Not the cup of tea for all of you, but very much that for some, so I am going ahead with it. No generic response here, but we had an exceptionally fine time doing the mathematics this past summer, so that seems worth the recounting. The tale gets off to a rocky beginning, because we set a record for false starts. Instant answer was that that's what the teachers had requested, because the wave of the K-12 future includes more geometry, earlier. But that didn't account for the enthusiasm of the planners, and it certainly didn't provide any kind of coherence for the whole set-up.
Geometry encourages developing the ability to relate two different cognitive functions.
Likewise interaction between the visual and the computational, and between the visual and the algebraic. As we designed and as we taught we kept checking in on the GFG's, and they provided exactly the glue we needed. Day One was dedicated to Pythagoras. Some groups came up with one proof, some with four, but no thumbs were twiddled. In the afternoon they tackled a batch of applications of the Pythagorean Theorem, ranging from railroads in the sun to maypoles.