Units of measure and scale

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Metric Units and Prefixes

The metric system uses base units such as the metre (m) for length, kilogram (kg) for mass, and second (s) for time.
Prefixes show multiples or fractions of a unit: kilo means ×1000, centi means ×0.01, milli means ×0.001.
For example, 1 kilometre = 1000 metres, 1 centimetre = 0.01 metres, and 1 milligram = 0.001 grams.
The International System of Units (SI) is the modern metric system, with seven base units including metre, kilogram, second, ampere (A), kelvin (K), mole (mol), and candela (cd).
Derived units are combinations of base units, such as the newton (kg·m/s²) for force and the hertz (cycles per second) for frequency.
The litre (L) is a widely used non-SI unit for capacity; 1 litre = 1000 cm³ and 1 m³ = 1000 litres.

Converting Between Metric Units

To convert between metric units, multiply or divide by powers of 10 according to the prefix.
Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm.
Mass: 1 tonne = 1000 kg, 1 kg = 1000 g, 1 g = 1000 mg.
Capacity: 1 litre = 1000 ml, 1 cl = 10 ml, 1 litre = 100 cl.
Time: 1 hour = 60 minutes, 1 minute = 60 seconds; these are not decimal conversions.
When converting, check the direction: from a larger unit to a smaller unit, multiply; from smaller to larger, divide.

Area and Volume Unit Conversions

Area units are squared, so conversions involve squaring the length conversion factor.
1 m² = 10 000 cm² because 1 m = 100 cm, so 100 × 100 = 10 000.
1 cm² = 100 mm² because 1 cm = 10 mm, so 10 × 10 = 100.
Volume units are cubed, so conversions involve cubing the length conversion factor.
1 m³ = 1 000 000 cm³ because 1 m = 100 cm, so 100³ = 1 000 000.
1 cm³ = 1000 mm³ because 1 cm = 10 mm, so 10³ = 1000.
1 cm³ is the same as 1 millilitre (ml), and 1 m³ = 1000 litres.

Metric–Imperial Equivalents

Approximate conversions between metric and imperial units are useful for everyday estimates.
1 inch ≈ 2.5 cm; 1 foot ≈ 30 cm; 1 mile ≈ 1.6 km.
1 kilogram ≈ 2.2 pounds (lb); 1 pound ≈ 0.45 kg.
1 litre ≈ 1.75 pints; 1 gallon ≈ 4.5 litres.
These are approximations; exact values differ slightly.

Reading Scales

To read a scale, first work out the value of each small division by counting the intervals between labelled marks.
For example, if a scale from 0 to 10 has 10 intervals, each small division is 1 unit.
If a scale from 0 to 5 has 10 intervals, each small division is 0.5 units.
Always estimate the reading to the nearest small division or half-division if the pointer lies between marks.
Check the unit label on the scale to ensure you record the correct unit.

Scale Drawings and Map Scales

A scale drawing represents a real object or distance using a ratio, such as 1 : 100.
A scale of 1 : 100 means 1 unit on the drawing represents 100 units in real life.
Map scales are often written as ratios, e.g. 1 : 25 000, meaning 1 cm on the map represents 25 000 cm (250 m) in reality.
To find a real distance, measure the drawing and multiply by the scale factor.
To make a scale drawing, divide the real length by the scale factor.
Ensure units are consistent when calculating with scales; convert if necessary.

Similar Shapes and Scale Factors

Similar shapes have the same shape but different sizes; corresponding angles are equal and corresponding sides are in the same ratio.
The scale factor is the multiplier that enlarges or reduces one shape to another.
If the scale factor is k, then lengths are multiplied by k, areas by k², and volumes by k³.
For example, if a shape is enlarged by scale factor 2, its area becomes 4 times larger and its volume 8 times larger.
To find a missing length in similar shapes, set up a proportion using corresponding sides.

Choosing Units and Estimating

Choose sensible units for the context: use millimetres for small lengths, metres for room sizes, kilometres for distances between towns.
Use grams for small masses, kilograms for body mass, tonnes for large loads.
Use millilitres for small volumes, litres for bottles, cubic metres for large volumes.
Estimating with approximate conversions helps check that answers are reasonable.
Always include units in your answer and round appropriately for the context.

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1.Which of these is a base unit in the International System of Units (SI)?
Easy
- Athe litre
- Bthe kilogram
- Cthe newton
- Dthe hertz
2.The prefix milli means a factor of:
Easy
- A1000
- B100
- C1/100
- D1/1000
3.A length is given as 3.5 km. What is this length in metres?
Medium
- A35 m
- B350 m
- C3500 m
- D35000 m
4.The area of a rectangle is 0.04 m². What is this area in cm²?
Medium
- A4 cm²
- B40 cm²
- C400 cm²
- D4000 cm²
5.The volume of a container is 2500 cm³. What is this volume in litres?
Medium
- A0.25 L
- B2.5 L
- C25 L
- D250 L
6.Which of the following are SI base units? (select all that apply)
Medium
- Asecond
- Bampere
- Cnewton
- Dkelvin
- Elitre
7.The litre is an SI base unit.
Easy
True or false?
8.Match each prefix to its meaning.
Medium
- kilo
- centi
- milli
- mega
- one thousandth
- one million
- one hundredth
- one thousand

複習筆記

## Units of Measure and Scale — KS3 Revision ### Metric units and prefixes A **system of measurement** quantifies physical qualities with numbers. The metric system uses **decimal (base-10) prefixes** on base units. - kilo → ×1000 (10^3): 1 km = 1000 m - centi → ×1/100 (10^{-2}): 1 cm = 0.01 m - milli → ×1/1000 (10^{-3}): 1 mg = 0.001 g **SI base units:** metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd). **Derived units** are combinations of base units, e.g. hertz (cycles per second), newton (kg⋅m/s^2). ### Converting units - Length: 1 km = 1000 m, 1 m = 100 cm, 1 cm = 10 mm - Mass: 1 kg = 1000 g, 1 g = 1000 mg - Capacity: 1 litre = 1000 ml, 1 ml = 1 cm^3 - Time: 1 min = 60 s, 1 h = 60 min **Area:** 1 m^2 = 10000 cm^2 (×100^2) **Volume:** 1 m^3 = 1000000 cm^3 (×100^3); 1 litre = 1000 cm^3 ### Metric–imperial approximations - 1 inch ≈ 2.5 cm - 1 mile ≈ 1.6 km - 1 kg ≈ 2.2 lb - 1 litre ≈ 1.75 pints ### Reading scales Count the intervals between labelled marks to find the value of one division, then read off the position. ### Scale drawings and map scales A **scale** is written as a ratio, e.g. 1 : 50000. This means 1 unit on the drawing represents 50000 of the same unit in real life. **Formula:** real length = drawing length × scale factor ### Similar shapes Similar shapes have equal angles and corresponding sides in the same ratio. The **scale factor** k relates lengths: - Lengths: multiply by k - Areas: multiply by k^2 - Volumes: multiply by k^3 Example: if k = 3, area factor = 9, volume factor = 27.

Metric system: A decimal system of measurement using base units and prefixes such as kilo and milli.
SI base units: The seven fundamental units: metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), candela (cd).
Derived unit: A unit formed from base units, e.g. newton (kg⋅m/s^2) or hertz (cycles per second).
Scale: A ratio comparing a drawing or model to the real object, e.g. 1 : 50000.
Scale factor: The multiplier k relating corresponding lengths in similar shapes; areas scale by k^2 and volumes by k^3.
Similar shapes: Shapes with equal corresponding angles and sides in the same ratio.

字卡

What is a system of measurement?

A frame in which physical qualities (such as length, weight, temperature, etc.) can be quantified with numbers.

What does the metric system standardise?

A set of base units and a nomenclature describing relatively large and small quantities using decimal-based multiplicative unit prefixes (such as kilo and milli).

What is the International System of Units (SI)?

A good example of the metric system, and there are other widely accepted systems (less popular than SI) mainly used in specific fields, such as the Gaussian units for electromagnetism.

List the seven SI base units and their symbols.

metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol), and candela (cd).

What is an SI derived unit? Give two examples.

A named combination of base units, such as the hertz (cycles per second) and the newton (kg⋅m/s²).

How is the degree Celsius defined in terms of the kelvin?

It is a shifted scale derived from the kelvin: 273.15 K is equal to 0 °C while 1 K interval is same to the 1 °C interval.

What does the prefix kilo imply?

A factor of 1000 (10³).

What does the prefix milli imply?

A factor of 1/1000 (10^-3).

Express the relationship between kilometres and metres using a prefix.

A kilometre is a thousand metres.

Express the relationship between milligrams and grams using a prefix.

A milligram is one thousandth of a gram.

Which country is a notable outlier in not fully adopting the SI system?

The United States (US). Although it uses the system in some contexts, the US has resisted full adoption, and continues to use different measurement systems.

What is metrication?

Adopting the metric system.

What is a base quantity?

A set of independent dimensions of nature selected in terms of which all natural quantities can be expressed.

What is a base unit of measure?

For each base quantity, a representative quantity is defined as a base unit of measure.

What is a derived unit? Give an example.

A unit derived from the base units used for expressing quantities of dimensions that can be derived from the base dimensions of the system—e.g., the square metre is the derived unit for area, which is derived from length.

What does it mean for derived units to be coherent?

They involve only products of powers of the base units, without any further factors.

What is the unit of time in the metric system?

The second.

What is the unit of length in the metric system?

The metre or a decimal multiple of it.

What is the unit of mass in the metric system?

The gram or a decimal multiple of it.

Name three other metric systems besides SI.

The MKS system, the centimetre–gram–second (CGS) system, and the metre–tonne–second (MTS) system.

What is the Gaussian system?

A still-popular blend of the CGS electrostatic (cgs-esu) system and the CGS electromagnetic (cgs-emu) system.

What is the katal?

A derived unit for catalytic activity equivalent to one mole per second (1 mol/s), added in 1999.

How are all base units defined today?

All base units are defined by physical constants – not by prototypes in the form of physical objects, as they were in the past.

What is the relationship between the metre and the kilogram in the MKS system?

The MKS system is based on the metre, kilogram, and second.

What is the relationship between the centimetre, gram, and second in the CGS system?

The CGS system is based on the centimetre, gram, and second.

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