The gradient of a line is the increase in the y-direction (y), when the x-direction (x) increases by one unit. 
Also it is easy to see that the gradient of a line is the trigonometric tangent of the angle which the line forms with the positive part of the X-axis: 

1.- If you change the value of x₀, you will see that at any point on a particular line, when the x increases by one unit, and from x₀ to x₀+1, the y always increases by the same, m units. 

2.-If you change the value of m, the slope of the line varies. 

3.-The y intercept, n, is the value of y when x=0, that is, the line cuts the Y axis at the point (0,n). You can test this by varying the value of n in the figure.

4.- Observe how a right angle triangle is formed, where the "cathetus" or short side opposite the angle a, is m, and the value of the adjacent side is 1. For this reason, the gradient also controls the direction of the line.

1.-  Observe the difference between the lines with a positive gradient, those with a negative gradient, and those with a gradient of zero. 

2.- Click on "clear", and change the value of the ordinate or y-intercept, n. You can observe the difference between lines with the same gradient but different values of n.

The gradient of the line which passes through P₁(x₁,y₁) and P₂(x₂,y₂) is:

In this figure you can change the points P₁ and P₂, and show that the gradient, m, remains the same for a particular line. 

Also you can change the value of m, obtaining lines with different gradients. 

1.- Go to the "init" button in the figure. 

2.- Copy the coordinates of the points P₁(x₁,y₁) and P₂(x₂,y₂)  into your exercise book and apply the formula given to find the gradient, m. 

3.- Check your value of m in the figure. 

4.- Calculate the gradient of the line which passes through the points P₁(1,2) and P₂(4,5). Introduce the value you have obtained into the figure and then the values of x₁=1 and x₂=4, in order to verify your calculations.