In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.

This lesson unit is intended to help teachers assess how well students are able to interpret exponential and linear functions and in particular to identify and help students who have the following difficulties: translating between descriptive, algebraic and tabular data, and graphical representation of the functions; recognizing how, and why, a quantity changes per unit intervale; and to achieve these goals students work on simple and compound interest problems.

The foundation of this lesson is constructing, communicating, and evaluating student-generated tables while making comparisons between three different financial plans. Students are given three different DVD rental plans and asked to analyze each one to see if they could determine when the 3 different DVD plans cost the same amount of money, if ever. (7th/8th Grade Math)

This task asks students to perform computations involving complex numbers using the given information.

A brief refresher on the Cartesian plane includes how points are written in (x, y) format and oriented to the axes, and which directions are positive and negative. Then students learn about what it means for a relation to be a function and how to determine domain and range of a set of data points.

This task presents a real world application of finite geometric series. The context can lead into several interesting follow-up questions and projects. Many drugs only become effective after the amount in the body builds up to a certain level. This can be modeled very well with geometric series.

This unit of four lessons highlights different aspects of students’ understanding and use of patterns as they analyze relationships and make predictions, as discussed in the Algebra Standard. In this cluster of activities, students use two interactive math applets (both catalogued separately) to learn about repeating and growing patterns. In the first part, students explore a two-square pattern unit and in the second part, students investigate repeating patterns with pattern units of three, four, and five squares. In Part 3, students analyze repeating patterns of colored cubes and lastly in Part 4, students create growing patterns of colored cubes and compare them to repeating patterns.

This lesson plan introduces the game Deep Sea Duel, which develops students' operation skills and strategic thinking, and can be played online or with cards. After playing several variations of the game, students attempt to identify a winning strategy and compare the game to other familiar games. Variations include whole numbers, decimals, fractions, exponents, and words. The lesson includes printable cards and a student worksheet, questions for student discussion and teacher reflection, assessment options, and extensions. The online game and the cited article are cataloged separately.

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.

This task does not actually require that the student solve the system but that they recognize the pairs of linear equations in two variables that would be used to solve the system. This is an important step in the process of solving systems.

This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols.

How many dots will be on the screen after the last bell? Through the use of videos and a couple of helpful clues, students are challenged to use counting, patterns, and algebraic thinking to determine how many dots (and even what colors!) will be on the screen (covered in the first Act) after the last bell.

" Double affine Hecke algebras (DAHA), also called Cherednik algebras, and their representations appear in many contexts: integrable systems (Calogero-Moser and Ruijsenaars models), algebraic geometry (Hilbert schemes), orthogonal polynomials, Lie theory, quantum groups, etc. In this course we will review the basic theory of DAHA and their representations, emphasizing their connections with other subjects and open problems."

This lesson unit is intended to help assess how well students are able to interpret and use scale drawings to plan a garden layout. This involves using proportional reasoning and metric units.

Elementary Algebra is a work text that covers the traditional topics studied in a modern elementary algebra course. It is intended for students who (1) have no exposure to elementary algebra, (2) have previously had an unpleasant experience with elementary algebra, or (3) need to review algebraic concepts and techniques.

This text respects the traditional approaches to algebra pedagogy while enhancing it with the technology available today. In addition, textual notation is introduced as a means to communicate solutions electronically throughout the text. While it is important to obtain the skills to solve problems correctly, it is just as important to communicate those solutions with others effectively in the modern era of instant communications.While algebra is one of the most diversely applied subjects, students often find it to be one of the more difficult hurdles in their education. With this in mind, John wrote Elementary Algebra from the ground up in an open and modular format, allowing the instructor to modify it and leverage their individual expertise as a means to maximize the student experience and success. Elementary Algebra takes the best of the traditional, practice-driven algebra texts and combines it with modern amenities to influence learning, like online/inline video solutions, as well as, other media driven features that only a free online text can deliver.

Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. y=bx ) to see how they add to generate the polynomial curve.

Learn about graphing polynomials. The shape of the curve changes as the constants are adjusted. View the curves for the individual terms (e.g. y=bx ) to see how they add to generate the polynomial curve.

This task asks students to use inverse operations to solve the equations for the unknown variable, or for the designated variable if there is more than one. Two of the equations are of physical significance and are examples of Ohm's Law and Newton's Law of Universal Gravitation.

This is a standard problem phrased in a non-standard way. Rather than asking students to perform an operation, expanding, it expects them to choose the operation for themselves in response to a question about structure. The problem aligns with A-SSE.2 because it requires students to see the factored form as a product of sums, to which the distributive law can be applied.