Factoring Quadratics REI 4b

[lhwalker]

To avoid mile wide factoring, I’m thinking CCSS intends for us to narrow down the toolbox but I need to know what to cut.  I taught factoring monomials from bi, then grouping, then turning trinomials into four factors for grouping and relating special cases to grouping, all the while motivating toward guess and check for shortcuts.  The CCSS do not seem to include grouping which wipes out the rest of my sequence.  Can someone please clarify what factoring tools will be used, particularly if don’t factor by grouping? Is there a connected sequence?

 

[Bill McCallum]

I would say that CCSS is not so much narrowing down the toolbox, as encouraging students to see that all the different tools work on the same principle, so that you don’t have to remember so many different tools. Your progression is fine, and can be understood as progressively more sophisticated applications of the distributive property. I don’t know exactly what you mean when you say that the “CCSS does not seem to include grouping.” If I understand correctly what you mean by grouping, it is an instance of the distributive property, which is fundamental from early grades (even if it does not go by that name).

[lhwalker]

I am still not happy with my “factor by grouping progression” because it ends in a trick.

Our curriculum currently requires “factor cubics by grouping,” so my progression has started off with a cubic like this: x^3+ x^2 -3x -3 which we factor by grouping :
(x3+ x2 )+(-9x -9)
x2 (x + 1) -9(x+1)
(x2 -9)(x+1)
(x-3)(x+3)(x+1)

The next step in my factoring progression has been to break up the middle term of a quadratic (with ANY integer coefficients)…and then factor by grouping:
4x^2 -11x – 3
4x^2 +x – 12x -3
x(4x + 1) -3(4x+1)
(x-3)(4x+1)

This method eliminates student frustration from guessing and checking, but in order to break up the middle term, the students use the AC trick, “What two numbers multiplied together give you AC and add to get B?” This Khan Academy video shows more examples. At the end of the video, he explains why the trick works, but of course 99.9% of the students will zone out during the explanation. https://www.khanacademy.org/math/algebra/multiplying-factoring-expression/factoring-by-grouping/v/factor-by-grouping-and-factoring-completely

After further study of the standards, I see no reason why we would have to teach factoring a cubic by grouping because we could use graphing calculators to find integer zeros for polynomials of degree greater than 2 and divide out factors from there. If that is true, I will encourage my district to remove the requirement for “factoring by grouping cubics” and focus on trial and error for simple quadratics and completing the square for more complicated cases. Am I correct that there is no need to teach factoring a cubic by grouping or did I overlook something?

[Cathy Kessel]

The Khan video seems to pull a rabbit out of a hat. To factor a quadratic, it begins (without explaining why) by asking students to find pairs of numbers A, B such that A + B is the coefficient of the linear term and A x B is equal to the product of the other two coefficients. This still involves guessing and checking, so I don’t understand Lane’s remark about eliminating it. (Maybe the frustration is meant to be eliminated by the choice of quadratics?)

It seems to me that working backward eliminates the rabbit (and the need to memorize the trick). If you have a quadratic 4x^2 + 25x – 21 (as in the Khan video), then if the quadratic could be factored, it could be written as (ax + b)(cx + d). Working backward, a x c = 4, so the possible values of a and c are (in some order) 4 and 1 or 2 and 2.

So, if it can be factored (using whole-number coefficients), 4x^2 + 25x – 21 = (4x + b)(x + d) or 4x^2 + 25x – 21 = (2x + b)(2x + d).

The second possibility can be ruled out because (2x + b)(2x + d) = 4x^2 + 2bx + 2dx + bd which would make 2(b + d) = 25.

Similarly, b x d = -21, so possible values are (in some order) -1 and 21, -7 and 3, 1 and -21, 7 and -3.

Trying them out in (4x + b)(x + d):

It seems unlikely that 4 x 21 or 4 x -21 would be involved, so trying other possibilities first.

(4x + 21)(x – 1) = . . . 21x – 4x . . . No.

(4x + 7)(x – 3) = . . . 7x – 12x . . . No. And switching the negative sign to the 7 won’t help.

(4x + 3)(x – 7) = . . . 3x – 28x . . . No, but switching the negative sign to the 3 would work.

Factorization is (4x – 3)(x + 7).

This example seems so long-winded that the idea of working backward might get lost in the mass of details. The same ideas about working backward to factor might be illustrated with 2x^2 – x – 1.

Such an example might serve as fodder for discussion of when one might want to use the quadratic formula instead (“Solve quadratic equations by inspection (e.g., for x^2 = 49), taking square roots . . . as appropriate to the original form of the equation” A-REI.4b). Or, it might point out how completing the square is a form of working backward that doesn’t involve checking several possibilities.

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