I have lots of ideas, which, it turns out, aren't going to fit into a comment.
If you don't need something too long, but want something that is pretty uniform and consistent, popsicle sticks (also sold as craft sticks) might work. The are pretty durable, but relatively cheap and, ultimately, disposable.
Others have suggested paint stirrers (which you ...
A cheap Venetian blind with thin plastic slats can be cut up with ordinary scissors and will yield many straightedges of any length you choose. The slight bow in the blade will flatten out when pressed against paper, and makes it easier for fumble-fingered people like me to pick up the straightedge from a flat surface.
Try shopping for a "paint guide".
These products have:
a straight plastic or metal edge
a grippable portion that is not on the edge
no ruler-like markings
Drafting triangles also have unmarked edges that are intended to be drawn along, but their accurate angles are too useful in your intended application.
Take an ordinary piece of paper and fold it. This must be the simplest way to produce an edge that is absolutely straight. A folded paper can be used to verify the straightness of a ruler, which depends on manufacturing precision.
You would be showing how to improvise a useful tool from readily available materials and you might be able to teach some ...
As for real world applications, most of the buildings around you older than 30 years were designed by an architect that manually used geometric constructions for their drawings. I learned geometric constructions in my Tech Drawing class before I learned them in maths class. Of course, CAD programs do all of that in the background now, but the reason that ...
I am a strong advocate for teaching constructions. At least when it comes to high school geometry. My experience in teaching geometry to freshmen and sophomore high school students has shown me how much can be gained from constructions. Of course, it also depends on how you teach it. You can micromanage it and take all the fun out of it. Or you can be ...
I suggest that emphasizing plane geometry problems and their solutions
could be more effective than a systematic introduction to constructive geometry.
Geometry in Problems.
MSRI Mathematical Circles Library.
Volume: 18; 2016.
Here us a ...
I've recently committed, in my college algebra classes, to presenting proofs for as many or all topics that I possibly can. This has made me much more aware of how often basic geometry constructions (transversals, etc.) are required. This includes: Pythagorean theorem, distance formula, midpoint formula, that any line is a linear equation, slope is constant ...
There are two websites that make geometric construction into a game. NiloCK mentioned euclidthegame.com in his comment. There is another construction game at sciencevsmagic.net, with a very different look and feel. I think some people will one enjoy one or the other much more, though I enjoy them both. I have only made it through about half the challenges on ...
I strongly recommend looking at http://www.geogebra.org it is free, available on almost every OS, and can be used to PLAY with geometric constructions. Playing with geometric constructions is probably the simplest way to "motivate" the general methods of geometric construction.
I question whether engagement is what is preventing students from grasping construction. I would argue that constructions are very much like geometric proofs in that they rely heavily on abstraction and formalism for something that students are already somewhat familiar with. In this thread about geometry sequencing, a colleague of mine makes the case that ...
you can use paper folding, which does not take many resources, to motivate elementary constructions. i think there is a book by johnson, d. a. published by maw that cover many basic constructions.
students don need to have heard of euclid to enjoy and benefit from solving construction problems. if the teacher practices construction, then it would be easier ...
It might be time to show them how algebra can support or deny constructions. I would have them start looking at Gauss's proof of the constructability of the heptadecagon. This could serve as an introduction to areas of abstract algebra and beyond.
Stock metal comes to mind if you want something slightly better than plastic/paper type things - in the u.s for example this can be acquired at hardware stores and cut to size relatively easily, here is a link for reference: https://www.homedepot.com/p/Everbilt-3-4-in-x-48-in-Aluminum-Flat-Bar-with-1-8-in-Thick-801917/204273967
Take a first step by forbidding straightedges also.
This will make them understand that every construction possible with straightedge and compass is also constructible with compass alone (Mohr-Mascheroni).
You can introduce with Napoleon's problem: Divide a circle in 4 equals arcs. First with straightedge and compass and show that the ruler method you ...
Aluminium carpet joining strips are another hardware store option. They make nice straight edges for use with craft knives and would also serve well here. They're a better shape than cheap metal bar stock.
I strongly believe in teaching construction problems. They can be used to stimulate lively discussions in the class. For example, see the several construction problems in Geometric Transformations, Vol I by Yaglom (MAA publication) (http://maa.org/press/ebooks/geometric-transformations-i). Also, given orthocenter, circumcenter and one of the vertices, is it ...
Although this has nothing to do with cross sections,
it fits the physical materials:
You could have the students cut out shapes
to illustrate constructive proofs of the Pythagorean theorem:
(Images from a Steven Strogatz NYTimes article.)
Another constructive proof is described ...