We just finished learning how to factor trinomials of the form axx + bx + c where a is not 1. My retired math teacher friend had taught me this trick (over chips and salsa and margaritas last year), and I tried it out with my class. I had to first prep the period with the following statements:
* I'm going to show you a trick today that's going to be your new BFF in math.
* First I'll predict how the class will go.
* You'll take notes on the process while we slowly go through it with an example and you take notes carefully like you're taking them for your best friend who's absent.
* You'll whine and say, "I'll NEVER like/use this trick. It'll NEVER be my BFF".
* After the 2nd problem you'll say, "okay, it's not so bad. I can be in the same room with this trick."
* After the third problem, you'll quietly wait for the 4th after you finish this current one a little faster.
* After the 4th problem, you'll be ready to marry this trick. Your NEW BFF.
We laughed and got down to business. And yes, I had to make fun of them for each of my predictions coming true. And yes, 2 or 3 classes later, when I came back to this kind of factoring, they almost all of them remembered it.
cool! I have always taught with tic-tac-toe but this might be even easier. Less rules to remember for the kids!
ReplyDeleteOooh, what's the tic-tac-toe? ... Also, I forgot to mention, that this only works when the 3 terms share no common factor. So 2xx + 7x - 4 would work, but 4xx + 14x - 8 wouldn't.
ReplyDeleteMs. Cookie
Don't forget GCF and it will work just fine.
DeleteI've taught this factoring using grouping lately, but just a few days ago Dave Cox posted about this on his blog (coxmath.blogspot.com). He calls it "bottom's up", because when you get that fraction after dividing through the "a", you bring the bottom up to the front of that factor. I love this! Definitely going to have to show my kids this.
ReplyDeleteThanks!
the "bottoms up" method is different. I might show that next year. I use the factor by grouping as well. I use it for all factoring problems even when the coefficient of 'a' is one. It seems to cut back on the confusion.
ReplyDeleteI love the prediction nature of the lesson, that's wonderful, and I plan to copy it.
ReplyDeleteHowever ...
4x^2 + 14x - 8
So we need ab = -32 and a + b = 14
That gives us 16 & -2
so
4x^2 -2x + 16x - 8 = (2x+8)(2x-1)
It works beautifully, despite the common factor, and there's absolutely no reason why it shouldn't.
I'd never seen this method before until, after about 6 years of teaching I was shown it by an Italian student in my IB mathematics SL class. Looked like some crazy wacko magic thingummy, but it worked. I can't stand 'magic methods', FOIL drives me crazy because it's just another shortcut (like telling kids to line up the numbers on the right hand side for addition) for teaching something that doesn't develop understanding required later to multiply out things like (5x^2+3x+2)(2x^4-7x^3+5x^2 - 4x + 4) so I sat down and analysed it algebraically and boy is it clever. Now I teach it all the time.
May I recommend that you go through the algebra for this with your students if you haven't done so already - it doesn't look like it from your blog post. Start from (ax + b)(cx + d) and work from there. In terms of developing an appreciation of the power of factorization, and in demystifying something 'magical' which in my opinion we must always do as far as mathematics is concerned (the magic in mathematics is in its power and its beauty, in understanding what lies behind the 'clever tricks') - this is a wonderful teaching moment.
I'm tutoring an 8th grader that was taught this method. It was working perfectly until we got to
ReplyDelete6x^2-11x+4
I could figure out the problem in my head but could not figure out how to find the answer using this trick. I know there is a common factor... so does this method just not work at all for this problem??
And ideas?
Well, the "6 x 4" goes on the "bottom" and the "-11" goes on top.
ReplyDeleteThen the factors of 24 that add to -11 are "-8 and -3". Then you have to divide both -8 and -3 by 6 and simplify to get: -4/3 and -1/2. This gives you (3x - 4)(2x - 1). Is that what you were getting? Maybe you forgot to divide by the x^2 coefficient (6)? Or .....
This trick doesn't work if all the terms have a common factor to begin with (then you have to factor out and then multiply back in later).
Let me know if this makes sense.
Okay.. yes this does make sense. I'm not sure now what we were doing wrong. But Great! Thank you so much! This is extremely helpful! :)
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ReplyDeletecool! I have always trained with tic-tac-toe but this might be even simpler. Less guidelines to keep in mind for the kids!
ReplyDeleteLove it!! Awesome!!
ReplyDeleteI'll use it with my students!!
Thanks for sharing it!!
This is great! My daughter is having the hardest time grasping factoring and I've been looking for ways to help her out. I think she would love this!
ReplyDeleteThis is awesome. Finally I have a new method that I can share to my students. Thanks so much for sharing.
ReplyDeletewhat about 4m^2-4m-8
ReplyDelete4xx + 14x -8 can be worked using bottoms up without grouping at the end.
ReplyDeleteLike the others rewrite as
xx+ 14x -32 then factor
(x + 16)(x - 2) Divide the 16 and -2
by the lead coefficient 4 and get
(x + 16/4)(x - 2/4) Bottoms up
(x + 4)(2x - 1)
Multiply either binomial by the common factor 4 and get
(2x + 8)(2x-1) or (x+4)(4x -2)
But 4xx + 14x -8 and xx+ 14x -32 have nothing in common with each other...they have different y-intercepts, minimums and zeros. How are these mathematically connected, ie. why does this lead to the correct zeros and factors at the end?
DeleteI like this.Please visit the website physicsisfunnyy.blogspot.com to improve your PCM
ReplyDeleteWhere did you get the 28 from?
ReplyDeleteHi Sierra. That's from 2 x 14, the a term and the c term.
DeleteWhat about -6v^2 - 11v -4 ? Help!
ReplyDeleteWhy does this work? I'm good with the multiply to ac and add to b, as I usually teach factoring by grouping. What I do not understand is dividing those numbers by a, reducing - why does that give you the opposite of the zeros? I am not a fan of tricks so want to understand why this works. Thanks!
ReplyDelete