I introduced exponent properties this year by writing this on the whiteboard and asking them to make observations about it.
Then I got more direct about the kind of observations I was after
There was some back and forth as we discussed what it means to simplify and how to simplify. I was short on answers and kept the discussion geared towards observations.
I lined up another example where there was a negative exponent
From here I gave them a little half sheet with six expressions on it – grouped in them in groups of two – and off they went simplifying (do I need to add that I had them working at VNPS???).
I spent less time this year than normal covering simplifying expressions and I didn’t notice that students were any worse at it – so I’d say this approach was a success. The next step I suppose is to apply it to other procedural concepts.
How many of the mathematical practices can we get at when we introduce concepts based in procedural fluency? That question is swimming through my head as I begin to teach factoring.
I know there are some people out there who could use this. Especially as finals near. It covers parallel lines cut by a transversal, vertical angles, complementary / supplementary, rhombus, special right triangles, polygon interior angles, and transformations.
This was created by my colleague Dave Casey – who is definitely at the top of my “I wish they had a blog, were on Twitter, and were presenting at conferences” list. This is just one of his worksheets – you should see his activities, amazing stuff.
The Goods (aka: the pdf)
1st sem review chart
#CMCN14 was lights out good this year. Amongst the many things I learned new – were a ton of reminders of things that I used to think about but had let slip. One of those things was the importance of an open middle, where students have a defined beginning and ending, but how they get there is largely up to them. During Dan Meyer’s talk he challenged us to find an open middle in the routine, procedural fluency building exercises students get. Most of the great problems have it – but it is a nice tool for tipping the scale for our procedural problems towards a deeper understanding.
Here’s the typical – pretty much closed middle – version of a problem about standard form:
Find the slope, y-intercept, and x-intercept of the following equation in standard form: 3x – 4y = 20
Here’s my one up
Write the equation of a line in standard form where the both intercepts are integers, and the slope is a fraction.
We could really be here all day playing with these
Write the equation of a line in standard form where the x-intercept is a fraction, the y-intercept is 7, and the slope is a negative fraction.
We can even get at MP3
Explain why it is not possible for the slope and x-intercept of a line to be an integer, but the y-intercept a fraction.
Lastly – the Asilomar conference grounds are so amazingly beautiful. Each tree, slightly beaten from the ocean breeze, stand in stillness as perfect landmarks to perseverance. And as the sun begins to set, and that air begins to cool, and those stars begin to show – it’s hard to believe that it’s all just the backdrop to a professional development experience. It’s humbling to be there – I mean you’re walking from presentation to presentation with a program booklet offering the intellects and energies of 200 amazing educators. But you only get to pick 5… good luck with that.
Some days you just need a worksheet. So here’s one. I took all the problems from math-drills and just spaced them out better. I love using matrix notation so always have my students do it.
The Goods: (aka: The pdf’s)
So Kristen and I are taking dance classes – so any three step process feels like a tango to me. This activity has nothing to do with dancing (sorry). This is one way that I use Vertical Non-Permanent Surfaces (VNPS) and Visible Random Groupings (VRG) for problems where we are working on our procedural fluency – whether it being solving equations with fractions, or factoring polynomials.
One down side of VNPS is that students believe they are finished when the correct answer reaches the whiteboard, rather than when everyone in the group understands how to do the problem. The one student who knows how to solve it just goes up to the board and solves it. There are a couple methods to help give students less places to hide in a group – here’s one of those methods:
Pick three problems that are similar. Tell students that there are going to be three rounds. In the 1st round they will be in groups of 3 (randomly assigned of course). Then in the 2nd round they will be in groups of 2 (randomly assigned again of course), and lastly they will be back in their seats working alone on paper. That’s it. Throw whatever standard you want at it. I believe it works because the thought of eventually being alone gives them the extra motivation to learn from their group.
Here are the three problems I started with in my geometry class, where my goal was to give them practice multiplying polynomials, solving by factoring, and using pythagorean theorem:
From what I’ve seen – the knowledge that they are eventually going to be alone makes those students who usually look for a place to hide in a group more apt to contribute and learn from the first two rounds.
I love to pose the question by waiting until they are in their groups and standing at whiteboards, then I pose the question on my whiteboard in the center of the room. It’s a nice math coach moment versus math teacher moment.
Here’s a smattering of whiteboard activity from this day: