8 February 2009

Baddie of the Month – Teaching “completing the square” for quadratics

Filed under: education,math — Tags: , , , , — admin @ 4:13 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.”

February 2009 Baddie of the MonthTeaching “completing the square” for quadratics

Yes, she is stuck on quadratics. But quadratics are so useful and can be fun and we keep teaching techniques that don’t give much value for the time and effort.

Why? These are reasons I have found online or been given "for" studying “completing the square”:

(1) one can then prove the quadratic formula , (2) one can find the vertex of the quadratic,
(3) one can graph quadratics using graph transforms and (4) one can solve integrals of the form dx/(x^2+bx+c)

My response:

  1. Students at this level cannot understand a mathematical proof. All they see is a manipulation of symbols/letters/numbers. Ask any non-math student. Ask any non-math adult. They didn’t get it. Period. So we teach them a technique in order for them to see a proof they don’t understand. (I am happy if they get accurate results from the quadratic formula with any numbers for a, b and c and can relate them to the graph of the quadratic and/or the answers to the question that was posed and judge the reasonableness of all.)
  2. It is much easier and useful to find the vertex of a quadratic by first realizing that every parabola is symmmetric and thus the vertex must be the value of the function at the half-way point between the roots*. Finding the vertex this way requires them to relate solutions from the quadratic formula to roots/zeros/x-intercepts, reinforces learning about midpoints, relates the vertex to the quadratic formula – more reinforcement  and it teaches them to find function values.  Win-win-win-win. Using "completing the square" teaches them to manipulate numbers.
  3. Does anyone actually graph a function using a graph transform – ever? After trying to teach graph transforms for over 20 years – I have decided that the best I can hope for is "a vague understanding" that "x^2+4" is "x^2 up 4" and 3sinx is "3 times taller than sinx" and cos2x is "2 times faster than cosx". Trying to combine all these is a total waste of time.
  4. Solve integrals – good grief. Maybe we should teach them partial fractions in algebra 1 too.

How to find the vertex: The x-value of the vertex is x= -b/2a. Substitute this value into the quadratic to get the y-value.

(BTW: nowhere did I find the semi-plausible defense that “completing the square” is also used to find the center of a circle and identify conic sections in the 2nd half of Algebra2. By then, I agree that completing the square is not an unreasonable technique to teach.

*roots found using the quadratic formula I am sure :) .

Related topics:

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

Future: Baddie – Teaching complex numbers in the same 2 month span as you teach graphing of quadratics.

Future: Goodie – Good questions for Quadratic Equations/Functions

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