Have you seen this meme and argument?
360 million dollars is 1 million dollars to 317 million people and have 43 million left? Hilarious and scary.
It would be a good “headache” to show the students and have them prove for/against using exponent rules. Perhaps a fun Socratic seminar.

The students usually already know the exponent rules by then, but the lesson starts by having them extend them to rationals by asking stuff like:

Rewrite 2*4 = 8 only twos as bases and whatever exponents are needed, so (2^1) * (2^2) = (2^3)

And then more complicated stuff like

16*(1/4) = 4

And then finally

2^? * 2^? *2^? = 2

Then from there I bring it down a notch and do my “Ice Cream Shoppe” view of functions. Which I’m currently trying to illustrate and will share here later. It’s cute and helps makes sense of what it means to have an input produce only one output.

Love to see it, though there seems to be something categorically different between “creating a headache” and “creating a metaphor.” Another example of a metaphor.

Joshua Greene:

Well, why do we care if something is a function anyway? For me the key is when I am making a definition and want to know that it is “well defined,” that there is a meaning to my definition and it is unambiguous.

As I work on my new blog sharing my experiences using “interactive student notebooks” for the first time in my classes I am pondering how to introduce functions with my 9th grade Algebra One that gives them more understanding than they had developed in 8th grade. I like the Khan Academy introduction https://www.khanacademy.org/math/algebra/algebra-functions/evaluating-functions/v/what-is-a-function to look at it mathematically (and circles are unambiguous and well defined so they kind of conflict with the above justification of functions…). Then from there I bring it down a notch and do my “Ice Cream Shoppe” view of functions. Which I’m currently trying to illustrate and will share here later. It’s cute and helps makes sense of what it means to have an input produce only one output.

It’s easy to know how many numbers are in the set {1, 2, 3, . . . n} so if I make a function (and make sure the inverse is a function) from any set to the counting numbers, the size of the set will be the destination of the last element. How many 5-digit multiples of 3 are there? Well, what function maps 10002 to 1 and 10005 to 2? One that divides by 3 then subtracts 3333. Then 99999 -> 30000, so there are 30000 of them. I can use functions to count all sorts of sets.

If I can define a function with an inverse from the integers to the counting numbers, I’d know those sets have the same number of elements, that is, the same cardinality, and if not, there must be ‘more integers’ than counting numbers. How about the rationals or the reals then? Is “infinity times infinity” from the AT&T commercial really mind-blowingly larger than infinity?

Historically, as I recall, the notion of “function” was developed for purely foundational reasons, centuries after differential calculus was practically all fleshed out without it. My suspicion is that the main reason the “function” idea is developed so carefully in algebra classes nowadays is simply because “they’ll need it for calculus.” (Which wouldn’t even be true, if we started calculus with Leibniz and Newton’s beautiful original vision, instead of the hyperformalized modern stuff.)

But cynicism aside, I think some basic calculus-like questions could be good headaches for the aspirin of functions.

1) The teacher is still essentially telling the rules – ‘impossible motion graphs are not functions, all others are’.

2) Creativity is discouraged – I love distance time graphs, I think a good cue that pupils really understand what is going on is when they can link elaborate stories to even the most bizarre looking graphs. When defining functions you need to limit some potential interesting interpretations, i.e. ‘no time travel allowed’ (one-to-many mapping), whilst permitting others, i.e. teleportation can be ok…as long as it’s continuously near-instant (step functions).

In this sort of situation when dealing with the formalities of mathematical definition I’m happy for that to be it’s own headache. My approach usually is along the lines of:

-A simple table, headed ‘is a function’ and ‘is not a function’
– A couple graph diagram examples in each to start
– A host of more graphs to try to predict and sort (make sure to include lots of nice subtle differences, e.g. step function vs step function with little gap, x^2 vs sqrt(x), sin/cos/tan etc.)

From there you can tease out what sort of rules pupils were trying to use, what did they notice and how can they describe it (you might get things along the lines of ‘there can’t be gaps’, or ‘it can’t go back on itself’). Once the final sorting has been decided get pupils to see if they can put into more formal definitions, little cues like writing ‘input’ and ‘output’ on the axes to start with and then later replacing with ‘x’ and ‘y’ might help. Get pupils to draw their own after (and maybe even try matching to a story!).

If you really want to get fancy then from here you might want to repeat similarly introducing different domains (e.g. only select integers, coordinate pairs, complex numbers), headings for ‘injective’, ‘surjective’, ‘bijective’ , matrix or 3D functions etc.

So let’s take a soda machine. If you put your money in and the press the button labeled coke, you expect a Coke to come out. -What if a Coke does come out? Do you know that the machine is functioning properly?
-Now imagine that you press that same button again (with a new dollar of course), and out comes a Sprite. Is the machine functional or not?
-Now imagine that the first time you press the Coke button you get a Sprite, but every subsequent time you press the Coke button you always get a Sprite. Is the machine functioning properly?
-What if both the button labeled Coke and the button labeled Sprite both consistently give you a Sprite? Is the machine functioning?

I generally let the students discuss which situations show the machine is functioning properly (in a mechanical sense).

The students then come up with the definition that it is a function if a certain input always gets you the same output. They also realize that it is okay if two or more inputs happen to yield the same output, as long as they each do so all the time. Parenthetically, they might also point out that human error can make a function look like it’s not a function. :)

This helps when we figure out what the vertical line tests is helping you to look for. It also connects later when we look at scatter plots and lines of fit, as it is our attempt to force a non-function into the closest function possible so that we can make more confident predictions about outcomes.

p.s. – I sometimes use the buttons on a cellphone, in addition to the button on a soda machine, to further the discussion.