An Ode to Prime Factorization (…and Whiteboards, and Error Analysis)

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August 27, 2012 by ray_emily

First off, a confession: I do not have much of a mathematics background (at all – I was a music major). So, I’ve far-too-recently discovered (don’t mock, please) that prime factorization is (a) awesome, (b) kinda fun, and (c) terrifically useful.

During my first year teaching, I taught my kids to find the greatest common factor and least common multiple by making these long, awful lists of factors and multiples. (Uh huh: My approach was somewhat like this. Yes, that’s a link to Khan Academy.) That was the only approach that my kids learned from me. No alternative methods or shortcuts – that was it. (Man, I am outing myself, tonight!)

Now, it is not news to you that prime factorization is an often more efficient path to finding both GCF and LCM. I’m still, however, totally psyched about it (several years after learning this), so, forgive me if I wax poetic for a bit – and share with you some of the reasons why I dig teaching kids to find GCF and LCM with prime factorization! (Also: Via a very informal analysis, I’ve gathered that this stuff gets glossed over, often – so this is not purely an expression of my love. I’m hoping maybe some other folks will get on board, too?)

1. It is an approach that is highly appealing to visual learners. (Let’s face it: Factor trees are pretty!)
2. It is an alternative – another option! – preferred by many to the drudgery of listing. (It also offers me the chance to reiterate this big idea that any number of paths can lead to a good solution.)
3. It builds a solid understanding of our number system, and of prime numbers as building blocks.
4. It is not always better or more efficient. (I like this, because it encourages my kids to assess the problem and consider, “which is the best course of action?” before jumping in.)
5. It requires a much deeper, more nuanced understanding of both numbers (in general), and GCF and LCM (specifically).
6. It is way more fun and interesting than listing, listing, listing.
7. When kids get comfortable finding GCF and LCM with prime factorization, they also become pros at reducing fractions using prime factorization. (That’s another thing I never considered teaching, back when I knew even less than I know now.)

Okay, okay. I’ll stop. But before I go, I want to share one more thing! Take a look, first, then I’ll describe, okay? (Click to enlarge.)

Last year, we were working on GCF and LCM when I broke out the big white boards for the very first time. (I didn’t have a blog, so I didn’t ask kids to refrain from writing their names; you’ll have to pardon the big black blotches.)  Each group was to find the GCF and LCM of a different pair of large-ish numbers, and share their solution (I’ve highlighted them in yellow) with the class.

Here’s the fascinating thing: The boards you see above all feature the same pair of numbers (252 and 120) – but only one board offers an entirely correct solution. (Hint: It’s not the super-organized, hyper-neat whiteboard. Sorry, my darlings.) Although all of the errors are fairly minor – no need to sound the alarms – it is fun to explore the teeny, tiny details (and omissions) that tripped up my kids. Also: What could I do to prevent this type of student error in the future?

PS: On these boards, no one made the common mistake of multiplying ALL of the prime factors together, to find LCM. I have definitely seen that one, before, and the result is a freaking ginormous solution.

PPS: I’m thinking that this could be a great skill to use for this Mistake Game that I keep hearing so much about. Still wary about trying that one out with my still-very-young crowd.

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