If we haven’t thought much about it, we could decide that we really do not use math very much. We would be wrong, of course. Just try to describe to someone else the precise difference between an elephant and a lump of coal without using any numbers, descriptive shapes (geometry), or units of measure. You could say that one has hair and the other hasn’t, but that doesn’t really give a clear picture of either the elephant or the lump of coal.

We need to, and do, communicate about shapes, sizes, and numbers many times every day, so whether we admit it or not, we use math a lot. Signs along the highway tell us speed limits, and speedometers in the dashboard tell us whether we are violating laws. The needle on the fuel gauge tells us whether we need to buy fuel, and those numbers printed on the green bills tell us how much fuel we can afford, but we have to do a bit of calculating first.

In our vocation, moving up the career ladder is almost always blocked if we cannot satisfy someone that we are qualified for more responsible and rewarding positions. The best evidence of such qualifications is usually certification that is granted after successful completion of one or more examinations — and math constitutes a significant fraction of such exams. We could debate the fairness of the importance of math, but after the debate is concluded, math’s significance is still there.

An Essential Part of Decision-Making

Whether we work as operators in a treatment plant or perform maintenance on a collection system, we frequently have to use observable quantitative data in calculations to generate new information, then use that information to make important decisions. For example, you observe that it is 11:50 a.m., and you realize that your lunch is three blocks away in the seat of the pick-up truck. You now must work the numbers to decide whether to walk or run to avoid being late for your salami sandwich.

Quite a few people in the water and wastewater vocations sincerely believe that they do not have the knack to solve math problems. Most folks are not motivated to study and master math until the opportunity has already passed. What they need more than anything else is confidence in their ability to learn — something this series of lessons hopefully will instill. In vocational math, you can do almost anything if you simply remember the rule of five “Ps” — “Patience and Practice Prevent Poor Performance.”

Quantitative Information Requires Both Numbers and Units of Measure

Examine the following three time periods and determine which is smallest and which is largest: 13 ####, 250 @@@@, or 2300 &&&&.

If you don’t know what the symbols represent, you don’t have a clue about the three magnitudes. If #### is years, @@@@ is hours, and &&&& is seconds, we can decide more easily the relative magnitude of each time period. The point is that almost any form of quantitative information we deal with must have both numerical values and units of measure in order to be of value. Take your own wage scale, for example. Does it make any difference to you whether you make $13.67 per hour or 13.67 drachmas per hour?

Develop the helpful habit of always using both numbers and units of measure for all quantitative information when you input information, throughout your calculations, and in expressing your answers. In fact, always try to let your solution determine the units of measure of your answers. This practice alone will catch and reveal a great many of the mistakes you might otherwise make. This will be shown throughout this series.

Logical Thinking Versus Memorized Formulas

Imagine the following situation. You have a Western Red Cedar picnic table with a top that is 1.5 in. thick, 5 ft long, 3 ft wide, and stands 34 in. off the ground. (In metric units it is 38.1 mm thick, 1.5 m long, 0.9 m wide, and stands 864 mm off the ground — it’s still a Western Red Cedar.) You are to determine the top surface area of this table, but you and everyone you know cannot remember the formula for the area of a rectangle. Solve the problem anyway using only the five pieces of given information.

You might immediately note that the fact that the table material is Western Red cedar has no logical effect of the surface area. Anything that does not logically affect the answer should not be included in the calculation of the answer — so leave this out of your solution.

By similar logic, you will not use the thickness of the top or the height of the table in your solution. But you sense that if the table was longer, it would have more top surface area. Also if it were wider, the table would have more surface area. You have identified two significant factors that need to be incorporated into out solution — length and width. What might you logically do with two significant quantitative factors so that if either or both of them were larger, the resulting answer would automatically get larger? Add, subtract, multiply or divide? You almost immediately decide to multiply — surface area = 5 ft × 3 ft = 15 ft2 (or in metric, 1.5 m × 0.9 m = 1.4 m2). You also discover that the formula “length × width = area” is very logical. As we look at many calculations commonly performed in our vocation, it should not surprise you to find that they are quite logical if we first understand why we need the answers.

Why Bother?

You say, “But I have already memorized the formulas that I need, so I’m set, right?” Be careful — sometimes we tend to standardize the way we do things before we have clearly thought about the way we do things. Let’s look at an example: the food to microorganism ratio (F:M) for the activated sludge process.

First, we must understand why we even care about the F:M. It is more than a value to put on a report. Actually, it is a measure of the unit organic loading intensity for the process. It indicates how much purification work must be done per unit of time by the working “bugs” in the process. We measure the work to be done as the biochemical oxygen demand (BOD) feed rate to the process and measure our crew of bugs in terms of the amount of mixed liquor volatile suspended solids (MLVSS) in the process, assuming that the bug count is closely related to the volatile suspended solids in the process.

It is quite logical to presume that the amount of purification work to be done each day will increase in direct proportion to the volumetric rate of flow to the process and to the BOD concentration of that flow, so we use both of these factors as multiplying factors (that is, in the numerator). It is also logical to presume that the amount of bugs (the biomass) on the job increases as the MLVSS concentration increases and also as the volume containing that concentration increases, and these terms constitute the denominator of our F:M formula.

Most references to F:M indicate that we calculate the microorganism portion as the product of the MLVSS concentration multiplied by the volume of the activated sludge aeration basin — the amount of biomass “under aeration.” This implies that those organisms in the process of settling out in the final clarifier to become return activated sludge are not participating in the purification process. It seems reasonable to me that even while these microorganisms are “sedimenting,” they are still actively metabolizing the organic matter they adsorbed and absorbed just a few moments before, so they are still “on the job” and working for us.

But let’s not quibble and blow a question on a certification exam. We will devote more time to various process loading calculations in future issues, including F:M.

Exercise to Build Your Thinking Muscles

Exercising the gray matter — through study and practice — is about the only way to develop math skills. Make yourself find problems to solve and then, if possible, solve them in a different way as a check on your work. Enlist others to dream up problems and check your solutions. This means that your solutions must be legible and organized so someone else can understand how you worked the problem — which is an excellent habit to develop anyway.

And remember, from now on it won’t help to criticize the teacher — because you are the teacher.

As you work, you need to remember that there is only a slight difference between being stubborn as a mule and stupid as an ass. You should not give up easily, but staying up all night making no progress will ruin both tonight and probably tomorrow as well.

Try to work as a team with someone else. Do not just copy each other’s work. You will be surprised how much you will learn while helping someone else understand how to work a problem.

If you have access to e-mail, the old coot who wrote this sincerely likes to help. So if you have a question that is bothering you, try to express it to Don Proctor at dproct@comcast.com.