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Program Y8 (Ages 12-13) Main premises Goals Ignite interest to math, specifically to research like activity, rather than simple persuasion for many conundrum problems Show the big picture emphasizing interconnections between topics, rather than picturing full details of each topic. Show Python and Wolfram Mathematics as tools that allow to investigate, test hypotheses, automate manual work, and visualize data. The more intuitive visualizations/ examples / tables / code the better. They can replace strict abstract definitions. “Multiplication table is symmetric, and each row and column has all elements” can be used as a shortcut for “commutativity” and “every non-zero element has an inverse” for a while. Focus We don’t have much time, so We have to limit scope: take only subsets of ideas from each topic, trying to get as many connections as possible All problems and examples should be multipurpose. For instance, explaining MMI it is reasonable to take problems for other topics in our scope We can’t provide a full picture of every topic, and actually, no need according to stated goals. We want to present some discrete examples for each topic, like points in a topic “circle” having maximum links to other selected points in other topic circles. We skip some very big and important topics, because they don’t have enough connections with core selected topics, or existing connections cannot be covered for year 7 students in a reasonable time. Also we sacrifice strictness of definitions and proofs: proof by illustration or intuitive example suffice. Mentioning some important concepts from skipped topics is ok unless it does not take much time and there is a high probability of building future connections, or connections with something students have heard about. More magic Because math is actually about magic. More “what if” ideas. What if we introduce sqrt(-1)? What if we take the Binomial theorem (1 + x)^n and use n = ½? What if we search for other objects like real numbers? Is it possible to have a finite set of objects like numbers? What if we take the sum of infinite geometrical series? … Topics Selected topics Logic Geometry Area: Pythagorean theorem Triangle and Quadrilateral Problems about compass and ruler. Combinatorics (probability) Pascal triangle. Fibonacci numbers Paths in graphs: number of ways to achieve a point A. Weights: number of way to achieve an imbalance n Learning curve to generating functions The partition function p(x) Functional analysis: The idea of good functions: polynomials and smooth functions Continuations Abstract algebra: Number fields: finite and complex numbers Skipped topics Group theory: Skip axiomatic definition, theorem about homomorphism. But we can mention and use the concept of “set of actions” and “composition of two actions”. Eg. plane motions. Limits and strict definition of continuation. The concept “curve that can be drawn by a pen w/o jumps” is considered as intuitive and strict definition is not required. Measure theory: like strict definitions of area and the idea Riemann integration Graph theory Affine and Projective geometry: very important, but now we don’t see shortcuts to present a sensible amount of concepts from them. Although this is discussible topic. Affine transform can be defined as “bijective map, mapping line to line” with quite intuitive first steps. Projective geometry is also something that everybody witnesses. Differential and Integration. But it is ok to mention the relation of circle area and circle circumference, ODO and speedometer, area under the curve and function value. Topics Landscape Geometry Triangle area: simple cases Pythagorean theorem Basic Approach: squares on checked paper = “squares with vertices in integer points”. Non need to write general equation c^2 = a^2 + b^2 Squares Table: S(a,b) = a^2 + b^2: search for regularities. Connection with k=0n(2k+1) = (n+1)2. What integer numbers can be represented by a^2 + b^2? Limit the problem to prime numbers. Introduce concept of multiplicative functions Triangle area: complicated cases Quadrilateral Problems about compass and ruler. Find the center of a segment. Divide a segment into 3 equal segments Find a given circle center. Given R and R2 draw a line, circle R1, and circle R2 touching each other Draw the right polygon with N vertices, N = 3,4,5,6,7,8,9,10, inscribed into a given circle. Rectangle split into a grid of M x N similar smaller rectangles: areas of some pieces are given, find the rest. Area of a small circular stripe. Sphere volume and area. Volume of a small spherical layer with thickness = epsilon. Logic Notations Methods: Proof by contradiction Extremal Principle Method of Mathematical Induction (MMI) Tower of Hanoi Basic problems: kn2k, knk 2k k=0n(2k+1) = (n+1)2, kn1k (k + 1)=1 - 1n+1, . Cn+1k+1=Cnk+Cnk+1 Plane with n mutually non parallel lines: Can be colored w/ two colors Combinatorics: What is the number of regions? Combinatorics Pascal Triangle Connections: Number of paths (a + b)^n Closed formula n! / (k! (n-k)!) Fibonacci numbers Paths for a grasshopper jumping 1 or 2 to the right. Connections: Fibonacci Snail Golden Ratio Compare growth rates with 2^n and 1.5^n Logarithm: Show Wolfram Mathematica: using logarithm to compare with 2^n and 1.5^n. Continuous fraction for F(n) / F(n+1) Permutations N! Number of derangements Weights and scales Decompositions w/ and w/o repetitions, w/ and w/o order. With order: paths in graphs. W/o order: weights. Weights 1,2,4,8, 16, for one-pan balance scales and binary system Weights 1,3,9,27, …for two-pan balance scales and Generative functions: Weights 1,3,9,27, …for two-pan balance scales and (1 + x + 1/x) (1 + x3 + 1/x3)... 1 / (1 - x + x^2). Show Series method in Wolfram Mathematica 1/(1+x)^2: what is this sequence Partition function p(x) Probability Two combinatorial problems: count all possible combinations and count good possible combinations. Number theory Prime numbers. Infinity of prime numbers Divisability criteria. Divisibility by 7. Modular arithmetics If p is a prime then the multiplication table is good: each row and each column contains all numbers, it is symmetric, the first row repeats elements from header Learning curve to quadratic reciprocity low Negative and positive numbers: “positive” = “can be represented as a^2”. Negative numbers: all others w/o zero. Investigate solutions of x^2 = -1 (mod p) x^2 = 2 (mod p) Number of points on a “circle” x^2 + y^2 = 1 (mod p) Number of points on a “circle” y^2 = x^3 + x + 1 (mod p) Learning curve to concept of multiplicative functions and more Number of dividers Sum of dividers Learning curve to Mobius mu function: TODO: What is the simplest problem with the answer = Mobius mu function ? R[sqrt[k]]: 1 / (a + b sqrt(5)) = d + b sqrt(5): how to solve for d and b, and when is it possible to solve? Complex Numbers 1 / (a + b sqrt(-1)) = d + b sqrt(5): how to solve for d and b, and when is it possible to solve? The main theorem of algebra Euler formula: e^ (i*a) = cos(a) + i sin(a) Functions and basic functional analysis: How can we define 2^x for rational x? What is 2^0? What is 0! Plotting functions. Solutions for x2-x-1 = 0 11-x=1 + x + x2+... Binom (1 + x)^n (1 + x)^½ Cracking a function: You have operations: take function value f(x), list all zeros, build new functions: g(x) = 1 / f(x), g(x) = (x - a) * f(x), Logarithm Log2: defined as an inverse function for 2^x. Idea of binary search History of logarithms: big tables with (1 + 0.000001)^n allowing to multiply any two numbers. Manifestation of e = 2.718281828459… in such tables Factorial n!: define using linear approximation between log(n!) and log((n+1)!) All rational points on a unit circle and Pyphagorian triples. Quadratic curves: conics