Is the game really impossible to play when there are multiple outputs? Seems like you could still figure out if the relation is a circle or whatnot.

Roy,

In New Zealand we don’t even teach graphing inequalities until we do linear programming.

That way the need for them arises naturally.

Re functions:

You can tease students by playing “Guess my number” with non-functions — I would normally offer a small lolly as prize for winning, but it’s not obligatory.

It goes: “my number squared is 16”. A student says 4 and I say, no, it’s -4.

Next one is “my number squared is 25”. They guess -5, and I say, no, +5.

They complain that I only allow one answer, so we change the rules to allow two. Next question is “sine of my number is 0.5”.

If they are up to complex numbers, “my number to the power 3 is -1” also works.

They can quickly work out which offered options give certainty of answer (“my number logged in 2”) and which do not.

A further example. The quadratic formula is a very useful formula it takes 3 numbers and spits out one or two. Given three input numbers here is no uncertainty about what it will split out. It’s not a function but hopefully you agree it is useful and interesting.

Why are you going to focus on a subset of formulae that exclude this one?

Surely there is a better motivation than some teacher wants to play a guessing game that arbitrarily excludes interesting cases such as, for example, the quadratic formula or circles.

When I get into functions and domain and range we talk about inputs and outputs. My favorite analogy for a function is a Pepsi machine (insert . We talk about the buttons on the machine and how the first button is Pepsi and so is the second. Then the third is Diet Pepsi and the fourth is Mountain Dew or whatever. The buttons are the input and the sodas are the outputs.

The idea of one input giving only one output is clear and students also learn quickly that two buttons (inputs) can produce the same soda (output) and that it is still a properly working soda machine (function).

We then talk about what it would mean if you pushed the Mountain Dew button and sometimes got a Mountain Dew, sometimes got a Cactus Cooler, and sometimes got a Dr. Pepper.

This then becomes a reference point for whenever we discuss if a relation is a function or not. “Remember the Pepsi machine.”

This isn’t a “headache” by itself, but for inequalities we try to stress the “truthiness” of something, just like equations:

The graph of 2x + 3y = 12 is the set of points that make the statement true

The graph of y > 2x – 5 is the set of points that make the statement true

One problem I like is having each kid pick a point, then running it through a “test” like y > x^2. They plot their point green or red depending on whether or not it passes the test — and a rough shape of the graph emerges. The entire concept that you can shade a full side of things once you know a point matches them isn’t obvious but becomes viable after these types of problems.

Even your version (test a point, shade a side) is better than what some texts do about shading “up” or “down” based on greater than or less than, rules which quickly vanish when pieces of equations move around or non-linear inequalities come about. One of my largest annoyances about linear inequalities is rules or instructions that work -only- for linear inequalities and nothing else, which strike me as completely useless.

My simplistic theory of learning is that WHAT we teach should respect the consensus of a larger community of experts (e.g., the Common Core) and that HOW we teach needs to incorporate explicit linkages to both the context of mathematics as a discipline and the multiple contexts in which non-mathematicians might use a specific skill or concept outside math class. I remember memorizing definitions of functions, domain and range as a high school student and believing at the time that the concepts had no value whatsoever either in math or in the broader world. I did not learn to appreciate the importance of the ideas (including the necessity of a single output) until Bob Chaney and I began using engineering control systems as a teaching tool for both math and science.
Automated control systems are an extremely important component of engineering, technical and medical careers. They can be as simple as the inequality which opens a door under one set of circumstance and closes it under another set. They can also be as complex as the automated decisions which guide a driverless car or as critical as the equipment which delivers the right amount of oxygen to a premature infant. In every case, the system design and operation demand a single output at any one time.
One of our early successes with our SAM vehicle (a calculator-controlled version of Papert’s Logo Turtle) came in asking trig students to find the two math functions which would guide SAM back home after it had moved randomly along 2 sides of a triangle. It is fairly easy when the first two sides were at right angles, although most students initially want to plug is the formula for the triangle’s interior angle while SAM needs to turn through the exterior angle. It gets harder when the angle (as well as the sides) were random, and the possibility arose of SAM being in any quadrant. Students experience the headache of realizing that their calculators were converting arctangents to a single output in a way that did not always bring SAM home. Amazingly enough, they are sufficiently motivated to keep working on the problem until they find a better way of giving SAM the single instruction it needs. The immediate physical feedback of watching SAM go astray increases the headache and is critical to maintaining motivation until they do get it right.

I think this is great. Those advocating starting with mathematical, abstracted examples and non-examples of functions (quadratic formula, equations of circles, etc), I think maybe don’t grok your typical 8th grader’s facility with quickly evaluating expressions in his head. (Some of them are great at it, sure, but most are iffy and some are still not good at it — and don’t we want every kid in the room to get what a function is, if we’re going to the trouble of teaching it?) For a brand new concept like “what’s a function?” it’s much better to hook it to something they can easily relate to, like Dan does here, rather than making evaluating the quadratic formula a prerequisite for understanding the new thing. PLUS, “what’s a function?” often comes chronologically before kids see equations of anything other than lines.

If someone’s going to take this lesson idea and build it out, I’d be interested in seeing what people come up with for what comes after this. How do you build the bridge from the understanding developed in this lesson to being able to approach all the types of function/not-a-function questions you want kids to be able to handle.

Back in the early ’80s, I found a book on precalculus that was comic-strip based (can’t recall the name off-hand, but it wasn’t one of Larry Gonick’s “cartoon guides”. I recall clearly that the author presented functions as guarantees (a little guarantee popped out every time the characters “ran the function machine.”

That really hammered home the notion of functions as predictable. I offer students example like getting paid an hourly wage of $20, working 40 hours in week 1 and grossing $800, working 40 hours in week 2 and grossing $800, and then working 40 hours in week 3 and grossing only $600. When they go to the payroll department (or directly to the boss) to complain, they’re told, “Oh, well, we had a bad week, so we can only afford to pay you $15 and hour this time.” Naturally, this would be an outrageous violation of a guarantee.

Then I give the example of a table of temperatures collected every hour on the hour from 1 AM to midnight on a certain day and ask whether, given the data, they could confidently state what the temperature was for each of the 24 hours that measurements were taken. Most quickly realize that they could do this since any given hourly measurement corresponds to exactly one number. Finally, I ask if they could reverse this process: given a temperature, could they (likely) be able to predict with certainty which of the 24 hours corresponded to that number.

Of course, the above is quite simplistic and doesn’t capture anything close to the full picture of functions or why mathematicians (and those in applied fields) find them to be important and worthy of study. But it does tend to get students to see the headache.

By the way: 1) when people say “Pearson book” or “McGraw-Hill book,” I’m at a loss. I have a pile of current Pearson intermediate algebra books on my floor that I got last fall when thinking about changing from the publisher I was using. I also looked at other texts for that course from that and other publishers. As the business has been reduced to perhaps three major publishers, they all have multiple texts, multiple (current) editions, from multiple authors. So saying “Pearson book” and the course isn’t describing a function;

2) the particular text from Pearson I’m using now for intermediate algebra (Bittinger, Beachy, & Johnson, 12th edition) does something on the topic we’re discussing that I don’t recall seeing elsewhere: it introduces and defines functions first and only then defines and explores relations.

Frankly, I find that quite bizarre. Seems to me that relations are the most general case, with functions being one natural restriction on them. Of course, there are special and important kinds of relations (equivalence relations being foremost in my mind) and subclasses of functions. But I can’t see why one would introduce functions before relations. So I don’t. I’ve never been shy about changing the order of topics to suit my thinking and purposes, though I’ve met many teachers, particularly in K-6, who find that to be terribly daring. Just wonder if anyone has found other authors/books teaching functions before introducting relations.

In New Zealand we don’t even teach graphing inequalities until we do linear programming.

Not even single variable inequalities?

We teach writing and solving single variable inequalities, starting in our Year 10 (US 9th Grade). That includes testing for meaning too in Year 11, such as writing inequations from word problems.

However we don’t test that skill graphically at all — though I imagine that many teachers use a number line to explain it, and some may test it internally.

The concept of graphing inequalities on a number line has few real world applications at all. This year, in fact, we have had instructions that the Algebra testing in Year 11 is going to move even further away from process skills like that and towards problem solving.

This is an amusing introduction. I like it. But, is it aspirin? Some students don’t know what letter to stand under in the last two examples. They are then given formal vocabulary for the situations. Isn’t this analogous to describing symptoms to the doctor and being told that what you are experiencing is called a headache?

The approach Bowen described for inequalities has worked well for me. Solutions & non-solutions can be plotted using different colors on Desmos. Ask if there are solutions in every quadrant? On the y-axis? Interesting solutions? Etc. If it doesn’t come up, at some point ask about making a fence between the solutions & non-solutions.

Likely a couple of points will be incorrectly coded as solutions (or non-solutions). Someone will eventually notice that they don’t fit the pattern, at which point you can fix them.

I wonder if students given the headache of not knowing where to stand would make the aspirin themselves. Like- I’m going to stand under the color of my shirt or the dominant color that I’m wearing. Could we teachers use this urge to naturally make things make sense to develop why functions are useful?

I am missing the fantastic motivation (headache) of the rule guessing games in motivating the distinction of functions from other relations.

It would be just as easy to play this game with a relation. I could draw on my piece of paper some polygon and you call out an x or y coordinate and I give you the corresponding coordinate. I could give you all of them or just one requiring you to try the same value several times to know if there is more than one. You have to guess what polygon I drew.

There would still be a rule to guess even with a non-function relation answer.

The game with functions is easier as you only have to try each x value once. But why is this particular way to make the game easier important? Dealing with only one to one relations would be even easier. But for some reason, not given here so far, exploring relations which are functions is very important.

What do you say to a student who says why should I care about the distinction between functions and other relations – because a coke machine?

Hmm. What if there’s a soda machine with an unmarked button. Joe puts money into the machine, hits that button, gets a Coke, walks off. June puts money in the machine 20 minutes later, pushes that button, gets a Sprite, walks off. Is the machine “functioning” correctly?

I’m afraid I have to vehemently disagree with your reasoning about the birthday example Dan. Using your own logic, the game “implicitly” restricts itself to only those students whose birthdays appear on the board. This doesn’t have anything to do with whether the game is a function or not. Something that would be of use to understand for students. Perhaps that’s what you were driving at in the first place, but from your response, I’m not so sure.

But you do actually have a function, given the domain is “implicitly” defined as Dan attempts to say.

The problem lies in this. ” This forces the students to restrict the domain for reasons that don’t really make sense.” That’s quite right – it doesn’t make any sense. Confusing for students as they’ll undoubtedly run up against such functions that do make sense(logarithms, etc.) in different contexts later in their studies, as any year 11/12 student knows who has been bored to death by the universality of Families of Functions, of which logs, etc. are certainly members.

(This by the way, is an excellent reasons to be very cautious around this issue, since it would seemingly directly fall under Dan’s original argument regarding “Rules That Expire”: http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue1/tcm2014-08-18a_pdf )

Far more powerful then would have been the birthday example with ALL months listed, to illustrate that(given n>12) a range may be much smaller than a domain and yet provide a function, resulting in domain pile-ups at somewhat regular range intervals(….. one might even chance to suggest a periodic distribution – hello trig).

That is, unless the entire intent was to spark an investigation into mathematical functions that have domain restrictions that DO make sense for very good reasons, but again, I’m fairly certain that was not the intent here.

Thanks for this. I was also thinking that you should check out CPM (College Preparatory Mathematics) at cpm.org. They often do a great job at motivating the need for new mathematics but I’ve only looked at one or two of their text books.

I use the mystery message to demonstrate both functions and one-to-one functions. I’ve used a modification of CPM as an illustration, and will eventually use it as a test question or basis for an exercise.

https://educationrealist.wordpress.com/2015/03/30/illustrating-functions/