Activities for Families

• Once children learn to play this verbally, it is a great travel game.
• Use a pile of objects. Players add one or two objects to the pile until the target quantity is reached.
• Use a number line. Advance a marker along the line one or two spaces during a move.
• Use subtraction. Players start at the target, say 10. On their turn, players choose whether to subtract 1 or 2. The first person to reach 0 wins.
• Use larger target numbers as your child’s skills improve.
• Instead of winning, the player forced to hit or go beyond the target number loses.
• Allow a player to add (or subtract) 1, 2, or 3 for each turn.

The Game Rules

A target number, say 10, is chosen. Let your child choose whether to go first or second. The total starts at 0. During a turn, a person chooses to add 1 or 2 to the current total. The first person to reach the target wins.

This game can also be played with subtraction. In this version, the starting total starts at the target, which in this example is 10. On a given turn the player chooses whether to subtract 1 or 2. The first person to reach 0 wins.

Another variation is that instead of winning, the player forced to hit or go beyond the target number loses. You can also experiment with what happens if you allow a player to add (or subtract) 1, 2, or 3 for each turn.

Understanding the Game

Without analyzing anything, this game is enjoyable to play and it provides good practice with adding or subtracting 1 and 2. We could leave it at that. However, it is also a great example of two problem solving techniques that you can show to your child when they are ready: 1) learning from simpler examples, and 2) looking for patterns.

Any of the versions can be studied this way. Let’s look at one: Subtracting starting at 10 and whoever gets 0 wins. The hard part of this game is that 10 is so far from 0. So, let’s look at a simpler version. When children are asked to do this, they often suggest starting at 5 or 6 – it seems absurd to them to start at 1, but that is actually what they should do! Often it is best to start as simply as possible – that means starting at 1. If it is your move and the count is 1, you win. Do the next few. If the count is 2, you win. If the count is 3, you must lose – whether you subtract 1 or 2, you will give your opponent a winning position. If the count is 4, you will win because you will subtract 1 and put your opponent in a losing position. Continuing in this way, build up a table of results:

1W – 2W – 3L – 4W – 5W – 6L – 7W – 8W – 9L – 10W

This table makes it obvious that there is a repeating pattern of 3. When starting at 10, you should want to go first and subtract 1. What is satisfying about this is that, once you decide to look at simpler versions of the problem, the analysis is quick and easy – no tricky analysis needed. Now you are a Master at this game and you know what to do starting at any number! Any version of this basic game is just as easy to analyze.

But wait, there is one last question. Why is there a repeating pattern of 3? Once one player is stuck on a losing number which is a multiple of three, every pair of moves after that can be made to add up to 3 – if the losing player subtracts 1 the other player subtracts 2, and if the losing player subtracts 2 the other player subtracts 1.

First and foremost, playing math games should be fun, like any other game your family plays together!

Please let your child make poor plays (mistakes) without correction, and resist the urge to tell them the best ways to play. Bit by bit, your child will get better at the game, and they will learn so much more if you let them figure things out. There is no hurry.

If you see your child make a mistake, ask them to describe why they decided to do what they did. If your child is stuck and doesn’t know which play to make, ask them to describe the pros and cons of their choices, or ask them about how they solved a similar situation in the past. If your child doesn’t remember how to do a calculation, discuss with them the methods they know for figuring it out. These conversations are important for helping your child to develop mathematically.

Through math game play and math conversations, you are helping your child learn to enjoy math and develop important problem solving skills!