The document provides information about rational numbers including their general form, examples of rational numbers, operations on rational numbers and their properties.
It defines a rational number as any number that can be expressed as a ratio of two integers. All integers and fractions are rational numbers.
The key properties of operations on rational numbers are discussed - addition, subtraction, multiplication and division can have closure, commutative, associative properties in some cases but not in others.
Several videos and activities are suggested to help teach and reinforce rational numbers concepts like representing them on a number line, comparing and ordering, finding rational numbers between two rationals and more.
Rational numbers can be expressed in the general form p/q, where p and q are integers and q ≠ 0. Rational numbers include all integers and fractions. Between any two rational numbers, there are an infinite number of other rational numbers that can be found, such as the average. Rational numbers can be compared and ordered by making their denominators common and comparing numerators. Certain arithmetic operations like addition, subtraction and multiplication are closed under rational numbers, while division is not.
The document provides information about algebraic expressions and polynomials. It begins by explaining that algebra uses a language of numbers, variables, and symbols to represent quantitative relationships. The document then covers topics like equivalent expressions, simplifying expressions, evaluating expressions, and different types of polynomials. It aims to help students acquire skills in performing operations with polynomials and using those skills to simplify, evaluate, and solve algebraic expressions and problems.
This document provides a daily lesson log for a Grade 9 mathematics class. It outlines the objectives, content, learning resources and procedures for lessons on the nature of roots of quadratic equations, including the sum and product of roots, and equations that can be transformed into quadratic equations. Key concepts covered are using the discriminant to characterize roots, the relationship between coefficients and roots, and solving various types of equations. Examples and follow-up questions are provided to discuss and practice the new skills.
Rational numbers can be represented as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, meaning these operations on rational numbers will always result in another rational number. Division of rational numbers is not closed as division by 0 is undefined. Rational numbers exhibit properties like commutativity for addition and multiplication but not for subtraction or division. They also demonstrate the associative and distributive properties. Every rational number has an additive inverse and a multiplicative inverse or reciprocal. Zero is the additive identity and one is the multiplicative identity.
College Algebra MATH 107 Spring, 2015, V4.8 Page 1 of .docxmonicafrancis71118
This document provides instructions and requirements for a PowerPoint presentation project on a student's chosen career. The presentation must include 10-12 slides covering: an introduction of the student and research topic, an outline, background on the career path and reasons for choice, educational requirements, pay ranges in three regions, and a summary and conclusion slide. References must be in MLA format. The presentation will be graded based on inclusion of required elements, design features like themes and visual elements, transitions and animations, and correct spelling and grammar.
This document provides a course breakdown for CFE 4th level mathematics. It includes 8 topics covering various mathematical concepts. For each topic, it lists the associated learning outcome, success criteria describing what students should be able to do, examples of questions, and a self-assessment key for students to rate their understanding as Secure, Consolidating, or Developing. The self-assessment will allow students to identify areas they need to focus more practice and revision on.
The document discusses various types of operators in the C programming language. It describes arithmetic operators like addition, subtraction, multiplication, division, and modulus. It provides examples of how each operator works and the order in which they are evaluated. It emphasizes using parentheses to change the order of operations and avoid ambiguity. It also discusses division by zero and good practices for writing clear code using operators.
This document contains notes from a math teacher for lessons on solving equations over several days. It includes the essential components of each lesson - learning goals, examples to work through, guided practice problems and homework assignments. The notes provide guidance to help students understand different methods for solving equations, avoid common mistakes, and assess their progress on the learning goals which involve solving various types of one-step and multi-step equations.
Rational numbers can be expressed in the general form p/q, where p and q are integers and q ≠ 0. Rational numbers include all integers and fractions. Between any two rational numbers, there are an infinite number of other rational numbers that can be found, such as the average. Rational numbers can be compared and ordered by making their denominators common and comparing numerators. Certain arithmetic operations like addition, subtraction and multiplication are closed under rational numbers, while division is not.
The document provides information about algebraic expressions and polynomials. It begins by explaining that algebra uses a language of numbers, variables, and symbols to represent quantitative relationships. The document then covers topics like equivalent expressions, simplifying expressions, evaluating expressions, and different types of polynomials. It aims to help students acquire skills in performing operations with polynomials and using those skills to simplify, evaluate, and solve algebraic expressions and problems.
This document provides a daily lesson log for a Grade 9 mathematics class. It outlines the objectives, content, learning resources and procedures for lessons on the nature of roots of quadratic equations, including the sum and product of roots, and equations that can be transformed into quadratic equations. Key concepts covered are using the discriminant to characterize roots, the relationship between coefficients and roots, and solving various types of equations. Examples and follow-up questions are provided to discuss and practice the new skills.
Rational numbers can be represented as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, meaning these operations on rational numbers will always result in another rational number. Division of rational numbers is not closed as division by 0 is undefined. Rational numbers exhibit properties like commutativity for addition and multiplication but not for subtraction or division. They also demonstrate the associative and distributive properties. Every rational number has an additive inverse and a multiplicative inverse or reciprocal. Zero is the additive identity and one is the multiplicative identity.
College Algebra MATH 107 Spring, 2015, V4.8 Page 1 of .docxmonicafrancis71118
This document provides instructions and requirements for a PowerPoint presentation project on a student's chosen career. The presentation must include 10-12 slides covering: an introduction of the student and research topic, an outline, background on the career path and reasons for choice, educational requirements, pay ranges in three regions, and a summary and conclusion slide. References must be in MLA format. The presentation will be graded based on inclusion of required elements, design features like themes and visual elements, transitions and animations, and correct spelling and grammar.
This document provides a course breakdown for CFE 4th level mathematics. It includes 8 topics covering various mathematical concepts. For each topic, it lists the associated learning outcome, success criteria describing what students should be able to do, examples of questions, and a self-assessment key for students to rate their understanding as Secure, Consolidating, or Developing. The self-assessment will allow students to identify areas they need to focus more practice and revision on.
The document discusses various types of operators in the C programming language. It describes arithmetic operators like addition, subtraction, multiplication, division, and modulus. It provides examples of how each operator works and the order in which they are evaluated. It emphasizes using parentheses to change the order of operations and avoid ambiguity. It also discusses division by zero and good practices for writing clear code using operators.
This document contains notes from a math teacher for lessons on solving equations over several days. It includes the essential components of each lesson - learning goals, examples to work through, guided practice problems and homework assignments. The notes provide guidance to help students understand different methods for solving equations, avoid common mistakes, and assess their progress on the learning goals which involve solving various types of one-step and multi-step equations.
The document provides instructions for a PSSA review for 8th grade mathematics that includes vocabulary reviews, practice exercises, and solutions for key math concepts tested. It outlines the five categories covered - Numbers & Operations, Measurement, Geometry, Algebraic Concepts, and Data Analysis & Probability - and provides sources for the content. Students are directed to work through the review independently using the provided answer keys and online reinforcement resources.
Rational numbers are a fundamental concept in mathematics, representing any number that can be expressed as the quotient or fraction
�
�
q
p
, where
�
p and
�
q are integers, and
�
q is not zero. This means that rational numbers include both positive and negative fractions, as well as whole numbers and zero. Essentially, if a number can be written in the form of a fraction, it is considered rational. This encompasses a broad range of numbers such as
1
2
2
1
, -3, 0.75, and 5, highlighting the flexibility and wide applicability of rational numbers in various mathematical contexts. The term "rational" itself is derived from the word "ratio," reflecting the idea that these numbers represent ratios of integers. Rational numbers are a subset of real numbers, and they can be precisely located on the number line. An important characteristic of rational numbers is that when expressed in decimal form, they either terminate or repeat. For example, the decimal representation of
1
4
4
1
is 0.25, which terminates, while
1
3
3
1
is 0.333..., which repeats. This property differentiates rational numbers from irrational numbers, which cannot be expressed as exact fractions and have non-terminating, non-repeating decimal expansions. Rational numbers play a crucial role in various areas of mathematics and everyday life. In algebra, they are used to solve equations and inequalities, and in geometry, they help in measuring lengths, areas, and volumes. In everyday life, rational numbers are used in financial calculations, such as determining interest rates, discounts, and budgeting, as well as in measurements and recipes. Understanding rational numbers is essential for mathematical literacy and is a foundational concept for more advanced topics in mathematics, such as calculus and number theory. Moreover, rational numbers are integral to computer science, particularly in algorithms and programming, where precise calculations and representations of numbers are necessary. The ability to manipulate and understand rational numbers enhances problem-solving skills and logical reasoning. The study of rational numbers also involves exploring their properties, such as the ability to perform arithmetic operations like addition, subtraction, multiplication, and division, which follow specific rules and patterns. This exploration extends to understanding how rational numbers relate to other types of numbers, such as integers and irrational numbers, and how they fit within the broader number system. Overall, rational numbers are a versatile and essential component of mathematics, with applications that extend far beyond the classroom, influencing various fields and everyday situations. Their study provides a strong foundation for mathematical understanding and practical problem-solving abilities.
Rational numbers are also significant in the study of number theory and real analysis, where they provide insight into the properties and behaviors of numbers
This document provides a mark scheme for a physics exam. It explains how examiners will award marks for questions on the exam. The mark scheme provides guidance for examiners, showing what is required to earn marks for different parts of questions. It also explains symbols used in the mark scheme and policies for awarding marks. The mark scheme will be used by examiners along with the exam questions to consistently and fairly award marks based on the requirements and guidelines laid out.
This document provides materials for a training session on English related to mathematics for mathematics teachers. It covers three units:
Unit 1 introduces the purpose of learning English for mathematics teachers and the objectives of the training session.
Unit 2 explains concepts related to numbers in English, including types of numbers, mathematical operations, fractions, decimals, percentages and ratios. Examples and exercises are provided.
Unit 3 discusses ratios, scales, proportions, percentages and interest in financial contexts. It provides more examples and exercises for participants to practice.
The materials are designed for independent study but also for use during the training session, where participants can ask questions. The overall goal is to help mathematics teachers improve their English comprehension of mathematics literature.
This document is a student workbook for 5th grade mathematics from Aptus Chile. It contains 9 pages of instructional material and practice problems related to fractions. The workbook is divided into units and classes, with examples and exercises for students. It covers topics like representing fractions, calculating fractions of numbers, proper and improper fractions, equivalent fractions using amplification and simplification, and comparing fractions. The material is copyrighted and for educational use by Aptus Chile.
This document provides a pupil with learning intentions, success criteria, examples, and a self-assessment key for various math topics. It includes 8 topics covering areas like number processes, expressions and equations, properties of shapes, data analysis, fractions and percentages. For each topic, the pupil must indicate if they feel secure, consolidating, or developing in achieving the identified math experience or outcome. This allows the pupil to self-evaluate their understanding of the essential concepts covered in level 4 math.
The document discusses rethinking the conceptual understanding of fractions. It proposes viewing fractions based on their location on the number line rather than as a part-to-whole ratio. This view of fractions as points on the number line makes their properties, such as equivalence and relative size, more intuitive and easier to understand. It also suggests this conceptual approach can help strengthen students' procedural knowledge of fractions and support their future learning of algebra.
This document provides the mark scheme and guidance for examiners marking the Cambridge International Examinations International General Certificate of Secondary Education Physics exam from May/June 2013. It explains the marking criteria for questions, including what constitutes method marks (M marks) and independent marks (B marks). The document provides examples of what is required to earn marks and clarifies policies on areas like significant figures and units. It also standardizes how examiners should approach issues like spelling, transcription errors, and 'error carried forward' situations.
The document provides information about fractions including definitions, examples, and formulas for adding and subtracting fractions. It discusses that fractions represent a part-whole relationship and can be placed on a number line between 0 and 1. Formulas are given for adding and subtracting fractions with the same or different denominators. The document also includes a class work assignment involving labeling fractions on a ruler and solving fraction equations.
This document provides an introduction to spreadsheets, including what they are, who uses them, and what they can do. It explains that spreadsheets allow users to organize and calculate data using rows and columns. Spreadsheets are commonly used by schools, sports teams, businesses, families, and government agencies. They can be used to answer "what is" and "what if" questions. The benefits of spreadsheets are that they are fast and accurate at performing calculations. The document then provides more details on the core components of spreadsheets, including cells, cell references, data types (labels, values, formulas, and functions), and examples of each.
This lesson plan is about solving linear equations and inequalities algebraically. Students will learn to find the solution of linear equations in one variable. They will practice translating between verbal and mathematical phrases and evaluating expressions. The lesson will review properties of equality like the reflexive, symmetric, transitive, addition, and multiplication properties. Students will learn steps to solve various types of linear equations using these properties. They will assess their understanding by solving sample equations on their own. For homework, students will solve equations and determine whether they have unique solutions, no solutions, or infinite solutions.
Unit 6 presentation base ten equality form of a number with trainer notes 7.9.08jcsmathfoundations
The document discusses concepts related to base ten, equality, and forms of numbers. It defines these concepts, examines how students develop an understanding through research on cognitive development, and provides classroom applications and strategies for teaching these concepts effectively. Diagnostic questions are presented to assess student understanding, and examples show how to respond to common student errors or misconceptions in working with numbers.
The document discusses the power of recursion and induction in mathematics, modeling, and technology. It provides examples of how recursion appears in definitions of natural numbers and functions. Recursion can also be used to solve complex problems by breaking them down into simpler subproblems. Spreadsheets are an example of how recursion naturally occurs in technology. Teaching recursion enhances modeling skills and helps move from complex to simple problems.
This document contains instructions and questions for an exam in Analog and Digital Electronics. It is divided into 5 modules. For each module, there are 2 full questions with multiple parts to choose from. Students must answer 5 full questions, choosing 1 from each module. They must write the same question numbers and answers should be specific to the questions asked. Writing must be legible. The questions cover topics like operational amplifiers, logic gates, multiplexers, flip-flops, counters, and more. Diagrams and explanations are often required.
1. The document is a daily lesson log for a Grade 9 mathematics class covering quadratic equations.
2. It outlines the objectives, content, learning resources and procedures used to teach illustrations of quadratic equations, solving by extracting square roots, and solving by factoring.
3. Examples are provided to demonstrate solving quadratic equations by extracting square roots and factoring. Students will practice solving equations using these methods.
1. The document is a daily lesson log for a Grade 9 mathematics class covering quadratic equations.
2. It outlines the objectives, content, learning resources and procedures used to teach illustrations of quadratic equations, solving by extracting square roots, and solving by factoring.
3. Examples are provided to demonstrate solving quadratic equations by extracting square roots and factoring. Students will practice solving equations using these methods.
Cs6660 compiler design november december 2016 Answer keyappasami
The document discusses topics related to compiler design, including:
1) The phases of a compiler include lexical analysis, syntax analysis, semantic analysis, intermediate code generation, code optimization, and code generation. Compiler construction tools help implement these phases.
2) Grouping compiler phases can improve efficiency. Errors can occur in all phases, from syntax errors to type errors.
3) Questions cover topics like symbol tables, finite automata in lexical analysis, parse trees, ambiguity, SLR parsing, syntax directed translations, code generation, and optimization techniques like loop detection.
The document provides information on rational numbers and operations involving fractions and integers:
1. It defines rational numbers as numbers that can be represented as fractions a/b where a and b are integers and b is not equal to 0.
2. Rules for addition, subtraction, multiplication, and division of integers and fractions are presented, such as signs of terms determine sign of sum/product.
3. Examples demonstrate applying the rules to evaluate expressions involving integers and fractions.
The document discusses the distributive property in mathematics. It provides examples of using the distributive property to evaluate numerical expressions and rewrite algebraic expressions. It also gives a step-by-step example of using the distributive property to find the total cost of baseball helmets when the price per helmet is given. The distributive property allows numbers and symbols to represent mathematical ideas by distributing a number across terms within parentheses during multiplication.
The document discusses properties of cube numbers and how to find cube roots. It defines perfect and non-perfect cubes, and explains that the cube of an even number is even, while the cube of an odd number is odd. It provides examples of finding cube roots through prime factorization, grouping factors of 3 to obtain the cube root. Finding cube roots of numbers not being perfect cubes results in decimal values.
Economic activities refer to those activities concerned with production, consumption, distribution and exchange of goods and services for monetary gain. Non-economic activities are not concerned with money and include social, political, religious and charitable activities.
The document discusses various definitions of economics including wealth, welfare, scarcity and growth definitions. It explores features and criticisms of each definition. The wealth definition by Adam Smith defines economics as the production and consumption of wealth. The welfare definition by Alfred Marshall shifts the focus to human welfare. Lionel Robbins' scarcity definition describes economics as dealing with scarcity of resources and unlimited wants. Samuelson's growth definition emphasizes efficient allocation of resources over time.
The document provides instructions for a PSSA review for 8th grade mathematics that includes vocabulary reviews, practice exercises, and solutions for key math concepts tested. It outlines the five categories covered - Numbers & Operations, Measurement, Geometry, Algebraic Concepts, and Data Analysis & Probability - and provides sources for the content. Students are directed to work through the review independently using the provided answer keys and online reinforcement resources.
Rational numbers are a fundamental concept in mathematics, representing any number that can be expressed as the quotient or fraction
�
�
q
p
, where
�
p and
�
q are integers, and
�
q is not zero. This means that rational numbers include both positive and negative fractions, as well as whole numbers and zero. Essentially, if a number can be written in the form of a fraction, it is considered rational. This encompasses a broad range of numbers such as
1
2
2
1
, -3, 0.75, and 5, highlighting the flexibility and wide applicability of rational numbers in various mathematical contexts. The term "rational" itself is derived from the word "ratio," reflecting the idea that these numbers represent ratios of integers. Rational numbers are a subset of real numbers, and they can be precisely located on the number line. An important characteristic of rational numbers is that when expressed in decimal form, they either terminate or repeat. For example, the decimal representation of
1
4
4
1
is 0.25, which terminates, while
1
3
3
1
is 0.333..., which repeats. This property differentiates rational numbers from irrational numbers, which cannot be expressed as exact fractions and have non-terminating, non-repeating decimal expansions. Rational numbers play a crucial role in various areas of mathematics and everyday life. In algebra, they are used to solve equations and inequalities, and in geometry, they help in measuring lengths, areas, and volumes. In everyday life, rational numbers are used in financial calculations, such as determining interest rates, discounts, and budgeting, as well as in measurements and recipes. Understanding rational numbers is essential for mathematical literacy and is a foundational concept for more advanced topics in mathematics, such as calculus and number theory. Moreover, rational numbers are integral to computer science, particularly in algorithms and programming, where precise calculations and representations of numbers are necessary. The ability to manipulate and understand rational numbers enhances problem-solving skills and logical reasoning. The study of rational numbers also involves exploring their properties, such as the ability to perform arithmetic operations like addition, subtraction, multiplication, and division, which follow specific rules and patterns. This exploration extends to understanding how rational numbers relate to other types of numbers, such as integers and irrational numbers, and how they fit within the broader number system. Overall, rational numbers are a versatile and essential component of mathematics, with applications that extend far beyond the classroom, influencing various fields and everyday situations. Their study provides a strong foundation for mathematical understanding and practical problem-solving abilities.
Rational numbers are also significant in the study of number theory and real analysis, where they provide insight into the properties and behaviors of numbers
This document provides a mark scheme for a physics exam. It explains how examiners will award marks for questions on the exam. The mark scheme provides guidance for examiners, showing what is required to earn marks for different parts of questions. It also explains symbols used in the mark scheme and policies for awarding marks. The mark scheme will be used by examiners along with the exam questions to consistently and fairly award marks based on the requirements and guidelines laid out.
This document provides materials for a training session on English related to mathematics for mathematics teachers. It covers three units:
Unit 1 introduces the purpose of learning English for mathematics teachers and the objectives of the training session.
Unit 2 explains concepts related to numbers in English, including types of numbers, mathematical operations, fractions, decimals, percentages and ratios. Examples and exercises are provided.
Unit 3 discusses ratios, scales, proportions, percentages and interest in financial contexts. It provides more examples and exercises for participants to practice.
The materials are designed for independent study but also for use during the training session, where participants can ask questions. The overall goal is to help mathematics teachers improve their English comprehension of mathematics literature.
This document is a student workbook for 5th grade mathematics from Aptus Chile. It contains 9 pages of instructional material and practice problems related to fractions. The workbook is divided into units and classes, with examples and exercises for students. It covers topics like representing fractions, calculating fractions of numbers, proper and improper fractions, equivalent fractions using amplification and simplification, and comparing fractions. The material is copyrighted and for educational use by Aptus Chile.
This document provides a pupil with learning intentions, success criteria, examples, and a self-assessment key for various math topics. It includes 8 topics covering areas like number processes, expressions and equations, properties of shapes, data analysis, fractions and percentages. For each topic, the pupil must indicate if they feel secure, consolidating, or developing in achieving the identified math experience or outcome. This allows the pupil to self-evaluate their understanding of the essential concepts covered in level 4 math.
The document discusses rethinking the conceptual understanding of fractions. It proposes viewing fractions based on their location on the number line rather than as a part-to-whole ratio. This view of fractions as points on the number line makes their properties, such as equivalence and relative size, more intuitive and easier to understand. It also suggests this conceptual approach can help strengthen students' procedural knowledge of fractions and support their future learning of algebra.
This document provides the mark scheme and guidance for examiners marking the Cambridge International Examinations International General Certificate of Secondary Education Physics exam from May/June 2013. It explains the marking criteria for questions, including what constitutes method marks (M marks) and independent marks (B marks). The document provides examples of what is required to earn marks and clarifies policies on areas like significant figures and units. It also standardizes how examiners should approach issues like spelling, transcription errors, and 'error carried forward' situations.
The document provides information about fractions including definitions, examples, and formulas for adding and subtracting fractions. It discusses that fractions represent a part-whole relationship and can be placed on a number line between 0 and 1. Formulas are given for adding and subtracting fractions with the same or different denominators. The document also includes a class work assignment involving labeling fractions on a ruler and solving fraction equations.
This document provides an introduction to spreadsheets, including what they are, who uses them, and what they can do. It explains that spreadsheets allow users to organize and calculate data using rows and columns. Spreadsheets are commonly used by schools, sports teams, businesses, families, and government agencies. They can be used to answer "what is" and "what if" questions. The benefits of spreadsheets are that they are fast and accurate at performing calculations. The document then provides more details on the core components of spreadsheets, including cells, cell references, data types (labels, values, formulas, and functions), and examples of each.
This lesson plan is about solving linear equations and inequalities algebraically. Students will learn to find the solution of linear equations in one variable. They will practice translating between verbal and mathematical phrases and evaluating expressions. The lesson will review properties of equality like the reflexive, symmetric, transitive, addition, and multiplication properties. Students will learn steps to solve various types of linear equations using these properties. They will assess their understanding by solving sample equations on their own. For homework, students will solve equations and determine whether they have unique solutions, no solutions, or infinite solutions.
Unit 6 presentation base ten equality form of a number with trainer notes 7.9.08jcsmathfoundations
The document discusses concepts related to base ten, equality, and forms of numbers. It defines these concepts, examines how students develop an understanding through research on cognitive development, and provides classroom applications and strategies for teaching these concepts effectively. Diagnostic questions are presented to assess student understanding, and examples show how to respond to common student errors or misconceptions in working with numbers.
The document discusses the power of recursion and induction in mathematics, modeling, and technology. It provides examples of how recursion appears in definitions of natural numbers and functions. Recursion can also be used to solve complex problems by breaking them down into simpler subproblems. Spreadsheets are an example of how recursion naturally occurs in technology. Teaching recursion enhances modeling skills and helps move from complex to simple problems.
This document contains instructions and questions for an exam in Analog and Digital Electronics. It is divided into 5 modules. For each module, there are 2 full questions with multiple parts to choose from. Students must answer 5 full questions, choosing 1 from each module. They must write the same question numbers and answers should be specific to the questions asked. Writing must be legible. The questions cover topics like operational amplifiers, logic gates, multiplexers, flip-flops, counters, and more. Diagrams and explanations are often required.
1. The document is a daily lesson log for a Grade 9 mathematics class covering quadratic equations.
2. It outlines the objectives, content, learning resources and procedures used to teach illustrations of quadratic equations, solving by extracting square roots, and solving by factoring.
3. Examples are provided to demonstrate solving quadratic equations by extracting square roots and factoring. Students will practice solving equations using these methods.
1. The document is a daily lesson log for a Grade 9 mathematics class covering quadratic equations.
2. It outlines the objectives, content, learning resources and procedures used to teach illustrations of quadratic equations, solving by extracting square roots, and solving by factoring.
3. Examples are provided to demonstrate solving quadratic equations by extracting square roots and factoring. Students will practice solving equations using these methods.
Cs6660 compiler design november december 2016 Answer keyappasami
The document discusses topics related to compiler design, including:
1) The phases of a compiler include lexical analysis, syntax analysis, semantic analysis, intermediate code generation, code optimization, and code generation. Compiler construction tools help implement these phases.
2) Grouping compiler phases can improve efficiency. Errors can occur in all phases, from syntax errors to type errors.
3) Questions cover topics like symbol tables, finite automata in lexical analysis, parse trees, ambiguity, SLR parsing, syntax directed translations, code generation, and optimization techniques like loop detection.
The document provides information on rational numbers and operations involving fractions and integers:
1. It defines rational numbers as numbers that can be represented as fractions a/b where a and b are integers and b is not equal to 0.
2. Rules for addition, subtraction, multiplication, and division of integers and fractions are presented, such as signs of terms determine sign of sum/product.
3. Examples demonstrate applying the rules to evaluate expressions involving integers and fractions.
The document discusses the distributive property in mathematics. It provides examples of using the distributive property to evaluate numerical expressions and rewrite algebraic expressions. It also gives a step-by-step example of using the distributive property to find the total cost of baseball helmets when the price per helmet is given. The distributive property allows numbers and symbols to represent mathematical ideas by distributing a number across terms within parentheses during multiplication.
The document discusses properties of cube numbers and how to find cube roots. It defines perfect and non-perfect cubes, and explains that the cube of an even number is even, while the cube of an odd number is odd. It provides examples of finding cube roots through prime factorization, grouping factors of 3 to obtain the cube root. Finding cube roots of numbers not being perfect cubes results in decimal values.
Economic activities refer to those activities concerned with production, consumption, distribution and exchange of goods and services for monetary gain. Non-economic activities are not concerned with money and include social, political, religious and charitable activities.
The document discusses various definitions of economics including wealth, welfare, scarcity and growth definitions. It explores features and criticisms of each definition. The wealth definition by Adam Smith defines economics as the production and consumption of wealth. The welfare definition by Alfred Marshall shifts the focus to human welfare. Lionel Robbins' scarcity definition describes economics as dealing with scarcity of resources and unlimited wants. Samuelson's growth definition emphasizes efficient allocation of resources over time.
The document discusses exponents and powers, including:
1) An introduction to exponents and powers, their history and common uses. Exponents were coined by Michael Stifel in 1544 to manipulate powers of 10.
2) The laws of exponents, such as the law discovered by Archimedes that 10a × 10b = 10a+b.
3) How to work with negative exponents as multiplicative inverses and convert between decimal and standard notations for large and small numbers using exponents.
Ch 10 Algebraic Expressions and Identities.pptxDeepikaPrimrose
This document discusses algebraic expressions and identities. It defines terms like monomials, binomials, trinomials, polynomials and their degrees. It also covers topics like addition, subtraction, multiplication and division of polynomials using different methods. The document then explains important algebraic identities like (a + b)2, (a - b)2, (a + b)(a - b) and their geometric proofs. It concludes by showing applications of these identities.
This document discusses squares, square roots, and properties related to them. It defines a square number as the result of multiplying a number by itself. It provides examples of perfect square numbers and discusses patterns in the last digits of squares based on the last digit of the original number. The document also covers finding square roots using repeated subtraction, properties of square roots based on the last digit, and other methods for finding squares and square roots. Additionally, it discusses Pythagorean triplets, prime factorizations, and other concepts related to squares and square roots.
This document contains a math quiz with multiple choice and free response questions testing various math skills including:
- Standard notation and place value
- Addition, subtraction, and finding differences
- Converting between minutes and hours
- Identifying numbers, patterns, and products in sequences
- Performing multi-step word problems involving addition, subtraction, and place value
1. The document contains a math quiz on decimals with multiple choice and short answer questions covering topics like place value, rounding, addition, subtraction, multiplication, and division of decimals. It also includes exercises on writing decimals as word names and ordering decimals from least to greatest.
2. Questions involve skills like identifying the place value of digits, operations with decimals, comparing and ordering decimals, rounding decimals, and writing decimals as word names.
3. The document provides practice with basic decimal skills and concepts through multiple representation questions.
Introduction to AI for Nonprofits with Tapp NetworkTechSoup
Dive into the world of AI! Experts Jon Hill and Tareq Monaur will guide you through AI's role in enhancing nonprofit websites and basic marketing strategies, making it easy to understand and apply.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
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3. INTRODUCTION
Why are there so many numbers?
Video Tutorial - https://www.youtube.com/watch?v=XN2Cj8GuVbY
Irrational - 22 trillion digits of pi (𝝅)
Visit the page to see the digits - https://pi2e.ch/blog/2017/03/10/pi-digits-download/
Why rational numbers? (Real life applications)
http://explainingmath.blogspot.com/2011/04/real-world-examples-for-rational.html
4. Children will be able to:
Describe properties of rational numbers & express them in general form
Consolidate operations on rational numbers
Represent rational numbers on a number line
Understand that between any two rational numbers there lies another rational
number (unlike whole numbers)
Generalize & verify properties of rational numbers (including identities)
Apply the logic on word problems
Objectives
5.
6.
7. A rational number is any number that can be expressed as a ratio of two integers
It can be written as a fraction
9. Standard Form of Rational Numbers
12/36 is a rational number.
But it can be simplified as 1/3;
common factors between the divisor and dividend is only one.
So we can say that rational number ⅓ is in standard form.
10. What are rational numbers? https://www.youtube.com/watch?v=SQ4cB9yXkHM
Examples of rational numbers & their decimal representation will be shown.
Decimal representation of irrational numbers will also be shown here to distinguish rational numbers from them.
Activity – Rational Maze
Description: A square maze will be given in a paper for each team. Each block in the maze contains numbers
(rational & irrational numbers)
Instructions to Students:
Students should sit in four groups, each group will receive a maze
The name of the group should be written on the paper given to them
The activity should begin only after the teacher instructs them to start
Students should identify only rational numbers and highlight (shade with highlighter) or circle with pen.
Connect all the highlighted/circled blocks in order to form a route from start to end points in the maze
Instructions to Teacher: (File link : ..Chapter 1 Rational NumbersRational_Maze_HW_Half copies.docx
Teacher should have the printout of all four different set of mazes in four different papers
Each set will be distributed to each group & each group will be named/numbered
Each group will use a different color to highlight or circle
The instructions to solve the maze should be given clearly, only after which the students should begin
The group which completes first gets a tally (to be updated in tally chart)
DELIVERY OF LESSON
11. Closure property of Rational numbers
What does it mean? https://www.youtube.com/watch?v=F0L2FENoJOo
Problems related to closure property verification will be given to students to solve as homework
Why a number divided by zero is UNDEFINED?
Watch to know: https://www.youtube.com/watch?v=J2z5uzqxJNU
Commutative, Associative & Distributive properties, Identity & Inverse (Additive & Multiplicative)
Students will be questioned for which operation these properties would fail
On the green board, teacher will be giving examples demonstrating each property over each operation
Rational numbers on a Number line
Watch to know how? - https://youtu.be/G9n8HbMdUIk?t=2300
On the green board, teacher will demonstrate how to plot rational numbers on the number line.
Word document (link) will displayed on smart board for the students to finish the exercise on plotting rational numbers.
Students completing all the questions will be rewarded a tally
DELIVERY OF LESSON
12. Compare & Order rational numbers
Watch the video to know how:
Worksheet will be given for the students to solve. (worksheet link - print to be taken)
Students completing the worksheet will be given a tally
Rational numbers between two rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=lg04THe8wfY
On the green board, teacher will consolidate methods to find single or many rational number for same or different
denominators with examples
Word document (link) will be displayed for the students to solve. Students completing all the questions will be given a tally
Chapter Practice Worksheet
Worksheet with all the topics on rational numbers will be provided for the students to solve in class.
Teacher may help/clarify during this practice session.
Students completing the worksheet will be given a tally.
DELIVERY OF LESSON
13. ADDITIONAL LEARNING OPPORTUNITY
Why is 𝜋 so irrational? Watch to know: https://www.youtube.com/watch?v=HSuqbqENIek.
Why scientists keep computing digits of pi?
History of 𝜋 - https://www.youtube.com/watch?v=1-JAx3nUwms
What to do with the digits of 𝜋? - https://youtu.be/zhqdIhYvFxU
Digits of 𝜋 & uses - What to do with the digits of π.docx
19. Operations on rational numbers and their properties
There are some properties of operations on rational numbers.
They are closure,
commutative,
associative,
identity,
inverse
and distributive.
20. Addition of rational numbers and their properties
Closure: If a/b and c/d are any two rational numbers, then (a/b) + (c/d) is also a rational number.
Example :
2/9 + 4/9 = 6/9 = 2/3 is a rational number.
Commutative: Addition of two rational numbers is commutative.
If a/b and c/d are any two rational numbers,
then (a/b) + (c/d) = (c/d) + (a/b)
Example :
2/9 + 4/9 = 6/9 = 2/3
4/9 + 2/9 = 6/9 = 2/3
Hence, 2/9 + 4/9 = 4/9 + 2/9
Associative: Addition of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers,
then a/b + (c/d + e/f) = (a/b + c/d) + e/f
Example :
2/9 + (4/9 + 1/9) = 2/9 + 5/9 = 7/9
(2/9 + 4/9) + 1/9 = 6/9 + 1/9 = 7/9
Hence, 2/9 + (4/9 + 1/9) = (2/9 + 4/9) + 1/9
21. Addition of rational numbers and their properties
Additive identity : The sum of any rational number and zero is the rational number itself.
If a/b is any rational number,
then a/b + 0 = 0 + a/b = a/b
Zero is the additive identity for rational numbers.
Example :
2/7 + 0 = 0 + 2/7 = 27
Additive inverse : (- a/b) is the negative or additive inverse of (a/b)
If a/b is a rational number,then there exists a rational number (-a/b) such that
a/b + (-a/b) = (-a/b) + a/b = 0
Example :
Additive inverse of 3/5 is (-3/5)
Additive inverse of (-3/5) is 3/5
Additive inverse of 0 is 0 itself.
22. If a/b, c/d and e/f are any three rational numbers,
then a/b x (c/d + e/f) = a/b x c/d + a/b x e/f
Example :
1/3 x (2/5 + 1/5) = 1/3 x 3/5 = 1/5
1/3 x (2/5 + 1/5) = 1/3 x 2/5 + 1/3 x 1/5 = (2 + 1) / 15 = 1/5
Hence, 1/3 x (2/5 + 1/5) = 1/3 x 2/5 + 1/3 x 1/5
Therefore, Multiplication is distributive over addition.
Multiplication is distributive over addition.
Multiplication is distributive over addition.
23. Subtraction of rational numbers and their properties
(i) Closure Property :
The difference between any two rational numbers is always a rational number.
Hence Q is closed under subtraction.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) is also a rational number.
Example :
5/9 - 2/9 = 3/9 = 1/3 is a rational number.
(ii) Commutative Property :
Subtraction of two rational numbers is not commutative.
If a/b and c/d are any two rational numbers, then
(a/b) - (c/d) ≠ (c/d) - (a/b)
Example :
5/9 - 2/9 = 3/9 = 1/3
2/9 - 5/9 = -3/9 = -1/3
And,
5/9 - 2/9 ≠ 2/9 - 5/9
Therefore, Commutative property is not true for subtraction.
24. (iii) Associative Property :
Subtraction of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b - (c/d - e/f) ≠ (a/b - c/d) - e/f
Example :
2/9 - (4/9 - 1/9) = 2/9 - 3/9 = -1/9
(2/9 - 4/9) - 1/9 = -2/9 - 1/9 = -3/9
And,
2/9 - (4/9 - 1/9) ≠ (2/9 - 4/9) - 1/9
Therefore, Associative property is not true for subtraction.
Subtraction of rational numbers and their properties
25. (ii) Distributive Property of Multiplication over Subtraction :
Multiplication of rational numbers is distributive over subtraction.
If a/b, c/d and e/f are any three rational numbers, then
a/b x (c/d - e/f) = a/b x c/d - a/b x e/f
Example :
1/3 x (2/5 - 1/5) = 1/3 x 1/5 = 1/15
1/3 x 2/5 - 1/3 x 1/5 :
= 2/15 - 1/15
= (2 - 1)/15
= 1/15 -----(2)
From (1) and (2),
1/3 x (2/5 - 1/5) = 1/3 x 2/5 - 1/3 x 1/5
Therefore, Multiplication is distributive over subtraction.
Subtraction of rational numbers and their properties
26. Multiplication of rational numbers and their properties
(i) Closure Property :
The product of two rational numbers is always a rational number. Hence Q is closed under
multiplication.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = ac/bd is also a rational number.
Example :
5/9 x 2/9 = 10/81 is a rational number.
(ii) Commutative Property :
Multiplication of rational numbers is commutative.
If a/b and c/d are any two rational numbers, then
(a/b) x (c/d) = (c/d) x (a/b)
Example :
5/9 x 2/9 = 10/81
2/9 x 5/9 = 10/81
So,
5/9 x 2/9 = 2/9 x 5/9
Therefore, Commutative property is true for multiplication.
27. (iii) Associative Property :
Multiplication of rational numbers is associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b x (c/d x e/f) = (a/b x c/d) x e/f
Example :
2/9 x (4/9 x 1/9) = 2/9 x 4/81 = 8/729
(2/9 x 4/9) x 1/9 = 8/81 x 1/9 = 8/729
So,
2/9 x (4/9 x 1/9) = (2/9 x 4/9) x 1/9
Therefore, Associative property is true for multiplication.
(iv) Multiplicative Identity :
The product of any rational number and 1 is the rational number itself. ‘One’ is
the multiplicative identity for rational numbers.
If a/b is any rational number, then
a/b x 1 = 1 x a/b = a/b
Example :
5/7 x 1 = 1 x 5/7 = 5/7
Multiplication of rational numbers and their properties
28. (v) Multiplication by 0 :
Every rational number multiplied with 0 gives 0.
If a/b is any rational number, then
a/b x 0 = 0 x a/b = 0
Example :
5/7 x 0 = 0 x 5/7 = 0
(vi) Multiplicative Inverse or Reciprocal :
For every rational number a/b, b ≠ 0, there exists a rational number c/d
such that a/b x c/d = 1. Then,
c/d is the multiplicative inverse of a/b.
If b/a is a rational number, then
a/b is the multiplicative inverse or reciprocal of it.
Example :
The multiplicative inverse of 2/3 is 3/2.
The multiplicative inverse of 1/3 is 3.
The multiplicative inverse of 3 is 1/3.
The multiplicative inverse of 1 is 1.
The multiplicative inverse of 0 is undefined.
29. Division of rational numbers and their properties
(i) Closure Property :
The collection of non-zero rational numbers is closed under division.
If a/b and c/d are two rational numbers, such that c/d ≠ 0, then
a/b ÷ c/d is always a rational number.
Example :
2/3 ÷ 1/3 = 2/3 x 3/1 = 2 is a rational number.
(ii) Commutative Property :
Division of rational numbers is not commutative.
If a/b and c/d are two rational numbers, then
a/b ÷ c/d ≠ c/d ÷ a/b
Example :
2/3 ÷ 1/3 = 2/3 x 3/1 = 2
1/3 ÷ 2/3 = 1/3 x 3/2 = 1/2
And,
2/3 ÷ 1/3 ≠ 1/3 ÷ 2/3
Therefore, Commutative property is not true for division.
30. (iii) Associative Property :
Division of rational numbers is not associative.
If a/b, c/d and e/f are any three rational numbers, then
a/b ÷ (c/d ÷ e/f) ≠ (a/b ÷ c/d) ÷ e/f
Example :
2/9 ÷ (4/9 ÷ 1/9) = 2/9 ÷ 4 = 1/18
(2/9 ÷ 4/9) ÷ 1/9 = 1/2 - 1/9 = 7/18
And,
2/9 ÷ (4/9 ÷ 1/9) ≠ (2/9 ÷ 4/9) ÷ 1/9
Therefore, Associative property is not true for division.
Division of rational numbers and their properties
32. SUMMARY
Rational number general form:
𝑝
𝑞
where 𝑝, 𝑞 are integers and 𝑝 ≠ 0
All integers and fractions are rational numbers
To find rational number between two rational numbers, calculate their average
There are infinitely many rational numbers between any two distinct rational numbers
Comparing and ordering of rational numbers are done by comparing their numerators after making their
denominators common (by taking LCM)
Decimal representation of rational number can either be terminating or non-terminating recurring
Operation
Property
Addition Subtraction Multiplication Division
Closure property ✓ ✓ ✓ ❌
Commutative ✓ ❌ ✓ ❌
Associative ✓ ❌ ✓ ❌
Existence of Inverse ✓ ✓
Existence of Identity ✓ ✓
33. Rational numbers on number line
Watch to know how? - https://youtu.be/G9n8HbMdUIk?t=2300
Compare & Order rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=CbrfJPv2qP8
Rational numbers between two rational numbers
Watch the video to know how: https://www.youtube.com/watch?v=lg04THe8wfY