And the end of every semester, instructors are often asked “What do I need on the final to make a ___ in the course?”, where the desired course grade is given. (I’ve never done a survey, but A appears to be the most desired course grade, followed by C, D, and B.) Here’s the do-it-yourself algorithm that I tell my students, in which the final counts for 20% of the course average.

Let be the grade on the final exam, and let be the up-to-date course average prior to the final. Then the course average is equal to .

Somehow, students don’t seem comforted by this simple algebra.

More seriously, here’s a practical tip for students to determine what they need on the final to get a certain grade (hat tip to my friend Jeff Cagle for this idea). It’s based on the following principle:

If the average of is , then the average of is . In other words, if you add a constant to a list of values, then the average changes by that constant.

As an application of this idea, let’s try to guess the average of . A reasonable guess would be something like . So subtract from all four values, obtaining . The average of these four differences is . Therefore, the average of the original four numbers is .

So here’s a typical student question: “If my average right now is an , and the final is worth of my grade, then what do I need to get on the final to get a ?” Answer: The change in the average needs to be , so the student needs to get a grade points higher than his/her current average. So the grade on the final needs to be .

Seen another way, we’re solving the algebra problem

Let me solve this in an unorthodox way:

This last line matches the solution found in the previous paragraph, .

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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