At various times when I tutor, I find myself explaining the same concept repeatedly over several weeks.
Recently it has been that way with — drumroll please … factoring out the GCF from polynomials.
One reason I’m getting so much “experience” with this is that many kids find this process very difficult. It’s not hard to see why. First of all, the process of finding the GCF is, in itself, somewhat tricky. Then too, factoring out the GCF from all terms in a polynomial is a multi-step process; students need to get each step right, and then they need to perform the steps in the correct order. If that alone were not enough to tax children’s minds, students also get confused by the difference between how to multiply pure numbers (constants and coefficients), and how to multiply variables.
Using my general approach of bringing in REAL-LIFE STORIES to help students grasp algebra, I’ve recently been using two real-life stories to help students nail down this process.
The first story helps students master the prerequisite skill of multiplying monomials. To help with this, I use the story of “rubbing your stomach and patting your head,” for reasons I’ll make clear in a moment. I show students that I can do this weird physical skill (my moment of great pride!), and ask them if they can do it, too. Good news … most students can still do this, even after all those years of just playing video games!
After students demonstrate this talent, I explain that multiplying monomials requires them to mentally “rub their stomach and pat their head.” What I mean is that, when multiplying monomials (like 3c^2 times 5c^4) students must do two very different things at the same time. When multiplying the pure numbers in monomials (like the 3 and the 5), they do pure MULTIPLICATION (3 x 5 = 15). But when multiplying the variable terms, they do ADDITION inasmuch as they must ADD the exponents of the variable terms (c^2 times c^4 = c^6, since 2 + 4 = 6). Giving students the story of patting their head while rubbing their stomach reminds them that they have to do these different operations in the same step. All they really need is that reminder, and then they can usually multiply monomials correctly. Though sometimes, I do need to remind students a number of times before the concept sticks.
Another part of the GCF process that gives students trouble comes after they’ve found the GCF. In this crucial next stage, they need to figure out what to multiply the GCF by to get each term in the original polynomial.
To help students surmount this next hurdle, I’ve begun using the story of giving someone a “boost.” Here I’m referring to the process of interlacing your fingers, palms up, to create a surface for someone to place their foot on while attempting to climb up, as in scaling a fence. I lace the fingers of my two hands together, giving kids a “visual,” and ask them if they’ve ever “given someone a boost” or had someone “give them a boost.” So far every student has known precisely what I’m talking about.
Then I explain that factoring out the GCF has a lot in common with giving someone a boost, except that when doing the algebraic process, instead of giving a “boost” to a person, they give a boost to numbers or algebraic terms.
As an example, suppose that a student needs to factor out the GCF for a binomial like this: 15x + 25x^3. After she finds that the GCF is 5x, she needs to “boost” the 5x so that it can rise up the the level of each of the two terms: first she needs to boost 5x to the level of 15x; after that, she must boost 5x up to the level of 25x^3.
I have students create what we call a “boosting box” on their paper. For this problem, the two “boosting boxes” — before working out the values — look like this.
I tell students that in the box they need to write the term by which they multiply 5x in order to get 15x. I urge students to break the task into two parts: 1) boosting the number through MULTIPLICATION, and 2) boosting the variable term(s) by ADDING exponents to get the value of the final exponent.
If students have learned how to multiply monomials as explained earlier, they can do this step fairly well, remembering to use MULTIPLICATION for the number terms, and ADDITION for the exponents. But don’t be surprised if you have to repeat the fact that you use different processes for these two parts. Here’s what the “boosting boxes” look like after being completed:
Once students know what should go in the boxes, tell them to start writing the answer. To do this, they first write down the GCF. Then to the right of that, inside parentheses, they put down the info they put in the “boosting boxes,” separated by the appropriate operation symbols of addition or subtraction.
And that is how I have been helping students factor out the GCF. Overall it generally works quite well. But you need to be patient, as this is a process that easily trips students up, so they need to work slowly and carefully to get the problems right, especially the first times they work through this process.
If you like this process of using real-life stories in algebra, I first developed this teaching strategy in my book, the Algebra Survival Guide. You can learn more about this book by clicking on the link in my blog’s sidebar. You can also just look it up on Amazon at this link.