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The methods I will consider are:
- Guess and Check
- Slide and Divide (a.k.a. "Bottoms up")
- Splitting the b (a.k.a. "Grouping")
- Area Model (a.k.a. "X and Box")
Before we jump in, I want to emphasize the importance of students knowing their multiplication tables. When teaching factoring we often ask students, "What are two numbers that multiply to be 48 and add or subtract to be 2?" Students that don't know their multiplication facts, will struggle to answer this question and will therefore struggle with factoring. For that reason, reviewing and reinforcing their multiplication tables is crucial if students are having difficulty with factoring.
Guess and Check
How does it work?
What's good about this method?
In my opinion, this method is preferred because it reinforces the concept of multiplying binomials and helps students understand that they're "undoing" the multiplying process (I still call it "F.O.I.L." even though some math teachers frown upon the "F.O.I.L." acronym).
This is the method that I learned back in the 90s when I was in high school, and I still find it useful.
What's wrong with this method?
One issue with this method is that students must have a high tolerance for frustration and fairly good number sense. In the example above, there are a handful of ways to get a product of $6x^2$ and likewise for $8$. Students often give up if they can't get the answer after two or three tries.
Another issue is that it can be time-consuming. If you were trying to factor $12x^2-19x-18$ through guess and check, you must consider all factors of 12 and all factors of 18. Checking all of these options can take several minutes.
What if a student forgets to take out the GCF first?
Fortunately, if students forget to take out the Greatest Common Factor first, they can take it out at the end. Try it with $6x^2+9x-6$!
Slide and Divide (a.k.a. "Bottoms up")
The Slide and Divide method has gained popularity in recent years, though it has its fair share of critics. Many people 🙋♀️ have taught this method but called it by a different name.
I first learned this method from a fellow teacher in the late 90s and taught it for years. However, my opinion has since changed, and I no longer advocate for teaching this method. My preferred method is "splitting the b."
How does it work?
In the example above, I started by multiplying the "a" value by the "c" value and rewriting the problem.
Then I factored "like normal" - looking for two numbers whose product is -4 and whose sum or difference is 3. Then, since I multiplied by the "a" value (2 in this case), I now go back and divide by the "a" value. The resulting fractions must be reduced. If they cannot be reduced or a denominator remains after reducing, move the denominator to the front of the x in the factor.
What's good about this method?
When using this method correctly (and taking out the GCF first), it will always lead to the correct answer.
What's wrong with this method?
It's more of a trick than a mathematically correct process. It requires no deep understanding of what the students are actually doing, and explaining why the method works is not easy.
Students often forget a crucial step in this method—the "divide" step. This is one reason why I stopped teaching it. If students forget the last few steps, it's not an effective method. For example, in my work above, students often stopped when they reached $(x+4)(x-1)$.
If the trinomial has a GCF and students forget to take it out first, they will have the wrong answer if they use this method. More on that below.
Another issue arises when students learn to solve quadratic equations by factoring. Some realize they don't need to fully factor before arriving at the answer. They end up saying that $2x^2+3x-2$ is equivalent to $(x+2)(x-1/2)$, which it is not (see image below). As a result, I penalize them for failing to follow the proper process.
In the above image, I would penalize the student for solving when the question asked them to factor. If the student had left their answer as $(x+2)(x-1/2)$, I would penalize them because multiplying those factors does *not* result in $2x^2+3x-2$.
What if a student forgets to take out the GCF first?
This is the second reason why I have stopped teaching this method. If students forget to take out the GCF at the beginning, they will likely end up with the wrong answer. See below:
Splitting the b (a.k.a. Grouping)
What's good about this method?
This method has fewer steps than Slide and Divide and the Area Model methods.
When using this method correctly, it will always lead to the correct answer. If students forget to take out the GCF at the beginning, they can take it out at the end. More on that below.
If you decide to teach this method, it's best to teach Factoring by Grouping first.
What's wrong with this method?
It's still a *method* and requires students to remember the steps of the method.
What if a student forgets to take out the GCF first?
Fortunately, if students forget to take out the GCF first, they can take it out at the end. See below:
Area Model (a.k.a. X and Box)
I used this method almost exclusively during the 2022-2023 school year with my Algebra 2 classes. I kept calling it the X Box method, and it wasn't until May that I realized I was saying the name of a gaming system! Hahaha!
The idea behind this is based on the area of a rectangle being length times width. When factoring, the trinomial represents the area, and students must find the two factors (length and width) that were multiplied to get the area/trinomial.
What's good about this method?
If you have taught the students how to multiply polynomials using the Area Model, this method is a smooth transition into factoring. Students will hopefully see the connection between multiplying and factoring.
Teaching this method also paves the way for dividing polynomials using the Area Model, which I successfully taught in lieu of the Long Division algorithm during the 2022-2023 school year.
What's wrong with this method?
It's still a *method* and requires students to remember the steps of the method. In 2022-2023, I found that when we had not done factoring for some time, students had forgotten how to set up the box.
I also noticed that some students became too attached to the Area Model and used it for factoring easier trinomials such as $x^2+12x+35$. I tried to discourage them since it's more time-consuming, but not all of them listened.
What if a student forgets to take out the GCF first?
Students must take out the GCF before jumping into factoring to avoid issues. The issue occurs if they blindly take out the GCF and forget to check the multiplication. They can avoid this issue by ensuring that the Area Model works when multiplying. See below:
Conclusion
My personal preferred method is Guess and Check, but for my students, I will be teaching the "Splitting the b" method in the future.
I prefer splitting the b to the other methods because there are fewer steps to remember and if a student forgets to take out the GCF at the beginning, they can take it out at the end.
However, as the teacher, you know your students best. Teach the method that is most appropriate for your students. If you have students who struggle with number sense, use a method they can handle effectively. The Guess and Check method might be a good fit for honors-level students. Just be cautious about introducing too many methods. Stick with one or two primary methods and then repeatedly reinforce the process.
Before you go, check out this fun fact:
When making a list of the factors of a number, start with 1 and itself, then proceed through the first few natural numbers. When the list "wraps around," you'll know you've exhausted all the possible factors.
For example, take the number 48.
First, start with 1 and 48, then 2 and 24, then 3 and 16, and 4 and 12. If 5 doesn't divide into 48, we move on to 6. Bingo! 6 times 8 equals 48.
There's no need to check if 7, 9, 10, or 11 divide into 48 since they would have appeared on the list with a smaller number if they were factors.
That's it for today! Happy factoring!
Need some factoring resources?
Looking to read more?
Check out this article from Math on Our Minds: Thinking Tasks: Factoring Quadratics.
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