Tiles make rectangles
Algebra tiles for products and factors
A tile kit
A tile kit has three kinds of tile: a big square x by x (the x²-tile), a strip x by 1 (an x-tile) and a small square 1 by 1 (a unit tile). The length x can be any number, so a strip never matches the unit squares exactly.
Can 1 x²-tile, 7 x-tiles and 12 unit tiles be packed into one rectangle with no gaps?
What this lesson covers
The idea
With x²-tiles, x-tiles and unit tiles, the product of two linear expressions is shown as a rectangle; arranging a quadratic's tiles into a rectangle gives its linear factors as the rectangle's dimensions.
A tile kit
A tile kit has three kinds of tile: a big square x by x (the x²-tile), a strip x by 1 (an x-tile) and a small square 1 by 1 (a unit tile). The length x can be any number, so a strip never matches the unit squares exactly.
Can 1 x²-tile, 7 x-tiles and 12 unit tiles be packed into one rectangle with no gaps?
- Yes
- No
- Only if x is 3
Build the rectangle
Put some of the x-tiles to the right of the x²-tile and the rest below it. The unit tiles must exactly fill the corner. Find the split that makes a rectangle.
Sides are factors
The product of two linear expressions is a rectangle of tiles. If the tiles of a quadratic fit into a rectangle, the sides of the rectangle are its linear factors.
x² + 7x + 12: put 3 x-tiles to the right and 4 below, and the 12 unit tiles make a 3 by 4 array. The sides are x + 3 and x + 4, so (x + 3)(x + 4) = x² + 7x + 12. Read it both ways: multiply the factors to get the tiles, or arrange the tiles to find the factors.
Notes
Tiles show that (x + 3)(x + 4) = x² + 7x + 12, and that a quadratic whose tiles make a rectangle has the rectangle's sides as its linear factors.
Check yourself
Build the rectangle with sides x + 2 and x + 5. How many x-tiles does it need?
Answer: 7
2 + 5 = 7 x-tiles. (It also has 2 × 5 = 10 unit tiles, so (x + 2)(x + 5) = x² + 7x + 10.)
The tiles of x² + 6x + 8 form a rectangle. What are its sides?
How many unit tiles are in the rectangle with sides x + 3 and x + 6?
Answer: 18
3 × 6 = 18 unit tiles.
Why can the tiles of x² + 6x + 7 not form a rectangle with whole-number sides?
- x + 2 and x + 4 — correct. Yes: 2 + 4 = 6 x-tiles and 2 × 4 = 8 unit tiles.
- x + 1 and x + 8. 1 + 8 = 9 x-tiles, but we have only 6.
- x + 3 and x + 3. 3 + 3 = 6 x-tiles, but 3 × 3 = 9 unit tiles, not 8.
- Because 7 is an odd number. It is not about odd or even. The corner of unit tiles and the number of x-tiles must agree.
- 7 unit tiles can only make a 1 by 7 corner, and that needs 8 x-tiles, not 6 — correct. Yes. The corner and the x-tiles must agree: the two sides add to 6 and multiply to 7, and no whole numbers do both.
- Because there are too many unit tiles. Unit tiles are never too many by themselves. The corner must match the x-tiles, and here it cannot.