Tile Tactics
How it works
Imagine you're trying to cover a 3 x 3 grid with "domino" (2 x 1) tiles and one square (1 x 1) tile. Can you always do it, no matter where the square tile is, or are there only certain places you can put the square tile so that it's possible to cover the rest of the grid with domino tiles? What about on a 3 x 5 or a 5 x 5 grid? What if you are trying to cover a 4 x 4 grid with domino tiles and 2 square tiles? Where can the square tiles go, and where can't they go for this to be possible?
In this activity, students start by exploring grids with odd numbers of squares (3 x 3, 3 x 5, and 5 x 5), which they try to cover with domino tiles and one square tile. On these grids, they try to determine which grid squares are "possible" squares for their one square tile (that is, if they place the square tile on that square, they can cover the rest of the grid with domino tiles) and which are impossible (if they place the square tile on that square, it's impossible to cover the rest of the grid with domino tiles) and try to use logic and patterns to predict which squares will be possible or impossible. Then they explore grids with even numbers of squares (4 x 4, 4 x 5, 4 x 6, etc.), which they try to cover with domino tiles and *2* square tiles.
Why we like this activity
It’s fun! Students enjoy tiling the grids and trying to figure out when it's possible or impossible.
It helps students to develop spatial reasoning.
It helps students to develop logical reasoning.
It requires students to engage in mathematical habits of mind:
- Finding and using strategies, looking for patterns, and making and testing predictions when trying to determine which squares are possible and which are impossible
It has a low floor and a high ceiling: Students can start identifying possible and impossible squares by trial and error, but finding ways to do this more efficiently than checking each square one by one requires more advanced reasoning and strategy.
