This session for Association of Teachers of Mathematics 2021 Annual Conference will give you some ideas for using origami for teaching and learning mathematics.
It covers fractions, volume, curves from straight folds and more. It is https://www.atm.org.uk/write/MediaUploads/Conference/2021%20Conference/785_Day_2_Programme.pdf
Two basic folds
A valley fold is concave. A mountain fold is convex. Most origami diagrams used a dashed line for valleys. Mountain fold is usually dash-dot-dot.
Fractions
Folding in half, quarters, eighths, sixteenths, etc is easy(ish)
How about dividing the edge of a square into thirds? See end of webpage https://www.foldworks.net/home/diagrams/
Puzzle purse/menko from a square, creased into a grid of thirds. In the Geogebra file, drag the slider for ang3 to go vary the shape between a square and a pinwheel/star https://www.geogebra.org/m/dzdhg2er
http://www.origamiheaven.com/historyofpuzzlepurses.htm
If you have paper that has different colour on each side, try making the iso-area version (by T. Kawasaki in Origami for the Connoisseur by Kasahara, Kunihiko and Takahama, Toshie ISBN: 9784817090027. An excellent book that features plenty of mathematical origami and some origami mathematics).
Tematebako (or treasure chest), which is made by combining six menko (with cuts and glue) http://www.origamiheaven.com/historyofthetematebako.htm
Returning to folding in quarters:
A4 creased into a 4 by 4 grid
Make a magazine box. What size paper do you need to make two smaller magazine boxes fit inside? What are the ratios of lengths, surface areas and volumes?
Dave Mitchell’s Two-way Tube – which configuration has the greater volume, or are they the same?
https://www.atm.org.uk/Shop/Learning-Mathematics-with-Origami-book-and-pdf/act0101pk
Next origami needs scissors or careful tearing: Hexatetraflexagon – 2 by 2 grid (square or oblong). Cut out the central portion (for A4, cut out the central A5 portion). Start at the top, fold down. Fold the right across, then the bottom up. Fold the left in, but at the top left inside reverse fold so that the result has rotational symmetry of order 2. https://byopiapress.wordpress.com/2019/10/20/it-works-it-works/
Curves from straight folds
Pleat a rectangle all valleys. Turn over then fold diagonals of rectangles to make a helix. If you use a square and increase the number of divisions, what’s the limiting shape?
Try modelling the helix in dynamic geometry software. How good is your model? How well does it generalise? https://www.geogebra.org/m/yjkbmd4m
Try the same with a triangle. The example below uses an isosceles right-angled triangle (a square folded on the diagonal will do). The angles are the same in each section, so the pleats become closer together the higher up they are (geometric progression). What is the limit of the apex of triangle? https://www.routledge.com/downloads/K16368/ProjectOrigami-Handouts.pdf
Hyperbolic paraboloid: Crease diagonals of a square. Then crease between diagonals only: on one side valley fold eighths. Then turn over and valley fold sixteenths. To shape the result, start from the outer edges and refold the creases – when you reach the centre, the results is an X shape. Let the paper expand so that the adjacent corners alternately go up and down. http://www.make-origami.com/HelenaVerrill/parabola.php
Try using other shapes e.g. A4 – what creases will you make? Regular Hexagon. Harder: a circle with concentric circles – you can cut out the central circle.
https://origamisimulator.org/ Examples > Origami > Hypar; Curved Creases > Circular Pleat.
The Workshop has a couple of origami model to make:
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Waterbomb Corrugation Origami
This is a difficult single-sheet origami tessellation that needs careful and accurate precreasing. The results are well worth the effort: organic and flexible. The collapsing stage is hard but becomes easier with practice. Use a square cut from 80 gsm A4 or A5 for your first attempt. You can then use larger paper or more divisions.
Video https://youtu.be/07sCcs4tTuQ
Step-by-step PDF instructions:
http://foldworks.net/wp-content/uploads/2021/04/WaterbombCorrugation.pdf
and some Geogebra simulations:
https://www.geogebra.org/m/jqd8hz23 Half Waterbomb Base
https://www.geogebra.org/m/pvrfpzwu Waterbomb Corrugation (2 by 2)
https://www.geogebra.org/m/ajhzqkws Waterbomb Corrugation (6 by 4)
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WXYZ Origami
An intermediate 3D modular origami model of four intersecting equilateral triangles. Use twelve squares, ideally in four colours. Memo cube paper 7.5 cm or larger works well.
Video https://youtu.be/W8W5h3rkMtk
Diagrams are at http://foldworks.net/wp-content/uploads/2018/06/wxyz.pdf