Teaching Three-Dimensional Geometry, I

I have recently had a rather unusual opportunity.  I’ve talked a bit over the last few months about my consulting work producing online videos for a flipped classroom; I’ve been working busily on the Geometry unit.

Now the last section of this unit is on three-dimensional geometry, and I’ve been given pretty free reign as to what to cover in this 20-lectures series of 5-7 minute videos.  And given my interest in polyhedra (which I could focus on exclusively with no shortage of things to discuss!), I felt I had a good start.

But the challenge was also to cover some traditional topics (cones, cylinders, spheres, etc.) — as well as more advanced topics — while not using mathematics beyond what I’ve used in the first several sections of the Geometry unit.

There is, of course, no “correct” answer to this problem.  But I thought I’d share how I’d approach this series of lectures, since geometry is such a passion of mine — and I know it is for many readers as well.  The process of reforming high school geometry courses is now well underway; I hope to contribute to this discussion with today’s post.

Where to start?  Cones and cylinders — a very traditional beginning.  But I thought I’d start with surface areas.  Now for cylinders, this is pretty straightforward.  It’s not much more difficult for cones, but the approach is less obvious than for cylinders.

Earlier in the unit, we derived the formula for the area of a sector of a circle, so finding the lateral surface area of a cone is a nice opportunity to revisit this topic.  And of course, finding the lateral surface area of a cylinder involves just finding the area of a rectangle.

Now what do both of these problems have in common?  Their solution implies that cones and cylinders are flat.  In other words, we reduce what is apparently a three-dimensional problem (the surface area of a three-dimensional object) to a two-dimensional problem.

This is in sharp contrast to finding the surface area of a sphere — you can’t flatten out a sphere.  In fact, the entire science of cartography has evolved specifically in response to this inability.

So this is a nice chance to introduce a little differential geometry!  And no, I don’t really intend to go into differential geometry in any detail — but why not take just a minute in a lecture involving spheres to comment on why the formulas for the surface areas of cones and cylinders are fairly easy to derive, and why — at this level — we’re just given the formula for the surface area of a sphere.

I try to mention such ideas as frequently as I can — pointing out contrasts and connections which go beyond the usual presentation.  Sure, it may be lost on many or most students, but it just may provide that small spark for another.

I think such comments also get at the idea that mathematics is not a series of problems with answers at the back of the book…on the face of it, there is no apparent reason for a student to think that finding the surface area of a cone would be simpler than finding the surface area of a sphere.  This discussion gets them thinking.

Next, I’m planning to discuss Archimedes’ inscription of a sphere in a cylinder (which involves the relative volumes).  This is a bit more straightforward, and it’s a nice way to bring in a little history.

I also plan to look at inscribing a sphere in a right circular cone whose slant height is the same as the diameter of the base, so that we can look at a two-dimensional cross-section to solve the problem.  In particular, this revisits the topic of incircles of triangles in a natural way — I find it more difficult to motivate why you’d want to find an incircle when looking at a strictly two-dimensional problem.

Now on to calculus!  Yes, calculus.  One great mystery for students is the presence of “1/3” in so many volume formulas.  There is always the glib response — the “3” is for “three” dimensions, like the “2” in “1/2 bh” is for “two” dimensions.

When deriving these formulas using integration, this is actually exactly a fairly solid explanation.  But for high school students who have yet to take calculus?

It is easy to approximate the volume of a right circular cone by stacking thin circular disks on top of each other.  If we let the disks get thinner and take more and more of them, we find the volume of the cone as limit of these approximations.  All you need is the sum

\displaystyle\sum_{k=1}^n k^2=\dfrac{n(n+1)(2n+1)}6.

I plan to prove that

\displaystyle\sum_{k=1}^nk=\dfrac{n(n+1)}2,

and then prove (or perhaps just suggest — I’m not sure yet) the formula for the sum of squares.

I think a fairly informal approach could be successful here.  But I do think such discussions are necessary — in calculus, I’ve routinely asked students why certain formulas they remember are true, and they struggle.  As a simple example, students can rarely tell me why the hypotenuse of a 30-60-90 triangle is twice as long as the shorter leg.

When teachers just give students formulas and ask them to plug numbers in to get answers to oversimplified word problems, of course there is a sense of mystery/confusion — where did these formulas come from?  I’m hoping that this discussion suggests that there is a lot more to mathematics than just a bunch of formulas to memorize.

As usual, I realize I have much more to say on this topic than I had originally supposed…I’ve only discussed up to the fifth lecture so far!  Since I have not had extensive experience teaching more traditional topics at the high school, it has been an interesting challenge to tackle the usual geometry topics in a way that grabs students’ attention.  It’s a challenge I enjoy, and of course I’ll have much more to say about it next week….

 

Guest Blogger: Scott Kim, IV

Well, this is the last installment of Scott Kim’s blog post on transforming mathematics education!  These are all important issues, and when you think about them all at once, they seem insurmountable.  It takes each of us working one at a time in our local communities, as well as groups of us working together in broader communities, to effect a change.  What is crucial is that we not only discuss these issues, but we do something about them.  Those of us who participated in the discussion a month ago at the Bay Area Mathematical Artists Seminar are definitely interested in both discussing and doing.

Scott suggests we need to move past our differences and find constructive ways to act.  No, this isn’t easy.  But we need to do this to solve any problem, not just those surrounding mathematics education.  It’s time for some of us to start working on these issues, and many others of us to continue working.  We can’t just sit and watch, passively, any more.  It’s time to act.  What are you waiting for?

Level 4. Resistance from SOCIETY (quarreling crew)

Sailing is a team sport. You can’t get where you want to go without a cooperative crew. Similarly, math education reform is a social issue. You can’t change how math is taught unless parents, teachers, administrators and policy makers are on board. Most adults cling to the way they were taught as if it were the only way to teach math, largely out of ignorance — they simply aren’t aware of other approaches.

Here are three ways society needs to change the way it thinks about math and math education in order for change to happen.

4a. Attitude. The United States has an attitude problem when it comes to math teachers. First, we underpay and under-respect teachers. And the situation is only getting worse as math graduates flock to lucrative high-tech jobs instead of the teaching profession. The book The Smartest Kids in the World and How They Got That Way describes how FInland turned their educational system around — they decided to pay teachers well, set high qualification standards, and give teachers considerable autonomy to teach however they think is best, with the remarkable result that student respect for teachers is extremely high.

Second, it is socially acceptable, even a badge of honor, to say that you were never good at math. You would never say the same thing about reading. Many people do not in fact read books, but no one would publicly brag that they were never good at reading. Our society supports the idea that parents should read to their kids at night, but perpetuates the idea that being no good at math is just fine.

Solution: respect teachers by paying them well, and value math literacy as much as we value reading literacy.

4b. Vision. The national conversation about math education in the United States is locked in a debate about whether we should teach the basics, or the concepts. As a result we see over the decades that the pendulum swings back and forth between No Child Left Behind and standardized testing on one extreme, and New Math and Common Core Math on the other extreme. As long as the pendulum keeps swinging, we will never settle on stable solution. The resolution, of course, is that we need both. In practice, schools that overemphasize rote math find that they must supplement with conceptual exercises, and schools that overemphasize conceptual understanding find that they must supplement with mechanical drill. We need both rote skills and conceptual understanding, just as kids learning to read need both the mechanical skills of grammar and vocabulary, and the conceptual skills of comprehension and argument construction.

Solution: We need a vision of math education that seamlessly integrates mechanical skills and conceptual understanding, in a way that works within the practical realities of teacher abilities and schoolday schedules. To form a vision, don’t just ask people what they want. A vision should go further than conventional wisdom. As Henry Ford is reported to have said (but probably didn’t), “If I had asked people what they wanted, they would have said faster horses.” Or as Steve Jobs did say, “It’s really hard to design products by focus groups. A lot of times, people don’t know what they want until you show it to them.”

4c. The will to act. As a child I grumbled about the educational system I found myself in. As a young adult I started attending math education conferences (regional meetings of the National Council of Teachers of Mathematics), and was astonished to find that all the thousands of teachers at the conference knew perfectly well what math education should look like — full of joyful constructive activities that challenged kids to play with ideas and think deeply. Yet they went back to their schools and largely continued business as usual. They knew what to do, but were unwilling or unable to act, except at a very small scale.

Solution: Yes, a journey of a thousand miles starts with a single step. And change is slow. But if we’re to get where we want to go, we need to think bigger. Assume that big long lasting change is possible, and in the long term, inevitable. As Margaret Mead said, “Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it’s the only thing that ever has.” I’m starting my small group. Others I know are starting theirs. What about you? 


Guest Blogger: Scott Kim, III

Today, I’ll post the third installment of Scott Kim’s blog on transforming mathematics education.  But before jumping into that, I want to share a little about Bridges 2018, which just took place in Stockholm, Sweden.  Because of my move and career shift, I decided not to go — at the time I would’ve needed to make travel arrangements, I didn’t even know whether I’d be living on the West coast or the East coast when I’d need to catch my flight!

In any case, my Twitter feed has been buzzing recently with tweets from Stockholm, and some have featured participants in the Bay Area Mathematics Artists Seminars.  Monica Munoz-Torres tweeted about Frank Farris’s talk on vibrating wallpaper patterns, which you may recall he gave at our March meeting of the BAMAS at Santa Clara University.

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And the Bridges Program Committee announced that Roger Antonsen won Best in Show for 2-dimensional Artwork for his piece, “Six Perfect In-Shuffles With 125 Cards and Five Piles.”

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Congratulations, Roger!

OK, now we’ll move on to Scott Kim’s commentary on transforming mathematics education.  His next point addresses a prevailing issue in mathematics education:  advances in technology relevant to teaching mathematics are moving along at a rate which outpaces curriculum development.

And it’s not just that.  Even if curriculum could be reimagined at a pace to keep up with technology, teachers would need to be retrained to use the new curriculum with the new technology.  Not just retrained on the job, but while students at university — meaning that institutions of higher education would need to have their faculty keep up as well.  This means resources of time and money, and the willingness and ability of mathematics and education faculty, as well as school districts, to embrace change.  A tall order, to say the least.

I could go on at length about this topic, but let’s give Scott a chance.  Again, if you just can’t wait for the fourth installment, feel free to go to Scott’s blog, where you see the post in its entirety.

Level 3. The wrong MATH (sailing in the wrong direction)

The mathematics we teach in school is embarrassingly out of date. The geometry we teach is still closely based on Euclid’s Elements, which is over 2000 years old. We continue to teach calculus even though in practice calculus problems are solved by computer programs. Don’t get me wrong: geometry and calculus are wonderful subjects, and it is important to understand the principles of both. But we need to re-evaluate what is important to teach in light of today’s priorities and technologies.

Here are three ways to update what we teach as mathematics.

3a. Re-evaluate topics. The Common Core State Standards take small but important steps toward rebalancing what topics are taught in math. Gone are arcane topics like factoring polynomials. Instead, real world mathematics like data collection and statistics are given more attention. As Arthur Benjamin argues in a brief TED talk, statistics is more important than calculus as a practical skill.

Solution: give kids an overview of mathematical topics and what they are for, long before they have to study them formally.

3b. Teach process. The widely used Writer’s Workshop program teaches the full process of writing to students as young as kindergarten. The process accurately mirrors what real writers do, including searching for a topic, and revising a story based on critique. We need a similar program for the process of doing mathematics. The full process of doing math starts with asking questions. Math teacher Dan Meyer argues passionately in his TED talk that we do students a terrible disservice when we hand them problems with ready-made templates for solution procedures, instead of letting them wrestle with the questions themselves. Here is my diagram for the four steps of doing math. Conrad Wolfram created a similar diagram for his Computer-Based Math initiative.

Solution: give kids an explicit process model for problem solving.

3c. Use computers. In an era where everyone has access 24/7 to digital devices, it is insane to teach math as if those devices didn’t exist. In his TED talk, Conrad Wolfram points out that traditional math teachers spends most of their time teaching calculating by hand — the one thing that computers do really well. By letting students use mathematical power tools like Mathematica and Wolfram Alpha, teachers can spend more time teaching kids how to ask good questions, build mathematical models, verify their answers, and debug their analysis — the real work of doing mathematics. And students can work on interesting real-world problems, like analyzing trends in census data, that are impractical to tackle by hand.

Solution: build and use better computer tools for doing math. Revamp the curriculum to assume the presence of such tools. Emphasize solving interesting problems, de-emphasize or delay learning about the mathematical mechanics for carrying out the computations. In other words, teach mechanics on a need-to-know basis.

Next week will feature the Level 4 of Scott’s remarks.  Until then!

Guest Blogger: Scott Kim, II

Today, I’ll continue with reblogging Scott Kim’s in-depth post about transforming mathematics education.  You might want to read last week’s post to get caught up.

I will say that the discussion generated quite a bit of interest.  Participants have been actively responding to each other in a very lively email thread.  The comments and discussions are still ongoing — I am having a hard time keeping up with them!  But in a later post, I’ll summarize some key ideas and observations made by members of the group.

But for now, I’d like to turn it over to Scott Kim.  Again, if you’re anxious to read the entire post, please feel free to go to his blog.  Or just be patient….  But you can see by looking at the heading that Scott is addressing a very important issue next.  I can still recall — when teaching gifted high school mathematics and science students — really understanding where the question “When am I going to ever use this?” comes from.

The answer is pretty simple.  Bright students want to know.  When I first started teaching at university, I thought it was the students’ job to find motivation for doing mathematics — after all, they were paying a lot of money for their education.

But I eventually realized that there are only about three months between the end of high school and the beginning of college.  Nothing magical happens to students to transform them into self-motivated human beings, hungering for knowledge for its own sake.

Actually, one of my goals is never to hear the question “When am I ever going to use this?” again.  If I do a good job teaching and motivation concepts, students will already be able to answer that question, and won’t need to ask it any more.

Yes, it’s a more challenging way to teach.  But I can tell you, for me, it has been worth it.

Now I’ll let Scott take over.  Enjoy!  We’ll look at the third level next week.

Level 2. Lack of MEANING (leaks)

The most common complaint in math class is “when are we ever going to use this?” And no wonder; the closest most kids get to using math meaningfully is word problems, which are typically dull mechanical problems, dressed up in dull mechanical narratives.

Traditional mathematics education focuses on teaching rote computational procedures — adding, dividing, solving quadratic equations, graphing formulas, and so on — without tying procedures to meaningful situations. Unfortunately most adults, including many teachers and administrators, think this is how it must be. But teaching only the rote procedures of math is like teaching only the grammar and spelling of English, without explaining what words mean, or letting kids read books. Mechanics without meaning is not just deathly boring, it is much harder to learn.

Here are three ways to plug the leaks of meaningless math.

2a. Use math. In our increasingly digital society, kids spend less and less time playing with actual physical stuff. All the more reason to get students out of their desks and into the world, where they can encounter math in its natural habitat, preferably integrated with other subject areas. My friend Warren Robinett told me “a middle-school teacher I knew would, after teaching the Pythagorean Theorem, take the kids out to the gym, and measure the length and width of the basketball  court with a tape measure. Then they would go back to the classroom and predict the length of the diagonal. Then they would go back to the gym, and measure the actual diagonal length. She said some of the kids would look at her, open-mouthed, like she was a sorceress.”

Solution: use problems that kids care about, and excite student interest.

2b. Read about math. Before we learn to speak, we listen to people speak. Before we learn to write, we read books. Before we play sports, we see athletes play sports. The same should apply to math. Before we do math ourselves, we should watch and read about other people doing math, so we can put math in a personal emotional context, and know what the experience of doing math is like. But wouldn’t reading about people doing math be deadly boring? Not if you are a good story teller. After all, mathematics has a mythic power that weaves itself into ancient tales like Theseus and the Minotaur. My favorite recent math movie is a retelling of the classic math fable Flatland, which appeals as much to my 7-year-old daughter as to my adult friends. Here’s a list of good children’s books that involve math.

Solution: read good stories about math in use.

2c. Ask your own questions. In math class (and much of school) we answer questions that someone else made up. In real life questions aren’t handed to us. We often need to spend much time identifying the right question. One way to have students ask their own questions is to have them make up their own test questions for each other. Students invariably invent much harder questions than the teacher would dare pose, and are far more motivated to answer questions invented by classmates than questions written by anonymous textbook committees. Mathfair.com goes further to propose that kids build and present their own physical puzzles in a science-fair-like setting. Kids can apply whatever level of creativity they want. Some focus on art. Some on story. Others add new variations to the puzzles or invent their own.

Solution: Give kids freedom to ask their own mathematical questions, and pursue their natural curiosity.

If we plug the leaks of meaningless math, we will grow a generation of resourceful mathematicians who understand how to solve problems. But are we teaching the right mathematics?  (To be continued….)

Transforming Mathematics Education: BAMAS, X

This past Saturday marked the tenth meeting of the Bay Area Mathematical Artists Seminars.  You might recall (see the post about Bay Area Mathematical Artists Seminars, VI) that at a recent meeting, we had a very stimulating dinner conversation about the future of mathematics education, with Scott Kim helping to guide the conversation.

Everyone was so engaged, it was unfortunate that the conversation had to come to an end.  So I invited Scott to lead a more formal discussion at a later meeting of the BAMAS.  We met at BAMAS member Stacy Speyer’s place — thanks for hosting, Stacy!

The discussion was quite animated.  Scott prepared a handout based on a lengthy blog post he wrote about various issues revolving around mathematics and mathematics education.  He graciously gave me permission to reblog his ideas.  The post is rather lengthy, so I’ll share it in installments.  You can go to Scott’s blog yourself if you can’t wait to read more.  So without further ado, I’ll let guest blogger Scott Kim take the wheel.  His original post was dated July 6, 2014.

Navigating Math Education

boatfire.png

 

Imagine that you are a sailor on a leaky boat that is on fire, sailing in the wrong direction, with a quarreling crew. Which problem would you fix first?

Well, that depends. If the leak is slow and the fire is raging, then you would put out the fire first. If the leak is gushing and the fire is small and contained, you would fix the leak first. It makes sense to fix the most urgent problem first.

What you would NOT do is fix one problem and declare victory. If your goal is to get to your destination safely, then you must fix ALL the problems, no matter how difficult. Anything less will not get you where you want to go.

Such is the situation with math education. The problems are so difficult and so numerous that it is tempting to fix one problem, and give up on the rest. And certainly we have to prioritize if we are to make progress. But if we are to get the ship of math education back on course, then we, collectively, must fix ALL the problems of math education. Nothing less will get us where we want to go.

Fixing all of math education may sound impossible or impractical. And indeed it is a formidable challenge. Well-meaning entrepreneurs who have launched successful businesses frequently grind to a halt when they try to start their own innovative schools. Resistance comes from all sides — standardized testing, textbook publishers, parents, administrators, government officials, and the students themselves trying to get into college.

But change is in the wind. America is losing its competitive edge, colleges are becoming impractically expensive, and the internet makes us dream of free education right now for everyone. I say we face the problem with eyes wide open, assess the full range of challenges we face, and look for the smartest moves that get us where we want to go.

With that in mind, here is my survey of the problems plaguing math education, and steps we can take to fix them. I’ve grouped the challenges into four levels that range from the tactical to the strategic: Mechanics, Meaning, Math, and Society.


Level 1. Faulty MECHANICS (fire)

The most obvious and urgent problem is that the mechanics of math are taught as a series of blink and you’ll miss it lessons, with little opportunity to catch up.

This one-size-fits-all conveyor belt approach to education guarantees that virtually everyone gradually accumulates holes in their knowledge — what Khan Academy founder Sal Khan calls Swiss cheese knowledge. And little holes in math knowledge cause big problems later on — problems in calculus are often caused by problems in algebra, which in turn are caused by even earlier problems with concepts like fractions and place value.

Here are three ways to fight the fire of poor pacing.

1a. Self-paced learning. The Khan Academy addresses the urgent problem of pacing by providing short video lectures that cover all of K-12 math. While the lectures themselves are fairly traditional, the online delivery mechanism allows students to work at their own pace — to view lectures when and where they want, and to pause and rewatch sections as much as they need. All lectures are freely available at all times, so kids can review earlier concepts, or zoom ahead to more advanced concepts. Short online quizzes make sure that kids understand what they are watching. And with an online dashboard that shows exactly how far each child has progressed, teachers can assign lectures as homework, and use class time to tutor kids one on one on exactly what they need.

Solution: the “flipped classroom.”

1b. Visual learning. I love the Khan Academy. My son hated it, because he, like many students, is a visual learner, and Sal Kahn’s lecture stick largely to traditional symbolic math notation. He would have done better with a visual experiential curriculum. Some kids are primarily audio or kinesthetic learners, some learn best socially. The bottom line is that different kids learn in different ways, and no one way is right for everyone. Education needs to address all learners, not just kids who learn in words.

Solution: teach every lesson three different ways.

1c. Testing for understanding. Nothing can change in education unless testing changes. Traditional standardized tests born of the No Child Left Behind era use multiple choice tests that assess only rote memorization of routine math facts and procedures. The new Common Core State Standards for mathematics, now entering schools across the nation, replaces standardized multiple choice tests with richer tests that include essay questions graded by human beings — a better way to assess mathematical understanding.

Solution: better assessment.

If we douse the fire of poor pacing in math education, we will increase test scores and student confidence. But there is more to mathematics than teaching the mechanics well.

I hope your interested is piqued!  Scott will continue next week….

Circle Geometry

Today, I thought I’d share a little more about things learned along the way with my curriculum consulting.  As I mentioned before, I’m creating a series of online lectures for the Geometry unit.  This past week, the section I was working on (and will still be working on into next week) is Circle Geometry.

As I also remarked earlier, I’m using the University of Chicago School Mathematics Project’s textbook on Geometry as a reference.  In this text are many theorems about the measure of the angle between two intersecting lines in terms of the measures of the intercepted arcs.

This image is certainly familiar.

Day153Circle1.png

The question I had to consider was how to organize all these results in a coherent 5–7-minute lecture.  It turns out that there was too much for just one lecture, so I did spread it out into two.  But I still needed a flow.

Although the results were not new to me, I had never taught this topic before.  My main experience teaching geometry at the high school level was designing and teaching a course on spherical trigonometry as it applies to studying polyhedra.  So this gave me an opportunity to stand back and just think about putting it all together.

I was happy with what I came up with — an approach which could be classified under “combinatorial geometry.”  I decided to pose the following question:

Day153Circle2.png

In other words, if you have two intersecting lines, and you draw a circle so that it intersects both lines, what configurations are possible?

Looked at in this way, there are just two considerations:  whether the intersection of the lines is outside, on, or inside the circle, and whether the lines are secant or tangent lines.

It’s not difficult to make the enumeration, so I’ll just give it briefly here.  There is only one configuration if the intersection of the lines lies inside the circle, since both lines must be secant lines.

Day153Circle3.png

When the intersection of the lines is on the circle, one of the lines may be tangent, although both cannot be since there is a unique tangent at any given point on a circle.  And when the intersection of the lines is outside the circle, zero, one, or two of the lines may be tangent to the circle.

This enumeration allows for a systematic approach.  If you’ve ever worked through find the angle measures, you know that starting with the arrangement in the upper right corner is the way to begin.  I won’t go through all the details, but I will just indicate that the following figure is all you need:

Day153Circle4.png

This simple case is analyzed by considering \angle QOR as an angle exterior to \Delta POQ.  The analysis of all the other cases builds from this.

I decided to include a discussion not found in the UCSMP text — continuity.  Of course this is not a topic which can be rigorously discussed at say, the 10th-grade level.  But why not give students an intuition of the idea?

Let’s start with a few diagrams.

Day153Circle5.png

This series illustrates the case that the intersection of the lines is outside the circle, and one of the lines is tangent.  We look at this as the limiting case of a series of pairs of secant lines.

This argument depends upon the fact that the measurement of all arcs and angles varies continuously as S moves around the circle.  While, as mentioned, this cannot be addressed rigorously, it is a very intuitive argument.  Moreover, there are many different software packages you could use to make an animation of this process, and display all the arc and angle measurements as S moves around the circle.

There is no reason not to introduce this argument.  In my pair of lectures, I used more traditional geometrical arguments as well.  It doesn’t hurt students to be exposed to a wide range of proof ideas.

I summarize all of these results in the following graphic.

Day153Circle6.png

The measure of the angle indicated with the red dot is half the measure of the intercepted arc, or the sum/difference of the measures of the intercepts arcs, shown in red and blue.  An arc in blue indicates its measure is to be subtracted rather than added.  I was very happy with this graphic.  I think that if a student followed the lecture, they could state every result just by looking at it.

This also proved to be a great segue into looking at the power of a point.  I thought I’d begin with the figure in the upper left, proving the usual theorem using similar triangles.

And now for another continuity argument!

Day153Circle7.png

This is a nice way to see that the power of any point on the circle is 0.  It is also a nice contrast to the theorems about the angle between the intersecting lines:  when PT and RT eventually reach 0, you’re not able to conclude anything about a relationship between QT and ST.

This means that there is no theorem relating lengths of segments for the two cases when the intersection of the lines lies on the circle.   I use the following graphic to indicate this, with two cases grayed out when the power of the point offers no conclusion.

Day153Circle8.png

Now all of these results are the usual ones found in high school geometry textbooks; nothing new here.  But for me, just having to step back and think about how to put them all together was a fun challenge.

Again, I am surprised at how much I’m learning even though I’m just putting together a few slides on elementary geometry.  The process of writing these lectures is an engaging one, and I hope the students who will eventually watch them will benefit from a perspective not found in more traditional textbooks.

What Is…A Polygon?

A haven’t made a post in quite some time in my “What is a Geometry?” thread.  In working on my online lectures in the section on Polygons, I of course needed to define just what a polygon is.  This turned out to be a little more challenging than I had imagined.  I thought that the issues that arose would make this discussion an interesting continuation of the “What is a Geometry?” thread.

In general, I think that the Wikipedia does a good job with mathematics — but specifically, the definition of a polygon leaves quite a bit to be desired.  I’ll reproduce it here for you:

In elementary geometry, a polygon is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain, or circuit.  These segments are called its edges or sides, and the points where two edges meet are the polygon’s vertices (singular: vertex) or corners.

OK, maybe not so bad of a start.  There are lots of examples given which fit this definition, but many which do not.  For example, this definition allows consecutive segments to lie on the same line, which is typically disallowed in most other definitions of polygons.

So maybe a clause may be added to the definition which does not allow this.  But then we encounter a polygon like this (I’m using screenshots from my lecture as illustrations):

Day152Polygon1

My definition begins with a list of vertices — but the problem is still the same.  The vertex labeled “4” is on the edge joining the vertices labeled “1” and “2.”  Again, this is usually avoided.

And what about the following figure?

Day152Polygon2.png

With the Wikipedia definition, a vertex can be an endpoint of more than two edges of a polygon.  Again, problematic.  There would be no way to distinguish this figure from a single polygon and two different triangles sharing a vertex.

Moreover, there is no condition saying that the straight line segments need to be distinct.  So the same segment might occur multiple times as an edge of a polygon.

None of these behaviors is illustrated anywhere on the Wikipedia page.  I’ve done some Wikipedia editing a while back, and would be interested in working on this page when I have more time to devote to such things.

So what is the fix?  I’m using the Geometry text of the University of Chicago School Mathematics Project as a reference, which is one of the most rigorous geometry texts around.  Here is their definition:

Day152Polygon3.png

They remark that this is the definition used in 23 out of the 45 geometry text they surveyed.  And in fact, it is the rewrite of a definition in previous editions:  “A polygon is the union of segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.”  This definition was problematic, though, since by this definition, the following is actually a polygon!

Day152Polygon4.png

Now this revised definition solves all of the problems above — but I couldn’t use it.  Why not?

One of the sections I’ll be writing lectures for is three-dimensional geometry — and (of course) I’ll be saying a lot about polyhedra in this section.  There are Platonic and Archimedean solids, as well as the Kepler-Poinsot polyhedra, like the small stellated dodecahedron shown below.

Fig122.png

The faces of the small stellated dodecahedron are pentagrams, five meeting at each vertex.

But the UCSMP definition does not allow edges to cross.  Each edge meets exactly two others, at each of its endpoints.  So that means that an edge cannot cross another in its interior.

Now I just can’t talk about polyhedra without talking about nonconvex examples.  Sure, it is possible to talk about pentagrams with edges crossing as decagons without crossing edges.

Day152Polygon5.png

But this would be the height of absurdity.  Besides the fact that none of the dozens of books and articles I’ve read on polyhedra in the past few decades ever do such a thing — and I’m sure none ever will.

So I had to go it alone.  I’ll share with you my definition — but I can’t say it’s the best.  The difficulty lies with being mathematically precise while still making the definition accessible to high school students.  Here it is:

A polygon is determined by a list of its vertices. Edges of the polygon connect adjacent vertices in the list, and there is also an edge connecting the last vertex in the list to the first one. All vertices in the list must be different. Finally, no three consecutive vertices of the polygon can lie on the same line, and no vertex can lie in the interior of another edge.

I don’t think this is too bad.  But there is still a subtle glitch, which I haven’t worked out yet, and which doesn’t necessarily need to be worked out at this level.  When I talk about triangles, for example, I allow cases where the sides have lengths 3, 4, and 7, for example.  But I qualify such a triangle by calling it a degenerate triangle.

Since a triangle is a polygon, a degenerate triangle should be a degenerate polygon, right?  The problem is that calling something a “degenerate polygon” gives the impression that it is actually some type of polygon.  But a degenerate triangle, by my definition, is not a polygon.  So when I use the term degenerate polygon, I’m not actually talking about a polygon….

So I’ll let you think this over.  I just wanted to share how surprised I was at how subtle the definition of something so “simple” could be.  An ordinary polygon.

If you find this sort of question intriguing, you might go online and research all the various definitions of polyhedron.  Convex polyhedra are easy to define (as are convex polygons), but when you get into the different types of behavior possible in the nonconvex cases, well, it becomes problematic.  In fact, no one, as far as I know, has ever come up with a satisfactory definition for “polyhedron.”  Might even do a blog post on that some day….