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I’ve Switched To

I decided to buy the url  If you follow this blog please change over to rather than as I will not be posting to this site anymore.


– B


Grazing Goat Problems (Sector Area / Trig)

I used to give this worksheet but now I draw the picture on the whiteboard and give the question orally.  Then I provide them a couple of minutes of silent time to work alone in their notebooks.  From there I group them and they work the problem at vertical erasable surfaces.  Here’s the basic question:

A goat is tethered to a 25 foot rope attached to a rectangular barn at point A.  What is the total grazing area available for the goat?


I leave them room to experiment – especially as to how to handle the situation when the goat starts to walk around the barn.  Last year I took them outside, gave a student a rope and they played the goat, with a picnic bench as the barn, and modeled the situation.  I didn’t do that this year for reasons I hope aren’t my own laziness.

But my goal is not just sector area, it’s also trig functions and thus onto to the triangular barn!  I mean – most barns are triangles aren’t they?  Here’s the next problem:


Some student work examples

IMG_2433 IMG_2434 IMG_2435 IMG_2436 IMG_2437 IMG_2438

The different iterations on this problem are pretty much endless:


I did this lesson the day after the students learned how to find the area of a sector.  I followed it up with another grazing goat problem the next day where they were working on paper by themselves.   It’s not like they knew exactly how to do the problem the next day  – but it was that they all persisted, asked questions, and got to a final conclusion.  So I’m happy.

I like this question because although it is not real world – it is about real things that students understand.  I also think it finds itself into the flow of a classroom where students find it appropriately challenging – they believe they can do it but yet the solution doesn’t seem immediately obvious.

Oh, and I teach in a community with a lot of agriculture and the students told me that you really don’t tether goats.  Lucky for me that didn’t stop anyone from solving it.

Letting (making?) Them Figure It Out

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.

Rotations Worksheet

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)


VNPS & VRG Tango!

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: