The challenges of chapter 5
The 5th chapter of Glencoe's Algebra 2 is a jumble of related topics, which are all critical to this class. My challenge here is to make the students understand the connections between all the topics, rather than learning them as a rote collection of isolated mechanical symbol manipulations.
And, simultaneously, I need to ensure that my students have the prerequisite skills as we go along. Yes, you would have hoped they already knew how to plot points, square negative numbers, or multiply binomials by FOIL by the time they get to Junior year, but a startling number of them either never "got" the skill, or have forgotten it. I get theatrically exasperated in class whenever a student punches "-5^2" into the calculator and dutifully reports that the square of negative 5 is negative 25.
Graphing quadratic functions, solving quadratics by graphing, solving factoring ... These are all connected. The zeros of a quadratic function (Say, f(x)=x^2-x-6) are the values of x making the function evaluate to zero. (x=-2, and x=3, in this case) These are also the points at which the graph of the function cross the x axis. And furthermore, if you factor the function and get = x^2-x-6 = 0 --> (x+2)(x-3) = 0, you can see that either x = -2 or x = 3 will make either the left or right term evaluate to zero, and make the whole equation true.
How do I make students see the connection between all of these topics? It seems like one good picture could show them ... and then I could close the loop with a sort of "factor-by-graphing" calculator activity where they graph equations, find the zero crossings, and then show that they can reproduce the original original by doing FOIL on the zeros of the function.
And once these are done, we're barely a third of the way into chapter 5. Next come two related topics (completing the square and the quadratic formula), three nominally related topics (the discriminant, parabolas in the form y = a(x-h)^2 + k, and graphing quadratic inequalities), and a topic which seems to come out of the left nullspace. Complex numbers? What are they doing here? Don't we already have enough to deal with in chapter 5?
I think I'll make up a "evergreen" presentation on complex numbers, run off the worksheets in advance, and keep it around in my classroom for a day when I find myself not completely prepared and want to essentially sub in my own class.
And, simultaneously, I need to ensure that my students have the prerequisite skills as we go along. Yes, you would have hoped they already knew how to plot points, square negative numbers, or multiply binomials by FOIL by the time they get to Junior year, but a startling number of them either never "got" the skill, or have forgotten it. I get theatrically exasperated in class whenever a student punches "-5^2" into the calculator and dutifully reports that the square of negative 5 is negative 25.
Graphing quadratic functions, solving quadratics by graphing, solving factoring ... These are all connected. The zeros of a quadratic function (Say, f(x)=x^2-x-6) are the values of x making the function evaluate to zero. (x=-2, and x=3, in this case) These are also the points at which the graph of the function cross the x axis. And furthermore, if you factor the function and get = x^2-x-6 = 0 --> (x+2)(x-3) = 0, you can see that either x = -2 or x = 3 will make either the left or right term evaluate to zero, and make the whole equation true.
How do I make students see the connection between all of these topics? It seems like one good picture could show them ... and then I could close the loop with a sort of "factor-by-graphing" calculator activity where they graph equations, find the zero crossings, and then show that they can reproduce the original original by doing FOIL on the zeros of the function.
And once these are done, we're barely a third of the way into chapter 5. Next come two related topics (completing the square and the quadratic formula), three nominally related topics (the discriminant, parabolas in the form y = a(x-h)^2 + k, and graphing quadratic inequalities), and a topic which seems to come out of the left nullspace. Complex numbers? What are they doing here? Don't we already have enough to deal with in chapter 5?
I think I'll make up a "evergreen" presentation on complex numbers, run off the worksheets in advance, and keep it around in my classroom for a day when I find myself not completely prepared and want to essentially sub in my own class.