When working with a tenth grader on imaginary numbers, we used the graphing calculator to study the contrast between y = x^2 - 4 (that has 2 zeros -- at 2 and -2) and y = x^2 + 4 that hovers up above the x-axis and has no real roots only imaginary. When students are able to visualize this difference and then tie it together with the quadratic formula (seeing that the discriminant b^2 - 4ac will result in a negative number for x^2 + 4 -- and that there is no sqrt of a negative number), they can then understand and process the concepts and procedures that are key to success in working with imaginary numbers.

Imaginary numbers are best studied and remembered by sequentially (and boringly) rewriting each component of the equation. For example,
sqrt(-4) x sqrt(-36)
sqrt 4 sqrt (-1) sqrt 36 sqrt(-1)
2 i 6 i
12 i^2
Then from up above drop in a -1 for i^2
12 (-1) =