Teaching Fractions According to the Common Core
... in grades 3-7 may be taught. The specific standards that are addressed in this article are listed at the beginning of each grade in san serif front. Students’ learning of fractions may be divided roughly into two stages. In the initial stage — that would be grade 3 and part of grade 4 in the Common ...
... in grades 3-7 may be taught. The specific standards that are addressed in this article are listed at the beginning of each grade in san serif front. Students’ learning of fractions may be divided roughly into two stages. In the initial stage — that would be grade 3 and part of grade 4 in the Common ...
The Goldston-Pintz-Yıldırım sieve and some applications
... course, such gaps are sometimes less than log x, and one might well ask whether it can happen infinitely often that the gap between consecutive primes is much shorter than average. Indeed, in 20051 , D. Goldston, J. Pintz and C. Yıldırım [15] succeeded in proving that, given any > 0, there exist a ...
... course, such gaps are sometimes less than log x, and one might well ask whether it can happen infinitely often that the gap between consecutive primes is much shorter than average. Indeed, in 20051 , D. Goldston, J. Pintz and C. Yıldırım [15] succeeded in proving that, given any > 0, there exist a ...
Collatz conjecture
The Collatz conjecture is a conjecture in mathematics named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanisław Ulam), Kakutani's problem (after Shizuo Kakutani), the Thwaites conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after Helmut Hasse), or the Syracuse problem; the sequence of numbers involved is referred to as the hailstone sequence or hailstone numbers (because the values are usually subject to multiple descents and ascents like hailstones in a cloud), or as wondrous numbers.Take any natural number n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1. Repeat the process (which has been called ""Half Or Triple Plus One"", or HOTPO) indefinitely. The conjecture is that no matter what number you start with, you will always eventually reach 1. The property has also been called oneness.Paul Erdős said about the Collatz conjecture: ""Mathematics may not be ready for such problems."" He also offered $500 for its solution.