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A group of children from Manorfield Primary School, Stoney Stanton sent in lots of ideas:
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)
Now, Alice from St Thomas' Church of England Primary School, wrote;
On B (how many pebbles there are round the edges), the pattern starts on 4 adds 2 which makes 6 adds another 2 which is eight then adds 4 which makes 12 adds 4 again (16) adds 8 (24) adds 8 adds 16, and so on.
So it adds one thing twice then doubles the adding number and starts to do that again.
Amelia from NLCS sent in her ideas;
First double forward then double sideways and so on
Pebbles added: 2,3,6,10,20,36,72
Pebbles Outside: 4,6,8,12,16,24,32
Squares: 1,2,4,8,16,32,64,128
Well done all of you - you obviously worked hard on this problem.