## Wednesday, February 24, 2016

### Rationals Slap Jack Review

This year more than ever, I have a wide range of learners in my room. The individual needs of these kiddos means that I need to choose activities that can "run themselves" so that I can take small groups or individuals aside and work with them. Test review days, especially, seem to run this way.

Here is an activity that I created for our class review today so I could pull together some small groups. I printed them front to back (they line up when printed), then cut a set for each small group:

The rules are this:
• Students play in 2 vs. 2 games (so 4 students per group)
• Students look at the 1st card and work with their partner to solve the problem completely on their whiteboard or in their notes
• First group to have a complete solution written can "slap" the deck to indicate they're done. They then have to defend their answer. If they're right, they get to keep the card. If not, the other group gets a chance to "steal" by getting the question right.
• If no one gets it right, the whole group has to work to figure the problem out. No one gets the card, but they're still responsible for the topic.
• Group with the most cards at the end wins!
I also gave each group a stack of red, yellow, and green cups to indicate if they needed help or not. Thank you, Math=Love for this great idea!

What worked?
• It's easy to differentiate by removing certain questions from the deck for some groups or by adding in enrichment cards (which I need to get around to making) for the stronger groups
• I can move around the room or pull small groups aside and the activity flows pretty smoothly
• Some groups chose to not actually play the game. Instead, they decided to sort them by what was easy and what was hard and go through the harder ones.
• The cups!! The cups!! Seriously, if you don't already do this, start. It will stop you from hearing your name frantically 47,214 times in the same period.

What needs improvement?

• I will definitely group more strategically next time. I had 1 or 2 kids who were fish out of water with their group and were scared to use the cup to indicate they were confused if no one else was.
• I honestly needed a prize to rush them a little bit more to "play"- some kids take it so seriously, they can't have fun with it
Overall, another awesome day with my kiddos!

## Friday, February 19, 2016

### Tasks vs. Projects in Calculus

So it's Friday afternoon and in an effort to really knock one out of the park (and avoid the pile of ungraded tests staring back at me), I have some serious thoughts going on about my next Calculus PBL, which I am trying to design with a fellow AP Calculus teacher at my school. I am lucky enough to teach in a STEM program that received a huge federal grant and, as a result, has access to lots of things others don't. Too often I'm thinking of ideas without enough time to actually go through the enormous process that is ordering anything through aforementioned federal grant. So here goes nothing:

Disclaimer: This is a brainstorm....brain dump....random collection of ideas that I hope other amazing AP Calc teachers might be able to expand upon? I offer no solutions in this post...just questions.

We are getting 2 new 3D printers next week and have a training on using them (YES!) and there is obviously a direct relationship to all the applications of volume we discuss in AP Calc. I've done some searching....which always just leads to more questions and ideas. I started from watching these 2 videos:

Solids of Revolution
This possibility seems like more of a logistical issue than a conceptual one. I know how to make this a demanding task for kids and have seen amazing applications- from the "Goblet Design" project from my wonderful coworker or the vase volume performance task I had the opportunity to play around with at NCTM Nashville (shout out to Brett Doudican from Coordinated Achievement on this one). It would be fun to 3D print these to scale and then use displacement to check the accuracy of our volume calculations. There are so many ways to give kids constraints and have them develop something amazing. It's task-based and, more importantly, not a project for the sake of doing a project.

Cross Sectional Volumes

Logistically, I actually think this is easier. In about 15 minutes of playing around I was able to create 2 designs that would model different cross sections. My kids have experience with 3D printing...this wouldn't be mind blowing for them.  My struggle is having them print 3D models seems like the thing I hate most....a project for the sake of doing a project. Might it be a fun way to spend some time after the AP exam? Sure. But if I want to invest any time in this, I want it to be something that's worthwhile mathematically as well. I want something that can be low-floor, high-ceiling since our school ranges in ability level from Honors Calculus to BC Calculus (to a kid who is taking an MIT Quantum Physics MOOC for fun....seriously). Can you tell that I have lofty goals here?

So here's where I need my MTBoS friends...

What sort of tasks and applications do you associate with this topic?

I want to do more than just create a model, so how can I challenge my kids with creating a model within given constraints and do it meaningfully? My coworker and I have been trying to think of how to get kids to apply this concept in the real world and really use the model to serve a purpose.

Please feel free to share, comment, and question!! We promise to tell you where this road leads us and share any resources along the way!

## Monday, February 15, 2016

### The Real Housewives of Pre-Calculus: The Rational Root Theorem

Over the years, I've always found that Rational Root Theorem has been a struggle for kids. They struggle to get all combinations of factors and struggle to remember which goes where. I don't blame them...they can be cumbersome problems with a lot to keep straight.

Each semester I feel like I did a slightly better job conveying it to the kids, but I think I've finally hit my groove with it.

Inspiration: Reality TV?

Yes, I said it. I was watching Bravo and marveling at one of the real housewife's closet. Like any proverbial math nerd, my inner "notice and wonder" started coming out....
How many of my salaries would it take to have a closet like that?
(the age old teacher question)
How did they decide the layout?
I bet we could make that more efficient
What cost more...the jewelry, the clothes, or the shoes?
How many different combinations could she make with all of it?

That was it!! How many combinations could she make with her outfits?!
It's a problem our kids have been doing since 5th, 6th, 7th grade!
Here's why:

I start off with a polynomial and we give him a name- Fred?
We mention that we are going to define f(x) as Fred's feet....f must stand for feet, right? And naturally, feet go on the ground. Now, Fred is still working on building his professional wardrobe so he has a limited selection of shirts and pants. Since we've oriented Fred with his feet on the ground, the constant is now his shirts and the leading coefficient is now his pants.

Let's find all the factors of each.

So we can see here that Fred has 4 shirts and 2 pairs of pants. We then proceed to make every "outfit" we can out of them.  Add a plus or minus and BAM- we have all our possible rational roots!

Small? Gimmicky? Yup.
It's been the only way I've been able to get kids to remember it though! And they know better than to put shirts on their legs or pants on their head. It's dramatically reduced the student error of putting factors in the wrong place. I saw a big difference on their individual practice. Unit test is tomorrow...let's see if it makes a difference there!