Illustrating Functions

Function definitions aren’t usually tested on either the SAT or the ACT and since I never worked professionally with math, functions were something I’d barely considered in algebra a billion years ago. So for the first few years of teaching, I kind of went through the motions on functions: unique output for each input, vertical line test, blah blah. I didn’t ignore them or rush through them. But I taught them in straight lecture mode.

Once I got out of the algebra I ghetto (which really does warp your brain if that’s all you do), I accepted that a lot of the concepts I originally thought boring or unimportant show up later. So it’s worth my time to come up with the same third way activities and lessons for things like functions or absolute value or inverses that I do for binomial multiplication and modeling linear equations and inequalities.

So every year I pick concepts to transfer from pure lecture/explanation to illustration. Sometimes it’s spur of the moment, other times I plan a formal curriculum change. In the case of functions, the former.

Last year I was teaching algebra II/trig and–entirely in passing–noted a problem in the Holt book that looked something like this:

and had two simultaneous thoughts: what a boring question and hey, I could really do something with that.

So the next day, I tossed this up on the board without comment.


I’ve given these instructions three times now–a2/trig, trigonometry, algetbra 2–and the kids are always mystified, but what the heck, the activity seems simple enough. No student ever reads through the entire list of instructions first. They spend a lot of time picking the message, with many snickers, then have fun translating the code twice.

But then, as they all try to translate someone else’s message using the cell phone code, bam. They realize intuitively that translating the whole-alphabet code would be an easy task. And with a few moments of thought, they realize why the cell phone code doesn’t offer the same simple path. They don’t know what it means, exactly. But the students all realize that I’ve demonstrated a difference that they’d never considered.

From there, I graph the processes, which is usually a surprise as well. The translation process can be graphed?



At this point, I can usually convince kids to remember the Vertical Line Test, which they were taught in algebra I. At that point, I go through the definitions for relation, function, and one-to-one function, using a Venn diagram (something like this with an addition inner circle for one to ones). Then I go back through what the students vaguely remember about functions and link it to the correct code example.

Thus the students realize that translating the message into code is a function in either code key. I hammer this point home, because the most common misconception kids get from this is that all functions must be one to one. Both are functions. Each letter has one and only one number assigned, and the fact that one translation key puts several letters to the same number is irrelevant for its determination as a function. Reversing the process, going from numbers to letters, only one of them is a function.

Then I sketch parabolas and circles. Are they both functions? Are either of them one-to-one functions?

In Algebra 2, I do this long before the inverse unit. In Trig, I introduce it right before graphing the individual functions as part of an overview. In both classes, the early intro gives them time to recognize the significance of the difference between a function and the more limited case of the one-to-one function–particularly in trig, since the inverse functions are very limited graphs for exactly the reason. In algebra II, the graphs reinforce the meaning of the Horizontal Line Test.

I haven’t taught algebra I recently, but I’d change the lesson by giving them a coded message and ask them to translate with the cell phone code first.

This leads right into function and not-function, which is all they need in algebra I.

I have periodically mentioned my mixed feelings about CPM. Here’s a classic example. The CPM book introduces functions with the following example.

Okay. This is a terrible example. And really boring. Worst of all, as far as this non-mathie can tell, towards the end it’s flat out wrong. A relation can be predictable without being a function (isn’t that what a circle is?). But just looking at it, I got an idea for a great test question (click to enlarge):


And I could certainly see some great Algebra I activities using the same concept. But CPM just sucks the joy and interest out of the great starting ideas it has.

Anyway. I wanted to finish up with a push for illustrations. What, exactly, do the students understand at the moment of discovery in this little activity? I’ve never seen anyone make the intuitive leap to functions. However, they do all grasp that two tasks that until that moment seemed identical…aren’t. They all realize that translating the message in the whole-alphabet code would be a simple task. They all understand why the cell phone code translation doesn’t lend itself to the same easy translation.

I look for illustrative tasks that convince kids to think about concepts. As I’ve said before, the tasks might kick off a unit, or they might show up in the middle. They may demonstrate a phenomenon in math, or they might be problems designed to lead the students to the next step.

The most common pushback I get from math teachers when I talk about this method is “I love the idea, but I don’t have enough time.” To which I respond that pushing on through just means they’ll forget. Well, they’ll probably forget my lessons, too, but–maybe not so much. Maybe they’ll have more of a memory of the experience, a recollection of the “aha” that got them there. That’s my theory, anyway.

There’s no question that telling is quicker than illustrating or letting them figure it out for themselves. Certainly, the illustration should be followed by a clear explanation with much telling. I love explaining. But I’ve stopped kidding myself that a clear explanation is sufficient for most kids.

That said, let me restate what I said in my retrospective: The tasks must either be quick or achievable. They must illustrate something important. And they must be designed to lead the student directly to the observations or principles you want them to learn. It’s not a do it yourself walk in the park. Compare my lesson on exploring triangles with this more typical reform math example. I resist structure in many aspects of my life, but not curriculum.

In researching this piece, I stumbled across this really excellent essay Why Illustrations Aid Understanding by David Kirsh. I strongly recommend giving it a read. He is only discussing the importance of visual illustrations, whereas I’m using the word more broadly. Kirsh comes up with so many wonderful examples (math and otherwise) that categorize many different purposes of these illustrations. Truly great mind food. In the appendix, he discusses the limitations of visually representing uncertainty.


On reading this, I felt like my students did when they realized the cell phone message was much harder to translate: I have observed something important, something that I realize immediately is true and relevant to my work–even if I don’t yet know why or how.

About educationrealist

38 responses to “Illustrating Functions

  • anonymousskimmer

    I like reading these posts too, even though they’re irrelevant to my life as I’m not a teacher.

    The problem is trying to remember the nomenclature of years past.

    “A graph mapping selection to delivery” I *think* this means that, in the form “F(x)=y” that selection (button) is the x-axis, and delivery (soda type) is the y-axis. And that therefore this does pass the vertical line test?

    And I’m relatively good at math. It’s no wonder if many parents have difficulty helping their students with their school work, and yet another argument for an inverted classroom.

    • anonymousskimmer

      So: A & B are true?

      • educationrealist

        Full disclosure. I whipped this together just for fun, so the wording is probably ambiguous. That’s another problem with my early tests.

        a is ambiguous, looking at it again. A necessary criteria for function is not one-to-one input to output, but only one output for each input. However, a student could argue that it is a function because all one to one functions are, by definition, functions.

        b, d, and e are true. I’d intended a to be false, but I’ll have to mull the wording. Glad you like the posts!

      • anonymousskimmer

        Thanks. I understand how D is true now (forgotten nomenclature). I thought E incorrect because there is no way the student can know the first sentence is true (that the vending machine attendent mixed up the sodas), and I recall word problems which would make unsupported statements and ask if the student can know if they are true or not.

      • educationrealist

        I took that from the CPM problem (above). Basically, if pressing Coke always gives Sprite, then it’s a function. It’s not a function if sometimes you get Sprite, sometimes Coke. But I see you’re looking at the big picture. Fair enough.

  • anonymousskimmer

    404 Not found on the link ” Why Illustrations Aid Understanding”

    But here’s one that seems to work:

    And the corollary is that not everything that can be expressed in an illustration can be adequately described in writing. Which is why naturalists drew too. 🙂

  • anon

    This probably wouldn’t be much help to your students, but it’s fun for me:

    Consider the function from the elements (x,y) of R^2 to the color space {black, white}, defined so: if x^2 + y^2 = 4, black; otherwise, white. This function can be perfectly represented by drawing with a black pen (or marker) on a white two-dimensional surface, and is impeccably deterministic: every coordinate pair maps to a single color.

    When graphed, this will appear oddly similar to the non-function y = plus or minus sqrt(4-x^2). Is it different in any important way?

    • educationrealist

      Is a circle a function, though?

      • anonymousskimmer

        But is anon saying that the vector space itself is R^2, thus making this a graph of an arc between (0,2) and (2,0)?

        Otherwise I agree, I thought this was the definition of a circle?

      • Mark Roulo

        “A relation can be predictable without being a function (isn’t that what a circle is?)”

        “Is a circle a function, though? ”

        In cartesian coordinates, no a circle is not a function (because each input can map to two ouputs).

        In radial (is that the correct term?), a circle can be a function. The unit circle function, for example, looks like this:
             f(r) = 1

        This is a bit like the up/down facing parabolas are functions in x (e.g. f(x) = y^2), but the sideways parabolas are not … *HOWEVER* the sideways parabolas are functions in y (e.g. f(y) = y^2).

        [I really like your encoding example, BTW!]

      • educationrealist

        Right. That’s my understanding–a circle is not a function because it doesn’t pass the vertical line test. But a circle is predictable. I was quibbling with the CPM terminology.

        My cell phone code, you mean? Thanks!

      • Mark Roulo

        “That’s my understanding–a circle is not a function because it doesn’t pass the vertical line test.”

        No, no, no! No.

        A circle is not a function “in cartesian space.”

        Saying “a circle is not a function” with no qualifiers is like saying that you cannot subtract a larger number from a smaller number (e.g. 3 – 8). You *CAN* subtract a larger number from a smaller number, you just need to be using Integers (or Rationals or Reals or …) as your number type rather than Natural numbers. Circles are functions (or can be …), just not in cartesian space.

        Now … for your Algebra I class the right thing may well be to tell them that circles aren’t functions (because cartesian space is assumed). But you (as the teacher) are better off clear on the distinction.

        To be clear … I’m not harping on what you should present. Lots of things get simplified in math class (negative number don’t have square roots … oh, really? What about complex numbers? Well, we don’t want to introduce those, yet. Ah … fine). This might be one of them. I wouldn’t introduce the unit circle in Algebra I, either. But it still exists 🙂

      • educationrealist

        No, I appreciate the correction. Would it be accurate to say that the definition of a function is for Cartesian space?

      • Mark Roulo

        “Would it be accurate to say that the definition of a function is for Cartesian space?”

        We are rapidly heading above my pay grade 🙂

        *The* definition of a function (in the math sense) is a mapping between sets (good old “domain” and “range”) where the rule is that each member in the domain maps to only one member in the range (though, as with your example in this post, multiple domain values can map to the same range value). It doesn’t have anything to do with graphs at all. And doesn’t even have to be about numbers 🙂

        Now, by STRANGE COINCIDENCE the common sets that we get for domain and range in Algebra I happen to be the sets of Real numbers and if those are our domain and range then using a cartesian plane and having ‘x’ be our domain and ‘y’ be our range gets what you (and your class) want.

        But consider a ‘straight line’ like this:

            f(x) = y = x.

        A nice straight line at a 45 degree angle. This is a function.

        But what if we change our labels a bit and write this:

            f(t) = r = t.

        Same mapping, but now in polar coordinates (where, Wikipedia tells me ‘t’ is the angle [‘t’ for theta] and ‘r’ is the distance from the origin [‘r’ for radius]). Same function, same mapping, but now we get a spiral [the ‘Spiral of Archimedes’ if I understand correctly]. Still a function … in fact the same function … but now the vertical line rule doesn’t work because we are in polar coordinates.

        I don’t know where this typically gets taught. It isn’t in my Algebra I Dolciani book, but I’m pretty sure that polar coordinates showed up before Calculus (maybe Algebra II?).

        The mapping matters, too, for Cartesian.

        Lets start with a polar coordinate unit circle:

            f(t) = r = 1

        Basically, a nice circle around (0,0) with a radius of 1. Now we want to map this *function* [not the shape!] to a cartesian view. We can get one of two mappings [remember, we’re mapping the function]:

            A) f(x) = y = 1


            B) f(y) = x = 1

        The first is a nice straight line at y = 1, and the ‘vertical line’ rule works. The second is a nice straight line a x = 1 and the vertical line rule doesn’t work … because this is a function in ‘x’ but not in ‘y’.

        Moving the *graph* of the unit circle function to cartesian coordinates gets us non=functions involving sines and/or cosines because we’ll need two outputs for most inputs.

        [I think I’m probably creating more confusion here … the real key ideas are that functions don’t have to be about numbers and graphing functions is only one way to look at them]

      • anonymousskimmer

        “I don’t know where this typically gets taught. It isn’t in my Algebra I Dolciani book, but I’m pretty sure that polar coordinates showed up before Calculus (maybe Algebra II?).”

        Back in the early/mid 90s there might have been a very brief mention in Trig or Geometry, but I believe Calc I was the first real exposure I got to it.

        Here they seem to be aligning it (Common Core) to Geometry, “Integrated Mathematics 4” (Senior year?) or Pre-Calc:

        And here it’s 10th and 11th grade honors math:

      • anon

        The circle certainly is a function, and in fact in a mathematical sense “A relation can be predictable without being a function” is not true. If given the input you can determine the output, then the relation is a function.

        The circle x^2 + y^2 = 4 is not a function of x, but it is a function.

      • anon

        All right, I’ve got more time now and can explain my original comment at greater length for those interested.

        A circle is not a function “in cartesian space.”
        This isn’t correct; if you use the third dimension to spin the parabola z = x^2 – 25 around the z-axis, you’ll get the paraboloid z = x^2 + y^2 – 25. The zeroes of that function form a circle of radius 5 in the x,y plane, but the space the paraboloid exists in, R^3, is as Cartesian as can be, and the paraboloid is a function by any standard. It will pass the vertical line test, where a vertical line is defined as a line parallel to the z-axis (we use a line parallel to the z-axis because we defined the function as z = x^2 + y^2 – 25, that is, you put in x and y and you get z out). Restricting the domain to only part of the x,y plane doesn’t make the result, a circle, any less of a function, in the same way that, since “f(x) = 2x” is a function of x, “f(x) = 6 when x = 3; undefined otherwise” is still a function of x.

        *A* definition of a function, in the math sense, is that it’s a set of ordered pairs (a, b), with the constraint that if (a,b) and (c,d) are both pairs belonging to the function, and a = c, then b must equal d. (There are other ways to define a function, but this is mathematically equivalent. Speaking of “*the* definition of a function” is misleading.) a and b could themselves be multidimensional values, so it’s more common to think of a function as a set of ordered n-tuples, with an invisible line dividing the dimensions in the domain from the dimensions in the range. The illegitimate circle you’re thinking of is conceived of as a defective function from R to R, a two-dimensional function that accepts a single real number as input and produces a single real number as output. For the defective function x^2 + y^2 = 25, we can immediately see that (3,4) and (3,-4) both belong to the “function”, which violates the definitional constraint. But the circle I mentioned originally isn’t a two-dimensional function, it’s a three-dimensional function. It accepts a point (x,y) from R^2, and produces a color value. Like the paraboloid I mentioned earlier, it is defined at all points in R^2; some example pairs belonging to this function are ((3,4), black), ((3,-4), black), ((0,5), black), and ((2,8), white).

        It happens, though, that if you graph this function, it will look like a circle of radius 5. Fundamentally, that’s what it is. All we’ve really done is added a third dimension to the original non-function so that we can satisfy a definition we imposed on ourselves; that third dimension is of no particular interest in itself.

      • anonymousskimmer


        That is a really neat explanation.

        It reminds me of a personal homepage of an RPI (I think) student in the late 90s who described how to mentally visualize higher-dimensional shapes via the use of colors and textures overlaid onto 3-D shapes.

  • Mark Roulo

    “f(x) = y^2” —-> “f(x) = x^2”

  • NewarkTFA

    I am an English teacher today, but I liked math in high school. I just wanted to tell you that I enjoy reading this blog generally, and that posts such as this one make me think you must be a wonderful high school math teacher. So many high school students–even students with decent skills and grades–don’t experience math as a potentially fun and intriguing opportunity for intellectual play. I think you make that experience possible for at least some students who might otherwise miss out on it altogether.

  • Michael

    I kicked off functions using pop culture with the rule that assigns each man to his gf.. i.e, is it a function if Justin Bieber is assigned to two of his gf’s? It certainly held their attention, but I like your approach better. Nicely done!

    • educationrealist

      When I go through relation to function, after the explanation, dating is always the analogy. First they just know each other, then they date, then they get all exclusive…


      • Michael

        Also, some more questions. What if one of the coke buttons stopped working, would it be a function then? No, because it doesn’t have an output (unless you consider do nothing an output)… or one of the coke buttons only returns your change, is it a function (yes, now it has a unique output from the other coke).

        Can you expand a bit upon your sentiments with CPM. I’m tempted to get online versions of the book to see how they introduce topics, which I give them credit for based on comparison to other texts.

      • educationrealist

        I think do nothing is an output, since it would do that as consistently as return change. Either way it’s a consistent output.

        You’ve hit on exactly what I like about CPM–they introduce stuff well. They also come up with great analogies that they ruin with cheese.

      • Michael

        Yeah I was writing quickly before my professional ended, in fact, neither of my examples would be functions. I was trying to think of an example that would be in the domain, but not get mapped to an output (to test every input has a unique output). Likewise, it would be interesting to have something like “You notice that there is some other drink available but no button to select it (its okay if something in the range isn’t used). Either way, good stuff. Keep the curriculum stuff coming.

  • Jim

    Mark Roulo – Your remarks demonstrate the utter futility of centuries of effort to clear up the conceptual foundations of mathematics.

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