- Miscellaneous external utilities
- Classroom-of-the-future student-teacher communication software (in progress)
- Sampling of Material for the elementary grades
- New York State Regents Mathematics Examination Archive (This archive will be expanded systematically, but just a few initial samples are available now)
- Middle School Materials
- Preface, Linear equations in one and several variables
- Sets and mappings
- Algebraic expressions and the basic rules of algebra
- Properties of Polynomials and their roots; Functions and graphing
- The logarithm and exponential
- Quadratic equations and complex numbers
- Inequalities
- Prime numbers and their properties
- Properties of polynomials in several variables and root finding for polynomial systems
- Basic geometry of the plane
- Parallel Lines
- Similar triangles and Pythagoras Theorem; basics of trigonometry
- Applications of trigonometry
- Trigonometric identities
- Radian measure and properties of small angles
- Geometric miscellany and applications; convexity; inside and outside; basics of area
- Combinatorics and the Binomial Theorem
- Basics of probability
- State transition models
- The 'normal distribution'
- Queuing theory
- Statistics
- Determinants
- Calculator collection

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- student_side_comm
- teacher_side_reply
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- student_side_reply
- future_classroom_top
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- Count dots to 20
- Count to 20
- Count to add
- Make an addition table
- Great big numbers
- More great big numbers
- Fancier great big numbers
- Speed addition
- Speed multiplication

- What is 'Animated Algebra'?
- What does animated algebra expect of you?
- How will algebra be explained?
- How is animated algebra organized?
- Basic Instructions for Use
- Calculators in animated algebra
- Styles of text in Animated Algebra
- Variables and values
- Practice with variables and values
- Variables as 'unknowns'
- Solving for unknowns
- More easy equations
- Harder equations
- Rules for stepping-stone equations
- More rules for stepping-stone equations
- Transposition
- Equations involving two variables
- Equations involving two variables, II
- Equations involving three variables, or more
- Practice with equations in 4 unknowns
- If some of the coefficients are fractions
- Exceptional cases of the subtraction procedure
- A linear equations calculator
- Who invented the subtraction method for solving equations containing several variables?
- Fast solution of easy equation systems
- Different languages can be used to say the same thing
- Advantages of the language of numbers
- Translating 'word problems' into algebraic formulas
- Word problem variations and the equation variations they imply
- Recognizing quantity-related words
- English words vary more than their translations into algebra
- Problems involving implicit facts: rates and proportions
- Problems involving implicit facts: averages
- Problems involving implicit facts: ratios and conversions between units
- Percentages and discounts
- Standard formulas and ratios

- Setting the values of variables
- Combining the values of variables
- Using combination values to form new combinations
- Combining variables with constants and constants with constants
- 'Box and arrow' diagrams for showing dependencies between combinations
- When we are only interested in the last of a series of combinations
- Algebraic expressions
- How to read algebraic expressions
- Evaluating algebraic expressions manually
- Common errors in reading and evaluating algebraic expressions
- Evaluating algebraic expressions automatically
- Algebraic identities
- Substitution in algebraic expressions and identities
- Basic algebraic identities: the 'laws of algebra'
- Why the laws of algebra must be true, I: the commutative law of addition
- Why the laws of algebra must be true, II: the associative law of addition
- What does 'multiplication' mean?
- Who invented multiplication?
- Why the laws of algebra must be true, III: the commutative law of multiplication
- Why the laws of algebra must be true, IV: the distributive law
- Why the laws of algebra must be true, V: the properties of 0 and 1
- Why the laws of algebra must be true, VI: the associative law of multiplication
- What does subtraction mean?
- Why the laws of algebra must be true, VII: properties of subtraction
- What does division mean?
- Why the laws of algebra must be true, VIII: properties of division
- Rules for writing fewer parentheses in algebraic expressions
- Powers of a number. Positive and negative exponents
- Polynomials containing just one variable
- Polynomials containing several variables
- Adding and subtracting polynomials
- Multiplying polynomials
- Dividing polynomials
- The degree of a polynomial
- A cancellation rule for polynomials
- Adding and subtracting polynomials involving several variables
- Multiplying polynomials involving several variables
- Rational expressions and formal fractions
- The greatest common factor of two polynomials
- Finding the common zeroes of two polynomials
- A polynomial version of the remainder theorem.
- The difference quotient and the derivative of a polynomial. Derivatives of sums and differences
- Properties of the derivative, II: Derivatives of polynomial products
- Properties of the derivative, III: the chain rule for substituted polynomials
- Rise, fall, maxima, minima, and derivatives
- Derivatives of rational functions and of quotients
- Double roots, multiple roots, and the derivative
- Sturm's theorem
- The unique prime factorization therem for polynomials containing a single variable
- The unique prime factorization therem for polynomials containing several variables
- Combining variables
- Showing numerical values
- Multiple use of a variable's value
- Fancier combination values
- Formulas name the outputs of combinations

- Polynomial equations with one variable; Polynomial degree and coefficients
- Quadratic equations
- Functions
- Names of functions
- Coordinates in the plane
- Practice with plane coordinates
- Practice with plane coordinates, II
- Plotting several points at a time
- Graphing a function
- Experimenting with polynomial graphs
- A modified way of graphing functions
- Condensed 'function snapshots'
- Finding the roots of polynomials, and why roots are important
- Double roots and multiple roots
- Zooming and panning function snapshots
- Precise location of roots
- Precise location of roots, II
- Precise location of roots for functions which are not polynomials
- Where roots might, must, and cannot appear
- Where roots might, must, and cannot appear, II
- The remainder theorem and the root theorem
- Fermat's law of signs

- The area squash principle
- Definition and basic properties of the logarithm
- Basic properties of the logarithm, II: logarithm of powers, graph of the logarithm.
- Definition and basic properties of the exponential
- The constant 'e'
- The general power function
- Logarithms and exponentials with other bases
- Graphs of the logarithm and exponential
- Area under the graph of the exponential
- Area under the graph of the logarithm
- Computing exponentials and logarithms

- Quadratic equations
- Completing the square
- Finding complex roots, I: A way of coming close to at least one complex root
- Finding complex roots, II: Coming close to the complex roots
- Avoiding repeated finding of roots that have already been found
- A more automatic version of the complex root finder
- Improving the precision of complex roots once roughly found
- Complex roots on the unit circle, and polynomials of higher degree
- Finding complex roots that lie far from z = 0
- Experimenting with a whole-plane complex root finder
- Another experiment with a whole-plane complex root finder
- The 'gradual change of polynomial' method for finding the roots of complex polynomials
- Finding complex roots of polynomials whose values, insted of their coefficients, are known
- Root-finding as a way of factoring polynomials

**Prime numbers and their properties** __Back to Top__

- Quotients (review)
- Divisors and primes
- This longer list shows the primes up to 2500
- The smallest proper divisor of n is always a prime
- Any number can be broken up into a product of primes
- The prime factorization theorem
- Proving the unique prime factorization Theorem
- Proving the unique prime factorization Theorem, II
- Proving the unique prime factorization Theorem, II
- Quotients (review)
- Proof of the Divisor Theorem
- More about the Divisor Theorem
- The Divisor Theorem as a computational procedure
- There are infinitely many primes
- A consequence of the prime factorization theorem
- Factoring extremely large integers is hard
- The Prime Number Theorem
- An extension of the prime number theorem
- Twin primes

**Basic geometry of the plane** __Back to Top__

- Shifting and turning objects
- Congruent shapes
- Shapes that are not congruent
- Measuring Angles
- Angles can vary in size
- Measuring Angles with a Protractor
- Measuring Angles
- Protractor Use, contd
- Practice in measuring angles
- Practice in measuring larger angles
- Congruent angles
- Congruent angles
- The sum of two angles
- Some important special angles
- 'Opposite' angles
- Equality of opposite angles
- Geometric reflections and Symmetry
- Geometric reflections and congruence
- 'Left-' and 'Right-handed' figures
- Proof by symmetry
- Another proof by symmetry
- A third proof by symmetry
- Triangle congruence by side-angle-side
- Triangle congruence by angle-side-angle
- The false SSA congruence principle
- Triangle congruence by side-side-side
- Two intersecting circles
- Triangle congruence by side-side-side
- A line intersecting a circle
- A line touching a circle at one point
- A special congruence rule for right triangles
- The area of plane figures
- Using smaller tiles
- The area of general figures
- The area of general figures, II
- Why areas add
- Some changes of shape don't change area
- An area comparison principle
- The area of a rectangle
- Uniform distortion of areas
- An ellipse

- When straight lines cross
- Parallel Lines
- The axiom of parallels
- Parallel and perpendicular lines
- Two lines both parallel to a third line
- The perpendicular distance between parallel lines
- Opposite sides of a rectangle are equal
- A line crossing two parallel lines
- Proportional division of segments by parallel lines
- Proportional division of segments by parallel lines, II
- The sum of the angles in a triangle
- The angles formed by a line crossing two parallel lines
- Opposite sides of a parallelogram are equal
- Another property of parallelograms
- The sum of the interior angles of polygons in general
- The existence of interior crossings
- Existence of interior crossings, II
- A comparison principle for the sides and angles of a triangle
- Another proof of the comparison principle for triangle sides and angles
- The shortest distance from a point to a line
- The shortest distance from a point to a circle
- The angle between lines meeting on a circle
- The angle between lines meeting inside a circle
- The angle between lines meeting outside a circle
- Geometric 'Locus'
- The points equally far from two lines
- The point equally far from all three corners of a triangle
- The point equally far from all three sides of a triangle
- The points p with which a given segment forms a triangle having an angle of given size at p
- The points p with which a given segment forms a triangle having a
*large*angle of given size - The points p with which a given segment forms a right triangle
- The area of a triangle

- Similar triangles
- Practice in recognizing similar triangles
- Defining triangles by two angles and a side
- Defining similar triangles by angles only
- Proportionality of similar triangles
- Equivalent conditions for triangle similarity
- Similar right triangles
- Heights and areas of similar triangles
- Pythagoras Theorem
- Proof of Pythagoras Theorem
- Proof of Pythagoras Theorem, contd
- Pythagoras' Theorem is
*extremely important.* - Pythagoras Theorem can be extended to 3 dimensions
- Pythagoras Theorem in 3 dimensions
- Something about Pythagoras
- Using Pythagoras' theorem to measure the world
- Dependence of horizon distance on observer height
- The Radius of the Earth
- Another proof of Pythagoras theorem
- Calculating the length of a semicircle of radius 1
- Calculating the length of a semicircle of radius 1
- Calculating the length of a semicircle of radius 1
- This table shows the number of digits of π that can be found by Archimedes' repeated-doubling method
- Here we have p computed to 800 digits on a computer by Archimedes' method
- How well did Archimedes himself do?
- Sine, cosine, and tangent of similar right triangles
- Trigonometry applied: a simple example
- Practice with sines, cosines, and tangents
- Sine, cosine, and tangent
- The angles in a few important special triangles have interesting sines, cosines, and tangents
- Five symmetry rules for sines and cosines
- Calculating the length of a semicircle of radius 1
- The Law of Sines
- Practice with the law of sines
- The Law of Cosines
- Practice with the law of cosines
- The Trigonometric Addition Laws for sines and cosines
- Finding the values of sines and cosines
- Trigonometric identities
- Some basic trigonometric formulas
- Checking trigonometic identities mechanically
- Trigonometric identities involving fractional angles, or more than one angle

- Measurement of triangles has many important applications, and has had a great influence on human history
- Another group of applications of trigonometry is to coastal navigation.
- Solving costal navigation problems
- More trigonometric formulas
- Aristarchos and the spirit of science
- Aristarchos'and the spirit of science
- So, what is a scientist?
- Things about Samos, Aristarchos home island, that may interest you
- Things about Samos Island that may interest you

- Some basic Trigonometric Formulas
- Some basic Trigonometric Formulas
- Some basic Trigonometric Formulas
- Deriving Other Trigonometric Formulas Mechanically
- Deriving Other Trigonometric Formulas Mechanically

- Measuring angles in terms of 'Radians' instead of 'Degrees'
- 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
- Converting between radians and degrees
- Sines and
- Cosines of small angles, and more details about sines
- Cosines of small angles, and more details about sines
- Examining the sines and cosines of small angles

- Right triangles whose side lengths are integers
- The ancient Egyptian 'rope stretching' method for measuring out square corners in a field or when starting to build a house
- Finding all the Pythagorean triangles
- Finding all the Pythagorean triangles
- Finding all the Pythagorean triangles
- Now we have found all the Pythagorean triangles
- Play around with polygons
- Convex and non-convex polygons
- A distinctive property of convex polygons
- The 'convex wrapper' of a polygon
- More about the convex wrapper
- Self-crossing and simple polygons
- Inside and outside
- Inside and outside, contd
- Inside and outside, III
- Inside and outside, IV
- Three polygonal arcs meeting at two common end-points
- Division of the plane by 3 arcs
- The three houses and three wells puzzle
- The houses and wells puzzle, contd
- Another characteristic property of convex figures
- The idea of 'area'
- The area of general shapes
- Why areas add
- Changing shape without changing area
- Why areas add
- An area comparison principle

- Exploring the very large
- The 'nested boxes' principle
- Using the 'nested boxes' principle
- Using the 'nested boxes' principle to count some very large sets
- The 'list choice' principle
- Some applications of the 'list choice' principle.
- Review of counting methods presented so far.
- Some applications of the 'list choice' principle, II. The number of arrangements of n different things.
- Applications of the 'list choice' principle, III. The number of ways of Dividing a set of n things into parts
- Using the 'list choice' principle backwards. The number of ways of choosing k things out of n.
- Using the 'list choice' principle backwards. Dividing n things into groups of given sizes.
- Circular arrangements.
- Circular arrangements allowing repetitions.
- The binomial theorem.
- The multinomial theorem.
- Formulae involving binomial coefficients.
- Counting by 'inclusion and exclusion'
- Counting by 'inclusion and exclusion', II. The inclusion-exclusion formula
- Counting by 'inclusion and exclusion', III. The general inclusion-exclusion formula
- Proof of the inclusion-exclusion formula
- The number of mappings of one set onto another
- Ramsey's theorem
- The 'multicolor' version of Ramsey's theorem
- Single-valued mappings of a set S into itself.
- An application of the cycles principle: Fractions are always expressed by repeating decimals.

- Urn models
- Cheating
- Successive drawings from urns
- Successive drawings from urns II. A simple case of the multiplication rule for probabilities
- Rolling a pair of dice
- Probabilities for sets of outcomes
- Probabilities and fair odds
- Casino dicing
- Basic properties of probabilities for sets
- Casino dicing, II. Bets on the order of events
- Sets defined by conditions on single drawings
- Casino dicing, III. The probability of 'making a point'
- Casino dicing, IV. 'Odds on Come' bets
- Casino dicing, V. 'Dont Come' bets
- Casino dicing, VI. The 'Place' bets and other bets
- Fair division of stakes when a game must end early
- Fair division of stakes among more than two players
- Some other order-of-events probabilities
- Sampling without replacement
- Sampling without replacement II. the multiplication rule for probabilities
- Sampling without replacement III. Bets on the order of events
- Sampling without replacement IV. A proof that the simplified method of calculating order-of-events probabilities is correct
- The number of dice that must be rolled or cards drawn in order to reach a given total
- Probabalistic inference Problems
- Probabalistic inference Problems II. Distinguishing between a pair of populations
- Casino blackjack
- Casino blackjack, II: Player probabilities and strategies
- Casino blackjack, III: Additional rules
- Casino blackjack, IV: Player probabilities and strategies in the 'Wild Aces' case
- Casino blackjack, V: Player probabilities and strategies given all the other rules
- Casino blackjack, VI: Card-counting and Beat the dealer

- Random transitions
- Transition probabilities
- The population view of the situation
- Advantages of the 'population' view
- The 'population bar' view
- Stability of the 'Statistical balance'
- Other unstable cases
- Fluctuations around the 'Statistical balance '
- Using 'population bars' to calculate equilibria directly
- Equations for the equilibriun occupancy percentages
- A summary
- How state transition models are applied
- Who invented state transition models?
- Pals from the beginning
- 'Gamblers ruin' problems

- A mechanical model
- A 'population' view of the situation
- A 'population bar' view of the situation
- A better stabilized population bar view of the situation
- Other gain/loss patterns
- Other gain/loss patterns, II
- Patterns statistically biased up or down
- The two parameters of the bell curve shape
- Mixed populations
- When your winnings depend on how much you have won already
- Different end-conditions
- Random whole-number gain/loss patterns'> first steps toward a formal analysis
- Random whole-number gain/loss patterns'>

binomial coefficient formula for the outcome - Approximate expressions for binomial coefficients
- The deMoivre/Gauss formula for the ideal bell curve
- A bit of history
- Bell curves and state transition processes
- Multiproperty gains and losses
- parameters of 2-dimensional bell surfaces

- What is 'queuing theory'?
- States and transitions in queuing problems
- A second assumption
- Simulating arrival and service-completion events
- The 'load ratio' and queue lengths
- A 'population bar' view of the situation
- The load factor in the 'population bar' view
- The equilibrium values of the queue-length probabilities
- Dynamics of the overload case
- Recovery from overload
- Waiting-time probabilities
- Waiting-time probabilities, II
- How long a line at a busy post office should you join?
- At what time should you plan to go to a busy post office?
- How much time are the people on waiting lines wasting?
- Other queuing disciplines
- Dynamics of the overload case
- Dynamics of the overload case

- Histograms
- A life-expectancy histogram
- Simplifying and generalizing the life-expectancy histogram
- Testing ideas for improving life expectancy
- Shape-related numerical properties of histograms
- Measures of deviation from a histogram's average
- Measures of deviation from a histogram's average, II. The standard variance and standard deviation
- Estimating averages and estimating medians
- Estimating other percentiles
- The Bell Curve of probabilities
- Other cases in which normal distributions emerge
- What happens when randomly chosen numbers are multiplied rather than added?
- Examples of normally distributed data
- Testing for statistical identity
- Using samples to see if data is normally distributed
- Mathematical properties of the standard deviation and variance
- Calculating the variance of normally distributed sums
- Testing ideas for improving life expectancy II. Estimating averages
- The bell-curve distribution and the estimation of averages
- Measuring other numerical properties of histograms
- Combined use of several histogram properties
- Relating two numerical properties of one population: the coefficient of correlation
- Linear relationships between two numerical properties of one population: estimating the 'regression line'
- Testing for statistical dependency by rank correlation
- Statistics of life insurance
- The present value of a future payment
- Statistics of life insurance, II. Term life policies
- Statistics of life insurance, III. Cash surrender values of life insurance policies
- Statistics of annuities

**Calculator collection** __Back to Top__

- One op integer calculator
- Two ops integer calculator
- Four function integer calculator
- Six function integer calculator
- Nine function integer calculator
- Ten function integer calculator
- Full integer calculator
- Four function rational calculator
- Four function rational complex calculator
- Expression evaluator
- Function snapshot generator