Conversations that get me thinking very deeply about math = A++ way to return from a luxuriously long winter break (spent being sick. Boo sinus infections!)

So. My precalculus team was discussing trig identities today – pacing our unit and then just sharing bits about how we’ve experienced teaching this topic in the past. A colleague brings up the piece about working both sides independently, vs. manipulating the entire equation. He shares an idea I hadn’t considered before, neatly captured here as part of a larger discussion on the issue (click to read entire thread)

I sat at lunch staring into space like a dummy, trying to more clearly parse the discomfort I’m still feeling with this approach. I think the beginnings of my difficulty lie here:

So one place I’m feeling the discomfort is: students could start with a false statement (something that is not an identity) and operate on both sides to produce a true statement. Worrisome, but hopefully they could realize that they couldn’t work backwards, starting with true statement, to produce the false statement. Maybe I’m worried they’ll get so excited about true statement they’ll forget to really justify the proof they’ve GOT by showing it “backwards.” It just leaves me uneasy.

Second place I’m feeling the discomfort: are there true statements – identities – where it’s possible to operate on both sides to get two identical expressions… but then in trying to write the “work backwards” part, run into trouble? I don’t know how to more clearly ask this question. I should probably think about this more when I’ve had some sleep.

Really, you should all just read all of: this

Really, I should appreciate the return to school. These teenagers are endlessly energizing. Their insights surprise me. Their struggles challenge me to adjust my approach.

Really, I should appreciate that I have colleagues who want to dig in so deeply on these ideas. I crave more! More conversation! More chances for me to fail in articulating my questions! More chances for me to refine my thinking about how to engage with students who ask “why can’t I…?”!

Really, I should be sleeping. Happy Friday! Happy weekend! Happy start of 2016.

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I just read the linked stuff. Here are my thoughts:

The question is “Does expression A have the same value as expression B for all values of the argument?”.

There is no need to make any assumptions about equality or truth. All that is needed is to find a sequence of reversible operations which when applied to both A and B yield the same expression. Apart from avoiding “multiplying by zero” the main source of problems is “squaring”.

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howardat58on 01.09.2016at 7:07 am

This one is not about this post but about your blog subtitle. You might find my 3 posts on math and music interesting. Here’s the first, the rest follow:

https://howardat58.wordpress.com/2015/07/31/the-mathematics-of-harmonic-means-and-the-beat-frequency/

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howardat58on 01.09.2016at 8:56 pm