# 9.1 Exploring Patterns and Processes

## Activity: Tile Pattern

Suppose you are tiling a bathroom or kitchen, and the last row of square tiles is to be a decorative border made up of blank tiles and patterned tiles as shown below.

A friend has offered to help. How would you describe the pattern and how to arrange the tiles?

There are lots of ways of tackling this. For example, you might say that you will need some blank tiles and some patterned tiles with the “bridges” on. Start with a bridge tile, and then put a blank tile next to it. Take another bridge tile, but turn it around so that the bridge is upside down, like a smile, and put it next to the blank tile in the same line. Then put another blank tile next to the smile tile. This is the pattern: Bridge, blank, smile, blank, bridge, blank, smile …

Or you may have decided to draw a picture of the tiles or demonstrate the pattern with the tiles themselves. Whatever you do, it’s probably easier to remember and apply if you have recognized that the pattern is a four-tile repeat with the two types of tile.

The decorative border is an example of a type of geometric pattern that has many applications in art, crafts, and design.

The next example is a number pattern which appeared in China and Persia over 700 years ago, but is still used by students in mathematical and statistical problems today, and it even appears in chemistry. It is known as Pascal’s triangle and the first part of it is shown below. [ Blaise Pascal was a seventeenth-century French mathematician who studied the properties of this triangle. ]

You can continue the triangle indefinitely by following the pattern.

Each row of numbers starts and ends with the number 1. Look at each pair of numbers in the last line of the triangle. For each pair, add the two numbers together and write their sum on the line below as shown. This process generates the next row of the triangle.

You can watch a larger version of Pascal's triangle being built in this video:

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9.0.1 What to Expect in this Unit

9.1.1 Pascal’s Triangle