Trominoes
A tromino is a $3 \times 1$ block:
What sized rectangles may be made using trominoes? You can print off and cut out trominoes from this sheet . Alternatively, if you have a set of the game Jenga, then use the blocks as trominoes (but check they are $3 \times 1$ first)
Can you cover $63$ squares of an $8\times 8$ chessboard using trominoes? (Why can't you cover $64$?). If so, which square remains uncovered, and are there other arrangements of the trominoes which would lead to a different square being left uncovered?
Think of some other squares which cannot be covered with trominoes. Can you "almost" cover them, leaving only one hole? When can an $n \times n$ square be covered or "almost" covered?
Think about colouring in the $8 \times 8$ chessboard with three colours.
How many squares of each colour are there?
Why do this problem :
This problem can be a good way to demonstrate mathematics emerging out of play.Possible approach :
Using wood blocks (trominoes from three squares, like dominoes from two), arranged as a tower, a game of coordination and experimentation begins as players try not to topple the stack.
Pick up one of the pieces of wood and begin posing some questions about it.
- Can these blocks form a square ? - other than the three side by side used for each layer in the stacking game.
Let the questions flow and get pursued. Try to encourage adjustment rather than abandonment when a line of questioning appears to run out. For example a five by five square can't be made because each block contributed three squares of area and there are 25 to be covered, so there must be a hole. OK, given there's a hole, can you make a square ?
Key questions :
- What questions can you pose ? (offer an example : what rectangles can and cannot be made ?)
- What others ?
- Which seem like good questions to pursue ?
- (later) What were our questions ? Where has each question taken us in what we now see or understand ? What came out of this that we didn't know already or didn't expect ?
Possible extension :
Equal Equilateral Triangles is a good next step.Possible support :
Create a competition to produce all rectangles up to side length 10.
Perhaps use a digital camera or the camera in a mobile phone to catch arrangements quickly. A drawing record may sap energy that might be used more effectively pursuing the task.