I’ve written several times of my interest in becoming a certified EMT, the intrinsic rush of pulling a damaged body away from the alluring blankets of effortless death that would smother and asphyxiate.

I would also like to substitute teach high school math and physics. In anticipation of the inevitable lament “When are we ever gonna have to use this in real life?”, I came up with an application of math that I hope will be of interest to most students. The answer to their question is, of course, you’ll never have to use math if you don’t want to, but there are lots of fun and worthwhile things you can do with it if you choose to.

It was obvious early on that I should incorporate money into the example, since many people assign money an inexplicably high value in their lives. I decided on paychecks; specifically, the non-linear nature of taxes. This topic not only meets the money criteria, but is also something that the students would hopefully be somewhat familiar with if they had had summer jobs and had looked at their pay stubs.

By the non-linear nature of taxes I mean, for example, that a person’s after tax income (ATI) for 80 hrs of work is less than twice as much as their ATI for 40 hrs of work. FICA and Medicare combined are a fixed 7.65% of gross, but federal withholding introduces the non-linearity.

The example I worked out here is for a non-married person paid $15/hr bi-weekly who declares 1 on their W-4. The picture below has two graphs of the same plot, where the top graph has a linear trendline to visually verify the non-linearity and the bottom graph has a trendline defined by the 3rd order polynomial given, which matches the plot far better.

It’s necessary to obtain such a polynomial equation through some kind of curve fitting algorithm in order for the data to be mathematically workable and useful. Using this equation, which serves as a model of good approximation, you can find the point on the curve where its slope is highest (about 50 hrs). This point could be significant in terms of max gain for least effort, but I’m hesitant to come to any conclusions about it after not having looked through a functions textbook since 10th grade. A more straightforward graph is shown below.

[Image lost to the digital aether]

Here, the after tax wages are graphed vs. hrs. After tax wages are simply after tax income divided by the corresponding number of hours. Yes, the terminology is of my invention.

As you can see, for a single person declaring 1 on their W-4, for any less than 28 hrs worked in the two week pay period, no government withholding is paid. This is why, for a $15/hr wage, their after tax wage is found simply by subtracting the FICA/medicare 7.65% of their gross from their gross and dividing by the corresponding number of hours:

[$15 – (7.65% of $15)]/1hr = $13.85

for 10 hrs:

[$150 – (7.65% of $150)]/10hrs = $13.85

At 28 hours, deductions for government withholding begin and the straight line succumbs to an exponential decay. I was curious how many extra dollars a person could earn if instead of a single full-time job they contributed 26.67 hours per pay period to each of 3 jobs, thereby avoiding government withholding. Contributing 20 hours per pay period to each of 4 jobs would be an equivalent option, you get the idea. The result, in this $15/hr case, is:

3 or 4 jobs: $13.85/hr * 80 hrs = $1108
1 job: $12.44/hr * 80 hrs = $995.20
difference = $112.80 = 11% of single job after tax income.

Elliot brought up a point that likely forces this gain to remain in the short-term domain: from the point of view of the IRS, working 20 hours for 4 jobs is probably the same as working 80 hours for 1 job, so that even when none of the checks from your 3 or 4 jobs ever have government withholding deductions, you’ll be handed an annual bill for all the money you owe, instead of receiving what normally is a check from the IRS. Still, I see no harm in withholding this information from the children if it gets them interested in mathematics. Given the itinerant nature of the substitute teacher, I’ll be out of the scene before any turmoil resulting from the telling of the half-truth develops.