#MTBoS30 (Maybe) Don't Go it Alone

I have been gathering evidence for my fantasy "Don't go it alone," series. We all work so hard, it is crazy that we feel so compelled to do so much ON OUR OWN. I subbed for a colleague teaching Algebra 1. She is so dang creative! Her class was working on graphing quadratics in factored form.
Unfortunately, working solo, she missed an entire page of graphing written as quadratic  = 0 (instead of y = or f(x) =)

I have done this too many times myself. Would this teacher have caught her mistake if I did not happen to cover for her? Sometimes we find 'em, sometimes we don't. With 15 sections of Algebra taught at our HS, the going it alone is too common place. I believe this because I am guilty of going it alone and I know the benefits of collaboration.

Mr. H. My BTSA Participating 1st year teacher is a systematic guy. He cares so much that the students get what they need step by step. He is careful not to overwhelm them with too much information.
Here is his pool problem for our unit on solids:

My Brain exploded with extensions: Will there be 1/2 the volume if the diameter is cut in half? Are there other dimensions that use the same amount of water? What is the largest pool you can have that amount of water? 

Yeah, I know that might not have been the intention in this lesson...but I loved the premise and wanted Mr. H. to give the students something to juicy to discuss and share. 

I had all these cool ideas for exploring circles in Geometry. Circles are a great bridge topic between Geometry and Algebra.
I have lots of good ideas, I am just not the most organized person in the world. Lucky for me, I have colleagues that are. Here was my lesson:
Me: Multiply  (x+1)^2
                      (x+12)^2      Stop when you find the pattern and just write the product.

Me: What did you notice?
       How would it change if all the sums were differences?
Me: Can we work backwards? Can we find a magic number that makes a perfect square trinomial, etc...
Me: How are these equations related to the circles we have been graphing?
Me: Can you find the center and radius of this circle in standard form? x^2 + 8x +y^2 = 9?

My students were happy enough and there were "oh, I get its." We got into fractions, and even radii whose lengths were square roots.

My lovely colleague asked me how I was approaching complete the square for circles. And I shared the above. Here is how she interpreted my lesson:

To see her entire development get the file here: file

I just love how she saw the whole thing. How she was able to put the lesson into a form I could not.

I have many things I want to do with my students that I am not sure how to structure, so I shy away. The more I share with colleagues, the more we create rich and inviting experiences for the students.

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