Despised Babies Are Eaten by Illogical Crocodiles

Lewis Carroll, who wrote Alice in Wonderland, was a mathematics professor at Oxford University in England. Besides his popular books, he is also known for his puzzles.

Today, our second and last day of the THSP mathematics conference, Job for the Future's Robert Knittle, our presenter, included sample Lewis Carroll puzzles in a booklet he gave us this afternoon. Puzzle #1, supposedly one of the famed professor's simpler ones, aroused my interest.

The puzzle has three implications:

1. All babies are illogical.
2. Nobody is despised who can manage a crocodile.
3. Illogical persons are despised.

Using the three implications listed above, what is the "inescapable" conclusion?

The same booklet offered a solution, but I had the urge to work on it myself. I teach my geometry students about conditional (if-then) statements. Instinctively I thought that the so-called implications are but conditional statements waiting for a proper makeover. I would love for them to know that everything that I had taught them on conditional statements was enough to solve a Lewis Carroll puzzle. :p

Here's how I played it:

I rewrote all three implications into the following if-then statements:

4. If it is a baby, then it is illogical.
5. If you can manage a crocodile, then you are not despised.
6. If you are illogical, then you are despised.

I applied the Law of Syllogism on statements 4 and 6, and it resulted to:

7. If it is a baby, then it is despised.

I compared statements 5 and 7. Since the contrapositive of a conditional statement is logically equivalent to the statement, I decided to write the contrapositive of the former:

8. If you are despised, then you cannot manage a crocodile.

I applied the Law of Syllogism, once again, on statements 7 and 8 to arrive at my "inescapable" conclusion:

9. If it is a baby, then it cannot manage a crocodile.

It is the same as the one presented in the booklet. Yayyy!

In Lewis Carroll's words, that would probably be rephrased as:

No despised baby is spared by illogical crocodiles.

Not really! :D

Proof of Joy

The tutoring period had already started. There were four students in the room who were already busy working collaboratively on exercise problems I have assigned to them.

I was erasing the doodlings Berenice and Leslie did on the dry-erase board during advisory period earlier when Danny came in. He was panting.

"Sir, I do not know how to do proofs, and I want you to help me." There was urgency in his voice.

I handed him the list of exercises we used in class earlier. "Choose any one that you think you could or want to do, and do it on the board."

"Okay."

As Danny copied his chosen problem on the board, he read aloud every word and symbol. He also said out loud what he was thinking about the problem.

"For my first step, I'm gonna write in the first given which is..."

He took a few steps away from the board, knelt on one leg, wrapped his left arm around his chest, and touched his chin with his right hand.

I took a quick check on the four students. Kari had a grin on her face. She had been watching Danny since he came in.

"You always say that it is a good thing to write the last step and just fill in the middle steps afterwards." He mumbled.

"Okay, I think the second step is..." He started writing again.

"Is it right, Mr. Jope?"

"No." I said. But before I had the chance to explain myself, he erased what he just wrote.

"No? Then it must be this... I must be right this time."

"Yes, you are."

He continued on his own while still talking to himself aloud. Occasionally, I made comments on what I was hearing.

"I'm done. Sir, did I do it right?

While I took a moment to review the two-column proof he had written on the board, I could tell he was intently studying my face for any trace of approval.

When I said yes, the young man threw his arms up, his face glowed like the neon orange shirt he was wearing.

"Oh, my God! I can do proofs now!"

I asked him to do two more problems, which he happily solved with minimal errors.

That afternoon, Kari and I had complimentary front row seats to the most entertaining geometric proof presentation we both have ever witnessed!

Daniel's joyful feet and Kari's beguiled face: priceless!

A Classifying Triangles Activity

Here's an activity that I have been using to help my students achieve mastery on classifying triangles by side lengths and angle measures.

The primary objectives of this activity are (1) to master vocabulary, and (2) be able to correctly relate the different triangle classifications with each other.

The activity involves a 4 X 5 table. The top row will feature the triangle classifications based on angle measures: acute, right, obtuse, and equiangular. The left column will feature the triangle classifications based on side lengths: scalene, isosceles, and equilateral.

In this activity, the students, individually or in groups, will determine if triangles can fall under two such classifications. For example: Is it possible to draw a triangle that is both acute and scalene? If so, the students will draw that triangle and label it appropriately. If not, then they may write NP for not possible.

I find this activity great for reviewing vocabulary (scaffolding). It is also great for classroom talk and more questioning.

What I like the most about this activity is its "friendliness" to diverse learners, from the learning-challenged to the gifted, and from the color-blind to the most artistic.

This poster was created by Kendra Cobos, Edgar Devora,

Vincent Grimaldo and Rey de Leon in my Geometry class.

Freshmen Irma Mata, Alex Tenopala, Kaela Garcia and Amber

Hernandez created this poster in my Geometry class.

Poster created by Dalia Gutierrez, Victoria Gomez and Berenice Pacheco.

Mnemonic Devices for Points of Concurrency in a Triangle

I have new terms to add to the already-crowded vocabulary bank of geometry.

The terms are PuBliC, BAsIN, CEmeNT, and ALTo.

These four terms I have coined myself are but mnemonic devices to help students and teachers remember the points of concurrency involving a triangle.

The P and B in PuBliC stand for perpendicular bisector. Each of the three sides of any triangle has a perpendicular bisector. All three perpendicular bisectors intersect at a point of concurrency, called circumcenter, the C in PuBliC.

Angle bisector, or bisector of an angle, is represented by B and A in BAsIN. Each of the three angles in any triangle has an angle bisector. All three angle bisectors have a point of concurrency, called incenter. The IN in BAsIN represents incenter.

Together, CE and NT in CEmeNT represent centroid. Centroid is the point of concurrency of a triangle's three medians, the "me" in CEmeNT. The word "middle" is associated with the word median, so I was very excited to find a term that has "me" in the middle.

ALT in ALTo simply means altitude. The lines that pass through each of the three possible altitudes of a triangle have a point of concurrency also. This point of intersection is called orthocenter, the "o" in ALTo.

I came up with these mnemonic devices on my second year of teaching Geometry. I realized it wasn't only my students who had difficulty remembering and associating the points of concurrency. "Mr. Jope" had the same problem, too! :D

Kaela Garcia, one of my freshmen, made this

poster as a project in my Geometry class.

This was my student Samantha Cerda's project.

This was done by Alexis Siller, one of my students.

Victoria Gomez turned in this poster as a project in my Geometry class.