Thinking Skills: Critical Thinking and Problem Solving

220 Unit 5 Advanced problem solving

5.3 Carrying out investigations

An investigation is a problem where a set of

information is given and the student is asked

to consider various scenarios, either to find

which is the best, or just to consider the results

of various options. Investigations are closely

related to modelling, in that a model may be

developed to help with the investigation.

Alternatively, a set of rules may be formulated

which determine a set of possibilities. Some

investigations can be quite open-ended,

meaning some students will be able to take

problems further, extract more detail,

illustrate the results better and so on.

The example below is investigative: you

have to consider various options and their

effect on the result. It uses the skills of spatial

reasoning and searching. Investigation

questions can use any of the skill types covered

in Unit 3 or any combinations of them.

Activity

A company making decorative wall tiles is

introducing a new range and their designer

has decided to base them on a 2 × 2 grid

system so they can be fitted together in

various combinations. She has decided to

make all possible tiles that can be created by

choosing half of one edge (starting from a

corner) and joining this to any other half-edge

on the tile, filling in the enclosed area with

colour. The tile below is one example.

12

3

4

56

7

8

The eight half-edges of the tiles are

numbered 1 to 8 on the diagram above and

any one may be joined to any other. In this

case, half-edge 1 is joined to half-edge 5.

The entire set of tiles consists of all

combinations excluding any which may be

rotated into each other; so, for example, the

two options shown below (1 connected to 5

and 3 connected to 7) are the same tile.

Reflections which result in different tiles are

included in the set. The set includes half-

edge 1 joined to itself, which will be a blank

white tile.

How many different tiles are there in the

set?

Make a symmetrical 4 × 4 pattern of tiles

which includes at least one of each and with

colours matching at all joining edges.