Geometric Construction (bisecting an angle with a compass and straightedge)

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  • #1
mpresic3
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In discussing flight mechanics with a (15 years younger) co-worker with a doctorate in Aerospace Engineering. We examined some angles and I happened to mention bisecting an angle. I told him in High School in the early 1970's we learned how to bisect an angle with compass, and straightedge. He was really mystified and said all angles could be bisected or trisected with a protractor.

Is it likely he is just forgetting he (probably) learned the process of geometric construction once? How common was geometric construction in the curriculum of past and present.

It is interesting to note that in my HS, we had an accelerated math sequence that unified geometry, algebra, and some calculus together, (and this program treated geometry lightly). One friend of mine who was in the accelerated sequence confided in me that he did not learn enough geometry. Did educators teaching HS math in the 1980's treat geometric constructions or geometry too lightly.
 

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  • #2
Ssnow
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It is possible, historically geometric constructions were important but it is possible that they were not present in that curriculum ... In the other side they are general math culture to do the bisection of an angle...
Ssnow
 
  • #3
bob012345
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Perhaps he misunderstood your comment and thought you never heard of a protractor? :biggrin:
 
  • #4
Ssnow
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Ok, when I speak about geometrical construction I consider only constructions involving the use of compass and ruler...
Ssnow
 
  • #5
brainpushups
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He was really mystified and said all angles could be bisected or trisected with a protractor.
But a protractor is a measuring tool, not a construction tool. If we allow measurement then angle trisection is trivial. Interestingly, angle trisection is not possible with a ruler and compass alone.

How common was geometric construction in the curriculum of past and present.
I'm teaching HS geometry this year and the text we use has a chapter dedicated to constructions and construction problems. They also come up in later chapters as part of developing conjectures and proofs. I'm not sure how common it is across the board.
 
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  • #6
mpresic3
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I looked at the curriculum in some states and it looks like geometric constructions are still being taught.
A older friend of mine who went to school in Michigan in 1960 told me they had to learn the elements of solid geometry. By the time I went to school in New York in 1971, there was no solid geometry in the math curriculum in geometry. I suppose things change.
It also looks like current curricula emphasize symmetry, reflections etc. I do not think these were emphasized in my program.
 
  • #7
brainpushups
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Interestingly, angle trisection is not possible with a ruler and compass alone.
I meant to stay straight edge and compass. A ruler is (obviously!) a measuring tool, not a construction tool.
 
  • #8
symbolipoint
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A healthy Geometry course will have geometric constructions using compass, straight-edge, & protractor. Some courses also include some very helpful paper-folding exercises. Whether Geometry course is in high school, or college, not matter. These constructions instruction and exercises must be present in the course.
 
  • #9
mathwonk
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All this and much more is in Euclid, which stopped being taught widely in the US apparently some 100 years ago. Having finally read (much of) it in my 60's, I am now of the opinion it should be taught in high school as the basic geometry course. It treats not only standard elementary plane and solid geometry, but also geometric versions of quadratic equations, trigonometry (including law of cosines), theory of equal area, proportionality and similarity, number theory including the famous and super important Euclidean algorithm. The statement in Prop. 16, Book III, even includes a version of the limiting definition of a tangent later generalized by Newton in his invention of the calculus. Limits are also utilized in the discussion of volumes of solids, as I recall. After decades of dumbing down the curriculum in the US, even in my private high school course in the 1960's much of the most significant content was omitted, (and I still placed second in the mid state in the geometry competition). In subsequent decades, the decline continued, with the adoption of such lamentable books as Discovering Geometry by Michael Serra, or anything by John Saxon. To be sure, even these books are useful to some learners, but in my (experienced) opinion are a disaster when chosen as the standard for widespread adoption for average and/or gifted students. (They were forced on my own children by their otherwise excellent school.) To give a couple of examples, angle bisecting occurs as Prop. 9, Book I of Euclid, on page 8 of my copy. It occurs on page 158 of Discovering Geometry. Pythagoras occurs as Prop. 47, Book I, p.35 in my Euclid, and on about page 417 of Discovering Geometry. Saxon's books seem aimed at students who dislike mathematics and even dislike understanding it. Thus, it is not surprising to me to hear of high school geometry curricula in the US which omit almost any topic of geometry, even the whole course, replacing it with some cookbook "precalculus" course.
 
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  • #10
gmax137
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All this and much more is in Euclid, which stopped being taught widely in the US apparently some 100 years ago. Having finally read (much of) it in my 60's, I am now of the opinion it should be taught in high school as the basic geometry course.


I had geometry in 9th grade more than 50 years ago. I have no idea what book we used. It sure wasn't Euclid. You're making me interested in looking into it though. I wonder if there's a "best" translation / version?
 
  • #11
bob012345
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I think Coxeter covers Euclid.
 
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mathwonk
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  • #14
gmax137
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I think Coxeter covers Euclid.
I see six or seven books by Coxeter listed on Amazon.
 
  • #15
brainpushups
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Here's an online resource that I'd recommend, especially if you're using something like the Green Lion Press version which has no commentary. Navigate to each book by the Roman numerals at the top.

I highly recommend Hartshorne:
That is a great book.

with the adoption of such lamentable books as Discovering Geometry by Michael Serra
I'm using Serra's book this year and I like the spirit of it. It is hard to take an axiomatic approach to a subject if you do not yet have an intuitive feel for it. I find that almost all high school students in 9th and 10th grade do not have an intuition for geometry. Many of the investigations in the book (especially early) are very simple most would understand the concept without doing them. But I think there is something to be said for how manipulating things and seeing it work helps concepts sink in.

My current section is mostly students who have struggled in math. Two of them have commented that they were dreading geometry and are surprised they are enjoying it.
 
  • #16
mathwonk
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I also like using such an approach with students who struggle. My objection is taking students who do not struggle, and who are even capable of reading and understanding say Jacobson's Geometry while in 3d grade, and forcing them in the 10th grade to use such a book. This is what I have seen happen. I also have friends who found Saxon's books a godsend for their struggling students. But my own children's private school adopted them exclusively for everyone for several years, until ultimately concluding that after using them, (in the words of the head teacher): "(with Saxon) we found the students didn't understand anything". Then they dropped them. By then it was too late for many children, including mine, to receive a more appropriate course.
 
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  • #17
mathwonk
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By the way, if the Serra book works well with students with no intuition of geometry, what do you think of using it for students in grades somewhere between 3rd and 8th, so that they do not reach 10th grade in that situation?
 
  • #18
brainpushups
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I think elementary school would be too early because algebra I skills are assumed for much of the text. For middle school I think it can be excellent. Our middle school math teacher uses Serra for a geometry course for students who are already comfortable with algebra and I think it has worked well. Those students don't usually get a second course in geometry in high school. They move to Alg II, Precalc or potentially statistics, Calc, and then potentially an advanced course either offered at the school if there are enough students to offer one, or they take an introductory college class off campus.
 
  • #19
mathwonk
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My friends in the north of the US (Providence Rhode Island) told me decades ago that elementary algebra 1 was taught there in 5 and 6th grade, compared to maybe 9th grade in my southern school. It sounds reasonable to me to offer Serra to middle schoolers, but not to follow up with a more substantive course afterwards seems to miss the opportunity to actually learn geometry. As a professional geometer, it is puzzling to me that our schools offer calculus to people who do not know rigorous euclidean geometry. The beauty of the presentation in euclid himself is there are no prerequisites at all, indeed one learns elementary algebra facts in a geometric way, and also elementary facts about proportions and even limits. Nonetheless the derivation is beautifully logical and, except for one or two very subtle topological assumptions that are hidden, quite rigorous.

To be somewhat more precise, except for tacitly assuming that two circles that should intersect in fact do so, that a line meeting a triangle away from the vertices must meet exactly two sides, clarifying the concept of "betweenness" for points on a line, and assuming that rigid motions of the plane exist, essentially everything needed is there. Thus the widespread impression that euclid is not up to modern standards of rigor, seems to me quite mistaken, and only possible to maintain if one has not read him. As you know, Hartshorne makes all this very clear.

In a course from Jerome Bruner, Harvard professor of psychology of learning, I first saw the geometric illustration of the algebra formula (a+b)^2 = a^2 + b^2 + 2ab, decomposing a square into two smaller squares and two equal rectangles. He was making a case for how to teach such things to young children. I wondered why it had never been shown to me that way in high school. That would in many cases cure the problem we have of students who think this formula has a^2 + b^2 on the right side, or maybe a^2 + b^2 + ab. Some 40 years later I learned this appears in euclid. This is in Prop 4 Book II. After a couple more pages, in Props 11-14, euclid even shows how to solve quadratic equations by completing the square, and proves the law of cosines, all with geometry! Even my favorite "modern" (1979) high school elementary algebra book, by Harold Jacobs, treats quadratic equations only on about page 500, with the "formula" on p. 540. By contrast, the great text "elements of algebra" by Euler, written over 200 years earlier for his mathematically illiterate butler, and which starts by explaining what a "quantity" is and how numbers can be used to measure them after a suitable choice of unit, has already treated quadratic as well as cubic equations in about half that many pages.

but i digress..

I just noticed however that one of the topological subtleties euclid tacitly assumes, arises in the OP's chosen topic, bisecting an angle. Namely when you make the compass construction of an equilateral triangle, it is assumed that the two arcs you construct really will meet. In fact most of the subtleties occur in the first 5 or 10 propositions (take a good look e.g. at Prop 1 and Prop 4). After that, I think it is pretty clear sailing.

Actually the hard part of getting started in euclid is the wacky "definitions" of points, lines and so on. One big advance in modern times is that you don't need to understand the definitions as they don't matter! Everything is in the axioms. So skip them, or take with a grain of salt, and start with the propositions. My mistake as a young man was to try to grasp the definitions, not succeed, and give up before getting to the propositions! It took me another 40 years to find out I gave up too soon. It is sort of like thinking you have to read the tedious scholarly preface of a great book before starting the book itself.
 
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  • #22
sysprog
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Okay if you say so. Any truly good Geometry textbook will give instruction on how to do geometric constructions.
True; however, that page has easy-to-find content that is specific to angle bisection.
 
  • #23
mpresic3
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When I started this thread, I did not mean to imply there is something wrong in the teaching of geometry in the high schools. The fact is that after teaching freshman physics recitations and labs in quite good universities, I found the students to be quite proficient in geometry, as evidenced by their SAT/ ACT of the incoming freshman. Their geometry knowledge did not seem to be inadequate.

I get quite tired of those that say we should teach advanced concepts in lower elementary schools. I tend to think those that present that idea do not spend much time in third grade classrooms. If we teach geometry to third graders, do we teach the multiplication tables to the tenth graders? Are the third grade teachers adequately prepared to teach Euclid to children (in perpetual motion), and unable to sit still in their seats.

One point in geometric construction that did not get mentioned when I was taught is that if you bisect a angle with straightedge and compass, the bisector drawn is more accurate than your eye can do with a protractor, although some may argue this point. Some will say it depends on the dexterity of the constructor. Also I do not remember after constructing the bisector, any point was made as to proving the said construction in this manner, really led to the angle bisector. Clearly the compass sweeps out equal lengths
 
  • #24
symbolipoint
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One point in geometric construction that did not get mentioned when I was taught is that if you bisect a angle with straightedge and compass, the bisector drawn is more accurate than your eye can do with a protractor, although some may argue this point. Some will say it depends on the dexterity of the constructor. Also I do not remember after constructing the bisector, any point was made as to proving the said construction in this manner, really led to the angle bisector. Clearly the compass sweeps out equal lengths
Upon reading your quoted passage, that makes me wonder, too. At some extent of teaching, we need to stop trying to prove, and proceed with common intuition - which may not be so common for everyone.
 
  • #25
mpresic3
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My point in the last sentence was really the opposite view. I feel after the student constructs the angle bisector, he or she should have to prove that their construction does indeed present the angle bisector. Because the compass sweeps out equal lengths, similar triangles demonstrate the angles of two similar triangles are equal.
 
  • #26
mathwonk
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@brainpushups: Since you are apparently now teaching high school and finding most 9th and 10th graders have little or no geometric intuition, I wanted to show you this link:

https://www.amazon.com/dp/0134392795/?tag=pfamazon01-20

In this book, apparently the leading American text for teaching mathematics to prospective elementary teachers, the author recommends, on page xiii of the preface, presenting areas of rectangles in 3rd grade, and other ideas of geometry gradually in subsequent grades until, building on these concepts, one can explain the pythagorean theorem in grade 8.

In your experience is this recommendation actually being implemented? Do your 9th graders seem to have experienced this much geometry from 3rd through 8th grades? I.e. do they know about area of triangles and rectangles and enough properties to state and explain the pythagorean theorem?
 
  • #27
mpresic3
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I seem to remember understanding how to take areas of lawn in 5 or 6 grade by breaking it up into rectangles and right triangles. I remember rote memorization of volumes of spheres cylinders and cones in the 7 grade curriculum. In most cases, the emphasis was on memory and calculation, but not on proof or full understanding. I knew the pythagorean theorem and the fact that sum vof angles in a triangle was 180 degrees by sixth grade, but that was because I had a big sister in high school who struggled through math. I understood the facts after reading a bit in her books, but I did not understand the proof or the necessity of proving the fact.
I think to some extent, mathematical proofs of the nature of high school geometry requires an adolescent cognition, which is less common in pre-teens
 
  • #28
brainpushups
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In your experience is this recommendation actually being implemented? Do your 9th graders seem to have experienced this much geometry from 3rd through 8th grades? I.e. do they know about area of triangles and rectangles and enough properties to state and explain the pythagorean theorem?

My impression is that the Common Core is pretty widely adopted. I like the Common Core in principle. In reality there is too much to learn everything well and, because I see students that come from a wide variety of different districts, the things that (I can infer) are omitted from the curriculum isn't always the same.

That said, I find that area is generally understood well, and students universally seem to have used the Pythagorean Theorem. Whether or not most can explain it depends on what you mean by 'explain.'
 
  • #29
mathwonk
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Given your answer, which seems to confirm some basic geometric knowledge, (the topics on area mentioned, plus some facts about triangles, consist basically of book 1 of euclid), what do you mean by saying "almost all 9th and 10th graders do not have an intuition for geometry"? I.e. what are they missing?
 
  • #30
brainpushups
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what do you mean by saying "almost all 9th and 10th graders do not have an intuition for geometry"?
That comment was driven mostly by the opinion you expressed here:
All this and much more is in Euclid, which stopped being taught widely in the US apparently some 100 years ago. Having finally read (much of) it in my 60's, I am now of the opinion it should be taught in high school as the basic geometry course.
Which I took to mean that Euclid should be the primary text for HS geometry.

By 'intuition for geometry' I meant the ability to take separate simple facts and use them to develop a proof (i.e. to do what Euclid does). I think that Euclid would be confusing/boring for most students this age. Perhaps 'intuition' isn't the right word because developing proof belongs to part of the rational brain, not the intuitive part. But, developing this skill includes carefully checking the validity of one's intuition and is a primary focus of most geometry work in HS.

Students seem to struggle with this for a couple of reasons. It could be that they may not be familiar with some of the terms/definitions. For such students, the work in Serra can be valuable. For example, I don't think any students in my class knew what vertical angles were before the class. One approach (less time consuming) is to simply tell them and perhaps have them write the term in their notes. Serra's approach is to provide figures of examples of vertical angles and examples of angles that are not vertical angles and then have the students try to define the term on their own. I'm pretty confident that more students remember the term after doing this than would remember if I just told them.

Another reason students may struggle with proof is they are inexperienced with forming a clear argument. This is (usually) the first time they are doing work like this.

The first proof in Serra is to show that vertical angles are congruent using the conjecture that linear pairs of angles sum to 180 degrees. Students actually did pretty well with this, but most needed a little hint. And it helped to use algebraic expressions (which are absent in Euclid, of course). The limited intuition in this example is just 'seeing' the two separate linear pairs.

Interestingly, students had a harder time with the second 'developing proof' exercise which was to prove that alternate interior angles are congruent. They were allowed to use the unproven conjectures that alternate exterior angles and corresponding angles are congruent. I think the argument for this one is more straightforward than the vertical angles proof, but the figure is slightly more complicated. Maybe(?) that's why more students needed help. Another example of what I mean by limited intuition for geometry.

In Serra, students had at least had some experience with physically comparing the angles using folding or superimposition. I guess I don't know if they would have struggled more or not we skipped over this.

I read and studied (most of) Euclid about 5 years ago and I agree that it is a beautiful work. For the right students it could be great, though even for the high fliers I'd want to supplement it to enhance what is a dry presentation.
 
  • #31
mathwonk
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Interesting. I have taught from Euclid in two different environments, both times supplemented by Hartshorne, and it went well in both settings. In both cases we covered mainly the first 4 books, which I am primarily familiar with, and which I consider most fundamental.

The first was in an upper level university course for prospective high school teachers, undergrads as well as graduate students. For several years I had taught this class from an advanced perspective assuming familiarity with high school geometry, which seemed reasonable in a class of upperclass university students, and focused on subtleties that are usually omitted in high school: i.e. foundational, axiomatic, and logical aspects. Eventually it became clear that even these prospective math teachers did not actually have a working grasp of basic high school geometry, and I considered using an actual high school level geometry book, but was reluctant partly for fear I would be ridiculed for teaching high school material in university.

Then I read Hartshorne's essay and the first part of his companion book book for Euclid. I thought if Hartshorne taught this to students at Berkeley, I should not be embarrassed to use it at my state school. And then, after being enticed into actually reading Euclid, I became entranced at how insightful it was. I began using it in this college course and had the best experience I ever had with these students. One student even invented a new proof of the concurrence of the medians of a triangle.

The second setting, after retiring from university, was in a 2 week summer course for bright, roughly 9, 10, and 11 year olds, using the same materials. Indeed these well above average elementary age students were even more appreciative than many of the college students. This also went well, but I supplemented the dry theoretical stuff with hands on constructions of colorful cardboard models of polyhedra, culminating in making a truncated icosahedron, which they recognized as a model of the soccer ball they played with every day outside. This practical and visually appealing stuff was key to giving everyone something to enjoy, even if the more theoretical material was elusive. For these motivated students, giving them challenging tasks to do, was also more fun than having them sit and listen to me present.


I myself find many parts of Euclid tedious in the extreme, and never advocate a rigid plodding through it or any book. But some parts of it are to me much more lucid than anything I have encountered covering the same ideas in trig or algebra books. Two particular examples stand out for me:

1) the expression of the algebraic formula (a+b)^2 = a^2 + b^2 + 2ab, as a simple decomposition of a square of side (a+b), into two smaller squares, an a-square and a b-square, plus two congruent rectangles of sides a,b, is the first. I suspect that a student who has seen this will be vey much helped in his/her algebra course to remember why the algebraic version of the formula is what it is. This is everywhere even on the internet nowadays, but I had not seen it in high school. I suspect this geometry is less abstract than the algebra, and hence accessible to younger students.

2) the law of cosines. Essentially no college students I have ever had knew this formula, (honestly I barely knew it myself), and many did not know what a cosine was. Euclid makes it clear geometrically as the error term in pythagoras' theorem, when non right triangles are considered. And the concept of a cosine is present as the length of a projection.

So although Euclid is famous for giving proofs, to me a big advantage of his book is the clear geometric representations of many crucial mathematical concepts, which arise later in algebra, trig, and even calculus. The interplay also cuts both ways. After using the geometric pictures to see why the algebraic formulas are true, since my students knew algebra, we then used the shorter algebraic versions as easier to manipulate than the geometric figures.

Although my 9-11 year olds were very strong and math oriented, I had similar success with the cardboard polyhedra models in an invited presentation to an average second grade class. E.g. almost everyone in the class was able at least to count up the numbers of faces, edges, and vertices ("corners"), and Indeed one of the (obviously very bright) 7 year old girls in the class actually guessed Euler's formula from looking at the data recorded on the board!

@brainpushups: Do you think there might be an audience, among the math oriented students in your high school, for a suitably supplemented presentation of the content of the first 4 books of Euclid? More importantly, do you think it would be useful to them?

One reason I raise this issue at all, is that I myself worked very hard to try to understand algebra, (especially) trigonometry, and calculus, finding them all quite difficult. After finally encountering Euclid, I felt I would have been helped greatly if I had seen such a really substantive presentation of geometry, much earlier.


@mpresic3: I was roughly in your situation wrt memorizing area and volume formulas in 7th grade, but a bit behind you in 5th, 6th grades. I don't remember how curious I was as to why they were true. As to proving them, now I like any explanation that is convincing, whether it is an actual airtight "proof" or not, at least as a first start. I.e. as to the necessity of proof, In my own case, with a faulty memory, I need some way of verifying that the formula or statement I have remembered is actually correct. And that is the value of some form of proof to me, at least now.
 
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  • #32
brainpushups
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Do you think there might be an audience, among the math oriented students in your high school, for a suitably supplemented presentation of the content of the first 4 books of Euclid? More importantly, do you think it would be useful to them?

Yes, if supplemented I think that could work. I agree with you that seeing a (typically) algebraic representation in a geometric way can offer insight and the example you described above is one of the ones that our small department has shared with students.

Useful? I suppose that depends on what the student(s) end up doing. Certainly for anyone pursuing math or science that approach could be useful. I'm not sure if it is better than other approaches, but that's the problem with education - there isn't a one size fits all approach.

A side note on constructions: one thing that drives me nuts about Serra is that he emphasizes that the classical construction tools are a straight edge and compass and says that measurement cannot be used for constructions. Then, the first construction (duplicating a segment) is done my using the compass as a measuring tool! Rather than take the approach of Euclid's second (or third) proposition from Book I Serra simply has students open the compass to the size of the segment and translate the compass to some other point/line.

I wonder if Euclid, once he had proven the method works, he bothered to draw the numerous arcs every time or if he'd have just been happy with translating the compass. I assume he wouldn't have bothered with every arc.
 
  • #33
mathwonk
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by the way @brainpushups, it is interesting that what I conveyed by my link to the elementary math teachers book was not what I had intended. The first edition of this book was written well before common core was introduced, and I myself taught teacher candidates from it, possibly a preliminary version, in 2001. I linked the fifth edition because that was the one now openly available in an online link. I myself was suprized at the apparent emphasis in it on common core standards since I did not recall them in the book I taught from. From my research this seems to be the most well regarded book, at least in math circles, for elementary math teacher training, and it has been for years. At any rate, this book was not originally written to comply with common core standards, but the author of this book did contribute to them later, and apparently emphasizes them now. I am curious to see if the earlier book has the same geometry topics. I don't know because I only taught the arithmetic part. But I seem to recall there was a geometry course for elementary teachers also taught from it at our school as early as 2001.
 
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  • #34
mathwonk
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@brainpushups: Re post #32: Probably you know Hartshorne discusses the very point you raise about copying line segments on page 20 of his book. I.e. he suggests that apparently Euclid used only a collapsible compass that cannot be translated, but then gave Prop I.2 to establish that it could in fact be used to do what a rigid compass can do. Hence Hartshorne says he will in future behave as if the compass is rigid. This is a subtlety I had never appreciated until reading Hartshorne and Euclid. I posed this proof as a challenge to my class and gave them 5 minutes to find their own proof of essentially Prop.I.2, and several of them succeeded! This was a lesson to me about (not) lecturing to bright kids, as they obviously got much more satisfaction out of doing this themselves than from listening to me talk. The issue at hand is one of the reasons I prefer to at least try to use books by great authors - the presentation is just more intelligent.

This is also related to a point made by mpresic3, using a compass to make measurements, as this really requires the side -angle -side theorem, (Prop. I.4) to prove that ruler measurement is the same as compass measurement. I.e. if you only keep the sides of the compass the same and the angle between them the same, how do you know the length of the line joining the two points stays the same on your surface? This may seem obvious using our intuition about flat surfaces, but if you imagine using it on a curved surface like that of an egg, you will see it is no longer true. It does not work to measure actual walking distances on the (curved) Earth for example. If people are expected to be able to bring their classroom knowledge out into thje world and apply it, it helps if they examine the assumptions they are making, i.e. when the things they are using are actually true.
 
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  • #35
symbolipoint
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@brainpushups: Re post #32: Probably you know Hartshorne discusses the very point you raise about copying line segments on page 20 of his book. I.e. he suggests that apparently Euclid used only a collapsible compass that cannot be translated, but then gave Prop I.2 to establish that it could in fact be used to do what a rigid compass can do. Hence Hartshorne says he will in future behave as if the compass is rigid. This is a subtlety I had never appreciated until reading Hartshorne and Euclid. I posed this proof as a challenge to my class and gave them 5 minutes to find their own proof of essentially Prop.I.2, and several of them succeeded! This was a lesson to me about (not) lecturing to bright kids, as they obviously got much more satisfaction out of doing this themselves than from listening to me talk. The issue at hand is one of the reasons I prefer to at least try to use books by great authors - the presentation is just more intelligent.
Do you really mean "bright" kids or actually mean "motivated" kids? Or maybe you meant, "the interested" kids?
 

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