A-REI.4 factoring Quadratics

Solving versus factoring

What does the student have right, and what does the student have wrong? How would you help?

6 replies on “Solving versus factoring”

The student seems to have the right idea, but got stuck once they factored the equation. It seems they just weren’t sure how to finish the problem.

I would encourage the student to set both expressions equal to 0 and solve. So they will have (a – 5) = 0 and (a + 3) = 0. They factored the original equation correctly, so they probably won’t have much trouble solving for ‘a’ now.

I think encouraging them to set both expressions equal to 0 is a recipe for getting correct answers without necessarily understanding what they’re doing.

I wonder what they would do with x^2 – 2x – 15 = 9. Would their attempt still be (x-5)(x+3)? I certainly have seen plenty of students who have the “see a quadratic, try to factor it” reflex without any idea of why factoring might be good. It’s like those kids who take (x+2)^2 = 9 and their first step is to square it out, then subtract 9, then try to factor …

My first questions to the student would be something like “What is your goal here — what would your final answer look like if you could get there?” and if they understand that what they want is to find the values of x that make the equation true, then “How can factoring like this help you get that answer?”, which might well lead to the x^2 – 2x – 15 = 9 digression as a way of seeing if they understand that 0 has some special properties that are really useful here.

I like “What is your goal?” You are asking what does “solve” mean in a way that the student can self-rescue. Generally, a habit of always writing expressions underneath each other would allow the student to check that s/he has not accidentally mislaid the equality statement (or “fence” as we call it).

This student knows how to factor quadratic polynomials (awesome!!), but I would guess he doesn’t really know what “solve” means. I think this goes back to some important vocabulary distinctions: solve vs simplify. These words are often misunderstood/misused by students, and it takes some diligent instruction + practice to sort it out.

I ran into this problem/mistake a lot last year. I think I focused SO much on the factoring part (because that is the newer + more difficult skill) that students were not very successful in consistently SOLVING for the variable.

I’d like to try a chunking strategy: do 3 examples of only the factoring step together. Then, using the same examples, finish all three problems by splitting + solving for the variable. The idea is that students see the same steps of a process all at once, before they put all of the steps together to factor and then solve.

The student knows how to factor. Yaaay.

As a teacher, I would guess that this student is so used to procedural knowledge that s/he’s gotten to the point of saying “I see a quadratic, it’s factorable, I factor, I’m done.” With no other information, my first instinct is that this student is so dependent on procedural/algorithms that the meaning behind most of what s/he has been learning in math is missing. (For example, what’s an equation versus an expression? Why does the zero-product property make sense? What does the solution to an equation actually mean?)

(Of course the kid could also have been super tired, and that would explain it too.)

Josh made me think about perhaps having students solve x^2 – 2x – 15 = 9, and actually find out when does it equal 9! Why does it only have to be zero? Having them factor the left side, then finding two factors whose product is 9, would be a great task for students to explore…

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