Systems of units
Why are there 12 inches in a foot but 1,000 meters in a kilometer, and how do you change any amount from one unit to another without changing the amount?
Where you are 6 inside
What this area is about
Units come in families built to fit together. The customary family grew out of old trade measures, so its steps are 12, 3, 16 and so on. The metric family uses one base unit per attribute and steps of ten. This area describes both families, then gives one method that converts any unit to any other.
How these fit together
The systems. The customary system is the inch, pound and gallon family. The metric system has one base unit for each attribute. Metric prefixes such as kilo, centi and milli name a size, and mean the same beside any base unit.
The method. Conversion factors are fractions such as \displaystyle \frac{12 \text{ in}}{1 \text{ ft}} that equal 1, so multiplying by one changes the unit and not the amount.
A result. Converting square and cubic units uses the length factor twice for area and three times for volume, so 1 \text{ m}^2 = 10{,}000 \text{ cm}^2.
Problems. Multi-step conversion problems change a unit partway through a longer problem.
A path through it
The customary system follows customary length units; the metric system follows metric length units, and metric prefixes follow it. Conversion factors need customary conversions and unit rates. Square and cubic units come after conversion factors. Multi-step problems follow measurement word problems.
Next: reasoning with units and metric conversions.