Equilateral Triangles Puzzle (from Twitter)

Mike Lawler issued a call to respond to a nice challenge Matt Enlow posted on twitter:

Equilateral ΔABC inscribed in a circle. 2nd equilateral ΔPQR such that P is on segment AB and Q & R are on arc AB. Locus of center of ΔPQR?

— Matt Enlow (@CmonMattTHINK) November 17, 2016

My scribbles

I will admit that, initially, I had no good ideas about how to approach this puzzle. Intuitively, I was drawn to the idea of specifying the point P on the chord AB, building the equilateral triangle PQR, and then finding the center of that triangle. In the spirit of doing something, I decided to set up a coordinate system and think about the resulting equations:

Now, these are a mess and didn't really go anywhere, but, I had two thoughts as I was writing these out:

1. Since we are just intersecting lines and circles, we are probably looking for a conic section
2. The fact that triangle ABC is equilateral gives us a relationship between the side length and the radius of the largest circle, but I doubt whether it is critical for this problem. My suspicion is that we will get something similar for any chord.

Now, I'd had the suspicion that the coordinates for the center of an equilateral triangle was the average of the coordinates of the vertices, so I did a little work to confirm that:

At this point, I felt that I wasn't really making progress. In particular, I felt that I didn't really have the right ideas about how to find Q and R, given P.

I turned to geogebra to build the picture. Given Matt's request not to send screenshots, I felt that this was a bit of a cheat, but anyway...

I still didn't have a good idea about how to find Q and R, so I first built a worksheet where Q was a free point, from which R would be constructed so that the triangle PQR is equilateral. Then, I played around with Q until it and R sat on the original circle.

After doing this, I had two observations/conjectures. Taking C as the center of the original circle:

1. we know the angle CPQ is 150 degrees
2. we know that the line CP is the perpendicular bisector of segment QR

Neither is really tricky, so I was a bit chagrined to have missed them earlier, particularly the second point.

With that I now had a method to (a) draw more realistic diagrams and (b) reduce my Geogebra sheet so that only P's location on AB was an open degree of freedom. Here's a link to that Geogebra work.

Where I am now

Playing around with the Geogebra sheet, I made and tested conjectures about what conic shape could describe the locus. For simplicity, I hoped it was a circle, but testing three points shows that doesn't look right. Instead, it looks like a hyperbola. With that, my plan is to return to the cartesian plane, define a friendly reference variable, and check that I get the right relationship for the coordinates of our center points.

Geogebra FTW

Putting aside the fun of working on a nice problem, this served me as a great reminder about Geogebra. As a result, we'll soon be playing Euclid the Game at home with the kids.

Politics, math, and dog-whistles

In Frank Herbert's Dune, there is a cool idea of a military language with a flexible structure so that any pair of people can speak to each other in a way that they will understand, but which no listener will understand.

Modern politics and social media are moving closer to realizing this idea, through dog-whistles.

For example, I was really struck by this ad for a math curriculum package (see the second paragraph):

For the author and their intended audience, Common Core means something very particular and particularly bad.

Personally, I find the fragmentation of language very troublesome. Among other things, it contributes to a certain type of magical/fallacious thinking, nicely exemplified by the popularity of ObamaCare (according to some survey results):

1. The Affordable Care Act gets broader support than ObamaCare (they are two names for the same thing)
2. Individual provisions get significant majority (more than 50% of respondents) support, while ObamaCare does not only earns minority (less than 50%) support.

This first point is silly, and I don't see any logical way to redeem that combination of beliefs. For the second point, while it is possible to logically reconcile the two observations, the most likely explanation is that people surveyed were mis- or under-informed.

Puzzling puzzlers

Preparing for classes today, we went through our lists and links to gather a list of puzzles for the kids. Hopefully, this will help short circuit the work we have to do next time.

• Eric Friedman Puzzle Collection (highly recommended by Mathpickle)
• Cryptarithm: shape substitution version, basic ones, 
• Funny dice 
• SolveMe Mobiles
• Make your own coin system
• Multiplication: Miller's Puzzle from Dudeney
• Find the Factors
• Multiples of 7 and asymmetric information: What's my number from Tanya Khovanova
• Nonagrams and battleship puzzles
• Logic Grid puzzles: Two dimensional starters, Logic-Puzzles, 
• Distortions logic puzzle (nice twist on truth tellers or liars)
• Questions from an adult, general public, math contest: Math Mama Writes
• Math is Fun collection
• Of course, there are a lot of great puzzles on Mathpickle
• Sudoku
• Number arrangements: magic squares, "magic" triangles and stars
• Zukei geometry puzzles (find the "hidden" shape)
• Area maze, K-maze, B-cross, Shikaku, Dokoeq (all from Naoki Inaba)
• Verruca Values

Election analysis

This isn't exactly a kids-learning post, so apologies to anyone disappointed. This is my attempt to organize some thoughts and analysis around the results of the 2016 US election. The questions I'm posing are "Why did Trump win/Clinton lose the presidential election?" and "What are the lessons for 2018 and 2020?"

This analysis is not complete, so apologies to anyone who wants a nicely packaged story.

I welcome data, analysis, and different perspectives supported by evidence.

Range of theories

CNN starts us off with a nice collection in their article: 24 theories why Trump won.

Before I summarize their list, I'll note that I intentionally use the combination "Trump won/Clinton lost" rather than focus on a single side. What I'm thinking is that there are factors specific to both sides as well as relative factors at play. It isn't appropriate to look at one group in isolation. That idea helps me with a simple taxonomy of their theories:

Trump-focused theories

1. Media. There are several sub-theories: (a) social media and fake new echo chamber: Trump supporters were ill-informed because of systematic problems with modern media. Facebook is particularly cited as a key culprit. (b) celebrity plays stronger than substance, 
2. Trump appealed to white males, so they supported him. Subtheories are (a) xenophobia/racism/sexism, (b) backlash to political correctness.
3. Trump appealed to voters who have been struggling economically

Clinton-focused theories

1. Voter suppression directed toward minority voters who would typically support the Democratic candidate. 
2. Leaked information related to scandals. Some sub-theories: (a) Russia provided hacked information, (b) FBI.
3. Third party candidates drained support from Clinton.
4. Clinton was a weak candidate who did not appeal sufficiently to those who formerly voted for Obama (a) saddled with too much negative baggage (b) not properly tested or vetted through the primary process

Relative theories

1. anti-establishment fervor

The most popular story

My impression is that the most popular theory is Trump 3: Trump won the votes of those who are struggling economically. One strong example from before the election is the Guardian's My Journeys in Trumpland.

Now, empathy is good and I applaud people trying to understand each other. That said, is the economic plight of the non-elite white voter the reason for this election outcome? It is a popular story with a nice human/humanizing face, but doesn’t ring true to me. Here are the reasons I’m skeptical.
First, from my own experience, this group has been struggling since the 1980s (maybe earlier). The story is not new.

Second, it looks like Trump’s overall support was about what we should expect from a “generic republican,” maybe underperforming a little. I’m basing this on the comparison against the past two elections (see here for example).

Third, the evidence that Trump’s supporters (on average, whatever that means) are fairly well-off (see 538 primaries and the income table from CNN exit polls.)

Confusion about voter turn-out

I am currently confused about voter turn-out. These articles from CNN and 538 seem contradictory, but I haven't had a chance to work through their numbers and reconcile.

Anti-establishment

Perhaps one useful test here is the degree to which incumbents were re-elected compared with past elections. I haven't yet gathered the statistics for congressional elections, but my impression is that incumbents were overwhelmingly re-elected, basically in line with past experience, if not more. See how few districts and states are cross-hatched in these maps, (marking that shows a flip in the party controlling that seat):

2016 House of Representatives results from NY Times as of 14 Nov 2016

2016 Senate results from NY Times as of 14 Nov 2016

Multiplication & Fractions Math Games from Denise Gaskins (a review)

I really like Denise Gaskins's new book: Multiplication & Fractions Math Games (links to paperback edition and accompanying printable.) How much do I like it? Well, I had already written a lengthy review that, somehow, I managed to lose and am now back writing another one.

I'm going to forego my preferred Good (what I liked), Bad (arguable weaknesses) and Ugly (unforgivable sins) because I don't really have anything to say in those two negative categories. Instead, let me just talk about who would find the book useful and why.

Group 1: Parents who feel their own math skills are weak.

Maybe you never really understood what multiplication means or what fractions are? As long as you start with an open mind and are willing to engage playfully, the activities in the book can help you as you help your kids. It starts with models that are visual explanations of the concepts. Gaskins also breaks learning these concepts into comfortable steps that emphasize patterns and relationships, the real ideas that are behind properly understanding multiplication and fractions (indeed, math generally). The sequence of games in each section starts by building familiarity and then fluency (speed) to solidify all of that work.

Group 2: Parents who worry about their kids struggling with these concepts

Anecdotally, these two areas are the first major stumbling point for students in their math studies. As I noted above for parents themselves, the sequencing in the book will help kids develop a strong foundation, beginning by understanding what multiplication means (and what fractions are). Beyond that, playing the games will make these concepts familiar and, I believe, lead them to recognize examples around them in their daily lives.

Group 3: Families who like to play games

Kids (and parents!) find these games fun. I've been field testing math games for the last 18 months and keep seeing how engaged kids get when playing math games. I have played many, though not all, the games in Multiplication & Fractions and strongly believe the games in the book will be winners with most kids.
Now, let's face it, you might not be thrilled with every game. For example, I wasn't so excited by the idea of playing War variations. However, a lot of other games in the book that are strategically and mathematically rich. Also, truth be told, my kids and students have really enjoyed playing multiplication war!

So, there's really nothing weak in this book?

There is only one worry I have about making a blanket recommendation: parents who start with a completely wrong mindset. If you believe in speed over understanding or mathematical gifts instead of effort, then this book is the wrong place to start. Instead, read Dweck's Mindsets and spend time with Jo Boaler's website. Maybe also re-watch Karate Kid (no joke, this is what I'm currently playing.)

A disclaimer, sort of

I'm friends with Denise Gaskins and got a review copy of this book. However, you should understand that we're friends because I'm a fan of her math teaching work and not the other way around. We've never even met in real life and, in fact, live in different continents. I know of her because advocacy of play-based math learning. I admire her because she is one of the best at creating resources that bring this material into the reach of the typical parent.