Units in equations

[Sarah Stevens]

The last two summers I have participated in a science modeling workshop to increase my own content knowledge for applying to analytic modeling and to experience pedagogy techniques to encourage conceptual understanding. Throughout these courses, the instructors (science teachers) were very clear about the necessity of labeling all numbers- even those in equations. For example, an equation to calculate the position as a function of time might look like
d = 65miles/hour * t -12miles
instead of our familiar math equations d=65t-12.

Through-out the experience, I learned how lax some math teachers (including myself) have become with units. I gained an appreciation for how labels on numbers kept the focus on the context and the meaning of the numbers in the context of the situation. So I left the training, determined to shift our math teachers to this “scientific” way of writing equations. We would be supporting science and scaffolding our students as we transition into a modeling focus.

However, at my first meeting with math teachers who hadn’t attended the trainings, I experienced quite a bit of questioning from my colleagues. The modeling progression came out the same week as my math training. The teachers quickly noted that the equations in the progression didn’t contain labels. I still argue that it does no harm, helps another content area, and keeps students anchored in the context of the problem.

I continued researching and noticed that the beginning of the High School “Number and Quantity” defines quantities as numbers with units. When I look at the Functions standards, I see the word quantity used in most of the standards- which I think further bolsters my thinking about units in equations. I would like to continue with our shift towards units on all measured numbers; but, with a lack of examples, my teachers are hesitant. We are very interested in your opinion on the matter, both personal and the requirements of the standards.

[Bill McCallum]

Physicists and mathematicians have different approaches to this. You are right that the tradition in physics is to include units in the equations themselves. In mathematics we tend to put the units in the definition of variables. The important thing is to put the units somewhere. So, if you write $d = 65 t -12$, you’d better have said beforehand that $d$ is distance in miles and $t$ is time in hours (and not “let $d$ be distance and $t$ be time”). Then you can deduce from the equation that the units of 65 are miles/hour. There are advantages to the physicists’ approach, but the mathematicians’ approach also had its virtues. For example, when the units are not present in the equation itself, it is easier to see the structure of the equation.

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