Electronic – Purpose of rheostat in measuring resistance

I suspect you ran into a calculation accuracy error. Let me explain.

If \$R_1\$ and \$R_2\$ are much less than the scope/DMM input resistance, the circuit you showed is practically unloaded (i.e. we assume infinite \$R_{dmm}\$).

Therefore, since \$R_1 = R_2\$, then \$V_{BB}=0.5 V_{AA}\$ (equal branch voltage divider) and \$ R_1 + R_2 = 2 R_1\$. Then look at the denominator of that formula:

$$ V_{in} R_2 - V_{out} (R_1 + R_2) = V_{AA} R_2 - V_{BB} (R_1 + R_2) = V_{AA} R_1 - 0.5 V_{AA} \times 2 R_1 = 0 $$

In other words, the denominator would be theoretically zero, giving an infinite \$R_{dmm}\$, as expected.

The problem is, that in practice the denominator won't be exactly zero, and any slight approximation of the values of \$R_1\$ and \$R_2\$ or any slight error in the measurement of \$V_{AA}\$ or \$V_{BB}\$ will greatly affect the result!

In other words, when the denominator approaches zero, it becomes much more numerically sensitive to the errors of the parameters used to calculate it.

Some metals are 'difficult' to solder, they have tenacious oxides that demand a more active flux that you usually get in cored solder.

I bought a short length of solder for aluminium for joining, amongst other things, aluminium. I tend to 'tin' the alli with the aggressive solder, and then use ordinary solder to make the joint to other metals, to economise on my use of the special stuff.

However, it's also good for soldering K-type thermocouple wire, which is nearly impossible with conventional solder, and for some (not all) of my collection of various resistance wires.

Try sourcing some 'alli' solder, it may work for your particular wire. If you want the kids to be able to solder the wire, it will be good for you to tin it first, as the flux fumes are also nastier than those of conventionally fluxed solder.

You might be better off introducing them to 'croc clips', and just leave it with them that some wires are difficult to solder.