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Section 1. An Introduction to the Problem-Solving Process

I believe that my plan addresses several contemporary problems: The plight of divorced, widowed or otherwise single women who have little or no experience in the business world, the simultaneous stress on personal finances and the likely stagnation of real estate development for the foreseeable future.

Student
MAT 222 Week 4 Assignment
Instructor’s Name
Date
QUADRATIC FUNCTIONS 2
Real World Quadratic Functions (title required on first line)
Quadratic functions are perhaps the best example of how math concepts can be
combined into a single problem.

But less time and attention than is required by a problem not well solved.
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solution essay and problem solving, and problem solving

Lesson Plans for Creativity, Problem Solving, and …
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There is much talk about the relationship between mathematics and music, which mostly consists of speculation by those on the outside, concerning some of the obvious things they have in common. Yet there isn't much said by those on the inside, and as a former student of mathematics and a lifetime musician, I will attempt to shed some light on the subject. I think that the degree that you can understand the relationship between music and mathematics is proportional to your understanding of both music and mathematics. The more you know of both, the more you will know of the relationship, and attempts to peer into the shadows from the outside will not yield much more than some wonderment. The important thing to realize is that numbers and math are not cold and lifeless, and that music, which is a tangible incarnation of numbers, reflects in its beauty and emotion some of the beauty and emotion in the world of mathematics.

I graduated from the University of Maryland with a Magna Cum Laude Phi Beta Kappa degree in mathematics in 1974. I had an aptitude for mathematics, and I must confess that something made me choose the life of the musician instead. The two have many similarities, in that they have strong intellectual, spiritual and creative foundations. I think I chose music because I can participate in the world. When you are really doing mathematics, the people, places and events in the world are distractions from your work. When you are really doing music, you can be just as deeply involved in the mathematical beauty of the music and the theory, yet you can be at the party. You can only really share mathematical beauty with other mathematics people, yet you can share the equivalent music beauty with anyone, and they can enjoy it on some level. There is an essential element of communication in music that I think is what made me choose that life.

Most of us I think have more of an idea of what music is than we know what math is. If we were to poll people on the street, they would probably associate math with numbers and calculators-- things that really are arithmetic. My guess is that as many mathematicians don't balance their checkbooks as non-mathematicians, though they can prove that it can be balanced. Math is about thinking. Math is about problem solving. Math is about working with what you do know to give you a framework and a method of exploring and understanding what you don't know, about seeing relationships and patterns. Mathematics is a mind-set, and an attitude when you face something you do not understand. But there is also a beauty and a wonder about mathematics that only insiders know about. Words like elegant and beautiful are used constantly by mathematicians to describe paths of reasoning and proofs.

Certainly many tasks in the life of a musician fall into this category. Arranging a melody on an instrument and finding fingerings that correspond to certain sequences of notes is definitely a type of math problem. Playing the same melody on different instruments is math, as is playing a stringed instrument and changing the tuning. And when you find the best key to play a certain melody on a guitar, for example, there is a sensation that is known to math insiders as elegance. Mathematicians praise each other for the elegance of a proof, referring to the esthetic beauty of it. When you write a new piece of music, when you find the best fingering on a stringed instrument for a sequence of notes, or when you arrange a piece of music for an ensemble, you can experience nearly identical sensations of elegance. As you learn about music and about chord theory, you learn to recognize chord changes, and you experience a mathematics of musical structure also. Playing harmonies, playing the same song in different keys, taking solos on unfamiliar songs-- these things all involve recognizing the structure of a piece of music. Good musicians can often listen to a song, observe the musical structure, and play along with it, without really knowing it or rehearsing it, because they recongnize patterns and familiar shapes. This type of thinking is very much like the way you think when you study mathematics.

Both music and math have concepts, and special symbols. What is a musical key? What is a number? The definitions of things in both disciplines are somewhat circular and vague, unless you understand what they are. You cannot define a number, but you know what they are much of the time and you can use them. It's no different with a musical notion like a minor key. Once you know what it means you can spot one, though you cannot really define it rigorously.

There are many things in music that are obviously math-related, and many musical notions can be explained in numbers. But it is important to note that numbers are not some way to describe music-- instead think of music as a way to listen to numbers, to bring them into the real world of our senses.

The ancient Greeks figured out that the integers correspond to musical notes. Any vibrating object makes overtones or harmonics, which are a series of notes that emerge from a single vibrating object. These notes form the harmonic series: 1/2, 1/3, 1/4, 1/5 etc. The fundamental musical concept is probably that of the octave. A musical note is a vibration of something, and if you double the number of vibrations, you get a note an octave higher; likewise if you halve the number of vibrations, it is an octave lower. Two notes are called an interval; three or more notes is a chord. The octave is an interval common to all music in the world. Many people cannot even distinguish between notes an octave apart, and hear them as the same. In western music, they are given the same letter names. If you blow across a coke bottle and it produces the note F, and you drink enough so that the air remaining in the bottle is twice as much, the note will be also an F, but an octave lower. If you shorten a string exactly in half, it makes a note an octave higher; if you double its length, it makes a note an octave lower. You can think of the concept of octave and the number 2 as being very closely associated; in essence, the octave is a way to listen to the number 2.

If you shorten a string to 1/3 its length, a new note is produced, and the second most fundamental musical concept, that of a musical 5th emerges. We call it a 5th, because it is the 5th scale note of the Western do-re-mi scale, but it represents the integer 3. (Incidentally, the 5th is the only interval other than the octave that is common to all musics in the world.) Strings of a violin are tuned a 5th apart. Men and women often sing a 5th apart, and most primitive harmony singing involves octaves and fifths. In fact, they say that when you are learning to tune a stringed instrument, you can only trust your ear to hear octaves and fifths, and you should not rely on your ability to compare other musical intervals properly. The next note in the harmonic series corresponding to the number 4 is 2 times 2 and thus a second octave. The number 5 produces a new note, called the musical 3rd. The 3rd is the other note in the fundamental chord, called the major triad, which is made up of 1st, 3rd and 5th notes of the Western scale. The number 6 produces a note an octave higher than the 5th, and it is also a very harmonious note. The number 7 produces the first dissonant note in the harmonic series, which has some numerological and religious significance. Also of spiritual and numerological interest-- the next dissonant overtones are the 11th and the 13th.

If you build a musical system out of these integer notes, it is what is now called the Pythagorean scale, as used by the ancient Greeks. If you bore holes in a flute according to integer divisions, you will produce a musical scale. Oddly enough, if you try to build complex music from these notes, and play in other keys and using chords, dissonances show up, and some intervals and especially chords sound very out of tune. Our Western musical scale paralleled the evolution of the keyboard, and finally reached its modern form at the time of J.S. Bach, who was one of its champions. After a few intermediate compromise temperings, as systems of tuning are called, the so called even-tempered or well-tempered system was developed. Even-tempering makes all the notes of the scale equally and slightly out of tune, and divides the error equally among the scale notes to allow complex chords and key changes and things typical of western music. Our ears actually prefer the Pythagorean intervals, and part of learning to be a musician is learning to accept the slightly sour tuning of well-tempered music. Tests that have been done on singers and players of instruments that can vary the pitch (such as violin and flute) show that the players and singers tend to sing the Pythagorean or "sweeter" notes whenever they can. More primitive ethnic musics from around the world generally do not use the well-tempered scale, and musicians run into intonation problems trying to play even Blues and Celtic music on modern instruments. Old bagpipes had the holes drilled in places that sound "sour" to modern ears. The modern musical scale divides the octave into 12 equal steps, called half-tones. 12 is an important number on Western music, and it is oddly also an important number in our time-keeping and measurement systems. The frets of a guitar are actually placed according to the 12th root of 2, and 12 frets go halfway up the neck, to the octave, which is halfway between the ends of the strings. On fretted instruments we are actually playing irrational numbers! And any of you who have trouble tuning your guitars might get a clue as to why they are so hard to tune. Our ears don't like the irrational numbers, but we need them to make complex chordal music. The student of music must learn to accept the slight dissonances of the Western scale in order to tune the instrument and to play the music.

Studying mathematics can also assist you in daily life as a musician. I cannot tell you how many times I have actually needed to solve an equation or refer to one of my math textbooks, but the answer is a very small integer. I think the only time I ever needed to do that was to compute how many combinations of a guitar capo that allows you to selectively capo any combination of strings at a given fret, rather than just clamp across all the strings as capos have traditionally done. I am not talking so much about solving the little algebra problems of life like changing money when you tour in foreign countries. Though it is a good exercise to go to England, pay British pounds to buy liters of gasoline, and try to figure out 1) what miles per gallon you are getting 2) what you are actually paying in US $ for a gallon of gasoline. That's kind of tricky, though it is junior-high school math involved.

Being a former math student makes it easier I think for me to use and understand my computer, which is an essential tool for a working musician today. We have to have mailing lists, and print out mailing labels to advertise our concerts using various Boolean and/or statements. Print out a list of everybody who has signed up in the last 2 years who either lives in northern Mass, coastal NH or Southern Maine, but only if they are media, and sort them by zip code. It's a math problem.

When you are setting up a sound system for a band, you might have a 16 channel mixer, with a monitor send and an effects send. How can you plug in your wires to send a mix to the main amps to send to the audience, send another mix to the monitors for the band to hear, and maybe a 3rd mix to a radio feed or a tape recorder. The wiring of sound systems and the routing of signals is a type of mathematics. The noise in a signal is determined by a theory called gain structure, where it passes through from 5 to 15 different devices and wires of different lengths through pre-amps, delays, choruses, reverbs, mixers, tuners; learning to understand and optimize your use of these things is definitely a math problem. Troubleshooting a sound system 30 minutes before the gig is a math problem. One of the speakers is not working. Why? Is it the speaker? Is it a bad wire? Is it the channel of the amp? A fuse? Is it the connector jack, or the mixer? Solving these kinds of problems is a form of mathematics, where you systematically eliminate possible problems and de-bug the system. Do you switch wires, speakers, or amp channels to find the broken one, and in what order?

The phone calls between band members to book your gigs and arrange rehearsals are a math problem. How do you notify everybody with the fewest calls? How do you all get together to rehearse? Do you take one or 2 or 3 cars to the gig? This is the hardest part of all. When the gig is close to home, and everybody lives near each other, it does not matter if everybody drives their own car, because the number of miles is small. If you are driving 800 miles on a tour, then the answer is simple, since you all travel together, But what if you live an hour apart and the gig is 2 or 3 hours away? Should you car-pool? And sometimes only some vehicles are big enough, and as a musician, you are forced to become an expert at routing theory.

Because I studied math, I know about the mailman problem, and Euler's bridge problem, and the famous brain teasers about the fox, the chicken and the cabbage, and the cannibals and the missionaries. These all involve traveling to several destinations, or transporting things in boats across a river, and problems of their type have been around in various forms for centuries. Some of these are very tricky problems, and you face versions of them every day as a musician. These old brain-teasers are exactly the kinds of things musicians do all the time. How do you get all the people and their gear to the gig most efficiently, if the string bass ony fits in one of the vehicles, and one of the drivers has another gig earlier in the day and they need the power amp and speakers? I used to study math puzzle books when I was a child, and the solutions to some of the problems of how to get the people or foxes and chickens across the river in the boat are not simple at all.

Each year, students come into my classroom who are supposed to possess skills that are prerequisites for the math activities that I teach. Usually, most of them don't. Most of the time, there is a great difference between what the students need to know to "get started" and what they actually know. Of course, I have to begin my instruction "where they are." This means that I will not have the advantage of merely working on the concepts and strategies. I will have to teach my students the fundamental parts of geometry, nature and shapes. If they master these skills, I will need to teach them how to approach geometry in an investigative manner using such techniques as collaborative learning; exploration and problem solving to formulate, test, and locally prove or disprove conjectures; and written and oral assignments to develop effective communication skills; and such tools as physical manipulatives, models, and software.

A creative problem-solving lesson plan for grades K-3.

Many students have difficulties when attempting to solve geometry problems. Many reasons are suggested or put forward for the students' lack of success in this area. These reasons include students' lack of exposure to life outside of their neighborhoods, minimal visual skills, and difficulty in understanding basic geometric concepts. I do not dispute these reasons for students' failure, but I propose that there is a way to ensure that geometric concepts, especially transformations and symmetry, permeate the mathematics classroom while, at the same time, maintaining student interest. Wallpaper is an example of a real world item that is seen everyday in most homes across the nation. It is also one of the most important when it comes to the topic of geometry. Some people look at wallpaper and go "wow! That's an incredible pattern" but mathematicians see wallpapers and go "hmm! What's the fundamental domain? What type of symmetry can be found here?" among others. I have tried to develop a series of lessons that will help teachers develop various strategies to teach geometry, with the help of symmetry, in their classrooms. It is my hope that implementing this curriculum unit will help teachers to teach geometry in a way that will excite students, assist their connection and application of "real world" scenarios to the concepts, aid their use of various strategies, and extend students' abilities to solve math problems in other contexts.

in which students use problem-solving skills to address a real-world problem
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