Direct Instruction vs. Inquiry

This is my 18th year teaching middle school math- all in Maryland, but across several different counties and schools.  I've been at my current school for the last 10 years.  My focus in my classroom is helping students to build their understanding of concepts through a mix of inquiry based learning and direct instruction.  Growing up, direct instruction was "lecture and take notes".  I truly believe that students don't gain true conceptual understanding unless they have the opportunity to "discover" it.  When I plan lessons, I really try to think about how I can get students to figure it out...what questions can I ask, what problems will allow them to "happen" upon some aspect that will bridge what they know to what they need to learn.

I love the "3Acts" type of lesson and have really felt true student engagement during those activities.  Next year our district is going back to 47 minute periods after many years of a block schedule at the middle school.  I am so disappointed because I'm concerned that many teachers who were willing to dabble in inquiry will go back to lecture/notes because students can "handle" that in a short period.

I guess my question is this...does it have to be all one or the other?  Or can lessons be engaging and yet still directed in some ways.  Is it still direct instruction when groups are exploring different aspects of a problem.  For example, on Tuesday I'll be trying out the following lessons with my Algebra students...     

Graphing Systems Lesson    

Each group will get the same initial problem...graph the system and identify the solution.  We haven't talked about systems at all...students know how to graph an equation so I'm thinking that they will very quickly figure out that the solution is the point where the lines intersect. 

Next each group goes in a different direction...one focuses on justifying the solution, another explores what a system might look like that has no solutions, etc.  My plan is that after groups have had time to explore their concept they will share what they have discovered with the rest of the group.

The end goal is that students will be able to graph a system of linear equations, identify a solution (if there is one, and justify that it is a solution to the system. 

So is this direct instruction? Inquiry? Somewhere in between?