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History of Mathematics: Academic Content Standards Timeline Michelle Yee

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Number, Number Sense, and Operations Origin of Zero How would math be different without zero? 2000 B.C. Babylonians represented zero by leaving gaps between wedge marks on clay. The gaps were not considered a number. –used base 60 numeration system 350 B.C. Greeks were unsure about the idea of zero –Did not use a positional system, therefore zero was not necessary. 600 A.D. Hindu-Arabic Decimal System- Zero is officially introduced. –AL' KHWARIZMI – wrote math book that introduced zero to the world A.D. Hindu-Arabic Decimal System reaches Europe.

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Number, Number Sense, and Operations Prime Numbers A whole number greater than one that has exactly two factors; 1 and itself. 325 B.C. EUCLID proved there is an infinite number of primes. - Author of The Elements- used to teach geometry for more than 2,000 years. 276 B.C. ERATOSTHENES –Sieve of Eratosthenes- an algorithm for finding prime numbers GAUSS proved any number is a product of primes.

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Number, Number Sense, and Operations Fractions 3000 B.C. Egyptians recognized that fractions begin with reciprocals of whole numbers. Fractions were written as a sum of unit fractions. An eye was placed over the integer to represent the reciprocal AKA unit fraction. Horus- Egyptian God who fought forces of darkness. His eye is the symbol of Egyptian unit fractions.

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Measurement Pi The ratio of the circumference to the diameter of a circle. Used by Egyptians, Babylonians, Greeks, Chinese and Hebrews Greeks discovered Pi was an irrational number B.C. ARCHIMEDES OF SYRACUSE- Made the first theoretical calculation of Pi. – used circles inscribed and circumscribed by polygons to find the value of Pi to be between 223/71 and 22/7.

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Measurement Volume and Surface Area B.C. ARCHIMEDES OF SYRACUSE Greek Mathematician Found the surface area and volume of a sphere: 4πr 2 4/3πr 2 Found volume of a cone: 1/3πr 2 h

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Measurement Metric System Invented in France in French National Assembly told a committee from the Academy of Sciences of Paris to standardize the units of measurement. Some mathematicians on the committee: –Jean Charles de Borda ( ) –Joseph-Louis Comte de Lagrange ( ) –Pierre-Simon Laplace ( ) –Gaspard Monge ( ) Meter comes from the Greek word metron, meaning measure.

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Geometry and Spatial Sense 569 B.C. PYTHAGORAS OF SAMOS Everything is number. Proved theorem which is named after him, Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Discovered that notes played in music correspond to ratios with small numbers. Example: Halving the length of a musical string gave a music note one octave higher than the first.

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Geometry and Spatial Sense Area of =(a+b)(a+b) a b b c One of the many proofs for the Pythagorean theorem: Area of a b c =ab/2 Area of=c 2 area +4 = ( ) area c 2 + 4(ab/2) = (a+b) 2 c 2 + 2ab = a 2 + 2ab + b 2 c 2 = a 2 + b 2

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Geometry and Spatial Sense Cartesian grid RENE DESCARTES French philosopher and mathematician Descartes was sick and lying on his bed staring at a fly. The ceiling had square tiles, and Descartes noticed he could describe where the fly was in relation to the points formed from the tiles.

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Patterns, Functions, and Algebra History of Algebra Egyptian Algebra 1850 B.C. Solved problems equivalent to a linear equation in one unknown. 300 B.C. Solved problems equivalent to a system of two second degree equations in two unknowns. Did not use symbols.

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Patterns, Functions, and Algebra History of Algebra Babylonian Algebra More advanced than Egypt. Had a general procedure equivalent to solving quadratic equations Considered some problems involving more than two unknowns and a few equivalent to solving equations of higher degree. Problems solved were taught through examples and no reasons or explanations were given. Little use of symbols No negative or irrational numbers

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Patterns, Functions, and Algebra History of Algebra Hindu Algebra Introduced negative numbers to represent debts A.D.BRAHMAGUPTA-first known to use negative numbers. Developed correct procedures for working with irrational numbers. Used some symbolism. Steps of problems were stated but reasons or proofs were not given. Included negative as well as irrational roots.

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Patterns, Functions, and Algebra History of Algebra Arabic Algebra Improved the Hindu number symbols and the idea of positional notation. Worked with irrational numbers Rejected negative numbers and solutions Solved quadratic equations and recognized two solutions Algebra is named after a book written by Al Khwarizmi. Algorithm and arithmetic are from modifications of Al Khwarizmis name.

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Patterns, Functions, and Algebra LEONARD EULER Swiss mathematician Contributions to all branches of mathematics including –Calculus –Geometry –Algebra –Number theory Inventor of graph theory. Developed several notations used today: –Π for pi –i for -1 –y for change in y –f(x) for a function – for summation

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Data Analysis and Probability BLAISE PASCAL French mathematician Wrote a math book at the age of 16. Pascals Triangle: –Triangular array of numbers studied in China and India. – Pascal discovered new properties of the triangle and solved problems using it. Pascals Triangle has many uses. Example: Evaluating combinations –If we need to know the number of combinations of n things taken r at a time, (the # of subsets of size r in a set of size n) we read entry number r of row number n from Pascals Triangle.

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Data Analysis and Probability BLAISE PASCAL Probability Theory Originated over a dispute between Pascal and French mathematician Pierre de Fermat about the problem of the points: Involved how to divide points between players of a game if the contestants scores and the score needed to win were known. Fermat Probability theory became an important tool for scientists studying the physical world.

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Data Analysis and Probability Inventor of the box and whisker plot Inventor of the stem and leaf plot 1977 JOHN TUKEY American Statistician

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Mathematical Processes B.C. ARISTOTLE Greek Philosopher and Mathematician Logical thinking- conclusions must be supported by observation Aristotle systemized deductive logic –From general to specific The modern scientific method is a combination of –Deductive reasoning –Inductive reasoning

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Mathematical Processes George Polya Hungarian mathematician Author of How to Solve It (1945) describes methods of problem solving. Developed Polyas Four step Problem-Solving Process 1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

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Sources Barrow, J. D. (1992). Pi in the sky: Counting, thinking, and being. Oxford: Clarendon Press. Gianopoulos, A., Langone, J., & Stutz, B. (2006). Theories for everything: An illustrated history of science from the invention of numbers to string theory. Washington, DC: National Geographic. Heeren, V.E., Hornsby, J., & Miller, C.D. (2001). Mathematical ideas (9th ed.). Boston: Addison Wesley Educational Publishers. Lewinter, M. & Widulski, W. (2002). The saga of mathematics: A brief history. Upper Saddle River, NJ: Prentice Hall, Inc. Windelspecht, M. (2002). Groundbreaking scientific experiments, inventions, & discoveries of the 17 th century. Westport, CT: Greenwood Press. Online Source OConnor, J. J. & Robertson, E.F. The mactutor history of mathematics archive retrieved October & November 2007 from history.mcs.st-andrews.ac.uk/history/index.html. Michelle Yee

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