## Epiphany Part II: The Return of the Tranformation

So the perfect storm happened at TMC. In case you missed it (because it was only in my head), here’s what happened:

1) I had an algebraic epiphany last month about using flowcharts to solve equations:

At the very end, I had an inkling of an idea to tie it into function transformations:

but I didn’t really know where to go from there or if it was a viable way to think about it.

2) I had a conversation with two people about how confusing it must be for kids to do transformations in Geometry where x + 2 means “move two to the right,” then move to algebraic transformations where seeing f(x +2) means “move two to the left.” We wondered how we could make the transition easier for students, but came up with nothing. (Side note: I can picture sitting at a table with a boy and a girl having this conversation, but I cannot picture the boy and the girl. I’m now thinking it was not at TMC, but maybe another workshop I was at this summer? But if you are reading this and it was you, let me know so it doesn’t keep me up nights anymore.) (Further side note: Um, somehow I never taught transformations in Geometry? Is that weird?)

3) I worked with Sheri Walker (@sheriwalker72) in the Going Deeper with Desmos morning session. We were tasked with making a new lesson using Desmos and she immediately turned to me and came up with an awesome idea because she knew that we both loved function transformations (who doesn’t, amirite?).  I also brought in a copy of my handouts from my session to share with her and then she just casually mentions how she approached function transformations and jots down something like this for the equation y = 5|2x – 6| + 7:

(I know, it’s weird that her handwriting is the same as the Chowderhead font, but that’s just how cool she is.)

Ok, so at this point I begin freaking out because Sheri can obviously read minds. I mean, I just had this epiphany three weeks ago. So I was so amazed at her mind-reading ability that it was not until later that I realized the elegance and awesomeness of her next step.

I really think you should be sitting down.

If you’re reading this on a tablet, make sure you have a soft spot for it to land when you drop it.

Here it is…

Ok, she called it something other than “fancy form” (programming form? transformation form? input/output form?) but you get the idea. Or maybe you’re like me and you think it’s pretty but “getting the idea” will suddenly hit you three hours later. SHE JUST TIED ALGEBRAIC AND GEOMETRIC TRANSFORMATIONS TOGETHER.

Added bonus? No more worrying about whether the form is in f(bx – h) or f(b(x -h), because this takes care of that. Want to see it action? Let’s take that same equation, y = 5|2x – 6| + 7. Except this time, let’s just find (x, y) pairs from the original absolute value function and use the fancy form to transform them:

What? That just happened. Are you as amazed as I was?  Now pondering how to fit this in to my already over-long function transformations unit….

## The Things I Do For MTBoS (Posters & Function Transformations)

Casey (@cmmteach) says, “MAKE ME A POSTER!”  and I say, “HOW SPACEY?”

.pdf  and .doc and yeah, because that would use a crapton of ink, bwpdf and bwdoc.

Then Julie (@jreulbach) says, “I NEED THAT IN A POSTER!” and I say, “I FEEL VAN GOGH IS SORT OF PICKY ABOUT USING HIS STUFF”

.pdf and .doc

Then no one says “MAKE ME A POSTER OF THIS QUOTE, TOO!” but I think they just didn’t realize their classroom was incomplete without it so YOU’RE WELCOME, EVERYONE.

.pdf and .doc

Updated: I also made a color version of the above poster: .pdf

Then Friday night on twitter we got into quite the discussion of (h, k) and function transformations, because that’s what cool kids do on Friday nights.  Julie was lamenting at kids not sure what to do first…reflect, dilate, shift?  So I mentioned what I call the S.S. method of graphing after one of the best math teachers with whom I’ve ever had the privilege to work.  I’m not going to lie to you; I didn’t trust it for a while. But then after a couple of years (!) she finally convinced me and it is beauty and efficiency all at once.  Then Julie says “GIVE ME AN EXAMPLE” and I say, “WELL HERE’S A WHOLE BLOG POST ABOUT IT ARE YOU HAPPY NOW?”

So you spend time talking about what the various parts do, maybe with some discovery desmos, or some crazy function match game with worksheet.  (hint: don’t use the t-tables! it just confuses the issue!), or the beautiful HOLY COW SO THAT’S WHY f(2x) COMPRESSES activity (which yes, here’s a quick showme video because Elissa said “CAN ANYONE HELP?” and I said “I WILL SHOWME YOU” even though other tweeps answered her much better than I did).

After all of that, it finally gets down to the nitty gritty of graphing a real function with all sorts of exciting things happening. And yes, yes, you could do all the shifting and dilating and reflecting, and substituting (M, N), but then I don’t know how that would be any quicker than just plugging points into the original equations after all that work (don’t get me started on the people that have them draw a graph for EACH TRANSFORMATION like we’ve got all the time in the world) and the Cal teacher would like them to be quick and efficient at these graphs.

But the good news is I can multiply and divide really well and I’m also really good at counting from the origin.  And that’s all that’s needed.

1) Mark (h, k) with a small x.  This is your new origin.

2) Take your basic t-chart (-2 to 2 normally does it for me, unless there’s domain or excitement issues).  Multiply/divide x’s/y’s as needed.  Feel free to multiply/divide by negatives in the SAME STEP if a flip is involved because we are that CRAZY GOOD AT MATHING.

3) Graph your t-chart from your new origin.  SHAZAAM.  Feel free to label points counting from the original origin if your teacher’s into that sort of stuff.

Here are some examples:

The one catch–a negative INSIDE the function.  We just have to do some reorganizing:

(and some of us need to learn the difference between flipping horizontally and vertically. I know which way to flip; I just always call it the wrong thing. I’ve started going with “flip across x-axis” or “y-axis”).

Anyway, maybe this is what most of you do anyway?  But I always have some teachers that are freaking AMAZED by it when I start graphing these babies at workshops so maybe you were, too. If you want more, here is the notetakermaker and the filled-in version.

Full disclosure: Of course y’all know Casey, Julie, and Elissa are three of the sweetest people on twitter and would never ask for things in the manner in which I implied. So I hope y’all consider this post just a feeble start to paying back all that MTBoS has done for me!