The goal of this lesson is to introduce students who are interested in human biology and biochemistry to the subtleties of energy metabolism (typically not presented in standard biology and biochemistry textbooks) through the lens of ATP as the primary energy currency of the cell. Avoiding the details of the major pathways of energy production (such as glycolysis, the citric acid cycle, and oxidative phosphorylation), this lesson is focused exclusively on ATP, which is truly the fuel of life. Starting with the discovery and history of ATP, this lesson will walk the students through 8 segments (outlined below) interspersed by 7 in-class challenge questions and activities, to the final step of ATP production by the ATP synthase, an amazing molecular machine. A basic understanding of the components and subcellular organization (e.g. organelles, membranes, etc.) and chemical foundation (e.g. biomolecules, chemical equilibrium, biochemical energetics, etc.) of a eukaryotic cell is a desired prerequisite, but it is not a must. Through interactive in-class activities, this lesson is designed to spark the students’ interest in biochemistry and human biology as a whole, but could serve as an introductory lesson to teaching advanced concepts of metabolism and bioenergetics in high school depending on the local science curriculum. No supplies or materials are needed.
The focus of the course is the concepts and techniques for solving the partial differential equations (PDE) that permeate various scientific disciplines. The emphasis is on nonlinear PDE. Applications include problems from fluid dynamics, electrical and mechanical engineering, materials science, quantum mechanics, etc.
This course asks students to consider the ways in which social theorists, institutional reformers, and political revolutionaries in the 17th through 19th centuries seized upon insights developed in the natural sciences and mathematics to change themselves and the society in which they lived. Students study trials, art, literature and music to understand developments in Europe and its colonies in these two centuries. Covers works by Newton, Locke, Voltaire, Rousseau, Marx, and Darwin.
This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.
This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.
This course focuses on the Great Depression and World War II and how they led to a major reordering of American politics and society. We will examine how ordinary people experienced these crises and how those experiences changed their outlook on politics and the world around them.
This course provides a basic history of American social, economic, and political development from the colonial period through the Civil War. It examines the colonial heritages of Spanish and British America; the American Revolution and its impact; the establishment and growth of the new nation; and the Civil War, its background, character, and impact. Readings include writings of the period by J. Winthrop, T. Paine, T. Jefferson, J. Madison, W. H. Garrison, G. Fitzhugh, H. B. Stowe, and A. Lincoln.
This is a seminar course that explores the history of selected features of the physical environment of urban America. Among the features considered are parks, cemeteries, tenements, suburbs, zoos, skyscrapers, department stores, supermarkets, and amusement parks. The course gives students experience in working with primary documentation sources through its selection of readings and class discussions. Students then have the opportunity to apply this experience by researching their own historical questions and writing a term paper.
This course is a seminar on the history of institutions and institutional change in American cities from roughly 1850 to the present. Among the institutions to be looked at are political machines, police departments, courts, schools, prisons, public authorities, and universities. The focus of the course is on readings and discussions.
Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.
Continues 18.100, in the direction of manifolds and global analysis. Differentiable maps, inverse and implicit function theorems, n-dimensional Riemann integral, change of variables in multiple integrals, manifolds, differential forms, n-dimensional version of Stokes' theorem. 18.901 helpful but not required.
The purpose of this video lesson is to expand the student's knowledge about enzymes by introducing the antioxidant enzymes that are intimately involved in the prevention of cellular damage and eventual slowing of the aging process and prevention of several diseases. Students will learn that natural antioxidant enzymes are manufactured in the body and provide an important defense against free radicals. The topic of free radical action is introduced, covering how they are constantly generated in living cells both by ''accidents of chemistry'' and also by specific metabolic processes.
Laszlo Tisza was Professor of Physics Emeritus at MIT, where he began teaching in 1941. This online publication is a reproduction the original lecture notes for the course "Applied Geometric Algebra" taught by Professor Tisza in the Spring of 1976. Over the last 100 years, the mathematical tools employed by physicists have expanded considerably, from differential calculus, vector algebra and geometry, to advanced linear algebra, tensors, Hilbert space, spinors, Group theory and many others. These sophisticated tools provide powerful machinery for describing the physical world, however, their physical interpretation is often not intuitive. These course notes represent Prof. Tisza's attempt at bringing conceptual clarity and unity to the application and interpretation of these advanced mathematical tools. In particular, there is an emphasis on the unifying role that Group theory plays in classical, relativistic, and quantum physics. Prof. Tisza revisits many elementary problems with an advanced treatment in order to help develop the geometrical intuition for the algebraic machinery that may carry over to more advanced problems. The lecture notes came to MIT OpenCourseWare by way of Samuel Gasster, '77 (Course 18), who had taken the course and kept a copy of the lecture notes for his own reference. He dedicated dozens of hours of his own time to convert the typewritten notes into LaTeX files and then publication-ready PDFs. You can read about his motivation for wanting to see these notes published in his Preface below. Professor Tisza kindly gave his permission to make these notes available on MIT OpenCourseWare.
The main objective of this lesson is to illustrate an important application of mathematics in practical life -- namely in art. Most of the pictures selected for this lesson are visible on the walls of Al-Hambra – Granada (Spain), which is one of the most important landmarks in the Islamic civilization. There are three educational goals for this lesson: (1) establishing the concept of isometries; (2) giving real-life examples of groups; (3) demonstrating the importance of matrices and their applications. As background for this lesson, students just need some familiarity with the concept of a group and a limited knowledge about matrices and the inverse of a non-singular matrix.
This learning video deals with a question of geometrical probability. A key idea presented is the fact that a linear equation in three dimensions produces a plane. The video focuses on random triangles that are defined by their three respective angles. These angles are chosen randomly subject to a constraint that they must sum to 180 degrees. An example of the types of in-class activities for between segments of the video is: Ask six students for numbers and make those numbers the coordinates x,y of three points. Then have the class try to figure out how to decide if the triangle with those corners is acute or obtuse.
The purpose of this learning video is to show students how to think more freely about math and science problems. Sometimes getting an approximate answer in a much shorter period of time is well worth the time saved. This video explores techniques for making quick, back-of-the-envelope approximations that are not only surprisingly accurate, but are also illuminating for building intuition in understanding science. This video touches upon 10th-grade level Algebra I and first-year high school physics, but the concepts covered (velocity, distance, mass, etc) are basic enough that science-oriented younger students would understand. If desired, teachers may bring in pendula of various lengths, weights to hang, and a stopwatch to measure period. Examples of in- class exercises for between the video segments include: asking students to estimate 29 x 31 without a calculator or paper and pencil; and asking students how close they can get to a black hole without getting sucked in.
The subject of enumerative combinatorics deals with counting the number of elements of a finite set. For instance, the number of ways to write a positive integer n as a sum of positive integers, taking order into account, is 2n-1. We will be concerned primarily with bijective proofs, i.e., showing that two sets have the same number of elements by exhibiting a bijection (one-to-one correspondence) between them. This is a subject which requires little mathematical background to reach the frontiers of current research. Students will therefore have the opportunity to do original research. It might be necessary to limit enrollment.
The aim of this video is to introduce high school students to the engineering concept of road construction and to the reasons why problems might arise in road construction. Presentation of this concept is made more accessible to students by comparing road construction to the art of baking a layer cake. This simple comparison can serve to emphasize how important it is to follow proper procedures and to use proper materials for successful road construction. The approach used is highly correlated with the common knowledge of baking layer cakes in Malaysia. Students should be able to relate the procedure of baking a layer cake to the importance of following the correct methods of road construction. An understanding of basic statistics is necessary before starting this lesson. This lesson will take almost 60 minutes to complete. During activity breaks, students are required to answer questions and complete assigned tasks related to the subject.
The Art of the Probable" addresses the history of scientific ideas, in particular the emergence and development of mathematical probability. But it is neither meant to be a history of the exact sciences per se nor an annex to, say, the Course 6 curriculum in probability and statistics. Rather, our objective is to focus on the formal, thematic, and rhetorical features that imaginative literature shares with texts in the history of probability. These shared issues include (but are not limited to): the attempt to quantify or otherwise explain the presence of chance, risk, and contingency in everyday life; the deduction of causes for phenomena that are knowable only in their effects; and, above all, the question of what it means to think and act rationally in an uncertain world. Our course therefore aims to broaden students’ appreciation for and understanding of how literature interacts with--both reflecting upon and contributing to--the scientific understanding of the world. We are just as centrally committed to encouraging students to regard imaginative literature as a unique contribution to knowledge in its own right, and to see literary works of art as objects that demand and richly repay close critical analysis. It is our hope that the course will serve students well if they elect to pursue further work in Literature or other discipline in SHASS, and also enrich or complement their understanding of probability and statistics in other scientific and engineering subjects they elect to take.
We will explore images that pertain to the emergence of Japan as a modern state. We will focus on images that depict Japan as it comes into contact with the rest of the world after its long and deep isolation during the feudal period. We will also cover city planning of Tokyo that took place after WWII, and such topics as the 1964 Tokyo Olympics.