Question

Dear Mitch,

Here's my question: What are the problems that have a number, a little triangle and then another number mean? I've never seen one of them before, at least I don't think so. Can you tell me what it is?

Thank you,

Emily

Answer

Dear Emily,

From your brief description I'm afraid I can only narrow down the possibilities that come to mind. If none of the ones I present ring the bell you hope to hear, please feel free to send another email – though on that one please include either your age or the grade level of your math class, as there are a number of mathematical symbols that stand for one concept/procedure at one level and a completely different one at another level. One example of such a situation is the letter 'x', which has long been used to mean "times" (or "multiply") at the elementary school level, but then turns into one of the two most famous variables of all time -- half of the (x, y) pairing that comprises the standard coordinate geometry system. Also, while I'm at it, a "dot" is a 'decimal point' in one context and a multiplication sign when it's lifted a tenth of an inch ... so, as you can see, the "triangle" to which you refer leaves *me *wanting to ask *you *questions.

Here goes:

** Possibility #1**:

If there is a sentence just above the triangle that states something like "for all A ^ B, is defined as 2A – 3B... ", then the triangle is indicating what's called a 'function'. And 'function' is just a fancy word for 'do this'. In other words, the definition of this particular function, which I made up to describe the procedure, simply expresses that when you see one number followed by a triangle, which is then followed by another number, you do what they tell you to do in their 'definition'. Function questions sometimes intimidate students (and even some teachers!), but once you get past the fact that they look odd compared to whatever you might have been studying, they are actually among the most straightforward of all the questions one is likely to encounter on a standardized test. Again, you did not include your grade level, but I'm going to venture a guess that you are one of the thousands of students who are in the midst of the panic for the upcoming SAT exam. The reason that the function idea pops into my head first is that you mention it looks unfamiliar. Believe it or not, when there are function questions on important exams they are *intentionally* made to look as unfamiliar as possible. However, this is done FOR YOUR SAKE, so that you are alerted to the fact that it is not something you have studied; no, functions are made up for the test and typically are not seen or heard from again. So, believe it or not, it is one of the few favors the test company does for you on the exam; after all, if, instead of a triangle they had used a simple horizontal line, you might, in the heat of the big sweaty test, take it to be a minus sign or a fraction bar. So, they go out of their way to make it clear you need to read the 'definition' telling you the 'do this' part, and, like following a recipe in a cookbook, you carefully do whatever they instruct you to do..

Again, to represent functions, the authors of the exam exert impressive effort to draw your attention to the fact that you are seeing a new symbol, and, on recent standardized exams they have used the musical notation known as the treble clef, an upside-down question mark, a large 'dot' in the middle of a square, the @ sign, and a solid octagon. On an otherwise serious exam that I was hired to write I decided to lighten things up a bit in the middle of the long test and, to indicate a function, I designed a smile-face type image with noticeably large ears and a pony tail that stuck straight up and extended above the line that is supposed to contain the question. There were laughs but, when marking the papers I did not notice an increase in correct responses to the big-eared-skull-with-the-excited-hairdo, even though it was a remarkably bright class. So thereafter I returned to a dot in a triangle.

** Possibility #2**: With most computer and laptop keypads, unless the teacher goes through the added procedure necessary to make an exponent into a superscripted number, it comes out of the printer as an upside down V, which did initially frustrate both teachers and students but people became accustomed to seeing this arrangement and it quickly became part of what I think of as mathematical slang. (And I'm not being derogatory because I am a prime example of someone who is satisfied enough with the upside down v that I usually let it go and have received surprisingly few follow-up questions to indicate people don't recognize it as a common but inelegant way of indicating that the number appearing after that inverted v is the exponent of the number preceding that v).

** Possibility # 3**: If you take a closer look at the symbol you describe, you might notice that instead of a perfectly crafted little triangle, one of the three lines extends slightly in both directions. This could be the symbol for angle, with the third line to distinguish it from the sideways caret of greater/less-than, or in another context a similar looking symbol is used to indicate two angles that are not congruent.

** Possibility # 4:** If you are in elementary school or higher and reviewing earlier material, triangles with two numbers are often used to indicate that the teacher is looking for what's often called a factor family, meaning that those are two of a set of three numbers that can have a relationship which forms at least one easy equation, and almost always a set of two such easy equations, one of which is a reversal of the other. For example, if you see a 9 on one side of a triangle and a 3 on an adjacent side of the triangle, you would mot likely (and correctly) put another 3 on the third side; the idea is to be able to read the numbers in order going around the triangle one way AND be able to read the numbers going around the triangle in the other way with merely changing the operation. So you might say 3 x 3 = 9, and rotating the triangle the reverse direction you would say 9 divided by 3 = 3.

** Possibility # 5**: The printer's cartridge or typewriter's ribbon is leaving too much ink, and rather than a triangle it was intended to be a caret or the upside down v that is used in set theory to mean "AND" (as opposed to the regular v which is used to indicate "OR".

** Possibility # 200**: (Or so it's starting to seem, anyway) is that this 'triangle' is not the equilateral or isosceles I somehow envisioned, but is a right triangle. If that is the case, then I'm almost certain the teacher is hoping you will come up with the third number in a Pythagorean triplet. Remember that? A-squared plus b-squared = c-squared, where c is the hypotenuse. Note: I avoided having to use any notation by substituting words. More time consuming, perhaps, but at least it communicates. And for me, that is an achievement!

Finally, if your math question is related to the world of chemistry, then a triangle can mean many other things, but most often it is there to indicate the Greek symbol 'delta', which relates to a measurement of CHANGE in something such as temperature or entropy.

Hope this helps,

Mitch