Shortcuts–for teacher's private pleasure?
deflashing to do
There are only a few quantities in these three element circuits containing either only series or only parallel connections:
V: potential difference across the battery
R: effective resistance of the circuit
I: current in the battery
R1: resistance of first resistor
I1: current in first resistor
V1: potential difference across first resistor
R2: resistance of second resistor
I2: current in second resistor
V2: potential difference across second resistor
But you can still do some rather involved algebra to give yourself shortcuts to finding any one of these quantities, given a number of others. These shortcuts, once committed to memory, enable you to avoid working through all of the steps necessary to analyse the circuit, starting from the physical behaviours.
Deriving the shortcuts might give you pleasure, which is why they are included here, but these shortcuts are only useful if you know the lie of the land and where you are going. Children are not in this position, so it's probably better to keep working through all of the steps. This has the further advantage of keeping the classroom discussions close to the physical behaviours.