Some sort of number talks with J3

Based on conversations about the dots pictures from Math4Love:

Day 3
I notice

1. there's a number 3, but the number of dots isn't the same (it isn't 3)
2. five over here (pointing to dots) and zero on the down part (the bottom half of the 10 frame)
3. J0: I see some letters...
4. I even noticed that. I noticed there's this plus (points to dash - )
5. J0: I noticed this square
6. I noticed it was a line (bottom row of the 10 frame)
7. I noticed these triangles (the white space in 10 frame sections that have dots)
8. I noticed these are 5 and an extra one (on second page of day 3)
9. I noticed that there are four left (empty cells on second page)
10. J0: you saw 5+1, I see 2 + 4
11. Those two are together. The other ones are lonely.

I wonder

1. Why didn't they make 10 dots?
2. Why did they only cover the middle of the square (points to a dot in the upper left square of teh 10 frame)?
3. I wonder, how do numbers talk? (after I read the title of the slide to her)
4. I wonder, why do they only put 1 on the bottom row?
5. I wonder, can we arrange them so none are lonely

Day 4

I notice

1. Five on the top and five on the bottom
2. Ten
3. five and four, nine

Day 5

I notice

1. This doesn't have a box to go in (a 10 frame)
2. It has a dot in the middle
3. we can count them 2, 2, 2 (pairing them up)
4. if we take 2 away, there will be four
5. the sides are the same (it has a line of symmetry in the middle)
6. it looks like an animals footprint
7. the top four make a diamond
8. If we turn our body to the side, the top four make a rectangle
9. taking out the two in the middle, we have a square
10. it has 8 dots.
11. the number of dots doesn't match the day number

I wonder

1. is it a real footprint?
2. I wonder, if we take the bottom five, it would be 3?

Solving contest problems (challenge from Mike Lawler)

These are my notes working through problems posted by Mike Lawler on his blog. You'll have to go there to see the problem statements.

The intended value of this write-up is to show examples of the actual problem solving thought process someone has followed, not just a polished solution.

Problem 19
Since the circle has area 156 π the radius squared is 156, which is 4 * 3* 13. That seems like a strange number, I'm curious to see where it will make calculations come out nicely, later in the problem.

We're told OA has length 4 sqrt(3). Squaring that only gives 48, so A is well within the circle. That means triangle ABC has vertex pointing toward O along the perpendicular bisector of side BC.

This lets me set up a picture of a right triangle with legs length $x$ (half the side of the equilateral triangle), $x$ sqrt(3) + 4 sqrt(3) (the altitude of the equilateral triangle plus OA) and hypotenuse r (2 sqrt(3*13)).

Applying the pythagorean theorem and simplifying along the way:

$$x^2 + 3(x+4)^2 = 156$$
$$4x^2 + 24x + 48 - 156 = 0$$
$$x^2 + 6x - 27 = 0$$

Visually factoring gets me $x$ is either -9 or 3.  3 is much more reasonable for half the length of the side of a triangle, so my answer is 6.

Note: If this weren't a contest problem, I might try to think about the -9 root a little more carefully.

Equiangular hexagon
I had already solved this problem from a recent tweet of Mikes, so I can't fully recreate my thought process.  Here were some of the highlights:

• wonder why 70%
• draw a picture: fail to notice that triangle ACE is equilateral
• Split the hexagon into two trapezoids by line CF
• Calculate the area of those two trapezoids
• Calculate the length from C the intersection with AE and segment CF.
• Calculate the area of ACE based on the two subtriangles split by CF.
• Obtain the quadratic equation for r based on the formulae for the two areas and the given 70% parameter.

School Competition

Let's call Andrea's rank $m$ for median. Since she is the unique median, there must be $2m - 1$ total contestants. We also know:

• $m \leq 36$ because Andrea scored higher than Beth who was ranked 37th
• $64 \leq 2m -1 $ because Carla ranked 64th, so there were at least that many competitors
• $3 \mid (2m - 1)$ since every school sent three competitors

The first two inequalities tell us that $33 \leq m \leq 36$. Because $2m -1$ is a multiple of three, $m$ can't be a multiple of 3, so it has to be 34 or 35. Calculating mod 3, we can quickly check both and see that $m$ has to be 35.

That means the competition had 69 competitors from 23 schools.

Using multiple choice
After putting together these notes, I saw a commenter on Mike's blog use "solution by multiple choice." When I was doing timed tests/contests, I would make use of the options as part of my strategy. However, I don't do that now, since I'm much more interested in exploring the mathematics of the problems than getting the final answer.

Leftorvers with 100 game

In Grades 3 and 4, we played a nice game that (I think) we got from Marilyn Burns. Looking for a reference after the fact, I see it explained in her book Lessons for Extending Division.

Basic play

• Start with a target number (we used 100) and collection of available divisors (we used integers 1 to 20)
• Players take turns choosing a divisor from the remaining available options. They divide the current target by that divisor and keep the remainder as their score for the turn. They also subtract the remainder from the target to create a new target for the next player.
• Each divisor gets crossed out when it is used, so it can only be used once.
• The game ends when the target is reduced to 0 or when all available divisors are exhausted.
• We played as a two player game.

Here's an example of a game play:

Player 1 chooses 17. 100 = 17 * 5 + 15, so player 1 scores 15 points, the target is reduced to 85, and 17 is no longer available as a divisor.

Player 2 chooses 20. 85 = 20 * 4 + 5, so player 2 scores 5 points, the target is reduced to 80, and 20 is no longer available as a divisor.

Player 1 chooses 14. 80 = 14 * 5 + 10, so player 1 scores 10 points, the target is reduced to 70, and 14 is no longer available as a divisor.

Player 2 chooses 18. 70 = 18 * 3 + 16, so player 2 scores 16 points, the target is reduced to 54, and 18 is no longer available as a divisor.

Player 1 chooses 19. 54 = 19 * 2 + 16, so player 1 scores 16 points, the target is reduced to 38, and 19 is no longer available as a divisor.

Player 2 chooses 13. 28 = 13 * 2 + 12, so player 2 scores 12 points, the target is reduced to 26, and 13 is no longer available as a divisor.

Player 1 chooses 15. 26 = 15 * 1 + 11, so player 1 scores 11 points, the target is reduced to 15, and 15 is no longer available as a divisor.

Player 2 chooses 16. 15 = 16 * 0 + 15, so player 2 scores 15 points, the target is reduced to 0 and the game ends.

Player one wins 52 to 48.

Our experience

We found this to be a fun, interesting, and engaging game. The practice with dividing and remainders was pretty obvious. In addition, it opened up some opportunities for strategic thinking, particularly at the end-stage of the game. I think there are also several good extension explorations.

Extensions

First, I created a simple pencilcode program for two players to play this game against each other. Here's a playable version (and here's the code).

Second, you'll notice that the first player in our sample game followed a "greedy strategy."  At each stage, that player chose the divisor that would give the most points on that turn. If you look closely, that isn't the best strategy at the end of the game.

So, a natural exploration is to find the best strategy for different starting targets. One specific point of about which we're curious: is it ever desirable to skip your turn (choosing 1 as the divisor is effectively a turn skip)? 

Some other areas for investigation:

• must the game always end on 0 or can we run out of divisors?
• given a target and collection of starting divisors, what is the shortest (number of turns) game possible? What is the longest game (number of turns) that does end at 0?

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