This is my 3rd post in the series on the #MTBoS blogging initiative:
When teaching quadratics in Accelerated Algebra 1, I teach solving by all the methods. Sometimes they ask, why so many methods? I try to show them the advantage of one method over the other. I want them to analyze the quadratic before they just jump in and try it. I would say they are 50-50. 50% will try factoring and 50% will try quadratic formula. However, over the past few years, some kids are turning to completing the square as their first mode.
On my quiz for this unit, I have a favorite question. I give them three quadratic equations and I ask them to solve them using factoring, completing the square, and the quadratic formula. They must use each method once and explain why they chose that method.
The first quadratic equation is a factorable quadratic with a = 1. It is the only one that is factorable. Kids will mess up by not looking at all of the problems first before deciding on a solving method. They will do the first with the quadratic formula because they like it. Then, they try one of the others and it is not factorable and they write - no solution or not factorable. Okay, well, if it isn't factorable, then use another method to actually solve it!
By this quiz, we have also learned complex numbers. The second equation has "a" not equal to 1. I would choose to do this one with the quadratic formula because I think I am more likely to make a mistake doing completing the square when "a" is not 1. The last equation is not factorable, but "a" is equal to 1, so do completing the square here. I try to give the kids a hint and have the constant on the other side of the equation. I do accept either method on the last two as long as they can explain their reasoning. The second one is a complex solution.
They are allowed to use their calculator on this part of the quiz, so if they really made the connections, they could check these by graphing it on their calculator, but I don't think they do that yet.