Fractions, factors, and functions.
A large portion of my career to date has been spent musing over how to engineer classroom environments that infuse meaning into these three mathematical structures. When it comes to polynomial factoring, the area model has provided the most success. After connecting 2-digit by 2-digit multiplication, the area model becomes a beautiful visual to make sense of the “adds to ___; multiplies to ___” phrase that echoes around the room.
But we don’t keep the area model around forever. Once we’ve used the model to build meaning, we mobilize that understanding in more symbolic situations in a careful, deliberate march toward mathematical abstraction.