(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

"Los estudiantes conectan la aritmética polinomial con los cálculos con números enteros e enteros. Los estudiantes aprenden que la aritmética de las expresiones racionales se rige por las mismas reglas que la aritmética de los números racionales. Esta unidad ayuda a los estudiantes a ver conexiones entre soluciones a ecuaciones polinomiales, ceros de polinomiales,, y gráficos de funciones polinómicas. Las ecuaciones polinomiales se resuelven sobre el conjunto de números complejos, lo que lleva a una comprensión inicial del teorema fundamental del álgebra. Los problemas de aplicación y modelado conectan múltiples representaciones e incluyen situaciones de mundo real y puramente matemáticas.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
"Students connect polynomial arithmetic to computations with whole numbers and integers.  Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers.  This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions.  Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra.  Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

"En este módulo, los estudiantes sintetizan y generalizan lo que han aprendido sobre una variedad de familias de funciones. Extienden el dominio de las funciones exponenciales a toda la línea real (n-rn.a.1) y luego extienden su trabajo con estas funciones a incluir la resolución de ecuaciones exponenciales con logaritmos (F-le.a.4). Exploran (con herramientas apropiadas) los efectos de las transformaciones en gráficos de funciones exponenciales y logarítmicas. Notan que las transformaciones en un gráfico de una función logarítmica se relacionan con el Propiedades logarítmicas (F-BF.B.3). Los estudiantes identifican tipos apropiados de funciones para modelar una situación. Ajustan los parámetros para mejorar el modelo y comparan los modelos analizando la idoneidad del ajuste y las juicios sobre el dominio sobre el cual un modelo es un buen ajuste. La descripción del modelado como, el proceso de elegir y usar matemáticas y estadísticas para analizar situaciones empíricas, comprenderlas mejor y tomar decisiones, está en el corazón de este módulo. En particular, a través de oportunidades repetidas para trabajar a través del ciclo de modelado (consulte la página 61 del CCLS), los estudiantes adquieren la idea de que la misma estructura matemática o estadística a veces puede modelar situaciones aparentemente diferentes.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics ".

English Description:
"In this module, students synthesize and generalize what they have learned about a variety of function families.  They extend the domain of exponential functions to the entire real line (N-RN.A.1) and then extend their work with these functions to include solving exponential equations with logarithms (F-LE.A.4).  They explore (with appropriate tools) the effects of transformations on graphs of exponential and logarithmic functions.  They notice that the transformations on a graph of a logarithmic function relate to the logarithmic properties (F-BF.B.3).  Students identify appropriate types of functions to model a situation.  They adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit.  The description of modeling as, “the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions,” is at the heart of this module.  In particular, through repeated opportunities in working through the modeling cycle (see page 61 of the CCLS), students acquire the insight that the same mathematical or statistical structure can sometimes model seemingly different situations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics."

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

Los estudiantes crean una comprensión formal de la probabilidad, considerando eventos complejos como sindicatos, intersecciones y complementos, así como el concepto de independencia y probabilidad condicional. La idea de usar una curva suave para modelar una distribución de datos se introduce junto con el uso de tablas y tecnología para encontrar áreas bajo una curva normal. Los estudiantes hacen inferencias y justifican conclusiones de encuestas de muestra, experimentos y estudios de observación. Los datos se usan de muestras aleatorias para estimar una media o proporción de población. Los estudiantes calculan el margen de error y lo interpretan en contexto. Dados los datos de un experimento estadístico, los estudiantes usan la simulación para crear una distribución de aleatorización y lo usan para determinar si hay una diferencia significativa entre dos tratamientos.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
Students build a formal understanding of probability, considering complex events such as unions, intersections, and complements as well as the concept of independence and conditional probability.  The idea of using a smooth curve to model a data distribution is introduced along with using tables and technology to find areas under a normal curve.  Students make inferences and justify conclusions from sample surveys, experiments, and observational studies.  Data is used from random samples to estimate a population mean or proportion.  Students calculate margin of error and interpret it in context.  Given data from a statistical experiment, students use simulation to create a randomization distribution and use it to determine if there is a significant difference between two treatments.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This undergraduate level course follows Algebra I. Topics include group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

In this module, students reconnect with and deepen their understanding of statistics and probability concepts first introduced in Grades 6, 7, and 8. Students develop a set of tools for understanding and interpreting variability in data, and begin to make more informed decisions from data. They work with data distributions of various shapes, centers, and spreads. Students build on their experience with bivariate quantitative data from Grade 8. This module sets the stage for more extensive work with sampling and inference in later grades.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En calificaciones anteriores, los estudiantes definen, evalúan y comparan las funciones y las usan para modelar las relaciones entre las cantidades. En este módulo, los estudiantes extienden su estudio de funciones para incluir la notación de la función y los conceptos de dominio y rango. Exploran muchos ejemplos de funciones y sus gráficos, centrándose en el contraste entre las funciones lineales y exponenciales. Interpretan funciones dadas gráfica, numérica, simbólica y verbalmente; traducir entre representaciones; y comprender las limitaciones de varias representaciones.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

(Nota: Esta es una traducción de un recurso educativo abierto creado por el Departamento de Educación del Estado de Nueva York (NYSED) como parte del proyecto "EngageNY" en 2013. Aunque el recurso real fue traducido por personas, la siguiente descripción se tradujo del inglés original usando Google Translate para ayudar a los usuarios potenciales a decidir si se adapta a sus necesidades y puede contener errores gramaticales o lingüísticos. La descripción original en inglés también se proporciona a continuación.)

En módulos anteriores, los estudiantes analizan el proceso de resolver ecuaciones y desarrollar fluidez en la escritura, interpretación y traducción entre varias formas de ecuaciones lineales (Módulo 1) y funciones lineales y exponenciales (Módulo 3). Estas experiencias combinadas con el modelado con datos (Módulo 2), preparan el escenario para el módulo 4. Aquí los estudiantes continúan interpretando expresiones, crean ecuaciones, reescriben ecuaciones y funciones en formas diferentes pero equivalentes, y gráficos e interpretan funciones, pero esta vez utilizando polinomial funciones y funciones más específicamente cuadráticas, así como funciones de raíz de raíz cuadrada y de cubos.

Encuentre el resto de los recursos matemáticos de Engageny en https://archive.org/details/engageny-mathematics.

English Description:
In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

Find the rest of the EngageNY Mathematics resources at https://archive.org/details/engageny-mathematics.

This lesson is about trying to get students to make connections between ideas about equations, inequalities, and expressions. The lesson is designed to give students opportunities to use mathematical vocabulary for a purpose to describe, discuss, and work with these symbol strings.The idea is for students to start gathering global information by looking at the whole number string rather than thinking only about individual procedures or steps. Hopefully students will begin to see the symbol strings as mathematical objects with their own unique set of attributes. (7th Grade Math)

" This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry."

This research-oriented course will focus on algebraic and computational techniques for optimization problems involving polynomial equations and inequalities with particular emphasis on the connections with semidefinite optimization. The course will develop in a parallel fashion several algebraic and numerical approaches to polynomial systems, with a view towards methods that simultaneously incorporate both elements. We will study both the complex and real cases, developing techniques of general applicability, and stressing convexity-based ideas, complexity results, and efficient implementations. Although we will use examples from several engineering areas, particular emphasis will be given to those arising from systems and control applications.

This lesson focuses on the observation of properties and the classification of objects to build ideas about variables. Learners compare buttons to identify similarities and differences. In addition, learners are introduced to Venn Diagrams by reading "The Elves and the Shoemaker" and sorting their shoes in a giant diagram.

In this game activity students practice comparing shapes and naming something that is alike or different about them.

This unit consists of four lessons in which students explore several meanings and representations of multiplication, including number lines, sets, arrays, and balance beams. They also learn about the commutative property of multiplication, the results of multiplying by 1 and by 0, and the inverse property of multiplication. Students write story problems and create pictographs. The unit includes activity sheets, assessment ideas, links to related applets, reflective questions for students and teachers, extensions and a bibliography of children's literature with a multiplication focus.

Students learn about linear programming (also called linear optimization) to solve engineering design problems. As they work through a word problem as a class, they learn about the ideas of constraints, feasibility and optimization related to graphing linear equalities. Then they apply this information to solve two practice engineering design problems related to optimizing materials and cost by graphing inequalities, determining coordinates and equations from their graphs, and solving their equations. It is suggested that students conduct the associated activity, Optimizing Pencils in a Tray, before this lesson, although either order is acceptable.

Acting as engineering teams, students take measurements and make calculations to determine the specific strength of various alloys and then report their data to the rest of the class. Using this class data, students write data-based recommendations to NASA regarding the best alloy to use in the construction of the engine and engine turbines for the Space Launch System that will eventually be used to transport astronauts to Mars.

In this data analysis activity students investigate data in connection with recyclable materials and develop plans to help the environment. Students collect data about aluminum can usage and graph that data in a line plot. The lesson includes student worksheet and extension suggestions.

In this 5-lesson unit primary students identify attributes of buttons and other objects in their classrooms. Throughout the unit students learn to collect data about objects, display the data (Venn diagrams, pictographs, and bar graphs), and analyze the data to determine what this tells us about buttons and/or other objects. Each of the five lesson plans includes necessary student/class worksheets, questions for students, extension ideas, and assessment ideas. Lessons 2 and 3 reference stories that are an important part of the skill development: "The Button Box" by Margarette Reid and "The Elves and the Shoemaker" story.

Analysis I covers fundamentals of mathematical analysis: metric spaces, convergence of sequences and series, continuity, differentiability, Riemann integral, sequences and series of functions, uniformity, interchange of limit operations.