Tangential

People have told me that it's often better to answer student tangential questions than to continue the planned lesson, because their questions are what interest them, so they're more likely to remain engaged if done correctly.

The lesson today was based on something from the CPM Core Connections Algebra 2 text, related to domain and range of functions. The lesson involves exploration of a few functions, realizing how to adjust the graphing window in a handheld calculator, locating points of interest (such as intercepts and local extrema), and noting the relationship between the window settings and domain and range.

The lead-in referred back to a few functions from previous lessons. One involved the negation of a squared expression, and another involved the square root of an expression. Some students noted that the negation of a squared expression would always result in a non-positive value. Other students asked why. Boom! Tangent. Let's go.

One thing I still struggle with is answering questions too readily, rather than guiding students to answering their own questions. Fortunately in this case, a student took over before I could ruin things. He explained to his mates that the square of an expression is always positive. (I did have to butt in and get them to allow a 0, amending the statement to be never negative.) The student then continued that line of logic to say that the negation of non-negative value must be a non-positive. Hooray!

I started moving things back to my plans when another student questioned the square root function. Why can't we ever find the square root of a negative. I think I was warmed up by the first tangent, so I was able to allow students, who felt they understood, to explain. But their explanations fell short (in my estimation) amounting to either hand-waving, or false statements. So I began guiding as follows.

What is the definition of square root?
(silence)
I wrote on the board a=sqrt(b) (except I used the radical symbol, which I don't know how to product here in a blog).
Students struggled with this, and eventually got to saying that it means that a*a=b. So I continued to first line to include the implication.
Then I moved to concrete. Underneath the first line I wrote 2=sqrt(4) --> and asked what that implied.
Students complied by saying 2*2=4. Exactly the setup I wanted.
Underneath I wrote a=sqrt(-1) --> a*a=-1, and asked what value of a could make that a true statement.
Students struggled. Some offered some values for a. We tested, and didn't get -1 after multiplication, or I pointed out that they were giving me two different values for a at the same time. But shortly, I saw the realization and acceptance -- there isn't any (real) number that could make it true.

I think this class really understands these two ideas now, rather than simply memorizing them as facts.