# Part I of Machines: A Useful Algebra 2 Representation

So last year, I started teaching Algebra 2 again after years of not teaching it. I worked with a colleague on the curriculum, and one thing we really wanted to make sure kids were continually exposed to were various representations of functions and relations. Of course this includes equations, graphs, tables, and words.

But in addition to these representations, I was inspired to include a fifth representation. It has a few drawbacks, but I can’t even express to you how many positive aspects it has going for it. It is the “machine.” I remember seeing images of these machines in middle school textbooks, and they really emphasize the idea of an input, output, and rule. Here’s one I randomly found online:

In this blogpost, I’m going to share how I introduced this representation, and how I subsumed the others in it. In future blogposts, I’ll share all the ways I’ve exploited this representation. It’s pretty magical, I have to say. So stay with me…

At the very start of the course, I introduced this machine representation also. Just not as fancy and cartooney.

I thought a lot about whether I wanted the machine to allow multiple inputs or allow multiple outputs. In my first iteration of drafting these materials, I did that, but then I backtracked. Things started to get pretty complex with an expanded definition for a machine, and I wanted to start the course simply. And, of course, I really wanted to emphasize the idea of a function and a non-function. So I started with the definition above. And started with things like this…

Notice these are “non-mathy” examples of machines. They eased kids into the idea, without throwing them into the deep end.

What was nice is that we got to understand and interrogate the idea of domain and range from this… where I described the domain as the “the bucket of all possible items that can be put into the machine and give you an output” and the range as “the bucket of all possible items that comes out of the machine.”

So for the sandwich one, we know the range is {yes, no}. And the domain might be {all foods} or {every physical thing in the universe}. We talked about the ambiguity and how for these non-math ones, there might be multiple sets of domains that make sense. But then for the math-y ones, we saw there was only one possible domain and range.

In fact, to really drive home the idea of inputs, outputs, domain, and range, I created an activity. I paired up the kids and one kid was the machine, and one kid was the guesser.

The machine got a card like this, with the rule:

The guesser got a card with the domain and range:

And the guesser would give words to the machine, and get a result. And their goal: figure out the rule. Then I would switch the machine and guesser, and give a new set of cards. It was crazy fun! I did it a long time ago, but I distinctly remember kids wanting to play longer than the time I had allotted. (If you want the cards I made, here’s a PDF I created Domain and Range Game.)

Next I showed how the “rule” in the machine could take a number of different forms — tables, words, equations, and graphs — and this is how I introduced the various representations. Kids were given these and asked to fill in the missing information…

… and then they were asked to find the domain and range for these same rules…

To drive home the various representations, I gave kids questions like these, where kids were given one representation and were asked to come up with the equivalent other representations.

So this was the gentle introduction my kids had to machines. I’ll explain where we went from there in future posts… and I promise you it’s going to be good… I’m really proud of it!

1. I have some objection to “we saw there was only one possible domain and range.” Almost all mathematical functions that apply to real numbers can be used with both smaller and larger domains and ranges. Diophantine equations are about what the domain of a function is given that the range is integers. Functions like square root have different domains depending whether the range is real or complex (which is a concept that algebra 2 students should become familiar with). A lot of functions that apply to real numbers can be applied to vectors and matrices, producing results that are vectors and matrices.

1. Mr K says:

I was about to post the same objection.

Exponential functions are completely different when the domain goes from reals to complex.

One of the wonders of algebra is that there are so many isomorphic groups, which means that closure for functions doesn’t necessarily determine their domain or range.

2. Jamie R Nordstrom says:

Overall a similar idea to a representation called “arrow diagrams” that I first encountered in “Mathematics: Modeling Our World” in the late 90’s. There are several videos on YouTube (search terms Nordstrom, arrow diagram) demonstrating inverses, composition, graphing, transformation, etc. I will now add your idea of non-mathematical domain and range to what I do.

3. I used something very similar for my Alg 1 students to build their definition/understanding of functions. I first saw it in the book Burn Math Class by Jason Wilkes. (If you haven’t read it, you would love it!) thanks for the card set- we are going to review functions as we move into exponents next unit. This will provide a memorable return to the idea. Thank you!

4. Mary says:

I love how you pulled so many different representations together. I use that basic idea but not with nearly as many connections. I especially love the idea of introducing domain & range with those non-numeric examples.How did you make all your arrow machines? I’d like to recreate something similar.