Learners study a sequence of drawings. They are first challenged to describe in words how they ‘see’ the general structure of the ‘drawings’, and how that structure gives the number of a component in each drawing. They then express their different generalisations algebraically. Having arrived at different equivalent algebraic expressions for the same thing, they feel the need to understand how the expressions are the same.
An interesting way to help learners appreciate that algebraic manipulation is both useful, and derived from 'ordinary' thinking about numbers, is to present them with a structure that can be seen in different ways. Then challenge them to describe the structure as they see it, at first in words and then algebraically. When they arrive at different expressions for the same structure, they feel the need to be able to convert one expression to the other.
If the challenge is to describe the general structure of drawings in a sequence of drawings, it leads naturally to 'the nth term' of a sequence.
For example, the sequence might be a sequence of 'hollow' squares made with small squares (such as square ATM MATs) :
Discussion about the general structure of the 'drawings' might develop like this:
T: Suppose you are making the sequence with square mats.
Starting with any 'drawing' in the sequence, how will you make the next drawing?
A: Add one square to each side?
T: How?
Pause
A: Each 'ring' of squares has four corners. So I pull the corners apart and then add one square to each middle bit.
B: I don't do it like that I just add a square on to the end of each side, then I move the positions of the sides.
C: The third one is a ring around the first one.
D: And the fourth one is a ring round the second, and so on!
Pause
T: Can you see how the number of squares in a drawing is related to its position in the sequence?
Pause
A: Each 'ring' is made from four corner squares and four 'middle' squares that complete the sides. The number of squares on each side is one less than the position of the ring in the sequence.
So the number of squares in a ring is four, plus four times one-less-than-the-position-of-the-ring.
B: You can split a 'ring' into four equal bits. It's just four times its position!
D: They are proper whole squares, with smaller proper whole squares removed from the middle.
So the number of squares in a ring is the square of 'er' the number that is the position plus one, minus the square of 'er' the number that is the position minus one.
T: Choose a letter to be the number that gives the position of the ring in the sequence.
What is your letter?
D: p
T: What is the number of squares in the pth ring?
D: p plus 1 squared, minus p minus one squared.
T: Can you write that without words?
D eventually writes: (p + 1)2 - (p - 1)2
T: So the number of squares in the pth ring is that (pointing to (p + 1)2 - (p - 1)2)?
A: But my way it's not that complicated! It's four, plus four times one less than p.
T: Can you write that without words?
A eventually writes: 4 + 4(p - 1)
T: So the number of squares in the pth ring is that (pointing to 4 + 4(p - 1)?
B: But it's just 4p! That was what I said.
E: But it's - they're all the same! How can it be (p + 1)2 - (p - 1)2 and also 4 + 4(p - 1) and also just 4p?