chocolate oranges

Every year around this time, I open up a chocolate orange and begin to wonder the same thing. As I remove wedges of the chocolate treat, is there any point at which the surface area of the partially-eaten chocolate orange is greater than the surface area of the intact orange? In other words, given a piece of foil that just covers the full orange (with no foil left over), will removing wedges render the foil insufficient to cover the remaining orange?

some sums

Had trouble falling asleep last night, so I began adding up numbers in my head. "Discovered" this interesting property of integers, though I'm sure it already exists more formally in the mathematics world:

Let n be a positive integer. Let S be the sum of all integers from 1 to n, inclusive.

If n is odd, then S is a multiple of n. However, the same property does not exist for even numbers.

It will take me some time to try to even wrap my head around this, and who knows if I even have the ability to do so. But pretty cool nonetheless. And a bit frightening that this is what goes through my head in the wee hours of the morning...

correlation?

I have noticed more white hairs on my head since getting married...

counting birthdays

happy (26th) birthday, nick!

Because math, in so many ways, in grounded in counting, I spend a lot of time thinking about how we quantify things in our lives. I've also been thinking a lot about birthdays recently, as I find myself surrounded by many friends and family members who were born in this tenth month.

What has always bothered me most is that a person can truly have only one birthday, to be celebrated on the day of one's birth. Everything that comes after represents an anniversary of one's birth day. But this is a minor point, an argument that I concede I will never win.

The argument I will win, however, is the following: We almost always incorrectly count birthdays. For example, the date on which one turns 25 is actually one's 26th birthday! But look at the Hallmark cards that arrive in the mailbox and at the writing on the cake and they will all say "Happy 25th Birthday!" Has anyone ever looked at these and said, "Hm... These are a year late."

anti-discounting

A few friends and I came across this parking meter in Amherst, MA a few weeks ago. We have not doctored this image at all.



1. What's wrong with the picture?
2a. If the meter accepted pennies, how many minutes would a penny buy you?
2b. If the meter accepted half dollars, how many minutes would 50 cents buy you?

a complete fail

I often turn to Entanglement when I'm looking to procrastinate. Unfortunately for me, I encountered this tile the other evening to start off the game:



It's as if someone knew my intentions and wanted to stop me dead in my tracks. Well, message received and understood clearly!

As for Entanglment and tiles, I now wonder how many different tiles exist in the game, and the probability of landing this tile again as the first move. Anyone know?

geometry's ironic assumption

I have emphasized over and over again to my Geometry students that "eyeballing" is never an acceptable justification for any argument. Today, I tried to make my stand a bit more compelling by presenting a series of optical illusions, a move to point out how unreliable our perceptions may be. Two lines that look congruent may actually be different lengths, or vice versa. An angle that looks like a right angle may not measure 90 degrees at all. Is that quadrilateral really a square? "Well, it looks like it is!" they tell me, exasperated with my unwavering prompts of "But do you know for sure?"



The reality is that our eyes and the sense that our brains try to impose on the world around us are completely unreliable. How we interpret the world may not be an accurate representation of the world. Or rather, what we "see" is simply that -- a representation. The images presented here are just two of a series of optical illusions I showed to my students, though these two most closely address the types of incorrect assumptions they might make with their work.

Ironically enough, students are allowed to make the assumption that a line is straight. Yet the following illusion suggests that even with straight lines, what we see is not actually truth.


Are the two red lines in the middle straight or curved?

a tea party of ill logic

Read the follow excerpt from Lewis Carroll's Alice in Wonderland. Then answer the questions that follow.

A Mad Tea-Party from Alice in Wonderland, by Lewis Carroll

There was a table set out under a tree in front of the house, and the March Hare and the Hatter were having tea at it: a Dormouse was sitting between them, fast asleep, and the other two were using it as a cushion, resting their elbows on it, and talking over its head. "Very uncomfortable for the Dormouse," thought Alice; "only, as it's asleep, I suppose it doesn't mind."

The table was a large one, but the three were all crowded together at one corner of it: "No room! No room!" they cried out when they saw Alice coming. "There's plenty of room!" said Alice indignantly, and she sat down in a large arm-chair at one end of the table.

"Have some wine," the March Hare said in an encouraging tone.

Alice looked all round the table, but there was nothing on it but tea. "I don't see any wine," she remarked.

"There isn't any," said the March Hare.

"Then it wasn't very civil of you to offer it," said Alice angrily.

"It wasn't very civil of you to sit down without being invited," said the March Hare.

"I didn't know it was your table," said Alice; "it's laid for a great many more than three."

"Your hair wants cutting," said the Hatter. He had been looking at Alice for some time with great curiosity, and this was his first speech.

"You should learn not to make personal remarks," Alice said with some severity; "it's very rude."

The Hatter opened his eyes very wide on hearing this; but all he said was, "Why is a raven like a writing-desk?"

"Come, we shall have some fun now!" thought Alice. "I'm glad they've begun asking riddles.--I believe I can guess that," she added aloud.

"Do you mean that you think you can find out the answer to it?" said the March Hare.

"Exactly so," said Alice.

"Then you should say what you mean," the March Hare went on.

"I do," Alice hastily replied; "at least--at least I mean what I say--that's the same thing, you know."

"Not the same thing a bit!" said the Hatter. "You might just as well say that "I see what I eat" is the same thing as "I eat what I see"!"

"You might just as well say," added the March Hare, "that "I like what I get" is the same thing as "I get what I like"!"

"You might just as well say," added the Dormouse, who seemed to be talking in his sleep, "that "I breathe when I sleep" is the same thing as "I sleep when I breathe"!"

"It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute, while Alice thought over all she could remember about ravens and writing-desks, which wasn't much.

Now that you have read this passage, think about and answer the following questions:

1. What do you make of the debate that Alice has with her new friends? Are "I say what I mean" and "I mean what I say" the same thing, as Alice claims? What about the statements that the Mad Hatter, the March Hare, and the Dormouse propose? Do those statements mean the same thing? Why or why not -- provide counterexamples if you think they mean different things.

2. Come up with two statements of your own, such that the original statement is true, but when you switch the first and second parts of the statement, it is no longer true.

3. Give an example of a statement that does mean the same thing forward and backward (i.e., when its two parts are switched,as Alice and friends do above).

making roots make sense

A remnant of pre-calculator days, the topic of rationalizing denominators remains entrenched in algebra textbooks and finds its way into standardized tests of all levels. And so I teach it, with the caveat that my students will never have to rationalize any denominators in my class unless they choose to do so.

I teach this unit with a bit of a history lesson, describing the choice that mathematicians once made (though it was not a difficult decision to make): should I divide by radical 2 (or some other irrational root), or should I divide by 2 (or some other rational denominator)? I talk to kids about square root tables and how they, along with tables of squares, trigonometric ratios, logarithms, etc., were once a mainstay of the mathematician's toolkit.

So I showed them slides with these images:







It took them a second to realize that I was joking, but still several students bewilderedly exclaimed, "Oh, man! At first I thought that mathematicians used to use those tables to help them do math!"

(photos from http://www.behance.net/gallery/square-root-table/282218)

october math joke!

What do you get when you divide a pumpkin's circumference by its diameter?

(you can find the answer in the comment section)

geometry around the house

What comes to mind when you think of geometry? How do you define it? What is it that people study when they study geometry? How does an understanding of geometry inform our everyday experiences?



For this assignment, simply look around your house or neighborhood. Search for things that rely in some way on geometry for their function, design, or aesthetic appeal. For example, you might point out a manhole cover in the ground; the fact that it is circular in shape is very important to its function. On the other hand, pointing out a stop sign as an octagon does little to help us understand why, if at all, it is interesting.



Give me a sense of your approach to geometry by answering the questions that I posed at the start of this post. Then, list three things that you found, and for each, write a short paragraph about the ways in which you think geometry plays a role in its function, design, and/or aesthetic appeal.

Don't forget to check out the "Geometry at Home" gallery!

umass gets the academic year rolling

I'm awaiting the results of this world-record-breaking attempt, which took place today. The previous record for longest sushi roll is 330 feet.

Surely there is a math problem in here somewhere. Post your suggestions as a comment!

UMass looks to break sushi roll record
The University of Massachusetts is celebrating the start of the academic year with an attempt to make the world’s largest sushi roll. Three-hundred volunteers, along with celebrity chefs, will try to build the 400-foot sushi roll today using 650 sheets of nori seaweed, 200 pounds of rice, 200 pounds of seafood, 100 pounds of avocado, 100 pounds of cucumber, 2 pounds of sesame seeds, 5 gallons of soy sauce, and 6 pounds of wasabi. (AP)

math words: radical

What does the word radical have to do with roots? Why do we interchange "radical 2" with "the square root of 2"?

A trip to the dictionary shows that "radical" comes from the Latin radix, which means root.

The use of the term "root" in math goes back even earlier, finding its origins in the Indo-European word werad. Initially used to refer to the roots or branches of plants, mathematicians used werad to refer to a number used to build up another number. For example, 2 is the square root of 4 because 2*2 (two squared) gives us (builds up to) 4.

When translated into Latin, werad became radix, thus giving us our modern usage of radical in math!

counting up

I am fascinated by the use of the number zero as a relative quantity as opposed to an absolute quantity. Following a recent unit on operations with integers, I engaged my seventh graders in a discussion about ways in which we use zero not to signify the absence of any thing, but rather to simply assign a starting point for counting.

One familiar example involves temperature: 0 degrees Fahrenheit, 0 degrees Celsius, and 0 Kelvin. How confusing is this: three different temperatures, each having a zero point, mean different things!* For those students working with quadratic equations, it's not uncommon to set some nonzero height equal to zero in order to simplify the problem.

Time provides yet another case: we use zero to indicate the commencement of some event (t=0). We often think of the seconds leading up to a rocket launch as a count-down: ten, nine, eight.... However, this is actually shorthand for T-minus-ten, T-minus-nine, T-minus-eight, ... , in which T, representing the time of launch, takes place at T=0. The engineers are actually counting up, using negative numbers leading up to zero!

As I discussed the count-down (which is actually a count-up) with my students, I made a connection that I had never noticed before. In our daily lives, we count up all the time! How do we indicate to everyone that it's time to starting singing "Happy Birthday" to the party honoree? One, two, three. Happy birthday... How do we coordinate a group's efforts to lift a heavy object? One, two, three. Lift.

Contrary to intuition, we decide to start our event at four in these two cases! How did this practice evolve, such that our count-down actually involves a count-up, the event taking place when we reach an arbitrary number that we never consciously consider a starting point?

* This example of temperature differs from other cases in which we deal with a change of units. With length, weight, volume, etc., zero indicates nothingness across all units.