Sal discusses the conditions of matrix dimensions for which addition or multiplication are defined. Created by Sal Khan.
The primary purpose of this task is to illustrate certain aspects of the mathematics described in the A.SSE.1. The task has students look for structure in algebraic expressions related to a context, and asks them to relate that structure to the context. In particular, it is worth emphasizing that the task requires no algebraic manipulation from the students.
Students learn about the mathematical characteristics and reflective property of ellipses by building their own elliptical-shaped pool tables. After a slide presentation introduction to ellipses, student “engineering teams” follow the steps of the engineering design process to develop prototypes, which they research, plan, sketch, build, test, refine, and then demonstrate, compare and share with the class. Using these tables as models to explore the geometric shape of ellipses, they experience how particles rebound off the curved ellipse sides and what happens if particles travel through the foci. They learn that if a particle travels through one focal point, then it will travel through the second focal point regardless of what direction the particle travels.
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Ana Donevska Todorova Humboldt Universitﾊt zu Berlin Mathematisch-Naturwissenschaftliche Fakultﾊt II Institut fﾟr Mathematik, Didaktik der Mathematik
This dynamic worksheet illustrates the relationship between the DETERMINANT, the Area of the pre-image, and the Area of the image.
Ana Donevska Todorova Humboldt Universitﾊt zu Berlin Mathematisch-Naturwissenschaftliche Fakultﾊt II Institut fﾟr Mathematik, Didaktik der Mathematik
Developing the Concept of a Determinant using DGS
This site has two PreCalculus Trigonometry textbooks and a PreCalculus Essentials textbook
Differential Equations are the language in which the laws of nature are expressed. Understanding properties of solutions of differential equations is fundamental to much of contemporary science and engineering. Ordinary differential equations (ODE’s) deal with functions of one variable, which can often be thought of as time.
The laws of nature are expressed as differential equations. Scientists and engineers must know how to model the world in terms of differential equations, and how to solve those equations and interpret the solutions. This course focuses on the equations and techniques most useful in science and engineering.
Course Format
This course has been designed for independent study. It provides everything you will need to understand the concepts covered in the course. The materials include:
Lecture Videos by Professor Arthur Mattuck.
Course Notes on every topic.
Practice Problems with Solutions.
Problem Solving Videos taught by experienced MIT Recitation Instructors.
Problem Sets to do on your own with Solutions to check your answers against when you’re done.
A selection of Interactive Java® Demonstrations called Mathlets to illustrate key concepts.
A full set of Exams with Solutions, including practice exams to help you prepare.
Content Development
Haynes Miller
Jeremy Orloff
Dr. John Lewis
Arthur Mattuck
This 10-minute video lesson provides an introduction to 2nd order, linear, homogeneous differential equations with constant coefficients. [Differential Equations playlist: Lesson 13 of 45]
This 8-minute video lesson continues the discussion from the previous video and shows how to find the general solution. [Differential Equations playlist: Lesson 14 of 45]
This 6-minute video lesson continues the discussion from the previous videos and introduces some initial conditions to solve for the particular solution. [Differential Equations playlist: Lesson 15 of 45]
This 9-minute video lesson continues the discussion from the previous videos and provides another example using initial conditions. [Differential Equations playlist: Lesson 16 of 45]
This 10-minute video lesson looks at what happens when the characteristic equations has complex roots. [Differential Equations playlist: Lesson 17 of 45]
This 10-minute video lesson continues the discussion of what happens when the characteristic equation has complex roots. [Differential Equations playlist: Lesson 18 of 45]
This 10-minute video lesson concludes the series on complex roots of characteristic equations and demonstrates an example with initial conditions. [Differential Equations playlist: Lesson 19 of 45]
This 18-minute video lesson provides an introduction to the Dirac Delta Function. [Differential Equations playlist: Lesson 41 of 45]
This 12-minute video lesson gives the first example of solving an exact differential equation. [Differential Equations playlist: Lesson 6 of 45]