10 January 2009

Baddie of the Month – Factoring a Quadratic with a≠1

Filed under: applets,education,math,sloodle — Tags: , , , , , — admin @ 5:13 am

Goodie: “A technique/question that can be applied in many places and teaches thinking.”

Baddie: “A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

I am going to try to blog a baddie and a goodie per month. We shall see and of course – this is my opinion.

January 2009 Baddie of the MonthFactoring a quadratic with a≠1 “by hand”.

Okay, I can mostly understand learning to factor “by-hand”:   x²+3x+2  or   x²+x-2.  

Once you understand the principles and get the technique (my scheme), factoring a quadratic by hand with a=1 is faster than using the quadratic formula.

But, I absolutely and totally do not understand the reasoning behind other factoring-by-hand techniques!

Why not: Factoring techniques

  • serve no useful purpose – once factored, with a≠1 you must still solve the individual factors.
  • don’t always work – MOST quadratics even with a=1 and real roots CANNOT be factored by hand.
  • are hard to learn, there are many “special cases”, they take alot of time to teach, …

What to do and why: Use the quadratic formula for all your factoring needs.

  • We are going to teach them the quadratic formula anyway.
  • It always works – either we get real roots and can factor or we get non-real roots and know we cannot factor.
  • By using the quadratic formula all of the time, the connection between quadratics, roots, x-intercepts, graphs of quadratics becomes clear.
  • Repetition of a single technique is much more likely to stay in their heads.

Conclusion: Don’t teach factoring by hand except when a=1. Use the quadratic formula.

Here’s how: ax²+bx+c=a(x-x1)(x-x2) where x1=(-b+D))/2ax2=(-b-D))/2a,  D=√(b²-4ac)

(Here “by hand” means looking for the factors without a formula like when you say “The factors of 2 are 1 and 2 and oh yes, they add to 3 (first expression) or the factors of 2 are 1 and 2 and oh yes, they subtract to 1 (second expression)”.)

Related topics to come in future blogs

Please, OMG please don’t teach completing the square – an even worse waste of time than hand factoring.

Please, please don’t teach complex numbers in the same 2 month span as you teach graphing of quadratics.

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