His first paradox involves sports statistics -- in particular, batting averages. I actually discussed batting averages just over a year ago on the blog (in the context of comparing batting averages to grades, as in why a .400 average is great, yet 40% in math class is an F). In particular, I said last year that Ted Williams was the last batter to hit .400, and this was about 70 years ago (by now, it's actually closer to 75 years). So Kung asks, why is it that in most sports, athletes perform better as the decades go on (think Usain Bolt again), yet no one has hit .400 in three-quarters of a century? His answer is that both pitchers and hitters have improved, and there is less variation among hitters, so even though the mean batting average is around .260, the standard deviation is much smaller.
Kung moves on to some medical paradoxes -- such as a medicine whose effectiveness can be given as either 4 percentage points, 40%, or 1 out of 25, as well as false positive tests. He then jumps from the doctor's office to the courtroom, where false positives show up in fingerprinting or DNA evidence.
The next paradox is known as Simpson's Paradox. This paradox, as Kung points out, is not named after O.J. Simpson (although he does mention the Simpson trial in discussing legal paradoxes), but rather the British mathematician Edward Simpson (who is still alive -- he just turned 93). Kung talks about a basketball game between the Cavaliers and the Thunder in which Kevin Durant has a better first half shooting and a better second half shooting than LeBron James -- yet LeBron has a better game overall shooting than Durant.
Simpson's Paradox applies to the educational world as well. Kung's biography as given by Great Courses states that the professor "is deeply concerned with providing equal opportunities to all math students" -- and so am I (as I mentioned back during my first winter break post). He mentions how at UC Berkeley back in the 1970's, more men were admitted to grad school than women, yet in each department, either about the same number in each gender were admitted, or even slightly more females in some departments. The same thing happens when we consider race instead of gender. Kung alludes to a situation where test scores in every race have gone up, yet overall scores have gone down. Here is a link to a specific example -- Texas beats Wisconsin within every ethnicity, yet Wisconsin has the higher test scores overall:
Now let's get to the Quick Conundrum for this lesson -- and it's somewhat relevant to Geometry. I mentioned the recreational mathematician Martin Gardner in yesterday's post -- recall how Kung would mention him several times throughout this series. Well, Kung's already mentioned Gardener for the second time. This time it's about the subject of Gardner's first Parade column -- the hexaflexagon.
To find out what a hexaflexagon is, we can watch a mathematician whose videos I posted here on the blog several times -- Vi Hart:
Vi Hart's videos show only hexaflexagons with three or six sides, but Kung's hexaflexagon actually has twelve sides. Just as with her dragon fractal video series, Hart's hexaflexagon series also consists of three videos (four if you count the hexaflexamexagon video -- the flexagon that you eat).
Hart suggests that since Martin Gardner was born on October 21st, 1914, math classes can celebrate October 21st as Hexaflexagon Day, or even the entire month of October as Hexaflexagon Month. So it's a day during the first semester that serves the same purpose as Pi Day does in the second -- it allows teachers to break from the usual curriculum, yet still do a math-related activity.
Indeed, several teachers have taken Hart's suggestion. One such teacher is the Oklahoma high school teacher whose blog I've linked to several times before, Sarah Hagan:
(Notice that the blog post is dated November 12th, but Hagan writes that she actually performed the activity on October 21st.)
Another blog mentioning hexaflexagons belongs to Kathryn Huxtable -- even though Huxtable, apparently, is not a math teacher:
Huxtable shows what is called a Tuckerman traverse for a 12-sided hexaflexagon. The same traverse appears in Kung's video as well, while Hart's video shows the traverse for the 6-sided flexagon. We see that a traverse is in fact a network -- the same sort of networks we saw back in Lesson 1-4 of the U of Chicago text, all the way back on the first day of school.
By now the readers of the blog are wondering -- if Hagan can have a hexaflexagon activity in her Algebra I class, why can't I include the activity in my Geometry class? After all, hexagons, as regular polygons, appear to be more relevant to Geometry than Algebra I. And if I can bend over backwards to make sure that I have a Pi Day lesson, then why not do the same for Hexaflexagon Day?
Well unfortunately, I'm just barely learning about hexaflexagons myself. I'd have to make sure that I can fold one myself before expecting students to create them. Then again, it's only January, so I have plenty of time to practice folding them myself before October rolls around.
Notice that two-and-a-half months ago, on October 21st, I posted a worksheet for Lesson 7-1, on Drawing Triangles (in preparation for SSS, SAS, and ASA Congruence in Lesson 7-2). Well, there are indeed triangles as part of the hexaflexagons, so it was possible for me to justify having a flexagon activity that day.
What sounds good to me would be to tie hexaflexagons to hexagons. In particular, we recall how the hexagon is one of three regular polygons that the Common Core expects students to be able to construct inscribed in a circle. Well, if we look at how to fold a hexaflexagon (in Hart's second video, as well as a link from Huxtable's page), we see that we actually don't need hexagons when we create the flexagon. It appears that we do need an equilateral triangle, though -- once we fold the first equilateral triangle, the rest of the folding looks easy.
Let's think back to Lesson 4-4, where we perform the construction of an equilateral triangle. We found out that this has nothing to do with the Common Core construction, since even though the construction involves circles, the triangles aren't inscribed in the circle. But notice that the triangle is one-sixth of a regular hexagon -- and this hexagon is inscribed in the circle.
So we can connect hexagons to Common Core constructions as follows -- first follow Lesson 4-4 to construct an equilateral triangle. Then we use that equilateral triangle as a guide to help us fold a strip into a hexaflexagon. Finally, we can see that the hexaflexagon fits perfectly into the original circles, corresponding to a hexagon inscribed in the circle. Hopefully students will be able to use this to remember how to construct the regular hexagon -- and they can have more fun doing this than playing Euclid: The Game, since they will be able to enjoy playing with the hexaflexagon afterwards.
I obviously haven't planned out the blog calendar for next year yet, but it's possible that we could be near Lesson 4-4 by October 21st -- recall that we had jumped up from Chapter 4 to Chapter 7 in the text last October.
Of course I could go on about hexaflexagons all day, but unfortunately we must get back to our regularly scheduled lesson. Lesson 7-8 of the U of Chicago text is on the SAS Inequality. I've chosen just to repeat the worksheet from last year. On that worksheet, I decided to add another inequality -- the SSS Inequality. This theorem is somewhat of a converse to SAS Inequality. It doesn't appear in the U of Chicago text, but does appear in both Glencoe and Dr. Franklin Mason.
This is what I wrote last year about the SAS Inequality. Notice that I made a brief reference to the Geometry student I was tutoring last year -- recall that I was going over Glencoe's Chapter 5 with him right around the time I was posting these to the blog:
Now let's get back to Lesson 7-8 of the U of Chicago text and the SAS Inequality in preparation for my tutoring session with my geometry student tonight. As I mentioned before, this section gives a proof of the SAS Inequality using the Triangle Inequality. As usual, I have decided to convert the proof to two-column format. The figure accompanying the proof gives two triangles, ABC and XYZ.
SAS Inequality Theorem:
If two sides of a triangle are congruent to two sides of a second triangle, and the measure of the included angle of the first triangle is less that the measure of the included angle of the second, then the third side of the first triangle is shorter than the third side of the second.
Given: AB = XY, BC = YZ, angle B < angle Y
Prove: AC < XZ.
1. AB = XY, etc. 1. Given
2. exists isometry T s.t. A'B' is 2. Definition of congruent
XY, C' same side of
3. C'Y = ZY 3. Isometries preserve distance
4. Let m symmetry line C'YZ 4. Isosceles Triangle Symmetry Theorem
5. m perp. bis.
6. QC' = QZ 6. Perpendicular Bisector Theorem
7. A'C' < A'Q + QC' 7. Triangle Inequality
8. AC < XQ + QZ 8. Substitution
9. XQ + QZ = XZ 9. Betweenness Theorem (Segment Addition)
10. AC < XZ 10. Substitution
We see that the proof is similar to that of SAS Congruence Theorem, except that this isometry puts C' on the same side of
Both Dr. M and Glencoe state a converse to SAS Inequality -- Glencoe calls it SSS Inequality. Dr. M hints at this proof -- we can use the same strategy that we used to derive Unequal Angles Theorem from its converse, Unequal Sides Theorem. We prove it indirectly:
SSS Inequality Theorem:
If two sides of a triangle are congruent to two sides of a second triangle, and the third side of the first triangle is shorter than the third side of the second, then the measure of the included angle of the first triangle is less that the measure of the included angle of the second.
Given: AB = XY, BC = YZ, AC < XZ.
Prove: angle B < angle Y
Assume not. Then angle B is either less than or equal to angle Y.
Case 1: angle B = angle Y. Then triangles ABC and XYZ are congruent by SAS Congruence, and so AC = XZ, a contradiction.
Case 2: angle B > angle Y. Then AC > XZ by SAS Inequality, a contradiction.
In either case we have a contradiction of AC < XZ. Therefore angle B < angle Y. QED
For Euclid, the SAS Inequality is his Proposition 24. Dr. M's first proof is based on Euclid:
and the converse, the SSS Inequality, is Euclid's Proposition 25:
Kung ends the lecture by describing two medicines that help cure kidney stones. Medicine A is better at dissolving small stones, and it's also better at dissolving large stones. So which medicine should the patient take? Of course, the best medicine to take is Medicine B -- just ask Edward Simpson why.