Ok, so the answer is 27. Are units really that important? Wouldn’t chemistry be more fun without them?

It is amazing how frequently students turn in answers without units as if they don’t matter. You know that they matter. If you have to walk 27 miles, 27 blocks (a city term) or 27 feet, well those are just entirely different experiences. The unit matters. If your salary is 27 dollars, 27 pesos, or 27 lira, those are entirely different salaries. The unit will determine whether or not you take the job. Why then would you ever turn in an answer without units? Either the units are so obvious that they do not need to be stated or you just may have no idea what you are doing.

You might be measuring the mass of something and not understand what mass is. You may come up with an answer 27 and not realize that it’s kilograms because you don’t know what a kilogram is or even that your answer has any meaning. If a 27 kilogram mass fell on your head, you’d know exactly what it means. But it hasn’t, and you haven’t stopped to think about it. So your answer is 27. No units.

Some numbers from measurements are unitless. In ratios, the units may cancel and leave you with just a unitless number. I was travelling at 30 mph. You were going at 60 mph. The ratio of our speeds was 60 mph/30 mph = 2.0. You were travelling twice (2.0) as fast as I was. No units.

When a chemistry teacher sees a problem that should have units but doesn’t, this is a sign that the student is going through some memorized routine for generating answers that means nothing. The kind of thing you did in the third grade when you completed a multiplication worksheet. Omitting the units tells us that your mind has not yet entered the process of finding a solution.

The problem is, of course, that you haven’t yet grasped that mathematics is a language and it is, in fact, our language of choice in the physical sciences. You have learned English, and understand that simple sentences have a simple structure like pronoun-verb-preposition-noun. But you also realize that a sentence of this type can actually communicate a particular idea such as “I go to school.” You have learned the grammar of mathematics and now it is time to apply this language to particular ideas. For example, density = mass / volume. The measured mass is 54 grams. The measured volume is 2.0 liters.

The density calculation is not	It is
d = 54/2.0 = 27	d=    54 g =  27 g 2.0 L            L

The first solution has no more meaning than the sentence “pronoun-verb-preposition-noun.” It is worth no credit. The second solution conveys the essence of the problem. We divided a mass by a volume and discovered that each liter of the substance has a mass of 27 g. This solution is worth a lot. The essence of density is captured in the “noun” part of the answer “g/L.” The adjective (27) is useful for conveying a particular fact, namely that the density of this object is 27 g/L. Leave out the g/L and you have omitted the density from your solution.

The next time you solve a problem without any units and get no credit for it, you might object. No credit at all? Which is why I am asking you to save this paper. We will be advising you to reread it from time to time.

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