## Coordinates, Sequences and Graphs

### Coordinates on Maps

Maps often use a grid or system of coordinates to fix the position of a place on a map. On this map the approximate locations of the three places can be given as: Northtown B10 Landville A11 Hare's Rocks B11

### Coordinates in Mathematics

In mathematics, the position of points can be given much more accurately.

Two number lines are drawn at right angles. This grid is called a number plane or a Cartesian graph. The horizontal axis is called the x-axis. The vertical axis is called the y-axis. The x-axis and the y-axis cross at the origin, O A point is given by a pair of coordinates (x, y). Coordinates are sometimes called ordered pairs. The first number is the position along the x-axis.The second number is the position along the y-axis. e.g. Point A is given by (3, 5) Point B is given by (-3, -2)

### Sequences In mathematics, numbers are often arranged in a sequence.
Often a sequence follows a rule or pattern. Each number in a sequence is called a term.

Sequence − Example One
Some sequences are obvious: 2, 4, 6, 8, 10... This is the sequence of EVEN numbers.
To continue writing the sequence simply keep adding 2.

Comparing this sequence to the set of counting numbers N = {1, 2, 3, 4..}

 n 1 2 3 4 5 ... y 2 4 6 8 10 ...

The sequence can be shown as a graph: The formula for this pattern is 2n. The next term, the sixth, would be 2 × 6 = 12

Sequence − Example Two
The sequence 1, 4, 7, 10, 13... is the sequence starting at 1 with three added each time.
To continue writing the sequence simply keep adding 3.

Comparing this sequence to the set of counting numbers {1, 2, 3, 4..}

 1 2 3 4 5 ... 1 4 7 10 13 ...

The sequence can be shown as a graph: The formula for this pattern is 3n − 2 . The next term, the sixth, would be 3 × 6 − 2 =16

Sequence − Example Three
Another kind of sequence is1, 4, 9, 16, 25... This is the sequence of square numbers.
To continue writing the sequence keep squaring the whole numbers, 62 = 36 etc.

Comparing this sequence to the set of counting numbers {1, 2, 3, 4..}

 1 2 3 4 5 ... 1 4 9 16 25 ...

The sequence can be shown as a graph: The formula for this pattern is n2 . The next term, the sixth, would be n236

### Patterns

Sequences can also be of diagrams or geometric patterns.
For example:

### The next pattern in this sequence would be a 6×6 square with 16 yellow squares in the middle surrounded by a border of pink squares.

### Graphs

As well as graphs for sequences, there are many other types of graphs.
All graphs must have certain features:

• A title
• Labelled axis indicating units used
• Even scales

### Line Graphs

Line graphs show how quantities are changing. Below are four examples: This graph shows the water heating up steadily till it boils after 6 minutes. Notice the changes in the boy's height as he moves through his teenage years. The number of containers changes considerably from month to month. Line graphs are often used to show distance and time.

Other types of graphs such as Cartesian graphs (x-y graphs) and scatter graphs are studied in more detail in later years.

### Statistical Graphs

These graphs are detailed in Topic 44.