Familiar Shapes:

Use a triangle, circle, square, pentagon, trapezoid, etc, for your shape. Be sure to name the shape and talk about it: "Now let's try a triangle with 3 sides that are all different!"

Obstacles:

Draw a line coming in part way from the side of the paper. Shapes aren't allowed to overlap with the line. Or, cut the paper into a shape itself. Remember, ideally your shape will fit around 5-8 times on the piece of paper. If it fits more than that, the game can be tedious for young tracers.

Roll Out the Dough:

Instead of paper and cardboard, use cookie cutters to cut out shapes from actual cookie dough (or use play dough)!

Symmetry Showdown:

Choose a shape that is asymmetrical. Then, Player 1 traces the shape brown-side-up, and Player 2 traces the shape color-side-up. (Or, if you aren't using a cereal box, write each player's name on their side of the shape.)

Solitaire Puzzles:

Pick your favorite shape, and then see how many times you can fit it on a single page. Try again and see if you can fit even more. Or, try to trace the shape the fewest number of times so that it can't be traced on the paper any more without overlapping.

Symmetry:

Play the Symmetry Showdown variation. Notice that the two sides of this shape are mirror images of each other. To see this better, fold a paper in half, then trace the shape on either side of the paper, so that the middle fold acts as a "mirror," and the shapes are reflections of each other on either side of the fold.

You can check to see if you got the shape oriented right by tracing in a dark color, then folding the paper in half and holding it up to the light. How closely do the shapes overlap?

Strategy:

A classic math anecdote involves the following game: Take turns placing pennies on a square table. Whoever can place the last coin wins. Player 1 can guarantee a win by placing the first penny in the exact center of the table. Then, wherever Player 2 places a penny on the table, Player 1 responds by placing a penny at the point on the exact opposite side across the initial center penny. On every turn, this "exact opposite" spot will always correspond to the place Player 2 put their penny, and so there will always be an opening for Player 1, until the table is full and Player 2 loses. (Of course, this strategy works better in theory than it would in practice. It would be very difficult to always put a penny at the exact opposite point. See our game Carronade to try something similar!)

Such a strategy wouldn't be useful for Cookie Cutter (unless your shape were something like a circle) but there are some choices that players can make to thwart their opponents. When there is room for 2 or 3 more shapes on the paper, ask your child how many more times they think the shape will fit. Whatever their answer, ask them to figure out who would win if their guess is correct. Their answers to these questions can affect their next move.

For example, if they guess 2 more shapes can fit, and it is their turn to trace next, let them figure out that they will be able to draw one more shape, and then their opponent will draw the last shape to win. In such a situation, it would be best for them to try to draw their shape in the middle of the remaining space, in such a way that their opponent will not have room for another tracing.

Tiling the Plane:

Some shapes fit together more snugly than others. If you play Cookie Cutter with any triangle or quadrilateral (4-sided shape), ask your child to try to trace the shape on the paper as closely as possible. Let them discover that these shapes actually fit perfectly together, like tiles on a kitchen floor. When shapes fit together precisely, with no space between, and can cover an entire piece of paper (no matter how big), this is called "tiling the plane." (A "plane" is the math term for a flat 2-dimensional surface, like a piece of paper.) What other shapes tile the plane?