Using Recycled Caps to Teach Prime Factorization

Note to educators: To demonstrate this lesson to the class at the whiteboard, I adhered pieces of a magnetic strip (only around $4 for a roll at Michaels!) to the tops of caps. I found the magnet to adhere better to the metal lids and Gatorade caps. The students shared their own cut-out pieces of paper that contained several prime factors. This is why I'm collecting caps so that ALL of my students will be able to use the caps along with me.


Review of terminology
Prime number: a number that has exactly two factors, just 1 and the number (e.g. 2, 3, 5, 7, 11, 13, ...)

Factors: numbers that multiply to get a number (e.g. the factors of 6 are 1, 2, 3, and 6)

Multiple: the result of multiplying by a whole number (e.g. the multiples of 4 are 4, 8, 12, 16, 20, ...)

GCF (greatest common factor): the highest number that divides evenly into the given numbers (e.g. the GCF of 4 and 12 is 4)

LCM (least common multiple): the lowest number that the given numbers divide into evenly (e.g. the LCM of 4 and 12 is 12)

Note: The GCF has to be NO GREATER than the lowest number in the set. And the LCM must be AT LEAST the highest number in the set. For example, in the pair of numbers 4 and 12 mentioned above, the GCF is 4 and cannot be greater than that lowest number. The LCM had to be at least the highest number 12. We'd never ask for the lowest common factor, for that would just be 1. And we'd never ask for the greatest common multiple... what are the common multiples of 4 and 12? (underlined)

4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, ...
12: 12, 24, 36, 48, ...

See? 4 and 12 have TOO MANY common multiples so we'd just ask for their LEAST common multiple. Now, here's how to reuse some caps to learn math...


The "Cap Method" for listing prime factors, finding the GCF and the LCM

First, define each cap. Here I let the silver cap = the prime factor 2, the blue cap = the prime factor 3, and the green cap = the prime factor 5.

To find the prime factorization of the numbers 18 and 24, use the factor tree or tower/ladder method (see the Pencast featured to right of screen). Write each number as a product of their primes with the caps:

Now, you can use the caps in the prime factors above to find the GCF of the numbers:

Next, you can use the prime factor caps to find the LCM of the numbers. One way to do it is to ask yourself a couple questions:

OR, we can refer back to the prime factorization and find the LCM another way:

In class today we used these methods to find the prime factorizations, GCF and LCM of the numbers 120 and 180. Using the caps seemed to simplify the process!

Where is this going? GCF can be used to reduce fractions. We also need to know how to find the GCF for factoring polynomials in algebra. LCM is used to find the LCD of fractions. We also need to know how to find the LCM and LCD for operations with rational expressions in algebra. These are such important topics.