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When starting compass and straightedge geometric constructions in my grade 8-9 maths classes, I usually begin by mentioning a little about Euclid and the fact that constructions have been done for thousands of years. The problem is that my students haven't heard about Euclid before, don't really understand how a geometric construction differs from a sketch drawing, and don't continue constructions beyond triangles and quadrilaterals in grade 10-12. Moreover, as far as I know there aren't any "real world" applications of geometric constructions.

What hook can I use to engage students in learning about geometric constructions?

Ideally the hook should be somewhat informative and very engaging, such as a story, video, article, starting problem, etc.

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    $\begingroup$ Coincidentally, I constructed a line on my wall parallel to the ceiling last night in order to put up a cabinet. Using a piece of string as a compass and a yardstick as a straightedge really seemed like the easiest way to accomplish this. I am pleased with the results. $\endgroup$ – Steven Gubkin Dec 14 '14 at 15:06
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    $\begingroup$ this might sounds stupid, but my students love constructions simply for the fact that it means no notes, no exercises, and they get do actually create something :) $\endgroup$ – celeriko Dec 14 '14 at 17:20
  • $\begingroup$ I don't think that this is really suitable for beginners, but this web implementation of Euclid's constructions based on GeoGebra got me motivated to play with constructions - Euclid: The Game $\endgroup$ – yoniLavi Dec 19 '14 at 19:44
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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 Euclid chose the axioms that he did for his constructions was that it was about the real world. Otherwise, why include the parallel postulate?

So as for a story, how about using a compass and straightedge (no rulers if you want to be pure) to design a net of an interesting building (ie. not a cube). Starting from a single length to define the height of a wall, you could build a gabled square house, a hexagonal round(ish) house, or a geodesic dome (soccer ball) all from simple geometric constructions. You would end up with a little Euclidean village.

If you have studied symmetry, the structures could be decorated with various wallpaper and frieze patterns, maybe replicated with similar techniques. One of your students could invent a recursive/fractal wallpaper!

If you have the opportunity, using software like Geogebra can extend the scope of design in purely geometric terms. The infinite scalability from changing the first and only measurement is also amazing to watch. A revolutionary tool for any part of geometry if you have computers or tablets available.

Though not using Euclidean constructions, I had some able students create nets for every Platonic and Archimedean solid (and samples from the two infinite series of prisms and skew prisms) in Geogebra, discovering Euler's theorem for plane networks and then using that to discover the more difficult combinations. A couple of the most complicated Archimedean solids were hard to lay out and they needed a helping hand, but it was a worthwhile activity, though it takes some time.

Constructions can be very rich, especially when combined with other areas of mathematics. I hope your students have fun with whatever you choose.

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  • $\begingroup$ I like the thrust of this response, especially the conclusion in your last sentence which is supported by examples like nets and Frieze patterns. I'm a little skeptical though about statements like your opening paragraph. My students won't be using compasses and straightedges to design any buildings ever, and even if they do design buildings on paper, they won't really be doing constructions in the Euclidean sense. $\endgroup$ – David Ebert Dec 21 '14 at 19:13
  • $\begingroup$ @DavidEbert I suspect that there may be 3 issues here. First, if your school system does not teach constructions as part of the technology/design curriculum, then there is little motivation for them, except historical reasons. Second, I suspect that the majority of our maths curriculum is outdated because computers do it now. However, it may do some good to understand how the computers do it, and then use programs like Geogebra to do the fiddly work. Third, if you don't like model buildings for a city of the future, how about creative packaging for commercial products? $\endgroup$ – Richard Dec 22 '14 at 2:45
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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 to pass that joy along. when was the last time you used a pair of compasses?

i have taught a week long class on polyhedral construction. at the beginning we will cover lots simple construction: construction simple angles, square roots and so on. eventually we learn how to construct a regular pentagon. student build all the platonic and archimedian solids not from cut outs but their o

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    $\begingroup$ Johnson, D.A. Paper Folding for the Mathematics Class. (Click Download full text at the given ERIC link to see the full 32 page book.) $\endgroup$ – Benjamin Dickman Dec 14 '14 at 2:12
  • $\begingroup$ @BenjaminDickman, thanks for finding the correct reference. $\endgroup$ – abel Dec 14 '14 at 2:13
  • $\begingroup$ I want to emphasize that the paper-folding activities work best with patty paper (i.e. for hamburger patties) or "wax paper," as mentioned in Johnson's book. The transparent quality of the patty paper makes it easy to copy and bisect line segments and angles, so "learning to do constructions" doesn't turn into "getting stuck learning to use a compass." $\endgroup$ – Xi Yu Dec 14 '14 at 4:48
  • $\begingroup$ I think that origami can motivate at least some geometric constructions. When I was first learning, I struggled to decipher a several different folds that all in come way boiled down to constructing angle bisectors. Once I discovered this, it made all of these folds much easier. Origami also motivates precision. Small imperfections in folding lead to a poor final product. Unfortunately, at the moment, I don't have any other examples of geometric constructions I have gotten out of origami. $\endgroup$ – Will Schwartz Dec 15 '14 at 1:42
  • $\begingroup$ See also this MESE posting on "Secondary Geometry Curriculum," which mentions a particular geometric folding construction. $\endgroup$ – Joseph O'Rourke Dec 15 '14 at 13:33
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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 proof should be taught at the end of the year. I agree with this. In order for students to write a geometric proof they need to be fluent in requisite geometric knowledge so that they can reason about these ideas and connect them. I believe that if we want a student to learning mathematical reasoning, they can't simultaneous learn the content. For example, if we want students to prove something about triangles, they should be able to fluently be able to do this with concrete numbers because generalizing. To this end, I think that students need to develop fluency in creating accurate sketches of what they are constructing before they can engage with construction. If I had a nickle for every angle bisector that was off by more than 10%, I would be a rich man. If students don't value precision in a concrete setting, they certainly won't be able to carry that value over to an abstract one.

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    $\begingroup$ I love how this site can facilitate the discovery of different points of view. Your statement "I believe that if we want a student to learning mathematical reasoning, they can't simultaneous learn the content" is the opposite of what I generally think! I aim to teach content through mathematical reasoning. I wonder if you could expand on this? $\endgroup$ – Steven Gubkin Dec 15 '14 at 3:10
  • $\begingroup$ I have been writing a post about this on and off for a while. I hope to post in more detail over break, so keep a look out for it. $\endgroup$ – Will Schwartz Dec 15 '14 at 20:17
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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.

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