Imagine trying to bake a cake using a recipe that measures flour in tonnes instead of grams. To make measurements practical, scientists use a for different types of scale. The International System of Units defines seven fundamental , 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 . These act as a naming system that applies powers of ten to .
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 . When moving from a smaller unit to a larger unit, you divide.
Calculate the mass of in grams.
Step 1: Multiply by the factor to convert tonnes to kilograms.
Step 2: Multiply by the next factor to convert kilograms to grams.
A cubic metre sounds like a manageable box, but it actually holds a massive 1,000 litres of water. This is because represent three-dimensional space, and their scale drastically compared to straight lines. Similarly, measure two-dimensional area.
The general rules for calculating these conversions are:
To convert area, you must square the linear . For example, since , a square actually contains . A is a special metric area unit equal to .
Calculate the area of a microchip in .
Step 1: Identify the linear between cm and mm.
Step 2: Square the linear factor to find the area .
Step 3: Multiply the original area by the squared factor.
Calculate the volume of a shipping crate in .
Step 1: Identify the linear .
Step 2: Cube the linear factor to find the volume .
Step 3: Divide the original volume by the cubed factor (moving from smaller to larger unit).
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 seconds.
Step 1: Divide by 60 to convert seconds to minutes.
Step 2: Divide by 60 to convert minutes to hours.
Step 3: Divide by 24 to convert hours to 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.
Step 2: Divide by 100 to convert pence to pounds.
Step 3: Add the costs together.
Understanding 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 in .
Step 1: Convert the distance from km to m.
Step 2: Convert the time from hours to seconds.
Step 3: Divide the new distance by the new time.
Why does adding simple algebraic terms cause so many lost marks in exams? Because examiners deliberately mix units to test . 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 . By converting all terms to a single common unit, you create valid that can be accurately simplified.
A rectangular garden has a length of metres and a width of centimetres. Calculate a simplified algebraic expression for the perimeter of the garden in centimetres.
Step 1: Identify the units that need changing for .
Step 2: Homogenise the units using the .
Step 3: Write the formula for perimeter () and substitute the homogenised terms.
Step 4: Expand the bracket to calculate the fully simplified expression.
Students often use linear conversion factors for area or volume (e.g., thinking 1 m² = 100 cm²). Always explicitly write out the squared or cubed factor (e.g., 100²) in your working to secure method marks even if your final calculation is wrong.
When answering time calculations, do not assume a calculator decimal is a direct measure of minutes (e.g., 1.5 hours is 1 hour 30 minutes, not 1 hour 50 minutes).
For 'Show that' algebra questions involving mixed units, examiners expect your very first working step to be homogenising the units before forming any equations.
In OCR Higher Tier papers, be prepared to write final answers for very small conversions (such as mm³ to m³) in standard form.
Standard unit
A base unit of measurement defined by a system, such as the metre for length.
SI base units
A fundamental unit of measurement, such as the kilogram or second, that is not derived from any other unit.
Metric prefixes
A naming system used in scientific notation to represent powers of 10 applied to standard units.
Conversion factor
A numerical multiplier or divisor used to convert a quantity from one unit of measurement to another.
Capacity
The internal volume of a container, typically measured in litres or millilitres for liquids.
Cubic units
Units of measurement used for volume, representing a three-dimensional space.
Squared units
Units of measurement used for area, representing a two-dimensional space.
Hectare
A metric unit of area primarily used in land measurement, equal to 10,000 square metres.
Compound units
Measurements derived from two or more different base units, such as speed or density.
Dimensional consistency
The requirement that all terms in an equation or sum must represent the same physical units before they can be added or compared.
Homogenising units
The process of converting all algebraic terms in a mathematical problem to a single common unit.
Like terms (unit context)
Algebraic terms that share the exact same variable and represent the exact same unit of measure.
Put your knowledge into practice — try past paper questions for Mathematics
Standard unit
A base unit of measurement defined by a system, such as the metre for length.
SI base units
A fundamental unit of measurement, such as the kilogram or second, that is not derived from any other unit.
Metric prefixes
A naming system used in scientific notation to represent powers of 10 applied to standard units.
Conversion factor
A numerical multiplier or divisor used to convert a quantity from one unit of measurement to another.
Capacity
The internal volume of a container, typically measured in litres or millilitres for liquids.
Cubic units
Units of measurement used for volume, representing a three-dimensional space.
Squared units
Units of measurement used for area, representing a two-dimensional space.
Hectare
A metric unit of area primarily used in land measurement, equal to 10,000 square metres.
Compound units
Measurements derived from two or more different base units, such as speed or density.
Dimensional consistency
The requirement that all terms in an equation or sum must represent the same physical units before they can be added or compared.
Homogenising units
The process of converting all algebraic terms in a mathematical problem to a single common unit.
Like terms (unit context)
Algebraic terms that share the exact same variable and represent the exact same unit of measure.