A few weeks ago, we had a lesson on one of my favorite topics to have little kids play around with – functions! When covering this topic with little ones, a very important first step is to create a cool function machine:

(Last year we made one by putting two chairs back to back and covering them with a blanket – while not as cool looking, it was quite functional, other than the blanket occasionally falling off.)

“Play” proceeded as follows. One person got inside the machine and was the operator. He/She then came up with a rule that everyone else had to guess experimentally, by putting in various inputs into the machine and receiving/recording the outputs. With a few exceptions, our functions all dealt with numbers. With the youngest group (4-5 yo), we used craft sticks for input/output, and with the older groups (6-10 yo) we had them write numbers on small pieces of paper.

The inputs and outputs were recorded on the board until someone guessed the rule. Most kids, when playing the role of the operator, wanted to make up rules that involved addition. However, we encouraged/helped them come up with more exciting ones. This is what the board looked like after a few of the more involved ones. Can you figure them all out?

The boy (6 yo) who came up with the divide by 2 function in the first picture did a great job of illustrating the concept of domain. When someone tried to put in an odd number (amazingly the first few were all even) a voice came from the machine declaring, “Only even numbers allowed.” When a 9 yo from a different group wanted to come up with the same function, I helped him widen the domain by modifying what was done to the odd numbers (last picture).

As I already mentioned, many kids came up with functions that involved addition, but I was surprised that only one kid tried doing subtraction (and that one didn’t go too well). His function was minus 1, but unfortunately the very first input he got was 0. He wasn’t sure what to do with this, so he decided to switch to a function that returned random answers. When we caught on and told him that he had to have a consistent rule that others could guess, we ended up with one crying kid. Any advice on the best way to handle those type of situations?

Overall though, the kids really got into the activity. I wasn’t surprised that everyone enjoyed being the machine operator (among other things, it certainly made them feel important), but it was nice to see that they were equally engaged in the “guessing” part. I loved hearing exclamations such as “It’s impossible!”, “What??”, “No way!”. And even “This is hard!” showed that they were thinking.

Next time we have a lesson on this topic I’d like to discuss inverses with them (had two great discussions with Katie on the subject a little while back, here and here) as well as compositions of functions. Other suggestions?

I really feel for the one crying kid… I bet he thought that all measurable functions were fair game:)

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And I like leaving room for functions that seem random. Sure he came up with it out of panic, but he made them think. And the best yet, they guessed his “rule” anyway. The real trouble with this function is remembering all your previous responses in case the audience asked a number again. Maybe a quality control on the teacher’s part of a description of the function at the beginning would help. Either is words, algebraic, or with a picture mapping one number line to another.

One reason for addition only was probably fear of making a mistake. While mathematical objects like functions cant make errors, humans, machines and even computers do occasionally make errors. The audience needs a chance to see this and have a plan for these situations. Like asking the same input again, or asking another function machine with the same rule.

The broader and more free we allow functions to be when we introduce the idea the better. Then when we deal with restricted domains, piecewise, or discontinuous functions the easier it will be for the students.

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As I tweeted, this has come at just the right time for me – as we’re planning a couple of function lessons on Monday and Tuesday. We’ve drawn pictures of machines, but now I’m hoping there’s a big box around somewhere! I might scale up the pictures on the photocopier and stick them on.

I like your idea of looking at inverses. That could be part of the second lesson…

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Ooh, I like the idea of kids drawing their own machines. Need to remember that for next time.

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Here they are. I found it a little worrying that a few of them seemed to have waste product as their output!

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We did this with older kids in our programming class: Manual Function Game and Function Machines. For us, the latter worked a bit better as one of the students delighted in giving his classmates difficult calculations when they were the function operator.

Truthfully, I always feel a bit of ambivalence about these guess the rule or pattern matching because it seems to fit the “One Right Answer” myth of math class, even though we all know that any finite collection of input-output pairs can be fit by an infinite number of rules with a great degree of flexibility for any input that hasn’t already been tested. Anyway, kids love it, so maybe I shouldn’t worry so much.

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I don’t think of it as guess “the” rule, but rather as guess “my” rule. I have a secret and you have to guess it based on certain clues, sort of like breaking a code. This way, it does not suggest that there is only one possibility, but there are more likely and less likely ones based on how we (humans) think. And as you said, kids really love this activity, and I think that the secrecy aspect has a lot to do with it.

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