## Saturday, August 17, 2019

### First Day, Once Again

I've seen some great advice for the first day of class.  (Here's the one I've read more than once. I've seen other great ideas, but I don't see them now.) I'd summarize my take on this article as:
• (Community) Start learning names, and get them learning each other's names.
• (Expectations) Don't spend much time on the syllabus; there are more important things to do. (Have them read it, and then you can quiz them on it the next day, or just ask for their questions.)
• (Learning, Expectations) If you use group activities (which are a very effective way to help groups of students learn), then you want to introduce students to this on day one.
• (Curiosity) An activity that helps them see what's coming in the course would be especially nice.

Our classes start in ten nine days. I am getting ready...

First day activity for Calculus I
I have them graph a parabola (y=x2), then draw a tangent to it at x=3. (Some don't know what that means, so I walk around checking.) And then estimate its slope. After they're done, I get to talk about what makes actually finding the slope hard - such a good intro to the course. And they've had time to review graphing a parabola.

First day activity for Linear Algebra
I have them solve a system of 3 equations in 3 variables. I  ask them to:
• Write down a description of the process,
• Solve the system,
• Now figure how to check whether your answer is correct. (Naysayers, has the group done enough to be sure that the answer is correct?)
• Extra: What does each equation represent geometrically? What does the solution represent geometrically?
Once again, a great introduction to the themes of the course.

First day activity for Geometry
I taught geometry earlier this summer for the first time. I had them draw a triangle (and make sure it was different than their neighbor's). Find the midpoints of each side (they could measure or fold). Connect each midpoint to the opposite vertex. I hoped most would do it well enough that the 3 connecting lines would intersect at one point. My goals were to highlight: vocabulary, shapes, construction (which we were not doing with straightedge and compass - yet), conjecture, and the possibility of proof.

First day activity for Precalculus
I've been thinking for the last few weeks about what I'd like to do for Precalculus. I have found exciting activities in the past that turned out to be way too hard, and intimidated the students. I have a lovely fractions activity, but that doesn't represent what we'll do going forward.

I am working hard to create an activity that looks at functions (and circles too) from 4 perspectives: equations, graphs, tables of values, and stories. I have 7 types of relationships (linear, quadratic, polynomial, rational, exponential, periodic, and circles). I don't have stories yet for the polynomial and rational. (My eternal gratitude to anyone who can give me a story I like for either of these.) And I won't show an equation for the periodic. (A trig function wouldn't make sense yet. But we'll get to discuss that.) So that makes 25 "clue sheets".

I have 40 students, who I'll put in 10 groups of 4. Each group will start with two clue sheets. [So 5 of the sheets will not be handed out at first. I can label those as graphs on the back, keep them at the front, and let student turn them over once they're pretty sure they didn't find a graph match to their set.] Each will describe a different type of function/relation from one perspective and ask them to do a few things. Then they pair up with a clue sheet for each pair, and go looking for the matching clue sheets (same function/relation, different perspective). They go back to their group and explain to each other what they found. (I'll have extras up front, so anyone done early can work on a 3rd function/relation.) When we're done, we'll have a summary of the function types we'll be studying all semester.

I dreamed some of this up late last night. When I started working today, I worried that it would be too hard. (I make up some crazy stuff sometimes when I'm falling asleep.) So my goal as I put this together has been to scaffold it enough. I am assuming some comfort with linear functions, and some familiarity (but not comfort) with quadratics and exponentials. They may not have encountered the others. (And most will not know any trig.)

I put my first second draft into a google doc here. Your suggestions may help me improve it. (I decided to leave out the rational function. 6 functions with 4 clues each would be 24. One story and one equation are left out. That's 22. The last two clues will sit up front.)

Edited (8/17): This is a great activity, but too complex for day one. I will do it on day two. On day one, we will review linear functions in a similar, but much simpler way. Here is my handout.

My Goals:
• Review plugging values in for x to find y. (Do they remember that b0=1? What else might trip them up?)
• Review graphing.
• See functions/relations in the context of modeling a situation.
• Identify functions/relations by type.
• See precalculus as a place to strengthen their understanding of all of this.

If you use this activity, please let me know what changes you decide to make and how it goes.

## Monday, June 17, 2019

### Teaching Geometry - Journal Post #1

I usually try to write blog posts with a point. I don't often have the energy for that lately. But I do want to keep track of how this class goes...

Today was the first day of my geometry class. (It is the first time ever for me to teach geometry. I have done lots and lots of prep.) It goes for 6 weeks, and meets 3 1/2 hours a day. (Yikes!) My students are mostly high school students, though this is a college class. The college screwed up registration. I had 28 students on my list, but a bunch thought they were on a waitlist (only happens when a class is full with 40 students), and came in not yet registered. I now have 35 or 36 students. Of course I ran out of handouts.

I gave what I thought were good directions for our first activity, and students managed to do strange other things.

My directions:

Welcome and Triangle Activity

This activity is an introduction to the themes of the course. The more thought and care you put into it (the activity and the course), the more learning and joy you will get out of it.

With a straightedge, draw a big triangle on this page. Make it as weird as you want as long as it is a triangle. Make sure it looks different than the triangle your neighbor draws.

Now you are going to draw three lines inside it. Each line will go from a corner (aka vertex) to the middle of the opposite side (aka midpoint). These lines are called the medians of the triangle. How will you find the midpoints? Do this as accurately as you can. Now look at what others have drawn. What do you notice?

[Can you guess how this activity connects with the themes of the course?]

***

I should have had them find the midpoints first. And then talk about connecting those to opposite vertices. (Fixing the handout now.) Some people connected the midpoints to each other. Some drew a new triangle inside the first triangle. (Pretty but nothing like what I asked for.)

So the activity took lots longer than I expected. Sadly, I didn't note the time. (I have a tutor working with me in class. I will ask her to keep track of how long things take.)

I thought I could manage to lecture on 4 sections of our textbook in about an hour and a half. No way. I enjoyed what I did with the students, but we only got through 2 sections.

Some of what I did was impromptu. I asked them to each write down their own definition of angle. Then I asked for volunteers to read their definitions. I talked about courage. I waited. Finally I got a volunteer. And then another. I did not criticize their definitions, even though I didn't like them much. The third one was closer. He was struggling for a word for how far the one side is from the other. I repeated his definition, with a big pause where he had been stuck. And then I said rotation. We talked about things that rotate.

One of the book problems we did just asked them to measure a line segment against a ruler printed in the book, and angles on a protractor. It was hard for them. And that was good for me to see. It tells me where we are.

I hated our textbook when I first started reading it, but eventually it seemed fine to me. I think the first chapter has more problems than the rest of the book.

1.1 Reasoning. They talked about types of reasoning and included 'intuition' (along with inductive and deductive reasoning). No. Intuition helps us guess what we might want to try, and helps us see things that might be true. But I don't like them calling it one of 3 types of reasoning.

1.2 on measurement turns out to be more useful than I expected. Looking it over now, there's a lot more there that I haven't gotten to. Good thing I've asked them to turn in their notes on their reading. I hope that helps. (They are supposed ot read the first 4 sections.)

The last hour and a half is devoted to labtime. We will be using lots of labs from Henri Picciotto's Geometry Labs (free). I had hoped to do 3 of them today. We didn't even finish one. (Lab 1.1 involves putting different shapes of blocks around a point.) I had also hoped to introduce them to euclidthegame. We'll do that tomorrow.

Quiz first thing tomorrow. I am hoping to keep them on top of things by quizzing daily.

We are part of a program called High School STEM Connection. 20 of my 35 students are part of that program, and are required to come to tutoring after class. Most of them did. They got most of their homework done. Yay them. I love spending time with them where I'm not in charge. I'm just there to help out when they need me.

I am hoping to complete 5 sections of the book tomorrow. I'm tempted to throw out the cool group activities, but I know I shouldn't. I think I'll just see how far we get, like today. And then tomorrow evening I'll have to think again about how to adjust.

## Sunday, May 19, 2019

### Teaching Basic Geometry

Our college offers a course that is pretty much the same as high school geometry. I have never before taught it because we only offer it in the summer, and summer semester is squished into 6 weeks. I haven't wanted to teach math on that sort of schedule because it seems like it would be hard for  students to learn well that quickly.

But I'm under load and I love geometry, so I will be teaching this course in a month. I've been preparing ever since January - reading the text, making plans and handouts, doing constructions. Geometry doesn't have the same flow for me that algebra does. I thought that might be because I've been teaching algebra for 30 years. But other math educators have also noticed how much harder geometry can be. Brain teasers everywhere. I like that, but I also need to figure out how to make this doable for the average student. I have gone through the book twice now, thinking about what they'll be learning, and hoping to notice the places they'll get stuck.

We will be doing lots of compass and straightedge constructions. Even though I loved geometry in high school, I hated that compass. I'd press too hard and open it further in the middle of an attempted circle. Or not hard enough, and then the pencil wouldn't draw. Arghh. Computer tools make that a problem of the past. I've mentioned two sites in previous posts, sciencevsmagic.net and euclidea.xyz. But euclidea won't let you unlock a higher level until you succeed at the lower level, and I found that frustrating. Henri Picciotto pointed out that euclidthegame.com is much like euclidea, but doesn't require you to solve one level before doing the next.  After checking it out, I'm not sure I have any reason to go back to euclidea; euclidthegame is much more satisfying.

Translating a Line Segment
So one of the tools I'll be using with my class is euclidthegame. I had a blast working through the levels back in January, and wanted to redo them now to see which sorts of things might be hard for the students. I got stuck on level 7, translating a line segment. I replicated two solutions given in the comments, but I didn't understand why the steps worked.

When I looked up 'translate a line segment', all of the sites I found said to hold your compass at the right size and move it over. That doesn't work on euclidthegame, and wasn't part of the original Greek compass and straightedge protocol. I almost gave up on understanding this construction, and thought about telling students they could skip this one, since our book does it the easy way too.

People say believing you can do something helps you do it, but I often succeed right after giving up. Ornery, I guess. I doodled a bit, and realized that thinking about a parallelogram was the key. After that, I was able to solve it quickly. I want to share my solution here. (Turns out, this idea was in the comments at euclidthegame, I just hadn't seen it.)

The idea is to make a line segment congruent and parallel to AB, with C as one endpoint of the new segment. If you imagine making a parallelogram with the old segment and the new one, and if you know that diagonals of a parallelogram intersect at their midpoints, it becomes relatively straightforward.
1. Sketching parallelogram ABDC,
2. We know that we can construct the diagonal BC,
3. And find its midpoint (E).
4. Then make a ray from A through E,
5. And a circle centered at E through A.
6. Where that circle and ray meet will be D.  (Because E is now the midpoint of AD.)
7. CD is the translated line segment we wanted.
Lovely.

The students won't have enough information when they get to this construction in the book (chapter 1) to do it the right way. We'll have to wait until chapter 4, where they will find out why those diagonals bisect each other. I sure am glad to know that ahead of time.

Planning the Course
We will be meeting from 8 to 11:35 am four days a week. Who is going to get there on time at 8am if they don't have to? So I will start with a quiz every day. If they don't do well, they can retake it (new questions, same topics) outside of class time. [So a real sleepyhead could still ace the quizzes, even if they came late. But that will take some serious dedication. That's cool.]

Then I'll do my usual combination of lecture, pairwork, groupwork, and guided discussion from 8:15 to 9:45. After they take a break they will have lab time. Partly because I don't believe anyone can concentrate for 3 1/2 hours in a regular classroom, and partly because I think they really need to get their hands on some of these tools. I'll have geoboards and dot paper to record their geoboard results. I'll be using most of Henri Picciotto's lovely Geometry Labs book. (Free. Thank you, Henri!) And they'll be using both euclidthegame and geogebra to do their constructions. Lab projects are 20% of their grade. They'll have to do constructions, geoboard activities, proofs, and one activity of their choice.

Anyone teaching geometry who'd like more details, please ask. And anyone who would like to share tips, I'm all ears.

## Monday, April 1, 2019

### Playful Math Ed Blog Carnival (aka Math Teachers at Play) #126

126 = 6*21 = 2*3*3*7

If you want to choose 4 chicks randomly from 9 total chicks, there are 126 ways to do it.

Students learn more if they make up the stories for story problems themselves. Can your students make up stories for these ways of making 126?

126 = 27 - 21 (difference of powers of 2)
126  = 42 + 52 + 62 + 72 (sum of consecutive squares)
126 = 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 (sum of consecutive even numbers)

This blog carnival has evolved from being mainly contributions to being mainly items the blog host has discovered. Since my passion lately is geometry, this issue is dedicated to geometry. (Which of the 3 ways of making 126 above has a geometric interpretation? Hint: There's a picture of it here... somewhere...)

Constructions
I have been intrigued for the past few years with Archimedes' method of finding pi. He didn't have the square root symbol, so he approximated using fractions, getting pi between 3 10/71 and 3 1/7. But if we follow his steps, and keep the square roots, we get a lovely pattern for our answer.You can try it. Construct a hexagon in a circle. If the radius of the circle is 1, then the hexagon's perimeter is 6. Perimeter over diameter = 6/2 = 3. Now create a dodecagon (12-sided polygon) from the hexagon. You can find the side lengths from repeated use of the Pythagorean theorem, and then find perimeter over diameter. Your result will be closer than for the hexagon. You can repeat this process until a pattern emerges.

If you want to get better at geometric construction (straightedge and compass style), play with it at sciencevsmagic or euclidthegame.

You can improve your geometric reasoning skills with the puzzles in Geometry Snacks (and More Geometry Snacks), by Ed Southall and Vincent Pantaloni. There are more puzzles at his blog. If you like them, the book is a treasure trove.

Because I've fallen in love with geometry, I decided to teach it this summer, for the first time ever. So I'm doing a lot to prepare. Henri Picciotto is an expert geometry teacher who graciously offered me his time over breakfast. He advised me to download his Geometry Labs book (free) from his Math Ed Page site. There is so much more there than this. But this alone was a huge gift. I think it may transform my course.

I've been collecting geometry mysteries. Medians are the lines from midpoints of the sides of a triangle to the opposite vertices. The 3 medians seem to always cross at one point. Why is that? I tried for weeks to prove it, and just couldn't. I finally gave up and looked at the proof. (And told my students how much fun I had failing!)   I then found another proof that followed a very different path. Can you prove it?

Here's a simpler mystery: If you make a 5-pointed star (perfectly even, I can't do that without digital help...), what is the angle at each point?

One of my favorites for seeing the geometry in math topics you didn't know were geometric is  Magic Pi - math animations. I hate that they're only on Facebook because I am not comfortable linking to facebook in class. But they are amazing. (I linked to one that's pure geometry. So cool.) They apparently do most of their animations in geogebra. I am a complete novice next to them. Here's a geogebra sketch I made today. It might be my first in their 3D mode.

At the beginning, I mentioned having students make up their own story problems.  Here's a lovely post from Arithmophobia No More about just that. Here's another angle on teaching story problems, from Jen at Math State of mind. Leaving out the numbers helps students to slow down.

This blog post, by Amy at When Life Gave Us Lemons, is about her son making up his own math games. And John Golden has a whole class make up variations on a game he shared with them.

Denise Gaskins, founder of this carnival, pulls together so many books and ideas I love in this post. I don't know how she does it! The (surface) topic is fractions, but more than that, it made me think about how we can help students learn by saying less. The video she includes, with a teacher asking the two boys questions, and never telling them they're wrong, is fabulous. One of the commenters at Denise's post linked to a discussion of his own with a student. And that made me think about Bob Kaplan's guide to 'becoming invisible' (or not giving away the math). (What math delights have you found lately by following your nose? Bunny hops rock!)

You can check out the Carnival of Mathematics here. And if you'd like to host this carnival (we need help next month!), you can learn more and sign up here.

## Friday, March 22, 2019

### Coming soon: Math Teachers at Play (aka Playful Math Ed) blog carnival

I'll be posting the blog carnival here sometime late next week. Right now I'm beginning to gather links to lovely, playful math ed posts (and sites and videos and ...). If you know of something I should include, please email me at mathanthologyeditor on gmail.

## Tuesday, February 12, 2019

### When Math Tells a Story

On the Living Math Forum group, I claimed that Algebra 1 tells a story more than Algebra 2 does. N asked me to explain what I had in mind. Here's my reply (with a few revisions):

Lately I've been saying this sort of thing in my pre-caclulus class, not about the course, but about individual equations. We say math is a language. If it is, then we should be able to tell stories in it. Each equation makes a statement, and sometimes those statements tell stories.

The equation for a circle is (x-h)2 + (y-k)2 = r2 . Many students see each equation like this as separate from any other equations/formulas they know. I try to get them to look at this deeply. From the structure of it (square plus square equals square), I see that it's really the Pythagorean Theorem. Why would something for right triangles show up in the equation of a circle?! (That blew me away a few years back. I've been teaching math for 30 years, but that question seemed deep.)

It's because our coordinate system has the two axes perpendicular to each other. So the distance from the x-coordinate of a point on the circle to the x-coordinate of the center is measured horizontally and the similar y distance is measure vertically. You can build a right triangle from the center to (almost) any point on the circle. The constant radius is the hypotenuse of that right triangle.

So this equation tells a little story.

How does a whole course tell a story?

Algebra is about solving equations and about graphing. We want to see how real life situations (anything with data that has two components, like time and height) can be represented with equations and with graphs. In Algebra 1 students learn to solve simple equations and to graph. And hopefully they learn how the two skills are connected. There's a bit more. Systems of equations allow us to model slightly more complex situations, using more variables. And in my (community college) Beginning Algebra course (which is pretty much equivalent to a high school Algebra 1), our grand finale (after factoring) is graphing and recognizing equations of parabolas.

Near the beginning of the course, I introduce my favorite problem. I bought a tree and planted it. It was one foot tall at first. It grows two feet a year. Let's make a data table for height versus time, and a graph, and an equation. What does the input variable (let's use t instead of x) mean? What does the output variable (let's use h instead of y) mean? What does the slope mean? What does the y-intercept (or h-intercept) mean? The graph and the equation both tell the story of the tree. Linear growth is modeled with lines, which have equations of the form y = mx + b (or, in our tree story, h = 2t + 1). You can come at that from so many angles.

The course can tell the tree's story, or any story that can be told through data, graphs, and/or equations. It tells the story of using math to help us think quantitatively about problems we care about. (And of course there are plenty of things we care about that cannot be quantified.)

## Sunday, February 10, 2019

### Links to Good Math Posts

I am closing tabs, so that maybe my browser will move faster. Here are all the links I couldn't stand losing:

## Thursday, January 10, 2019

### Geometry and Visual Thinking Puzzle Collection - Help Needed

Math lovers, I need your help. (Math-likers welcome too. Math-haters who are into torturing themselves are also quite welcome, if you know any.) If you love puzzles, that might be enough to suck you in. I'm hoping...

I am pulling together a collection of problems meant to entice students who aren't all that into math. (So, relatively easy puzzles.) I think we focus too much on algebra and not enough on the visual skills that come along with geometry. So each problem in this collection includes visualizing. Most of them involve geometry.

I'm calling them PPODs - Puzzle Problem Of the Day. I plan to post them in my classrooms and in the math department of the community college I work at.

I have found quite a few good problems in Geometry Snacks, by Ed Southall and Vincent Pantaloni. (I bought the e-version of the book yesterday.) I just now discovered Ed's blog (solvemymaths.com), and found more goodies. My other favorite source is the beautiful by-hand drawings done by Catriona Shearer, @cshearer41. I couldn't find who to credit for about a quarter of the problems I've collected.

I'm aiming for 60 problems, one for (almost) each day (M-Th) of the 16-week semester. I have about 50 already. What I need is help determining the difficulty level of each one. I need volunteers to help me figure that out. I'm using a scale of 1 to 5 myself, and anything I'd call a 4 or 5, I'm throwing out. (I've gotten stuck on a few problems. Those are not in the collection, of course.)

If you're interested, I'll send you my collection. And then you post your ratings here.
Just let me know in the comments if you'd like to check these out. (Or email me at mathanthologyeditor@gmail.com.) Also, I'm interested in your ratings and thoughts even if you just end up doing a few of the puzzles.

## Wednesday, November 28, 2018

### Logarithms: How Math Helps Science

I'm teaching my algebra students about logarithms today. It is likely the hardest algebra topic there is. When I started at Contra Costa College, after having taught full-time at Muskegon Community College for 6 years, I think I still was not 100% solid on logs. (I could do the problems, but proving those properties was a tangled mess.)

I had developed my murder mystery project because my officemate at MCC was a chemistry teacher, and she had complained that teachers weren't teaching logs well enough in Intermediate Algebra.I hoped that project would help. I think it does, but I don't have anything solid to base that on.

For many years now, I've told students a bit of history in my introduction to logarithms. John Napier invented them in 1614 as a way of making hard calculations easier. (Can you imagine finding the square root of 192.7 without a calculator? You'd have to guess, check, and modify repeatedly...)

Around that time Kepler came up with his three laws of planetary motion. Over the years, I have repeatedly wondered out loud whether Kepler was using logarithms while he figured these things out.
Today when this came up, I was able to turn to our tutor and ask him to look it up. He found a fascinating article on it.

Turns out Kepler came up with the first two "laws" before playing with Napier's new idea. But his third came after.

I. Planets move in ellipses with the Sun at one focus.
II. The radius vector (line from sun to planet) describes equal areas in equal times.
III. "The proportion between the periodic times of any two planets is precisely one and a half times the proportion of the mean distances."

This is an improper use of the word proportion. He was making a proportion from the logs of the ratios, and saw that log(T1/T2) = 1.5*log(r1/r2). In modern terms, we say something like: The square of the ratio of the times (that it takes two planets to go around the sun) equals the cube of the ratio of their mean distances from the sun. The fact that he was thinking in terms of logs shows how helpful the new (in 1614) concept was for helping scientists see patterns. (Read that article for more on Kepler and Napier.)

I love that math can help us see new things.

## Saturday, September 8, 2018

### Geometric Construction of the Regular Pentagon

I never particularly enjoyed geometric constructions when I was younger. It may be because I had a tendency to press down too hard, and mess up the width my compass was set to.  Like so many things, technology has a fix for that.

You can do geometric constructions on a number of sites, and in geogebra. My favorite site, which I discovered about 5 years ago, is sciencevsmagic.net. I having been telling students about it for years, and decided this summer to play with it again, to see how much I still like it. (A lot, it turns out.)

There are 40 challenges, in sets of 4. This time around I got every shape but the pentagon. I asked for hints (from math friends on facebook) which I then avoided reading because I wanted to do it on my own. I got nowhere that way, and finally followed instructions for how to construct it. Every other shape I have constructed makes sense on its own, but the construction of a regular pentagon must be proved with algebra. Ok, I don't feel so bad about 'cheating'.

I kept constructing it over and over, as I tried to understand it better. The first time (which you see above) I used these instructions (which I found by following links from wikipedia) and constructed it at sciencevsmagic, achieving my final shape challenge. Then I did it again using geogebra. I needed to figure out how it worked, and was at a loss. Cut-the-knot, a site I've gone to many times with my math questions, had a different construction - which I followed on both sciencevsmagic and geogebra - and an explanation (by Scott Brodie) that I kept working through as I wrote this.

[Sadly, Alexander Bogomolny, the creator of the cut-the-knot site, has died. I hope the math blogging community can find a way to maintain his amazing site. We miss you, Alexander.]

I will explain this as I understand it. My explanation will mostly just restate what I learned from these sites. But it may help some people, since some of the reasoning steps in Scott's explanation were hard for me to follow.

The constructions are worth doing first, so you can get a feel for what's happening as you do it. But when you're done, the question is still open - is this really a regular pentagon? The following explanation proceeds in two major steps. First it looks at relationships in the regular pentagon and the pentagram (which is a regular pentagon with a five-sided star inside it). Then it looks at the relationships built by the construction method, and we finally see that the side length given in the two ways is the same. Bingo!

The Construction
If you're ready to follow this, I trust that you already know how to construct the perpendicular bisector of a line segment. I will leave out the construction details for each of those.
• Starting with a circle with center O, construct a horizontal diameter, AB.
• Now construct its perpendicular bisector, CD.
• Construct the perpendicular bisector of AO, intersecting it at E.
• Construct the circle with center at E, through C, labeling its intersection with AB as F.
• Construct the circle with center at C, through F, labeling its intersection with the original circle (above B) as G.
• Go around the original circle, constructing this same size circle from each new point. (Center at G, through C, gives H at the other intersection with the original circle. Etc.)
The question remains: Does this really construct a regular pentagon, or is it maybe just pretty close?

Relationships in a regular pentagram
• The angles in a regular pentagon are each 108°, so each external angle, like ∠HMN, is 72°. That makes each angle at a point, like ∠NHM, 36°. Since ∠HMG is 108°, ∠MHG and ∠MGH must each be 36°.
•  Since angles ∠GLH and ∠LGH are each 72°, ΔHGL has two equal sides, HL and HG (which also equal CM, HI, etc).
• ΔCHI is similar to ΔIHM. So corresponding sides are in proportion. We get CH/HI=HI/HM, giving us HI2 = CH*HM = CH(CH-CM) = CH(CH-HI).
• Let CH = x*HI. Then the equation above gives us HI2 =x*HI(x*HI-HI), which gives us a (perhaps) familiar equation: 1 = x2-x, whose solution (in this case, where x is clearly greater than 1) is x = (1+`√5`)/2 aka φ. (At this point, I get excited. φ, aka the golden ratio, shows up in such different contexts!)
• So CH = φ*HI, and (similar triangles) HI = φ*HM and HM = φ*MN.
• We now have relationships between all the sides of the pentagon, star, and line segments on the star, but these are not yet connected to the radius of the circle.
• Now consider ΔODH. Two sides are radii, and the angle at O is 36°. Another triangle similar to all the others we've found. So OD = φ*DH.
• Also note that ∠DHC (inscribed angle on the diameter) is a right angle.
• Let's switch to simple variables names now. Let r=OD, s=CG, t=DH, and d=CI=CH.
• Then d= φ*s, r= φ*t, and t2+d2 = (2r)2, giving (r/φ)2 +(φ*s)2 = 4r2.
• This gives us s2 = 1/φ2*(4r2 - (r/φ)2) = (4/φ2 - 1/φ4)*r2.

Analyzing our construction
• We will find the length of s = CG = CF.
• OE=1/2*r. And EC = `√(OE``2+OC``2) = ``√(5/4)*r = ``√5`/2*r = EF. OF = (`√5`/2- 1/2)*r.
• So s2 = CG2 = CF2 = OC2 + OF2 = (1+(`√5`/2- 1/2)`2)*r2.`
• We have two very messy expressions for s2, one from the regular pentagon/pentagram and one from our construction. Are they equal? It may help to write out positive and negative powers of φ to help simplify the first expression. Yes! They are both equal to (5-`√5`)/2*r2, making the side length of the regular pentagon inscribed in a circle of radius 1 equal the square root of (5-`√5`)/2.
• And that proves that our construction created a regular pentagon. (Whew!)

I've completed 32 of the 40 challenges at sciencevsmagic. I still haven't figured out how to make some of the shapes within the original circle, and still haven't found the least moves for some of the shapes. I'm so glad there's still something to work toward the next time I revisit this.

[Blogger doesn't deal well with superscripts and square roots. Once again, I am noticing that I ought to learn some LaTex. Sigh.]