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For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
Formative vs summative assessments. Formative assessment, formative evaluation, formative feedback, or assessment for learning, [1] including diagnostic testing, is a range of formal and informal assessment procedures conducted by teachers during the learning process in order to modify teaching and learning activities to improve student attainment.
[2] [3] Historically, the concept of a proof and its associated mathematical rigour first appeared in Greek mathematics , most notably in Euclid 's Elements . [ 4 ] Since its beginning, mathematics was primarily divided into geometry and arithmetic (the manipulation of natural numbers and fractions ), until the 16th and 17th centuries, when ...
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
The multiplicative order of a number a modulo n is the order of a in the multiplicative group whose elements are the residues modulo n of the numbers coprime to n, and whose group operation is multiplication modulo n. This is the group of units of the ring Zn; it has φ (n) elements, φ being Euler's totient function, and is denoted as U (n) or ...
The Cognitive Abilities Test (CogAT) is a group-administered K–12 assessment published by Riverside Insights and intended to estimate students' learned reasoning and problem solving abilities through a battery of verbal, quantitative, and nonverbal test items. The test purports to assess students' acquired reasoning abilities while also ...
In the Whyte notation for the classification of steam locomotive wheel arrangement, a 2-6-6-4 is a locomotive with a two-wheel leading truck, two sets of six driving wheels, and a four-wheel trailing truck. All 2-6-6-4s are simple articulated locomotives. Other equivalent classifications are: UIC classification: (1'C)C2 ' (also known as German ...
Bloom's taxonomy is a framework for categorizing educational goals, developed by a committee of educators chaired by Benjamin Bloom in 1956. It was first introduced in the publication Taxonomy of Educational Objectives: The Classification of Educational Goals. The taxonomy divides learning objectives into three broad domains: cognitive ...