2.1 The International System of Units and Unit Conversion
Key Takeaways
- The SI base units tested at grades 4-8 are the meter (length), kilogram (mass), second (time), kelvin (thermodynamic temperature), and mole (amount of substance); the ampere and candela complete the set of seven.
- Metric prefixes are powers of ten, so every metric-to-metric conversion is a decimal-point move: kilo- is 10³, centi- is 10⁻², milli- is 10⁻³, micro- is 10⁻⁶, and nano- is 10⁻⁹.
- Dimensional analysis converts units by multiplying by fractions equal to one, arranged so unwanted units cancel — the single most reliable method for a timed, no-calculator-needed conversion item.
- 1 mL = 1 cm³ exactly, and 1 L of pure water at 4 °C has a mass of about 1 kg, which is why water's density is 1.00 g/cm³ and why displacement volumes convert directly to milliliters.
- Kelvin is the SI temperature unit and has no degree symbol: K = °C + 273.15, while °F = (9/5)°C + 32; a kelvin and a Celsius degree are the same size, so temperature *changes* are numerically identical in both scales.
Why the Measurement System Is Its Own Competency
Competency 002 names the international system of measurement and unit conversion as a distinct expectation. Exam items rarely ask "what is the SI unit of mass?" in isolation. They embed a conversion inside a density, speed, or concentration problem, or they ask which student error produced an answer that is off by a factor of 1,000. Fluency here protects your score across Domains II, III, and IV, because every calculated item on the test arrives in units.
SI Base Units and the Derived Units Built From Them
| Quantity | SI base unit | Symbol | Grades 4-8 classroom instrument |
|---|---|---|---|
| Length | meter | m | Meter stick, metric ruler, tape measure |
| Mass | kilogram | kg | Triple-beam balance, electronic balance |
| Time | second | s | Stopwatch, photogate |
| Thermodynamic temperature | kelvin | K | Thermometer (usually read in °C, converted) |
| Amount of substance | mole | mol | Calculated, not directly measured |
| Electric current | ampere | A | Ammeter |
| Luminous intensity | candela | cd | Light meter (rare at 4-8) |
Derived units are combinations of base units and are where most classroom measurement actually lives:
- Volume: cubic meter (m³) in strict SI; the liter (L) is accepted for classroom use, where 1 L = 0.001 m³ = 1,000 cm³.
- Density: kg/m³ in strict SI; g/cm³ or g/mL in the classroom.
- Speed: m/s. Force: newton (N) = kg·m/s². Energy: joule (J) = N·m. Power: watt (W) = J/s.
- Pressure: pascal (Pa) = N/m².
Notice that a newton, a joule, and a watt are all shorthand for combinations of meters, kilograms, and seconds. Being able to unpack them is what lets you check whether an answer is dimensionally sensible.
The Prefix Ladder
| Prefix | Symbol | Multiplier | Example |
|---|---|---|---|
| giga- | G | 10⁹ | 1 GW = 1,000,000,000 W |
| mega- | M | 10⁶ | 1 MJ = 1,000,000 J |
| kilo- | k | 10³ | 1 km = 1,000 m |
| hecto- | h | 10² | 1 hL = 100 L |
| deka- | da | 10¹ | 1 dam = 10 m |
| (base) | — | 10⁰ | 1 m |
| deci- | d | 10⁻¹ | 1 dm = 0.1 m |
| centi- | c | 10⁻² | 1 cm = 0.01 m |
| milli- | m | 10⁻³ | 1 mm = 0.001 m |
| micro- | µ | 10⁻⁶ | 1 µm = 0.000001 m |
| nano- | n | 10⁻⁹ | 1 nm = 0.000000001 m |
The classroom mnemonic "King Henry Died By Drinking Chocolate Milk" (kilo, hecto, deka, base, deci, centi, milli) helps students place the decimal point, but the exam expects you to know the two-step prefixes as well: micro- is three more places past milli-, and nano- is three past micro-.
Dimensional Analysis: The Reliable Method
Dimensional analysis multiplies the starting quantity by conversion fractions that each equal one, arranged so that unwanted units cancel.
Worked example 1 — a two-step metric conversion. Convert 2.5 km to centimeters.
2.5 km × (1,000 m / 1 km) × (100 cm / 1 m) = 2.5 × 100,000 = 250,000 cm = 2.5 × 10⁵ cm
Worked example 2 — a compound unit. A cyclist travels 54 km/h. Express this in m/s.
54 km/h × (1,000 m / 1 km) × (1 h / 3,600 s) = 54,000 / 3,600 = 15 m/s
The shortcut worth memorizing: to convert km/h to m/s, divide by 3.6; to go the other way, multiply by 3.6.
Worked example 3 — density with a unit trap. A rock has a mass of 87.5 g and displaces water from 40.0 mL to 65.0 mL. Find its density in g/cm³.
Volume = 65.0 − 40.0 = 25.0 mL = 25.0 cm³. Density = 87.5 g ÷ 25.0 cm³ = 3.50 g/cm³. Because 1 mL = 1 cm³ exactly, no numerical conversion is needed — only the recognition that the units are interchangeable.
Water: The Reference That Ties the Units Together
Pure water at 4 °C has a density of 1.00 g/cm³. That single fact links three units:
1 cm³ = 1 mL of water = 1 g of mass, and therefore 1 L of water ≈ 1 kg.
This is why a full 2 L soda bottle has a mass of about 2 kg, and why an object with density greater than 1.00 g/cm³ sinks in fresh water while one below it floats. Exam items use it as a fast plausibility check: a student who reports a pebble's density as 0.045 g/cm³ has almost certainly made a decimal error.
Temperature Conversion
| Reference point | Celsius | Kelvin | Fahrenheit |
|---|---|---|---|
| Absolute zero | −273.15 °C | 0 K | −459.67 °F |
| Water freezes | 0 °C | 273.15 K | 32 °F |
| Room temperature | ~22 °C | ~295 K | ~72 °F |
| Water boils (at 1 atm) | 100 °C | 373.15 K | 212 °F |
Conversions: K = °C + 273.15 and °F = (9/5)°C + 32, with °C = (5/9)(°F − 32) in reverse. Kelvin uses no degree symbol — write 295 K, not 295 °K. Because a kelvin interval and a Celsius degree are the same size, a temperature change of 15 °C is also a change of 15 K, a distinction the exam tests when a specific-heat problem gives ΔT.
A student measures a metal cube as 2.0 cm on each edge and finds its mass to be 21.6 g. What is its density, and what everyday check tells the student the value is plausible?
Convert 72 km/h to meters per second.
A student heats a water sample from 18 °C to 46 °C and must report the temperature change in kelvins for a specific-heat calculation. What should the student write?