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Equal Differences Over Equal Intervals 1
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An important property of linear functions is that they grow by equal ... More

An important property of linear functions is that they grow by equal differences over equal intervals. In this task students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope. In F.LE Equal Differences over Equal Intervals 2, students prove the property in general (for equal intervals of any length). Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
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Equal Differences Over Equal Intervals 2
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An important property of linear functions is that they grow by equal ... More

An important property of linear functions is that they grow by equal differences over equal intervals. In this task students prove this for equal intervals of length one unit, and note that in this case the equal differences have the same value as the slope. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
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Equal Factors Over Equal Intervals
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In this task students prove that linear functions grow by equal differences ... More

In this task students prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Exploring Linear Equations
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This multimedia mathematics resource shows how math is used at the Calgary ... More

This multimedia mathematics resource shows how math is used at the Calgary Zoo to calculate how much it costs to feed the animals. An interactive activity allows students to change variables in linear equations to create unique ways of obtaining the same solution. A print activity is provided. Less

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Subject:
Mathematics
Algebra
Material Type:
Instructional Material
Interactive
Provider:
NSDL Staff
Provider Set:
Key Concepts in Algebra
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Exponential Functions
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In this task students prove that linear functions grow by equal differences ... More

In this task students prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. Less

More
Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Exponential Growth Versus Linear Growth I
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This problem illustrates how an exponentially increasing quantity eventually surpasses a linearly ... More

This problem illustrates how an exponentially increasing quantity eventually surpasses a linearly increasing quantity. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Exponential Growth Versus Linear Growth Ii
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In this task students observe using graphs and tables that a quantity ... More

In this task students observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Exponential Growth Versus Polynomial Growth
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This problem shows that an exponential function takes larger values than a ... More

This problem shows that an exponential function takes larger values than a cubic polynomial function provided the input is sufficiently large. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Finding Parabolas Through Two Points
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In this task students have the opportunity to construct linear and exponential ... More

In this task students have the opportunity to construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Gone Fishing: My, My Little Fish, How You've Grown!
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Opening with a cartoon showing the weights of three combinations of fish, ... More

Opening with a cartoon showing the weights of three combinations of fish, this activity challenges students to determine the weight of each fish. This activity is part of the Figure This! collection of challenges emphasizing real-world uses of mathematics. The introduction discusses algebraic reasoning and notes its importance to scientists, engineers, and psychologists. Students are encouraged to begin by adding the weights on all three scales. The answer page describes three strategies for solving the problem. Related questions invite students to use the strategies to solve similar problems. Answers to all questions and links to resources are included. Less

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Subject:
Education
Mathematics
Algebra
Material Type:
Activities and Labs
Homework and Assignments
Images and Illustrations
Instructional Material
Lesson Plans
Provider:
National Council of Teachers of Mathematics (NCTM)
Ohio State University College of Education and Human Ecology
Provider Set:
Figure This!
Middle School Portal: Math and Science Pathways (MSP2)
Author:
National Council of Teachers of Mathematics (NCTM)
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Homerun Hoopla
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This lesson is designed for students to gather and analyze data about ... More

This lesson is designed for students to gather and analyze data about baseball figures. The student will use the Internet or other resources to collect statistical data on the top five home run hitters for the current season as well as their career home run totals. The students will graph the data and determine if it is linear or non-linear. Less

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Subject:
Material Type:
Lesson Plans
Provider:
University of North Carolina at Chapel Hill School of Education
Provider Set:
LEARN NC Lesson Plans
Author:
Anne Walters
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Identifying Functions
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This task emphasizes the expectation that students know linear functions grow by ... More

This task emphasizes the expectation that students know linear functions grow by constant differences over equal intervals and exponential functions grow by constant factors over equal intervals. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
In the Billions and Exponential Modeling
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This problem provides an opportunity to experiment with modeling real data. Populations ... More

This problem provides an opportunity to experiment with modeling real data. Populations are often modeled with exponential functions and in this particular case we see that, over the last 200 years, the rate of population growth accelerated rapidly, reaching a peak a little after the middle of the 20th century and now it is slowing down. Less

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Subject:
Education
Mathematics
Functions
Material Type:
Activities and Labs
Instructional Material
Provider:
Illustrative Mathematics
Provider Set:
Illustrative Mathematics
Author:
Illustrative Mathematics
Less
Introduction to Partial Differential Equations
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Partial differential equations (PDEs) describe the relationships among the derivatives of an ... More

Partial differential equations (PDEs) describe the relationships among the derivatives of an unknown function with respect to different independent variables, such as time and position. Experiment and observation provide information about the connections between rates of change of an important quantity, such as heat, with respect to different variables. Upon successful completion of this course, the student will be able to: State the heat, wave, Laplace, and Poisson equations and explain their physical origins; Define harmonic functions; State and justify the maximum principle for harmonic functions; State the mean value property for harmonic functions; Define linear operators and identify linear operations; Identify and classify linear PDEs; Identify homogeneous PDEs and evolution equations; Relate solving homogeneous linear PDEs to finding kernels of linear operators; Define boundary value problem and identify boundary conditions as periodic, Dirichlet, Neumann, or Robin (mixed); Explain physical significance of boundary conditions; Show uniqueness of solutions to the heat, wave, Laplace and Poisson equations with various boundary conditions; Define well-posedness; Define, characterize, and use inner products; Define the space of L2 functions, state its key properties, and identify L2 functions; Define orthogonality and orthonormal basis and show the orthogonality of certain trigonometric functions; Distinguish between pointwise, uniform, and L2 convergence and show convergence of Fourier series; Define Fourier series on [0,pi] and [0,L] and identify sufficient conditions for their convergence and uniqueness; Compute Fourier coefficients and construct Fourier series; Use the method of characteristics to solve linear and nonlinear first-order wave equations; Solve the one-dimensional wave equation using d'Alembert's formula; Use similarity methods to solve PDEs; Solve the heat, wave, Laplace, and Poisson equations using separation of variables and apply boundary conditions; Define the delta function and apply ideas from calculus and Fourier series to generalized functions; Derive Green's representation formula; Use Green's functions to solve the Poisson equation on the unit disk; Define the Fourier transform; Derive basic properties of the Fourier transform of a function, such as its relationship to the Fourier transform of the derivative; Show that the inverse Fourier transform of a product is a convolution; Compute Fourier transforms of functions; Use the Fourier transform to solve the heat and wave equations on unbounded domains. (Mathematics 222) Less

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Subject:
Functions
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Provider:
The Saylor Foundation
Provider Set:
Saylor Foundation
Less
Linear Algebra II
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Linear Algebra is both rich in theory and full of interesting applications; ... More

Linear Algebra is both rich in theory and full of interesting applications; in this course the student will try to balance both. This course includes a review of topics learned in Linear Algebra I. Upon successful completion of this course, the student will be able to: Solve systems of linear equations; Define the abstract notions of vector space and inner product space; State examples of vector spaces; Diagonalize a matrix; Formulate what a system of linear equations is in terms of matrices; Give an example of a space that has the Archimedian property; Use the Euclidean algorithm to find the greatest common divisor; Understand polar form and geometric interpretation of the complex numbers; Explain what the fundamental theorem of algebra states; Determine when two matrices are row equivalent; State the Fredholm alternative; Identify matrices that are in row reduced echelon form; Find a LU factorization for a given matrix; Find a PLU factorization for a given matrix; Find a QR factorization for a given matrix; Use the simplex algorithm; Compute eigenvalues and eigenvectors; State Shur's Theorem; Define normal matrices; Explain the composition and the inversion of permutations; Define and compute the determinant; Explain when eigenvalues exist for a given operator; Normal form of a nilpotent operator; Understand the idea of Jordan blocks, Jordan matrices, and the Jordan form of a matrix; Define quadratic forms; State the second derivative test; Define eigenvectors and eigenvalues; Define a vector space and state its properties; State the notions of linear span, linear independence, and the basis of a vector space; Understand the ideas of linear independence, spanning set, basis, and dimension; Define a linear transformation; State the properties of linear transformations; Define the characteristic polynomial of a matrix; Define a Markov matrix; State what it means to have the property of being a stochastic matrix; Define a normed vector space; Apply the Cauchy Schwarz inequality; State the Riesz representation theorem; State what it means for a nxn matrix to be diagonalizable; Define Hermitian operators; Define a Hilbert space; Prove the Cayley Hamilton theorem; Define the adjoint of an operator; Define normal operators; State the spectral theorem; Understand how to find the singular-value decomposition of an operator; Define the notion of length for abstract vectors in abstract vector spaces; Define orthogonal vectors; Define orthogonal and orthonormal subsets of R^n; Use the Gram-Schmidt process; Find the eigenvalues and the eigenvectors of a given matrix numerically; Provide an explicit description of the Power Method. (Mathematics 212) Less

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Subject:
Algebra
Functions
Material Type:
Assessments
Full Course
Homework and Assignments
Readings
Syllabi
Textbooks
Provider:
The Saylor Foundation
Provider Set:
Saylor Foundation
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