If I were teaching math at the beginning algebra level (an occasional fantasy in which I indulge), maybe I would have a take on this.
It would go along these lines: Numbers are conceptual thingies, so we can always play mental games with them. But they are also tools that we use for specific tasks: Describing various things about the real world.
And different kinds of numbers are useful for describing different sorts of real things that exist in the real world.
Whole number are useful for counting whole things.
Fractions are useful for measuring amounts of things that aren’t whole: Distances, weights, time intervals, areas, etc.
Note that negative numbers aren’t useful for the above tasks. You can’t have -5 apples in your hand. If Johnny has 5 apples, he can’t give 7 of them to Susie. You can’t have a basket with -5.37 bushels of wheat. This is why negative numbers are confusing to beginners: They seem abstract and nothing but abstract.
BUT: Here is my introductory lecture on negative numbers. And guess what, it has nothing to do (at first glance) with numbers less than zero!
Suppose an airplane is flying at 35000 feet. The pilot changes his altitude by 400 feet. Now how high is he flying? The astute student will ask: Did he go 400 feet UP or 400 feet DOWN? The number 400, by itself, doesn’t tell us enough. So we have to add a qualifier (UP or DOWN) to the number. Many similar example could be given: The temperature, at first 40 degrees, changed 10 degrees. Up or down? Countable or measurable quantities can have “before” and “after” values, and they can change. (This concept, that variables are variable should, I think, be taught more thoroughly very early in algebra. I would even include teaching the Δx notation and terminology.)
So numbers can have direction to them. Mathematicians have decided that it is useful and convenient to think of the direction as part of the number – The airplane changes its altitude by 400↑ feet or 400↓ feet. The numbers 400↑ and 400↓ are two distinctly different numbers. Note that this explanation says nothing about any numbers being less than zero; nothing about Johnny and his -5 apples or Farmer Bill and his -85.7 bushels.
Once we get that understood, this we can discuss the notation: We write 400↓ as -400 and call it “negative 400” – and then I would discuss other useful interpretations. Now we can discuss why 0 (zero) isn’t really always 0 (as in nothing). What we call zero may just be an arbitrary or convenient starting point, like 0 altitude (sea-level) or 0 degrees (temperature), which aren’t really at the “bottom” of their scales. Thus, we can have numbers even smaller. Thus, negative numbers less than zero. Right here, in the real world! It’s not just an obscure abstract concept!
Okay, teach them all that early in their Algebra I class. Now they have all the right basic concepts: That numbers are useful to describe things in the real world; and that different kinds of numbers can be useful (or not) for describing different kinds of things in the real world. This will leave their minds open for the evil day that we must teach them even more weird kinds of numbers to describe even more weird kinds of things that are found in the real world. Thus, when the time comes to introduce imaginary numbers, their minds are already primed for the idea.
All you need to add are some plausible (understandable to 9th graders) kinds of situations that can be described neatly with imaginary numbers. Here, I would make some vague remarks to the effect that they are useful in electronics problems. And I would mention that they can be useful as directed numbers in a two-dimensional plane, not just as ↑ and ↓ numbers. At this level of their education, I doubt that I could give any actual examples in detail.
But a key thought here is to introduce all the earlier concepts (like fractions, negative numbers, variables, etc.) in ways that prime them to be ready for new ideas to be added as we go. I’m not so sure that this is typically done very well in most Algebra classes.
Here’s a thought: Beginning algebra students are taught that variables like x or y, as seen in equations, represent specific numbers. They are NOT taught until much later (if at all, in Algebra classes) that variables represent quantities that change. If a problem involves “before” and “after” numbers, two different variables are used to represent them.
So, early on, challenge your students with this question: If variables represent specific numbers, why are they called “variables”? Let them think about that for a while.
And, early on, challenge them with this question: Why are the Real numbers called Real numbers? Are there also Unreal numbers? Let them wonder about that for a while too.