Fun with Functions: To Infinity and...Back!

What is a function?  Before yesterday I had no earthly idea except that it's clearly an important mathematical concept and one that young children can easily experience at the concept level.  Based on my search for function machine activities last night, it seems that functions are essentially about rules and change, both of which seemed to dominate my little experiment with the kid today.

It all started yesterday when I was looking for math games at the library and found the book Anno's Math Games II.  The first chapter is about a marvelous, magical machine that turns one thing into something else.  A chicken into a chick, a butterfly into a caterpillar, and...can you guess the rule?   Later in the chapter Anno moves the rules toward transforming numbers but when I suggested to my six year old that we make a magical machine of our own she jumped up and literally shouted:

"Yay!  Let's make one!  I'll go get Lucy [our cat] and we'll turn her back into a kitten!"

Um.  Well, not really a real machine, right?  Sort of a machine in our heads?  With numbers?  We left it there for the day.

Last night I got to searching.  It seems that a function machine is actually pretty straightforward.  You put a bunch of numbers in and try and figure out from a number of results what 'the rule' (function) is.  So, I made my own function machine using clip art of a washing machine and a dryer (dirty in and clean out...) and some arrows.  I figured that we could play around with it a little and see what happens. 

What happened was that it basically flopped.  I made the first rule, we put a number in and wrote down what came out.  We did it a couple times and then she successfully guessed what the rule was.  Not exciting.  Then she got to make up a rule, and the whole thing collapsed into an activity that essentially boiled down to practicing familiar math facts.  Plus, she was really, really disappointed about the machine itself.  I think partially because it was a picture, and we all know a picture of a machine is not a real machine, right?

No worries, though.  At that point she took matters literally into her own hands and made her own...

There were buttons for adding, subtracting, multiplying and dividing.  There were buttons for stop, go and compute. 

There was a little door where you could put in a slip of paper with the input written on it.  There was an emergency brake. (If you can't tell by now, these were all her ideas.  I was just grateful she had salvaged the activity -- even if it meant being demoted to the position of scribe as she dictated the rules of how to operate the thing.)

There was also a screen: "Numbers can be a bit dangerous," she said. "The screen is for if you don't want to go on into infinity." There was a steering wheel "in case we go into infinity, or a really, really hard problem.  Press the steering wheel to pause, so you can think about it.  Or, if you get into infinity steer the wheel backward and you'll be back where you started."

Apparently, the transformation of numbers is not only tricky but a bit dangerous as well.  

When it was time to try out the machine I encouraged her to put her results on my nice little worksheets, but her energy was really more about the slips of paper she wrote on and put into her machine.  Fair enough.  Here are some of the rules we played around with:

One round of multiplying by three.  Four rounds of 'multiply by 2, take away 1' (on another sheet, not pictured).

We also put the machine on 'geometry setting' and steered "the wheel backward to get it to change the number of sides" of a shape.  We started with a triangle and the rule of 'add one side'.  Interestingly, the kid turned the triangle into a rhombus, not the square I expected, which made a lot of sense given the angles.  And, when she tried to add a side to the square she initially just divided the square with a diagonal line.  A nice moment to reinforce 'side' and 'edge'. 

Also, because we recently explored the concept of half-ness and double-ness we did some doubling using the legos.  I suppose we were on the 'geometry' setting then, as well, because of the 3D shapes.

Reading this post over, I realize I forgot one really important idea: that you can run the 'rule' both forward and backward through the function machine.  Except that...the machine she built does have a steering wheel.  As she did say, "...you can steer the wheel backward and you'll be back where you started."  I guess that's an inverse operation right there!

I've been thinking about how we might do this next time.  Harnessing my kid's imagination and her need for the concrete/physical seems like the key.  Maybe we'll figure out how to make bigger machines, some toy-sized, some human-sized.  The little plastic cats could go in one door and a different amount of bigger stuffed cats could come out the other?  Or, a dress-up function where you go into the machine (a tent?) dressed normally and come out with hats, scarves or necklaces added?  Hmmm...  Maybe we could make up simple Jump Patterns (ala Math in Your Feet) and change them somehow? 

It's easy enough for me to come up with ideas for physical/concrete functions, but harder to think of ones that create some complexity and/or interesting results. I also can't tell which is more important -- doing the computation of the rule, or figuring out what the rule is.  I've obviously got some thinking and learning to do here so if you are able to elucidate this topic for me, I'd be really grateful!

In the end, it's clear from watching her make her own machine that she's basically got the concept: you can change something into something else using different kinds of rules.  More importantly (for her, anyhow) is that you don't have to settle for a representation of a machine -- you can make your own real one!