The document provides a summary of the US Mathematics Core Standards for middle school grades 6-8. It outlines the core concepts covered in each grade including: numbers and operations; ratios and proportional relationships; expressions and equations; geometry; statistics and probability; and functions. The standards progress from understanding ratios and proportional relationships, working with rational numbers and expressions, to applying concepts like the Pythagorean theorem and working with linear equations and functions.
This document provides a reference guide for the 4th grade Common Core standards addressed by the Investigations curriculum. It lists the domains, clusters, and individual standards covered in each of the 9 units. The domains include Operations and Algebraic Thinking, Number and Operations in Base Ten, Number and Operations - Fractions, Measurement and Data, and Geometry. The guide also includes descriptions of the Mathematical Practices for 4th grade. In 3 sentences or less: This reference guide maps the 4th grade Common Core standards to the Investigations curriculum units, showing which standards are addressed in each unit for the domains of number, operations, measurement, data, geometry and the mathematical practices.
This document provides an overview of sets of numbers and interval notation in algebra. It defines key terms like natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as a ratio of integers, while irrational numbers cannot. The document explains how to classify numbers based on these sets and represents sets of numbers visually on a number line. It introduces inequality symbols like <, >, ≤, ≥ and uses them to compare numbers. Finally, it explains how to use interval notation, like (a, b], [c, d), to represent sets of real numbers algebraically.
This document provides an overview of key concepts in algebra including:
- Evaluating algebraic expressions and using variables, formulas, and mathematical models.
- Foundational concepts of sets such as intersections, unions, and subsets of real numbers.
- Properties and applications of real numbers including the number line, inequalities, absolute value, and distance.
- Simplifying algebraic expressions by combining like terms.
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
The document discusses the distributive property in algebra. It provides definitions of key terms like term, coefficient, and like terms. It gives examples of using the distributive property to simplify expressions and solve problems involving perimeter. The distributive property allows multiplying a number by the sum of two other numbers, distributing the number factor across the addition.
This document discusses methods for solving inequalities, including:
1) Determining if a number is a solution of an inequality.
2) Graphing inequalities on a number line.
3) Using the addition principle and multiplication principle to solve inequalities algebraically, such as isolating the variable.
4) Applying the addition principle and multiplication principle together to solve more complex inequalities.
This document refers to solving linear inequalities and expressing the solutions in interval notation. It provides homework problems from page 136 involving solving linear inequalities for x and expressing the solutions sets in interval notation, with problems 1 through 9 being odd numbered problems and problem 10.
This document provides a reference guide for the 4th grade Common Core standards addressed by the Investigations curriculum. It lists the domains, clusters, and individual standards covered in each of the 9 units. The domains include Operations and Algebraic Thinking, Number and Operations in Base Ten, Number and Operations - Fractions, Measurement and Data, and Geometry. The guide also includes descriptions of the Mathematical Practices for 4th grade. In 3 sentences or less: This reference guide maps the 4th grade Common Core standards to the Investigations curriculum units, showing which standards are addressed in each unit for the domains of number, operations, measurement, data, geometry and the mathematical practices.
This document provides an overview of sets of numbers and interval notation in algebra. It defines key terms like natural numbers, integers, rational numbers, and irrational numbers. Rational numbers can be written as a ratio of integers, while irrational numbers cannot. The document explains how to classify numbers based on these sets and represents sets of numbers visually on a number line. It introduces inequality symbols like <, >, ≤, ≥ and uses them to compare numbers. Finally, it explains how to use interval notation, like (a, b], [c, d), to represent sets of real numbers algebraically.
This document provides an overview of key concepts in algebra including:
- Evaluating algebraic expressions and using variables, formulas, and mathematical models.
- Foundational concepts of sets such as intersections, unions, and subsets of real numbers.
- Properties and applications of real numbers including the number line, inequalities, absolute value, and distance.
- Simplifying algebraic expressions by combining like terms.
This chapter reviews real numbers including:
[1] Classifying numbers as natural numbers, integers, rational numbers, irrational numbers, and real numbers. Rational numbers can be written as fractions while irrational numbers cannot.
[2] Approximating irrational numbers like π as decimals to a given number of decimal places by rounding or truncating.
[3] How calculators handle decimals by either truncating or rounding values based on their display capabilities. A scientific or graphing calculator is recommended for this course.
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
The document discusses the distributive property in algebra. It provides definitions of key terms like term, coefficient, and like terms. It gives examples of using the distributive property to simplify expressions and solve problems involving perimeter. The distributive property allows multiplying a number by the sum of two other numbers, distributing the number factor across the addition.
This document discusses methods for solving inequalities, including:
1) Determining if a number is a solution of an inequality.
2) Graphing inequalities on a number line.
3) Using the addition principle and multiplication principle to solve inequalities algebraically, such as isolating the variable.
4) Applying the addition principle and multiplication principle together to solve more complex inequalities.
This document refers to solving linear inequalities and expressing the solutions in interval notation. It provides homework problems from page 136 involving solving linear inequalities for x and expressing the solutions sets in interval notation, with problems 1 through 9 being odd numbered problems and problem 10.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
This document provides a suggested list of mathematical language terms for grade 6. It includes terms related to problem solving, reasoning and proof, communication, connections, representation, number sense and operations, algebra, geometry, measurement, and statistics and probability. The list contains over 150 mathematical terms organized into these conceptual categories.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
The document discusses Godel's incompleteness theorems and Hilbert's program. It provides background on key figures like Hilbert, Godel, Russell and Cantor. It then explains Hilbert's program to formalize all of mathematics and prove its consistency. Godel showed that any theory capable of elementary arithmetic cannot be both consistent and complete. Specifically, for any formal theory T including basic arithmetic truths, T includes a statement of its own consistency if and only if T is inconsistent.
The document provides a list of tier 2 math vocabulary words that 7th grade students should know for their Unit 3 on properties of operations and integers. It defines 22 words including additive identity, reciprocal, additive inverse, and perimeter. For each word, it provides a simple definition and example to help explain the meaning in a math context.
This document outlines key concepts and skills related to rational numbers including: representing addition and subtraction on a number line; identifying opposite numbers and describing situations where opposites add to zero; computing absolute value and understanding addition as the sum of a number and its distance from another; rewriting subtraction using additive inverses; applying properties of operations to add and subtract rational numbers; multiplying and dividing rational numbers while interpreting real-world situations; and converting rational numbers to decimals. The overall skills involve representing, comparing, adding, subtracting, multiplying, and dividing rational numbers, and applying these concepts to solve real-world problems.
The document outlines the syllabus for Class IX mathematics for the academic session 2011-2012. It is divided into two semesters. The first semester covers chapters on number systems, polynomials, coordinate geometry, introduction to Euclid's geometry, lines and angles, triangles, and Heron's formula. The second semester covers chapters on linear equations in two variables, quadrilaterals, areas of parallelograms and triangles, circles, constructions, statistics, probability, and surface areas and volumes. Mental maths practice is scheduled every Monday based on the concerned topic. Related activities are provided at the end of each chapter.
Darmon Points for fields of mixed signaturemmasdeu
This document discusses Darmon points for fields of mixed signature. It begins by reviewing the history of constructing Darmon points, both in the non-archimedean and archimedean cases. It then outlines the goals of the talk, which are to sketch a general construction of Darmon points, give details of the construction, explain algorithmic challenges, and illustrate with examples. The document provides context on Darmon points and their relation to elliptic curves, number fields, and conjectures like the Birch and Swinnerton-Dyer conjecture.
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
This document provides a lesson on solving equations by addition and subtraction. It begins by introducing key terms like variables, expressions, and equations. It then shows how to solve equations using the addition and subtraction properties of equality. Examples are provided of solving equations step-by-step and translating word problems into mathematical expressions. The document concludes with practice problems for students to assess their understanding of solving equations by addition and subtraction.
1) The document summarizes research into generating conjectures for upper bounds on the domination number of bipartite graphs. 14 conjectures were produced, of which 3 were previously known to be true.
2) For the 11 false conjectures, smallest counterexamples were found to disprove the conjectures. An example of finding a smallest counterexample is shown.
3) The goal of the research was to determine a collection of upper bounds that could accurately predict the domination number of any bipartite graph. Such a collection could help compute domination numbers more efficiently.
1) Biased Normalized Cuts presents a modification of Normalized Cuts that incorporates priors to allow for constrained image segmentation.
2) It seeks solutions that are sufficiently "correlated" with noisy top-down priors, like an object detector, and can be computed quickly given the unconstrained solution.
3) The algorithm constructs a "biased normalized cut vector" that linearly combines eigenvectors such that those correlated with a user-specified seed vector are upweighted while inversely correlated ones have their sign flipped.
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
1. The document contains an unsolved mathematics paper from 1985 containing multiple choice, multiple answer, true/false, and fill in the blank questions.
2. The questions cover topics like complex numbers, vectors, geometry, functions, and series.
3. Answers to the questions are not provided, but the document directs readers to a website for solutions.
We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
Segments and Properties of Real Numbers (Geometry 2_2)rfant
1) The document discusses properties of real numbers related to equality and applying them to measure segments between points on a line.
2) It defines betweenness for three collinear points and shows examples of using subtraction to find distances between points given two point distances.
3) The key properties of equality for real numbers discussed are reflexive, symmetric, transitive, addition/subtraction, multiplication/division, and substitution.
The document summarizes the key areas of focus for 5th grade mathematics based on the Common Core State Standards. It outlines three critical areas of instruction: (1) developing skills with fractions, multiplication of fractions, and limited cases of division of fractions; (2) extending skills with decimal fractions and operations with decimals to hundredths; and (3) developing an understanding of volume. It then provides more details on the learning expectations within each of these critical areas.
This document provides a teaching guide for a 7th grade math lesson on sets. It introduces concepts like well-defined sets, subsets, universal sets, and the null set. Students will use Venn diagrams to represent sets and subsets. The lesson defines terms like union and intersection of sets and teaches students to perform set operations and represent unions and intersections using Venn diagrams.
Gráfico diario del ibex 35 para el 25 08 2015LaboratorioCFDS
Este documento presenta un gráfico diario del índice bursátil IBEX 35 con medias móviles simples de diferentes períodos. Explica que las tendencias se definen por las medias móviles y son niveles de soporte y resistencia. Proporciona previsiones de que el mercado podría dirigirse a la zona de mínimos de 2009 y luego a la zona máxima de 2010 antes de perder los mínimos de 2009, probablemente en 2013. También analiza posibles escenarios del IBEX 35 dependiendo de si recupera o
O Grupo Bruno Miguel Pegado apresenta sua nova marca Pegado, que produz bicicletas, motocicletas e quadriciclos em vários tamanhos de motores para o mercado angolano e mundial. A empresa tem uma loja, armazém, linha de montagem e assistência técnica em Angola e objetiva expandir para outras províncias, oferecendo produtos de qualidade a preços acessíveis e garantia de um ano.
The document is a scanned receipt from a grocery store purchase on June 15th, 2022 totaling $58.37. It lists items bought including ground beef, chicken breasts, tortillas, cheese, and produce such as tomatoes, lettuce, and onions. The receipt shows the item prices, taxes, and total amount due.
This document provides an overview of probabilistic reasoning and uncertainty in knowledge representation. It discusses:
1) Using probability theory to represent uncertainty quantitatively rather than logical rules with certainty factors.
2) Key concepts in probability theory including random variables, probability distributions, joint probabilities, marginal probabilities, and conditional probabilities.
3) Representing a problem domain as a probabilistic model with a sample space of possible variable states.
4) Independence of variables allowing simpler computation of probabilities.
5) The document is an introduction to probabilistic reasoning concepts to be covered in more detail later, including Bayesian networks.
This document provides a suggested list of mathematical language terms for grade 6. It includes terms related to problem solving, reasoning and proof, communication, connections, representation, number sense and operations, algebra, geometry, measurement, and statistics and probability. The list contains over 150 mathematical terms organized into these conceptual categories.
The document discusses the natural logarithm function ln(x) and the natural exponential function exp(x). It begins by defining ln(x) as the area under the curve y=1/t from 1 to x, and noting that its derivative is 1/x. It then defines exp(x) as the inverse of ln(x). It is shown that for rational r, exp(r) = er, and this definition is extended to irrational r. The derivative of exp(x) is then shown to be exp(x) itself.
The document discusses Godel's incompleteness theorems and Hilbert's program. It provides background on key figures like Hilbert, Godel, Russell and Cantor. It then explains Hilbert's program to formalize all of mathematics and prove its consistency. Godel showed that any theory capable of elementary arithmetic cannot be both consistent and complete. Specifically, for any formal theory T including basic arithmetic truths, T includes a statement of its own consistency if and only if T is inconsistent.
The document provides a list of tier 2 math vocabulary words that 7th grade students should know for their Unit 3 on properties of operations and integers. It defines 22 words including additive identity, reciprocal, additive inverse, and perimeter. For each word, it provides a simple definition and example to help explain the meaning in a math context.
This document outlines key concepts and skills related to rational numbers including: representing addition and subtraction on a number line; identifying opposite numbers and describing situations where opposites add to zero; computing absolute value and understanding addition as the sum of a number and its distance from another; rewriting subtraction using additive inverses; applying properties of operations to add and subtract rational numbers; multiplying and dividing rational numbers while interpreting real-world situations; and converting rational numbers to decimals. The overall skills involve representing, comparing, adding, subtracting, multiplying, and dividing rational numbers, and applying these concepts to solve real-world problems.
The document outlines the syllabus for Class IX mathematics for the academic session 2011-2012. It is divided into two semesters. The first semester covers chapters on number systems, polynomials, coordinate geometry, introduction to Euclid's geometry, lines and angles, triangles, and Heron's formula. The second semester covers chapters on linear equations in two variables, quadrilaterals, areas of parallelograms and triangles, circles, constructions, statistics, probability, and surface areas and volumes. Mental maths practice is scheduled every Monday based on the concerned topic. Related activities are provided at the end of each chapter.
Darmon Points for fields of mixed signaturemmasdeu
This document discusses Darmon points for fields of mixed signature. It begins by reviewing the history of constructing Darmon points, both in the non-archimedean and archimedean cases. It then outlines the goals of the talk, which are to sketch a general construction of Darmon points, give details of the construction, explain algorithmic challenges, and illustrate with examples. The document provides context on Darmon points and their relation to elliptic curves, number fields, and conjectures like the Birch and Swinnerton-Dyer conjecture.
Integral Calculus. - Differential Calculus - Integration as an Inverse Process of Differentiation - Methods of Integration - Integration using trigonometric identities - Integrals of Some Particular Functions - rational function - partial fraction - Integration by partial fractions - standard integrals - First and second fundamental theorem of integral calculus
This document provides a lesson on solving equations by addition and subtraction. It begins by introducing key terms like variables, expressions, and equations. It then shows how to solve equations using the addition and subtraction properties of equality. Examples are provided of solving equations step-by-step and translating word problems into mathematical expressions. The document concludes with practice problems for students to assess their understanding of solving equations by addition and subtraction.
1) The document summarizes research into generating conjectures for upper bounds on the domination number of bipartite graphs. 14 conjectures were produced, of which 3 were previously known to be true.
2) For the 11 false conjectures, smallest counterexamples were found to disprove the conjectures. An example of finding a smallest counterexample is shown.
3) The goal of the research was to determine a collection of upper bounds that could accurately predict the domination number of any bipartite graph. Such a collection could help compute domination numbers more efficiently.
1) Biased Normalized Cuts presents a modification of Normalized Cuts that incorporates priors to allow for constrained image segmentation.
2) It seeks solutions that are sufficiently "correlated" with noisy top-down priors, like an object detector, and can be computed quickly given the unconstrained solution.
3) The algorithm constructs a "biased normalized cut vector" that linearly combines eigenvectors such that those correlated with a user-specified seed vector are upweighted while inversely correlated ones have their sign flipped.
This document discusses integral exponents and how to evaluate expressions with zero and negative exponents. It provides examples of simplifying expressions with zero and negative exponents by using the definition that a negative exponent means to take the reciprocal of the base and raise it to the positive value of the exponent. It also explains that any nonzero number raised to the zero power is equal to 1, and expressions should be rewritten with only positive exponents.
1. The document contains an unsolved mathematics paper from 1985 containing multiple choice, multiple answer, true/false, and fill in the blank questions.
2. The questions cover topics like complex numbers, vectors, geometry, functions, and series.
3. Answers to the questions are not provided, but the document directs readers to a website for solutions.
We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
Segments and Properties of Real Numbers (Geometry 2_2)rfant
1) The document discusses properties of real numbers related to equality and applying them to measure segments between points on a line.
2) It defines betweenness for three collinear points and shows examples of using subtraction to find distances between points given two point distances.
3) The key properties of equality for real numbers discussed are reflexive, symmetric, transitive, addition/subtraction, multiplication/division, and substitution.
The document summarizes the key areas of focus for 5th grade mathematics based on the Common Core State Standards. It outlines three critical areas of instruction: (1) developing skills with fractions, multiplication of fractions, and limited cases of division of fractions; (2) extending skills with decimal fractions and operations with decimals to hundredths; and (3) developing an understanding of volume. It then provides more details on the learning expectations within each of these critical areas.
This document provides a teaching guide for a 7th grade math lesson on sets. It introduces concepts like well-defined sets, subsets, universal sets, and the null set. Students will use Venn diagrams to represent sets and subsets. The lesson defines terms like union and intersection of sets and teaches students to perform set operations and represent unions and intersections using Venn diagrams.
Gráfico diario del ibex 35 para el 25 08 2015LaboratorioCFDS
Este documento presenta un gráfico diario del índice bursátil IBEX 35 con medias móviles simples de diferentes períodos. Explica que las tendencias se definen por las medias móviles y son niveles de soporte y resistencia. Proporciona previsiones de que el mercado podría dirigirse a la zona de mínimos de 2009 y luego a la zona máxima de 2010 antes de perder los mínimos de 2009, probablemente en 2013. También analiza posibles escenarios del IBEX 35 dependiendo de si recupera o
O Grupo Bruno Miguel Pegado apresenta sua nova marca Pegado, que produz bicicletas, motocicletas e quadriciclos em vários tamanhos de motores para o mercado angolano e mundial. A empresa tem uma loja, armazém, linha de montagem e assistência técnica em Angola e objetiva expandir para outras províncias, oferecendo produtos de qualidade a preços acessíveis e garantia de um ano.
The document is a scanned receipt from a grocery store purchase on June 15th, 2022 totaling $58.37. It lists items bought including ground beef, chicken breasts, tortillas, cheese, and produce such as tomatoes, lettuce, and onions. The receipt shows the item prices, taxes, and total amount due.
El cronograma del primer semestre para la asignatura de Educación Tecnológica de los cursos 4°ABC incluye seis clases que cubren la presentación y creación de presentaciones en PowerPoint y hojas de cálculo en Excel, así como la evaluación de las actividades realizadas en dichos programas.
This document discusses the design and testing of flexible coaxial cables for use in a cryogenic dark matter detection experiment. Measurements show that a flexible cable transmits more noise than a standard rigid vacuum cable, especially in the 1-10 kHz frequency range important for signals. However, the flexible cable shows similar susceptibility to microphonic noise. Further design improvements could include sealing the graphite coating with varnish and using a vacuum dielectric with plastic spacers to allow flexibility. Removing the bulky copper tower used currently could simplify installation and reduce heat leaks and cooling requirements.
The document is a scanned receipt from a grocery store purchase on January 15th, 2023 for $58.46. It lists the items bought which include milk, eggs, bread, cereal, orange juice, bananas, and ground beef. The payment was made with a credit card ending in 4321.
A b s t r a c t
Throughout many centuries, the musical structure has had numerous modifications. We can observe the constant use of digits for convenience of notation of the music sounds, for example : digital organ bass, lute tablatures, guitar jazz ciphers. At nowadays the digital system of music teaching is absent in the curriculum and is not applied in practice because of teacher's insufficient professional knowledge in the sphere of child's neurophysiology . The findings of our scientific investigations have permit us to understand the most delicate mechanisms of child’s mental activity and to detect new creative abilities.Application of the information technologies will help schoolmasters to improve the quality, speed and efficiency of music teaching for beginners
This document provides an overview of the 5th grade common core standards in mathematics. It lists the domains, clusters, and individual standards. The domains covered include operations and algebraic thinking, number and operations in base ten, number and operations - fractions, measurement and data, and geometry. For each standard, it indicates which unit(s) in the 5th grade Investigations curriculum cover that standard.
This document outlines the aims and content of the IGCSE Mathematics - Additional Standards syllabus. The aims are to consolidate elementary mathematical skills, develop knowledge of mathematical concepts, foster problem solving abilities, and apply mathematics to real-world situations. The content includes set theory, functions, quadratic functions, indices and surds, polynomials, simultaneous equations, logarithmic and exponential functions, straight line graphs, trigonometry, permutations and combinations, binomial expansions, vectors, matrices, differentiation, and integration. Assessment objectives are to recall and apply techniques, interpret mathematical data, comprehend concepts and relationships, and formulate and solve problems.
This document provides an overview of high school math concepts related to rational numbers and fractions, as outlined in the Common Core State Standards. It includes:
1) A breakdown and comparison of specific standards for The Real Number System (N-RN) and Arithmetic with Polynomials and Rational Expressions (A-APR) and their alignment with Washington Performance Expectations.
2) Examples of common student misconceptions related to these standards and potential resources to address them.
3) Sample problems and online lessons for practicing skills such as rewriting expressions with rational exponents, adding and multiplying rational expressions, and creating equations to represent real-world situations.
sets of numbers and interval notation, operation on real numbers, simplifying expression, linear equation in one variable, aplication of linear equation in one variable, linear equation and aplication to geometry, linear inequaloties in one variable, properties of integers exponents and scientific notation
This two semester algebra 1 course covers linear, quadratic, and other foundational algebraic concepts. Students will learn to perform operations with real numbers, simplify expressions, graph and solve equations and inequalities, work with functions and polynomials, and apply algebraic skills to probability and data problems. Assessment is based on completing homework, projects for each unit, and passing unit tests with a proficiency of 70% or higher. The course addresses state standards for algebra 1.
The document provides an overview of operations and algebraic thinking standards from kindergarten through 8th grade. It shows that in the early grades, standards focus on representing numbers, addition, subtraction and basic multiplication/division. In later grades, standards expand the scope of numbers and introduce concepts like ratios, proportions, expressions and patterns. Students are expected to apply mathematical operations to increasingly complex word problems and equations over time.
Algebra 2 Standards Math Draft August 2016.pdfssuserbdee04
The document provides an overview and standards for an Algebra 2 course. It includes 4 sections:
1) An overview of the critical areas of instruction including extending real numbers to complex numbers, solving various equations, analyzing different function types, and conditional probability.
2) Standards for number and quantity, algebra, functions, and statistics/probability.
3) Detailed explanations of the 4 critical areas covering complex numbers, various equation types, diverse function families/models, and conditional probability.
4) Specific mathematics standards within each domain covering topics like complex numbers, polynomials, equations, functions, and data analysis.
This document outlines a 2nd quarter math plan for 8th grade students covering topics in equations, ratios/proportions, percents, graphing linear equations, and collecting/analyzing data. It includes objectives, standards, and assessments for each of the multi-day topics from November through January. Key concepts include solving one-step, two-step, and multi-step equations; writing and solving proportions; graphing linear equations using various methods; finding slope and using slope-intercept form; and performing linear regression and making predictions from data.
This document provides an overview of solving linear equations, formulas, and problem solving techniques. It begins by introducing the basic properties of equality used to solve linear equations, such as distributing terms and adding/subtracting terms to isolate the variable. Examples are provided to demonstrate solving equations with fractions and solving literal equations for a specified variable. The document also discusses identities, contradictions, and using a general formula to solve families of linear equations. It concludes by outlining a problem solving guide to organize the steps of reading, visualizing, and developing an equation model to solve word problems.
The document discusses relations and their application to databases in the relational data model. It defines binary and n-ary relations, and explains how databases can be represented as n-ary relations with records as n-tuples consisting of fields. Primary keys are introduced as fields that uniquely identify each record. Common relational operations like projection and join are explained, with examples provided to illustrate how they transform relations.
This document introduces some key concepts in set theory and mathematical language. It defines what a set is, and introduces ways to represent sets using roster notation. It discusses well-defined sets and the symbols to denote whether an element is or is not part of a set. Examples are provided to illustrate definitions of natural numbers, integers, rational numbers, and real numbers. The document also covers universal and existential statements, and how to rewrite them using variables.
The document discusses unit rates and proportional relationships. It states that students should be able to determine if two quantities are proportional by checking for equivalent ratios in tables or graphs. It also says students should be able to identify the constant of proportionality or unit rate in tables, graphs, equations, diagrams or verbal descriptions of proportional relationships.
Guia periodo i_2021_-_matematicas_9deg_-_revisada_(1)ximenazuluaga3
The document provides information about mathematics courses for 9th grade students at Institución Educativa INEM "Jorge Isaacs", including the names of teachers for each course and section. It also includes criteria for evaluating student work, standards, and learning objectives for the subject of real numbers.
This document discusses algebraic expressions. It defines an algebraic expression as an expression involving variables, and notes they originated from Arabic mathematics. It then provides definitions and examples of terms, variables, and how expressions are formed by combining terms using operations. The document asks questions about representing word problems as expressions and evaluating expressions for given variable values. It discusses forming expressions from word problems and evaluating them. Finally, it gives practice problems asking to evaluate expressions for given variable values.
The document discusses the basics of R, including its history, advantages, and disadvantages. It describes the R console as the main interface where commands are executed. Key concepts covered include variables for storing values, the workspace for accumulating defined variables, and performing basic arithmetic and assignments. The document also introduces basic data types in R like logicals, numerics, integers, characters, and how to determine the class of each. Vectors and matrices are discussed as the main data structures, including how to create, name, and perform operations on them element-wise.
Igcse international mathematics extended standardsRoss
This document outlines the aims, assessment objectives, and curriculum content for the IGCSE International Mathematics exam. The aims focus on developing mathematical skills and applying them to other subjects and real-world problems. The assessment objectives evaluate students' ability to apply mathematical concepts, solve problems, recognize patterns, draw conclusions, and communicate mathematically. The curriculum content covers topics in number, algebra, functions, and geometry including operations, equations, graphs, sequences, trigonometry, and geometric shapes and their properties.
The document discusses the key differences between the English and mathematics languages, including how words, symbols, and concepts have different meanings or representations. It explains how mathematics has developed a precise symbolic language to concisely express relationships, operations, and concepts in a way that is internationally understood regardless of spoken language. Precise definitions and notations are provided for important mathematical concepts like sets, relations, functions, and equations.
This document provides a quarterly budget of works for Grade 7 Mathematics that outlines the most essential learning competencies and objectives to be covered over 4 days each week for 4 quarters. It includes topics such as sets, integers, rational numbers, real numbers, measurement, algebra, geometry, and statistics. Some key objectives are to illustrate operations on integers and rational numbers, write algebraic expressions and equations, graph linear equations, and calculate measures of central tendency and variability for statistical data. The budget aims to help learners master important mathematical concepts and skills through hands-on activities over the course of the school year.
M.S. 442 Carroll Gardens School For Innovation 6th Grade Math Info SlideCGSI
The document provides information about the 6th grade math curriculum at CGSI, including learning outcomes, a curriculum map, and teacher bio. The curriculum map outlines 6 units covering topics like fractions, ratios, expressions, equations, geometry, and statistics. Each unit lists essential questions, common core standards, key concepts, projects/assessments. The teacher bio section is blank.
Plane-and-Solid-Geometry. introduction to provingReyRoluna1
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1. SFS MATHEMATICS SCOPE & SEQUENCE DOCUMENT
Note: Listed below is a summary of the US Mathematics Core Standards. Please consult standards for full details
MIDDLE SCHOOL
Gr. 6 Gr. 7 Gr. 8
NUMBERS &OPERATIONS NUMBERS & OPERATIONS
RATIOS & PROPORTIONAL RELATIONSHIPS (RP) RATIOS & PROPORTIONAL RELATIONSHIPS (RP) The NUMBER SYSTEM (NS)
6. RP Understand Ratio Concepts to solve problems 7. RP Analyse proportional relationships to solve 8. NS Know that there are no.s that are not rational, and approximate them by rational no.s
Understand ration & use ration lang. to describe relationships between problems Understand that every no. has a decimal expansion; the rational no.s are those with decimal expansions that terminate in 0s or eventually repea
two quantities Compute unit rates associated with ratios of fractions, length, areas, etc Know that other no.s are called irrational.
Use ration and rate reasoning to solve problems Recognise and represent proportional relationships between quantities. Use rational approx..of irrational no.s to compare their size, locate them on a no. line, and estimate the value of expressions
Solve unit rate problems including those that involve unit pricing & Decide if two quantities are in proportional relationship
constant speed. Identify the constant of proportionality (unit rate ( in graphs, tables, equations,
Find per cet. Of a quantity as a rate / 100 diagrams
Use ration reasoning to convert measurement units Se proportional relationships to solve ratio & per/cent problems.
The NUMBER SYSTEM (NS)
6. NS Apply & Extend x, ¸
offractions by Fractions The NUMBER SYSTEM (NS)
Interpret & compute quotients of fractions & solve word problems 7. NS Apply, Extend Operations with Fractions to +, -, X,
involving div. of fractions by fractions
¸
+, -, x, multi-digit no.s
¸ rational no.s
+,- of rational no.s & represent on a number line
Greatest common factors; lowest common multiple of no.s less than 100. Show that a no. and its opposite have a sum of 0
Use the distributive law. e.g. 1 + -1 = 0
6. NS Apply & Extend Previous Understandings of ¸
+, - , x , of rational no.s
Numbers to rational No.s x of negative nos. creates a positive no. e.g. -1 x -1 = 1
+, - nos. express quantities having opposite directions e.g. temp, integers can be divided – with the quotients being rational no.s
elevator convert rational no.s to decimals using long division
rational nos. can be expressed on the number line. Solve real problems.
Opposite of the opposite is the no. itselfeg. –(-3) = 3
Ordered pairs indicate location in quadrants of the coordinate plane.
Plotting location on integers & rational nos. on a number line
Ordering & absolute value of rational no.s
Inequalities refer to position on a number line.
Solve real problems by graphing points on a number plane
ALGEBRAALGEBRA
EXPRESSIONS & EQUATIONS (EE) EXPRESSIONS & EQUATIONS (EE) EXPRESSIONS & EQUATIONS (EE)
8. EE Work with radicals & integer exponents
6. EE Apply & extend previous understandings of 7. EE Use properties to generate equivalent expressions Apply properties of integer exponents to forming equivalent numerical expressions
Use square & cube root to represent solutions in the form x2=p , x3 = p where p is a pos. rational no.
arithmetic to algebraic expressions Apply properties of operations as strategies to +, -, factor and expand linear
whole no. exponents expressions with rational coefficients know that 2 is irrational
expressions where letters stand for numbers expanding no.s using powers of 10
identifying sum, product, factor, quotient, coefficient perform operations with no.s expressed in scientific notation
7. EE Solve real-life mathematical problems using use scientific notation & appropriate units to rep. very large & small quantities.
order of operations
apply the properties of operations to generate equivalent expressions numerical & algebraic expressions & equations 8. EE Understand the connection between proportional relationships, lines & linear equations.
Solve multi-step real-life problems involving pos. & neg. rational no.s in any form Graph proportional relationships interpreting the unit rate as the slope of the graph. Compare diff. relationships e.g. dist& time graph to a dist- time equation to determine wh
– whole no.s, fract. &dec. objects has the greater speed.
6. EE Reason about & solve one variable equations Use similar triangles to explain why the slope m is the same between two distinct points on a non-vertical; line in the coordinate plane; show the equation y = mx for a line
Use variables to rep. quantities in a real-world prob. – construct simples through the origin and the equation y = mx +b for a line intercepting the vert. axis at b
& inequalities equations & inequalities to solve problems.
Solve word problems in the form px + q = r, p(x + q) = r where pro-numerals are
8. EE Analyse & solve linear equations and pairs of simultaneous linear equations
using substitution to solve equation & inequalities Solve linear equations in one variable
rational no.s Give examples of linear equations in one variable with one solution
use variables to represent numbers and expressions when solving real-
world problems Solve inequalities in the form of px + q > r, px +q < r where pro-numerals are Solve linear equations with rational no. coefficients include expansion of expressions using the distributive law
rational no.s Solve pairs of simultaneous linear equations
solve problems in the form of x + p – q, px = q in which pro-numerals are Graph solutions Understand that solutions of pairs of linear equations correspond to the points of intersections of their graphs
all positive rational no.s Solve pairs of linear equations by graphing their equations e.g. 3x + 2y = 5
Solve real-world problems using linear equations in two variables e.g. given coordinates fr two pairs of pt,s determine whether the line through the first set of points intersec
solving inequalities in the form of x > c or x< c – such equations have
a line through the second set.
many solutions which can be represented on a number line.
6. EE Represent & analyse quantitative relationships FUNCTIONS (F)
8. F Define, evaluate & compare functions
between dependent & independent variables A function is a rule that assigns one output to every input. A graph of a function is a set of ordered pairs consisting of an input and output
use variables to represent & compare variables in real world problems Compare properties of two functions represented in different ways – i.e. graphically, algebraically, numerically in tables
Analyse relationships between variables using graphs & tables – e.g. – Interpret the equation y =mx + b as a defining linear function whose graph is a straight line as opposed to other functions that are not linear.
motion at constant speed, list and graphs ordered pairs of distances, 8. F Use functions to model relationships between quantities
times and write the equation d=65t to represent relationship between Construct a function to model a linear relationship between two quantities
dist. & time Describe qualitatively the relationship between two quantities by analyzing a graph
2. GEOMETRY (G) GEOMETRY (G)
6. G Solve real- world & mathematical problems 7. G Draw, construct & describe geometrical figures & 8. G Understand congruence & similarity using physical models, transparencies, or geometrical
involving area, surface area, and volume describe the relationships between them. software.
Find the area of triangles, quadrilaterals, polygons by breaking them into Solve problems involving scale drawings of geometrical figures – including Verify properties of rotation, reflections, and translations
known shapes i.e. triangles computing actual length and areas from scaled drawing & representing the scale 2D shapes are congruent if they can be obtained from each other by a series of rotations, reflections, translations & dilations
Find the volume of a rectangular prism using unit cubes to show that the drawing at a different scale. use formal arguments to est. facts about the angles sum and exterior angles of triangles, about angles created by parallel lines cut by a transve
formula for volume is L x B x H. Apply the formula
Draw (freehand, with devices & technology) geometrical shapes with different
Draw polygons in a coordinate plane given the coordinates for the 8. G Understand & apply the Pythagorean Theorem
conditions.
vertices,
Represent 3D shapes using nets for rectangles, triangles, use the nets Describe 2D shapes made from slicing 3D shapes – rectangular prism, square explain a proof for the Pythagorean Theorem and its converse
to find the area of these figures based pyramid Apply the Pythagorean Theorem to finding the unknown side of triangles
Apply the Pythagorean Theorem to find the distance between two points in a coordinate system – number plane.
7. G Solve real-life & mathematical problems involving
angles measure, area, surface area, and volume. 8. G Solve real-world & mathematical problems involving cylinders, cones and spheres.
Know the formulas of the area & circumference of a circle and solve problems; Know and apply the formulas for the volume of cons, cylinders, and spheres
Use facts about supplementary & complementary, vertical and adjacent angle to
solve multistep equations for an unknown angle.
Solve real-world problems involving area, volume & surface area of 2D & 3D
objects – composed of triangles, quadrilaterals, polygons, cubes, and prisms
STATISTICS & PROBABILITY(SP)STATISTICS& PROBABILITY (SP)
6.SP Develop understanding of statistical variability 7. SP Use random sampling to draw inferences about a 8. SP Investigate patterns of association in bivariate data.
Recognise that statistics anticipates variability in data population Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe
Understand that statistics can be used to gather information about a population patterns such as clustering, outliners, positive or negative association, linear association, and nonlinear association.
Understands that data has a distribution described by its centre Know that straight lines are widely used to model relationships between two quantitative variables.
(median), spread (range) and overall shape (frequency distribution) by examining a sample of the population, generalisations about the population
from a sample are valid only if the sample is representative of that population. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
Understand that random sampling tends to produce representative samples and Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a tw
6.SP Summarize & describe distributions. support valid inferences. way table. Construct such a table that summarises data on two categorical variables collected from the same subjects. Use relative frequencie
Display data in plots on a number line – i.e. dot plots, histogram, box Use data from a random sample to draw inferences about a population with an calculated for rows or columns to describe possible association between two variables.
plots unknown characteristic of interest.
Summarises data according to: no. of observations; nature of the
attribute, how it was measured and its units of measure; giving 7. SP Draw informal comparative inferences about two
quantitative measures of centre (median, mean) and variability or populations.
deviation. As well as describing any pattern or deviations from the
pattern. Informally assess the degree of visual overlap of to numerical data distributions
Relating choice of measures of centre and variability to the shape of the with similar variables.
data distribution and the context in which the data was gathered. Use measures of centre and measures of variables for numerical data from
random samples to draw informal comparative inferences about two populations.
7.SP Investigate chance processes and develop, se and
evaluate probability models.
Understand that the probability of chance expresses the likelihood of the event
occurring – o indicate unlikely event; ½ indicates neutrality; 1 indicates event is
likely to occur.
Approximate the probability of chance by collecting data and observing its
frequency and predict the relative frequency given the probability
Develop a probability model and use it to find probabilities of events.
Develop a uniform probability model by assigning equal probability to all
outcomes and then use the model to determine probabilities of events.
Find probabilities of compound events using lists, tables, tree diagrams, and
simulation.
Understand that the probability of a compound evetn is the fraction of outcomes
in the sample space for which the compound event occurs.
Represent sample space for compound events using lists, tables, tree diagrams,
Identify the outcomes in the sample space which compose the event.
Design and use a simulation to generate frequencies for compound events