Inequality Card Tricks

So, yesterday was Thanksgiving, and the family was all gathered. What a perfect time to engage your loved ones in a fun, educational, math game! Unfortunately, the only person who was interested was my father, and I'm sure he was just being nice. But what transpired was great material for a blog post!

The game we played was called "Greater Than," and it was created by Professor Golden. I had played the game before with my mother for my teaching class, and it was very straightforward. It's a card game for 2 players or teams and it's designed to help students practice inequalities. The dealer gives 4 cards to each team/player. Then each team/player selects a card from their 4, which is their starting value, and reveal it at the same time. Then, the non-dealer plays a turn, and the operation and value of that card gets applied to both player's cards. Then, it's the dealer's turn, and so on, until all the cards are played. The team/player who has the greatest value at the end wins! The operations were very simple: adding and multiplying by positive or negative numbers. (To see them in more detail, click the link above.)

After I explained the rules to my father, we started a game. I dealt 4 cards, and my father and I flipped our first card. We both flipped a positive 10. At this point there was no need to continue the game, since whatever cards that were played afterwards would be applied to both cards, and they would end up being equal. So, I dealt another hand, and we flipped both our cards.... We both flipped -11! At this point, we decided to break one of the rules, and thus the first variation was born! The first of three. Here is how it all went...

Variation 1: When a player plays a card, it only applies to their own card. It doesn't apply to both cards.

I dealt 4 cards to each of us. And when we looked at our hand to select our first card, it took longer than the previous two hands, since we could strategize the order of our cards to figure out the greatest value. Before, strategizing was harder, since the value of your hand was controlled by the cards that the other team/player used. Now, each player had sole control over the value of their hand. My hand consisted of +(10), x(-3), +(-4), x(10). If I played the positives first, I would be left with x(-3), which would turn my value negative. And the more negative the number, the lesser the value. So, I played the +(-4) first, followed by the x(-3), which gave me a positive 12. Then, I played the +(10), which gave me 22, and then I played the x(10), which gave me a positive 220. If I had played x(10) before the +(10), I would have gotten 12x10 = 120 + 10 = 130. And 220 > 130, and I want the greatest value possible. So, the order and combinations is very important to think about! Unfortunately, my father had the better hand with 310 (220 < 310). We played 2 more rounds, and my father won all of them. And then I decided to make the math game more challenging and changed the rules again. So, we moved on to play Variation 2.

Variation 2: Adding division to the mix.

So far, we've been adding and multiplying positive and negative numbers. But division is out of the mix. So I decided to include it, and I let the Face cards (not including the Ace, the Ace is still 11) be the dividing or multiplying-by-a-fraction cards. Kings were 1/2, Queens were 1/3, and Jacks were 1/4. With the new rules in place, I dealt 4 cards to each of us. I got x(1/3), +(-11), x(9), x(-5). The +(-11) had to be played first, so I could then use x(-5) to get +(55). Then I used x(9). This math was a little harder to do in my head. First, I added 55 +55 to get 110. Then I multiplied 110 by 4 to get 440. 2x4=8, and I needed to multiply by 9, so I had one more 55 to add, which gets me 495. Then, I divided by 3 or multiplied by 1/3, which got me the final value of 495/3. This is also equal to 165, which I did by figuring out how many 3's go into 4, drop the next number, and so on. My father this time didn't do so well, this round. He got -11/4. We played 2 more rounds, all of which I won. But then my father wanted to try his variation of the game. So, we moved on to Variation 3.

Variation 3: Diamonds now represent division.
My father wanted a whole entire suit to be division, instead of the Face cards. The rest of the rules would be the same. I pointed out that this would mean that we could only divide by a negative number. My father was fine with that. So, I dealt 4 cards to each of us again, and I got +(-10), +(-7), x(-1/2), +(3). I wanted to add as many negatives as possible, so I added (-10) and (-7) to get (-17). Then I multiplied by (-1/2) to get (17/2). Finally, I added the (9), which I converted to (18/2). 8+7 = 15, so I carried the 1, and 1+1+1=3. So, I got (35/2). My father got (-3/11), so I won this hand. We played 3 more games, which my father won all of them. At this point, Thanksgiving dinner was ready, and the game playing was over!

Reflection: All the versions of the "Greater Than" Game will be useful to teach students about inequalities. Depending on the confidence level of my students, I would have them do the original version first, followed by Variation 1. Variation 1, I believe, will teach them more about strategy and how to think a few steps ahead in order to figure out a problem. In this case, it's which card combination and order will achieve the greatest value. Then, if they are getting really comfortable, I would tell them to move on to Variation 2, and let them use a paper and pencil to help them divide/add/multiply/subtract fractions until they felt comfortable enough to do the math in their heads. I would also allow the students to use paper and pencil to figure out difficult multiplication problems with the original version, especially if I saw them struggling. All in all, the "Greater Than" Game is a great way to get students to practice math, learn about inequalities, and learn how to think steps through ahead of time to solve a problem.