A group of children from Manorfield Primary School, Stoney Stanton sent in lots of ideas:

T.K and A.H said:

The pattern: As you double it each time, the shape changes from a square to a rectangle repetitively as each time you add a square to make a rectangle (square $+$ square = rectangle) and add a rectangle to make a square (rectangle $+$ rectangle = square).

A.H and E.R said:

We have noticed that each shape changes from a square to a rectangle because if you add a square to a square you get a rectangle e.g.:
$4\times4 + 4\times4 = 8\times4$ - a rectangle

S.B. and N.L. produced the following table of results:

 Number Shape Number of pebbles on side Area of shape Perimeter of shape 1 Square 2x2 1cm$^2$ 4cm 2 Rectangle 2x3 2cm$^2$ 6cm 3 Square 3x3 4cm$^2$ 8cm 4 Rectangle 3x5 8cm$^2$ 12cm 5 Square 5x5 16cm$^2$ 16cm 6 Rectangle 5x9 32cm$^2$ 24cm 7 Square 9x9 64cm$^2$ 32cm 8 Rectangle 9x17 128cm$^2$ 48cm

PATTERNS AND FORMULAE
It was said in whole class discussion that the pattern for the size of the shapes was:
Squares: each side is the same length as the longest side of previous rectangle
Rectangle: one side is the same length as the side of the previous square, the other side is a "new length"

A.H and E.R also said:

The formula for the area = $2$ to the power of $(n-1)$.
The pattern for the new side of the rectangles is $+1$, $+2$, $+4$, $+8$, $+16$ ... (it doubles)
The pattern for the perimeter is $+2+2$, $+4+4$, $+8+8$, $+16+16$ ... (it doubles)

PREDICTIONS
S.W. and L.B filled in their table for the first six shapes, and then predicted:
For shape number seven, the area will be double the area of shape six and the perimeter will be $8$ more than the perimeter of shape six. And for shape eight, the area will be double the area of shape seven and the perimeter will be $16$ more than shape seven.

Well done all of you - you obviously worked hard on this problem.