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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

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.

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.

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)

RotationsWS

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:

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You don’t have to write great lessons.  But you do have to teach them.  The best lessons in the world – if delivered by an untrained hand, are sure to fall well short of their potential.

So let’s say you find the perfect 3-Act lesson from Dan Meyer, or real-world application from Mathalicious.  And you are stoked.  Now what?  Well, filling this out is a start I am implementing this year:

I recommend not leaving any of this to chance.  Your natural teaching skills will have plenty of time to shine during the lesson – but I recommend formally addressing (which to me means write it down) the following important aspects of allowing a great lesson to reach its potential:

1. Focus Question:
What is the main question(s)  the lesson meant to address.  This just focuses you.

2. Reason(s) I Think This Assignment Is Great
Remind yourself why you choose this particular assignment out of the great sea of possibilities.  What makes this lesson so great?  And as a bonus, you should start to see a pattern as to what you value in a lesson.

3.  What I Want Them To Learn
Write the skill, or standard, or whatever it is you want students to take away from this lesson. Again – it is important to formally confront it here so you make sure it happens.

4. How I Am Going To Set It Up
What’s the hook? What’s the scaffold? How is the question going to be formulated?  Give this the time it needs because if you mess this up it’s hard to recover.

5. How I Will Help Students Who Don’t Know Where To Begin
I started prepping myself for this a couple years ago and it pays off big time.  Have your response down so you can give the student the right hint and then move on. Maybe have a visual print out ready to go.

6. These Are The Most Likely Mistakes
List them out and also talk about how you will address them.

7. The Extension Question(s)
We have to have somewhere for the top students to get the question to ensure everyone is challenged. Be specific about what the extension is and how you are going to pose it. Be very careful of anything that sounds or feels like busy work.

On the back of the paper I would reflect on the lesson as a whole.  Here are the two I filled out for my 1st day of classes this year:

Need an interesting way to teach circle properties?  Me too!  And as a bonus this lesson also has the Pythagorean Theorem and extension questions that provide some advanced geometry for those students who are hard to challenge.

And don’t get intimated by satellites – just go here and read up.

The Hook (aka: Act 1)

A stunning time-lapse over the Swiss Alps.

[vimeo 61451883 w=500 h=281]

Chalandamarz of geostationary satellites in the nightsky over Diavolezza from Roland Stalder on Vimeo.

See anything interesting?

Spoiler Alert!  Some of the stars are not moving.  Students will pick up on that and it will cause a sense of intrigue.  I’ll bet you a beer that it will.  The reason the stars aren’t moving is because they are in a very special orbit called “geostationary orbit” so they are matching the earths rotation exactly.

Me:  “Any explanation about what these stars that don’t move are?”

Students give some opinions about what is happening here.  Address them and discuss them.  No fear – spend some time googling and you will be able to talk intelligently about that.

Me:  “What else do we notice about these stars that don’t move?”

The key here is that not only do some stars not move, but the ones that don’t move also appear to be in a straight line rather than randomly dispersed about the night sky.    Now let’s make the point a bit more dramatically with another video – this video gives away the answer about what these crazy unmoving stars are in the title:

[vimeo 39536582 w=500 h=281]

Geostationary satellites in the Swiss Alps from Michael Kunze on Vimeo.

The goal of this intro is to pull the students in and get them interested in this amazing, kind of creepy fact, that some of those stars in the night sky are actually satellites looking down on us.  Now I get into the slide show and talk about satellites.  I can give you that if you want.

Take a look at this cool image of the earth as a beehive of satellites.  That outer ring is geostationary orbit.  It’s literally a ring that is 22,336 mi above the earth.  Safe to say retail space up there is limited and in high demand.

Oh what the hell – let’s just crack up our speakers and really draw the students in by showing them the launch of a geostationary satellite.

http://youtu.be/Gz40PihX690

OK, table set.  Here’s the two questions we’re serving up:

1.  What percentage of the earth can a single satallite cover?

2.  How many satellites are needed to cover the entire earth?  By the way – a system of satellites is called a satellite constellation.

I’m also going to have them make a scale drawing of their satellite constellation and write a generate equation for the amount of earth a satellite can cover based on it’s height h above ground.

The Data: (aka: Act 2)

I made a handout which a bunch of satellite information on it that I gives students.  It has more than they need, so it forces them to filter out the unhelpful things.

geostationary orbit – 22,336 mi

earth’s diameter – 7920 mi

This Diagram Is From American Academy of Arts & Sciences

Perhaps more specifically here – we are figuring out how many communication satellites in geostationary orbit we will need to provide communications to the entire earth.   The complication here is that communication satellites are line of sight with each other, and with people on earth.  Meaning you have to be able to see the satellites to use them – and they need to be able to see each other to transfer the signal.

The Answer:  Act 3

The math says 3.

Thanks for the nice image Purdue

But Lockheed actually uses 6!  Don’t take my word for it – check the last paragraph here from spacedaily.com:

So why do they send 6 satellites when they only need 3?  Great question!  The 4th satellite is needed for “full capacity” because due to the earths shape (stupid non-perfect sphere) there a small portion of it that needs the 4th, but 3 covers most of the earth. The 5th and 6th are to “add capacity”, which is engineering speak for increasing the amount of computing power the satellite constellation is capable of providing.

 The Extensions:

This problem can be pretty much get as difficult as you want it to be.  Here are some ideas:

1.  Line of sight is actually not sufficient to ensure communication with satellites.  The elevation angle from the observer to the satellite must also be greater than 10 degrees.  How many satellites will we need to cover the earth given this restriction?  For more detailed information on this question, go to page 19 of this document from the American Academy of Sciences.

These Diagrams Are From American Academy of Arts & Sciences

2.  Can the constellation ever cover 100% of the earth?

Umm, nope.  Geostationary satellites have to be located somewhere in a ring directly above the equator.  If students derive the general formula for the amount of coverage for a given height it will look like this:  

No matter how big h gets, cosine will tend to 1, but it will never be exactly 1 and thus we can never hit 100%.  The north pole can never be reached by a satellite in geo orbit.

3 . What is the worst case communication delay between someone on one side of the earth and someone on the other side?

4.  Write a general expression for the percentage of the earth that a satellite can cover based on it’s height.

The Goods 

Here is the worksheet I used – I don’t necessary like how it’s formatted, but here it is:

GeostationarySatellites