The physics
A voltage pushes a current through a resistance. Keep a metal at a steady temperature. Then the current grows in step with the voltage. This is Ohm's law.
V = I R
A wire's resistance grows with its length and falls with its area. Its material sets how much, through the resistivity ρ. The conductivity is σ = 1 / ρ.
R = ρ L / A 1 mm² = 10⁻⁶ m²
Pull a wire longer, and it gets thinner too. Its volume stays the same, so R grows as the square of the length. Two wires of one mass have R ∝ 1 / r⁴.
Two metals that both run from end to end sit side by side. Let a rod's radius change evenly from r₁ to r₂. Then R = ρL / (π r₁ r₂).
A metal's resistance grows with temperature. The rise is nearly a straight line.
R = R₀ (1 + α ΔT)