In my practical experience, it is VERY difficult to find evidence for the law of one price. However, this is taught in introductory courses in economics. Very strange.
Even for a simple thing, such as a cup of coffee, we find multiple prices all the time. The single price for an item is a very illusive idea in practice.
I agree that the law of one price is an abstract idea as it requires the absence of friction. Physicist also use models ignoring friction for mechanics, although friction is ubiquitous in real life (macroscopic) mechanics. However, the models without friction serve a purpose, as they help to make the underlying mechanism visible.
The law of one price is rather to be understood in the sense that price differences are tolerable only to an extent where friction effects come in. E.g., the buyer of a cup of coffee may simply be to busy to compare prices and accepts a higher price in the café at hand.
I think they should reverse the order. First teach the Law of Multiple Prices, which is consistent with what people observe and experience in reality. Then they can ask: under what assumptions and conditions, would we observe a single price? This seems more logical to me.
From a didactic point of view, I would start with the simpler approach. That is the model without friction. Then, ask the question why the model fails in such a central aspect.
Keep in mind, there may be multiple prices, but that doesn't mean prices are arbitrary. Behind still lurks the law of one price as soon as frictions become small compared to price gradients.
In teaching, the problem is that I can give very few common and practical examples of single prices for goods. The students challenge me, and then I am stuck.
when I'm teaching price theory in micro economics, I clearly point out that all models fail to describe reality in detail. And to start with, we need the most simple approach which clearly has the least accuracy. However, if the simple approach helps the students to find an economic perspective the model did its job. Therefore, I'm less concerned that reality is far more complex and sometimes in open conflict with simplified abstract concepts.
I doubt my students would understand the term market failure unless they know what a market is, how it should work and what economists expect the market to do.
I guess I would respectfully disagree. From a logical point of view, it may make sense to start from the simple and move to the more complex. However, when the lectures are inconsistent with what they observe, the students become restless. I usually use the example of a cup of coffee, but it is not a good example. Then I talk about a bag of rice, but in the market they know that there is no single price for rice. Then I tell them to wait for next semester to understand reality. What common example do you use for a single price?