The featured Math Game this week involves no arithmetic, no counting, and no calculation. But it is an excellent (and enjoyable) introduction to the concept of plotting points on the Cartesian plane. Granted, the orientation of the vertical axis in the game is inverted from the usual Cartesian y-axis: the index increases, rather than decreases, as one moves downward. And points are plotted in the open squares, rather than on the intersections of the grid lines. But these are small issues, and we've found they cause no problems for children as they transition to the real Cartesian plane.
Now, if you really want to build familiarity with the actual Cartesian plane, one variation is to use a paper printout of the xy plane. (They're available on TeachersPayTeachers.com for $2.00 or less.) Or, you can make your own xy plane, with coordinate limits of your choice. You can make it as large as you want. And I would recommend you go ahead and center your printout at the origin, so that both positive and negative coordinates are used. It's never too early to introduce the concept of negative numbers.
When you're ready to play, simply draw an oval of the appropriate size around the string of points where you want your ship to be. When your opponent scores a "hit", place an X on the point within the oval where the "hit" occurred.
To begin with, your children will fire in a somewhat random fashion, naming points as the mood strikes them. But as they mature, you can help them develop a more systematic approach. Teach them to cover the board with a coarse "net" of shots. Then, if the small ships still escape undetected, gradually refine the "net" by subdividing the open sections.
When your children are somewhat more sophisticated, start getting creative with the game, and customize your own weapons. I've played the game sometimes where each player has several mines available, of varying size. A mine has an area effect. When using it, you specify the mine's location, and any portion of a ship within the detonation radius of the mine is blasted. Of course, your opponent has to notify you of the mine's effect, and where the "hits" occurred.
I've also allowed a certain number of sonar "pings" per player. When a using a sonar "ping", you specify a square region on the board, and your opponent has to notify you of the presence of any ships within such a square. They don't have to give you the coordinates of the ships. They just have to say "Yes, I have a ship somewhere in there", or "Nope, keep searching". And if they have multiple ships in the square, they have to tell you how many ships there are.
In summary, Battleship is a wonderful game. Lots of fun, lots of variations. And it's great for teaching the plotting of points in the plane.
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Brian and Melanie Fulton both earned doctoral degrees in mathematics at Virginia Tech. They formerly taught math at the university level, and now run a hobby farm while accuracy-checking collegiate mathematics texts. They homeschool their four children, frequently employing the aid of chicken, dairy goat, cat, and dog tutors.