# Functions vs. Equations: f(x) is y and more

I wanted to talk about function algebra, which naturally would include a reference to function notation.

So here’s the frustrating thing about writing this blog. I try to include links to other sites that explain a concept, so that I don’t have to reinvent the wheel for my reading audience. But a google gives me these results: useless links that do little more than say “f(x) is the same as y”. That’s not math. That’s test prep. And there’s nothing wrong with test prep, but every one of these sites purport to be math teaching sites, and hey, I’m not a mathematician, but shouldn’t we be explaining what f(x) means?

Someone somewhere is saying “See, this is why we need teachers to be math majors, instead of English majors who get 800 on the GRE quant section. You can’t substitute math understanding that comes with the study of these important principles.” That someone somewhere is wrong. I used to think that in my early days, until I had too many conversations like this:

Me, to AP Calculus teacher WHO MAJORED IN MATH: Hey, what do you tell your kids about function notation?

AP Calculus teacher WHO MAJORED IN MATH: f(x) is the same as y.

Me, nonplused: Well. Yeah. But I mean about why we developed function notation, what it serves that can’t be served by….

AP Calculus teacher WHO MAJORED IN MATH: It’s just notation. Don’t be confused.

Me: I’m not confused. But they serve different purposes, and I’m just trying to be sure I accurately capture…

AP Calculus teacher WHO MAJORED IN MATH: They don’t serve different purposes. It’s just notation f(x) is the same as y.

Me: Ok.

In my experience, very few math teachers WHO ACTUALLY MAJORED IN MATH care about these things either. My beer drinking buddy is an exception (and he’s now department head), and he’s the only math teacher I’ve found so far who was interested in my work on this subject.

Textbooks? McDougall Litell, CPM has a lot of those function machines. But no explanation. Holt does a little better but I didn’t understand that until I understood what I was looking for.

So I spend more time looking for a good link. Otherwise, I have to spend a lot of time figuring out how to explain function notation accurately, or at least inoffensively, so that people reading this blog don’t make me remind them that, for chrissakes, I’m an English major not a mathematician! That takes time. It’s not time I wanted to spend. I don’t want to tell you what function notation is, in a way that will pass expert muster. I want to tell how I build on function notation to teach function algebra. But I can’t do that well without explaining function notation, which I didn’t set out to do. This leads to many blog entries taking much more time than they should. The original intent for my function algebra post was to be just a quick little throwaway.

I began writing this post nearly a month ago, and got stalled looking for a way to characterize the explanation. You may be wondering why I would explain something I don’t understand—but that’s not it, really. I just don’t know what to call it. And that’s fine for teaching, not so much for writing, and so I spend hours trying to figure out the correct query. Which took me, literally, up until today.

Just fifteen minutes ago (as I write this sentence) I finally found the kernel in this discussion on function notation before Euler, in which someone writes:

but [Newton] refers to these as equations, not functions, and admittedly (written the way they are) that is exactly what they are. It seems anything that we would today write as a function, Newton described in words, such as:

HA. I learned something I hadn’t quite understood completely before–a function and an equation are not the same thing. Googling “what is the difference between an equation and a function” led me to the right websites. I realize now that I wasn’t just looking for an explanation of function notation, but rather why and when we use functions vs. equations.

Here’s an explanation that covers what I was trying to say.

So my research paid off. In practice, what I’ve been doing in this lesson is introducing function operations and function notation as a way to overcome a constraint in using equations.

******************************************************************

Sami needs $15 more to buy the new hoodie that he wants. But if Sami skips the hoodie, he needs just three more dollars to buy a ticket to the pizza feed on Friday. If Sami has x dollars, how much money, in terms of x, does Sami need if he wants both the hoodie and the ticket to the pizza feed? The first thing the kids think is that Sami needs$18 more.

I say okay, Sami has $20. How much does the hoodie cost?$35. How much does the pizza feed cost? $23. How much ….oh. Huh, say the kids. He needs a lot more than$18.

Depending on how goofy I feel, I might get out some fake money. I count out $20, give it to a quiet student. How much more for the hoodie? Count out another$15. Now how about the…Right about then, a student gets it: you need the $20 twice. So then we go to the board and model the two different equations for each purchase. y=x+15 y=x+3 So if we are getting both things, what are we doing? Adding, the class choruses. Ah, now there’s a new wrinkle. The kids have been adding equations for a while now, in systems. So I say, let’s try to add these equations. 2y=2x+18. Is that right? We test it with$20 and the kids realize that the right side “works” (that is, we get $68) but the left side says we still need to divide by 2, which would be…wrong. “So what’s happening is that we are running into the limits of an equation. An equation tells us that two expressions occupy the same point on a number line–that is, after all, what “equal” means.” “But when we use multiple variables in equations, then the equation becomes a relationship between two variables, an if-then. If y=x + 15, then the point (3, 18) is a solution because setting x=3 and y=18 creates an equation that has both sides occupying the same point on the number line. If 3x + 2y=12, then (2,3) is a solution because setting x=2 and y=3, etc.” But in an equation, the variables are values. So in the Sami case, we can’t treat y as a collection point. We can’t keep track of the dependent variable because it varies, obviously. The y in the first equation has a different value from the y in the second equation. If we wanted to keep them separate, we could use two different variables, like z = x + 15 and y = x + 3. Or we could number the ys: y1 = x+15, y2 = x+3. “Using the language of functions makes a lot of these constraints disappear.” “First, logically. Functions are different in a key way from equations: a function is an output. An equation is a relationship between variables. Yes, y=x+3 and f(x)= x+3 yield the same results, which is why we teachers always tell you to remember that ‘y and f(x) are the same thing’. However f(x) isn’t a variable, but an output. So when we add two functions, we’re adding outputs. Remember, too, that a function doesn’t even have to be an equation, like in the cell phone code example. Then there’s function notation, invented by Euler. Function notation enables unique names, usually a single letter. But it doesn’t have to be. You can get creative with the letter names and the input values.” “Function notation is just more elegant and efficient, too. Instead of saying ‘if x=7’ you can just say f(3). Once you define the function named ‘f’, anything can be input, even another expression, like f(a+7). And then, instead of saying ‘y=’ and solving for x, write f(x)= 3.” “So let’s call Sammy’s cash on hand c, and then create a function h for hoodie, and p for pizza feed. h(c) = c+15 p(c) = c+3 In both cases, c represents the money Sami has, so the input value is the same. But the output value varies based on the function used.” “Now, this is a small difference. But how many have you been told that f(x) is the same as y?” Bunch of hands raised. “Yep. And in a lot of ways, it is. But you have to be wondering why, if they’re the same thing, we bother teaching you about function notation.” Lots of nods. “So as you move on into advanced math, you’ll start to learn other reasons why we sometimes use functions and other times use equations. For now, it’s enough to know that function notation allows us to keep track of our different outcomes. “Once we can do this, we can actually create an entire math with functions. They can be added, subtracted, multiplied. They have inverse operations.” “But then why do we use equations?” “Well, for one thing, functions don’t do systems well. Remember, when we solve systems, we are expecting both the x and the y (and any other variables) to be equal. Functions don’t handle that well. So you’ll see that we switch back and forth between equations and functions as needed.” When you need to add expressions, functions are great. So now we can add h(c) and g(c). h(c) + p(c) = (x + 15) + (x + 3) = 2x + 18 “Because we are adding outcomes, and have a unique way of tracking each outcome, we can add them properly. Remember, too, that since a function doesn’t need to be an equation, I can add or subtract outcomes without even having an equation. If a(x) = 9 and b(y) = 17, then b(y) – a(x) is 8, and I don’t have to care if a(x) and b(y) are generated by an expression or a rule or a code or a random happenstance—provided, of course, that random happenstance is only one per input.” ****************************************************************************** I know. You’re wondering why I don’t just follow the AP Calculus teacher’s “f(x) is the same as y”. Well, it turns out that function operations are a big part of pre-calc, so they’ll use this later. In the meantime, I give them some practice with function notation (I stole this at random). Not enough. Kids don’t really know it later. But at least they’re exposed to it. Then I go on to linear function addition and subtraction. I usually just put problems on the board. Sample quiz: Here’s a test question: And from here I go on to linear function multiplication (aka quadratics) and, eventually, rational expressions (linear function division). Like teaching congruence with isometries, I can’t argue that using functions to further our work in linear and quadratic equations is better. I find it more…elegant, maybe? But the execution isn’t quite there. This is the first year I’ve really taught this whole sequence: introducing functions, function addition/subtraction/notation, function multiplication, inverse functions, rational expressions. Writing it up has revealed an obvious improvement. Up to now, my function illustration has been a quick standalone lesson. Then later I introduce the notion of function addition and in doing so, bring up function notation. This is goofy, now that I look at it. In the future, I’ll introduce functions and then go into function notation. I can spend a day or two on that, quiz that early. Then I can go back into linear equations or inequalities (the placement is flexible) and then bring up function addition and subtraction, with function notation already covered. You know what’s irritating? The huge effort described at the beginning of this post to figure out how to describe what I was teaching led me to this. The huge effort underwent solely in order to write this post. Which I was griping about. In learning how to describe function notation for my readers, I learned that the proper way to characterize my work is as a difference between functions and equations, and that led to an idea for better sequencing. This is kind of a placeholder post. Obviously, I’m in flux about this right now. My linear equations unit has been in good shape for a while. This gives me plenty of room to add flourishes, introduce more complicated topics onto a subject the students know well. Meanwhile, linear function multiplication has proven to be a great introduction to quadratics. So now I’m involved in putting it all together. Next up in this sequence: the post that I really wanted to write, on my quadratics introduction. Sorry for the slow rate of posts lately. I did five in April, then got lazy. Advertisements ## About educationrealist #### 80 responses to “Functions vs. Equations: f(x) is y and more” • franklindmadoff E.R., You and your readers may be interested in some recent updates to my blog. I took 4 years worth of test score data in California (star/cst results) and combined it with Census/ACS data on median family income, parent education levels, and poverty rates by race/ethnicity, along with various other metrics in school funding and the like. On concentrated poverty and its effects on academic outcomes Update #1 Update #2 [The first post is much too long…. I’ll revise it later] – FDM • anon As I read the differencebetween link you provide, I don’t get why the Sami example is supposed to lead to functions. It’s clearly in the domain of equations; you’ve got three fixed quantities and two relationships among them. There is no “set of possible inputs”. The implication of saying “ok, if x is Sami’s cash on hand, then the price of a pizza is p(x) = x + 3” is that no matter how much money Sami scrounges up, he’ll never be able to afford a pizza because its price will rise as Sami gets richer (and fall as he gets poorer). It’s true that modeling the system with the equations y = x + 15 y = x + 3  doesn’t work, but that problem isn’t solved by relabeling with functions: f(x) = x + 15 f(x) = x + 3  is just as impossible, and will prove 0 = 12 just as quickly when you check the value of f(-3). The problem isn’t that you set up a system of equations, it’s that you gave the same name to two different, unrelated things. • educationrealist I don’t think you understood anything I said. Particularly since I said that the problem was giving the same name to two different things. And you spent a whole bunch of time explaining something that was irrelevant. • anon I tend to agree with you; I don’t understand what you’re saying. The Sami example seems to be a problem where functions are (1) unnecessary, (2) unhelpful, and (3) conceptually inappropriate. What makes it a lead-in to functions? What is the constraint in using equations that using functions will overcome? • educationrealist If I want to know how much money Sami needs to buy both, I’ll need to add the expressions. So definitely they aren’t unnecessary, unhelpful, or conceptually inappropriate. Yes, I could create two variables. However, the kids are used to modeling with x and y. So at that point, I can either say “you need a new variable” or I can introduce them to functions, because functions enable addition of expressions. Since I want them to become familiar with functions, this seems like a logical place to introduce function addition. • anon So, what’s bothering me here is that the example works if read as a system of equations, but not if read as function definitions. “So let’s call Sammy’s cash on hand c, and then create a function h for hoodie, and p for pizza feed. h(c) = c+15 p(c) = c+3 In both cases, c represents the money Sami has, so the input value is the same. But the output value varies based on the function used.” If we say that h is a function giving the cost of a hoodie, p is a function giving the cost of a pizza, and c is Sami’s cash on hand, then these are correct as equations, but definitely incorrect as function definitions. You can see this easily by imagining that Sami’s friend, Eino (how are they choosing these names?), also wants to buy a hoodie. Eino has cash on hand in the amount e. If h is a function that tells us the price of a hoodie, what is h(e)? The answer, of course, is h(e) = c+15. h is a constant function giving c+15 no matter what the input is. If it had been defined as “h(x) = c+15”, this wouldn’t be a problem, but if you define h as “h(c) = c+15” then h(e) is e+15, which is wrong. So what you’ve done here is define 3 variables, named c, h(c), and p(c). This isn’t an unusual thing to do, if for example you’re already working with some functions and you want to fix a particular value as the result of applying one — but you introduced these functions simultaneously, just so you could label the variables h(c) and p(c) instead of h and p. The variable name h(c) might look like a function, but since c is constant, that’s an illusion. I don’t know what you mean by “functions enable addition of expressions”. It doesn’t sound like your kids had problems adding (c+15) to (c+3) and getting (2c+18) at the beginning of the lesson. • educationrealist As I understand functions, h(e) is still e+15. As long as the outside rule is h, then whatever is in the parenthesis is substituted in. So if h(c) is the function, and Josie has (2a) dollars, then what she needs for the hoodie is h(2a) = (2a) + 15. Now, I grant you that this isn’t a particularly useful function. but it’s certainly consistent. As for the rest, I went through it in the text. • anon The following things cannot be simultaneously true: – h is a function that gives the price of a hoodie – h is defined as h(x) = x+15, where x is the amount of money the person desiring to buy the hoodie has – there is an amount of money that would enable Sami to buy a hoodie So if you’re going to say that h(x) takes the input “how much money do I have” and gives the price of a hoodie, the question “how much money would Sami need in order to buy a pizza and a hoodie” is ill-formed; he can’t do it. (You seem to be advancing this model, given the claim “if h(c) is the function, and Josie has (2a) dollars, then what she needs for the hoodie is h(2a) = (2a) + 15” — but you also seem to believe that Sami could buy a hoodie if he had more money, which isn’t compatible with this.) If your model of the situation is instead that the price of a hoodie doesn’t change, then there’s not much of a point to having a function that gives it. You can, but if you do use a function for this, it’s incorrect to define the function as h(c) = c+15, because that function is sensitive to its parameter, but the price of a hoodie is constant. I think the problem exposes a good issue, and I like your method of walking your class through seeing that “$18 more” is wrong, because you need the original amount twice. But functions don’t do any work here.

I’m very curious how your kids would respond if you presented the problem with the slightly different phrasing

Sami needs $15 more to buy the new hoodie that he wants. But if Sami skips the hoodie, he needs just three more dollars to buy a ticket to the pizza feed on Friday. Sami’s friend Eino wants to buy both. If Sami has s dollars, how much will it cost Eino to buy a hoodie and a pizza ticket? • educationrealist “I think the problem exposes a good issue, and I like your method of walking your class through seeing that “$18 more” is wrong, because you need the original amount twice. ”

If you see this, then you’re wasting your time. Definitely wasting mine.

• educationrealist

“But functions don’t do any work here.”

I know. They aren’t supposed to. You are fixated on the application.

• lumofan

Frankly, I think you are kind of on the wrong side of the argument here in original purpose.

I think other people you have talked with have got the general point correct that students really should just understand something like “f(x) is the same as y” for the purpose of standardized testing. They are not actually being taught more rigorous or relevant information in the sense you propose or worry about. In a physics class the onus is on the teacher to be sure to not confuse students over problems with a time dimension as well. Just like “a vector has both direction and magnitude” is the sort a stupid definition that high school students memorize, so is this obsession with “kid” “functional notation.”

Lack of proper education in sets, algebra and so forth (meaning “real” algebra, not “kid” algebra) may be depressingly the status quo in American K12 schools, but this approach isn’t really meaningful. Not your fault, don’t mean to be rude at a personal level in this blog post, it’s just an ugh factor.

Wikipedia is a plenty useful default resource for math like this, though on the net surely some enterprising teachers might have developed engaging, age appropriate curricular materials on these topics. I can’t provide that offand but I can recommend that you might find material or advice from other teachers cross-displicine from CS

Imho high school students, of varying ability levels, absolutely could be taught introductory concepts to Abelian group theory, cardinality, and whatnot, though not part of the standard curriculum. It would take some effort and understanding of pedagogy to figure out the most promising ways and real world interests to tie in for a particular cohort of students. I think calculus or statistics class, possibly computer science class are the most natural place, where teachers have the time for a unit, students have more aptitude and such. Not underestimating or lacking care for lower level students it’s likely more a time issue on what is officially part of the curriculum and what concepts must be cut.

Ultimately from what you’ve personally described of the classes you teach it might not be the time/place to teach such material, but it’s also a waste of time to browbeat students with lobotomized descriptions of functions and mappings beyond what you know is required for standardized testing. I have to concur with that view.

Also, one last personal request, if you aren’t caring for everything else I said, please please please learn to present what “one-to-one” and “onto” mean if you’re going to teach anything you outlined in this blog post to students. Whatever the local high school curriculum or textbooks has or lacks on relations or functions and so on, anyone who knows what they are talking about would point there as a good place to start.

The worst thing is to give students information that is wrong or incomplete relative to a standardized test they might have (like not knowing how for “high school kid” functions might be defined in terms of one-to-oneness. Though I believe your state testing materials, textbooks, whatever probably could have mentioned things, like a vertical line test, maybe not realizing and conveying to students broader concepts. ) Honestly the calculus teacher ought to know and teach students reasonably relevant definitions for the material those students will learn, given things like the fundamental theorem, though I know those are older and more advanced students.

• educationrealist

You’re kind of nuts if you think I’m browbeating information into kids. And you’re even nuttier if you think I give my kids information that won’t work for standardized tests. And what’s really funny is that you think I’m going into too much detail while obsessing over whether or not I use the word “one-to-one” or “onto”. I cover that in an earlier post.

• Dan

Most CAS packages, and Maple in particular, are also quite picky about the differences between functions and equations so that they can offer a more refined discretization of the “software power” to “software complexity” continuum. Thus, I think there are quite a few very good explanations of function notation and equation notation (and the reasons for the differences) in some of the online guides to function creation within said CAS packages. I mention this mainly because the differences between the two was something I had to recently relearn (which, given that I am a phd student in nuclear engineering, was more than a little embarrassing) as I was attempting to teach myself to use Maple. An example of one of the guides I used is below.

• educationrealist

Hey, thanks for understanding.

It’s hard for me to explain this without taking the length of another post, but in terms of what I teach, it goes something like this:

1) Definition of a function
2) Function notation
3) Advantage of function notation over equations for certain operations (this is the one that I’ll be modifying in the future based on what I’ve learned)
4) Function operations (addition, subtraction, multiplication).

I do all of this in the context of linear and quadratic equations, which takes precedence. I just want to give my kids an understanding of *why* we teach functions in terms that make sense to them.

Will be checking that link out. Thanks!

• Joseph Nebus

I remember as a mathematics major feeling bewildered when I was just getting used to the pace of college instruction when suddenly some but not all my instructors started writing functions as, say, “f:t in R -> R^3” or the like. I need time to warm up to novelties like that.

• educationrealist

So it’s not just me? The switch from equations to functions isn’t usually explained, is it? Yet the PSAT is testing function notation in ways that aren’t instantly obvious.

• DensityDuck

Huh. I made it all the way through a rocket-science Master’s degree program without ever having such a formal statement of function notation as you present here. I mean, obviously it showed up all the time (anything involving differential equations, obviously, but also stuff like variation of parameters to solve integrals) but there was never a point where I can recall someone saying “you can treat a function just like a number and do things with it”.

I don’t know whether that means anything regarding this post. And I do have to mention that the grad program I was in had about a 40% attrition rate, and PDE class was responsible for most of that.

• educationrealist

What’s PDE?

And you can also treat a variable just like a number. But you can’t treat an equation just like a number. Is that true? (Asking, not asserting.)

• DensityDuck

Partial Differential Equations. Something about “taking a derivative in only one variable of a two-variable function” just blew some people’s minds.

One day I tried to illustrate a problem for my classmates, using colored chalk to indicate the different stages of the solution. When I got finished they were looking at me as though I’d drawn a diagram on the board using bits of string, cut-up road maps, and clippings of newspaper headlines.

• educationrealist

Oh, I see. Yes, I think if people don’t really understand what a function is, that could add to confusion.

I like colors to distinguish stages and parts. Good approach.

• Mark Roulo

“But you can’t treat an equation just like a number. Is that true?”

For many values of “just like,” I believe that you can treat a function just like a number.

You can add, subtract, multiply and divide constants by functions (and the reverse). You can add, subtract, multiply and divide functions by each other. You can find the inverse for functions (that have an inverse …).

In fact, one of the *HUGE* mental hurdles in Calculus is learning that you can operate on functions, not just numbers [limits are another huge mental hurdle….]

What did you have in mind?

• educationrealist

“For many values of “just like,” I believe that you can treat a function just like a number.”

No, I know that you can treat a function just like a number. The question is whether you can treat an equation like a number in the same way.

In fact, that’s how I got into this, because I wanted to teach students how to add and subtract functions.

• Mark Roulo

“…The question is whether you can treat an equation like a number in the same way. ”

I’m pretty sure that you cannot (at least not how you want to).

An equation (e.g. x = 5, 2 = 3, x = x + 1) can be either true or false, but doesn’t have a value per-se other than that. So I don’t think you can add, subtract, etc. I mean, you *CAN* if you claim to be doing boolean algebra on the true/false values… but that isn’t what you are talking about here (or what most people would mean).

What, for example, would it mean to “add one” to “x = 5”?

So, I’m pretty sure the answer is, “No.”
[But I am not a mathematician …]

• educationrealist

Right. You can change an equation but you are bound by the properties of equality. So you can say f(x) + 1, but not y=x+5 + 1.

This got a huge discussion online. I need to link it in; it was really helpful and interesting. As were the comments here.

• lumofan

See, just to jump in on this, nothing was learned here other than whether DensityDuck likely had a lazy or irresponsible calculus teacher; remaining commenters can be given the benefit of the doubt for not having any specific formal studies.

One can do certain things to certain functions (such as continuous functions of one variable on the real plane, we know that’s mostly what these high school students will be working with) but in other contexts one cannot apply various operations. Whether or not various operations and properties apply to a function depends on more than just the definition of a function. Generically “treating something like a number” is not going to lead to a good description of a function, you have commenters here that are confused with ideas like how to treat a variable in a computer program. Differentiability is not an inherent property of functions, for instance, everyone in this comment chain should get that.

A word you are looking for, at least what Mark Roulo was looking for because several of his sentences make no sense otherwise, is expression. A better offhand description, and possibly vaster more useful to give your students, is that an expression can be treated like a number, and when a function, defined by an expression, satisfies various properties, relevant at the depth of what the students are learning (say again a function of one variable on the real plane) certain operations can then apply. And with equations you do have the obviously right conclusion, that students should know they can’t change the underlying equality, they have to do things to “both sides.”

• educationrealist

See, if you want to know why I dismiss your comments, it’s because you are utterly not grasping what is always taught and what isn’t. And so when you spend 3 paragraphs explaining how to teach something that is standard, without understanding what isn’t, it’s kind of a waste of space.

• anon

So you can say f(x) + 1, but not y=x+5 + 1.

Sure, but f(x) isn’t a function, it’s a value. The function is just f, and f+1 doesn’t make sense. You can’t say “[f(x)=x+5] + 1” either.

• educationrealist

It’s an output, not a value.

• anon

That’s not right. For one thing, saying “it’s an output” has no mathematical meaning. But here, consider the Caesar cipher, a function defined piecewise like so:

c(A) = D
c(B) = E
c(C) = F
[...]
c(Y) = B
c(Z) = C


This function maps letters of the english alphabet to letters of the english alphabet (it shifts them forward three points). I’ve capitalized the letters to distinguish them from variable names. You can still work with expressions like f(x), and extend the function by defining (using . for concatenation) c(x . y) = c(x) . c(y) — but you definitely can’t say “c(x) + 1”, because adding numbers to letters is nonsensical, and c(x) is a letter. The expression f(x)+1 is legal when f takes numeric values, because in that case f(x) is a numeric value, and can be manipulated as such. Saying this again: when f is a real-valued function, the expression “f(x) + 1” is legal because f(x) is a real number. It’s nothing more or less. If f is some other kind of function, all bets are off. The only operations you can do on functions *because they’re functions*, rather than with reference to the values they take, are composition and inversion.

• educationrealist

Okay, I’m kind of done. You’re writing a lot, and it’s pointless.

In computer programming, there is a useful distinction between named functions and anonymous functions. When we write f(x)=2x+3, we are not only defining a function, but we are giving it a name. The function is named f, and it maps x to 2x+3. We give the function a name so we can refer to it later, even if the problem involves many different functions.

• candid_observer

I guess I shouldn’t jump in without reading all the post and the comments, but I’m not going to let that stop me. (If I had to do so, I’d probably not make a germane point or two.)

Look, functions have a mathematically rigorous definition, often introduced in some kind of formal setting. They are a mapping from one domain into another. You take an object (or, getting technical, an ordered n-tuple) from one domain, and there is exactly one object corresponding to it in the other domain (often called the range). None of this has anything to do with equations per se. An equation might define the mapping, so that if one says that f(x) = x+5, then one knows, given a number x, what number corresponds to it in the range. But a function need not have an equation corresponding to it; indeed there are functions that provably don’t. (In fact, getting a bit technical, there are only countably many equations, but uncountably many functions from numbers to numbers.)

You’re obviously right on the point that functional notation is pretty important in getting past the limitations of using just “y” — as if there is only one variable name one might ever have a need to use. Of course, mathematicians and others don’t in any case restrict themselves to just “y”, but can use “z”, or “y1” or “y2” or anything that strikes their fancy to correspond to the object mapped into by different possible functions. But often functional notation is the best, partly because that notation can be ornamented to carry with it its meaning.

I think though that the real problem that you have here in is just in your discomfort using functional notation. The thing is, functional notation is in many ways whatever its contriver specifies it to be. Certain common practices, however, are pretty much assumed. Not sure where you can go to develop a sense of those common practices.

• educationrealist

This is not the introduction to functions. I have another post on that which I link in. Beyond that, you aren’t really understanding the point.

• lumofan

For the record this guy appears to making the same general point I made. Good for him for not reading any of the comments. :p I absolutely promise we’re independent readers who are just seeing the same issue with your blogpost here.

If your prior musings on this lesson plan are as wandering as the top post to this comment section the whole structure of this teaching unit you’re talking about is off. You’re not going into too much or too little detai per se; you don’t understand what you are talking about in the first place, at least as it matters to teaching students a proper foundation for further study in mathematics, even as far as calculus.

You seem concerned here with a general pedagogical idea that’s getting more at the issue students might have with understanding something like variables, pointers, or references in a computer programming language. Sure, some students might get confused with notation as purely notation, as naming things, but I’m not sure that’s the real issue you wanted to teach. For students who aren’t even taking a computer science class or certainly haven’t been given proper rigorous definitions it’s dangerous and confusing if you’re proceeding as in these other comments threads to conflate concepts like “a name which refers to things shorthand” or “a property or structure in some programming language” with a definition in mathematics. I appreciate the drive to find better overall material for your students, the quality of textbooks isn’t up to you, having to put together material from different sources that wasn’t prepared in a coherent text isn’t your fault, &c

Except candid_observer is completely right that, again, functions have a formal definition used everywhere beyond cheap or hastily written high school textbooks at least. If you are required to teach students certain shortcuts to meet the standards of some state test, absolutely do that up to what is necessary. (which, again, is what you started the issue with, y=f(x) for basically functions of one variable on a real-valued, 2D plane) Otherwise this unit and lesson plan description does not seem that useful to the students and doesn’t provide much clarity for their future classes. Certainly, if you’re concerned about lower level or slower students in particular, I’d worry a lot about confusing them further with imprecise definitions.

And the reason we can tell is that the material you are considering for excercises is not germane to the underlying issue. It’s mostly just general practice of high school algebra without reinforcing any understanding of functions, operations, sets and so forth. Well, also a little of what you say and with comment responses like to DaveRadcliffe, just a few sentences that suggest you might not know much about, and at least don’t teach to these level of students things like coordinate systems and parameterization.

But what you’re setting up here is a situation where, at best, the students learn more rigorous definitions from a future course or book (say in the calculus class) and finally think “Oh, that’s what these terms really mean.”

Again, please don’t think this is all personal. If anything, for the general reader, it illustrates structural problems for American schools, with state curricular orgs, with insane insular notions of IP and so forth. Every teacher every year in different classes from algebra to calculus obviously shouldn’t be stuck rehashing certain material, or worse, in the dark about standardized tests out of their control and more. I know you’re not a first year teacher and the topics, books, tests that are extant for any teacher are often out of their control. You mention this is a first in presenting or focusing on some of this material though, perhaps without the paper, non-electronic books and resources being valuable to the students. And for that matter it is completely on the shoulders of the computer science teachers and book authors to take responsibility for their material and be clear that constraints of programming langauge functions, return types, call structure and all are distinct from mathematics. As you’ve mentioned it’s great to do your best to coordinate with other teachers at your school locally and discuss how the best progression for teaching topics from class to class works among you.

• educationrealist

Again, you’re not making sense because you don’t appear to understand what limited issue I’m teaching. On Twitter and here, I’m getting enough feedback from mathematicians and college professors that tell me I’m accomplishing the goal.

• Jim

I’m completely lost.

• Jim

Throughout much of the discussions here there is a persistent confusion of language and metalanguage. A term like “function” belongs to the language but a term like “variable” or “equation” belongs to the metalanguage.

• Jim

I think that I remarked on the futility of centuries of effort to make things clear. In one of Kleene’s books on mathematical logic he remrked on the importance of not confusing mathematics with meta-mathematics. He said something to the effect that someone who could not keep these two things separate should give up on mathematical logic and take up something else – he suggested bee-keeping. .

• anon888

You are being pedantic. At a high school calculus level, y=f(x), is reasonable. I enjoy abstract algebra, and homomorphisms and symmetries, but I wouldn’t annoy a high school teacher with some nitpick.

• E.

I’m a college math professor, and I’m becoming more doubtful about the necessity of having math majors as primary or secondary school teachers. A typical math major, even at a good college typically takes calculus, differential equations, and then proceeds to major in linear algebra taking it many times. Maybe they’ll take real analysis. It’s not as if they have significant exposure to rigorous mathematical concepts. More likely, they hit a conceptual brick wall three courses away from completing the major.

I’m not sure if my observations generalize to the high school level, but I’ve seen a lot of (non-Ph.D.) lecturers at the college level who are incredibly proficient at certain tricks in the sense that they know how to break down any problem a student might see into steps and teach those steps. Then the class becomes about a set of problems. Even if those problems were initially designed to test conceptual understanding. I really wish they wouldn’t teach like this and simply be content with teaching less although a good number of students can’t solve the harder problems.

I don’t even think this phenomenon is about teaching to the test. It’s just that some people don’t have an understanding that math could be about more than solving problems.

I’m not sure that non-math majors are better. But perhaps someone who majored in a difficult liberal arts subject (philosophy or English at the right university) and is open to learning math might be better in some respects. For example, my mathematically incompetent father was great as a teacher because he simply couldn’t remember any tricks. He honestly probably doesn’t know the cross-multiply method of adding fractions. But he could draw a pizza pie and break it up into a certain number of equal pieces.

To try to disambiguate some of the issues raised in this post (which I haven’t read that carefully):
1. A function should be thought of as an oracle that given certain input, gives certain output.
2. Functions are often expressed in terms of elementary functions tied together by the usual operations of arithmetic and function composition (this is your function algebra); not all functions have an expression in this way. Most don’t.
3. The function algebra expression of a function f(x) is code for: to evaluate the function at a certain value, substitute in that value for x.
4. More conventionally in mathematics, there is an algebra of polynomials. In this case, x is merely a symbol called an _indeterminate_. The substitution of a value for x is a specific operation called _evaluation_. It’s actually a function on the algebra of polynomials. There is also a more rigorously defined algebra of functions if differential algebra.
5. Indeterminates which are symbols should be distinguished from variables which are stand-ins for a value that you intend to substitute later.
For example, the equation x=x^2 is false in the polynomial algebra. But if x is allowed only to range over the set {0,1}, then this is true if both sides of the equations are viewed as functions.
6. The use of “=” in high school algebra is often ambiguous:
a. We may write solve the equation: x=x^2 which means “find the subset of values of x (as a subset of a previously given set, usually the reals) for which the equation is true.
b. We may also define a new variable by writing y=x^2. This means that we introduce a new variable y and restrict to the set of (x,y) that satisfy y=x^2. The old-fashioned name for this was “dependent variable.” I’m not sure about the pedagogical history of this, but I imagine that it went out of fashion with the new math.

• educationrealist

Thanks, this is helpful. We still use the phrase dependent variable, but mostly in science.

I’m going to print this out. I appreciate it.

BTW, I do teach less, and try to teach it conceptually. I do understand what you are saying about introducing a concept not to work problems, but simply to understand a concept.

• E.

I have no doubt that you’re doing good work.

I went to my bookshelf to try to find a book that might explain these concepts carefully. Unfortunately I had no luck. The best books for lay people are 50-100 years old and are just as careless.

For an interesting criticism about how math is taught, have a look at Lockhart’s Lament. The original article is available online although he has expanded it into a nice, short book.

The conceptual/trick dichotomy is why I’m agnostic about Common Core. Real mathematicians have come up with the standards. Unfortunately, it looks like their implementation will involve teachers explaining a new set of tricks, ones even further removed from actual conceptual mathematics.

• Jim

Indeterminates are actual mathematcal entities and talk of them is part of mathematics. Talk of variables is part of meta-mathematics.

• E.

I don’t think the distinction makes any sense. Sure, variables are meta-mathematics in that they stand for elements of sets. But we have ways of manipulating sets and their elements. Provided that we are sufficiently careful (like picking a universe ahead of time), we can import those ways into mathematics.

• Jim

An indeterminate is just an ordinary element of an algebraic system on the same ontological level as any other entity in mathematics. Its no more symbolic than pi or the hypergeometric function. But variables are part of the language of mathematics. Comparing functions with equations or variables seems confusing to me. They are on different semantic levels.

• Jim

An equation is a statement in the language of mathematics. It makes sense to say that it is true or false. One wouldn’t ask if a function were true or false..

• E.

I can’t believe that I am taking the bait here. Here,
y=x^3+x
is a statement involving variables. Suppose we have already specified x to range over a set S. Then the statement defines a function in a sloppy but not meta-mathematical way.

If you want to be incredibly pedantic, a function from a set S to a set T is a subset U of S\times T such that for any s\in S, there is exactly one element t\in T such that (s,t)\in U. Now, take this statement:
let U be the subset of S\times S given by
{(x,y)|y=x^3+x}.
This defines the same function as before. In fact the original statement was just shorthand for this unpleasant bit of pedantry.

There is no meta-mathematics here.

I stand by my comment about a little learning.

• Jim

An equation is a statement in the language of mathematics. One can say that it is true or false. One wouldn’t speak of a function as being true or false.

• Jim

Your function definition uses variables and equations but it does not mention them. To compare functions and equations don’t you have to mention them? If you use variables and equations you’re not doing meta-mathematics but if you mention them you are.

By the way if I were a public school teacher I wouldn’t talk about this stuff but I would try to avoid creating confusions that might have to be unlearned later.

• Jim

I certainly agree that your definition of the function x^3 + x is not meta-mathematical but my original comment concerned treating things like functions which are mathematical as if they were the same sort of things as variables and equations which are essentially linguistic.

• Iamthep

I really enjoy most of your postings, but this one is really bad. A function establishes a relationship between two sets. The cost of the hoodie is independent of the money that Sami has. You should not use h(x) to determine the cost of the hoodie. You are telling me that I have to pay more for a hoodie when Sami gets more money?

I really wish my teachers had gotten rid of y and x as variable names.

The appropriate solution is to say that hoodie_cost = 15 + current_sami_money. pizza_cost = 3 + current_sami_money. Thus we can say that hoodie_cost + pizza_cost = 2 * current_sami_money + 18. Sami currently has current_sami_money. So he needs 18 + current_sami_money more to get both.

Feel free to change the variable names if you find better ones. That is the whole point!

My math teachers/professors crippled my math ability for years by only using constant variable names. Do your kids a favor and never use x and y as variable names again. ( and include units whenever possible!) This was especially bad when they started in on calculus and using dx, dy, etc. I thought d was an operation!

• educationrealist

I hate units, which is not to say they shouldn’t be used, but it’s not something I focus on.

Like another commenter, you are entirely focused on something minor. The point of the Sami question was not to solve the problem. It’s not a system. The lines are, in fact, parallel. The point of the Sami question was to show that their normal tools wouldn’t get them there.

So stop focusing on that part.

• Iamthep

Why don’t you follow through with:
[quote]If we wanted to keep them separate, we could use two different variables, like z = x + 15 and y = x + 3. Or we could number the ys: y1 = x+15, y2 = x+3.[/quote]

Much of what you said about functions is the same for equations. Nothing stops you from using unique names when needed as you showed above.

In fact, p(x) = 3 + x is an equation. But I can arbitrarily say that p(x) = y. Which means that y = 3 + x. There is nothing illegal about those operations. Or if it freaks you out to have a function that takes x and does nothing with it, i can say p(x) = y + 0 * x. Better yet, we know p(x) is the price of pizza. So we can say that p(x) = price_of_pizza. Thus price_of_pizza = 3 + x.

This confusion is why I hope you get away from x and y. The use of x and y is just shorthand notation. It makes it easy to write on a chalk board. It also lets you read” y = axe” as y equals the constant ‘a’ times variable ‘x’ times the natural number. Of course I could have also meant that y is equal to the variable ‘axe’. We just assume no one would be that evil.

[quote]“Well, for one thing, functions don’t do systems well. Remember, when we solve systems, we are expecting both the x and the y (and any other variables) to be equal. Functions don’t handle that well. So you’ll see that we switch back and forth between equations and functions as needed.”[/quote]
This is a really strange thing to say.

if you have y = 17 + x and y = 2*x then this is no different than saying that f(x) = 17 + x and f(x) = 2 * x; Maybe you are thinking, “How can you have two of the same functions equal the same thing?” And you are right, but then again you have y equaling two different things and it doesn’t cause you a problem. So let’s say that f(x) = 17 + x and g(x) = 2 * x. All you are asking is “For what input of x does f(x) equal g(x)?”

But we could have also written z = 17 + x and y = 2 * x. To solve this we just ask, for what value of x does z = y become true?

Back to the original problem.
How to solve this with a system of equations:

1. price_of_pizza = sami_current + 3
2. price_of_hoodie = sami_current + 15
3. sami_current + extra_money = price_of_pizza + price_of_hoodie.

price_of_pizza + price_of_hoodie = 2 * sami_current + 18

Set to be equal to 3.
sami_current + extra_money = 2 * sami_current + 18.

extra_money = sami_current + 18.

So what seems to be the problem?

• educationrealist

Missed this, but again, I was not trying to solve the system, so I’m not sure why you’re obsessing on that part.

• Renato

A function is *something that you do* with something that produces a single, well-defined something else.

When the functions are simple enough, you can talk about them without using variables. Examples: the function “to increase a number by one”. Or the function “to square a number and then increase it by one”, which is clearly the result of a composition. Using variables/equations to talk about sets/functions (and vice-versa) is very convenient, so it’s natural that they get mixed up.

To me, that confusion became paralyzing when I tried to learn change of variables on partial differential operators (which are defined on abstract functions but, as written, work on variables). Took me a long, long time to get comfortable with that.

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• David

This has been thoroughly discussed, but for what it’s worth it doesn’t appear anyone else has said exactly this:

Speaking informally, a simple advantage of f(x) notation over y= notation is that functions can take any old thing as input and give any old thing as output, while the equation notation restricts you to only expressing relations that can be sensibly written as an equation.

For example, if I can get razors at 10 for $5 and 50 for$20 I can happily define a minimum price function:
p(10) = $5 p(50) =$20
p(60) = $25 but it would be awkward to express this as y=something (you could do it, but you’d have to give y multiple definitions for different values of x). For a more abstract example, you can define f(n) = nth digit of pi. You don’t have to limit yourself to numbers either. You could define a price function q, with q(shirt) =$20
q(movie) = \$12

Or even a dinner function g:
g(Monday) = lentils
g(Tuesday) = stir-fried kangaroo

My own attitude to using f(x) instead of y is that you use f(x) when you want to direct attention to the relation f between x and f(x) as an important thing in its own right, while you might use y if you’re mainly interested in the value of y given some x. I feel like this might be a bit abstract for students brushing up against function notation for the first time, however.

• George

Historically, the idea of a “function” has been one of the more difficult concepts for mathematicians to nail down. Wikipedia has a nice article:

https://en.wikipedia.org/wiki/History_of_the_function_concept

In broad strokes, the function concept grew out of attempts to understand the concepts used by Newton and Leibniz in the calculus. There’s broad agreement among modern mathematicians about what the “correct” definition of a function is, but I think it’s important to realize that this is a vast generalization of the more intuitive idea used by, for example, Euler. The more formal, modern, definition is appropriate for a calculus class (at least a formal, college-level one), but I suspect (with admittedly no direct evidence) that the more informal intuitive idea of a function being defined as an algebraic (or, more generally, an analytic) formula that expresses a relationship between two variables would be better for a high school algebra class, especially one for lower ability students. This definition also works better with the intuitive idea of the graph of a function: a type of curve that you can draw in the plane.

Frankly, I tend to think that the more formal definitions rather than the intuitive ideas would not only not help with many high-school students, but would actually worsen their situation with respect to math.

• educationrealist

The trick, for me, is to describe the difference between an equation and a function. At some point kids need to understand how to use functions, which are relevant in inverses, transformations, and all sorts of other math they meet on the way to calculus.

• George

Yes, that’s an important and kind of subtle difference. I think that an equation is a property that a number may or may not satisfy; the emphasis is usually on trying to find which numbers satisfy the equation, i.e., solve the equation. On the other hand, a function describes a relationship between different quantities, such that if you know the value of one of the quantities, you can determine the other one.

A major difficulty, of course, is that we use the same notations for both ideas. Some of the comments above mention that in computer science, the equals sign that is used in equations is often differentiated from the equals sign used in function definitions. However, this still elides another difference: the use of variables is also a little different in the two. In equations, you’re looking for specific values for the variable that make the statement true. In functions, you can substitute in specific values in the expression that defines the function, but you often want to treat the variables as objects in their own right, and just leave them as variables. This is an additional level of abstraction, and a lot of students have a great deal of difficulty with it. I think that it takes time (and some innate ability and interest, too) to really make this step.

I’ve occasionally heard people say things like, “But the idea of a function is easy! It’s just…” (I’ve heard similar comments about sets, solving for equations in algebra, real numbers, complex numbers, and arithmetic, etc., too.) Whenever I hear this, I always want to reply: “Um, it took centuries of thought by absolutely brilliant people to settle on the modern understanding of this stuff. It can’t possibly be as easy and intuitively obvious as you seem to think it is.”

• educationrealist

Yes, that’s one of the really wonderful things that occurred as a result of this post. Many mathematicians, here and on twitter, responded and made it clear that this was complicated, and they were a bit nonplused to realize this wasn’t really covered extensively in high school. Teachers do cover transformations these days more, but there’s no attempt to clarify the confusion between equations and functions–not that it’s simple, just that there is a difference. My attempt to write this up, which had frustrated me for a month, really paid off.

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• Unkown

On the site differencebetween.info it states “A function is a relationship between two variables,” but in your write up you state “Functions are different in a key way from equations: a function is an output. An equation is a relationship between variables.”

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• Alex

Hi, I stumbled on your post after trying to program an inverse function procedure in SageMath, and realizing that I was getting hung up on some of the ambiguities that you bring to light.

There was a comment about the equals sign that I’d like to elaborate on.
The = sign can mean either assignment or equality, and programming languages use different symbols for each (in Python, for example, single = for assignment and double == for equality), because it’s essential to define variables before relating them. For example, if y = 2 and x = y, then x == y, but if x or y hasn’t been defined yet, then asserting that x == y will throw an error. When dealing with functions and equations in mathematical notation, the = usage tends to be less precise, because nothing will crash (except perhaps the students’ minds). But it’s often helpful to distinguish = and == in mathematical notation as well, because it clarifies the underlying logic and assumptions. For example, if h(c) = c + 15 and p(c) = c + 3 (function definitions), then h(c) + p(c) == 2c + 18 (a statement of equality based on the definitions, NOT an assignment).

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• howardat58

I just found this post. There is some confusion around the comments (I read almost all of them).
Do have a look at the DEFINITIONS section of my recent post

https://howardat58.wordpress.com/2018/02/17/the-chain-rule-and-the-theory-2/