Algebraic interpretation of ambiguous verbal expressions [closed]

Question

I am helping my daughter with Algebra. She has been asked to generate algebraic expressions for each of the following verbal statements:

1. "Five times the quantity of x squared plus m, minus two times the quantity of x squared plus m."

2. "The product of six and x squared, increased by the sum of four and the square of x."

In the first question, does the presence of "the quantity of" imply that the following two terms are grouped together. Alternatively, should the statement be intrepreted in strict left to right order? Is it 5(x^2+m) or 5(x^2)+m etc. etc.?

In the second question, is the first part (6x)^2 or 6(x^2)?

I think I must be complicating things, but I'm absolutely at a loss as to how to help my daugther learn rules for interpreting these types of questions. We've reviewed her textbook and none of the examples have any level of ambiguity to them.

Answer 1

There's no standard terminology for this, and it's indeed ambiguous. That's why we have unambiguous mathematical notation, using the PEMDAS rules; it would be very confusing if formulas were written in English (try reading Euclid's Elements to see how people explained math before this notation was developed).

But you can use heuristics to help. In the first example, the phase "quantity of x squared plus m" is used in both parts of the expression, so I would assume that this is a distinct expression, which should be translated into (x^2 + m). So the full result would be

5(x^2 + m) - 2(x^2 - m)

In the second example, notice that they use different phrasing: " squared" and "square of ". In the first use, I would guess that they intended everything before "squared" to be the quantity that's being squared, so it translates to

(6 * x)^2 + 4 + x^2

Answer 2

Both situations are ambiguous. Yes, it is poor writing. To deal with the grammar:

Five times the quantity of x squared plus m

We don't know whether the prepositional phrase "plus m" (interpreting "plus" as a preposition) modifies "five times the quantity of x squared" or only "x squared". The former interpretation would be a bit unusual, because normally we would simply write "x squared" instead of "the quantity of x squared", so I think that the latter interpretation is more likely, but it's certainly not entirely clear.

The product of six and x squared

We don't know whether the participle "squared" modifies "the product of six and x" or only "x".

In each case, the error is sometimes called "attachment ambiguity", and ELU has a tag for it (which I've added to this question) if you'd like to read about some other examples.

Answer 3

Is it 5(x^2+m) or 5(x^2)+m etc. etc.?

It's worse than that. It could be either of those things. Or (5x)^2+m. Or... It's hopelessly ambiguous. This is why real mathematicians never use this sort of language, and why we have mathematical notation. "Word problems" are a legitimate part of teaching mathematics, but this isn't a good word problem.

...how to help my daughter learn rules for interpreting these types of questions...

Yes, this is the real question. I hope your daughter gets a better mathematics teacher in the future!

Your daughter will struggle to make sense of the question because it doesn't actually make sense! The skill being taught isn't mathematics or English. It's how to get a better grade by reading the teacher's mind. And this is especially tricky if the teacher doesn't understand the subject they're teaching. I can think of at least two approaches.

1. Actually try to read their mind. Look for hints from previous assignments or other things they've said in class. For example, do they tend to set simple questions, or do they like to try and trick the students? If simple, I'd ignore "the quantity of" and put 5x^2+m as the answer. If tricky, they're more likely to want 5(x^2+m). You might find other types of hints in previous work.

2. Practice judgement as to when it actually matters. Will a low grade on this assignment actually have any bad consequences for your daughter? If not, then just write down the first answer that comes to mind, waste as little time as possible, and look for opportunities to learn real mathematics another time (or spend the time making music, playing games, doing something else more enjoyable). Cultivate the mindset that a low or high grade is a choice that you make, and it's OK to choose a lower grade sometimes; the results of this assignment don't reflect your daughter's intelligence or self-worth. Perhaps take it more seriously when you get to senior high school, at which stage you will hopefully meet a more competent teacher.