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 When Will I Ever Use This?

 James E. Lewis

Academic Setting

Rio Rancho High School is the largest high school in the state.   It is founded on the principle of academic excellence and strives diligently to attain that goal.  In partnership with INTEL, the students and faculty of Rio Rancho High School have constant access to modern technology.  The school serves students and families from a myriad of cultural and economic groups.   Rio Rancho High School’s demographic break down is 55% Anglo, 30% Hispanic, 7% African-American, 5% Native-American, and 3% Asian-Pacific Islander.  Rio Rancho High School is based on the academy system.  Traditionally, the high school was made up of a Humanities Academy, a Fine Arts Academy, a Business and Technology Academy, a Science Academy and the First Year/Freshman Academy.   With the start of the 2002/2003 school year, the Freshman Academy will be moved to a new mid-high school and the high school will only contain students from 10th to 12th grades.  Each academy specializes in its own field, but core courses are also taught.  Students spend most of the day in their own academy but may attend classes in one of the other academies if necessary.  Rio Rancho High School is on the four by four block schedule.  There are only four 100 minute classes per day and classes meet every day of the normal school week.  The accelerated schedule allows students to acquire four full credits per semester.   The extremely fast pace requires class attendance and dedication to their studies.  Additionally, over 50% of the school population engage in school supported activities such as athletics, band, theater, drama, science, and clubs.  Students from Rio Rancho High School received over $240,000.00 in grants and scholarships from the 2001/2002 Science Fair alone.  Parental support is outstanding and the school’s technological base allows weekly progress reports to parents.  Rio Rancho High School’s academic program is often studied by educational administrators  from other states and is currently in the vanguard of progressive education in New Mexico.

Class for Which the Unit is Designed

This unit is designed for any class where basic mathematical concepts are taught.  I  use it primarily for Algebra 1 and Algebra 2, but the ideas and procedures are easily adaptable to fit any class.  It is not designated as a given unit, but is to be used anytime a student questions the need for mathematics.  Several categories exist within the unit and material within the units may be used as a response to a variety of questions.

Goals and Objectives

The most important skill a student may learn is the ability to think.  Brute memorization and tools such as the slide rule or calculator can help a student complete assignments, but do not help with their thinking skills.  It is usually necessary to show students, “What is in it for them?” prior to students making any attempt to master a concept.  The goal of this unit is to inform students that there need not be a current use for the concepts that they are learning, but there may  be critical need for the concepts in the future.  There may not be any instantaneous reward for mastering a topic, but the ability to use what they have learned may benefit them in the future.  A secondary goal is to inform students that not all mathematical discoveries have been made and that they can make an impact on society through their ideas and discoveries.  With the advent of computers and other modern technological and communication devices, it is easy for students to feel that they cannot discover anything new or find use for their ideas that would bring them fame and fortune.  By showing students what others have done, I hope to encourage them to try to extrapolate new ideas and innovations.  Of course I will also attempt to show students current uses for the concepts that they are learning.

Context and Background

Rationale

           “If you can show me just one place that I might be able to use this, I might try
           and learn it.”  “My parents do not know how to do this and they are doing
           okay.”  “Who thinks of this stuff?”    --typical comments by high school
           students

          “How about tonight’s homework?”   “Fortunately your parents are not in my
           class or they would be failing too.”   “People like you and me that do not give
           up when they have an idea.”   --typical replies by high school teachers

Teachers have always complained about the quality of their students.   A common complaint is the current generation of students are not as dedicated as the previous one.  Given today’s environment of cheap technology, instantaneous communications, and the abundance of material goods, there may be some truth in the complaint.   Today’s students demand instantaneous rewards.  They are a part of what I call “Generation Me”  and they believe the world revolves around their needs.   Spoiled by the use of technology and enabled by their parents, today’s students do not always demonstrate the desire to learn.  They prefer to accomplish and go on to the next task.  Learning/mastery of the concept or task is not required.  Mathematics can be described as nothing more that a set of rules to be memorized or as a language in which the true beauty is hidden until one becomes fluent.   New Mexico State Educational Standards require mastery of mathematical concepts.  It is difficult to distinguish between mastery and short term ability to use concepts on normal high school tests or even on national tests such as the ACT or SAT.  Mastery is at the discretion of the facilitator.  I chose this unit as a means to instill a quest for learning into my students.  It is an attempt to get them to learn why a concept works rather than just how to use it.  Memorization does not lead to discovery, innovation comes from need and requires flexibility.   Mastery of concepts provide the needed flexibility.  This leads to the question, “How do we get the student interested enough in mathematics to look beyond short term use to long term mastery?”   I believe that by demonstrating that most of the mathematical concepts the students learn were originated by people no different than themselves, it will lead them to believe that they can make an impact.  They must realize that there are more concepts to discover and other uses for the concepts that they have mastered.  That makes it incredibly important to have answers for the questions given by the typical student. 

            To answer those questions it is necessary to have available facts and stories of many different concepts, the environment that led to their discoveries, and some of the different uses for the concepts.  Each example/concept delineated is an individual subunit and should be used by the teacher as the need arises.  They are not designed to teach individual concepts but to explain the how and why of the concept.  Demonstrating the historical uses and extrapolating possible future uses of specific concepts should increase the student’s desire for knowledge, rather than their need to get through something.  The fact that a mediocre college student became one of the richest men in the world just by finding a use for someone else’s discovery should get the typical student’s attention.   It is also important to show that mathematical concepts can be mastered without the use of modern technology.  It must be stressed that calculators are an aid, not an end, to mathematic calculation.  The human brain has a flexibility not found in most calculators.  With proper training and frequent use, students will find that the only calculator they truly need is found between their ears. 

Background 

Mathematics is involved in any endeavor we undertake.  Equations do not normally run up to ask and beg to be solved.  Most of the time we are not even aware that mathematics is a part of what we are doing.  Students must be given sufficient background on mathematicians and their concepts to recognize that they lived their lives in much the same manner as the students.  They had many of the same questions and shared many of the same concerns of today’s students.   The following information is not inconclusive of all mathematics.  It is a brief overview of selected mathematicians and concepts intended to demonstrate to students that even though the times have changed, they have the same opportunities to contribute to the field of mathematics.  It is intended to provide teacher support for use in the implementation of the units. 

Archimedes 

Archimedes (287 - 212 BC), the son of an astronomer, was born in Syracuse, Sicily.  With the exception of studies at Euclid’s school in Alexandria, Egypt, Archimedes spent his entire life at home.  Considered to be one of the three greatest mathematicians of all  time, he did not have the luxury of blackboards, modern day paper, or graphing calculators.  Archimedes was said to have used any available surface to complete his calculations.  Dusty table tops and even his own body were places he enjoyed calculating.

   Figure 1

             Archimedes (Figure 1) enjoyed both practical and theoretical mathematics.  He perfected a means of integration which allowed him to determine the area, volume, and surface area of many different bodies.  “Archimedes’ Screw” is a pumping device which helped irrigate fields and improve crop yields.  This pump is still used in many parts of the world.  He was also intrigued by pulleys and other mechanical devices.  Legend has it that King Hiero had commissioned a ship to be built for war.   Upon completion the ship-builder’s were unable to launch the ship.  Archimedes had once claimed, “that given a place to stand, he could move the earth.”    King Hiero challenged him to launch the ship and with a contraption comprised of numerous pulleys, Archimedes launched the ship using only the strength in one arm.  Another story attributed to Archimedes and King Hiero is the legend of the crown.  King Hiero commissioned a crown to be completed by a local goldsmith.  When he received the crown he was sure that the goldsmith had cheated him, but he had no way of proving it.   He gave the problem to Archimedes.  At first Archimedes was stumped at how to solve the problem, then one day when he got into his bath, the water overflowed.  Archimedes developed the theory that the volume of water displaced was proportional to the amount of an object submerged.  By submerging an amount of gold equivalent to what was supposed to be in the crown, he proved that the goldsmith had cheated the king.   

            Archimedes had many great inventions but he considered himself a theoretical mathematician.  His approximation of pi between 3-10/71 and 3 ½ was the most accurate estimation of his time.  He also determined a new method of approximating square roots.  Possibly his greatest innovation was his variety of techniques for what is now know as integral calculus.  Archimedes broke geometric figures such as parabolas or ellipses into a series of infinite rectangles.  When he added the area of the rectangles together he had the area bounded by the geometric shape.  Archimedes detested academic fraud.  He would often send examples of his discoveries to friends in Alexandria and was appalled to find that sometimes he would find his discoveries published under someone else’s name.  His method to rectify the situation was to include theorems he knew were false to his friends to embarrass them if they stole his ideas. 

            Archimedes was not fully recognized during his lifetime, but many of his innovations are still in use today.  As mentioned above “Archimedes’ Screw”  (Figure 2) is still used for irrigation, his principles of buoyancy are taught to personnel in the navy involved with damage control procedures, and his work with fluids was the beginnings of the science of Hydrostatics.  Archimedes is often considered the last of the great Greek mathematicians.  Archimedes is an important mathematician to study because he did not have all of the technological advances that students have available today.  His father, even though an astronomer, is not credited with any significant innovation during his lifetime.  Archimedes was fortunate to study and correspond with many of the important mathematicians in Alexandria, Egypt and depended upon the patronage of King Hiero

.

The Bernoulli Brothers   

 Jakob and Johann Bernoulli were part of the famous Bernoulli family.  Over three generations eight different members of the Swiss family made significant contributions to mathematics.  Jakob and Johann were primarily noted for their work in calculus but their work on fluid dynamics created some of the concepts necessary for the flight of “heavier than air”  objects.  Jakob (1654 - 1705) is also known as James or Jacque.  He is the older of the two brothers and the first to study mathematics.  Johann (1667 - 1748) is also known as John or Jean.  He was the tenth son of a successful merchant.  His father wanted him to be a merchant but he proved totally inept as an apprentice.  In 1683 he was given permission to enter the University of Basel, where his older brother Jakob (Figure 3) was already a professor.  Johann (Figure 4) pursued a degree in medicine, but was tutored in mathematics by his brother.  He soon developed a mastery of Leibnizian calculus and received his doctorate in mathematics.   

In 1691, Johann decided to visit Paris.  While in Paris he met France’s leading mathematician L’Hospital.  The two became friends and L’Hospital convinced Johann to teach him calculus.  The friendship came to an abrupt end when L’Hospital published a book on differential calculus.  Even though L’Hospital claimed all of the work published was his original work, most of the text was taken directly from John’s notes and letters.  Johann and Jakob were very jealous of each other.  Jakob never accepted Johann as his academic equal.  He claimed all of Johann’s academic achievements could be traced directly to his teaching.  The two competed at every opportunity.  Often the challenge resulted in discovery.  Johann could not take the academic chair at the University of Basil because Jakob already occupied it.  He was forced to take the chair at the University of Groningen.  In 1705, after the death of his brother Jakob, Johann eventually took the chair at the University of Basil.  By this point Johann had someone new to compete with, his son Daniel.  Johann was also very jealous of Daniel and he kicked Daniel out of the house when Daniel won a prize that he and Johann had competed for.  In another instance, Johann stole one of Daniel’s papers, changed the name on it, and claimed it for his own.  The resulting scandal tarnished Johann’s reputation, but did not discredit his work.   Johann’s influence over the European mathematical community was enhanced by some of his pupils.  One of the most important was Leonhard Euler.  Johann was an avid letter writer and was in correspondence with both Leibniz and Euler.  As a result of Johann’s support, Leibniz’  calculus gained precedence over Newton’s work in continental Europe. 

            The Bernoulli brothers are most remembered for the Bernoulli number series and their work with differential calculus.  The Bernoulli family that contained Johann and Jakob as siblings would be considered middle class by today’s standards.  As stated above, Johann’s inability to function as a merchant required he find another occupation.   Jakob’s superb tutoring provided Johann with a career after his lack of success in medicine and theology.  Their success played an important role in the opportunities for the next two Bernoulli generations. 

George Boole 

George Boole (1815 - 1864) was born the son of a shoemaker, John Boole, in Lincoln, England.  John’s hobby was constructing optical instruments and he provided George with his introduction to mathematics.  George (Figure 5) did not study for a degree and his primary interest was languages.  At age 12 he translated a poem written by the Latin poet Horace.  His father was so impressed he had the translation published.  Unfortunately, most of the educated people in the area thought it improbable that a 12 year-old could produce a translation of such depth and he was accused of fraud.   At the age of 16, George became an assistant school teacher.  His goal was to enter the church and he continued to study languages.  He was self-taught in French, German, and Italian.  Because of his ability with languages he was able to read scientific publications before they were translated allowing him to extend his knowledge beyond his peers.   

            When George turned 20 he had a change of heart and decided to open his own school.  He began to teach himself mathematics.  He later stated that he wasted five years trying to teach himself rather than learning from an accomplished teacher.  George was unable to attend a university as his family was dependent upon the income from his school.  He continued to study mathematics and his papers were published in the Cambridge Mathematical Journal.   He began to study algebra and his application of algebraic methods to solve differential equations brought him to mathematical fame.   George was appointed to the mathematics chair at Queens College, Cork, Ireland in 1849 and taught there for the rest of his life.    

            George died at a relatively young age.  His wife, Mary Everest  (niece of Sir George Everest, the man who Mt. Everest is named for), probably played an integral role in his death.  In 1864 George walked to the college in a pouring rainstorm.  He gave his lecture while still in wet clothes and eventually developed a fever.  His wife felt that the cure should resemble the cause.  She put George to bed and proceeded to throw cold water on him.   He died of pneumonia December 8, 1864.    

            George Boole is primarily known for his way of representing logic through algebraic methods.  He pointed out the analogy between algebraic symbols and those that represented logic forms and created a new algebra of logic.  This new form of algebra is called Boolean algebra and now finds application in computer construction and programming.  Boole developed the following axioms for use in Boolean algebra: 

            Premise:  There is a set B and two operators + and * that satisfy the following                             axioms: 

            Axiom 1:  Closure Property

                        If a and b are elements of B, (a + b) and (a * b) are in B

            Axiom 2:  Cardinality Property

                        There are at least two elements a and b in B where a is not equal to b

            Axiom 3:  Commutative Property

                        If a and b are elements of B, then (a + b) = (b + a) and

                                                                              (a * b) = (b * a)

            Axiom 4:  Associative Property

                        If a, b, and c are elements of B, then (a + b) + c = a + (b + c) and

                                                                                     (a * b) * c = a * (b * c)

            Axiom 5: Identity Property

                        B has identity elements with respect to + and *
                        0 is the identity for +, a + 1 = a
                        1 is the identity for *, a * 0 = a

            Axiom 6:  Distributive Property

                        * is distributive over + and + is distributive over *
                        a * (b + c) = (a * b) + (a * c) and a + (b * c) = (a + b) * (a + c)

            Axiom 7: Complement/Inverse Property

                        For every a in B there exists an element a’ in B such that a + a’  = 0 and a * a’ = 1

If we substitute “and”  for +, “or”  for *, and one addition operator “not”  we have the three operations used to make control gates for computers. 

      Boole developed his axioms to define logic operations.  Computers had not been invented, yet his development of Boolean algebra made computer programming possible.  Boole’s family would have been considered lower middle class today.  His path out of poverty was education and his mathematical and logical skills were essential to his success. 

Rene Descartes 

Rene Descartes (1596 - 1650) is considered first of the modern day school of mathematics and was a contemporary of Galileo.  He was born in La Haye, France and died in Stockholm, Sweden.  He started his scholastic life with the desire to become a philosopher.  At 8 years old he started studying the classics, logic, and traditional Aristotelian philosophy at the newly built Jesuit college of  La Fleche in Anjou.  He stayed at La Fleche for eight years.  Due to poor health, Descartes (Figure 6) was permitted to stay in bed until 11:00 in the morning.   He continued this practice almost his entire life.  Descartes moved to Paris and received a law degree from the University of Poitiers in 1616.  He felt that the only area where he had any competence was mathematics and began studying mathematics and science at the military school in Breda.   On the night of November 10, 1619, Descartes claimed to have had a dream  where all of the sciences were unified.  He claimed this was the critical point in his lifetime.  He developed a new philosophy and his innovations in analytical geometry from parts of that dream.  For the next ten years Descartes traveled all through Europe trying to find a country that would suit his nature.  He finally decided on Holland and he stayed there for the next twenty years.  Descartes is best know for his introduction of analytical geometry.  Analytical geometry is also called coordinate geometry or Cartesian geometry.  It is based on reducing  a geometry problem into a relatively simple algebra problem.  It uses the Cartesian plane (named after Descartes), a set of perpendicular axis broken into a grid.  By using coordinates from the grid  (Figure 7) it is possible to take curves and straight lines and convert them into algebraic expressions.  Descartes is also famous for his theories towards finding the roots, or zeros, of polynomials.  Descartes’  “Rule of signs”  is used to calculate the number of possible positive and negative roots in a given polynomial.  In Descartes’ third book, he fixed the custom of using letters at the beginning of the alphabet to represent known quantities and letters at the end of the alphabet for unknown quantities.                                                               

            Descartes’ father was a member of the local parliament.  His family was relatively influential and would be considered wealthy by today’s standards.  He was fortunate to have the financial flexibility to change his career following a strange dream.  He believed in comfort and sleeping late.  He died shortly after the Queen of Sweden convinced him to move to Sweden.  She desired 5 a.m. lessons on tangents and after a life sleeping late in warm climates, Descartes died after a few months in the cold northern climate. 

Leonhard Euler              

Leonhard Euler (1707 - 1783) is one of the most important mathematicians of the 18^th century.  He was born in Basel, Switzerland and died in St. Petersburg, Russia.  His father, Paul, was a student of Jakob Bernoulli and during his tenure at the University of Basel, he lived in the Bernoulli house with Jakob’s brother Johann.  Euler’s father Paul was a minister and he wanted Leonhard (Figure 8) to follow in his footsteps.  Even though Euler’s family had moved away from Basel, Leonhard was sent to school there.  His first school was not very good and did not offer any mathematics.  Euler’s introduction to mathematics came from his father.  He worked very hard and continued to study mathematics on his own.  Euler entered into the University of Basel at 14 to begin general instruction to prepare him for the church.  Johann Bernoulli soon discovered Euler’s great potential for mathematics following Euler’s request for tutoring.  Euler attained a master’s degree in philosophy, comparing and contrasting the philosophical ideals of Newton and Descartes.  With the support of Bernoulli, Euler was able to convince his father that he had a career in mathematics and shifted the emphasis of his studies.  He completed his studies in 1726 and began publishing mathematical papers.  He took second prize from the Paris Academy of Sciences in 1727 and was in desperate search for an academic appointment at a university.  The death of Nicholas Bernoulli opened a position at the St. Petersburg Academy of Science.  At the age of 19 he accepted a position with the mathematics and physics division.  Euler was appointed as senior chair of mathematics at the Academy in 1733.  The new position allowed him to get married.  He fathered 13 children but only 5 survived infancy.  He often claimed that some of his greatest discoveries were made while children were playing around his feet.  Euler also had health problems.  He almost lost his life due to fever in 1735 and by 1740 he had lost the use of one eye.  Euler was totally blind by 1771.  Euler became worried about political unrest in Russia.  He left St. Petersburg in 1741 and assumed a position with the Berlin Academy of Science.  During his 25 years in Berlin, Euler published over 380 articles and  books.  He was the most prolific mathematics writer of the 18th century.  He published books on algebra, calculus, optics, planetary motion, and ballistics.  The president of the Berlin Academy died in 1759 and Euler assumed leadership of the Academy.  He was not designated as president due to friction with Fredrick the Great, the King of Germany.  Eventually, Euler returned to his position at St. Petersburg.  He continued to publish until his death in 1783.  The St. Petersburg Academy of Science continued to publish works of Euler 50 years after his death. 

             Due to the sheer abundance of Euler’s work it is almost impossible to delineate his most important contributions.  He made decisive contributions to geometry, calculus, and number theory.  He was very influential in standardizing some of the mathematical notation we use today.  We owe Euler for the function notation  “f(x),” e as the base for natural logarithms, i for the square root of -1, p for pi, and ? for summation.  The equation ln(-1) = pi allows us to find the natural log of a negative number.  Euler made a significant impact on modern mathematics.  His poor health and many disappointments could have discouraged his output, but Euler worked on mathematics until the day he died. 

Bill Gates   

William (Bill) Gates III (1955 - present) is one of the richest men in the world.  Estimated to have a net worth of over 100 billion dollars, Gates came from a relatively normal background.   Gates”  father was a lawyer and wanted Bill to become a lawyer also.  Skinny, shy, and awkward, Gates seemed an unlikely successor to his overachieving parents.  He attended a public elementary school and a private high school.  There he discovered his interest in software and started programming computers.   He was 13 years old.  Gates entered Harvard University as a freshman in 1973.     While at Harvard, Gates developed a version of the programming language BASIC for use on a microcomputer (PC).  He never graduated, dropping out of college during his junior year to devote his efforts to his new company, Microsoft.  Gates (Figure 9) became and continues to be the chairman and chief software architect of Microsoft Corporation.     

            Microsoft was founded on the simple belief that the PC would be a very valuable tool and eventually be found on every desktop in business and at home.  Gates’ first five customers went bankrupt, but in 1980, IBM asked Microsoft to develop an operating system for their first PC.  Microsoft purchased a disc operating system (DOS), already in use by another company, renamed it MS-DOS and licensed its use to IBM.  IBM’s PC turned out to be the vanguard of the “PC revolution” and turned Microsoft into an extremely successful company with revenues exceeding 25.3 billion dollars in 2001.

             Gates never demonstrated significant mathematical talent.  He did not discover any new mathematical concepts.  He did demonstrate the foresight to find a use for concepts developed by others.  His family was considered upper middle class, but his current net worth places him in a class by himself. 

Blaise Pascal 

Blaise Pascal (1623 - 1662) like many of the early mathematicians devoted much of his life to the study of mathematics and theology.  Pascal was born in Clermont, France and died in Paris.  Pascal (Figure 10) was originally home schooled as his father did not want him overtaxed with trivial studies.  He believed that his son should study languages and would not allow him to learn mathematics.  Naturally, this excited Pascal’s curiosity and he endeavored to learn mathematics on his own.  At 12 he asked his tutor what geometry was.  He was so enthused by the reply that he gave up play time to study mathematics.  Within a few weeks, he had discovered for himself many of the properties of figures.  His father was so impressed that he gave him a book on Euclidian geometry, which Pascal read quickly and soon mastered. At age 14 he was admitted to a weekly meeting of French mathematicians.  From this weekly meeting sprung the French Academy of Science.  At 16 he published a paper on conic sections and at 18 constructed one of the first arithmetic calculators.  In 1650 Pascal suddenly gave up mathematics to study religion.  Fortunately for mathematics, his father died in 1653 and Pascal had to administer his father’s estate.  During this period he returned to his work in mathematics.  In November of 1654 Pascal had a near brush with death.  He considered it a “work of God”  that he was saved and returned to his religious studies until his death in 1662.   

            Pascal is best known for Pascal’s triangle and his work with Fermat on probabilities.  It is interesting to note that the work on probabilities originated and was directly related to gambling.  Pascal was the only son of a local judge.  His family would have been considered upper-middle class today.  He was fortunate to be financially secure to study anything he desired.  His reputation as one of France’s greatest mathematicians rests more on what he could have accomplished without his religious hiatus, than on what he actually did. 

Pascal’s Triangle 

Pascal’s Triangle (Figure 11) probably originated in China.   Pascal is attributed with its discovery as he was one of the first to demonstrate some of the interesting facets of the triangle.   Pascal’s triangle can be used to derive the Fibonacci number sequence and Sierpinski’s triangle.  If you add the elements in each row, the sum will always be a power of 2.  For example, sum of the elements in the third row is equal to 8 or 2^3.  The sum of the fifth row is 32 or 2^5.  In fact the sum of the nth row is equal to 2^n.  The apex of the triangle is 1 and is considered the 0th row.  2^0 is equal to one.  Another interesting fact is that if the second element in a row is prime, the first and last elements will always be one, then every other element of the row is divisible by the prime number.  For example the eleventh row is 1 11 55 165 330 462 462 330 164 55 11 1.  11, 55, 165, 330, and 462 are all divisible by 11.  If a row is made into a number using each element as a digit (carry the tens digit if the element has more than one digit),  the number will be equal to 11^n, where n is the number of the row.  Example: row 5 contains the elements 1 5 10 10 5 1.  Combining the digits, with carries, into a single number you attain 161051.  161051 is 11^5.   The Fibonacci number sequence can be found by adding the diagonals of the triangle (Figure 12).  The Fibonacci number sequence is one of most interesting sequences in mathematics.  From the Fibonacci numbers you can determine the “golden ratio”  and there are many instances where the sequence is found in nature.  One of the diagonals of Pascal’s triangle is the triangular number sequence (Figure 13).  This sequence is derived by adding consecutive integers to consecutive numbers in the triangle.  To go from element one to two, add two.  From element two to three, add three.  From element (n - 1) to element n, add n.  For this particular diagonal if you add any two consecutive elements together, their sum will always be a perfect square.  Example:  the diagonal sequence is 1 3 6 10 15 21 28...   The sum of 1 and 3 is 4.  The sum of 10 and 15 is 25.  The sum of 21 and 28 is 49.  All results are perfect squares.  An important use of the triangle is the ability to expand binomials using the elements of each row.  (x + a)^2 = x^2 + 2xa + a^2.  The coefficients of the expanded binomial are 1 2 1.  The elements of the second row are 1 2 1.  The elements of the sixth row 6 are 1 6 15 20 15 6 1.  If we expand (x + a)^ 6,  (x + a)^6 = x^6 + 6x^5a + 15x^4a^2 + 20x^3a^3 + 15x^2a^4 + 6xa^5 + a^6.  The coefficients are the same as the elements in the sixth row.   There are many other interesting ways to manipulate Pascal’s triangle to find other number patterns.

Step Functions 

Step functions are a group of discontinuous functions that can demonstrate linear properties.  Belonging to the category of functions known as “special functions,” they are often the bane of first year algebra students.  Figure 14 is an example of the step function [x].  As seen in the Figure, [x] is defined as the greatest integer less than or equal to a number.  For example:   [2.5] = 2, [2.0000001] = 2, [1.99999999] = 1.   The fact that the result is always an integer creates a series of steps.  Notice the open end on each of the step segments.  That open end corresponds to the first point in the next step keeping the relation a function.  Step functions are primarily seen in the world of commerce and often relate to billing.  The post office uses step functions to determine cost of mailing packages.  Your cell phone bill if plotted would be a step function.  Step functions can be graphed using curve sketching techniques.   The parent graph for the greatest integer function, [x] is f(x) = [x].  The function used for sketching the family of graphs is f(x) = a[x - h] + k.  Translation and expansion follow normal  curve sketching techniques.

  

Graph Theory

Graph theory is one of the most exciting forms of mathematics.   With the advent of computers, large amounts of data can be processed quickly.  Computers are mostly “brute force”  machines and are able to process the data because of their high operational rates.  As problems become complicated the number of calculations that must take place within the computer rise greatly.  Today graph theory is used to streamline the number of calculations made within the computer.   Graph theory is primarily the use of nodes and verticies (figure 15) to draw a picture of a particular problem.  One of the earliest uses of graph theory was Euler’s solution to the Konigsburg bridge problem.  Figure 16 is a drawing of Konigsburg bridge situation.  The premise of the problem was:  is it possible to cross all seven of Konigsburg’s bridges only once and end up where you started?  Using graph theory, Euler demonstrated that the task was impossible.  Graph theory can be used to find the most economical route that a traveling peddler might take, or the most efficient flight path from one destination to another.   Graph theory is particularly effective in determining least search algorithms for sorting arrays or finding elements within an array.

Probability 

The lottery (Figure 17) is a tax for people that have not learned math!  It is nice to dream about winning the lottery, breaking the bank in Las Vegas, or beating the odds to fill the “inside straight”  at your local Saturday night poker game.  Reality is that this is extremely unlikely to happen.  Since the advent of gambling, people have always tried to determine sure ways to win.  For gambling to be profitable to the organization running the games, those organizations take advantage of the mathematics branch known as probability.   There are many different definitions for probability.  For the purpose of this discussion probability is as follows:
 the number of correct solutions.
 the number of possible solutions

            Many people play the lottery Powerball.  When the jackpot gets near 100 million dollars people tend to get a bit crazy about things.  What is the chance of winning?  The number of correct solutions is one.  The number of possible solutions is C(50,5) * 50 or 50*50*49*48*47*46/(5*4*3*2).   The likelihood that you will win the lottery is one in one hundred five million, nine hundred thirty eight thousand.  If the jackpot were 100 million dollars and you bought a ticket for every possible combination, you would still lose over 5 million dollars and that does not include taxes.  In the movie Maverick, Mel Gibson draws the ace of spades to win the poker game.  What was the probability of that really happening?  There are 52 cards in a deck of which five were in his opponent’s hand, four were discarded and there were four cards in Maverick’s hand.  There is only one ace of spades (Figure 18) in the deck.  Maverick’s opponent might have already discarded the ace or may have had it in his hand, so we must add those eight cards back to the number of possibilities.  So Maverick had a one in forty seven chance of drawing the ace.  Another way of putting it was that he had a 98% chance of not drawing the ace.  However, as it was a movie, he drew the ace and won the game.  Gambling is not like the movies.  Figure the odds before you place your bet.

Binary Numbers

Most people think that computers (Figure 19) are very smart.  A point of fact is that a computer only knows one thing.  At any given time is there a charge or no charge present.  Being a yes/no answer we find that there are only two possible choices.  Binary arithmetic only has two possible digits, 1 and 0.  1 + 0 = 0, 1 + 1 = 10, 10 + 10 = 100.  In binary, 10 is equal to the decimal system’s 2, 11 is 3, and 111 is 7.  Binary only uses the digits 1 and 0 in combination. Coupled with Boolean algebra, binary arithmetic is what allows computers to process information.  For a computer to work, three things must be present:  a mechanism for determining charge/no charge, a clock to time the checks, and a compiler to interpret the charges.  Each charge/no charge is called a “bit.”  A group of bits make up a “byte.”  The number of bits in a byte depends upon a computer.  Half a byte is a “nibble”  and two bytes make a “word.”  Words are stored in registers that make up the computer’s memory.  The bits in registers can be added together using binary arithmetic.  To multiply, the computer adds the number being multiplied to itself the number of times in the multiplier.   The above is a very simplistic overview of how a computer works.  Many other things must take place within the equipment for us to get usable answers.  The computer does not know the answer, it only gives us the information we ask for.  If we do not ask the computer correctly, it will not know what we are talking about and will not be able to give us the correct answer.  Hence, garbage in, garbage out.  To use the computer for our benefit, we must have a complete mastery of the concept being used.  The computer will not replace the human brain nor make discoveries on its own. 

Minority Mathematicians  

            In the Eurocentric school of thought almost all mathematical discoveries are attributed to white European or American men.  German, Greek, and particularly the English, have made most of the remarkable contributions to mathematics.   We occasionally include a member of a minority group such as Srinivasa Ramanujan

(Figure 20) or  Ada Lovelace.  In fact, this school of thought is not entirely correct.  Many mathematical innovations came from minorities.  Ramanujan did outstanding work with series.  Lovelace (Figure 21) assisted Charles Babbage in the creation of the “analytic engine,” often considered the first computer ever built.  Lovelace determined a means to program the engine to calculate Bernoulli numbers.  Nina Bari was a 20th century Russian woman who discovered many trigonometric series.  She also translated many works by other mathematicians into Russian.  Benjamin Bannaker (Figure 22) was the son of a slave and and the daughter of an indentured servant.   He received very little education but was taught to read by his grandmother.  Bannaker had an almost photographic memory and played an important role in the cartographic survey of the United States.  He taught himself mathematics and astronomy from borrowed books and constructed the first striking clock made in America.  A major innovation of mathematics, algebra, is credited to al-Khwarizmi.  Prior to this point mathematical theory was mostly the geometric view of the Greeks.  The Eurocentric view of mathematics states that mathematical discovery almost ceased during the dark ages of Europe.  We now know that to be false as a great deal of mathematical discovery was taking place in northern and eastern Africa.  In addition to his work with algebra, al-Khwarizmi translated numerous Greek texts into Arabic.  For over 400 years, African mathematicians continued to contribute to mathematics.  Omar Kayyam derived a complete classification of cubic functions found by means of intersecting conic sections.  Al-Karaji is credited with being the first to completely free algebra from geometrical operations.   At the end of the dark ages, Spain was one of the first European areas for mathematical study.  This is easily understood due to its near proximity to Northern Africa.   Chou Pei, a Chinese scholar from the 3rd century BC provided the first written proof of the Pythagorean theorem (Figure 23).  Sun Tzu, a Chinese philosopher and mathematician, used mathematical theories in his writings and provided us with a very early proof of the “remainder theorem.”  Chinese mathematicians proved the theory known as “Pascal’s triangle” 300 years before Pascal was born (Figure 24). Mathematicians come from all races and economic backgrounds.  Anyone with imagination, determination and persistence can impact society through mathematics.

Conclusion  

To assist in mathematics mastery, I propose the following “commandments”: 

            1.  Thou shalt read thy problem carefully from alpha to omega, before thou tryist to come to a conclusion. 

            2.  Thou must never divide by zero err false results thou shalt experience. 

            3.  Thou mightest do anything whatsoever thou dost desire to one side of the equation as long as thou shalt do ye also to the other.

              4.  Thou shalt use thy “common sense,” else thou shalt discover elephants flying at 400 miles per hour.  Yea, cats and dogs living together, chaos.

              5.   Thou shalt ignore the teachings of false prophets and demons encouraging thee to do all thy assignment in thy head.

              6.  Thou shalt not accuse thy all-knowing teacher, nor thy book of knowledge of falsehood until thou hast communicated with highest authority and thou hast received written documentation that thou art correct. 

            7.  Thou must demonstrate the ability to walk with simple steps, before thou canst even consider skipping steps.

              8.  When thou turneth in an answer, without including the work, that answer is most surely wrong.  Even if thou hast the answer correct, thou hast not proven that thou has the correct answer.  Verily I say unto thee that thou must include thy calculations. 

            9.  A computer or calculator doest not think.  Their answer is only as correct as their input.  Placeth garbage into a machine and thou wilt receive only garbage as thy reward.

              10.If thou shalt learn to read, write and listen correctly to the language of mathematics, verily thou shalt reap rewards of A’s and B’s throughout thy journey from infancy through manhood.

      While the above rules are presented in a humorous manner, they are still important to the proper mastery of mathematics. To conclude, I offer two thoughts to take into consideration.  The first is: “People who do not learn math usually work for people that do.”   The second:  “The only thing better than learning mathematics is teaching mathematics.”

Implementation

           “This is the part our teachers told us about.  When algebra saves our lives.”
           Val Kilmer from the movie Red Planet.

This unit section describes possible scenarios where the background information may be used.  They are not designed as individual lessons, but as a means of inspiring students to master concepts.   Typical student questions have been broken into cases with suggested uses for the information provided. 

            Case 1:  When will I ever use this?  One of the most common questions asked at the introduction to a new lesson.   There are two simple answers and one innovative response  to the question.   The first answer is: never.  That is correct, never.  Equations do not run up to you on the street and scream “Solve me, solve me!”   Unfortunately, mathematics is used as a gage to determine if you have the capability of mastering difficult concepts.  Failure to master even those basics will not preclude you from a job at McDonalds, but will rule out other educational or most other job opportunities.  The second simple answer is:  always.   Mathematics is involved in almost everything we do, from making change to driving a car.  Often we are not aware that we are doing mathematics, we attribute it to common sense.   When we shop, we compare prices and determine the number of items needed.  Taking a nap, we calculate the time available and determine a wake up time.  Most of us would not even consider filling an eight ounce glass with a gallon of milk.  The third option is to take into account that there may not be a present need for the concept, but that it either leads to other mathematics tools or that use for the concept may be found in the future.  Bill Gates did not know that Boolean and binary arithmetic would change his life.  He found a use for them in programming.  Professional gamblers calculate probabilities to stay in business.   The key is that most people have to work for a living and it is much easier to use your head rather than your back to work.  Besides, you will need it for the test.  Use worksheet 1 from the appendix to augment this discussion.

            Case 2:  Who had time to think up this stuff?  Didn’t they have a social life?  Obviously we have more leisure opportunities today than at any other time in history.  We have developed many activities to take up that time.  Consider a world without television or radio.  A world where Nintendo is a form of oriental philosophy.  How would you fill your time?  The quest for knowledge is an important part of self-improvement.  If a mathematical education can change you from an incompetent merchant to the mathematics chair at a university, as it did Johann Bernoulli, is it not worth it to make an attempt?  Most of the mathematicians mentioned in the background lived lives very similar economically to your own.  While a few were privileged and did not have to work for their studies, most did.  For a great many mathematicians, their social life revolved around their work.  Mathematic societies and clubs were places that the educated could exchange ideas.  Use the lessons learned from history and you could be the next person worth over 100 billion dollars.  Use worksheet 2 from the appendix to augment this discussion.

             Case 3:  Why bother learning this?  Smarter people than me don’t use it.  This case is very similar to case 1.  The difference is the second statement.  Smart people are not necessarily intelligent.  Successful people are not necessarily smart.  Many students today have role models where wealth is the measure of success.  Unfortunately, most do not realize that for every one of their role models that achieve that wealth, there are thousands that do not.  It is essential that we give our students something to fall back upon.  If a mathematician is the role model, the student will still be employable even if they do not achieve mathematic stardom.  Skill, discipline and persistence are the primary traits looked for in job or educational candidates.  Encouraging students by demonstrating the success of many mathematicians will help instill those traits.  As mentioned in case 1, mathematics is involved in almost everything we do.  It is possible that you might never use some of the skills but it is better to be familiar with the concepts in case they are actually needed.  Use worksheet 3 from the appendix to augment this discussion.

             Case 4:  My calculator can do that, why should I bother?  An immediate reply to this question is to have the student divide four by zero on their calculator.  Ask the student to take the square root of negative 16.  An immediate follow-up would be to ask for the logarithm of negative ten.  All three activities should result in an error message.  An Algebra II student should be able to tell you that the square root of negative sixteen is plus or minus 4i.  A basic algebra student should know that to divide by zero is undefined.  Euler’s rule allows us to determine the logarithm of negative one as ln(-1) = pi.  Students must be taught that their calculators and computers are valuable aids to calculation.  They do not think.  Talk about Archimedes using his body as a chalkboard.  Encourage the students to think on their own.  Use worksheet 4 from the appendix to augment this discussion.

             Case 5:  All the good stuff has already been discovered.  Why should I try?  Time again to answer questions with questions.  Ask how many computers were available when George Boole originated Boolean algebra.   Question your students as to how many airplane rides the Bernoulli family took.  One of my students at Estancia High School developed a principle I now teach to all my classes.   I named it “Derek’s Law”   after the student that discovered it.  Derek had failed Pre-Algebra for two consecutive years and was in his third attempt in the class.  The class was learning to convert decimals into fractions.  For non-repeating decimals this is relatively easy, but to convert repeating decimals required thought and substitution.  “Derek’s Law”  simply states, count the number of digits that repeat in your decimal.  Take one series of the numbers and place it over the same number of 9 digits and reduce. Much easier than the standard method.  Derek’s Law works for any repeating decimal that consistently repeats after the decimal point.  Students should be challenged to come up with new uses for old concepts and new concepts for old tasks.   Find other uses of Pascal’s triangle.   Have them work with number series to see what they can find.  Use worksheet 5 from the appendix to augment this discussion.

             Case 6:  Who cares about this stuff?  It won’t help me get a date.  $100,000,000,000 looks like incentive to me.  Darwin proposed survival of the fittest as a means of evolution.  During previous times, when technology was not a factor, this often meant the biggest or strongest prevailed.  With today’s technology, it pays to use your head.  Find an innovative way to accomplish a task cheaply and you will have more dates than the average Olympic champion.  Save the company money and you will continue employment when others are being “down sized.”  The ability to use mathematics can mean the difference between taking your sister to the prom or Britney Spears.  Use worksheet 6 from the appendix to augment this discussion.

             Case 7:  Why did you teach me the hard way, when it is so much easier with one of the other methods?   What is easy for one person may not be easy for another.  There are many ways to accomplish the same task.  Students should use what is right for them.  Additionally, mathematical conc