Standard Units

The Foundation of the Metric System

Imagine trying to bake a cake using a recipe that measures flour in tonnes instead of grams. To make measurements practical, scientists use a standard unit for different types of scale. The International System of Units defines seven fundamental SI base units, which are unique because they are not derived from any other unit. These include the metre for length, the kilogram for mass, and the second for time.

Rather than inventing completely new words for very large or very small measurements, we use metric prefixes. These act as a naming system that applies powers of ten to standard units.

• Large multiples: Tera ($10^{12}$), Giga ($10^9$), Mega ($10^6$), and kilo ($10^3$).
• Small submultiples: centi ($10^{-2}$), milli ($10^{-3}$), micro ($10^{-6}$), nano ($10^{-9}$), and pico ($10^{-12}$).

Converting Linear Metric Units

How do you switch between measuring a microscopic cell and a marathon distance without changing the core number? Multiplying and dividing by powers of ten allows you to slide smoothly up and down the metric scale. When converting from a larger unit to a smaller unit (e.g., kilometres to metres), you multiply by the conversion factor. When moving from a smaller unit to a larger unit, you divide.

• Length: $10 \, mm = 1 \, cm$, $100 \, cm = 1 \, m$, $1000 \, m = 1 \, km$
• Mass: $1000 \, mg = 1 \, g$, $1000 \, g = 1 \, kg$, $1000 \, kg = 1 \, \text{tonne}$
• Capacity: This measures the internal volume of a container for liquids. The core factors are $10 \, ml = 1 \, cl$ and $1000 \, ml = 1 \, \text{litre}$.
• The critical link between linear dimensions and capacity is that $1 \, cm^3 = 1 \, ml$, meaning $1 \, \text{litre} = 1000 \, cm^3$.

Calculate the mass of $0.082 \, \text{tonnes}$ in grams.

Step 1: Multiply by the factor to convert tonnes to kilograms.

• $0.082 \times 1000 = 82 \, kg$

Step 2: Multiply by the next factor to convert kilograms to grams.

• $82 \times 1000 = 82{,}000 \, g$

Area and Volume Conversions

A cubic metre sounds like a manageable box, but it actually holds a massive 1,000 litres of water. This is because cubic units represent three-dimensional space, and their conversion factors scale drastically compared to straight lines. Similarly, squared units measure two-dimensional area.

The general rules for calculating these conversions are:

$$\text{New Area} = \text{Original Area} \times (\text{Linear Factor})^2$$ $$\text{New Volume} = \text{Original Volume} \times (\text{Linear Factor})^3$$

To convert area, you must square the linear conversion factor. For example, since $1 \, m = 100 \, cm$, a $1 \, m^2$ square actually contains $100 \times 100 = 10{,}000 \, cm^2$. A hectare is a special metric area unit equal to $10{,}000 \, m^2$.

Calculate the area of a $4.5 \, cm^2$ microchip in $mm^2$.

Step 1: Identify the linear conversion factor between cm and mm.

• $1 \, cm = 10 \, mm$

Step 2: Square the linear factor to find the area conversion factor.

• $10^2 = 100$

Step 3: Multiply the original area by the squared factor.

• $4.5 \times 100 = 450 \, mm^2$

Calculate the volume of a $3{,}400{,}000 \, cm^3$ shipping crate in $m^3$.

Step 1: Identify the linear conversion factor.

• $100 \, cm = 1 \, m$

Step 2: Cube the linear factor to find the volume conversion factor.

• $100^3 = 1{,}000{,}000$

Step 3: Divide the original volume by the cubed factor (moving from smaller to larger unit).

• $3{,}400{,}000 \div 1{,}000{,}000 = 3.4 \, m^3$

Non-Decimal Units: Time and Money

You can easily scale metric units by shifting decimal points, but non-decimal units require specific and unusual conversion chains. Time calculations rely on multiples of 60 and 24, while UK money relies on 100 pence per pound.

Calculate the total number of days equivalent to $302{,}400$ seconds.

Step 1: Divide by 60 to convert seconds to minutes.

• $302{,}400 \div 60 = 5040$ minutes

Step 2: Divide by 60 to convert minutes to hours.

• $5040 \div 60 = 84$ hours

Step 3: Divide by 24 to convert hours to days.

• $84 \div 24 = 3.5$ days

Calculate the total cost in pounds for 3 kg of apples at 85 pence per kg and a £2.50 bag of flour.

Step 1: Calculate the total cost of apples in pence.

• $3 \times 85 = 255$ pence

Step 2: Divide by 100 to convert pence to pounds.

• $255 \div 100 = 2.55$
• This gives £2.55.

Step 3: Add the costs together.

• £2.55 + £2.50 = £5.05

Compound Units

Understanding compound units explains why UK road limits are given in miles per hour, but scientific formulas require metres per second. These units are built by combining two or more base measurements, like distance and time for speed, or mass and volume for density. When converting a compound unit, deal with each measurement separately before combining them back into the final calculation.

Calculate a speed of $108 \, km/h$ in $m/s$.

Step 1: Convert the distance from km to m.

• $108 \times 1000 = 108{,}000 \, m$

Step 2: Convert the time from hours to seconds.

• $1 \times 60 \times 60 = 3600$ seconds

Step 3: Divide the new distance by the new time.

• $108{,}000 \div 3600 = 30 \, m/s$

Applying Unit Conversion in Algebraic Contexts

Why does adding simple algebraic terms cause so many lost marks in exams? Because examiners deliberately mix units to test dimensional consistency. This means all terms in a sum or difference must share the same units before they can be mathematically combined.

Before performing any algebraic simplification, you must go through the process of homogenising units. By converting all terms to a single common unit, you create valid like terms (unit context) that can be accurately simplified.

A rectangular garden has a length of $4x$ metres and a width of $y$ centimetres. Calculate a simplified algebraic expression for the perimeter of the garden in centimetres.

Step 1: Identify the units that need changing for dimensional consistency.

• The final expression must be in cm, so the length must be converted from metres to centimetres.

Step 2: Homogenise the units using the conversion factor $1 \, m = 100 \, cm$.

• Length = $4x \times 100 = 400x$ cm

Step 3: Write the formula for perimeter ($P$) and substitute the homogenised terms.

• $P = 2 \times (\text{Length} + \text{Width})$

• $P = 2(400x + y)$

Step 4: Expand the bracket to calculate the fully simplified expression.

• $P = 800x + 2y$ cm