Mea culpa about translation.

For a moment I was thinking in homogeneous coordinates - that's not the right thing to do in the introductory phase.

Thanks for catching the error and making an important point. I am letting my original comment stand unedited so that your point stands.

About rotations though, one need not let the cat out of the bag and explain what addition of rotation is *.

One simply defines addition of two linear operators as the addition of the vectors that each would have individually produced. This can be demonstrated geometrically with arrows, without fixing coordinates.

* In 2D it's a scaled rotation.