**I See It: The Power of Visualization**

**Marc Garneau, Chris Hunter**

This session started with a great exploration of divisibility using base 10 blocks. It isn’t strictly applicable to my grade level (high school) but was a good puzzle that I will try to keep in mind for all the crazy shortened schedule days in case one falls between units. Kids of all ages find divisibility rules fascinating because they seem like magic, I enjoy finding the math that make tricks work.

I’m most excited about the introduction to radicals. The approach is to focus on the geometric interpretation of a square root – it’s the root (side length) of a square! Take your basic perfect square, say 16. What side length gives the square an area of 16? Four, so four is the square root of 16. Now, what side length gives the square an area of 18? Not a whole number, but that number is called the square root of 18. Maybe we could simplify that a bit? Dividing the 18 into four squares doesn’t really help but nine squares comes out nicely – each side of the small squares is the square root of two, thus the side of the square with area 18 is three square roots of 2. I’m still amazed and I’ve talked through this a few times since the presentation. I was not looking forward to the radicals unit this year, but I am now!

I’m most excited about the introduction to radicals. The approach is to focus on the geometric interpretation of a square root – it’s the root (side length) of a square! Take your basic perfect square, say 16. What side length gives the square an area of 16? Four, so four is the square root of 16. Now, what side length gives the square an area of 18? Not a whole number, but that number is called the square root of 18. Maybe we could simplify that a bit? Dividing the 18 into four squares doesn’t really help but nine squares comes out nicely – each side of the small squares is the square root of two, thus the side of the square with area 18 is three square roots of 2. I’m still amazed and I’ve talked through this a few times since the presentation. I was not looking forward to the radicals unit this year, but I am now!

Finally, we looked at dynagraphs. I was just talking to my coworker today about comparing linear and exponential functions. I think this is a great way to see the relationship between input and output – especially since we spend so much time focused on comparing two outputs (many of us in attendance were momentarily confused when the linear graph with a slope not equal to one didn’t step along the way that x+3 does).

**Now What?**

**Danielle Maletta**

The majority of this session was a refresher on things I’d heard at other PD, but those reminders are important because anything not immediately applicable gets lost in the shuffle. She spent some time talking about how to give low status kids a boost. One way was to explicitly mention all the skills needed for a task – we’ll need help organizing and recognizing patterns and … so kids can recognize at least one skill as something they’re good at. Related to that is a “How are you math smart?” poster that I’d love to have in my classroom.

One task I loved was something Danielle calls a treasure hunt. She gives groups of students a pile of cards (equations, graphs, descriptions etc.) and gives them the first clue. Students work with their group to find all the cards that fit the clue and send one person up to the teacher with the clue and the cards. If they found all the solutions they get clue two, if not, they get sent back to their desks. I like this because students have to look at characteristics of your choosing (ex: Is 4 in the domain?) rather than matching by one or two aspects.

**Creative Integration: CCSS Math Practices, Quality Activities,**

**Excellent Questioning Techniques and Technology**

**Tom Reardon**

We started right off with a rich task:

In how many ways can you use only the digits of 8 and plus signs ( + ) to create an expression whose sum is 1000?I didn’t think it was a particularly rich task to start (a great sign - it was definitely low entry) but as we went through all sorts of patterns emerged. This is another problem that doesn’t fit particularly well in a particular spot in my curriculum, but this one is great for the last ten minutes of class when my students need something a little different.

The painted cubes problem was also fascinating.

A cube with dimensions n x n x n, that is built from unit cubes, is dipped into a can of paint. How many unit cubes will have paint on zero faces? one face? two faces? three faces?This one could fit more directly into my curriculum as there end up being four different patterns – a constant function, a linear, a quadratic and a cubic. It is approachable in Algebra 1 and if I use it during the quadratics unit it would reinforce the characteristics of each type of function.

**Modeling Mathematics Using Problem Solving Tasks**

**Andrew Stadel**

**Better Than Engagement**

**Dan Meyer**

• Dial up the math slowly.

• Create the need for notation and vocabulary.

We so often jump directly to the abstract and the precise. We need to specifically ask kids to estimate, to sketch, to predict. Then once they have a baseline, they’ll want to know if they are right. This allows us to dial up a notch and try some examples. Dan asked “If math is the aspirin, what is the headache?” The headache should be something tedious or repetitive or cumbersome (ex: so many examples they wish for a generalization). It’s the teacher’s job to create a headache, then offer strategies and vocabulary as aspirin. This process of exploring first and formalizing later is something I’m good at in geometry. I am still working to find explorations for some aspects of Algebra but I’m getting there – this conference certainly helped build my pool of resources!

**Bringing Problem-Solving Into Your Math Classroom**

**Fawn Nguyen**