Since I published a deck of factorization diagram cards last September, a few teachers have picked up copies of the cards and started using them with their students. I’ve started collecting ideas for games you can play using the cards, and want to share here a few game ideas from Alex Ford who teaches middle school in St. Paul, Minnesota.
If you want to get your own set you can buy one here! Also, if you have any other ideas for games or activities using the cards, please send them my way.
War
First, you can play a classic game of War. The twist is that while playing you should only look at the diagram side of the cards, not the side with the written-out number. So part of the game is figuring out which factorization diagram represents a bigger number. One could of course just work out what each number is and then compare, but I imagine students may also find tricks they can use to decide which is bigger without fully working out what the numbers are.
Variant 1: primes are wild, that is, primes always beat composite numbers. (If you have two primes or two composite numbers, then the higher one beats the lower one as usual.) This may actually make the game a bit easier, since when a prime is played you don’t actually need to work out the value of any composite number played in opposition to it.
Variant 2: like variant 1, except that primes only beat those composite numbers which don’t have them as a factor. For example, 5 beats 24, but 5 loses to 30: since 30 has 5 as a prime factor it is “immune” to 5.
As a fun follow-on activity to variant 2, try listing the cards in order according to which beats which!1
Set
Alex and his students came up with a fun variant on SET. Start by dealing out twelve factorization cards, diagram-side-up. Like the usual SET game, the aim is to find and claim sets of three cards. The difference is in how sets are defined. A “set” of factorization cards is any set of three cards that either
- Share no prime factors in common (that is, any given prime occurs on at most one of the cards), or
- Share all their prime factors in common (each prime that appears on any of the cards must appear on all three).
Here are a few examples of valid sets:
And here are a few invalid sets:
In order to claim a set you have to state the number on each card and explain why they form a set. If you are correct, remove the cards and deal three new cards. If you are incorrect, keep looking!
Alex and his students found that, just as with the classic SET game, it is possible to have a layout of twelve cards containing no set. For example, here’s the layout they found:
Just to double-check, I confirmed with a computer program that the above layout indeed contains no valid sets. As with the usual SET, if you find yourself in a situation where everyone agrees there are no sets, you can just deal out three more cards.
The natural follow-up question is: what’s the largest possible layout with no sets? So far, this is an open question!
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The Math Less Traveled 
Are you working on the open question at the end of the post: find the largest subset(s) of [1,…,n] that contain no three elements that are pairwise relatively prime and no three that share all their prime factors? I am tempted to ask this at http://math.stackexchange.com with a link to here – but you should if you want to, and can tell me not to.
No, I’m not working on it — post it wherever you like!
Another game idea with the factorization cards: put a card in the center of the group. FIrst one to identify the value takes the card. Continue until all the card are taken. The player with the most cards is the winner.
Nice and simple, I like it!