In Unit 2 of the online course, we highlighted a number of “contemporary obsessions” that have the potential to support effective learning but are easily sidetracked by features that have less to do with learning math.
We didn’t include games in our list of obsessions, though they would fit. Used well, games can support both conceptual development and procedural fluency. Used poorly, they can reinforce weak strategies and bad habits and perpetuate achievement gaps. We’d like to highlight some games we’ve found helpful—and the ways we’ve used them.
In this post, I’ll focus on Sleeping Queens. I’ll briefly outline the rules before getting into the mathematical possibilities.
On the surface, this game is about collecting five Queens or fifty points, whichever comes first.
As the game opens, twelve Queens lie sleeping face down, and each player has five cards. You get Queens by playing a King, using a Knight to steal one from an opponent, or sometimes by playing a Jester. You lose Queens when opponents steal them (with a Knight, unless you block the Knight with a Dragon) or put them back to sleep (with a potion, unless you block the potion with a wand). Every time you play a card, you draw to replace it, so you always have five cards.
Note that you can also adapt two decks of regular playing cards to a very close approximation of the commercial game. You can also switch the roles of Kings and Queens so that the Queens wake the sleeping Kings.
If you like, you can watch this video to get a better sense of how the game works* https://www.youtube.com/watch?v=MkfM4hsIzaY
*Note we do not endorse all of the side comments in the video. In particular, the narrator suggests that playing the game will develop working memory (P3), but research tells us that working memory isn’t like a muscle you can strengthen. However, developing high fluency with mental math does mean that you don’t have to use working memory for simple calculations, which frees up working memory for other tasks.
Where’s the Math?
There are several mathematical elements in the game, some of which can be used with widely varying levels of sophistication. This can be a nice feature provided that it is used to scaffold learning as opposed to sorting learners. In other words, so-called weaker learners should not be left to use weak strategies; they should be supported in developing stronger ones (P2*).
*Annotations like this refer to Math Minds principles and practices, which are described in [link to Blog 1] and summarized at the end of this post.
The Queens have values of 5, 10, 15, or 20. You win with five Queens—but it’s also possible to get fifty points with three or four Queens. The game doesn’t teach kids how to add fives and tens—but it does provide a fairly simple context in which to practice adding fives and tens. So how might you teach them to do this?
To begin with, they need to know that 5 + 5 = 10. They also need to be able to distinguish tens from ones: The one in 15 refers to one ten, the two in 20 refers to two tens, and the five in 50 refers to five tens (R-b). Given that, they might take a few minutes before playing to practice counting tens in some simple combos. Notice how the questions build on one another. This is an example of structured variation used to prompt attention to place value. Some of the variations use more fives or tens that would appear in the game itself, but they’re structured to prompt attention to tens (P-b):
10 + 10 = 20 (two tens, or twenty*)
10 + 10 + 10 = 30 (three tens, or thirty)
10 + 10 + 10 + 10 = 40 (four tens, or forty)
10 + 10 + 10 + 10 + 10 = 50 (five tens—and enough to win!)
5 + 5 = 10 (one ten)
5 + 5 + 5 + 5 = 20 (two tens)
5 + 5 + 5 + 5 + 5 = 25 (two tens and one leftover five)
15 + 5 = 20 (Do they still recognize the 5 + 5 as a ten?)
15 + 15 = 30 (Still?)
15 + 15 + 5 = 35 (Now there are three tens and an extra five, for a total of 35.)
15 + 10 = 25
15 + 20 = 35
15 + 15 + 20 = 50 (A win with only three cards!)
Can they find a way to win with four cards?
*or two-ty (followed by three-ty, for-ty, five-ty, six-ty, etc.); this language can help emphasize that you’re counting tens.
Patterns in Multiples
If you play a Jester, you must then draw the top card from the pile. If it’s not a number card, you just keep it. If it is a number card, you count around the circle (starting with yourself) to see who gets to pick a Queen (like eeny-meeny-miny-moe). With two players, this means that if you draw an odd number, you get a Queen; an even number means your partner gets the Queen. Some kids notice this pattern on their own, but many benefit from someone prompting attention to it (P-a), then offering them some structured practice requiring them to predict and test who gets the Queen for various numbers (P-b). If you have them consider who gets the Queen if they draw a one, two, three, four, five, six, etc., an alternating pattern emerges. Why? If they already recognize odd and even numbers, can they see the pattern in evens and odds? Some are surprised to note that if they play the Jester, odd gets them a Queen. If their opponent plays the Jester, however, even gets them a Queen. With three players, multiples of three become the desired card.
Aside from determining how many to count off when you play a Jester, number cards don’t really play a role in the game. But they can be traded in for cards that do (you can trade or play a card, not both). Trades may be simple or quite complex:
- You can trade a single card.
- You can trade any number of matching cards (e.g., two fives or three tens)
- You can trade cards that combine to form matches. This can take many forms:
- 3 + 4 = 7 (two numbers that add to a third; you can trade in the 3, 4, and 7)
- 1 + 2 + 4 = 7 (three numbers that add to a fourth)
- 1 + 2 + 3 + 1 = 7 (four numbers that add to a fifth)
- 3 + 5 = 6 + 2 (two numbers that have the same sum as two others)
- 3 + 5 + 1 = 6 + 3 (three numbers that have the same sum as two others)
- 3 + 2 = 1 + 4 = 5 (two pairs and a single with the same value)
Again, it’s important that kids are supported in moving to progressively more challenging skill levels. This doesn’t happen automatically just by playing. Even very young kids can play using #1 and #2. If shown how to count the objects on the cards—and then how to count on from larger numbers—they can start building fluency with #3a-d.
#3e-f are significantly harder than a-d. For e, you can sum all five cards and divide by two to find a target number. For f, you can sum all five cards and divide by three to find a target number. This requires more advanced adding strategies, awareness of what it means to divide into halves or thirds, and the ability to find a half or a third. Finding half of any number up to fifty is no small feat and, again, is one that’s more easily accomplished through structured variation (P-a) than through playing the game over and over.
Initially, manipulatives may be used to support learners in seeing the half: The blocks in the image below represent the total of all number cards in the hand, organized into groups of ten.
The pencil splits each chunk of blocks in half; each ten becomes five, and the eight becomes a four: 5 + 5 + 4 = 14. So we need to see if we can make two fourteens with the given cards (try it!).
9 + 5 = 7 + 3 + 4
I can trade in all my cards! Because this is kind of a big deal, we added a rule that if you discard your whole hand, you get to wake a Queen. You might include a similar rule for triples, even if they don’t involve the entire hand. Or perhaps this could allow you to choose a card (or any card except a king?) from the discard pile.
When attempting to find matches, it’s also helpful to be able to identify odd and even numbers and to understand how even and odd numbers combine. If I don’t have an even total, there’s no way I can break my hand into two equal parts. But I don’t have to add my whole hand to find out if the total is even or odd.
- Even + even = even
- Odd + odd = even
- Odd + even = odd
Why is that? And why does it matter? If you think of even numbers as paired numbers and odd numbers as those with one outside of the pairs, it’s easier to see what happens when you add them.
Here’s an odd number with another odd number (i.e., 5 + 7). Each has a protruding odd bit. When you combine them, the odd bits make a pair. Therefore, odd plus odd is even.
Every pair of odd numbers makes and even number. Let’s say I have 2, 3, 4, 6, and 8. Note that there’s only one odd card. Since it has no partner odd, the total is odd, and I can see there’s no point in trying to split my cards into two equal parts.
But if I have 3, 5, 4, 6, and 8, I can combine the 3 and the 5 to get an even number. Since the rest of the cards are also even, the sum is even. Now I know it makes sense to try and split the total in half. It doesn’t guarantee that it will work, but it’s worth trying. In this case, the total is 26, half of which is 13—and I can, in fact, make two groups of 13: 8 + 3 and 6 + 4 + 3.
What if I have 3, 9, 4, 6, and 8? The sum is even, so I total my cards and get 30. Half of 30 is 15, but I can’t make 15. My next best option is to try and get rid of 4 cards. Which one should I ignore? If I want to maintain an even sum, I should ignore an even card, which narrows my options to:
- 3, 9, 4, 6 (ignore the 8): (3 + 9 + 4 + 6) ÷ 2 = 11 (doesn’t work)
- 3, 9, 4, 8 (ignore the 6): (3 + 9 + 4 + 8) ÷ 2 = 12 (works: 9 + 3, 8 + 4)
- 3, 9, 6, 8 (ignore the 4): (3 + 9 + 6 + 8) ÷ 2 = 13 (doesn’t work)
This might seem like a lot of calculating, but if strategies are introduced gradually, they can become fluent.
Learners who can skip-count by threes will more easily recognize when the hand total is divisible by three, in which case total divided by three (i.e., divided into three equal groups) becomes the target number to look for.
You may have noticed that this game draws from a variety of skills that are not necessarily closely connected. In Math Minds terms, we would say that the game itself is not mathematically raveled (R-a). It offers multiple levels of engagement and meaningful contexts for practice, but it requires that teachers consider children’s background (R-b) and prompt attention to ways their current knowledge might be expanded.