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I'm having a "challenge" in my geometry classes getting students to avoid using rulers as measuring devices in constructions. As natural as that usage is, they're only supposed to use them to connect points to form line segments and extend line segments. It's very understandable when they use them to mark off a line segment congruent to a given line segment. But then they extend the idea to a sort of "guess and check" when they really should be using a compass instead. At the same time, I don't want to criticize their ingenuity or suggest that tools shouldn't be used as fully as possible to solve problems when appropriate.

So what I'm leaning towards now is having 25 or so unmarked straightedges to have students use, so they are constrained to using them as Euclid intended. I haven't ever seen them sold as such, so I'm wondering if people have any suggestions about common "blank ruler-like" products that could be used as substitutes. (I would like to have an actual product for them to use, and not, say, using the edge of another textbook or an unsharpened pencil or something like that.) Alternatively, I might get something like a simple piece of moulding cut into foot-long segments and sanded down.

Does anyone have any experience or suggestions along these lines?

For instance, here's an example of a brilliant but illegal construction I'd like to avoid seeing:

enter image description here

Here, we bisect $\angle BAC$ by swinging an arc with $A$ at the center that intersects the angle at $D$ and $E$. Then, using the ruler to measure $DE$, construct $F$ as the midpoint. Then $\overline{AF}$ is our angle bisector.


ETA: I visited my local craft store and found 10" craft sticks in a pack of 50 for $7. The edges are rounded like popsicle sticks, but they're long enough that students will be able to draw any line segments they want with just the straight part. I think this is going to work really well for me. Thanks to everyone for the suggestions!


ETA after the constructions unit: I'm really glad I steered away from the rulers. For instance, when we discussed as a class how to determine whether every point on the perpendicular bisector of a line segment was the same distance from the two endpoints of the line segment, I asked "What tool do we use in constructions to verify that two line segments are the same length?" and everyone held up their compasses. That surely wouldn't have happened if they had a ruler on their desk.

A few downsides along the way. First, not being lacquered, the craft sticks got written on. Sometimes, students would play with them in ways that made them more splintery than I'd like to have classroom supplies. Finally, and most surprising, both the students and I were constructing clearly parallel lines that were clearly skew to the degree that I'm not sure they're as uniformly straight as you would think. Not a huge deal, because I was verifying their work with a compass anyways, but it was a bit of a bummer.

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    $\begingroup$ On the rulers that I have, the markings come off just from normal usage, cheap junk. But fortunate to you, just sand the markings off, should not take long. Ask your students to do that — some vocational skills while inhaling microplastics, yay. $\endgroup$ – Rusty Core Sep 13 at 17:56
  • $\begingroup$ @RustyCore LOL, Unfortunately, I also need actual rulers from time to time too. But buying super-cheap rulers and then defacing them is an intriguing idea! $\endgroup$ – Matthew Daly Sep 13 at 20:41
  • $\begingroup$ Funny, your example is the exact exam question which I “solved” using a ruler in middle school geometry, and for which I got in trouble when fighting (rather violently) the teacher's “unjust” docking of points. So +1, very good question. $\endgroup$ – Konrad Rudolph Sep 15 at 12:51
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    $\begingroup$ @BobJarvis Euclid never mentions straightedge. He asks that you be able to draw a straight line given points (I. Post. 1) and to be able to extend a given straight line without making a gap or an angle (I. Post. 2). I guess a straightedge is considered a mechanical means to satisfy these postulates, although no one has ever denied it does so only approximately. $\endgroup$ – user1527 Sep 15 at 16:24
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    $\begingroup$ @user1527 I would also add that an arbitrary angle can be trisected using a compass and a ruler, and that has been understood through history as "cheating" as strict constructions go. $\endgroup$ – Matthew Daly Sep 15 at 16:27

12 Answers 12

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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[1] have suggested paint stirrers (which you are often given gratis at hardware stores when you are buying other things). I originally dismissed this idea, as they tend to be fairly rough cut and made from soft pine, which may not hold an edge for very long. However, some time with a belt sander (and possibly the application of some wood lacquer) might make this a very viable option.

  • Wood molding or other small pieces of wood can be cut down to size. I would suggest that the best strategy here is to go to the hardware / lumber store and see what they have. You might find something you like. This might be a little more expensive, but you can get the pieces cut down to whatever length you like (or do it yourself if you have the tools). If you go this route, you should probably use a harder wood (such as oak); it will hold an edge longer.

  • Going very low tech, take a piece of $8\frac{1}{2} \times 11$ inch printer paper, and fold it in half the long way two or three times (therefore obtaining a piece of paper which is $11$ inches long and one or two inches wide. If you put a sharp crease into the paper, it should last for an hour or two.

  • Pieces of cardboard, paperboard, or cardstock. Cardstock (which is, basically, a subset of paperboard) can be obtained for cheap from most office supply stores, or can be cut out of cereal boxes (take a cereal box, a good ruler, and (preferably) an X-acto knife, and cut the box into one inch wide pieces). Larger pieces can be cut down to size with a paper cutter or X-acto knife. If you need something more durable, cut several pieces to size, then glue them together.

  • If you are wealthy, there is always this guy. Or this. That's what grants are for, right?


[1] Tip o' the hat to André 3000 and Darrel Hoffman.

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    $\begingroup$ Hmmmm. Standard Popsicle sticks are apparently 4.5", but there are craft sticks that get to be 10" or longer, and some with squared ends instead of rounded. This looks like it could be more affordable than I had hoped for! $\endgroup$ – Matthew Daly Sep 13 at 20:57
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    $\begingroup$ my local hardware store sells strips of brass and aluminum metal in 3 or 4 foot lengths. Wouldnt take much work to cut them into shorter pieces and file down the edges to turn them into durable straightedges. $\endgroup$ – Johnny Sep 14 at 3:41
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    $\begingroup$ Thin strips of soft metal like brass or aluminum are easy to bend by accident, which makes them useless for the intended purpose. $\endgroup$ – alephzero Sep 14 at 13:17
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    $\begingroup$ You think those last two links are expensive... Well, this one comes with a calibration certificate: starrett.com/metrology/product-detail/380-24%20W~SLC $\endgroup$ – AJMansfield Sep 15 at 0:35
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    $\begingroup$ @alephzero A 1x10x1000 mm aluminium bar only easily bends in one direction. If it's bent then there is always one way that you can put it down and lightly press down on it so that it becomes straight. You can also cut it with sheet metal scissors. But for a set of 30 this would still come down to ~USD50 for the aluminium plus USD20 for the scissors. $\endgroup$ – Nobody Sep 15 at 16:08
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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.

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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
  • low prices.

Drafting triangles also have unmarked edges that are intended to be drawn along, but their accurate angles are too useful in your intended application.

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  • $\begingroup$ I find those triangles disconcerting---my mother (who is a freelance technical illustrator) has many such triangles, all of which are marked. The unmarked triangles are trippy. $\endgroup$ – Xander Henderson Sep 13 at 17:30
  • $\begingroup$ Actually, this reminded me of the fact that most places that sell house paint will give away stir-sticks for free whenever you buy a can, which would make pretty effective straight-edges. This is only useful if you also wanted a bunch of paint, of course, so if you were planning on any home improvement in the near future, it might be something to think about. (They might also give you extras if you ask nicely, so you can get enough for the whole class...) $\endgroup$ – Darrel Hoffman Sep 14 at 13:35
  • $\begingroup$ @DarrelHoffman My first thought was, honestly, paint stirrers. However, these tend to be somewhat rough cut, and I am not sure that they would work well as a straight edge. They are also made of a pretty soft pine, and might not hold an edge for very long. $\endgroup$ – Xander Henderson Sep 14 at 13:43
  • $\begingroup$ @XanderHenderson True enough, but they might be good enough for a few weeks of geometry lessons, and you can't beat the price... $\endgroup$ – Darrel Hoffman Sep 14 at 13:48
  • $\begingroup$ @DarrelHoffman Nothin' better'n free! :D $\endgroup$ – Xander Henderson Sep 14 at 13:53
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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 geometry by considering why the folded edge must be straight.

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  • $\begingroup$ Why is the edge of a fold absolutely straight? Is this a result of Gauss's theorema egregium? $\endgroup$ – Tom Anderson Sep 16 at 12:29
  • $\begingroup$ Some paper does have a little stretch to it, so an insistent origamist can create slightly curved folds... $\endgroup$ – paul garrett Sep 16 at 17:52
  • $\begingroup$ I've done this many times. $\endgroup$ – David Dubois Sep 17 at 0:44
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I hope you like this one...

String

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    $\begingroup$ ^_^ Actually, I think of string as making a better compass than a straightedge. $\endgroup$ – Matthew Daly Sep 15 at 8:51
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    $\begingroup$ Chalk line. :) $\endgroup$ – Xander Henderson Sep 15 at 16:14
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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

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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 mention is not accurate enough. Then you show that the last draw of a straight line, may be replaced by a (more precise) draw of the compass with the approriate radius you just have on hand.

This will open the appetite to replicate the basic Euclidean constructions. When your students are convinced of the Mohr-Mascheroni, then give back the ruler and mention that it is a far less precise instrument than a compass to "copy" a distance. A ruler is just good enough to mesure a distance, or to draw a straight line.

At this point your students should prefer starightedge and compass to rulers.

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  • $\begingroup$ I think you missed the mark here. The very wiki article you posted states "we are referring to figures that contain no straight lines, as it is clearly impossible to draw a straight line without a straightedge" -- OP wants the class to be able to draw line segments, so forbidding straightedges would seem to be anathema to that. $\endgroup$ – Doktor J Sep 17 at 14:59
  • $\begingroup$ I think OP wants "students to avoid using rulers". One way is to forbid their use, and other approach it to convince them it's useless. There is no need to draw lines complete to be convinced of the superiority of the compass over the ruler. $\endgroup$ – AlainD Sep 17 at 17:26
  • $\begingroup$ And by the way, it is perfectly possible to draw a straight line with a compass only, or at least a dense subset of it. And it's good for the students to understand there is a difference between "to draw any point" and to "draw all the points". $\endgroup$ – AlainD Sep 17 at 17:27
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You could use a beam compass and only let them use one side of the compass as the straightedge.

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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.

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Tongue Depressor - They are six inches long and you can get a pack of 100 on amazon for about five dollars. You can also get them at your local Walmart or Hobby Lobby.

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If you have a budget for this then drafting triangles are a easy solution, and they give you physical examples of two common triangles (more useful in introductory trig or analytic goemetry then in pure Euclidian geometry, but still).

By design they have nice, crisp edges designed exactly to use for line drawing.

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There are painting straight edges available for about $7 apiece. Here's one I found on Amazon:

12 inch painting straight edge

You might find them cheaper (and possibly smaller, when used for trim work/etc) at your local hardware store, and they generally don't come with ruled markings because they're intended to get covered in paint anyways. Additionally, unlike many of the other suggestions, these have a handle side that's intended to be gripped and held steady.

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