You may also like

problem icon

Fencing

Arrange your fences to make the largest rectangular space you can. Try with four fences, then five, then six etc.

problem icon

Two by One

An activity making various patterns with 2 x 1 rectangular tiles.

problem icon

Hallway Borders

A hallway floor is tiled and each tile is one foot square. Given that the number of tiles around the perimeter is EXACTLY half the total number of tiles, find the possible dimensions of the hallway.

Pebbles

Stage: 2 and 3 Challenge Level: Challenge Level:2 Challenge Level:2

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.


The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2012. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.