert 4.5 metres to millimetres.

Metres (m) are BIGGER than millimetres (mm). We multiply by 1000.

\(4.5 \times 1000 = 4500\) mm

Example: Convert 350 grams to kilograms.

Grams (g) are SMALLER than kilograms (kg). We divide by 1000.

\(350 \div 1000 = 0.35\) kg

Section 3: Time and Time Calculations

Time is a critical measure, especially when dealing with speed and scheduling. Unlike the metric system, time conversions are NOT based on 10s.

Standard Time Conversions

• 1 minute = 60 seconds
• 1 hour = 60 minutes
• 1 day = 24 hours
• 1 year = 365 days (or 366 in a leap year)

Handling Time Calculations

When solving problems involving time intervals, you must be careful not to treat minutes as decimal fractions of an hour unless explicitly asked to do so.

Common Mistake Alert!

If a journey takes 1 hour and 30 minutes, this is NOT 1.3 hours!

To convert minutes to a decimal of an hour, you must divide the minutes by 60.

Calculation: \(30 \div 60 = 0.5\). So, 1 hour 30 minutes is 1.5 hours.

Step-by-Step: Adding and Subtracting Time

Example: A film started at 19:45 and lasted 2 hours 25 minutes. What time did it end?

1. Add the hours: \(19 + 2 = 21\) hours.
2. Add the minutes: \(45 + 25 = 70\) minutes.
3. Convert the excess minutes into hours: 70 minutes is 1 hour and 10 minutes.
4. Add the extra hour to the hour count: \(21 + 1 = 22\) hours.
5. The final time is 22:10.

Key Takeaway: Time requires conversion factors of 60 and 24, not 10 or 100.

Section 4: Area and Volume Units

When we measure area (2D space) or volume (3D space), the units change significantly.

Area Units (Units Squared)

Area is measured in square units (e.g., \(\text{cm}^2\), \(\text{m}^2\)).

If you convert length units, you must apply the conversion factor twice (or square it!).

Did you know? 1 metre = 100 centimetres.

Therefore, to convert \(\text{m}^2\) to \(\text{cm}^2\):

\(1 \text{ m}^2 = (100 \times 100) \text{ cm}^2 = 10,000 \text{ cm}^2\)

Rule: If the length conversion factor is \(k\), the area conversion factor is \(k^2\).

Volume Units (Units Cubed)

Volume is measured in cubic units (e.g., \(\text{cm}^3\), \(\text{m}^3\)).

You must apply the conversion factor three times (or cube it!).

To convert \(\text{m}^3\) to \(\text{cm}^3\):

\(1 \text{ m}^3 = (100 \times 100 \times 100) \text{ cm}^3 = 1,000,000 \text{ cm}^3\)

Important Link: Capacity and Volume

There is a special connection between capacity (litres) and volume (\(\text{cm}^3\)):

• \(1 \text{ millilitre } (mL) = 1 \text{ centimetre cubed } (\text{cm}^3)\)
• \(1 \text{ Litre } (L) = 1,000 \text{ centimetres cubed } (\text{cm}^3)\)

This is often tested in problems involving filling tanks or containers!

Section 5: Imperial and Approximate Conversions

While the metric system is primary, you must know a few standard approximate conversions between the metric and imperial systems (like miles, gallons, pounds).

These approximations will always be given to you in the exam if needed, but it helps to memorise the common ones:

Measure	Metric	Approximate Imperial
Length	5 miles	\(\approx 8 \text{ km}\)
Length	1 inch	\(\approx 2.5 \text{ cm}\)
Mass	1 kg	\(\approx 2.2 \text{ pounds (lb)}\)
Capacity	1 litre	\(\approx 1.75 \text{ pints}\)
Capacity	1 gallon	\(\approx 4.5 \text{ litres}\)

Strategy: If you are given a length in miles and asked for the answer in kilometres, you must use the given conversion factor (e.g., 5 miles = 8 km) to set up a ratio or proportion.

Section 6: Compound Measures

Compound measures combine two or more fundamental units. The most common examples you will encounter are Speed and Density.

The units themselves tell you the formula! For example, if a unit is \(\text{m/s}\) (metres per second), the formula is metres (distance) divided by seconds (time).

1. Speed, Distance, and Time

Speed measures how fast an object is moving. Its units include \(\text{km/h}\) (kilometres per hour) or \(\text{m/s}\) (metres per second).

The Formulas (The DST Triangle):

• Speed = \(\frac{\text{Distance}}{\text{Time}}\) \(\implies S = \frac{D}{T}\)
• Distance = Speed \(\times\) Time \(\implies D = S \times T\)
• Time = \(\frac{\text{Distance}}{\text{Speed}}\) \(\implies T = \frac{D}{S}\)

Crucial Step: Consistent Units!

Before using the formula, ensure your units match the required answer unit. If the speed is in \(\text{km/h}\), your distance must be in km and your time in hours!

Example Problem Setup: A car travels 10 km in 15 minutes. Find the speed in \(\text{km/h}\).

We need time in hours. \(15 \text{ minutes} = 15 \div 60 = 0.25 \text{ hours}\).

Speed = \(\frac{10 \text{ km}}{0.25 \text{ h}} = 40 \text{ km/h}\)

2. Density, Mass, and Volume

Density measures how much 'stuff' (mass) is packed into a given space (volume). Its units are usually \(\text{g/cm}^3\) or \(\text{kg/m}^3\).

The Formulas (The DMV Triangle):

• Density = \(\frac{\text{Mass}}{\text{Volume}}\) \(\implies D = \frac{M}{V}\)
• Mass = Density \(\times\) Volume \(\implies M = D \times V\)
• Volume = \(\frac{\text{Mass}}{\text{Density}}\) \(\implies V = \frac{M}{D}\)

Analogy: Think of density like packing a suitcase. If you put a lot of clothes (mass) into a small bag (volume), the density is high! If you spread the same clothes into a giant trunk, the density is low.

Key Rule for Density: Always check if the units of mass and volume match the units required for density. If the density is in \(\text{g/cm}^3\), but your mass is in kg, you must convert the mass to grams first.