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 Month – Factoring 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))/2a, x2=(-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, please don’t teach complex numbers in the same 2 month span as you teach graphing of quadratics.
[...] Here is an interesting math blog: http://www.mathcasts.org/janita/?p=43 [...]
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Pingback by Baddie of the Month - Teaching complex numbers with the quadratic formula « SL2RL-Math247 — 8 March 2009 @ 12:14 am
A. HALIM…
Although your article about math graphs sounds interesting but i’m not sure if i could agree with you in 100%….
Trackback by A. HALIM — 19 March 2009 @ 8:21 pm
Formula for 13x^2 + 14x + 1 ? No. Tsk tsk.
If it takes less than a minute to check if an expression is factorable, why not? Before running the tedious formula.
Special cases? I have none. One method for all, allows to check if an expression is factorable before doing the work.
Try this: Teaching Factoring – Should we? and I’d be happy to point you towards a discussion of trinomial factoring by breaking the middle.
Comment by Jonathan — 12 April 2009 @ 4:02 pm
If only the problem was “Factor 13x^2 + 14x + 1″. My sample problems are: “Using either Lagrange’s or Newton’s interpolation method, find the polynomial of smallest order that passes through the points (-1,0), (0,1) and (1,28). Graph this function finding all intercepts and extreme values and show that it goes through these points.”. These kids don’t have time to mess around with factoring – which NO ONE ever remembers or uses after Algebra 1 or with completing the square. The quadratic formula is tedious? No way! It is an easy to remember formula that works every time.
Comment by admin — 26 April 2009 @ 10:20 pm