My Favorite One-Liners: Part 77

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

At 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 $F$ be the grade on the final exam (as I write a big F on the chalkboard). [groans] After all, final starts with $F$ , and it’s important to assign variable names that make sense.

Also, let $D$ be the up-to-date course average prior to the final. [more groans]

This gives us the course average. Just to be nice, let’s call that $A$ . [sighs of relief]

So $A = 0.2F + 0.8D$ .

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 $x_1, x_2, \dots x_n$ is $\overline{x}$ , then the average of $x_1 + c, x_2 + c, \dots, x_n + c$ is $\overline{x} + c$ . 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 $78, 82, 88, 90$ . A reasonable guess would be something like $85$ . So subtract $85$ from all four values, obtaining $-7, -3, 3, 5$ . The average of these four differences is $(-7-3+3+5)/4 = -0.5$ . Therefore, the average of the original four numbers is $85 + (-0.5) = 84.5$ .

So here’s a typical student question: “If my average right now is an $88$ , and the final is worth $20\%$ of my grade, then what do I need to get on the final to get a $90$ ?” Answer: The change in the average needs to be $+2$ , so the student needs to get a grade $+2/0.2 = +10$ points higher than his/her current average. So the grade on the final needs to be $88 + 10 = 98$ .