I have been a fan of origami since I was a youngster but never really thought of its application to geometry until the last few years. There are a ton of books and on-line resources for origami creations. One of the activities I like is the origami tetrahedron and cube shapes that you blow up. These can be hard for little children but older kids like to do those activities and like the flexegons manipulative there is conceptual and spatial intelligence involved with making these shapes. I do suggest using a large square of paper for beginners as some of the Japanese origami paper I have bought is too small to make a cube that can easily be blown up.
Another activity to show with squares of paper is fraction reductions and the Fibonacci Sequence Model.
This photo is a diagram of fractions made into a spiral by taking a pieces of different colored origami papers and cutting them in 1/2's reducing down and layering them by gluing them down in a spiral. It shows the spatial connection of fractions to whole numbers and the concept of how numbers form patterns and how these relationships in forms build the world around us. I used Mortenson Math blocks or tiles to make the model of the Fibonacci Sequence below. It could also be done with papers cut to scale too. Grid paper or graph paper is excellent to use by coloring blocks to show the concept.
In practical life context, our community has built a spring training facility, Riverside Park for the Chicago Cubs. We the people got a public playground too and this structure in the park is a big model of hands on geometry. Here again we are able to see the sacred geometrical shapes in building. We can see and discuss how the triangle shape is more stable in building because we can experience that strength by climbing on the structure where it is supported by the square shapes and the triangular shapes. We can feel a equilateral triangle and a isosceles triangle and see it's relationship to other shapes as building blocks. We can see the connections these archetypical forms become other shapes and how they connect to form hexagons and on to the Platonic Solids.
I have been thinking about how to teach to the senses in math. How to make my learner have a personal relationship with math and numbers. I think if we can develop that synesthesia muscle in our brains where we feel math perhaps it will serve us better as we learn and enjoy math. Synesthesia may be how we heal from our fears about math. We have to form more personal relationships between numbers. Learn more about Synesthesia at this link below.
How would Vi Hart do this? Well I can only wonder? Here is another of her open-ended ness videos.