This week, my students were excited about math. They were excited about what they came up with. They were creative. They were asking each other questions and trying to figure out why things worked. That’s DESMOS’s fault. Here’s what happened:

My Advanced Algebra classes are starting a unit on “Families of Functions”, and I discovered earlier this week that they had NO idea what the graph of x^3 looked like. Wanting them to make sense of it, I leaned hard on Desmos and I was not disappointed.

**PART I**

Wednesday, we did a black box activity. Students got a sheet with many blank graphs and I displayed this Desmos graph.

Starting with what they already know, I asked them to sketch the graph of 2x on the first blank graph. (Because I use Apple TV and Explain Everything to display from my iPad I get to walk around the room and peer over student’s shoulders this whole time.) Everyone drew a line. Lovely. I turn on the 2x graph, and we’re confirmed, that’s what it looks like. I mentioned briefly to make sure students had hit the origin, having noticed one or two kids had missed that on their papers.

Cool, okay, now draw x^2. No problem, we’d been talking about that a lot recently. I remind students that we’re not looking for perfection here, just one or two important points, and then then general shape of the graph. Most students drew a lovely parabola, and a couple needed reminding about the origin. I turn on the graph, and now my screen looks like:

We keep going like this, and students lose their minds when x^3 goes up. They had hypothesized that x^3 might look like a wider parabola, or a skinner one. One hypothesized it would be shifted up because, “all the numbers will be bigger”. In each period maybe 2-3 students guessed the correct shape of the graph. There was much discussion about “Why does it look like that?!” and we had great discussions as we went through with x^4 and x^5. Very quickly, I ended up playing this and students argued amongst themselves.

It was lovely.

## PART II

The next day, students came in and had a matching activity. I gave them these graphs, and equations, and had them cut them out and match them with NO technology.

Here’s the word doc so you can adjust to your liking:

Once they’re matched to the student’s satisfaction, they get to bust out a graphing calculator or a phone with Desmos to see if they’re correct.

**THE REALLY AWESOME PART**

Finally, students are asked to make conjectures based on the matching they’ve just done, and they did just what I wanted. All conjectures were to be of the form:

*If an equation has ______, then the graph has ______. *

Here are the most popular conjectures:

*If an equation has an***EVEN exponent**,*then the graph has***a parabola.***If an equation has an***ODD exponent**,*then the graph has***a twisty, schwoopy schwoop.**[We decided “schwoopy schwoop” is what the x^3 graph was called. For those who care, it’s officially called a “cubic parabola”, which I had to look up and is way less fun to say than “schwoopy schwoop”]*If an equation has an***x^x**,*then the graph has***a starting point of (0, 1).**

Then I get to give my very favorite directions to students.

**“Bad news, folks. Absolutely ALL of these conjectures are false. They’re true sometimes, but not all the time. I want you all to take out either a graphing calculator or Desmos and BREAK them. Prove to me these are false. BREAK THE CONJECTURES.” **

Students get super into it, right away. And man oh man did they not disappoint. I’m a moron and didn’t save everything they did, but here is what I remember of how they broke the conjectures. (Note: We spent a lot of time on the first one and not much on the last two just because we ran out of class time.)

*If an equation has an***EVEN exponent**,*then the graph has***a parabola.**

And what’s great is that while the purple graph doesn’t look like a parabola, we were able to use desmos to zoom in and see that it does actually get “curvy”. Awesome. Mistakes are great. Desmos is fabulous.

Also, Look at the green one! That’s the absolute value graph! But it’s not! But it is!!! That’s so awesome! Figuring out why it works like that is flipping fabulous.

*If an equation has an***ODD exponent**,*then the graph has***a twisty, schwoopy schwoop.**

*If an equation has an***x^x**,*then the graph has***a starting point of (0, 1).**

Bottom line – the way this activity worked, students were invited to play and they were invited to be creative. I love that. I was so proud of them and they were so excited. It was great to hear students “Ooh!”-ing and “Aah!”-ing over the graphs Desmos helped them create. They were constantly asking, “Why does it look like that?!” which is exactly the sort of thing I want happening in my classroom.

If you can improve on this, please tell me how. I already love it, but you’re all probably going to have lots of ideas for how it would be better. Or maybe how I can follow it up next week and keep building on this? Eh?

[**Update**: I should mention the structure of this activity was shown to me by Terry Wyberg at the University of Minnesota. Not sure where he stole it from. It’s called a “Type II”}

I really like your explanation on how to have students use Desmos in a classroom setting. I am a college student pursuing a teaching career in mathematics, and just recently discovered Desmos about two years ago. I really like your activity ideas for in the classroom to help students get a better grasp on graphing functions.

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