What Mrs. Anderson’s High School Chemistry Class Taught Me About Units

I found a bunch of posts I started writing years ago but never finished. Here’s one of them.... just finished.

I remember Mrs. Anderson at the blackboard in my high school chemistry class, writing out a conversion problem - something like converting a volume in liters to milliliters to moles. She worked it by stacking fractions, one after another, each one arranged so a unit in the numerator of one fraction matched the unit in the denominator of the next. Then she crossed them out in pairs until only the answer's units remained. "Learn a few equations," she said, "and you can solve just about anything." She was not talking about memorizing formulas. She was talking about what she called factor labeling.

Factor labeling - today more commonly called a more fancy dimensional analysis - treats units as algebraic objects. You multiply and divide them the same way you multiply and divide numbers. A conversion factor like 1,000 milliliters per liter is really just the number one, with some units, so multiplying by it changes the label without changing the quantity. Chain enough of these factors together and the units in between cancel, leaving you with exactly the unit you wanted.

The value of the method is not speed. It is error detection. In high school it helped in chemistry (not my favorite subject). In college physics (loved it) it became something I depended on, once problems started combining velocity, acceleration, force, and energy in the same calculation and a single wrong exponent could hide inside an otherwise reasonable looking number..

Here’s a simple example. In circuit analysis, say a resistor carries 25 milliamps (I) at 12 kilohms ® and you need to figure the power (P) dissipated in watts. Run the raw numbers straight through P = I²R without tracking units and you get 7,500 - off by a factor of a thousand from the real answer. Run it through factor labeling instead: convert 25 milliamps to 0.025 amps and 12 kilohms to 12,000 ohms before multiplying, and the units confirm the answer lands in watts: 7.5 watts.

I have used this same check in electrical engineering courses, in circuit analysis, and in grading student calculations for capstone projects. A wrong answer with clean unit cancellation is rare. A wrong setup almost always leaves a stray unit sitting where it should not be.

Fifty plus years after high school, I always check my units before I trust my numbers. Thanks, Mrs. Anderson!