# Math

### Example: What is the area of this rectangle?

The formula is:

Area = w × h

w = width

h = height

We know **w = 5** and **h = 3**, so:

Area = 5 × 3 = **15**

### Example: What is the area of this triangle?

Height = h = 12

Base = b = 20

Area = ½ × b × h = ½ × 20 × 12 = **120**

We are working on learning the coordinate planes which will lead us up to our 7th grade work of beginning algebra.

The plane containing the "x" axis and "y" axis

## Cartesian Coordinates

Using Cartesian Coordinates we mark a point on a graph by **how far along** and **how far up** it is:

The point **(12,5)** is 12 units along, and 5 units up.

## X and Y Axis

The *left-right* (**horizontal**) direction is commonly called **X**.

The *up-down* (**vertical**) direction is commonly called **Y**.

Put them together on a graph ...

... and you are ready to go

Where they cross over is the "0" point,

**you measure everything from there**.

- The
**X Axis**runs horizontally through zero - The
**Y Axis**runs vertically through zero

**Axis**: The reference line from which distances are measured.

The plural of Axis is * Axes*, and is pronounced

*ax-eez*

### Example:

Point **(6,4)** is

6 units across (in the **x** direction), and

4 units up (in the **y** direction)

So **(6,4)** means:

Go along 6 and then go up 4 then "plot the dot".

And you can remember which axis is which by:

x is A CROSS, so x is ACROSS the page.

## Play With It !

Now is a good time to play with

Interactive Cartesian Coordinates

to see for yourself how it all works.

## Like 2 Number Lines Put Together

It is like we put two Number Lines together, one going left-right, and the other going down-up.

## Direction

As **x** increases, the point moves further **right**.

When x decreases, the point moves further to the left.

As **y** increases, the point moves further **up**.

When y decreases, the point moves further down.

## Writing Coordinates

The coordinates are always written in a certain order:

- the horizontal distance first,
- then the vertical distance.

This is called an "**ordered pair**" (a **pair** of numbers in a special **order**)

And usually the numbers are separated by a comma, and parentheses are put around the whole thing like this:

**(3,2)**

**Example: (3,2) means 3 units to the right, and 2 units up**

**Example: (0,5) means 0 units to the right, and 5 units up.**

In other words, only 5 units up.

## The Origin

The point (0,0) is given the special name "The Origin", and is sometimes given the letter "O".

## Abscissa and Ordinate

You may hear the words "Abscissa" and "Ordinate" ... they are just the x and y values:

- Abscissa: the horizontal ("x") value in a pair of coordinates: how far
**along**the point is - Ordinate: the vertical ("y") value in a pair of coordinates: how far
**up or down**the point is

## "Cartesian" ... ?

They are called *Cartesian* because the idea was developed by the mathematician and philosopher **Rene Descartes** who was also known as * Cartesius*.

He is also famous for saying *"I think, therefore I am"*.

## What About Negative Values of X and Y?

Just like with the Number Line, you can also have negative values.

Negative: start at zero and **head in the opposite direction**:

- Negative x goes
**to the left** - Negative y goes
**down**

So, for a negative number:

- go backwards for x
- go down for y

For example **(-6,4)** means:

go **back** along the x axis 6 then go up 4.

And **(-6,-4)** means:

go **back** along the x axis 6 then go **down 4.**

## Four Quadrants

When we include negative values, the x and y axes divide the space up into 4 pieces:

**Quadrants I, II, III **and** IV**

*(They are numbered in a counterclockwise direction)*

In **Quadrant I** both x and y are positive, but ...

- in
**Quadrant II**x is negative (y is still positive), - in
**Quadrant III**both x and y are negative, and - in
**Quadrant IV**x is positive again, while y is negative.

Like this:

Example: The point "A" (3,2) is 3 units along, and 2 units up.

Both x and y are positive, so that point is in "Quadrant I"

Example: The point "C" (-2,-1) is 2 units along in the negative direction, and 1 unit down (i.e. negative direction).

Both x and y are negative, so that point is in "Quadrant III"

Note: The word **Quadrant** comes form *quad* meaning **four**. For example, four babies born at one birth are called *quadruplets*, a four-legged animal is a *quadruped*. and a *quadrilateral* is a four-sided polygon.

## Dimensions: 1, 2, 3 and more ...

Think about this:

**1**

**2**

**3**

The Number Line can only go:

- left-right

so any position needs just **one number**

Cartesian coordinates can go:

- left-right, and
- up-down

so any position needs **two numbers**

How do we locate a spot in the real world (such as the tip of your nose)? We need to know:

- left-right,
- up-down, and
- forward-backward,

that is **three numbers**, or 3 dimensions!

## 3 Dimensions

Cartesian coordinates can be used for locating points in 3 dimensions as in this example:

Here the point **(2, 4, 5)** is shown in

three-dimensional Cartesian coordinates.

In fact, this idea can be continued into four dimensions and more - I just can't work out how to illustrate that for you!