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.

Seen another way, we’re solving the algebra problem

88(0.8) + x(0.2) = 90

Let me solve this in an unorthodox way:

88(0.8) + x(0.2) = 88 + 2

88(0.8) + x(0.2) = 88(0.8+0.2) + 2

88(0.8) + x(0.2) = 88(0.8) + 88(0.2) + 2

x(0.2) = 88(0.2) + 2

x = \displaystyle \frac{88(0.2)}{0.2} + \frac{2}{0.2}

x = 88 + \displaystyle \frac{2}{0.2}

This last line matches the solution found in the previous paragraph, x = 88 + 10 = 98.

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