DESCRIPTION OF INTERACTIVE

Graphical Representation:
Graphs that are used to display patterns start with 2 number lines.
One number line is placed horizontally - the independent axis and the other is placed vertically - the dependent axis.
They are joined together and intersect at 0 on each axis creating the point (0, 0).
Depending on the numbers in the relationship, the scatter plot may use one quadrant or it may use all four quadrants.
The graph of a linear relation will be one straight line or a set of points that follow a straight line.
Linear relations can decrease as you move across the graph from left to right.  
Linear relations can increase as you move across the graph from left to right.  
Linear relations can be constant where there is zero change as you move from left to right. It is a horizontal line.
A vertical line is also a linear relationship. It does not move from left to right; there are many dependent values for the one independent value.
If a graph of a relation is not a line or a series of points that follow a line, then it is a non-linear relation. There are many types of non-linear relations. One type of non-linear relation is a quadratic relation.

Numerical Representation:
In a mathematical pattern, there are two quantities linked together, such as length and area. In the relation, the quantities may take on specific values. Because the two quantities can  change or vary, they are called the independent variable and the dependent variable. The dependent variable always depends on the independent variable.
The numbers that describe a pattern need to be organized to use them to create other representations such as a graph. It is important to know which numbers are connected.  
Numbers may be displayed in a vertical table of values.
The independent value (input) is in the left column and the dependent value (output) is in the right column.
A horizontal table of values may be used to save space on a page. The independent value is the top row.
Other times, the numbers may be organized as ordered pairs. The first number or coordinate is the independent value and the second number or coordinate is the dependent value.
A numerical representation of a pattern is a linear relation when there is a constant increase or decrease in the dependent values as long as the independent value has its own constant change.
An example of a linear relation is as the time increases by 1 hour, the temperature is dropping by 3 degrees.
There is a decrease of 3 degrees every hour.
Another example is as the base of the rectangle increases by 2 cm, the perimeter of the rectangle increases by 8 cm.
If the numbers in the input column are not in order, it is difficult to determine if the numerical representation is linear. The table of values needs to be reorganized.  
When the values are reorganized so that the independent value (width) is always increasing, you can tell if the relation is linear or non-linear. Since the dependent value (area) does not have a constant increase in value, it is non-linear.
Sometimes, the dependent values for a relation will be given as a string of numbers separated by commas. The independent values, term numbers or position numbers,  are understood to be 1, 2, 3, 4, 5, 6, ... when a pattern is presented in this manner.

Algebraic Representation:
Relations are often described using a pattern rule, word equations, formulas or equations using letters to represent the variables. We often see x and y chosen.  
Pattern rules, word equations, formulas or equations using letters are different ways to write a pattern rule; the style changes as we gain experience working with patterns. When a pattern rule is written in words only, the rule describes the dependent variable; the independent variable is understood to be the term numbers 1, 2, 3, 4, 5, ... similar to the sequence of numbers in the numerical representation.
When a rule is written as a formula or an equation, an equals sign (=) is included. In general, an equation or formula will be written with the dependent variable on the left side of the equals sign.
Pattern rules often involve arithmetic operations (arithmetic operations include +, - , x, /) along with the independent and dependent variables. Rules that represent a linear relation may only use the operations of addition, subtraction, multiplication and division.  Other operations such as squaring (multiplying a number by itself) cannot be used for linear relations. The equation will not show exponents on either variable.
If either variable is multiplied by itself, then the relation will be non-linear.  The equation for a non-linear relation may include exponents on the variable(s).
Mathematical equations with letters are most often written using the letters x and y.

Pictorial Representation:
Some say: “A picture is worth a thousand words.” Mathematicians, architects, engineers and others, will often use pictures or models to present their ideas. To illustrate a linear pattern, you can use blocks, tiles and other geometrical shapes to create 3D models.
Pictorial representations of linear patterns will show a constant increase or decrease in the total number of items as you progress through the terms. It is also important to indicate the position number or term number; some patterns will start at 0, others may start at 1 and sometimes they may start at different values. The starting point depends on the context of the question.
A linear pattern as a pictorial representation could include:
A pattern that begins at 0 with 2 yellow blocks; the total number of blocks increases by the addition of 3 blue blocks at each step.
A pattern that as the position number increases from 0, the number of blocks decreases by 4. The pattern is a linear relation.
A pattern where the term number starts at 1 and the number of blocks increases by 4 each step - 1 block is added to each of the 4 arms.

Verbal Representations:
Mathematics is easily shared with graphs, symbols, numbers and pictures but there are many situations where using written words can be a less complex way to share. Think of using the computer - it is easier to write a sentence than writing equations or making graphs. Also, mathematics is used to solve problems, and problems are usually communicated using words.
When reading or writing about a relation, it is important to use appropriate vocabulary; the words can help you decide if the relation is linear or non-linear. There is not a special list of words to use; you have to read the situation described and think about how the words relate to common choices that you will see.
As you read over the examples below, ask yourself:
What words are used to state a starting amount?
What is the value of the increase or decrease? How do you know it is an increase? What words would be used for a decrease?
Are there any words to show a constant change in the increase or decrease?  
Examples of linear relations:
Read each of the descriptions and think about why it is a linear relation.
Ramin has 2 hockey cards; he gets 5 more cards with every purchase to have a total number of cards.
The total cost for a banquet includes the $250 rental cost of a banquet hall and the $45 meal cost for every person that attends the banquet.
Examples of non-linear relations:
Read each of the descriptions and think about why it is a linear relation.
The height of a ball is measured over time. Jamal holds the ball in his hands and tosses it up. The ball stops at the maximum height and then returns to Jamal’s hands.
The maximum area of a rectangle with a set perimeter will have equal side lengths - it will be a square.

You have looked at a lot of information about the characteristics of a linear pattern in five different representations. You also read about and saw the differences between linear and non-linear relations