Why Skipping Units Costs You the Cheapest Check You Have

A speed of $20\ \text{m/s}$ sustained for $45\ \text{s}$ covers

$$20\ \frac{\text{m}}{\text{s}} \times 45\ \text{s} = 900\ \text{m}.$$

The seconds cancel; metres survive. The answer is a length because the algebra of the units says so.

Now suppose you grabbed the wrong number and multiplied the speed by a distance:

$$20\ \frac{\text{m}}{\text{s}} \times 45\ \text{m} = 900\ \frac{\text{m}^{2}}{\text{s}}.$$

The digits are identical; the quantity is not. Square metres per second is not a length, so it cannot answer "how far", and you know that without re-reading the question. Write only 900 and the two lines look the same.

The check survives rearrangement. From $d = vt$ you get $t = \frac{d}{v}$, and $\frac{\text{m}}{\text{m}/\text{s}} = \text{m} \times \frac{\text{s}}{\text{m}} = \text{s}$ — a time, as it should be. A rearrangement that leaves you with anything else is wrong.

A tank holds $48\ \text{L}$ and drains at $3\ \text{L}/\text{min}$. It is empty when

$$48 - 3t = 0, \qquad t = 16.$$

Sixteen what? The equation has already decided. You can only subtract a volume from a volume, so $3t$ must be a volume, and $t = \frac{48\ \text{L}}{3\ \text{L}/\text{min}} = 16\ \text{min}$. Substituting back: $48\ \text{L} - (3\ \text{L}/\text{min})(16\ \text{min}) = 48\ \text{L} - 48\ \text{L} = 0\ \text{L}$.

Had the tank drained at $3\ \text{L}/\text{s}$, the arithmetic would be identical and the answer $16\ \text{s}$. Nothing in the digits separates sixteen minutes from sixteen seconds; only the unit does, and it came from the equation, not from tidiness.

There is a reason the number, not the quantity, sticks. In a well-known set of experiments, people new to university physics sorted problems by the literal features of the problem as written^[1]. The experts sorted the same problems by the principle needed to solve them^[1].

Symbols come loose in the same way. When college students were asked to turn a sentence into an equation, a large number put the multiplying number on the wrong side, writing $6S = P$ where $S = 6P$ was meant^[2]. Researchers then worked through it with students one at a time to see how the mistake was happening^[2].

None of that work was about units^[1][2]. I read it as the same shape of error: a symbol handled on its own, detached from the thing it stands for. If a letter can come loose that way, a 900 certainly can.

In one study, 160 people, half of them ten- and eleven-year-olds and half of them adults, were asked whether finished answers made sense^[3]. Most of the children leaned on the rule they had just followed instead of checking the answer against the real situation^[3]. The children who did check against the situation got more answers right, but the two simply went together, and this kind of study cannot say which caused which^[3].

A unit is the smallest version of that check: not whether 900 is plausible, which is hard, but what 900 is a quantity of, which is easy.

Take a rectangle with sides of $5\ \text{cm}$ and $8\ \text{m}$. Multiply the bare numbers and you get 40, which is correct only in the unusual unit $\text{cm}\cdot\text{m}$. Converting first,

$$5\ \text{cm} \times 8\ \text{m} = 0.05\ \text{m} \times 8\ \text{m} = 0.4\ \text{m}^{2},$$

or in centimetres $5\ \text{cm} \times 800\ \text{cm} = 4000\ \text{cm}^{2}$, the same area written differently. Read that unlabelled 40 as square metres and you are out by a factor of one hundred; the unit makes the mismatch visible while you can still fix it.

The habit worth building is not "write the unit", it is "say what the number is". In one small trial, a single class of ten- and eleven-year-olds was taught to treat word problems as descriptions of real situations and to ask what their calculation assumed about them^[4]. Compared with two other classes they gave more answers that fitted the real situation, on new problems too, though the researchers presented this as a first exploration rather than a settled result^[4].

Explaining your own steps to yourself has broader support. A review that pooled the research on prompting students to do this found a modest improvement on tests straight after the lesson, and a bigger one when the explanation was given a structure rather than simply asked for^[5]. The same review says there is much less evidence that it works in ordinary classrooms, or that gains are still there later^[5].

A unit is about the smallest structure such an explanation can have: one word, in a fixed place, checkable by the writer. Three questions are worth asking of a line of working, not of a finished page:

• What quantity is this number a measure of?
• Do the two things I am about to add or subtract measure the same kind of quantity, and have I put them in the same unit yet?
• Does the unit I ended with match the kind of thing the question asked for?

If you cannot name the quantity yet, write "(units?)" in its place and carry on; the mark is what brings you back before you hand the work in.

Before changing anything, read back through marked work and note where the unit went missing. If it only happens on the last question of a set, the problem is attention at the end of an evening, not the habit.

Those questions fit what the studies above suggest, but no one has tested them on units, so treat them as sensible practice rather than a proven fix^[4][5]. If you work with a Learn4Less tutor, ask them to stop you mid-line and make you say what the number measures, not just check the boxed answer. The same instinct catches other quiet errors, covered in this post on catching calculation mistakes.

None of the studies behind this post studied units: not one counted unit errors, tested the habit of writing units in every line, or looked at what happens to a grade when the unit is missing^{[3][1][2][4][5]}. Every link from that work to units is my reasoning, not their finding. What a particular teacher gives or takes for a missing unit is not something this research touches; ask, because it varies.

The people studied were not high-school students sitting exams, either: people new to university physics^[1], college students^[2], ten- and eleven-year-olds and adults^[3], and one class of ten- and eleven-year-olds^[4]. Applying any of it to your own Sunday-night problem set is a reasonable guess, not a finding.

The advice half is the thinnest part. It rests on one small exploratory trial^[4] and on a review that calls its own evidence weak for real classrooms and for whether gains last^[5]. The mathematics needs none of that support. Units cancel, quantities of different kinds cannot be added, and a rearrangement that produces the wrong kind of quantity is wrong. You can check all of that on the page, which is the point of writing units down.