How does the resistance of a conductor change when the temperature is increased?

One of the factors that affects the resistivity of a metallic conductor is the temperature. As the temperature increases, the ions vibrate with greater amplitude. Because of this, the ions and the electrons are more likely to collide and the electron finds it more difficult to move through the conductor. Since there is an impede  in the electron flow as the temperature increases, the resistivity of metallic conductor increases. This relationship is shown in the formula of temperature dependence of resistivity which is:

`rho = rho_o[1+alpha(T-T_o)]`

where 

`rho`  is the resistivity at some temperature T (in Celsius degrees),

`rho_o`  is the resistivity at some reference temperature T0 (usually taken to be 20°C) and

`alpha`  is a parameter called the temperature coefficient of resistivity.

Moreover, resistivity of a conductor is directly proportional to its resistance as shown in the formula:

`R= rho *l/A`

Isolating the resistivity of the conductor, it becomes:

`rho = R * A/l`

Plugging this to the formula of temperature dependence of resistivity, the equation becomes:

`rho = rho_o[1+alpha(T-T_o)]`

`R*A/l =R_o *A/l[1+alpha (T-T_o)]`

And this simplifies to:

`R=R_o[1+alpha(T-T_o)]`

Notice that the resistance of the conductor is proportional to the temperature change.

Therefore, as the temperature of the conductor increases, its resistance increases.