It may have been G. H. Hardy who first stated that mathematics is about finding patterns. In his autobiography he writes, "A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas." It is an idea I often repeat to my students, and to myself when I am trying to present new ideas to them.
We are at that point in the year where we are working with the Fundamental Theorem of Algebra. We have gotten to the point that most of the students can graph an equation, y=f(x) and looking at the graph write the factored form if the roots are all rational. In the vernacular of the kids, the can see the roots. As we began working with complex numbers, one student lamented that they wished you could see the roots as well on a quadratic with non-real solutions. Several others murmured agreement, and it seemed like a teachable moment, so I turned and said, cryptically I hope, "Well, you can, but you have to recognize the pattern." It is hard to believe they thought that there was something simpler than the quadratic formula, but they are more willing to do ten minutes work with a calculator than two minutes with a pencil, and besides, it is pretty, so I was willing to lead them toward it.
I began by setting up some simple problems. graph the equation and find the vertex, then solve by the quadratic formula (never tell them they are actually practicing, we are discovering exciting, not in the book, kind of stuff here).
They do a few:
x2 -2x + 10, with a vertex at (1, 9) had solutions at x= 1 +/- 3i
x2 +4x + 5, with a vertex at (-2,1) had solutions at x= -2 +/- 1i
x2 -6x + 12, with a vertex at (3,3) had solutions at x= 3 +/- isqrt(3)
AHHH, now they had a clue.....
We do a few more, and they seem to be on top of it, but not one has noticed that ALL the problems had a leading coefficient of one... with only a few minutes to go, I gave them
2x2 -4x + 20 (no one notices, it seems, that this is twice the first problem we had done) ... they sketch the graph, trace to x=1 to find the vertex at (1, 18) and hands fly up, eager lips whisper to each other, 3 +/- i sqrt(18), and nods in return assure them they are right... so the teacher springs his trap... pointing, a student responds with certainty... and the teacher tilts his head, and gives him "the eye".... "Did you check?"
A timid young lady offers, "I got an answer of 1 +/- 3i." Heads lean toward each other. What happened. I was sure we had it... and then, an offering of insight... "It's half as much, I mean, its the square root of half of the y part...because of the two in front." But it is almost a question. The bell rings, and no one moves... "Is that it? Tell us."
He turns to the board to hide his smile as he erases.... painfully slowly in their mind..then turns back.... and shrugs... then offers... "Hey, have a great weekend."