In my Numbers, Operations, and Patterns course that I’m teaching, we discuss rational numbers a lot. We’re working through Teaching Fractions and Ratios for Understanding by Susan Lamon. It’s an indispensable book for the deep dive into rational numbers as well as the inherent glimpse into the development of elementary-to-junior-high mathematical thinking.
Anyway, we often talk about units and unitizing. You have your standard units, such as meters and seconds. You have your oft-chunked units such as m/s. Then you have your non-standard units which really makes for a challenge:
Whenever we deal with fractions or percents of stuff, we always like to note what the unit is: 95% of the fuel tank left, 60% of the job completed, 30% of six glasses of water, 50% of a case of 42 cans of soda.
What would happen if a student asked for the units on a percentage grade? What would you tell them? Would you tell them they learned 75% of the material? They completed 75% of the work? Would it differ for their percent grade on a test versus the end of a grading period?
The correct answer, of course, to all of these queries is this:
The unit on a percentage grade is “of the possible attainable points“ (excluding potential extra credit opportunities).
That’s our unit.
Then within that we have all these machinations around weighting points according to various scale factors.
The homework portion of your percentage grade is worth “total number of points obtained minus late penalties” over “total number of possible points” times 0.2
The tests portion of your percentage grade is worth “total number of points obtained minus penalties” over “total number of possible points” times 0.4
And so on.
That’s one of the reasons it’s tough to actually specify the “of” of a percent grade. It’s a mishmash of combines what a student did, what a student knows, and whatever pseudo random scale factor the teacher decided to assign to the category. And then on top we have moving definitions of what the phrase “in order to receive full credit you must show your work” means. On top of that, we have whatever items we decided to assign and assign a point score to.
All this isn’t to say don’t grade (though there are plenty who’d advocate for that). Many schools and districts make it impossible to not grade. This is a suggestion to think a bit more deeply about the “of” when students get a percentage total on an assignment, test, or grading period score. Approach student grades with humility. The percent assigned to a student is not a proper standard unit by any means, nor should you treat it as such.