Algebra tiles: factorising by rebuilding the rectangle

0:00Here is a rectangle. Nobody has told you how big it is, only what its sides are called. One side is x plus three. The other side is x plus four. x is a length, and we are never going to decide what it is. That is the point. So what is the area? You know the answer will be a quadratic. But the answer is not what I am after. I want you to see where every piece of it comes from.

0:29Because by the end of this, factorising will stop being a hunt for two numbers that happen to work, and become one question: do the pieces close into a rectangle? The ordinary way first. Multiply everything in the first bracket by everything in the second. Four products, because two times two is four. x times x is x squared. x times four is four x. Then three times x is three x, and three times four is twelve.

1:05Four loose products. Now look at where they land. One is an x squared. One is a plain number. And two of them, the four x and the three x, land on the same power of x. That collision is the only reason there is a seven at all. Seven is not something the brackets contain. It is what happens when two of the four products land on top of each other.

1:32x squared, plus seven x, plus twelve. Now build that same rectangle out of tiles. Three shapes, and only three. A square whose sides are both x. A strip that is x long and one wide. And a unit square, one by one. Watch the middle one carefully. A strip is not a square. If you draw x the same length as one, the picture stops meaning anything. Every tile comes out square and the three shapes collapse into one. So keep x visibly longer than one.

2:09Now lay them out into a rectangle x plus three across and x plus four up. One big square. Seven strips. Twelve units. Twenty tiles, and they fill the rectangle exactly. No gaps, no overlaps, nothing hanging over the edge. So where did those seven strips go? Not into a pile of seven. Three of them stand upright along the right hand side, filling a band three wide and x tall.

2:41The other four lie flat along the top, filling a band x wide and four tall. Three one way, four the other. Seven altogether. And now the seven has a reason. It is three because one bracket said three, and four because the other bracket said four. The strips do not know they are meant to add up. They are filling two bands, and the bands are as deep as the two numbers.

3:10The sum is not a rule you memorise. It is two edges. Now the twelve units, and this is the part everybody skates over. They are not a heap of twelve. They are a block, three across and four up. That corner is the only place left for them, what remains once the big square and the two bands have taken their share. And the leftover corner is three wide and four tall, because those are the two numbers.

3:41So the corner holds three fours. Twelve. The product is not a second rule sitting alongside the first. It is the area of one corner, and that corner's sides are the same two numbers that set the bands. One picture. Read it two ways. Read it outwards, from the sides in: this rectangle is x plus three, times x plus four. Read it inwards, from the tiles out: this rectangle is one x squared, plus seven x, plus twelve.

4:16Same rectangle, so those two things are equal. Left to right, that is multiplying out. Right to left, that is factorising. They were never two skills. They are one picture read in two directions. The direction that feels hard is hard only because you must work out where the cuts go. So let us go looking, and get it wrong on purpose first. Somebody hands you one big square, seven strips and twelve units, and says: make a rectangle.

4:51The seven strips have to divide into two bands. One and six? Two and five? Three and four? Take one and six. The corner comes out one by six, so it wants six units. You are holding twelve. Take two and five. That corner is two by five, so it wants ten. Ten of your units sit down, and two are left on the table. And here is the sting. Two and five does build a rectangle. It just builds the wrong one. Two and five is the rectangle for x squared plus seven x plus ten.

5:28Three and four. The corner is three by four. It wants twelve, and twelve is exactly what you have. It closes. Two conditions, people say. Add to seven, multiply to twelve. But look at what you just did. There was only ever one condition. The split has to add to seven because the strips fill two bands. It has to multiply to twelve because the corner is as big as those two bands are deep. One demand: the corner must come out the exact size of the pile of units you were given.

6:07And only one split can ever manage it. Watch the corners as the split evens up. One and six wants six. Two and five wants ten. Three and four wants twelve. They climb, and they never repeat. So the constant you were handed picks out one split, and no other. x squared plus eleven x plus thirty. Which pair? Five and six. So it is x plus five, times x plus six.

6:39Now break the easy case. Make the front of the rectangle not one x, but two. Two x plus three, across. Three x plus one, going up. The big squares stop coming singly. There is a block of them now: two across, three up. Six x squared tiles. The side band is three wide and three x tall, so nine strips go in there. The bottom band is two x wide and one tall, so two more strips go there.

7:10Nine and two. Eleven strips, arriving from two different places, which is why they are worth drawing in two colours. And three unit tiles in the corner. Six x squared, plus eleven x, plus three. So fill in the general blank. p x plus a, times q x plus b. The squares: p across and q up, so p q of them. The units: a by b in the corner, so a b of them.

7:41And the strips come from two bands that no longer match each other. The side band is a wide and q x tall, giving a q strips. The top band is p x wide and b tall, giving p b strips. Together, p b plus a q. Set p and q both to one and the middle collapses back to a plus b, where we came in. That was tested on nine hundred sets of numbers. The tiles were counted, the algebra worked out, and the two never disagreed.

8:17Try one cold. Somebody empties a bag on the table: one square, eight strips, fifteen units. Eight has to split. One and seven wants seven units in the corner. Two and six wants twelve. Three and five wants fifteen. Four and four wants sixteen. Fifteen. Three and five. So the rectangle is x plus three across and x plus five up, and all twenty four tiles go down with nothing left over.

8:49Notice what you did not do. You did not guess pairs and check them against the middle term. You asked a single question. What size corner do I need? And the pile of units answered it. Now, honestly, where this runs out. Every tile has a positive area. There is no such thing as a negative strip in a box of tiles. So x squared minus five x plus six is out of reach. You can factorise it perfectly well: x minus two, times x minus three.

9:25But you cannot build it, because there is no way to divide minus five into two whole bands of strips. Of six hundred and twenty five pairs I tested, exactly one hundred and sixty nine give an expression with nothing negative anywhere in it. And those are precisely the ones the tiles can build. Not most of them. All of them, and nothing else. So the tiles are not the whole story. But while they last, they are the reason the story is true.