Algebra 2 and Trig this year are a little tricky in NY State. How do you teach a rich course, but also prepare kids for a state test? If you’ve never seen the state test? If you suspect the test will cover way too many topics, and do so poorly? And did I say, also teach a rich course?

So, this teacher squeezes in extras. Go off topic, and onto other topics of value. Challenge questions that preview what they’ll see in a few weeks. And better, challenge questions that help their analytical skills, without practicing current topics (math class is for thinking and doing and performing!). A little extra graphing to tie topics together (more on that, later. Who’s going to teach me to take screen shots from the TI?)

And some days, I just squeeze so I have time for something else. And yesterday’s lesson, how to handle $a + n\sqrt{b}$, not a killer for squeezing. I threw what could have been a quickie worksheet on the board:

Model A:
Add: $(4 - 5\sqrt{10}) + (7 + \sqrt{10})$
$11 - 4\sqrt{10}$

Model B:
Multiply and simplify: $(4 + 3\sqrt{7}) (5 + 2\sqrt{7})$
$(4(5 + 2\sqrt{7}) + 3\sqrt{7}(5 + 2\sqrt{7})$
$20 + 8\sqrt{7} + 15\sqrt{7}+6(7)$
$62 + 23\sqrt{7}$

Try:
A2. $(1 + \sqrt{8}) + (5 - \sqrt{2})$
A2. $11 - (11 + \sqrt{10})$

B1.  $(8 - \sqrt{6})(5 + \sqrt{6})$
B2.  $(3 + \sqrt{10})^2$
B3.  $(3 + \sqrt{5})(3 - 3\sqrt{5})$
B4.  $(5 - \sqrt{2})(5 + \sqrt{2})$
B5.  $(6 + \sqrt{5})(2 - \sqrt{20})$

So they’re buzzing. I’m checking homework and walking around a bit. And I decide to use a question I’ve played with a little:

“What’s a question that someone else might get wrong?” followed by “Explain.” And then I got another kid to comment, or to answer the question, or to explain how to avoid the error. And back for another “question that someone else might get wrong?”

I ran, back to back, two of the best five minute discussions I have had all year.