Prime factorization with factor trees (actually, using factor trees in general), in my opinion and experience, is the WORST! Organized kids can usually keep track of what they need to, but many struggling students lose “branches” of their trees and forget that they still need to factor composite numbers like 6. If a kid learned factoring with trees, I’ll roll with it… but it hurts my soul just a little bit.
A few years back I adopted the ladder method, which lines the primes up nicely in a row, and students know *for sure* they’re done when they see the 1. The cake method is a nice little twist because anyone who’s anyone prefers the thought of making a cake over building a ladder. At least it sounds exciting!
Let’s backtrack a minute. Prime factorization:
- builds an understanding of the connections between a number and its factor pairs;
- can help with factoring trinomials… or polynomials in general;
- is useful in simplifying radicals, especially when students don’t have a strong grasp on perfect squares. I’ll be addressing simplifying radicals in a future entry (maybe even the next one!).
Before you begin, discuss primes with your students:
- Primes: what are they?
- What’s it called when something’s not prime (composite)?
- Can you come up with a list of the first ____ prime numbers?
- How do you know if one number is divisible by another (when you divide, you don’t get a decimal, etc.)?
The most important ones students usually need to know are: 2, 3, 5, 7, 11, 13… and each time we do this I ask my class to recite that list as I write it on the board.
Prime Factoring with Ladders
Now we can begin our ladders!
(1) If we’re factoring 980, begin with 980 at the top of your work space. Draw a ladder rung (left and lower side of a box). Start with the smallest primes first and check: does 2 divide evenly into 980? Yes. 490 times. Write 2 on the left side and 490 below.
(2) Check: is 490 divisible by 2? Yes. 245 times. (2 on the left, 245 below). Only write a prime number on the left once you know it works.
(3) Once we can’t divide by 2 any more, we move on to 3, and then 5. 5 divides 245 evenly 49 times. 5 only works once.
(4) Lastly, 7 divides 49, and then divides 7 again. Do not leave 7 hanging out at the bottom! You can only stop this process when you see a 1.
(5) Depending on the wording of the question, you can have your students use this ladder to write 980 as a product of factors (2*2*5*7*7), list the factors, etc. Or you can use the ladder to simplify √(980). *But please, oh please, insist that they use anything other than an x for multiplication! Dots, stars, parentheses, you name it. In Algebra, x suddenly has whole new meanings.*
Prime Factorization with Cakes
Really, the process is exactly the same, except you’re working upward. While making cakes is more exciting than building ladders, I’m torn. With factor cakes you have to be more conscious of your space and make sure you start low enough to have space for the “candle” at the top. Here’s a snapshot of it. The candle is even glowing!
One last thing, I promise – the best part about cakes and ladders is that you can use these to help students factor the greatest common factor out of a polynomial. I’ll be writing more on that later. (The ideas just keep flowing… 🙂 )
UPDATE: Check out my new posts on the awesome applications of factor cakes and ladders!
So… cakes or ladders? You choose.
– Miss Elsie