Let's break this down.

Let the three consecutive even integers be represented by:

• The middle integer is $n$.
• The even integer before $n$ would be $n - 2$ (since the integers are even and consecutive).
• The even integer after $n$ would be $n + 2$.

Now, the sum of these three integers would be:

$(n - 2) + n + (n + 2)$

Simplifying this expression:

$(n - 2) + n + (n + 2) = n - 2 + n + n + 2 = 3n$

So, the expression for the sum of three consecutive even integers where $n$ is the middle number is:

$\boxed{3n}$

If you have any questions or need further clarification, feel free to ask!

Here are some follow-up questions for further exploration:

1. What would be the sum if $n = 4$?
2. How would the expression change if we were dealing with consecutive odd integers?
3. Can you find a general formula for four consecutive even integers?
4. What happens to the sum if the integers were consecutive multiples of 3?
5. How can this expression be used in a real-world scenario?

Tip: Always ensure that you check if you’re dealing with consecutive even or odd integers, as this changes the approach slightly.