Why Units Matter In Mathematics And Science
- A measurement is a numerical value paired with a unit.
- The number tells you "how much", and the unit tells you "of what". Writing units is not decoration, it is part of the meaning.
Measurement
A value that combines a number and a unit to describe the size of a quantity (for example, 2.4 m, 15 g, 8 s).
Without units, you cannot reliably compare quantities, communicate results, or check whether an answer makes sense.
- In problem solving, units are also a quick error-check.
- If your calculation ends in $\text{cm}$ rather than $\text{cm}^2$ (area) or $\text{cm}^3$ (volume), something has gone wrong.
Systems Of Measurement Are Agreed Conventions
A system of measurement is a set of agreed base units from which other units are derived.
System of measurement
A structured set of units based on chosen base units, with rules for creating and converting other (derived) units.
Two systems are widely encountered:
- The metric system, formalized as the International System of Units (SI)
- The imperial system, still used in some countries for everyday measurements
The SI Base Units You Should Recognize
SI defines seven base units:
- meter (m) for length
- kilogram (kg) for mass
- second (s) for time
- ampere (A) for electric current
- kelvin (K) for temperature
- mole (mol) for amount of substance
- candela (cd) for luminous intensity
In standard mathematics, you will most often work with length, mass, time, area, volume, capacity, and speed.
Types Of Measurement And Common Units
- Many units naturally "belong" to certain types of measurement.
- Sorting by type is often the first step in a word problem.
Length And Distance (1D)
- Length measures one-dimensional size.
- Distance is length along a path.
- Common units include mm, cm, m, km, and in imperial, inches, feet, miles.
Length
A one-dimensional measure of size, describing how long something is.
Area (2D)
Area measures the size of a surface and uses squared units such as cm$^2$, m$^2$, km$^2$.
Area
The measure of the size of a two-dimensional surface, expressed in square units.
Volume And Capacity (3D)
- Volume measures three-dimensional space and uses cubed units such as $\text{cm}^3$, $\text{m}^3$.
- Capacity is the amount a container can hold, often measured in liters (L) and milliliters (mL).
Volume
The amount of space a 3-dimensional solid occupies, measured in cubic units such as cm$^3$.
Capacity
The amount of space available to hold something, often measured in liters (L) or milliliters (ml).
- In everyday speech, "volume" and "capacity" are often treated as the same idea.
- Mathematically it helps to remember: volume is about 3D space (cm$^3$), while capacity is often a liquid measure (mL, L).
Mass (Not Weight)
Mass describes the amount of matter in an object and is measured in g and kg (or lb in imperial).
- Do not confuse mass (kg) with weight (a force measured in newtons, N).
- In mathematics problems, "weight" is sometimes used informally when the unit is kg, but scientifically that is mass.
Time And Speed
- Time is measured in seconds (s), minutes (min), hours (h).
- A common compound unit is speed, measured as distance per time, for example m/s or km/h.
Compound unit
A unit formed by combining other units through multiplication or division, such as km/h or m/s.
Metric Conversions Use Powers Of Ten
The metric system is designed so that many conversions are based on powers of ten.
Metric Prefixes
- Common prefixes include:
- kilo- (k) means $\times 10^3$
- centi- (c) means $\times 10^{-2}$
- milli- (m) means $\times 10^{-3}$
- Converting between metric units is mainly shifting the decimal point.
Multiplying And Dividing By Powers Of Ten
- Multiplying by $10^n$ moves the decimal point $n$ places to the right.
- Dividing by $10^n$ moves it $n$ places to the left.
$$280 \times 0.001 = 0.28$$ (because $0.001 = 10^{-3}$)
$$34500 \div 10^2 = 345$$ (divide by 100)
A quick mental check: multiplying by a number less than 1 should make the answer smaller, and dividing by 100 should remove two zeros (or move the decimal two places left).
Converting Area And Volume Requires Squaring And Cubing The Scale Factor
- Length conversions use a scale factor (for example, $1\text{ m} = 100\text{ cm}$).
- But area and volume change faster:
- If length scales by $k$, then area scales by $k^2$.
- If length scales by $k$, then volume scales by $k^3$.
Since $1\text{ m} = 100\text{ cm}$, then
- $1\text{ m}^2 = (100\text{ cm})^2 = 10\,000\text{ cm}^2$
- $1\text{ m}^3 = (100\text{ cm})^3 = 1\,000\,000\text{ cm}^3$
- Do not convert $\text{m}^2$ to $\text{cm}^2$ by multiplying by 100 instead of 10\,000.
- Always square (or cube) the conversion factor.
Using Units In Geometry Problems
Geometry formulas produce units that match the dimension of what you are measuring.
Area Of A Square
For a square with side length $s$, area is: $$A = s^2$$
A square has side length 12 cm. Thus: $$A = 12^2 = 144\text{ cm}^2$$
Surface Area And Volume Of A Cuboid
- A cuboid (rectangular prism) has length $l$, width $w$, height $h$.
- Surface area: $$SA = 2(lw + lh + wh)$$
- Volume: $$V = lwh$$
- A cuboid measures $3\text{ m} \times 4\text{ m} \times 5\text{ m}$.
- Surface area: $$SA = 2(3\cdot 4 + 3\cdot 5 + 4\cdot 5)=2(12+15+20)=94\text{ m}^2$$
- Volume: $$V = 3\cdot 4\cdot 5 = 60\text{ m}^3$$
- Always write units in your final answer.
- In multi-step problems, keep units through the working to avoid mixing cm with m, or m$^2$ with m$^3$.
Choosing Sensible Units And Checking Reasonableness
Often the key skill is not converting, but choosing an appropriate unit and judging whether an answer is sensible.
Picking Units From Context
- Small objects (for example, matchbox dimensions) are usually in mm or cm.
- Paper length may be written in mm (297 mm) or cm (29.7 cm), both represent the same length.
- Room and house floor areas are often in m$^2$.
- Shipping container dimensions are commonly in m.
- Swimming pool length is often in m.
- A container's capacity is often in mL or L.
Explain why both statements can be correct:
- "The length of an A4 sheet is 297"
- "The length of an A4 sheet is 29.7"
Solution
- If the unit is mm in the first case, $297\text{ mm} = 29.7\text{ cm}$ in the second.
- The number alone is ambiguous, the unit resolves it.
- If you convert 3 m$^2$ to cm$^2$, do you multiply by 100 or 10\,000? Why?
- Which is more sensible for a giraffe's height: 5.2 m or 5200 m?
- A shoebox has a capacity of 6307. Suggest a sensible unit.
Converting Between Metric And Imperial Units Requires A Given Conversion
- Metric to imperial conversions are not powers of ten, so you must be given (or remember) a conversion factor, for example:
- $1\text{ inch} \approx 2.54\text{ cm}$
- $1\text{ mile} \approx 1.609\text{ km}$
- Then use multiplication or division, and keep units consistent.
- When converting, write the conversion as a fraction that cancels units. For example, $$5\text{ miles} \times \frac{1.609\text{ km}}{1\text{ mile}} \approx 8.045\text{ km}$$
- The unit "miles" cancels, leaving km.
Should The World Use One System?
- Using a shared system (like SI) supports global communication in science, trade, and engineering, which connects to globalization and sustainability.
- However, traditional systems (like imperial units) are deeply connected to culture and everyday practice.
- In high-risk contexts (such as aviation and medicine), standardizing units reduces mistakes.
- For example, dosage errors can occur if mg and micrograms ($\mu$g) are confused.
- Clear unit choices and careful labeling show how human-made measurement systems can influence community safety.
