Mathematics Project Preliminary Index (some of the sections and screens listed are incomplete)

http://www.settheory.com/Setl2/web_math/index_all_items.html

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Index of this index

  1. Miscellaneous external utilities
  2. Classroom-of-the-future student-teacher communication software (in progress)
  3. Sampling of Material for the elementary grades

  4. New York State Regents Mathematics Examination Archive (This archive will be expanded systematically, but just a few initial samples are available now)
    1. Math regents exam A July, 2005

  5. Middle School Materials

  6. Preface, Linear equations in one and several variables
  7. Sets and mappings
  8. Algebraic expressions and the basic rules of algebra
  9. Properties of Polynomials and their roots; Functions and graphing
  10. The logarithm and exponential
  11. Quadratic equations and complex numbers
  12. Inequalities
  13. Prime numbers and their properties
  14. Properties of polynomials in several variables and root finding for polynomial systems
  15. Basic geometry of the plane
  16. Parallel Lines
  17. Similar triangles and Pythagoras Theorem; basics of trigonometry
  18. Applications of trigonometry
  19. Trigonometric identities
  20. Radian measure and properties of small angles
  21. Geometric miscellany and applications; convexity; inside and outside; basics of area
  22. Combinatorics and the Binomial Theorem
  23. Basics of probability
  24. State transition models
  25. The 'normal distribution'
  26. Queuing theory
  27. Statistics
  28. Determinants
  29. Calculator collection

Miscellaneous external utilities

  1. Google calculator
Classroom-of-the-future student-teacher communication software (in progress)                Back to Top
  1. teacher_side_comm
  2. student_side_comm
  3. teacher_side_reply
  4. fut_clr_teach_controls
  5. student_side_reply
  6. future_classroom_top
  7. future_classroom_student
  8. class_server
  9. class_summary
Sampling of Material for the elementary grades                Back to Top
  1. Count dots to 20
  2. Count to 20
  3. Count to add
  4. Make an addition table
  5. Great big numbers
  6. More great big numbers
  7. Fancier great big numbers
  8. Speed addition
  9. Speed multiplication
  1. arrange_2_1
  2. arrange_2_2
  3. arrange_2_4
  4. arrange_3_2
  5. arrange_3_4
  6. arrange_5_9
  7. arrange_7_4
Middle School Materials Preface, Linear equations in one and several variables                Back to Top
  1. What is 'Animated Algebra'?
  2. What does animated algebra expect of you?
  3. How will algebra be explained?
  4. How is animated algebra organized?
  5. Basic Instructions for Use
  6. Calculators in animated algebra
  7. Styles of text in Animated Algebra
  8. Variables and values
  9. Practice with variables and values
  10. Variables as 'unknowns'
  11. Solving for unknowns
  12. More easy equations
  13. Harder equations
  14. Rules for stepping-stone equations
  15. More rules for stepping-stone equations
  16. Transposition
  17. Equations involving two variables
  18. Equations involving two variables, II
  19. Equations involving three variables, or more
  20. Practice with equations in 4 unknowns
  21. If some of the coefficients are fractions
  22. Exceptional cases of the subtraction procedure
  23. A linear equations calculator
  24. Who invented the subtraction method for solving equations containing several variables?
  25. Fast solution of easy equation systems
  26. Different languages can be used to say the same thing
  27. Advantages of the language of numbers
  28. Translating 'word problems' into algebraic formulas
  29. Word problem variations and the equation variations they imply
  30. Recognizing quantity-related words
  31. English words vary more than their translations into algebra
  32. Problems involving implicit facts: rates and proportions
  33. Problems involving implicit facts: averages
  34. Problems involving implicit facts: ratios and conversions between units
  35. Percentages and discounts
  36. Standard formulas and ratios
Logic, sets and mappings

  1. About logic
  2. About logic
Algebraic expressions and the basic rules of algebra                Back to Top
  1. Setting the values of variables
  2. Combining the values of variables
  3. Using combination values to form new combinations
  4. Combining variables with constants and constants with constants
  5. 'Box and arrow' diagrams for showing dependencies between combinations
  6. When we are only interested in the last of a series of combinations
  7. Algebraic expressions
  8. How to read algebraic expressions
  9. Evaluating algebraic expressions manually
  10. Common errors in reading and evaluating algebraic expressions
  11. Evaluating algebraic expressions automatically
  12. Algebraic identities
  13. Substitution in algebraic expressions and identities
  14. Basic algebraic identities: the 'laws of algebra'
  15. Why the laws of algebra must be true, I: the commutative law of addition
  16. Why the laws of algebra must be true, II: the associative law of addition
  17. What does 'multiplication' mean?
  18. Who invented multiplication?
  19. Why the laws of algebra must be true, III: the commutative law of multiplication
  20. Why the laws of algebra must be true, IV: the distributive law
  21. Why the laws of algebra must be true, V: the properties of 0 and 1
  22. Why the laws of algebra must be true, VI: the associative law of multiplication
  23. What does subtraction mean?
  24. Why the laws of algebra must be true, VII: properties of subtraction
  25. What does division mean?
  26. Why the laws of algebra must be true, VIII: properties of division
  27. Rules for writing fewer parentheses in algebraic expressions
  28. Powers of a number. Positive and negative exponents
  29. Polynomials containing just one variable
  30. Polynomials containing several variables
  31. Adding and subtracting polynomials
  32. Multiplying polynomials
  33. Dividing polynomials
  34. The degree of a polynomial
  35. A cancellation rule for polynomials
  36. Adding and subtracting polynomials involving several variables
  37. Multiplying polynomials involving several variables
  38. Rational expressions and formal fractions
  39. The greatest common factor of two polynomials
  40. Finding the common zeroes of two polynomials
  41. A polynomial version of the remainder theorem.
  42. The difference quotient and the derivative of a polynomial. Derivatives of sums and differences
  43. Properties of the derivative, II: Derivatives of polynomial products
  44. Properties of the derivative, III: the chain rule for substituted polynomials
  45. Rise, fall, maxima, minima, and derivatives
  46. Derivatives of rational functions and of quotients
  47. Double roots, multiple roots, and the derivative
  48. Sturm's theorem
  49. The unique prime factorization therem for polynomials containing a single variable
  50. The unique prime factorization therem for polynomials containing several variables
  51. Combining variables
  52. Showing numerical values
  53. Multiple use of a variable's value
  54. Fancier combination values
  55. Formulas name the outputs of combinations
Properties of Polynomials and their roots; Functions and graphing                Back to Top
  1. Polynomial equations with one variable; Polynomial degree and coefficients
  2. Quadratic equations
  3. Functions
  4. Names of functions
  5. Coordinates in the plane
  6. Practice with plane coordinates
  7. Practice with plane coordinates, II
  8. Plotting several points at a time
  9. Graphing a function
  10. Experimenting with polynomial graphs
  11. A modified way of graphing functions
  12. Condensed 'function snapshots'
  13. Finding the roots of polynomials, and why roots are important
  14. Double roots and multiple roots
  15. Zooming and panning function snapshots
  16. Precise location of roots
  17. Precise location of roots, II
  18. Precise location of roots for functions which are not polynomials
  19. Where roots might, must, and cannot appear
  20. Where roots might, must, and cannot appear, II
  21. The remainder theorem and the root theorem
  22. Fermat's law of signs
The logarithm and exponential                Back to Top
  1. The area squash principle
  2. Definition and basic properties of the logarithm
  3. Basic properties of the logarithm, II: logarithm of powers, graph of the logarithm.
  4. Definition and basic properties of the exponential
  5. The constant 'e'
  6. The general power function
  7. Logarithms and exponentials with other bases
  8. Graphs of the logarithm and exponential
  9. Area under the graph of the exponential
  10. Area under the graph of the logarithm
  11. Computing exponentials and logarithms
Quadratic equations and complex numbers                Back to Top
  1. Quadratic equations
  2. Completing the square
  3. Finding complex roots, I: A way of coming close to at least one complex root
  4. Finding complex roots, II: Coming close to the complex roots
  5. Avoiding repeated finding of roots that have already been found
  6. A more automatic version of the complex root finder
  7. Improving the precision of complex roots once roughly found
  8. Complex roots on the unit circle, and polynomials of higher degree
  9. Finding complex roots that lie far from z = 0
  10. Experimenting with a whole-plane complex root finder
  11. Another experiment with a whole-plane complex root finder
  12. The 'gradual change of polynomial' method for finding the roots of complex polynomials
  13. Finding complex roots of polynomials whose values, insted of their coefficients, are known
  14. Root-finding as a way of factoring polynomials
Inequalities

Prime numbers and their properties                Back to Top

  1. Quotients (review)
  2. Divisors and primes
  3. This longer list shows the primes up to 2500
  4. The smallest proper divisor of n is always a prime
  5. Any number can be broken up into a product of primes
  6. The prime factorization theorem
  7. Proving the unique prime factorization Theorem
  8. Proving the unique prime factorization Theorem, II
  9. Proving the unique prime factorization Theorem, II
  10. Quotients (review)
  11. Proof of the Divisor Theorem
  12. More about the Divisor Theorem
  13. The Divisor Theorem as a computational procedure
  14. There are infinitely many primes
  15. A consequence of the prime factorization theorem
  16. Factoring extremely large integers is hard
  17. The Prime Number Theorem
  18. An extension of the prime number theorem
  19. Twin primes
Properties of polynomials in several variables and root finding for polynomial systems                Back to Top

Basic geometry of the plane                Back to Top

  1. Shifting and turning objects
  2. Congruent shapes
  3. Shapes that are not congruent
  4. Measuring Angles
  5. Angles can vary in size
  6. Measuring Angles with a Protractor
  7. Measuring Angles
  8. Protractor Use, contd
  9. Practice in measuring angles
  10. Practice in measuring larger angles
  11. Congruent angles
  12. Congruent angles
  13. The sum of two angles
  14. Some important special angles
  15. 'Opposite' angles
  16. Equality of opposite angles
  17. Geometric reflections and Symmetry
  18. Geometric reflections and congruence
  19. 'Left-' and 'Right-handed' figures
  20. Proof by symmetry
  21. Another proof by symmetry
  22. A third proof by symmetry
  23. Triangle congruence by side-angle-side
  24. Triangle congruence by angle-side-angle
  25. The false SSA congruence principle
  26. Triangle congruence by side-side-side
  27. Two intersecting circles
  28. Triangle congruence by side-side-side
  29. A line intersecting a circle
  30. A line touching a circle at one point
  31. A special congruence rule for right triangles
  32. The area of plane figures
  33. Using smaller tiles
  34. The area of general figures
  35. The area of general figures, II
  36. Why areas add
  37. Some changes of shape don't change area
  38. An area comparison principle
  39. The area of a rectangle
  40. Uniform distortion of areas
  41. An ellipse
Parallel Lines                Back to Top
  1. When straight lines cross
  2. Parallel Lines
  3. The axiom of parallels
  4. Parallel and perpendicular lines
  5. Two lines both parallel to a third line
  6. The perpendicular distance between parallel lines
  7. Opposite sides of a rectangle are equal
  8. A line crossing two parallel lines
  9. Proportional division of segments by parallel lines
  10. Proportional division of segments by parallel lines, II
  11. The sum of the angles in a triangle
  12. The angles formed by a line crossing two parallel lines
  13. Opposite sides of a parallelogram are equal
  14. Another property of parallelograms
  15. The sum of the interior angles of polygons in general
  16. The existence of interior crossings
  17. Existence of interior crossings, II
  18. A comparison principle for the sides and angles of a triangle
  19. Another proof of the comparison principle for triangle sides and angles
  20. The shortest distance from a point to a line
  21. The shortest distance from a point to a circle
  22. The angle between lines meeting on a circle
  23. The angle between lines meeting inside a circle
  24. The angle between lines meeting outside a circle
  25. Geometric 'Locus'
  26. The points equally far from two lines
  27. The point equally far from all three corners of a triangle
  28. The point equally far from all three sides of a triangle
  29. The points p with which a given segment forms a triangle having an angle of given size at p
  30. The points p with which a given segment forms a triangle having a large angle of given size
  31. The points p with which a given segment forms a right triangle
  32. The area of a triangle
Similar triangles and Pythagoras Theorem; basics of trigonometry                Back to Top
  1. Similar triangles
  2. Practice in recognizing similar triangles
  3. Defining triangles by two angles and a side
  4. Defining similar triangles by angles only
  5. Proportionality of similar triangles
  6. Equivalent conditions for triangle similarity
  7. Similar right triangles
  8. Heights and areas of similar triangles
  9. Pythagoras Theorem
  10. Proof of Pythagoras Theorem
  11. Proof of Pythagoras Theorem, contd
  12. Pythagoras' Theorem is extremely important.
  13. Pythagoras Theorem can be extended to 3 dimensions
  14. Pythagoras Theorem in 3 dimensions
  15. Something about Pythagoras
  16. Using Pythagoras' theorem to measure the world
  17. Dependence of horizon distance on observer height
  18. The Radius of the Earth
  19. Another proof of Pythagoras theorem
  20. Calculating the length of a semicircle of radius 1
  21. Calculating the length of a semicircle of radius 1
  22. Calculating the length of a semicircle of radius 1
  23. This table shows the number of digits of π that can be found by Archimedes' repeated-doubling method
  24. Here we have p computed to 800 digits on a computer by Archimedes' method
  25. How well did Archimedes himself do?
  26. Sine, cosine, and tangent of similar right triangles
  27. Trigonometry applied: a simple example
  28. Practice with sines, cosines, and tangents
  29. Sine, cosine, and tangent
  30. The angles in a few important special triangles have interesting sines, cosines, and tangents
  31. Five symmetry rules for sines and cosines
  32. Calculating the length of a semicircle of radius 1
  33. The Law of Sines
  34. Practice with the law of sines
  35. The Law of Cosines
  36. Practice with the law of cosines
  37. The Trigonometric Addition Laws for sines and cosines
  38. Finding the values of sines and cosines
  39. Trigonometric identities
  40. Some basic trigonometric formulas
  41. Checking trigonometic identities mechanically
  42. Trigonometric identities involving fractional angles, or more than one angle
Applications of trigonometry                Back to Top
  1. Measurement of triangles has many important applications, and has had a great influence on human history
  2. Another group of applications of trigonometry is to coastal navigation.
  3. Solving costal navigation problems
  4. More trigonometric formulas
  5. Aristarchos and the spirit of science
  6. Aristarchos'and the spirit of science
  7. So, what is a scientist?
  8. Things about Samos, Aristarchos home island, that may interest you
  9. Things about Samos Island that may interest you
Trigonometric identities                Back to Top
  1. Some basic Trigonometric Formulas
  2. Some basic Trigonometric Formulas
  3. Some basic Trigonometric Formulas
  4. Deriving Other Trigonometric Formulas Mechanically
  5. Deriving Other Trigonometric Formulas Mechanically
Radian measure and properties of small angles                Back to Top
  1. Measuring angles in terms of 'Radians' instead of 'Degrees'
  2. The area of a slice of a circle cut by two lines drawn from its center is exactly half the length of the curved side of the sector
  3. Converting between radians and degrees
  4. Sines and
  5. Cosines of small angles, and more details about sines
  6. Cosines of small angles, and more details about sines
  7. Examining the sines and cosines of small angles
Geometric miscellany and applications; convexity; inside and outside; basics of area                Back to Top
  1. Right triangles whose side lengths are integers
  2. The ancient Egyptian 'rope stretching' method for measuring out square corners in a field or when starting to build a house
  3. Finding all the Pythagorean triangles
  4. Finding all the Pythagorean triangles
  5. Finding all the Pythagorean triangles
  6. Now we have found all the Pythagorean triangles
  7. Play around with polygons
  8. Convex and non-convex polygons
  9. A distinctive property of convex polygons
  10. The 'convex wrapper' of a polygon
  11. More about the convex wrapper
  12. Self-crossing and simple polygons
  13. Inside and outside
  14. Inside and outside, contd
  15. Inside and outside, III
  16. Inside and outside, IV
  17. Three polygonal arcs meeting at two common end-points
  18. Division of the plane by 3 arcs
  19. The three houses and three wells puzzle
  20. The houses and wells puzzle, contd
  21. Another characteristic property of convex figures
  22. The idea of 'area'
  23. The area of general shapes
  24. Why areas add
  25. Changing shape without changing area
  26. Why areas add
  27. An area comparison principle
Combinatorics and the Binomial Theorem                Back to Top
  1. Exploring the very large
  2. The 'nested boxes' principle
  3. Using the 'nested boxes' principle
  4. Using the 'nested boxes' principle to count some very large sets
  5. The 'list choice' principle
  6. Some applications of the 'list choice' principle.
  7. Review of counting methods presented so far.
  8. Some applications of the 'list choice' principle, II. The number of arrangements of n different things.
  9. Applications of the 'list choice' principle, III. The number of ways of Dividing a set of n things into parts
  10. Using the 'list choice' principle backwards. The number of ways of choosing k things out of n.
  11. Using the 'list choice' principle backwards. Dividing n things into groups of given sizes.
  12. Circular arrangements.
  13. Circular arrangements allowing repetitions.
  14. The binomial theorem.
  15. The multinomial theorem.
  16. Formulae involving binomial coefficients.
  17. Counting by 'inclusion and exclusion'
  18. Counting by 'inclusion and exclusion', II. The inclusion-exclusion formula
  19. Counting by 'inclusion and exclusion', III. The general inclusion-exclusion formula
  20. Proof of the inclusion-exclusion formula
  21. The number of mappings of one set onto another
  22. Ramsey's theorem
  23. The 'multicolor' version of Ramsey's theorem
  24. Single-valued mappings of a set S into itself.
  25. An application of the cycles principle: Fractions are always expressed by repeating decimals.
Basics of probability                Back to Top
  1. Urn models
  2. Cheating
  3. Successive drawings from urns
  4. Successive drawings from urns II. A simple case of the multiplication rule for probabilities
  5. Rolling a pair of dice
  6. Probabilities for sets of outcomes
  7. Probabilities and fair odds
  8. Casino dicing
  9. Basic properties of probabilities for sets
  10. Casino dicing, II. Bets on the order of events
  11. Sets defined by conditions on single drawings
  12. Casino dicing, III. The probability of 'making a point'
  13. Casino dicing, IV. 'Odds on Come' bets
  14. Casino dicing, V. 'Dont Come' bets
  15. Casino dicing, VI. The 'Place' bets and other bets
  16. Fair division of stakes when a game must end early
  17. Fair division of stakes among more than two players
  18. Some other order-of-events probabilities
  19. Sampling without replacement
  20. Sampling without replacement II. the multiplication rule for probabilities
  21. Sampling without replacement III. Bets on the order of events
  22. Sampling without replacement IV. A proof that the simplified method of calculating order-of-events probabilities is correct
  23. The number of dice that must be rolled or cards drawn in order to reach a given total
  24. Probabalistic inference Problems
  25. Probabalistic inference Problems II. Distinguishing between a pair of populations
  26. Casino blackjack
  27. Casino blackjack, II: Player probabilities and strategies
  28. Casino blackjack, III: Additional rules
  29. Casino blackjack, IV: Player probabilities and strategies in the 'Wild Aces' case
  30. Casino blackjack, V: Player probabilities and strategies given all the other rules
  31. Casino blackjack, VI: Card-counting and Beat the dealer
State transition models                Back to Top
  1. Random transitions
  2. Transition probabilities
  3. The population view of the situation
  4. Advantages of the 'population' view
  5. The 'population bar' view
  6. Stability of the 'Statistical balance'
  7. Other unstable cases
  8. Fluctuations around the 'Statistical balance '
  9. Using 'population bars' to calculate equilibria directly
  10. Equations for the equilibriun occupancy percentages
  11. A summary
  12. How state transition models are applied
  13. Who invented state transition models?
  14. Pals from the beginning
  15. 'Gamblers ruin' problems
The 'normal distribution'                Back to Top
  1. A mechanical model
  2. A 'population' view of the situation
  3. A 'population bar' view of the situation
  4. A better stabilized population bar view of the situation
  5. Other gain/loss patterns
  6. Other gain/loss patterns, II
  7. Patterns statistically biased up or down
  8. The two parameters of the bell curve shape
  9. Mixed populations
  10. When your winnings depend on how much you have won already
  11. Different end-conditions
  12. Random whole-number gain/loss patterns'> first steps toward a formal analysis
  13. Random whole-number gain/loss patterns'>
    binomial coefficient formula for the outcome
  14. Approximate expressions for binomial coefficients
  15. The deMoivre/Gauss formula for the ideal bell curve
  16. A bit of history
  17. Bell curves and state transition processes
  18. Multiproperty gains and losses
  19. parameters of 2-dimensional bell surfaces
Queuing theory                Back to Top
  1. What is 'queuing theory'?
  2. States and transitions in queuing problems
  3. A second assumption
  4. Simulating arrival and service-completion events
  5. The 'load ratio' and queue lengths
  6. A 'population bar' view of the situation
  7. The load factor in the 'population bar' view
  8. The equilibrium values of the queue-length probabilities
  9. Dynamics of the overload case
  10. Recovery from overload
  11. Waiting-time probabilities
  12. Waiting-time probabilities, II
  13. How long a line at a busy post office should you join?
  14. At what time should you plan to go to a busy post office?
  15. How much time are the people on waiting lines wasting?
  16. Other queuing disciplines
  17. Dynamics of the overload case
  18. Dynamics of the overload case
Statistics                Back to Top
  1. Histograms
  2. A life-expectancy histogram
  3. Simplifying and generalizing the life-expectancy histogram
  4. Testing ideas for improving life expectancy
  5. Shape-related numerical properties of histograms
  6. Measures of deviation from a histogram's average
  7. Measures of deviation from a histogram's average, II. The standard variance and standard deviation
  8. Estimating averages and estimating medians
  9. Estimating other percentiles
  10. The Bell Curve of probabilities
  11. Other cases in which normal distributions emerge
  12. What happens when randomly chosen numbers are multiplied rather than added?
  13. Examples of normally distributed data
  14. Testing for statistical identity
  15. Using samples to see if data is normally distributed
  16. Mathematical properties of the standard deviation and variance
  17. Calculating the variance of normally distributed sums
  18. Testing ideas for improving life expectancy II. Estimating averages
  19. The bell-curve distribution and the estimation of averages
  20. Measuring other numerical properties of histograms
  21. Combined use of several histogram properties
  22. Relating two numerical properties of one population: the coefficient of correlation
  23. Linear relationships between two numerical properties of one population: estimating the 'regression line'
  24. Testing for statistical dependency by rank correlation
  25. Statistics of life insurance
  26. The present value of a future payment
  27. Statistics of life insurance, II. Term life policies
  28. Statistics of life insurance, III. Cash surrender values of life insurance policies
  29. Statistics of annuities
Determinants                Back to Top

Calculator collection                Back to Top

  1. One op integer calculator
  2. Two ops integer calculator
  3. Four function integer calculator
  4. Six function integer calculator
  5. Nine function integer calculator
  6. Ten function integer calculator
  7. Full integer calculator
  8. Four function rational calculator
  9. Four function rational complex calculator
  10. Expression evaluator
  11. Function snapshot generator