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# From 21 to 500: Game & Math Salon

Here’s a fun, very simple classroom game you can play for multiplication.

You may know the game 21, aka blackjack. In classrooms, I like to play with a deck that only includes numbers from 1 to 10.

Twenty-one. Each player gets two cards (face up). They can “hit” to take another card, or “stay” to stick with what they have. Whoever gets as close to 21 without going over wins. (Traditionally this game is played against the dealer in casinos. It’s fine to play it that way as well.)

Here’s how 21 becomes 500:

Five Hundred. Each player gets two cards. As in 21, they can “hit” or “stay.” The difference in 500 is that you multiply the numbers on your cards together. The goal is to get as close to 500 as possible without going over.

So in the image above, the 19 in Twenty-One would be a 240 in 500. Worth sticking in either case.

I only just made this game up last week, and haven’t played too much, so please experiment. Is 500 the best number to have as the bust point? Still, the game makes kids estimate, make a single strategic decision, and multiply one digit numbers and two digit numbers. And it takes almost no time to teach it.

Announcement: Math Salon on August 16!

For Seattleites: we’re happy to announce that we have a Math Salon on the calendar. Supported and hosted at the Greenwood Library, this event is a great opportunity for you and your kids to spend a Saturday afternoon playing with math. If you’re interested in joining us, please rsvp here.

1. eddi

I’ve been fiddling with the bust point. In 21, the weighted average of the cards is 6.54, which means you hit 21 with 3.2 on average. Practically, this means on a given play, you hit once then have to think about it. If you do the same with 500, the average of the cards is 5.5; but you don’t want to divide (I think), you want to take the 5.5th root of 500, which is 3.09, which is about the same. But then I wondered if the average of the cards was the right measure, since we’re multiplying, so I used the geometric mean of 1-10, which is 4.5; when you take the 4.5th root of 500, you get 3.94, so on a play you’ll hit once but won’t want to hit twice. If you wanted to match the average of 21, you’d pick a target of 128 (4.528^3.2).
I’ve actually been thinking that this task, trying to design fun/fair/interesting games, is wonderful for playing with mathematical thinking…

Cheers,
eddi

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Dan

Hi Eddi,

What a dramatic demonstration of how tinkering with a game makes you think more deeply about the math than simply playing it! I love all your calculations 🙂

In this case, I think the most important metric for the game is average number of turns before busting, and what mathematics is required to play well. With 500 as the bust point, there’s a nice feature that it’s almost always worth hitting if you’re below 100, something 128 doesn’t have. This means that kids will move from one digit times one digit to one digit times two digit as they play the game. It also means that hitting on three digits is almost always a bad idea (something that kids will hopefully begin to see as they play).

2. eddi

Tinkering is one of the habits I love fostering, primarily because kids don’t think they have permission to do it in math. I teach high school and I’m always looking for ways to both practice skills and engage students with math in ways they don’t associate with “math”; that’s why I love your work 🙂
As to the bust point, I’m sure kids would hate 128, as there’s nothing obvious about it (unless you’re familiar with powers of 2), but I liked thinking about the metric…