I won’t lie… Quarter 3 completely got away from me. My last post was almost 4 months ago and Quarter 4 starts tomorrow! But better late than never, right?
Tomorrow I will begin factoring out the Greatest Common Factor (GCF) with my ESOL Algebra Part 1 kids. (You’re probably thinking… what?! Is she crazy?!) So far, they’ve tackled every challenge I’ve thrown at them this year with great success. I realize it’s not *actually* included in the Part 1 curriculum, but I’m trying to set them up for success next year. If they understand the basics of the math they’ll be learning next year in Algebra 1, then they will be able to focus more mental energy on the language acquisition & application questions, which we have also been tackling slowly throughout the year.
With factoring, I teach in the following order:
- Greatest Common Factor
- Factor By Grouping with 4 terms (see post here)
- Factor Trinomials By Grouping, STARTING with a leading coefficient other than 1. (see post here)
- Special Cases (Difference of squares, leading coefficient of 1, etc.)
My philosophy here is similar to my approach in solving equations. Start with the most complex examples. Then the special cases and leading coefficients of 1 are a cinch! Also, I realize that factor by grouping is *not* technically an Algebra 1 concept, but I find it to be the most mathematical and consistently successful method of factoring for students (that, and boxes). And Algebra 2 teachers tend to support it *because* it’s mathematical, unlike the slide and divide trick. So I just jump right into the nitty gritty of factoring with the students. They can handle it. Trust me. (And I can even let you know how tomorrow goes!)
Greatest Common Factor
First let’s talk about GCF. It’s a challenge when students don’t know their multiplication facts, for sure! But my team teacher, BV, and I came up with work around. It’s related to the factor ladders I use for prime factorization and simplifying radicals, so check out that post if you haven’t already!
(1) Ask: What number divides evenly into all 3 coefficients: 36, 42, & 78? If your students (like mine) can’t figure out that 6 works, start with prime factors.
2 works! Write the result for each term below the factor ladder:
3 works! Follow the same process.
(2) Ask: How many xs do all terms have in common? In this case, 2. Write x^2 out front.
I like to explain this concept to my students with the analogy of being a cookie thief. I steal “cookies” (or xs) from every “person” (or term), but I steal EQUALLY. I don’t want to be “unfair.” So if Jose has 4 cookies, Maya has 2 cookies, and Andre has 3 cookies, I would take 2 cookies from EVERYONE because that’s the most I can take and still be fair. Obviously the definition of fair and unfair in this situation is a totally different topic of conversation for a totally different day….
Then ask: How many ys do all terms have in common? 1 y. Write it out front, with the resultant of each term below.
(3) Write your final answer. The front of the factor ladder represents all factors that make up the GCF. Since they are factors, we multiply them. Then we write the resultant (what’s left below the factor ladder) in parentheses.
A few thoughts & modifications:
- Some students… may need to actually write out the number of xs and ys so they can physically cross out an equal number of variables from each term.
- Some students… may only need two rungs on their factor ladders – one factor rung for the number (6), and one rung for the variables (x^2y).
- Some students… may not need any scaffolding at all in order to be successful with GCF!
After all, the goal is ultimately to break away from the scaffolds so students can be more efficient in their mathematical processes as they reach Algebra 2 and beyond. The scaffolds just build understanding by helping students to visualize the work they are doing.
Go forth and factor only the greatest things… 😉