## Posts tagged ‘geometry’

### They’re Moving Second Base

When I first heard that baseball is moving second base, my first thought was, “My, goodness! Isn’t it enough that we’re dealing with a global pandemic, a Russian tyrant invading a neighboring country, a humanitarian crisis in Nicaragua, food insecurity in Somalia, Haiti, and Madagascar, an ever-widening wealth gap, an uptick in calls from unknown numbers, paper cuts, and excessively long lines at the Starbucks drive-thru? I mean, when’s it gonna stop?”

But my second thought was, “This is going to wreak havoc on the secondary textbook publishing industry.” Just look at all the problems that exploit the baseball context:

- Location of the pitcher’s mound (Q2)
- Throw from catcher to second base
- Throw from catcher to second base
- Distance from first base to third base (Q4)
- Runner’s speed in relation to second base

All of those problems are predicated on a consistent distance between bases. Won’t the relocation of second base cause inconsistency?

Well, actually, it won’t.

According to the official rules of baseball, one vertex from first base, third base, and home plate are to be coincident with three vertices of the infield square; but, the *center* of second base is to be coincident with the fourth vertex. With the rule change, second base will be moved so that one of its vertices will be coincident with the fourth vertex of the infield square, finally bringing a state of geometric consistency to the game that I, for one, believe is long overdue. The image above shows the new (white) and old (gray) locations of second base.

The question all fans should be asking isn’t why are they changing the layout of the infield. The more pertinent question is, what took so damn long?

As it turns out, second base doesn’t get all the credit for the previous configuration issues. To the contrary, it was the movement of the other bases that resulted in a problem. In the 1860s, it was generally agreed that all four bases should be positioned with their centers at the vertices of the infield square. And by “generally agreed,” I mean that there was consensus about this, but it wasn’t officially stated in the rules until 1874. Then in 1877, the rules changed so that the back corner of home plate — at the time, home plate was still a square, not a pentagon like today — coincided with the vertex of the infield square, positioning all of home plate in fair territory. A decade later, first base and third base were moved to be entirely within fair territory, too; but most folks didn’t even notice, because that same year (1887) a number of other rules changes garnered more attention:

- Pitchers were limited to just one step when delivering a pitch; previously, they could take a running start
- Batters were prohibited from requesting a high or low ball from the pitcher, as they had been allowed in the past
- The pitcher’s mound was moved back five feet (from 50′ to 55′)
- Five balls were required for a walk, reduced from six
- Four strikes were required for a strikeout, increased from three

With so many drastic rules changes happening simultaneously, it’s hardly a surprise that first and third were repositioned in relative obscurity while second was left floundering in geometric misalignment.

Just so you know, the rules change will only occur in the minor leagues this year. If it pans out, you can bet you’ll see it in the MLB in a year or two.

But why stop there? Here are some other rules changes in sports that should probably be implemented.

**Scoring system in football.** I mean, you can score 1, 2, 3, 6, 7, or 8 points depending on what you do. Isn’t that a little excessive? While we’re at it, let’s change the width of the field, too — who the hell thought 53⅓ yards was an appropriate dimension?

**College basketball uniforms.** Bring back 6, 7, 8, and 9. You may not have known that those digits are not allowed, because each of them requires two hands. Referees indicate the player who committed a foul using their fingers — for instance, holding up two fingers on the right hand and three fingers on the left to indicate that an infraction was committed by number 23 — and the digits 6‑9 would require more than five fingers.

**Frames in bowling. **Two balls ain’t enough. Give everyone three attempts to knock down all ten pins.

**Cheerleader weigh-ins. **Really, folks? The 15th century called, and they want their misogyny back. One anonymous NFL cheerleader wrote that she was banned from performing because she weighed more than 122 pounds. While we’re at it, ban weigh-ins for jockeys, too. The Kentucky Derby — which apparently has one of the more liberal weight allowances — caps the weight at 126 pounds; that includes 7 pounds for the jockey’s gear, so the jockey can’t tip the scale at more than 119 pounds.

**Taunting.** Allow it everywhere. In college football, the rule is just stupid. Admittedly, one player shouldn’t be allowed to stand over another player while making insulting comments about their mother; but “taunting” according to the NCAA Rule Book includes spinning or spiking the ball, choreographed acts, and the player altering stride when approaching the end zone. C’mon! Further, I’d like to see taunting *encouraged* a bit more at some events, such as math competitions. Wouldn’t it be great if one participant walked up to another and said, “You can’t even spell Q.E.D.!”

### 4 Folds, 40 Theorems, and Chinese New Year

If you asked a mathematician how to make an origami coin purse for Chinese New Year, you might receive the following instructions:

- Rotate a square piece of paper 45°.
- Reflect an isosceles right triangle about a diagonal of the square so that congruent isosceles right triangles coincide.
- Reflect the lower left vertex of the isosceles right triangle so it coincides with the opposite leg, such that the base of the isosceles right triangle and one side of the newly formed isosceles acute triangle are parallel.
- Reflect the other vertex of the isosceles right triangle to coincide with a vertex of the isosceles triangle formed in the previous step.
- Fold down a smaller isosceles right triangle from the top.
- Fill the coin purse with various denominations of currency, such as a $2 bill, a $1 gold coin, a quarter, or a penny.

Don’t like those directions? Perhaps these pictures will help.

To be truly authentic for Chinese New Year, you’ll want to use red paper instead of white.

The Common Core State Standards for Mathematics (CCSSM) have added rigor to geometry by requiring students to prove no fewer than a dozen different theorems. Further, the geometry progressions document formulated by Dr. Hung-Hsi Wu suggests that a high school geometry course should use a transformational approach, basing a majority of the foundational axioms on rotations, translations, and reflections.

What struck me about the origami coin purse was the myriad geometry concepts that could be investigated during this simple paper-folding activity. Certainly, the idea to use origami in geometry class in nothing new, but this one seems to have a particularly high return on investment. Creating the coin purse requires only **four folds**, and though perhaps I lied by claiming **forty theorems** in the title, the number of axioms, definitions, and theorems that can be explored is fairly high. (Claiming “forty theorems” wasn’t so much untruthful as much as it was archaic. The word *forty* has been used to represent a large but unknown quantity, especially in religious stories or folk tales, as in *Ali Baba and the 40 Thieves*, it rained “40 days and 40 nights,” or Moses and the Israelites wandered the desert for 40 years.)

For instance, the fold in Step 3 requires parallel lines, and the resulting figures can be used to demonstrate and prove that parallel lines cut by a transversal result in alternate interior angles that are congruent.

No fewer than four similar isosceles right triangles are formed, and many congruent isosceles right triangles result.

The folded triangle in the image above is an isosceles acute triangle; prove that it’s isosceles. And if you unfold it from its current location, the triangle’s previous location and the unfolded triangle together form a parallelogram; prove it.

The final configuration contains a pair of congruent right triangles; you can use side-angle-side to prove that they’re congruent.

But maybe you’re not into proving things. That’s all right. The following are all of the geometric terms that can be used when describing some facet of this magnificent creation.

- Parallel
- Congruent
- Similar
- Alternate
- Angle
- Square
- Reflection
- Rotation
- Translation
- Transversal
- Bisect
- Coincide
- Right Angle
- Perpendicular
- Isosceles
- Vertex
- Triangle
- Trapezoid
- Parallelogram
- Pentagon

And the following theorems and axioms can be proven or exemplified with this activity.

- Sum of angles of a triangle is 180°.
- Alternate interior angles formed by a transversal are congruent.
- Any quadrilateral with four congruent sides is a parallelogram.
- Congruent angles are formed by an angle bisector.
- Triangles with two sides and the included angle congruent are congruent.

There is certainly more to be found. What do you see?

### What’s Your Problem?

Problems in the MathCounts School Handbook are presented “shotgun style,” that is, a geometry problem precedes a logic puzzle and follows a probability question. (I worked for MathCounts for seven years and then served as a writer and chair of their Question Writing Committee, so I’m not unbiased.)

By comparison, textbooks often present 50 exercises on the same topic, each one only minimally different from the previous one. That tips the hand to students, methinks, and makes them realize, “Oh, I just need to do the same thing.” I prefer the MathCounts approach, where students have to dig into their bag of tricks to find a viable solution strategy.

With that in mind, here are a few problems I’ve encountered recently, each one not like the others.

**Problem 1.** The simple polygon is made from 73 squares, connected at their sides. What is the perimeter of the figure?

**Problem 2.** What is the expected number of times that a six-sided die must be rolled to get each number 1–6?

**Problem 3.** A wall is to be constructed from 2 x 1 bricks (that is, bricks that are twice as long in one direction as the other). A strong wall must have no **fault lines**; that is, it should have no horizontal or vertical lines that cut entirely through a configuration, dividing it into two pieces. What is the minimum size of a wall with no fault lines? The figure below shows a 3 × 4 wall that has both horizontal and vertical fault lines.

**Please share great problems you’ve recently encountered in the Comments.**

No answers, but here are some hints.

*Problem 1. *Look for a pattern.

*Problem 2.* Check out this simulation for the Cereal Box problem.

*Problem 3.* The smallest arrangement without a fault line is larger than 3 × 4 and smaller than 10 × 10.

### Questions and Answers

I’ve been teaching my sons how to tell time on an analog clock. The following is a recent conversation:

Me: The little hand is the hour hand, and the big hand is the minute hand.

Eli: What’s the third hand for?

Me: That’s the second hand.

Alex: Why is the third hand called the second hand?

Here are some questions that I’ve been asked recently for which I *did* know to respond…

How good are you at algebra?

Vary able.Why is simplifying a fraction like powdering your nose?

It improves appearance without changing the value.What do you get when you cross geometry with McDonald’s?

A plane cheeseburger.

### Observations

Some things I’ve noticed…

- Algebra is
*x*sighting. - Rational people are partial to fractions.
- Geometricians like angles… to a degree.
- Vectors can be ‘arrowing.
- Calculus teachers can go on and on about sequences.
- Translations are shifty.
- Complex numbers are unreal.
- Most people’s feelings about integers are positive.
- On average, people are mean.

### Polyhedra Play Things

This morning at the Battlefields of Northern Virginia Council of Teachers of Mathematics (BNVCTM) spring conference, I had the pleasure of meeting Art Stoner of A+ Compass. Art was wearing a t-shirt with the following joke:

ALGEBRA: It’s only fun till someone loses an

i.

Art introduced me to Magformers, winner of the 2007 Oppenheim Toy Portfolio Platinum Award. Magformers are plastic polygons with small magnets along the middle of each edge, which can be used to make Archimedean solids, Johnson solids, houses, tanks, stars, and various other polyhedra.

I bought a set of 18 squares and 20 triangles. If my sons like them, I’ll buy more. (I know, I know… who am I kidding? Whether I buy more depends mostly on whether *I* like them.)

Talking about geometric objects reminds me of two geometry jokes…

- Without geometry, life would be pointless.
- Think outside the tesseract.