courses are aligned Example of good learning objectives aligned with the common core state standards. What makes this a good learning objective: Sample Learning objective: “State theorem” implies memorization and recalling what you have memorized “Prove theorem” this is when you apply knowledge “Apply theorem” solving a problem by applying knowledge This objective is good because it is concise and accessible Common Pitfalls in planning effective lessons/Ways to avoid these pitfalls:
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. . . . . . . . . . . . . . . . . . . . . 1.3.3 Prime numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 The Euclidean algorithm . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The fundamental theorem of arithmetic: the unique factorization theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 1 3 3 5 7 7 8 9 10 11 11 12 12 14 15 17 2 Fields 2.1 Introduction to fields . . . . . .
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a2+b2=c2 is the formula for the Pythagorean Theorem. Let a=x‚ b=2x+4‚ c=2x+6 x2+ (2x+4)2=(2x+6)2 The binomials are plugged into the Pythagorean Theorem. The binomials are squared. x2+4x2+8x+16=4x2+12x+36 Notice the 4x2on both sides of the equation‚ I will
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Gerhard Gentzen‚ and others provided partial resolution to the program‚ and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets‚ although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in reverse mathematics)
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for m et n‚ we have [pic] (3) for [pic]‚ we have [pic] (4) for [pic]‚ we have [pic] Using the above formulas‚ we can easily deduce the following result: Theorem. Let [pic] We have [pic] This theorem helps associate a Fourier series to any [pic]-periodic function. Definition. Let f(x) be a [pic]-periodic function which is integrable on [pic]. Set [pic] The trigonometric series [pic] is called the Fourier
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Economics 201A: Economic Theory (first half ) Tu-Th 12:30–2:00 150 GSPP 1 Description Economics 201A is the first semester of the required microeconomic theory sequence for first-year Ph.D. students in the economics department. The first half of the fall semester focuses on choice theory‚ consumer theory‚ and social choice. The second half will be taught by Chris Shannon and will cover general equilibrium. (A separate syllabus will be distributed for the second half.) In the spring‚ the
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Twice the Angle - Circle Theorems 3: Angle at the Centre Theorem Definitions An arc of a circle is a contiguous (i.e. no gaps) portion of the circumference. An arc which is half of a circle is called a semi-circle. An arc which is shorter than a semi-circle is called a minor arc. An arc which is greater than a semi-circle is called a major arc. Clearly‚ for every minor arc there is a corresponding major arc. A segment of a circle is a figure bounded by an arc and its chord. If the arc is a minor
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should get the answer. (√3 – i)-10 We are using De Moivre’s Theorem to solve this problem. De Moivre’s Theorem: If z=r(cosθ + i sinθ)‚ then for any integer n‚ zn=rn(cos(nθ) + i sin(nθ)). So ‚ we have z = √3 – i‚ and we would like to evaluate z-10 = (√3 – i)-10. First‚ we need to express z = (√3 – i) into polar form. r = √(〖(√3)〗^2+1^2 )=2 tanθ = -1/√3 θ = 5π/6 So‚ z=2(cos(5π/6) + i sin(5π/6)) Apply De Moivre’s Theorem‚ z-10 = (√3 – i)-10 =2-10 (cos(10*5π/6) + i sin(10*5π/6)) = .
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Elements of Information Theory Thomas M. Cover‚ Joy A. Thomas Copyright 1991 John Wiley & Sons‚ Inc. Print ISBN 0-471-06259-6 Online ISBN 0-471-20061-1 Elements of Information Theory Elements of Information Theory Thomas M. Cover‚ Joy A. Thomas Copyright 1991 John Wiley & Sons‚ Inc. Print ISBN 0-471-06259-6 Online ISBN 0-471-20061-1 WILEY SERIES IN TELECOMMUNICATIONS Donald L. Schilling‚ Editor City College of New York Digital Telephony‚ 2nd Edition John Bellamy Elements of
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Enrollment and the “Third Law of Demand” A theorem proposed by Professors Alchian and Allen in their text‚ University Economics (1964) has had several rebirths of interest in the literature. The so-called “third law of demand‚” or “relative price theorem‚” holds that a fixed cost added to a good of varying quality causes the consumer to prefer the category considered of higher quality to the lower. Recently a number of studies‚ keeping this theorem in mind have looked into a relationship between
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