A Teacher's Guide to Teaching Factoring (Part 1): Defeat the Dread & Ditch the Gimmicks
In my first few years of teaching Algebra 1 and 2, there were few things that I dreaded more than -THAT- time of year. It quickly approached and loomed on my curriculum map like a dark cloud: FACTORING QUADRATIC EXPRESSIONS.
Why did I dread it?
😫 Factoring quadratic expressions is a complex process! There's a wide variety of special "shortcuts", and -believe me!- I've tried them all!
Here's a quick look at a few of the "shortcuts" that I personally tried in my classroom:
👉 THE X METHOD: In this method, students drew a large X on their paper. In the top and bottom "elbows" of the X, they wrote the value of the product of a and c, as well as the value of b (coefficients from the quadratic expression form ax² + bx + c). The left and right "elbows" of the X are meant for the factors that both MULTIPLY to the a×c and ADD to the b.Even typing how "The X" process works has me out of breath and likely has you confused! 😅 Here's a visual and example of this process:
✅PROS TO THIS METHOD:
✔ Quick method for students with strong number sense
✔ Didn't take up much space on student paper
🆇 CONS TO THIS METHOD:
✘ Does not work when a ≠ 1
✘ Students often forgot to consider the sign (+, -) of the a, b, and c when using this method
✘ Many students struggled to master this method
✘ The steps are too complex for students to retain throughout the year
✘ Students didn't really understand what factoring IS
👉 THE BOX METHOD: I doubt that you will be shocked when I tell you that this method involves drawing a BOX, or rectangle, divided into 4 equal spaces. Within these spaces, students would record the ax², the c, AND calculate and record the factors that MULTIPLY to the a×c and ADD to the b. At this point, this method is already like THE X method above, version 2.0.
Once the box was complete, students would identify the common factors horizontally across each row & vertically across each column.
Here's a look at the Box Method & an example of this process:
✅PROS TO THIS METHOD:
✔ Works for quadratic expressions of any a value
🆇 CONS TO THIS METHOD:
✘ Takes up a lot of space on student paper
✘ Takes students a lot of time to complete
✘ Students often forgot to consider the sign (+, -) of the a, b, and c when using this method
✘ In more advanced factoring problems, students often missed a Greatest Common Factor that could be factored from the expression
✘ The steps are too complex for students to retain throughout the year
✘ Students didn't really understand what factoring IS
👉 THE AIRPLANE METHOD: By the name of this method, you've probably already guessed that this method involves drawing an AIRPLANE...😏 JUST KIDDING! Instead of drawing a literal picture of an airplane as the name suggests, this method is taught via a narrative of "building" and "landing" an airplane and the *very loose* interpretation of the result looking like an airplane.
Once again, this method uses THE X method above (is this version 3.0?) to identify the two factors that MULTIPLY to the a×c and ADD to the b.
Once these factors were identified, students would write the expression (AKA, build their airplane):
(ax + Factor #1)(ax + Factor #2)
To "land the airplane", students must analyze each binomial of the "airplane" & divide all terms in each binomial by its greatest common factor. If a circle is drawn around the GCF of each of these binomials, they look somewhat like "wheels" of the "airplane."
Here's a look at the Airplane Method & an example of this process:
✅PROS TO THIS METHOD:
✔ Works for quadratic expressions of any a value
🆇 CONS TO THIS METHOD:
✘ In more advanced factoring problems, students often missed a Greatest Common Factor that could be factored from the expression
✘ The steps are too complex for students to retain throughout the year
✘ Students didn't really understand what factoring IS
Although these shortcuts worked in the short term, I found that I was reteaching these strategies each time factoring quadratic expressions was required in future units. My students just weren't retaining these skills because they didn't really understand what factoring is & how these shortcuts helped them identify the correct factored expression.
They just didn't GET IT, even though they thought that they did! I mean...these shortcuts gave them an answer, right? What more is there to understand? 😏
😩My students dreaded factoring, too! Since students didn't really understand what factoring IS, they had a level of discomfort & a lack of confidence when factoring quadratic expressions.
If I had randomly chosen one of my students all of those years ago and asked them to explain to me WHAT factoring IS, I might hear:
✘ It's the process of finding two factors that MULTIPLY to the a×c and ADD to the b. ✘
👉 Of course it makes sense that this would be their answer! It was a majority of what I had been teaching them! This, of course, was the part of the "shortcut" process that they were most actively using their brains. But this in itself is NOT factoring!
✘ It's a shortcut to rewriting a quadratic expression. ✘
👉 So close, but students were missing WHY it is mathematically sound. There is no "magic" in math. They didn't understand the connection between factoring and another mathematic process they had known & understood for quite some time: distribution!
✘ I DON'T KNOW! ✘
👉 This is the true problem: they didn't know. I had never built the bridge between what factoring really IS and WHY it helps us solve future problems like solving quadratic equations. I may have mentioned a time or two that students could check their answers by using distribution - but they didn't understand WHY.
By neglecting to make these connections for students, I failed to show them the value of factoring & to build their understanding of what factoring IS.
😩 I felt like I didn't have enough time or enough resources to build student understanding of factoring.
👉 Many can relate to this: I felt like I just didn't have the time to give my students the factoring practice time that they needed! With a full curriculum and limited time, I felt that I just couldn't spend the necessary time to truly BUILD student understanding of factoring.
My students needed to marinate on factoring. They needed to soak it in! They needed to chew on it for a bit until it tasted right! They needed opportunities to try & fail, to write down & erase, to use distribution to check their answer & try again if it wasn't right.
There were no shortcuts for my students if they were to MASTER factoring quadratic expressions.
👉 I honestly felt like I didn't have the resources I needed to make scaffolded instruction & practice for my students.
I'll start by saying that I've spent hours... ⏲AND I MEAN HOURS⏲...searching for (or pulling out of my own brain) enough quadratic expressions that are skill level appropriate for my students, that expose students to all special cases of factoring quadratic expressions, & that contain enough variation to keep students interested.
Additionally, most resources I had access to suggested introducing factoring quadratic expressions where a = 1. But I found that introducing factoring with these expressions exclusively caused issues in more advanced problems as students were often careless about situations when a ≠ 1. I thought: Is there a better way to structure the introduction of varying quadratic expressions so students could appropriately build their skills along the way?
In reflection, I decided 2 things:
1) I needed to schedule the time that my students needed to really understand factoring.
2) I needed to find a better way to structure the quadratic expressions I introduced to my students as they built their understanding of factoring.
In Part 2 of this blog series, I'll share more about how I structure my factoring instruction now & offer some advice about building student understanding of factoring quadratic expressions!
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