Math II (Несколько текстов для зачёта), страница 13

representing my plunging and leaping

personality.

--KRIS MCNAMARA

Fig. 1

Poem Incorporating Mathematics Terms

Write a poem, at least fifteen lines long, about any subject, in which you incorporate at least eight of the words on the following list. If you have Ms. Davidson for mathematics, you must use at least twelve of these words.

Congruent

Contrapositive

Coordinate

Cylinder

Equation

Equilateral

Exponent

Formula

Function

Parabola

Isosceles

Locus

Matrix

Midpoint

Parallel

Permutation

Perpendicular

Probability

Pyramid

Radical

Rotation

Translation

Vertex

(a)

The advanced writing assignment

Poem Incorporating Math Terms

Write a poem, at least twelve lines long, about any subject, in which you incorporate at least six of the words on the following list. If you have Ms. Davidson for mathematics, you must use at least ten of these words.

Add

Angle

Divide

Equation

Geometry

Graph

Line

Multiply

Negative

Parallel

Perpendicular

Point

Polygon

Positive

Probability

Quadrilateral

Subtract

Triangle

(b)

The regular writing assignment

PROJECTS

Source: Mathematics Teacher, May2001, Vol. 94 Issue 5, p430, 3p

Author(s): Isleb, Jo Ann; Albert, Maureen; Kasten, Peggy

Project CLIMB

Project CLIMB (Creating Links in Math and Business) is a teacher-developed project that was designed to help answer the students' question, When are we ever going to use this? The project allows precalculus students to communicate with people in the business world by using e-mail. Students are put into groups of three or four and assigned a business contact. The students determine from this contact person exactly what the company does, how teams are used in the company, and how specific mathematics topics are used by the contact person on the job. The student project includes six e-mail requests for information during a semester. The information requested centers on the precalculus topics of matrices, statistics, linear programming, logarithms, trigonometry, and probability. These broad topics are used by people in a variety of fields. The business contact uses e-mail to respond.

Project CLIMB was successfully initiated by starting small and expanding gradually. The first year of the project began with two classes and about twelve business contacts. The mathematics teachers who were developing the project carefully selected the business contacts. The contacts could easily access the Internet, used mathematics in their jobs, and could be relied on to give prompt and thorough responses. In the second year, the project was expanded from two to eight classes. An additional forty business contacts were located in a variety of ways, including referrals by individuals who were already involved in the project and solicitation over the Internet from businesses located in the community.

To participate in Project CLIMB, students must have an e-mail account. On the first day of the project, classes go to the computer lab and are taught how to use e-mail. At that time, one student in each group sends the business contact an e-mail message identifying himself or herself and asking for a description of the contact's job and company. As soon as the student receives a response, he or she shares it with the group and gives the teacher a copy. Another person in the group then takes responsibility for sending a second question. This procedure is followed until all questions have been asked and replies have been received.

The project was designed to make evaluation easy and to use little class time. Each group gets a folder to store paper copies of the e-mails sent and the e-mails received. Dates that the group sent or received e-mail are recorded on a cover sheet stapled inside the folder. Students have a few minutes of class time to discuss their progress. The assessment includes both individual and group components. Each student must send e-mail, communicate information to the group, prepare a summary describing what she or he learned, and participate in the group's presentation. The group's grade is based on the completeness of the folder.

The responses that students receive are rich and informative. For example, when asked by a student about real-life use of logs and exponents, a chemist replied as follows:

Chemists use pH as a measure of how acidic or basic a compound is. The scale for pH is from 1 to 14, where pH 1 is very acidic and pH 7 is neutral like water, pH is really a measure of how many hydrogen ions the compound releases into the water. Some compounds keep their hydrogen ions so that they are very acidic even though lots of the compound is in the water. Other compounds completely break apart, like hydrochloric acid and sulfuric acid, and so are quite acidic for the same amount of compound in the water. It also matters how much of the compound is in the water. It may be very dilute and not so acidic. For instance, the acetic acid in vinegar in your salad is pretty dilute and won't hurt you, but full-strength acetic acid will cause a very serious burn.

pH is really a negative log of the number of hydrogen ions.

pH = -log (hydrogen ions)

The base for the logs is base 10. If you have pH 7 water and add enough acid to make pH 6, it has 10 times as many hydrogen ions free in the water, so to go from 7 to 6 isn't a jump of just 1, but a jump of 10 when you consider the number of hydrogen ions.

Project CLIMB offers the following benefits to participating students:

• The content addresses national, state, and district goals in an innovative and interesting way. For example, the Illinois Learning Standards state that

• [s]tudents must have experiences which require them to make such connections among mathematics and other disciplines. They will then see the power and utility that mathematics brings to expressing, understanding and solving problems in diverse settings beyond the classroom.

• High school students learn to use e-mail before they are required to use this technology in college or in the workplace.

• Students learn from people in the real world how the mathematics that they are studying is applied.

• Students develop an ongoing relationship with their business contact, and they learn to communicate in a professional manner.

Students enjoy the project.

AN UNBREAKABLE CODE?

Source: Newsweek, 03/05/2001, Vol. 137 Issue 10, p45, 2/3p, 1 diagram

Author(s): Levy, Steven

The secret is a key that disappears when you use it

It may roundly be asserted..." wrote amateur cryptography maven Edgar Allan Poe, "that human ingenuity cannot concoct a cipher which human ingenuity cannot resolve." Harvard professor Michael Rabin begs to differ. Last week he revealed details of a scheme called "hyper-encryption" which purportedly delivers a means to protect information that's mathematically guaranteed to be unbreakable.

Rabin's system, which ignited heated debates on Internet discussion groups after an article in The New York Times, is far from being implemented. In fact, Rabin has yet to even publish a paper on the idea (developed with his doctoral student Yan Zong Ding). But it's worth thinking about because it addresses an important problem in protecting our private messages and conversations over the Internet and on mobile phones. How can we know that those systems can't be broken? It's true that there are plenty of ways to crack a code in addition to attacking the mathematical system that actually scrambles messages: there are plenty of potential pitfalls in implementation that may be exploited. And of course, if all else fails, you could pummel the sender or recipient until he coughs up the goods. But the mathematical formulas that make up the heart of those systems are critical, and Rabin's idea might cast light on how to make these permanently secure.

The idea begins with a source of an unending stream of random numbers, perhaps a satellite blasting huge volumes of bits in rapid fire. So many, in fact, that it's impossible for the most advanced storage systems imaginable to capture them all. When people want to communicate with hyper-encryption, their computers "agree" on a way to grab certain of those numbers, "like plucking raisins out of a vast pudding," says Rabin. Those random numbers (the equivalent of the normally impractical "one time pad," the only previous form of provably unbreakable cipher) are used to help the sender scramble the message--then the recipient uses them to help restore it to the original form. As the sender and recipient use those numbers, the computer discards them: think of the tape recording in "Mission: Impossible" when the message self-destructs. So even if a foe captures the scrambled message, then learns which pattern was used to grab numbers from the stream, the snoop won't be able to decode the message, because the crucial random numbers from the stream will be gone.

Policy issues, such as whether unbreakable codes will give terrorists an unbeatable edge in hiding their activities, can come later, when and if hyper-encryption is put into practice. For now, "we can prove secrecy," says Rabin, obviously delighted at going public with his brainchild.

How hyper-encryption works

Professor Michael Rabin's mathematically secure scheme allows people to pass secret messages that stay secret, forever.

A huge stream of random bits is broadcast by a source. Alice and Bob pluck bits in a secret, prearranged pattern from the stream. When Alice sends her message to Bob, those bits help scramble the message. Bob also knows which bits were taken so he can unscramble the cipher. Alice and Bob don't retain the random bits, and an eavesdropper can't break the code because the random stream can't be duplicated or completely stored.

THE MATHEMATICS OF HISTORY

Source: WorldLink, Mar/Apr2001, p10, 2p

Author(s): Matthews, Robert

What connects the number of war dead or the intensity of an earthquake with mounds of grain? Robert Matthews describes research that shows a numerical correlation among the three

A December night in New York City: a taxi stops on Fifth Avenue, near Central Park. The passenger gets out and starts to cross the road. He is English and for a moment forgets that in the US they drive on the right. It is already too late: he is struck by a car travelling about 50 kilometres an hour.

By some miracle, he is not killed, but it takes him months to recuperate fully. His recovery was more than a personal victory. The year was 1931, and the man was Winston Churchill. Had the car been travelling just a little faster, the 57-year-old future British prime minister almost certainly would have died.

That road accident in Manhattan 70 years ago was one of those hinges of fate on which the history of entire nations has rested. It is far from unique, of course. The torrential rain that allowed the lightly armed force of England's Henry V to defeat the far larger French army at Agincourt in 1415 is a famed example, as is the heap of unwashed dishes in the laboratory of Alexander Fleming that led to the discovery of antibiotics. Churchill himself wrote of how in 1920 the king of Greece died after being bitten on the nose by a monkey, setting into motion events that led to a war in which 250,000 Greeks and Turks died.

In the past, some historians have seen such anecdotes as evidence that historical events are the result of actions by just a few key people. Or, as the 19th century British historian Thomas Carlyle put it, "history is the biography of great men".

Most modern historians see this as too glib, and the hinge-of-fate view of history has fallen into disrepute. Events of historical significance are now typically viewed as the product of a host of interacting forces, none of which can be identified as indisputably crucial.