Georgia HomeroomHM.LMIR.6Late Modern Math Foundations
Engage in the mathematical and cultural accomplishments of the world’s societies in the late 17th century through the early 20th century in order to grasp the foundational aspects of modern mathematics.
Appears in: Mathematics, High School
Expectations
The origins of calculus
HM.LMIR.6.1Determine tangents to quadratic curves using the algebraic techniques of Fermat, Barrow, and Newton.
HM.LMIR.6.2Describe the influence the French Revolution had on mathematics education.
Non-Euclidean geometry
HM.LMIR.6.3Prove that the summit angles of an isosceles birectangle are congruent, but that it is impossible to prove they are right without referring to the parallel postulate or one of its consequences.
HM.LMIR.6.4Compare and contrast the hypotheses of the acute angle (Hyperbolic), the right angle (Euclidean), and the obtuse angle (Spherical).
HM.LMIR.6.5Prove that under the hypothesis of the acute angle, similarity implies congruence.
HM.LMIR.6.6Describe the societal factors that inhibited the development of non-Euclidean geometry.
Abstract algebra and number theory
HM.LMIR.6.7Add, subtract, and multiply two quaternions.
HM.LMIR.6.8Investigate abstract algebra and group-theoretic concepts.
HM.LMIR.6.9Identify whether a given set with a binary operation is a group.
HM.LMIR.6.10Explain how the ancient Greek pattern of material axiomatics evolved into abstract axiomatics.
HM.LMIR.6.11Solve simple linear congruences of the form $ax = b\bmod m$.
HM.LMIR.6.12Use Fermat’s Little Theorem and Euler’s Theorem to simplify expressions of the form $a^k\bmod m$.
HM.LMIR.6.13Use Gauss’ Law of Quadratic Reciprocity to determine quadratic residues of two odd primes; i.e., solve quadratic congruences of the form $x^2 = p\bmod q$.
HM.LMIR.6.14Verify that the real primes which can be expressed as the sum of two squares are no longer prime in the field of Gaussian integers.
The nature of mathematicians in the 17th, 18th, and 19th centuries.
HM.LMIR.6.15Describe the mathematical contributions of Newton, Euler, and Gauss.
HM.LMIR.6.16Explore the history of African American mathematicians in the 17th , 18th, and 19th centuries and describe their contributions to mathematics.
HM.LMIR.6.17Explore the history of female mathematicians in the 17th, 18th, and 19th centuries and describe their contributions to mathematics.
Georgia Milestones coverage
No Georgia Milestones blueprint cites this standard.