You can create:
- Function machines!
Imagine a machine that takes some numbers in, does something to them, then gives you the results. This is, quite possibly, the most popular metaphor in all mathematics: function as a machine! Working with function machines is simple. One person, the machine operator, makes up a machine. Other people offer numbers (or whatever input the machine takes). The operator gives machine's outputs back. Repeat, until you can guess what the machine does.
Here is some of the complexity in the activity:
- What if the machine's creator says the machine adds the number to itself, but your guess is the machine multiplies numbers by two? When are machines the same?
- Can you work your machines backwards? (Hint: you can do it with some of them). This is called "inverse functions."
- Can you connect several machines together? This is called "composition of functions."
- Can you attach the output exit of your machine to its input entrance and cycle it again and again? This is called "iterations."
- Come up with some non-number function machines. How about a machine that turns baby animals into adult animals (chicken-hen, foal-horse, kitten-cat...) or a machine that builds a traditional house for a person from each culture (tipi - North American Indians, igloo - Eskimos, round-windowed holes - hobbits...), or a machine that...
- Come up with funky ways your machine actually works. Does the machine that triples make clones, or is it based on reflections and mirrors?
You can do this activity in the car or on a walk. However, for more complex function machines, you will want to write down input-output tables and look for patterns to be able to guess. You can also plot inputs and outputs on a coordinate plane, using graph paper. Excel spreadsheets with hidden formulas provide a way to program your function machines on computers.
There are many function machine applets available on the web. Example 1, Example 2, Example 3... Natural Math is working on an applet that will allow people to create their own function machines. Of course, most computer algebra systems do it, but they require at least as much programming as Excel. It would be nice to have something more kid-friendly.
Because this activity is empowering, since people can make up functions at their own level. Also, multiple representations and the metaphor help people with asynchronous development.
As you go
- Notice patterns in your machines
- Support and celebrate mathematical arguments among activity participants
- If people want to spend a lot of time drawing detailed machines, instead of coming up with fancy math rules, support them
- Record your machines and share with the family multiplication study group
Higher and deeper
Functions are omnipresent in all branches of mathematics. Algebra, as a study of number patterns, has especially many functions.
Strewing and snippets
Once you set up the activity and everybody knows what's involved, you can play function machines on the go, a few minutes at a time. You can capture especially interesting machines somehow: ask authors to name them, write a few values in a table, describe the machine in words. You can put up these machine descriptions on your walls to remind people to play ("strewing").