Monday, February 15, 2016

The Real Housewives of Pre-Calculus: The Rational Root Theorem

Over the years, I've always found that Rational Root Theorem has been a struggle for kids. They struggle to get all combinations of factors and struggle to remember which goes where. I don't blame them...they can be cumbersome problems with a lot to keep straight. 

Each semester I feel like I did a slightly better job conveying it to the kids, but I think I've finally hit my groove with it. 

Inspiration: Reality TV?

Yes, I said it. I was watching Bravo and marveling at one of the real housewife's closet. Like any proverbial math nerd, my inner "notice and wonder" started coming out....
How many of my salaries would it take to have a closet like that?
(the age old teacher question)
How did they decide the layout? 
I bet we could make that more efficient
What cost more...the jewelry, the clothes, or the shoes?
How many different combinations could she make with all of it?

That was it!! How many combinations could she make with her outfits?!
It's a problem our kids have been doing since 5th, 6th, 7th grade! 
Here's why:

I start off with a polynomial and we give him a name- Fred?
We mention that we are going to define f(x) as Fred's feet....f must stand for feet, right? And naturally, feet go on the ground. Now, Fred is still working on building his professional wardrobe so he has a limited selection of shirts and pants. Since we've oriented Fred with his feet on the ground, the constant is now his shirts and the leading coefficient is now his pants. 

Let's find all the factors of each. 

So we can see here that Fred has 4 shirts and 2 pairs of pants. We then proceed to make every "outfit" we can out of them.  Add a plus or minus and BAM- we have all our possible rational roots! 

Small? Gimmicky? Yup.
It's been the only way I've been able to get kids to remember it though! And they know better than to put shirts on their legs or pants on their head. It's dramatically reduced the student error of putting factors in the wrong place. I saw a big difference on their individual practice. Unit test is tomorrow...let's see if it makes a difference there! 


  1. I was just about to introduce the Rational Root thm tomorrow in class. I may have to give Fred a try.

    1. Let me know if you think of any ways to improve it :)

  2. Amazing blog, great ideas. But I would like to add a bigger picture question. In the year 2016, should we still be teaching the rational root theorem? Does not the technology of the era make such a theorem obsolete? Think about why in the past, this theorem was important and why it is no longer?

    1. I completely see where you're coming from, especially at the level at which it's being taught. For those of us with a passion for the beauty of mathematics, it is a really interesting fact and can be quite elegant. However, it's a struggle for a lot of the kids who don't have that appreciation and see it as something tech could just "do for them"