Math 10C · Measurement
Two systems, one idea
Canada measures in metres and buys plywood in feet. Both systems are going to keep existing, so the useful skill is not preferring one — it is moving between them without losing track of what the number means.
- 1. The words, first
- 2. A conversion is a multiplication by one
- 3. Area and volume conversions square and cube the factor
- 4. Precision, and what a measurement is really saying
- 5. What costs marks
The words, first
The idea: A conversion factor is a fraction equal to one. Once that lands, every conversion is the same move.
| Word | What it means |
|---|---|
| SI | The metric system: metre, kilogram, second, litre, and prefixes that are all powers of ten. |
| Imperial | Inches, feet, yards, miles, ounces, pounds. The relationships are not powers of ten, so they have to be learnt: 12 inches to a foot, 3 feet to a yard, 5280 feet to a mile. |
| Referent | A familiar object used to estimate a measurement. A doorway is about 2 m; a thumb joint is about an inch. |
| Conversion factor | A fraction whose top and bottom are the same quantity in different units, so it equals 1. Multiplying by it changes the units and not the value. |
| Proportional reasoning | Setting up equal ratios. It is what a conversion factor is doing underneath. |
| Precision | How finely a measurement is recorded. 4.50 cm claims hundredths; 4.5 cm only claims tenths. |
| Accuracy | How close a measurement is to the true value. A precise measurement can still be inaccurate, and the two words are not interchangeable. |
A conversion is a multiplication by one
The idea: Write the factor so that the unit you are getting rid of cancels. If the units come out right, the arithmetic almost always is too.
100 cm and 1 m are the same length, so 100 cm / 1 m equals 1. Multiplying by it changes how a quantity is written without changing what it is:
3.5 m × (100 cm / 1 m) = 350 cm
Notice the m cancels top and bottom and cm survives, which is exactly what was wanted. Had you written the factor the other way up, the units would have come out as m²/cm — obviously wrong, and visible before any arithmetic.
Chain them. 90 km/h into metres per second is two factors in a row:
90 km/h × (1000 m / 1 km) × (1 h / 3600 s) = 25 m/s
The direction check. Before calculating, ask whether the number should get bigger or smaller. A centimetre is smaller than a metre, so the same length needs more of them. An answer that moved the wrong way is wrong regardless of how neat the working looks.
Area and volume conversions square and cube the factor
The idea: 1 ft = 0.3048 m, but 1 ft² is 0.3048² m². The factor applies once per dimension, and forgetting that is the most expensive error in this unit.
Worked example. A room is 12 ft by 15 ft. What is its area in square metres?
Route one — convert the sides first: 12 ft = 3.658 m and 15 ft = 4.572 m, so the area is 16.7 m².
Route two — convert the area: 180 ft² × (0.3048 m/ft)² = 180 × 0.0929 = 16.7 m² ✓
Both give the same answer, which is the point. Using 0.3048 once on the area would give 54.9 m², an answer three times too big — and it looks perfectly plausible if you are not checking.
Volume cubes it. 1 ft³ = 0.3048³ m³ ≈ 0.0283 m³. Same rule, one more dimension.
Precision, and what a measurement is really saying
The idea: Every measurement carries an uncertainty of half the smallest unit it was recorded to. That is not pedantry — it is what the number means.
A length of 7.0 cm recorded to the nearest tenth could truly be anything from 6.95 cm to 7.05 cm. Writing 7 cm instead claims much less: anything from 6.5 to 7.5.
So do not drop trailing zeros. 4.50 cm and 4.5 cm are different claims about the same instrument, and the zero is carrying information.
Referents for estimating. A metre is about one big step. A centimetre is about a fingernail's width. A kilogram is about a litre of water. An inch is roughly a thumb joint. Being able to estimate is what tells you a calculated answer of 54.9 m² for a small bedroom is wrong.
What costs marks
The idea: Three of the four are avoidable by writing the units down.
- The factor upside down. Write it so the unwanted unit cancels, then check the units in the answer.
- Forgetting to square or cube the factor for an area or a volume.
- Not converting before comparing. 40 inches and 1 m cannot be compared until they are in the same unit.
- Losing a trailing zero, which quietly downgrades the precision you were given.