What it shows
Square tiles of side x, strips of x by 1 and unit squares are arranged into a rectangle. One x squared tile, five x strips and six unit tiles fit exactly into a rectangle with sides x + 2 and x + 3, so x squared + 5x + 6 = (x + 2)(x + 3).
The science
The area of a rectangle is length times width, and the distributive law says (x + 2)(x + 3) = x squared + 3x + 2x + 6. The tiles make each term an area: the big square is x squared, each strip is x and each small square is 1. Arranging all the tiles into one rectangle with no gaps shows the two factors as the side lengths. The unit square side must not divide the x length exactly, otherwise students will wrongly treat x as a fixed number.
Materials
Cost: under Rs 50. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Thick cardboard or foam sheet in three colours
- A ruler and cutter
- A board with a right-angled corner frame
- Labels for x squared, x and 1
- A box to hold the tile set
Safety
For this project: An adult must cut the tiles with the cutter.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
Choose x = 7 cm (any length that is not a whole multiple of the unit) and cut one 7 cm square from the first colour.
Cut ten strips 7 cm by 1 cm from the second colour and twelve 1 cm squares from the third colour.
Glue two thin cardboard rails at right angles on the board as a corner to build rectangles against.
Place the x squared tile in the corner, then arrange 5 x strips: 3 along one side and 2 along the other.
Fill the remaining corner with the 6 unit squares; they should form a 2 by 3 block with no gaps.
Read the sides: x + 3 along the top and x + 2 down the side, so x squared + 5x + 6 = (x + 2)(x + 3).
Try x squared + 6x + 8 and find the rectangle (x + 2)(x + 4), and show that x squared + 5x + 7 cannot be made into a rectangle at all.
Write the expansion and the factorisation on cards and mount a finished rectangle as the display.
What you should see
Six unit tiles close the rectangle for x squared + 5x + 6 and eight for x squared + 6x + 8, giving sides (x + 2), (x + 3) and (x + 2), (x + 4), while 7 units leave a gap.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1What does the x strip represent?
A rectangle of area x times 1, which is the term x.
2Why must the strips be split 3 and 2?
Because 3 + 2 = 5 is the coefficient of x and 3 times 2 = 6 is the constant, which fills the corner exactly.
3Why can x squared + 5x + 7 not be made into a rectangle?
No two whole numbers add to 5 and multiply to 7, so no arrangement of strips leaves a corner that 7 unit squares fill.
4What is (x + 1)(x + 4) with tiles?
One x squared tile, 5 strips split 1 and 4, and 4 unit squares: x squared + 5x + 4.
Take it further
Make red tiles for negative terms and factorise x squared take away x take away 6 as (x take away 3)(x + 2).
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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