A material property times a shape

Resistivity $\rho$ comes from the atomic structure of the substance: how much the carriers are scattered as they drift. It is a table value for copper, and it says nothing about any particular piece of copper.

Turn it into an object's resistance with the geometry:

$$R = \frac{\rho L}{A}, \qquad \text{units of } \rho: \ \Omega\cdot\text{m}.$$

So a short thick copper wire and a long thin one are made of the same material and have very different resistances. Quoting the table value as the resistance of a wire mixes the two up, and the units give it away: $\Omega\cdot$m is not $\Omega$.

One dependence is direct and easy: double the length, double the resistance, because the carriers have twice as far to be scattered over. The other is not.

For a round wire $A = \pi r^2$, so the radius and the diameter reach $R$ through a square:

$$R = \frac{\rho L}{\pi r^2}.$$

Double the diameter and the area quadruples, so the resistance falls to one quarter, not a half. Run the factors one at a time and it is hard to get wrong: $r \to 2r$, $A \to 4A$, $R \to R/4$.

The same square runs backward. A $0.5$ mm pencil lead has roughly twice the resistance of a $0.7$ mm lead of the same length, since $(0.7/0.5)^2 \approx 2$, rather than the factor of $1.4$ a linear reading gives.

Length is the one dimension that scales in direct proportion, which is exactly why the two get blended. Keep them apart: length is linear, area is squared, and diameter enters through the area.

$I = \Delta V/R$ defines $R$ at any operating point. Ohm's law is the stronger claim that $R$ stays constant as the current varies, and it is true only of ohmic materials, which means most metals at a fixed temperature.

A lamp filament is not one. Push more current through it, it gets hotter, and its resistance climbs. So a lamp drawing $0.4$ A at $2$ V does not draw $0.8$ A at $4$ V; it draws noticeably less, because at the hotter operating point $R$ is larger.

The graph shows it directly. Plot $I$ against $\Delta V$ and an ohmic resistor gives a straight line through the origin, while a lamp gives a curve that bends toward the horizontal axis: each extra volt buys less extra current than the last one did. That curvature is the physics, not bad data.

Before treating $\Delta V/I$ as a constant that travels from one point to another, ask whether the device is ohmic.

Two arrangements are in common use and they mean opposite things.

1. $I$ on the vertical axis, $\Delta V$ on the horizontal. Slope is $1/R$. The steeper line is the smaller resistance, and $R$ is the reciprocal of the slope.
2. $\Delta V$ vertical, $I$ horizontal. Slope is $R$ itself, so the steeper line is the larger resistance.

The reliable habit is to say out loud what one unit of rise over one unit of run means physically. "More amps per volt" means the charges are getting through more easily, which is less resistance. That sentence survives any relabelling of the axes.

On a curved characteristic, note that two different numbers are available at a point: the ratio $\Delta V/I$ from the origin to that point, and the local slope of the tangent there. They are not equal, and the resistance at that operating point is the ratio, not the slope.