How to Confuse Kids about f(x)

Hello, mathematicians, puzzlers, and other curious people!

This post previously appeared as part of Mathematical Musings, so if it looks familiar, that could be why.

If you\’ve taken more than just a little algebra, you\’ve probably encountered f(x), \”f of x,\” good old function notation. You\’re probably comfortable with the idea y=f(x).

If you\’ve gotten a little further than that, you probably met g(x) and h(x), which crawl out of the woodwork when we want to deal with more than one function, or h(t), their relative that shows up when we launch or drop an object and then let gravity do its thing. If you got that far, you probably remember that f, g, and h are names of functions, rather than variables we can solve for.

(If you got a bit further than that you may have met f\'(x), f(x)\’s rate of change at x; you may also have met functions called by Greek letters. But that\’s further ahead than what we\’re going to use today.)

\"\" — This is neither a number nor a function, though you can find some numbers and functions in it (look for the spiral patterns in the leaves). CC BY-SA source

Now, here\’s the confusing bit. There\’s a group* of functions called direct proportions, which are functions in the form f(x)=kx where k is a constant that depends on what quantities you\’re dealing with. For example, if f(x) is the perimeter of a square with sides x units long, f(x)=4x…and if you\’re a person who hasn\’t yet wrapped your head around function notation, but you do know how parentheses-as-implied-multiplication work, you would have some excuse for thinking f=4. You would still be wrong, but not as wrong as if you thought f=artichoke.

*there\’s a rabbit trail I could run down here; I won\’t do it today, because it doesn\’t fit the topic at hand, but do not let me forget about this

Direct proportion functions are important, being the formulas used for most conversions, between quantities such as currencies (which vary daily or more often), units within measurement systems (like f(x)=16x where f(x) is the number of ounces in x pounds, or k(m)=0.001m, the number of kilometers in m meters), and between measurement systems (for example, one of relatively few exact conversion factors between US-standand measurements and metric, f(x)=2.54x where f(x) is the number of centimeters in x inches). (Temperatures in Fahrenheit and Centigrade are a notable exception, because they have nonequal zero levels. The boiling and freezing points of water, together with the slope and point-slope-form formulae, can be used to build up a conversion function.)

Because of this importance, direct proportion functions show up in lots of algebra problems. Because they\’re an easy type of function (you just multiply), they can show up early in the curriculum — even when the students have just met function notation. That\’s the problem. The f in f(x) looks to them for all the world like it could equal k, in those problems, but it doesn\’t because the teacher says so, so their takeaway lesson is \”math isn\’t supposed to make sense\” when we* want them to think the opposite of that.

*for some value of \”we\” that includes me, as well as every math teacher who gets it

So, what\’s the solution?

If you\’re a textbook writer, set up your curriculum with other types of functions for as long as possible before f(x)=kx.

If you\’re an algebra teacher, hold off on functions in the form f(x)=kx in examples or student work for a while, until they have a feel for f(x) being a name of a formula if possible. If you have to introduce them sooner, make sure you do lots of non-proportionate examples, and specifically show them that for those functions you can\’t come up with a number to plug in as f in f*x to get f(x).

If you\’re a student or former student who had been confused by this exact thing, know that it\’s not your fault, let go of any negative opinions about yourself that formed based on it, and pass this along to others who need it if you get the chance.

Calc You Later!

Math With Avrila

How to Confuse Kids about f(x)

— by

Response

Leave a Reply Cancel reply