Mathematics route · lesson 1 of 5
- 01Quantities, ratios, and scaleUnseen
- 02Functions as dependenceUnseen
- 03Vectors and coordinate changeUnseen
- 04Change and accumulationUnseen
- 05Probability as evidenceUnseen
Mathematics / Field trial lesson 01
Quantities, ratios, and scale
Learn to read a number as a measured relationship, preserve that relationship through conversion, and use units to expose impossible answers.
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- Awaiting learner observation
- Evidence
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A number cannot tell you what was measured
A room is 4.8. A medicine dose is 4.8. A journey lasts 4.8. The numeral is identical, but the claims are not. Each becomes usable only when a unit names the comparison: 4.8 metres, 4.8 milligrams, or 4.8 hours.
A quantity joins a numerical magnitude to a unit. The unit is not a label pasted on after the arithmetic. It determines which comparisons and operations make sense. Length can be added to length. Dividing length by time creates a rate. Adding length to time does not create a physical quantity at all.
Read the figure description
Three aligned number lines share the same left and right endpoints. The first runs from 0 to 2 metres, the second from 0 to 200 centimetres, and the third from 0 to 2000 millimetres. This shows that changing units changes the numeral and unit together while preserving the represented length.
A ratio compares; a scale factor transforms
A ratio compares two quantities in a stated order. If a map uses 1 centimetre for every 25 metres in the landscape, the map-to-real ratio is 1 cm to 25 m. Reversing it describes a different conversion.
A unit rate names how much of one quantity corresponds to one unit of another. A scale factor tells you what multiplication carries one representation into the other. The reliable question is not “multiply or divide?” It is “which factor has the units I need after cancellation?”
Worked example
Worked example — recover a model car's full length
A model car is 18 cm long and built at 1 to 24 scale. Find the full-size length in metres.
- Interpret 1 to 24 as 1 model centimetre corresponding to 24 real centimetres.
- Multiply 18 model cm by 24 real cm per 1 model cm. The model-centimetre units cancel.
- The product is 432 real cm.
- Convert with 1 m per 100 cm, giving 432 cm times 1 m per 100 cm, or 4.32 m.
- Check magnitude: a full car should be much longer than the model, so 4.32 m is plausible and 0.75 cm is not.
ConclusionThe full-size car is 4.32 m long. The unit chain records why multiplication, then division by 100, was appropriate.
Make the units cancel on purpose
A dimension describes the kind of quantity, such as length, time, or mass. A unit is a chosen standard for expressing that dimension, such as metre, second, or kilogram. Metres and centimetres are different units of the same dimension, so conversion can connect them. Metres and seconds are different dimensions, so no conversion factor can make their sum meaningful.
Dimensional consistency is a necessary check, not a proof that an answer is correct. An equation with mismatched dimensions must be wrong. An equation with matching dimensions may still use the wrong numerical factor or model.
Worked example
Worked example — convert a walking rate without losing time
A walker travels 1.5 kilometres in 18 minutes. Express the average rate in metres per second.
- Start with the measured relationship 1.5 km per 18 min, not two disconnected numbers.
- Multiply by 1000 m per 1 km. Kilometres cancel, giving 1500 m per 18 min.
- Multiply by 1 min per 60 s. Minutes cancel from numerator and denominator in the arranged factor.
- Compute 1500 divided by 1080, which is approximately 1.39.
- Check scale: ordinary walking is on the order of metres per second, so 1.39 m/s is plausible; 83.3 m/s would signal a missing minute-to-second conversion.
ConclusionThe average rate is about 1.39 m/s. Every conversion factor equals one, so the represented rate stays fixed while its description changes.
| Operation | Required relationship | Example | Diagnostic |
|---|---|---|---|
| Addition | Same dimension | 2 m plus 30 cm | Convert to a common unit first |
| Multiplication | Any dimensions with a stated interpretation | 3 m times 4 m | The result is area in square metres |
| Division | A meaningful comparison or rate | 120 km divided by 2 h | The result is speed in kilometres per hour |
| Equality | Same dimensions on both sides | distance equals speed times time | Units must reduce to the same dimension |
Hold the object still while its representation changes
Prediction before motion
A drawing is 12 cm long. Before moving the scale control from 1 to 10 toward 1 to 25, predict what happens to the represented real length and name the unit that should remain after the calculation.
- Invariant
- Real length divided by model length equals the selected scale denominator when both lengths use the same unit.
- Observe
- Compare the double number line, the ruler marks, and the dimensional check. Which marks changed, and which represented relationship stayed invariant?
- Static account
- At 1 to 10, a 12 cm drawing represents 120 cm or 1.2 m. At 1 to 25, the same 12 cm drawing represents 300 cm or 3 m. The drawing stays 12 cm; only the represented real length changes with the factor.
Audit a blueprint before trusting the answer
Scale reasoning matters when the representation and the built object cannot be compared directly. A blueprint, dosage chart, map, recipe, or scientific model compresses one relationship into another. The safety habit is the same: state what each unit refers to, orient the factor, cancel visibly, and test the magnitude.
The final task asks you to combine all three objectives. It is deliberately a different form from the immediate checks, so a correct response can contribute transfer evidence rather than repeating the same click.
The frozen 0.1-preview release record retains the earlier sketch and manifest.