## Trying to find math inside everything else

### The Factor Draft

Last year at #TMC13, I ran a session called Making Math Games. I stared off with an overview of what makes a game a good game, while still being good math pedagogy as well. Then we spent most of the session in two groups brainstorming idea for games for topics that are somewhat of a drag to get through. The other group worked on something in Algebra 2, though I don’t recall what – I must say both groups were supposed to write up what we did and neither did. (But I do think Sean Sweeney was in the other group, so maybe he remembers.)

My group worked on a game for factoring, focusing on Algebra 1. I took the ideas from the session and made a mostly operational game. Then, about 2 months ago, Max Ray came to visit me on the day I was unveiling that game in class. He saw it and it worked out…okay, but here was definitely improvements to be made. So we talked over lunch (about many things, not just the game – he’s great to talk to!) and then tried out some changes with my lunch gang. The changes seemed to work and I went forward with the new version in my afternoon classes to great success. By the end I think I had a really wonderful game, and so I wanted to share it with you.

### The Materials

A set of Factor Draft cards includes 3 differently-colored decks. Mine, pictured here, were green, blue, and yellow. One deck (green here) is the factor cards, with things like (x + 2) and (x – 1) written on them. Another deck (blue) is the sum cards, with numbers like 10x or -4x. The last deck (yellow) is the product cards, with numbers like +36 or -15.

### The Set-up

Lay out the cards as follows: make a 3 x 6 rectangle of factor cards, a 4×3 rectangle of sum cards, and a 4×3 rectangle of product cards, all face up. Place the remaining cards in separate piles next to the playing area.

In the cards I printed, I didn’t put the Xs on the blue sum cards. Max suggested I do because it’s easy to be confused on which is which.

### The Objective

The goal of the game is to collect 4 cards that can be used to complete a true statement of the following form: (factor card)(factor card) $= x^2$ + (sum card) + (product card).

### Gameplay

Each turn, a player may select any card from the playing field and place it face-up in front of them. They then replace that card with a new card of the same color from the deck. Play passes to the left. A player may have any number of cards in front of them, and may use any four cards to build a winning hand.

The cards I collected after turn 5. There’s two possible cards I could pull to win the game – can you see which ones?

If at any point a player achieves victory, if they had more turns than the other players, they must allow the other players additional turns to attempt to tie. Upon a tie, discard the winning cards and continue play as a tie-breaker.

A winning hand.

My co-teacher, when we were testing the game, said that it felt like Connect 4, in that with each move you have to decide whether to go on the offense to try and complete your four cards, or go on the defense and block the other players’ sets. But as each player gets more and more cards in front of them, it’s hard to see all of the connections and effectively block, so the game will always eventually lead to victory.

I may need to adjust the number of cards and type of cards in the decks, but I think what I currently have works well – if you have any feedback on the card distribution, let me know. The sum cards go from -10 to +10, with the numbers closer to 0 more common. The product cards go from -60 to +60, with each product card being unique. And the factor cards go from (x-10) to (x+10), also with the ones closer to 0 being more common. (There are no (x+0) cards.)

I did a whole little analysis to determine how many of each type of card to include…but maybe that’s a post for another day.

Sum:Product Deck – The first four pages are the sum deck, the next four are the product deck, the last four are the factor deck.

Factor Draft Play Mat and Rules – Players can use these mats to place their cards and check for a win.

#### Comments on: "The Factor Draft" (4)

1. Really nice James – I love drafting mechanics and open drafts are super cool!

I must admit it took me a while to grok why that was a winning hand until I realised you just assumed every player had a x^2 card to finish it off. Would some sort or player board for people to arrange their solutions, partial or otherwise, be helpful? The backside of the player card could be an ‘expanded’ version for (ax+b) cards?

I’m stealing this. Thanks :-)

• That’s a good idea – make a little placemat for the players to put their cards on, to show why they win and help them think it out (though part of the gameplay is to do a lot of it in your head so that your opponents can’t see what you need).

I’m definitely thinking about an (ax + b) expansion, especially because I’ll be teaching Algebra II next year.

• Bryan Bazilauskas said:

good idea. or maybe players get an x^2 card to begin the game with?

2. Bryan Bazilauskas said:

I really like this one, will use it and report back.