Although this task is quite straightforward, it has a couple of aspects …
Although this task is quite straightforward, it has a couple of aspects designed to encourage students to attend to the structure of the equation and the meaning of the variables in it. It fosters flexibility in seeing the same equation in two different ways, and it requires students to attend to the meaning of the variables in the preamble and extract the values from the descriptions.
The course consists of a sampling of topics from algebraic combinatorics. The …
The course consists of a sampling of topics from algebraic combinatorics. The topics include the matrix-tree theorem and other applications of linear algebra, applications of commutative and exterior algebra to counting faces of simplicial complexes, and applications of algebra to tilings.
The main aims of this seminar will be to go over the …
The main aims of this seminar will be to go over the classification of surfaces (Enriques-Castelnuovo for characteristic zero, Bombieri-Mumford for characteristic p), while working out plenty of examples, and treating their geometry and arithmetic as far as possible.
Topics vary from year to year. Fall Term: Numerical properties and vanish …
Topics vary from year to year. Fall Term: Numerical properties and vanish theorems for ample, nef, and big line bundles and vector bundles; multiplier ideals and their applications
This course provides an introduction to algebraic number theory. Topics covered include …
This course provides an introduction to algebraic number theory. Topics covered include dedekind domains, unique factorization of prime ideals, number fields, splitting of primes, class group, lattice methods, finiteness of the class number, Dirichlet's units theorem, local fields, ramification, discriminants.
Content varies from year to year. Introduces new and significant developments in …
Content varies from year to year. Introduces new and significant developments in algebraic topology with the focus on homotopy theory and related areas. Spring 2003: An introduction to higher algebraic K-theory.
In this activity, students explore how trebuchets were used during the Middle …
In this activity, students explore how trebuchets were used during the Middle Ages to launch projectiles over or through castle walls as well as how they are used today in events such as Punkin’ Chunkin’. Students work as teams of engineers and research how to design and build their own trebuchets from scratch while following a select number of constraints. They test their trebuchets, evaluate their results through several quantitative analyses, and present their results and design process to the class.
The purpose of this task is to emphasize the adjective "geometric" in …
The purpose of this task is to emphasize the adjective "geometric" in the "geometric" series, namely, that the algebraic notion of a common ratio between terms corresponds to the geometric notion of a repeated similarity transformation.
In this task students must investigate this conjecture to discover that it …
In this task students must investigate this conjecture to discover that it does not work in all cases: Pick any two integers. Look at the sum of their squares, the difference of their squares, and twice the product of the two integers you chose. Those three numbers are the sides of a right triangle.
Prepared with pre-algebra or algebra 1 classes in mind, this module leads …
Prepared with pre-algebra or algebra 1 classes in mind, this module leads students through the process of graphing data and finding a line of best fit while exploring the characteristics of linear equations in algebraic and graphic formats. Then, these topics are connected to real-world experiences in which people use linear functions. During the module, students use these scientific concepts to solve the following hypothetical challenge: You are a new researcher in a lab, and your boss has just given you your first task to analyze a set of data. It being your first assignment, you ask an undergraduate student working in your lab to help you figure it out. She responds that you must determine what the data represents and then find an equation that models the data. You believe that you will be able to determine what the data represents on your own, but you ask for further help modeling the data. In response, she says she is not completely sure how to do it, but gives a list of equations that may fit the data. This module is built around the legacy cycle, a format that incorporates educational research feindings on how people best learn.
This set of problems for Linear Algebra in the open-source WeBWorK mathematics …
This set of problems for Linear Algebra in the open-source WeBWorK mathematics platform was created under a Round Eleven Mini-Grant for Ancillary Materials Creation. The problems were created for an implementation of the CC-BY Lyrix open textbook A First Course in Linear Algebra. Also included as an additional file are the selected and modified Lyryx Class Notes for the textbook.
Topics covered include:
Linear Independence Linear Transformations Matrix of a Transformation Isomorphisms Eigenvalues and Eigenvectors Diagonalization Orthogonality
The principal purpose of the task is to explore a real-world application …
The principal purpose of the task is to explore a real-world application problem with algebra, working with units and maintaining reasonable levels of accuracy throughout.
Students act as food science engineers as they explore and apply their …
Students act as food science engineers as they explore and apply their understanding of cooling rate and specific heat capacity by completing two separate, but interconnected, tasks. In Part 1, student groups conduct an experiment to explore the cooling rate of a cup of hot chocolate. They collect and graph data to create a mathematical model that represents the cooling rate, and use an exponential decay regression to determine how long a person should wait to drink the cup of hot chocolate at an optimal temperature. In Part 2, students investigate the specific heat capacity of the hot chocolate. They determine how much energy is needed to heat the hot chocolate to an optimal temperature after it has cooled to room temperature. Two activity-guiding worksheets are included.
The purpose of this task is to give students practice writing a …
The purpose of this task is to give students practice writing a constraint equation for a given context. Instruction accompanying this task should introduce the notion of a constraint equation as an equation governing the possible values of the variables in question.
This task looks at zeroes and factorization of a general polynomial. It …
This task looks at zeroes and factorization of a general polynomial. It is related to a very deep theorem in mathematics, the Fundamental Theorem of Algebra, which says that a polynomial of degree d always has exactly d roots, provided complex numbers are allowed as roots and provided roots are counted with the proper "multiplicity.''
The intention of this task is to provide extra depth to the …
The intention of this task is to provide extra depth to the standard A-APR.2 it is principally designed for instructional purposes only. The students may use graphing technology: the focus, however, should be on what happens to the function g when x=0 and the calculator may or may not be of help here (depending on how sophisticated it is!).
For a polynomial function p, a real number r is a root …
For a polynomial function p, a real number r is a root of p if and only if p(x) is evenly divisible by x_r. This fact leads to one of the important properties of polynomial functions: a polynomial of degree d can have at most d roots. This is the first of a sequence of problems aiming at showing this fact.
This task continues ``Zeroes and factorization of a quadratic polynomial I.'' The …
This task continues ``Zeroes and factorization of a quadratic polynomial I.'' The argument here generalizes, as shown in ``Zeroes and factorization of a general polynomial'' to show that a polynomial of degree d can have at most d roots. This task is intended for instructional purposes to help students see more clearly the link between factorization of polynomials and zeroes of polynomial functions.
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