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

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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
 
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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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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.
 
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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?
 
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bob012345
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I think Coxeter covers Euclid.
 
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mathwonk
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gmax137
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I think Coxeter covers Euclid.
I see six or seven books by Coxeter listed on Amazon.
 
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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.
 
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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?
 
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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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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
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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 knowlege 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
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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.
 

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