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.

Geometry – First Semester Review Worksheet

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.

1st sem review chart2

1st sem review chart

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

Standard Form Equations With An Open Middle

#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.

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)