# 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.

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 ...

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!