http://www.oercommons.org/browse/keyword/differentiation daily 1 2000-01-01T12:00+00:00 http://www.oercommons.org/courses/planning-full-days-practicing-fast-facts Jen Saul explains how she keeps class active, engaged, and packed full of learning opportunities. She suggests mapping out your class schedule, keeping students moving, and emphasizing effort. She shows a few ways she engages and differentiates for students with her "fast facts" routine, which helps students with multiplication facts. Mathematics and Statistics 2012-11-01T12:47:10 Course Related Materials http://www.oercommons.org/courses/engaging-the-high-achievers In a multi-grade class of fourth, fifth, and sixth graders, students learn to work and communicate in teams. Through projects and a class structure that supports differentiation, Ms. Ehrke is able to keep students challenged and engaged. Her strategies for differentiation and communication can be used in any classroom. Arts Humanities Mathematics and Statistics Science and Technology Social Sciences 2012-11-01T12:47:09 Course Related Materials http://www.oercommons.org/courses/differentiation-with-chain-rule-self-practice-sheet Use this for class lesson demonstration or let the pupils use it for self learning (drill and practice) Mathematics and Statistics 2012-07-06T21:22:36 Course Related Materials http://www.oercommons.org/courses/differentiation-using-quotient-rule-self-practice-sheet Use this for class lesson demonstration or let the pupils use it for self learning (drill and practice) Mathematics and Statistics 2012-07-06T21:22:36 Course Related Materials http://www.oercommons.org/courses/basic-differentiation-self-practice-sheet Use this for class lesson demonstration or let the pupils use it for self learning (drill and practice) Mathematics and Statistics 2012-07-06T21:22:25 Course Related Materials http://www.oercommons.org/courses/the-idea-of-limit-in-differential-calculus Bringing two points on a parabola sufficiently close to yield the gradient. Mathematics and Statistics 2012-07-05T07:26:49 Course Related Materials http://www.oercommons.org/courses/derivatives-of-sine-and-cosine An illustration of the derivatives of the sine and cosine functions.If consider the numbers are complex numbers, it shows a connection with the Euler's formula. Mathematics and Statistics 2012-07-05T07:26:47 Course Related Materials http://www.oercommons.org/courses/graphing-derivative This applet is to help students to get the idea to sketch Derivatives graph for a given graph. Mathematics and Statistics 2012-06-22T16:47:52 Course Related Materials http://www.oercommons.org/courses/definition-of-derivatives This applet provides the construction to see the definition of Derivatives (as limit of the Newton quotient) in action Mathematics and Statistics 2012-06-22T16:47:29 Course Related Materials http://www.oercommons.org/courses/calculus-i-2 This course begins with a review of algebra specifically designed to help and prepare the student for the study of calculus, and continues with discussion of functions, graphs, limits, continuity, and derivatives. The appendix provides a large collection of reference facts, geometry, and trigonometry that will assist in solving calculus problems long after the course is over. Upon successful completion of this course, the student will be able to: calculate or estimate limits of functions given by formulas, graphs, or tables by using properties of limits and L’hopital’s Rule; state whether a function given by a graph or formula is continuous or differentiable at a given point or on a given interval and justify the answer; calculate average and instantaneous rates of change in context, and state the meaning and units of the derivative for functions given graphically; calculate derivatives of polynomial, rational, common transcendental functions, and implicitly defined functions; apply the ideas and techniques of derivatives to solve maximum and minimum problems and related rate problems, and calculate slopes and rates for function given as parametric equations; find extreme values of modeling functions given by formulas or graphs; predict, construct, and interpret the shapes of graphs; solve equations using Newton’s Method; find linear approximations to functions using differentials; festate in words the meanings of the solutions to applied problems, attaching the appropriate units to an answer; state which parts of a mathematical statement are assumptions, such as hypotheses, and which parts are conclusions. This free course may be completed online at any time. It has been developed through a partnership with the Washington State Board for Community and Technical Colleges; the Saylor Foundation has modified some WSBCTC materials. (Mathematics 005) Mathematics and Statistics 2012-04-16T16:20:41 Course Related Materials http://www.oercommons.org/courses/real-analysis-i This course is designed to introduce the student to the rigorous examination of the real number system and the foundations of calculus. Analysis lies at the heart of the trinity of higher mathematics algebra, analysis, and topology because it is where the other two fields meet. Upon successful completion of this course, the student will be able to: Use set notation and quantifiers correctly in mathematical statements and proofs; Use proof by induction or contradiction when appropriate; Define the rational numbers, the natural numbers, and the real numbers, and understand their relationship to one another; Define the well-ordering principle the completeness/supremum property of the real line, and the Archimedean property; Prove the existence of irrational numbers; Define supremum and infimum; Correctly and fluently manipulate expressions with absolute value and state the triangle inequality; Define and identify injective, surjective, and bijective mappings; Name the various cardinalities of sets and identify the cardinality of a given set; Define Euclidean space and vector space and show that Euclidean space is a vector space; Define the complex numbers and manipulate them algebraically; Write equations for lines and planes in Euclidean space; Define a normed linear space, a norm, and an inner product; Define metric spaces, open sets; define open, closed, and bounded sets; define cluster points; define density; Define convergence of sequences and prove or disprove the convergence of given sequences; Prove and use properties of limits; Prove standard results about closures, intersections, and unions of open and closed sets; Define compactness using both open covers and sequences; State and prove the Heine-Borel Theorem; State the Bolzano-Weierstrass Theorem; State and use the Cantor Finite Intersection Property; Define Cauchy sequence and prove that specific sequences are Cauchy; Define completeness and prove that Euclidean space with the standard metric is complete; Show that convergent sequences are Cauchy; Define limit superior and limit inferior; Define convergence of series using the Cauchy criterion and use the comparison, ratio, and root tests to show convergence of series; Define continuity and state, prove, and use properties of limits of continuous functions, including the fact that continuous functions attain extreme values on compact sets; Define divergence of functions to infinity and use properties of infinite limits; State and prove the intermediate value property; Define uniform continuity and show that given functions are or are not uniformly continuous; Give standard examples of discontinuous functions, such as the Dirichlet function; Define connectedness and identify connected and disconnected sets Construct the Cantor ternary set and state its properties; Distinguish between pointwise and uniform convergence; Prove that if a sequence of continuous functions converges uniformly, their limit is also continuous; Define derivatives of real- and extended-real-valued functions; Compute derivatives using the limit definition and prove basic properties of derivatives; State the Mean Value Theorem and use it in proofs; Construct the Riemann Integral and state its properties; State the Fundamental Theorem of Calculus and use it in proofs; Define pointwise and uniform convergence of series of functions; Use the Weierstrass M-Test to check for uniform convergence of series; Construct Taylor Series and state Taylor's Theorem; Identify necessary and sufficient conditions for term-by-term differentiation of power series. (Mathematics 241) Mathematics and Statistics 2011-11-11T11:22:52 Course Related Materials http://www.oercommons.org/courses/multivariable-calculus-2 Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. This course begins with a fresh look at limits and continuity, moves to derivatives and the process of generalizing them to higher dimensions, and finally examines multiple integrals (integration over regions of space as opposed to intervals). Upon successful completion of this course, the student will be able to: Define and identify vectors; Define and compute dot and cross-products; Solve problems involving the geometry of lines, curves, planes, and surfaces in space; Define and compute velocity and acceleration in space; Define and solve Kepler's Second Law; Define and compute partial derivatives; Define and determine tangent planes and level curves; Define and compute least squares; Define and determine boundaries and infinity; Define and determine differentials and the directional derivative; Define and compute the gradient and the directional derivative; Define, determine, and apply Lagrange multipliers to solve problems; Define and compute partial differential equations; Define and evaluate double integrals; Use rectangular coordinates to solve problems in multivariable calculus; Use polar coordinates to solve problems in multivariable calculus; Use change of variables to evaluate integrals; Define and use vector fields and line integrals to solve problems in multivariable calculus; Define and verify conservative fields and path independence; Define and determine gradient fields and potential functions; Use Green's Theorem to evaluate and solve problems in multivariable calculus; Define flux; Define and evaluate triple integrals; Define and use rectangular coordinates in space; Define and use cylindrical coordinates; Define and use spherical coordinates; Define and correctly manipulate vector fields in space; Evaluate surface integrals and relate them to flux; Use the Divergence Theorem (Gauss' Theorem) to solve problems in multivariable calculus; Define and evaluate line integrals in space; Apply Stokes' Theorem to solve problems in multivariable calculus; Properly apply Maxwell's Equations to solve problems. (Mathematics 103) Mathematics and Statistics 2011-11-11T11:22:52 Course Related Materials http://www.oercommons.org/courses/single-variable-calculus-i This course is designed to introduce the student to the study of Calculus through concrete applications. Upon successful completion of this course, students will be able to: Define and identify functions; Define and identify the domain, range, and graph of a function; Define and identify one-to-one, onto, and linear functions; Analyze and graph transformations of functions, such as shifts and dilations, and compositions of functions; Characterize, compute, and graph inverse functions; Graph and describe exponential and logarithmic functions; Define and calculate limits and one-sided limits; Identify vertical asymptotes; Define continuity and determine whether a function is continuous; State and apply the Intermediate Value Theorem; State the Squeeze Theorem and use it to calculate limits; Calculate limits at infinity and identify horizontal asymptotes; Calculate limits of rational and radical functions; State the epsilon-delta definition of a limit and use it in simple situations to show a limit exists; Draw a diagram to explain the tangent-line problem; State several different versions of the limit definition of the derivative, and use multiple notations for the derivative; Understand the derivative as a rate of change, and give some examples of its application, such as velocity; Calculate simple derivatives using the limit definition; Use the power, product, quotient, and chain rules to calculate derivatives; Use implicit differentiation to find derivatives; Find derivatives of inverse functions; Find derivatives of trigonometric, exponential, logarithmic, and inverse trigonometric functions; Solve problems involving rectilinear motion using derivatives; Solve problems involving related rates; Define local and absolute extrema; Use critical points to find local extrema; Use the first and second derivative tests to find intervals of increase and decrease and to find information about concavity and inflection points; Sketch functions using information from the first and second derivative tests; Use the first and second derivative tests to solve optimization (maximum/minimum value) problems; State and apply Rolle's Theorem and the Mean Value Theorem; Explain the meaning of linear approximations and differentials with a sketch; Use linear approximation to solve problems in applications; State and apply L'Hopital's Rule for indeterminate forms; Explain Newton's method using an illustration; Execute several steps of Newton's method and use it to approximate solutions to a root-finding problem; Define antiderivatives and the indefinite integral; State the properties of the indefinite integral; Relate the definite integral to the initial value problem and the area problem; Set up and calculate a Riemann sum; Estimate the area under a curve numerically using the Midpoint Rule; State the Fundamental Theorem of Calculus and use it to calculate definite integrals; State and apply basic properties of the definite integral; Use substitution to compute definite integrals. (Mathematics 101; See also: Biology 103, Chemistry 003, Computer Science 103, Economics 103, Mechanical Engineering 001) Mathematics and Statistics 2011-11-11T11:22:51 Course Related Materials http://www.oercommons.org/courses/calculus-teacher-s-edition CK-12 Calculus Teacher's Edition covers tips, common errors, enrichment, differentiated instruction and problem solving for teaching CK-12 Calculus Student Edition. The solution guide is available upon request. Dreyfuss, Andrew Narasimhan, Ramesh Prolo, Jared Mathematics and Statistics 2011-10-17T11:11:36 Course Related Materials http://www.oercommons.org/courses/calculus-student-s-edition CK-12 Foundation's Single Variable Calculus FlexBook introduces high school students to the topics covered in the Calculus AB course. Topics include: Limits, Derivatives, and Integration. Mathematics and Statistics 2011-10-17T11:11:36 Course Related Materials http://www.oercommons.org/courses/physics-principles-of-physics-a The first half of this course provides students with the essential tools and skills that are required for dealing successfully with physics at first-year university level. The three broad areas that are covered are (a) mathematical techniques and their relationship with physical phenomena, (b) experimental procedures and (c) communication skills, in particular report writing. The second half of the course covers material similar to that of the first half of PHY1004W. Second semester: Mechanics: vectors, kinematics, dynamics, work, energy power, conservative and non-conservative forces, friction, impulse, momentum, collisions, rotation, rotational dynamics, torque, rotational inertia, rotational energy, angular momentum, static equilibrium, gravitation. Properties of matter: elasticity, elastic moduli, hydrostatics, hydrodynamics. Thermodynamics: temperature, heat, kinetic theory of gases, thermodynamic laws, entropy. UCT PHY1023H. Andy Buffler Angus Morrison Science and Technology 2011-09-16T11:15:50 Course Related Materials http://www.oercommons.org/courses/vector-calculus-2 This is a two-semester course in n-dimensional calculus with a review of the necessary linear algebra. It covers the derivative, the integral, and a variety of applications. An emphasis is made on the coordinate free, vector analysis. Peter Saveliev Mathematics and Statistics 2011-03-23T10:24:29 Course Related Materials http://www.oercommons.org/courses/the-fountain-of-life-from-dolly-to-customized-embryonic-stem-cells-fall-2007 " During development, the genetic content of each cell remains, with a few exceptions, identical to that of the zygote. Most differentiated cells therefore retain all of the genetic information necessary to generate an entire organism. It was through pioneering technology of somatic cell nuclear transfer (SCNT) that this concept was experimentally proven. Only 10 years ago the sheep Dolly was the first mammal to be cloned from an adult organism, demonstrating that the differentiated state of a mammalian cell can be fully reversible to a pluripotent embryonic state. A key conclusion from these experiments was that the difference between pluripotent cells such as embryonic stem (ES) cells and unipotent differentiated cells is solely a consequence of reversible changes. These changes, which have proved to involve reversible alterations to both DNA and to proteins that bind DNA, are known as epigenetic, to distinguish them from genetic alterations to DNA sequence. In this course we will explore such epigenetic changes and study different approaches that can return a differentiated cell to an embryonic state in a process referred to as epigenetic reprogramming, which will ultimately allow generation of patient-specific stem cells and application to regenerative therapy. This course is one of many Advanced Undergraduate Seminars offered by the Biology Department at MIT. These seminars are tailored for students with an interest in using primary research literature to discuss and learn about current biological research in a highly interactive setting. Many instructors of the Advanced Undergraduate Seminars are postdoctoral scientists with a strong interest in teaching." Meissner, Alexander Science and Technology 2010-10-07T04:39:16 Course Related Materials http://www.oercommons.org/courses/space-mathematics-website This Website contains over 200 authentic math problems that cover solar physics, space physics, radiation dosimetry, and the human impacts of space weather. The problems range from pre-algebra to calculus and span the math skills appropriate for grade 8-12 students. The problems are taken from authentic applications of arithmetic, graph analysis, pre-algebra, and algebra. Mathematics and Statistics Science and Technology 2009-10-15T02:24:24 Course Related Materials http://www.oercommons.org/courses/street-fighting-mathematics-january-iap-2008 This course teaches the art of guessing results and solving problems without doing a proof or an exact calculation. Techniques include extreme-cases reasoning, dimensional analysis, successive approximation, discretization, generalization, and pictorial analysis. Applications include mental calculation, solid geometry, musical intervals, logarithms, integration, infinite series, solitaire, and differential equations. (No epsilons or deltas are harmed by taking this course.) This course is offered during the Independent Activities Period (IAP), which is a special 4-week term at MIT that runs from the first week of January until the end of the month. Mahajan, Sanjoy Mathematics and Statistics 2009-05-01T07:38:22 Course Related Materials