**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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[...] Past: January 2009 Baddie of the Month: Hand-factoring a quadratic with a≠1. [...]

Pingback by Baddie of the Month - Teaching “completing the square” for quadratics « SL2RL-Math247 — 8 February 2009 @ 4:13 am

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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