**becoming**something.

Long ago, in a grade 11 class far, far away:

Once we had done a few different functions, I would put this on my beloved smartboard (btw, miss you! love you!)

Usually, someone would assume I meant the function we had just done, but on a good day, another person would say "What type of function is it though?" and I would rejoice! Just that moment felt like a breakthrough - ooooooh it could be ANY function!

"Quadratic" I would say. Much unhappiness...ummm miss, we did that LAST year....hello, we're like 16 now.

Once I beat them into submission, we would get an equation that did indeed pass through the two points, like this:

**Looking at a bigger picture now:**

I said "Is there any

*quadratic function that goes through these two points?" Which brought us to the idea that there could be at least 2, depending on which one is the vertex. And on a great day, we would get into "What if*

**other****of them is the vertex?", and how there are infinitely many possible answers, because that's how many parabolas you could**

*neither***through those two points if neither one is the vertex.**

*draw***Since my first career choice was acting....**

I would then smack my forehead, and exclaim "Wait a minute! Did I say quadratic? Wow, you guys are like 16, whoah, sorry, my bad, I meant it was an absolute value function!" Knowingly rolling their eyes, they would tuck into that.

**And a bigger picture would emerge:**

Because now it was time to talk about a bunch of other stuff, like: compare your method for finding this rule with the quadratic? do you expect any of the parameters to be the same? which and why? are you making any assumptions this time?

After we had done the absolute value and all its possibilities, I did the head-smacking thing again, they rolled their eyes again, only to be shocked that now I was "realizing" I actually wanted a

**linear**function that went through these same two points.

Blink. Miss. That's, like, insulting. Oblige me, I would say, and after a quick review of slope, etc:

But then when I asked the same questions as before, like is there any other possibility, which is the vertex, compare methods, parameters, etc there was some wonderment. And on a great day, someone would notice that the slope matched part of the absolute value function....coincidence?

**The big picture that day was:**

When all went well, and I had enough of them hooked to reach critical mass, they could see that if there is enough information to DRAW only one function, that means there is enough information to find one and only one rule,and also that any time you are making assumptions about which function, which vertex, etc, your answer must include those assumptions. But I didn't want to say it! I wanted them to!

**And that's it, right, no bigger pictures out there?**

Well, that's the thing. From there, each year, I added more and more functions, and more and more questions, so that it evolved into something that got too big to do in one class, so I split it up into 2, then 3, so that I had a sort of cycle going, where after finishing a unit or 2, I would do a Big Picture class, and everyone got to know what that meant. But even that changed each year.

This year, with geogebra, it's all been ramped up about 20 pegs. Now

**get to be the ones typing**

*they**rule into the input, and seeing if they get something that goes through the given points. Instant feedback of the most authentic kind. They also get to colour things in! Which is hugely motivating, as it turns out, not really surprising, as I love to make things look nice too.*

**their**So currently, it is an assignment that comes in two parts, the geogebra part, which they get here on geogebratube, and a list of exploration questions in this word document.

But I'm still the one asking all the questions, which is where I hope the next change will come from. Just because I love the connection between the drawing and the algebra doesn't mean they do. Already in some of their work, I have read things that I never thought of, which is absolutely thrilling. And humbling. And enlightening.

As soon as all parent permission forms are in, I will post some of their work, which was astonishing, to say the least!

Happy New Year, can't wait to see what we'll all be doing/sharing/thinking one year from now!

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