Units as a Planning Tool

In this lesson you will: Multiply by a conversion factor — a fraction whose value is 1 — arranged so the unwanted unit cancels; Put the unit you want to lose on the opposite level of the fraction from where it sits now; Square the factor for an area unit; a length factor of 3 becomes an area factor of 9.

A conversion factor is a disguised 1

Since 1 mile=5280 feet, the fraction (5280 ft)/(1 mi) is a quantity divided by an equal quantity. Its value is 1.

Multiplying by 1 cannot change how far something is. It only changes the words the distance is reported in. That is the entire theory of unit conversion, and it means you never have to remember whether to multiply or divide — you only have to arrange the fraction so the unit you are done with cancels.

Units cancel exactly like common factors. A mi upstairs and a mi downstairs delete each other, the same way a 7 upstairs and a 7 downstairs do.

Aim the factor at the unit you want to lose

The rule is mechanical. Find the unit you want gone, notice whether it is currently upstairs or downstairs, and write the factor with that unit on the opposite level.

In 60 mi/h, the mi is upstairs, so the factor needs mi downstairs: (5280 ft)/(1 mi). The h is downstairs, so its factor needs h upstairs: (1 h)/(3600 s). Two factors, two cancellations, one answer.

Area squares the factor

Here is the conversion that goes wrong most often. A yard is 3 feet, so a square yard ought to be 3 square feet. It is not. It is 9.

A square yard is a square measuring one yard on each side, so it is 3 ft by 3 ft. Count the squares in the picture. To convert an area from square feet to square yards you divide by 9, not by 3.

The same logic runs the other way for volume: a cubic yard is 27 cubic feet. The factor is raised to whatever power the unit is.

Common mistake — The factor went in upside down

Asked to turn 90 minutes into hours, a student multiplies: 90* 60=5400 hours.

The reasoning is not lazy. The student knows the number 60 is the link between minutes and hours, and multiplying is what conversions usually feel like. The 60 simply went in the wrong way up.

Track the units and the mistake is visible before the arithmetic: 90 min*(60 min)/(1 h) has minutes on top twice and nothing to cancel, so the answer would carry min^2/h.

Turned over, 90 min*(1 h)/(60 min)=1.5 h, and the minutes cancel.

How to catch it: Look at what canceled. If the unit you wanted to remove is still there — or has doubled up — flip the factor and multiply again. A size check backs it up: an hour is a big unit, so a count measured in hours must be a smaller number than the same amount in minutes.