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Dr. Strangemath or: How I Learned to Stop Worrying and Love the Parenthetical Grouping

Lately I've had a number of students who have erred on homework and tests because they haven't used their calculator correctly. It's not for lack of understanding technology or their calculator's basic functions that causes the problem. Rather, the problem actually boils down to an overestimation of the calculator's ability to understand the user's intended meaning.

Let me explain: A common online "troll" question is "48÷2(9+3) = ?" which can be correctly evaluated in two ways, depending on how it is interpreted. Some calculators will return 2, while others will return 288. Strictly-speaking, both are technically correct.

Why? It's all about interpretation. We could interpret it as 48/[2(9+3)], which would give us an answer of 2, or we could interpret it as (48/2)*(9+3), which would give us an answer of 288. The question is posed in an ambiguous way such that we have to assume the author's intended expression in order to proceed.

What about the order of operations? PEMDAS? BEDMAS? Won't that give us a rule to get the definite answer? It actually doesn't matter. Following the order of operations would reduce the question to "48÷2*12 = ?" which doesn't fix the problem. Multiplication and division are actually the same operation expressed differently and thus hold the same precedence in mathematical equations. The answer to the question posed will change depending on which operation we perform first, and our selection is arbitrary.

So how does this involve calculators? Well, calculators (especially the fancy TI and Casio graphic ones) all utilize some type of operating system. They all use bits of computer code that translate numbers into other numbers. At some point, some engineers were developing the low-level code that runs your calculator and they made some design decisions. One of these involved determining how user input would be treated.

It was here, at this point in the design phase, that ambiguity was addressed. This is when the makers of your calculator decided on how they would assume your intent when you entered "48÷2(9+3)". Some treated it like "48/[2(9+3)]" while others treated it like "(48/2)*(9+3)". The user (in this case, the student) would need to be more specific if they wanted to ensure the result was actually the one they need.

The takeaway for students is that clarity is crucial when feeding numbers into your calculator. You understand what you mean, but the calculator isn't necessarily on the same page. It's much smarter and safer to include extra parentheses and group terms together in a way that makes it absolutely clear to your calculator what you mean.

For example, let's say we're working on the Ideal Gas law in chemistry. This law states that P*V = n*R*T. If your homework problem gives you P, V, R and T and asks you to find n, you would simply divide both sides of the equation by R*T. Algebraically, this would give n = (P*V)/(R*T) (notice how those parentheses group the terms clearly?).

Where students make their mistake is getting lazy with parenthetical grouping and simply entering in their calculator something like "P*V/R*T" Their calculator interprets it as "[(P*V)/R]*T" which yields a wrong answer. Or, even worse, they'll get really lazy and enter "P*V; Ans/R; Ans*T" which is also wrong.

Hopefully this lends some clarity to calculator issues. Always, always, ALWAYS use parenthetical grouping to clarify your entry to the calculator. Check and double-check that your entry makes your intended expression absolutely clear. Otherwise, expect to lose some points on an assignment/quiz/test.

4. (a fun one)