When introducing functions I always relate them to a machine. Specifically a toaster...I'm not sure why. I ask them what happens when you put bread into the toaster. You get out toast, duh. I ask if you'll ever get out eggs? Nope. We decide that there's certainly something wrong with that toaster if you're getting out eggs.
Then we took a look at this beautiful function machine I drew on the iPad...told them it's my math machine. For the most part, they're all familiar with function machines like this. I've always used this analogy but I've never really taken it so far and I think that's what made the difference. This time we stuck with the actual function machine to help give them something to relate to.
I wrote the numbers one at a time. 2 then 10, 3 then 15, 10 then 50. Asked what the machine is doing and they said times 5 so I wrote it in there.
Same thing with the second machine.
When I got the third one though I started the same way...5 then 8, 7 then 10, 3 then 6...they yelled out plus 3...but when I wrote 5 then 9 next they said they couldn't do it and the machine must be broken. We discussed that the (5,8) and (5,9) pairs were the ones that let us know the machine was broken.
In the next machine I put 1 in twice but it came out to be the same answer every time so all was well.
For the last one, they said at first they thought it was broken because they didn't know the rule, but then we talked about how sometimes we won't be able to find the rule but that's ok. Nothing about the number pairs indicated the machine was broken so everything was ok. We just called it "not broken" and moved on.
As far as notes I also took a different route this year and I have the ISN to thank for that because it helps me to streamline the idea that I'm trying to get across. In the past I've split the notes into four sections: from points, from a table, from a mapping diagram, and from a graph. I decided it felt like too much. Especially since I've always thought that points, tables, and mapping diagrams are all pretty much the same information.
So instead I decided to just go simpler. Also I wrote most of the notes out for them. I think class time is better spent doing practice than sitting and copying down what I have on the board. So I did some explaining and they ran through the examples. This is where I feel like it's never gone better. The broken/not broken analogy worked so well they flew through them. They actually asked if they could write broken/not broken instead of function/not a function. I said no, but said they could write it in parentheses.
The "from a graph" section required a little more explaining, but once I put estimated coordintes on the one graph and showed them why it was "broken" they got it. The explanation on the vertical line test I got from Sarah at mathequalslove and the function/not a function sorting activity (which I assigned for homework) came from Math Tales from the Spring.
The next day we worked on evaluating functions in function notation and it went equally as well (which was also a pleasant surprise). I decided to stick with the function machine analogy since it worked well the day before.
So for f(x) = 3x + 4, we drew a little function machine in their notebook to show what it meant.
When we did f(x) = x^2 and g(x) = x + 1 we just drew two machines and called one f and the other g. I told them that the letter just let us know which machine to put the number in. Even composite functions went well. Sweet.
All in all the function machine idea may seem a little elmentary, but if it helps them make sense of the idea does it really matter? None of them are drawing machines when doing the problems but it really seemed to help them understand the idea which I was very happy with.