Math Games: Lost Cities


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.

In our posts exploring the games Sleeping Queens and Froggy Boogie, we highlighted opportunities for learners to practice making several significant mathematical distinctions (P-b*). But neither game is designed to teach—that is, to prompt attention to—those distinctions (P-a).

Lost Cities is another game that offers opportunities for meaningful practice at varying levels of expertise—provided strategies are taught in meaningful ways. Again, we emphasize that the goal is not to allow so-called weak learners to engage using simple strategies while strong learners use strong ones. While learners may start with easier strategies, the goal is for all learners to gradually progress to strong ones.

The Rules

The game consists of five colored sets of cards numbered from two to ten. Each color also has three “investment cards” that look like handshakes. You get points by playing cards in front of you, one card per turn. Cards are worth face value without an investment, double with one investment, triple with two, and quadruple with three. High score wins. On your turn, you play one card and draw one card. When playing, you lay down cards from smallest to largest and sorted by color; investments (if any) must be played first. Whether you play one, two, or three investment cards, it costs twenty points to invest in a single color. Most of the time, you’ll want to lay down cards to collect points, but sometimes, you may wish to simply discard onto one of the five discard piles (one per color) and draw—just keep in mind that instead of drawing from the pile, your opponent can pick up the top card from any of the discard piles.

This player would score as follows:

Yellow: 3 × (2 + 4 + 5) – 20 = 13

Blue: 2 × (3 + 5 + 7) – 20 = 10

White: 4 + 5 = 9

Green: 4 + 7 + 9 = 20

Red: 4 × (4 + 8 + 10) – 20 = 68

Total: 13 + 10 + 9 + 20 + 68 = 120

If you like, you can watch this video to get a better sense of how the game works

Where’s the Math?

It’s probably obvious that there’s plenty of opportunity to practice basic facts while scoring this game. But how might the necessary background be raveled (R-b) and prompted (P-a)?

I’ve found it helpful to use a scoresheet that helps keep track of how many of each number have been played.

Given that, learners can eliminate costs of investment before adding; for example, if they had three investments, they would need to subtract three twenties, or six tens.

That can be done by crossing off tens or by crossing off twos with eights, threes with sevens, etc. That makes the rest of the skip-counting easier. But it doesn’t teach pairs that make ten.

They can also make scoring easier by grouping numbers before tallying. For example, if they have two red investments with a 4 and a 6, they could score them as three fours and three sixes, or they could combine them and score them as three tens. This could be used as an example of the distributive property of multiplication over addition.

Scoring offers a great opportunity for practicing skip-counting. Normally, learners are expected to count to a hundred by twos, tens, and fives. With this, they can score their fours by counting two twos for each and score their eights by counting three twos for each. In doing so, they are using an important relationship between twos, fours, and eights (P1). Eventually, they should also be encouraged to automatize those facts, but developing the relationship between them in this manner is a worthwhile intermediate stage that perhaps shouldn’t be rushed.

If they can also skip-count to a hundred by threes, they can similarly count every six as two threes and every nine as three threes. Although not typically done, it can be easier to learn threes if you go all the way to a hundred, because that offers enough space for the ones’ digit in multiples of three to repeat:

3          6          9          12        15        18        21        24        27        30

33        36        39        42        45        58        51        54        57        60

63        66        69        72        75        78        81        84        87        90

93        96        99        102…

When skip-counting by sixes, they can also think of every six as a five and a one: Count every “x” on their scoresheet as five, then (starting where they left off) go down the row again, adding another one for every x. Doing so can support developing understanding of the distributive property; e.g.:

8 × 6

= 8 × (5 + 1)     Instead of eight sixes, think of eight “five-plus-ones.”

= 8 × 5 + 8 × 1     Eight five-plus-ones, or eight fives and eight ones.

= 40 + 8

= 48

Similarly, when skip-counting by nines, they can count every “x” on their scoresheet as ten, then go down the row again and take off one for every x.

So how might they handle all those sub-totals? They could simply line them up and add them with the standard algorithm. To help keep them thinking in terms of place value, it can be helpful for them to do parts of this mentally. If they add up all the ones’ digits, they can put the total in the space marked “total ones.” If they add up all the tens’ digits, they can put the total in the space marked total tens—but it’s important that they put them in the tens’ place. Now they can add to find a total. Some find it helpful to highlight the tens’ digits in one color and the ones’ digits in another.

Although there are many different ways of organizing tens and ones into sub-totals, the scoresheet is designed to support engagement with with finding and counting tens, either before or after skip-counting. In other words, there is both a clear focus and room for individual choice in terms of how the goal of totaling the numbers is achieved. As a result, different approaches are similar enough to offer meaningful contrast in terms of how the tens and ones are grouped.

Once learners get good at scoring and get a sense of how the game works, they might also consider the conditions under which different investments are profitable. Consider the cards in the image below. Let’s take another look at our cards from above. How would each color have scored without the investments?

With Investments                              Without Investments

Yellow: 3 × (2 + 4 + 5) – 20 = 13          2 + 4 + 5 = 11 (slightly better with)

Blue: 2 × (3 + 5 + 7) – 20 = 10              3 + 5 + 7 = 15 (better without)

White: 4 + 5 = 9                                  (same)            

Green: 4 + 7 + 9 = 20                          (same)

Red: 4 × (4 + 8 + 10) – 20 = 68           4 + 8 + 10 = 22 (much better with)

Are investments always worthwhile? Which colors benefited and lost in this hand?

With Investments                                                      Without Investments

Yellow: 2 × (5 + 7 + 8) – 20 = 20                                  5 + 7 + 8 = 20 (same)

Blue: 2 × (4 + 9) – 20 = 6                                              4 + 9 = 13 (better without)

White: 3 × (2 + 3 + 5 + 6 + 8 + 9 + 10) – 20 = 119      2 + 3 + 5 + 6 + 8 + 9 + 10 = 43 (better with)

Note that taking time out from the game to consider these carefully juxtaposed possibilities (P-b) is important to support deeper understanding. This can be further explored by posing the question under what conditions is each type of investment (one, two, or three handshakes) profitable?

Structuring the variations necessary to do so could be strongly guided in the first example (one handshake). By highlighting use of systematic contrasts (P-a), learners might be invited to choose their own test cases for the two and/or three-handshake scenarios (P-b).

As with Sleeping Queens and Froggy Boogie, Lost Cities does not have a built-in raveled structure. However, in the context of a well-raveled program, it offers opportunities for meaningful engagement with mathematical ideas.

*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.

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