A rare day away from school today, at the Leicestershire Heads of Maths meeting run by the wonderful Jan Parry. She did a great session on concrete and visual approaches to algebra, which reminded me about the first time I saw the visual representation of the process of completing the square (it was actually on "The Story of Maths" starring Marcus Du Sautoy on Channel 4. I shared it recently when I presented at Maths Hubs and Spokes as part of my session on concrete approaches for conceptual understanding (technically not concrete I know, but still) and I thought I would share it here.
Du Sautoy showed that by chopping off half of the extra length at the end of the rectangle and moving it to the top, you created a near-square; just 3 short in each direction (i.e. 9 square units short). When I first saw this my mind just exploded - as a maths graduate I of course knew how to complete the square and what it told me about the graph/roots/minimum point etc of the quadratic but I hadn't even realised that I had no real understanding of what that process looked like in a non-abstract representation. Now I always start completing the square by showing this to pupils and getting pupils to work with similar images before going to the abstract. Hopefully, if you haven't seen this before yourself this will allow you to have that "Oh yeah!!" moment that I had when I first saw it - which will hopefully mean your pupils understand from the beginning what it means to physically "complete the square".