Isn’t summer vacation the best?! Clearly my writing goals didn’t happen so much this past school year, so I’m hoping to use some of my free time this summer to blog and add to my Teachers Pay Teachers account.
Factoring… Do you find that Algebra 1 kids stare at you with blank looks, Algebra 2 kids roll their eyes and groan, and Pre-Calc kids say, “oh yeah, I’ve learned that. Once.”? Upper level teachers are at the mercy of their peers, confronted with choosing from the myriad of methods that students have learned until now.
If you can work with your department to choose a consistent method of factoring (and for that matter, solving equations…), your students will be most successful. But PLEASE. Oh PLEASE. Don’t torture them with Guess and Check. There are so many more useful options and structures out there!
- Factor by Grouping
- Box Method
- Slide and Divide (I don’t recommend this for students on an AP track – from what I hear, AP graders will dock points for lines of work that are not equivalent. Plus, students often forget the slide back part of “slide and divide.”)
Clearly, I have my opinions. I’ve attempted teaching each one of these methods at some point in my career, and just about every year I learn something new from my IB Math students. My former team teacher, BVD, says she has yet another one I’d like. But the most mathematically successful method for my students *so far* has been Factor by Grouping, especially for students who struggle with number sense. You can add in structures as needed to support your struggling students.
Today I will address Factor by Grouping and the Box Method. But FIRST, you should take a look at my posts on Factor Ladders, Greatest Common Factor (Part 1), and Factoring 4 Terms by Grouping (Part 2). They will provide a little background on the progression I use for my Factoring unit.
Method 1: Factor by Grouping
Clearly, trinomials have 3 terms. Grouping requires 4. Let’s make it happen! Note: I start students out with leading coefficients other than 1 so it becomes really easy when they do have a leading coefficient of 1!
Pros: Easy to raise or lower the level of structure, based on student abilities. Upper level teachers support it because it’s mathematically sound!
Cons: The process takes longer than typical “guess and check” for trinomials with coefficients of 1. I show students this “shortcut” after they’ve had a chance to practice a ton.
(1) Write down the Multiply and Add numbers
- Draw the “multiply rainbow” from the first to the last term, multiplying those two coefficients together, and writing the result to the side (5 * – 4 = -20).
- Circle and write the “add” number off to the side as well.
- *This helps students to get in the habit of organizing these numbers in their brains.*
(2) Write down the two numbers that multiply to equal -20 and add to equal -8.
- Make sure students focus on the factors of -20 first! Some students just try to pick numbers that add to -8, but that list is infinite. The factor pairs of -20 are not.
- You can have students make a factor list for -20 and then circle the two that work. (Students should have been practicing this skill before today’s lesson.)
- Students can create a diamond and put the multiply number at the top of the x and the add number at the bottom. (Again…. you should already be having students practice this skill.)
- Read the end of this blog post for more information on both methods.
(3) Use these two numbers to split the middle term into 2x and -10x. Aaaaand voila! You now have 4 terms.
*As students become comfortable with factor by grouping, help them to remove the scaffolds! They can begin to skip the factor ladder part of factoring by grouping (see above), and jump straight to writing 5x (x – 2) + 2 (x – 2). Regardless, I find the squiggle in the middle of the four terms really helps students to clearly see the grouping.
Method 2: Factoring with BOXES!
This is how *I* learned to factor in high school. Shout-out to Mr. Don Bartley in Northport (if you happen to find this…): I still call them Bartley Boxes!
Pros: Similar to factor by grouping – just organizes the 4 terms in a stacked box. Students see the relationships between the terms and their factors more easily. Very connected to how I teach multiplying polynomials. ALSO, many Algebra 1 students have already seen the similar structure of Punnett Squares in Biology!
Cons: Students lacking strong number sense have difficulty finding the GCF of numbers like 8 and 20, but this may just take practice.
(1) Organize your terms into a 2 x 2 box:
- First Term: upper left corner of the box
- Last Term: lower right corner of the box
- Product of these terms: outside the lower right corner
- Middle Term: Under the middle of the box
(2) Find two numbers that multiply to equal -20 and add to equal -8.
- Use factor lists and/or diamonds as needed.
- Write each of these with an x (the variable) in the empty spaces.
(3) Factor the GCF out of each row and column, writing the answers to the left and above.
*CAM, one of my awesome coworkers, uses arrows to help students visualize GCF factoring.
(4) Finally, write the left result in a parentheses as one factor, and write the top result as a second factor. If there is no sign, tell students to write + in between the two terms.
*As you practice examples with your students, remind them of the connections between the factors and terms. While, yes, the GCF of -10x and -4 is -2, -10x/(5x) is ALSO -2. This may help them with their GCF difficulties in some cases.
*If students need to, they can expand xx instead of writing x^2 in the upper left box. This enables them to physically cross out the common variable factors.
– Miss Elsie