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
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
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.
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.
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:
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 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.
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.
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:
The Number Line can only go:
so any position needs just one number
Cartesian coordinates can go:
- left-right, and
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:
- up-down, and
that is three numbers, or 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!