So Part 2 was supposed to be about the big ideas in K-7 mathematics that students will need for high school. But that’ll have to wait for Part 3. Instead, more on times tables.
Three oft-used arguments for the importance of memorizing times tables:
- When learning higher levels of math, there just isn’t time to use calculators or strategies to determine basic facts.
- Besides, thinking taxes working memory which means by the time you’ve worked out the first part of the question, you will have forgotten the… Where am I?
- Because factoring.
1 & 2 are gospel. Well, so is 3; nevertheless, it’s the focus of this post. I have a couple of thoughts on times tables and factoring trinomials.
The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10. I can get away with this if we’re talkin’ Pythagoras. But factoring?! I mean, that’s all it is, right? To factor x² + 7x + 10, you just have to ask yourself, “What two numbers multiply to 10 and add to 7?”
HS math teachers, try this: give your students a quiz on factoring. Include both x² + 10x + 24 and x² + 25x + 24. Get back to me. For extra credit (yours, not theirs), throw x² + 6x + 5 in there. If your students are anything like mine, I bet x² + 25x + 24 gives them at least as much difficulty as x² + 10x + 24. What does this mean for these students? More practice multiplying by one?!
Of course, 1 × 24 falls outside most times tables. Recall of products to 10 × 10 gets us the factors of x² + bx + 60 – if b = 16. But x² + 17x + 60, x² + 19x + 60, x² + 23x + 60, and x² + 32x + 60 are fair game, right? Try c = 48. Or 72. Or 96. Or 100. What role does memorizing times tables play? What role does being flexible with numbers play?
My point, I think, is that these are different, albeit related, skills. In other words, the “it” they’ll need for factoring (trinomials) is factoring (numbers). And number sense. This has some implications for K-7: not necessarily more “What’s 4 × 6?” but more “A rectangle has an area of about 24 square units. What could its length and width be?” or even “The answer is 24. What’s the question?”; not thinking digits/standard algorithm but thinking – and talking! – factors/mental math strategies, e.g. 16 × 25 = (4 × 4) × 25 = 4 × (4 × 25) = 4 × 100 = 400 (via Sherry Parrish).
nothing to do with post but @k8nowak says put pictures in posts
Say you’re still asking, “How am I supposed to teach them factoring when they don’t even know their multiplication facts?” When I introduced polynomial division in Math 10, some of my high school students didn’t even know long division. So I taught division of numbers and polynomials side-by-side, highlighting connections. Can the same miiindset (channeling my inner Leinwand) be applied to factoring trinomials and times tables?
And what about something like x² − 2x − 24? If that – asking yourself, “What two numbers multiply to -24 and add to -2?” – is all it is, why not factoring trinomials to teach multiplication (and addition) of integers?
Chris Hunter 
The reason some students struggle with factoring trinomials is not because they haven’t memorized products to 10 × 10.
That’s kind of a general statement, isn’t it? I had a Pre-Calc student once where this was the exact reason why they had so much trouble factoring (they couldn’t come up with any numbers at all that multiplied to 24 other than 1 x 24). Just because there’s more to it doesn’t mean the multiplication table doesn’t sneak in there.
Jason,
I added the qualifier “some” to soften the initial statement I made in my draft.
I absolutely agree. But, often, this isn’t the conversation I’m having with my secondary colleagues. I want to have the conversation about the “more to it” part, but the times table argument is so appealing – whether we’re talking about something like factoring, where admittedly it probably does a little more than just “sneak in,” or something like surface area of composite objects, where it’s a stretch.