The document provides an overview of mathematics and algebra concepts through examples and word problems. It begins with an introduction to algebra, explaining how it involves using symbols and rules to represent mathematical relationships. Several word problems are then presented and solved step-by-step using algebraic expressions and equations. Key concepts covered include addition, subtraction, multiplication, division, ratios, equations, and solving for unknown values. The document emphasizes translating problems into algebraic form as the first step to finding their solutions.
The document summarizes a mathematics lesson taught by teacher Aswathy A about equations. The 40 student class focused on understanding different problems in equations through discussion, observation, and analyzing notes. Students learned about algebraic methods of solving problems and the concept of algebraic methods in equations. Through various classroom activities using word problems, students practiced forming and solving algebraic equations to find unknown values like the number of children among a group or the number of original boys and girls in a class. The lesson aimed to help students recall terms like algebra and algebraic form, explain equations, and learn to form and solve different types of algebraic equations.
The document is a digital textbook on mathematics for 8th standard. It contains explanations and examples on solving equations using addition, subtraction, multiplication, division and algebraic methods. It also includes index and exercises with word problems related to equations on different topics.
Detailed Lesson plan of Product Rule for Exponent Using the Deductive MethodLorie Jane Letada
The document outlines the procedures for a lesson on the product rule for exponent-like terms with exponents. It includes the objectives, subject content, materials, and steps of the lesson. The teacher leads the students in examples of applying the product rule to simplify expressions with the same bases and adds the exponents. Students then practice applying the rule to example expressions on their own.
The document provides information and steps to complete a three circle Venn diagram representing survey results from a class of 40 students on their pet preferences of cats, dogs, and birds. It gives some values to place directly on the diagram and other information to help determine missing values. It works through placing the known values and using the additional information to logically determine the remaining unknown values step-by-step to fully complete the Venn diagram.
This document introduces pairs of equations and how they can be used to solve word problems. It provides examples like using equations to find the price of a chair if four chairs and a table were bought for a total price, or setting up equations to determine the length and breadth of a rectangle given its perimeter. The document encourages the reader to practice setting up and solving their own word problems using pairs of equations. It provides 5 sample word problems for the reader to work through with the answers given.
(8) Inquiry Lab - Equations with Variables on Each Sidewzuri
This document discusses how to use algebra tiles to model and solve linear equations. It provides examples of modeling equations such as 4x + 2 = 2x + 8 and 3x + 3 = 2x - 3 using tiles. The key steps are to model the equation with tiles, then remove tiles from both sides until just one variable is left. This isolates the variable and reveals the solution. Properties of equality like adding the same amount to both sides are demonstrated through the tile manipulations.
The document is a slide presentation on mathematics learning in Singapore given by Yeap Ban Har from the Marshall Cavendish Institute in Singapore. It discusses Singapore's history of improving mathematics education over time, from achieving low passing rates on early exams to consistently high performance on international tests. It also describes Singapore's focus on visual and concrete learning approaches, as well as the country's emphasis on developing intellectual competence through mathematics.
Differentiates expression from equation, Translate word phrase to numerical e...April Rose Anin
This document outlines a lesson plan on teaching mathematical expressions and equations to 6th grade students. The objectives are for students to differentiate between expressions and equations, translate word phrases to numerical expressions, and write simple equations. The lesson procedures include a review game, discussion of new concepts like expressions and equations, practice exercises, and a group activity to identify examples. Student understanding is evaluated through practice problems writing expressions and solving equations. The teacher reflects on teaching strategies and seeks help from the principal on any difficulties encountered.
The document summarizes a mathematics lesson taught by teacher Aswathy A about equations. The 40 student class focused on understanding different problems in equations through discussion, observation, and analyzing notes. Students learned about algebraic methods of solving problems and the concept of algebraic methods in equations. Through various classroom activities using word problems, students practiced forming and solving algebraic equations to find unknown values like the number of children among a group or the number of original boys and girls in a class. The lesson aimed to help students recall terms like algebra and algebraic form, explain equations, and learn to form and solve different types of algebraic equations.
The document is a digital textbook on mathematics for 8th standard. It contains explanations and examples on solving equations using addition, subtraction, multiplication, division and algebraic methods. It also includes index and exercises with word problems related to equations on different topics.
Detailed Lesson plan of Product Rule for Exponent Using the Deductive MethodLorie Jane Letada
The document outlines the procedures for a lesson on the product rule for exponent-like terms with exponents. It includes the objectives, subject content, materials, and steps of the lesson. The teacher leads the students in examples of applying the product rule to simplify expressions with the same bases and adds the exponents. Students then practice applying the rule to example expressions on their own.
The document provides information and steps to complete a three circle Venn diagram representing survey results from a class of 40 students on their pet preferences of cats, dogs, and birds. It gives some values to place directly on the diagram and other information to help determine missing values. It works through placing the known values and using the additional information to logically determine the remaining unknown values step-by-step to fully complete the Venn diagram.
This document introduces pairs of equations and how they can be used to solve word problems. It provides examples like using equations to find the price of a chair if four chairs and a table were bought for a total price, or setting up equations to determine the length and breadth of a rectangle given its perimeter. The document encourages the reader to practice setting up and solving their own word problems using pairs of equations. It provides 5 sample word problems for the reader to work through with the answers given.
(8) Inquiry Lab - Equations with Variables on Each Sidewzuri
This document discusses how to use algebra tiles to model and solve linear equations. It provides examples of modeling equations such as 4x + 2 = 2x + 8 and 3x + 3 = 2x - 3 using tiles. The key steps are to model the equation with tiles, then remove tiles from both sides until just one variable is left. This isolates the variable and reveals the solution. Properties of equality like adding the same amount to both sides are demonstrated through the tile manipulations.
The document is a slide presentation on mathematics learning in Singapore given by Yeap Ban Har from the Marshall Cavendish Institute in Singapore. It discusses Singapore's history of improving mathematics education over time, from achieving low passing rates on early exams to consistently high performance on international tests. It also describes Singapore's focus on visual and concrete learning approaches, as well as the country's emphasis on developing intellectual competence through mathematics.
Differentiates expression from equation, Translate word phrase to numerical e...April Rose Anin
This document outlines a lesson plan on teaching mathematical expressions and equations to 6th grade students. The objectives are for students to differentiate between expressions and equations, translate word phrases to numerical expressions, and write simple equations. The lesson procedures include a review game, discussion of new concepts like expressions and equations, practice exercises, and a group activity to identify examples. Student understanding is evaluated through practice problems writing expressions and solving equations. The teacher reflects on teaching strategies and seeks help from the principal on any difficulties encountered.
Solving Equations by Factoring KTIP lesson planJosephine Neff
1) The document is a lesson plan for teaching 9th grade algebra students how to factor quadratic equations and use factoring to solve equations and real-world problems.
2) The lesson involves reviewing factoring patterns, teaching students to factor quadratic equations in standard form using various methods, and using factoring to solve physics problems involving height, speed, and time.
3) Formative and summative assessments are used to check students' understanding of factoring quadratic trinomials and using factoring to solve equations.
This document provides a lesson on solving sequence problems using Singapore bar models. It introduces consecutive numbers and examples. Students are asked to draw a picture to represent a story involving a bathtub of jello falling from a plane. The jello splits into more pieces at each mile, following a pattern of doubling. Students are then given practice problems to find three or four consecutive numbers that add up to a given total.
Srinivasa Ramanujan (1887-1920) was a renowned Indian mathematician. He showed early signs of genius in mathematics as a child. He was self-taught and made significant contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan developed his own theorems and results without any formal training. He was recognized by mathematicians in England and his work was found to be of extraordinary significance. However, he struggled with poor health and poverty, and ultimately died young at the age of 32 in India. Ramanujan is celebrated annually in India on his birthday as a brilliant mathematician who made major contributions despite facing disadvantages.
History of ramanujan's and his contributions in mathematics by, sandeepsandeep kumar singu
Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite his lack of formal training. He was born in 1887 in Tamil Nadu, India and showed an extraordinary aptitude for mathematics from a young age. Ramanujan's work attracted the attention of the English mathematician G.H. Hardy, who brought him to Cambridge University in England. There, Ramanujan continued his groundbreaking mathematical research until his untimely death in 1920 at the age of 32.
The document provides a mathematics curriculum guide for third grade students in the Isaac School District. It focuses on unit 8 which covers addition, subtraction, and number systems over 3 sessions. The unit teaches students that numbers can be represented in many ways and used to solve problems. Students will learn about relationships between numbers, place value, and comparing and ordering whole numbers. They will solve 2-step word problems using the four operations and identify arithmetic patterns. Students will also learn to fluently add and subtract within 1000 using strategies based on place value.
Srinivasa Ramanujan Date Of Birth 22.12.1887Padma Lalitha
In last slide I have mentioned Srinivasa Ramanujan D.O.B. as
22.12.1987. I am extremely sorry for that. Please read it as 22.12.1887. Thanks to my friend Smt. Indira, who brought it to my notice.
This document introduces students to using ratio tables and double number line diagrams to solve ratio problems. It provides an example of using these tools to determine how long it will take Katie to make 8 bracelets if it takes her 20 minutes to make 5 bracelets. The document guides students through setting up a ratio table, stretching it out, creating a double number line diagram, using the diagram to determine the time, and checking that the answer is reasonable. It concludes with an exit ticket problem for students to solve on their own using a double number line diagram.
Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He was born in 1887 in a small village in India and showed a strong aptitude for mathematics from a young age, teaching himself advanced mathematical concepts from books. Despite facing health and financial issues that prevented him from attending university, he gained recognition for his brilliant work on mathematical theories and continued to make significant contributions on his own.
Srinivasa Ramanujan was a renowned Indian mathematician born in 1887 in Tamil Nadu, India. He made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Some of his key achievements include formulating the Ramanujan prime, the Ramanujan theta function, and discovering highly composite numbers. The Indian government celebrates December 22nd as National Mathematics Day in honor of his birth anniversary. Ramanujan worked closely with mathematician G.H. Hardy at Trinity College, Cambridge and together they published many papers on advanced mathematical topics. Despite his short life and self-taught background, Ramanujan produced groundbreaking mathematical innovations and inspired many to pursue mathematics.
This lesson plan is about teaching students about quadrilaterals. It involves several hands-on activities and discussions to help students understand the different types of quadrilaterals, including squares, rectangles, parallelograms, rhombi, trapezoids, and isosceles trapezoids. The students are divided into groups to describe shapes and identify properties. The teacher also tells a story about "King Quadrilateral" and his family to reinforce the names and relationships between the different quadrilaterals. The lesson aims to help students comprehend the key concepts and properties of quadrilaterals.
This document contains a collection of mathematical problems that were historically used to discriminate against Jewish applicants during oral entrance exams for the mathematics department at Moscow State University in the Soviet Union. The problems were designed to have simple solutions but be very difficult to find. The document includes 21 such problems, along with hints and full solutions. It aims to preserve these problems and their solutions for historical and mathematical value.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
The document provides a detailed lesson plan for a mathematics class on evaluating algebraic expressions. It includes objectives, subject matter, instructional procedures, evaluation, and assignment. The lesson involves identifying steps to evaluate expressions, applying those steps through group activities, and showing appreciation for group work. Students work in groups to evaluate expressions when given values for variables. They are evaluated based on accuracy, timeliness, and cooperation. For an assignment, students evaluate expressions when given different values for two variables.
1) Srinivasa Ramanujan was one of India's greatest mathematical geniuses who made substantial contributions to analytical number theory, elliptic functions, and infinite series.
2) He was mostly self-taught and showed extraordinary talent from a young age, mastering advanced mathematical concepts from books he received.
3) Ramanujan struggled for recognition in India but eventually his work was brought to the attention of the English mathematician G.H. Hardy, who helped arrange for Ramanujan to travel to Cambridge University in 1914 where he spent five productive years collaborating before falling ill and returning to India, where he passed away in 1920.
The document provides a detailed lesson plan for a mathematics class on evaluating algebraic expressions. It includes objectives, subject matter, instructional procedures, evaluation, and assignment. The lesson involves identifying steps to evaluate expressions, applying those steps through group activities, and showing appreciation for group work. Students work in groups to evaluate expressions when given values for variables. They are evaluated on accuracy, timeliness, and cooperation. For an assignment, students evaluate expressions when given different values for two variables.
Srinivasa ramanujan a great indian mathematicianKavyaBhatia4
Srinivasa Ramanujan was an Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. He developed his mathematical abilities largely through self-study and had a natural genius for mathematics. Ramanujan received recognition during his lifetime, including being elected as a Fellow of the Royal Society, but he died young at age 32. Even on his deathbed, he was working on theorems. His notebooks contained thousands of results without proofs that mathematicians have since worked to prove. Ramanujan's life and accomplishments have inspired biographies and films that highlight his brilliance in mathematics that was largely self-taught.
(1) Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite having little formal training in pure mathematics.
(2) He was born in 1887 in India and showed an extraordinary aptitude for mathematics from a young age, mastering advanced mathematical concepts including trigonometry at age 13.
(3) Ramanujan received recognition for his genius and was invited to study at Trinity College, Cambridge in England. However, he struggled with the climate and culture in England and his health declined, and he ultimately returned to India where he passed away in 1920 at the young age of 32.
The document discusses various math problems and their solutions using algebraic methods. It introduces concepts like writing word problems algebraically by denoting the unknown as a variable like x. It explains how to solve problems by doing the inverse operations in reverse order, and discusses how this approach can be written algebraically. The document also discusses the history of algebra and its origins from ancient Egyptian and Arab mathematicians.
The document provides step-by-step solutions to 4 math problems involving rational functions, polynomials, and perimeter/area calculations. The first problem asks whether a farmer has enough fence to enclose a field given the perimeter needed and available fence. The next problems involve simplifying a rational expression, graphing a rational function, and finding the domain and range of a polynomial function. The reflection at the end discusses choosing these problems because they cover a variety of math concepts and allowed the author to feel confident explaining the material to others.
Solving Equations by Factoring KTIP lesson planJosephine Neff
1) The document is a lesson plan for teaching 9th grade algebra students how to factor quadratic equations and use factoring to solve equations and real-world problems.
2) The lesson involves reviewing factoring patterns, teaching students to factor quadratic equations in standard form using various methods, and using factoring to solve physics problems involving height, speed, and time.
3) Formative and summative assessments are used to check students' understanding of factoring quadratic trinomials and using factoring to solve equations.
This document provides a lesson on solving sequence problems using Singapore bar models. It introduces consecutive numbers and examples. Students are asked to draw a picture to represent a story involving a bathtub of jello falling from a plane. The jello splits into more pieces at each mile, following a pattern of doubling. Students are then given practice problems to find three or four consecutive numbers that add up to a given total.
Srinivasa Ramanujan (1887-1920) was a renowned Indian mathematician. He showed early signs of genius in mathematics as a child. He was self-taught and made significant contributions to mathematical analysis, number theory, infinite series, and continued fractions. Ramanujan developed his own theorems and results without any formal training. He was recognized by mathematicians in England and his work was found to be of extraordinary significance. However, he struggled with poor health and poverty, and ultimately died young at the age of 32 in India. Ramanujan is celebrated annually in India on his birthday as a brilliant mathematician who made major contributions despite facing disadvantages.
History of ramanujan's and his contributions in mathematics by, sandeepsandeep kumar singu
Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite his lack of formal training. He was born in 1887 in Tamil Nadu, India and showed an extraordinary aptitude for mathematics from a young age. Ramanujan's work attracted the attention of the English mathematician G.H. Hardy, who brought him to Cambridge University in England. There, Ramanujan continued his groundbreaking mathematical research until his untimely death in 1920 at the age of 32.
The document provides a mathematics curriculum guide for third grade students in the Isaac School District. It focuses on unit 8 which covers addition, subtraction, and number systems over 3 sessions. The unit teaches students that numbers can be represented in many ways and used to solve problems. Students will learn about relationships between numbers, place value, and comparing and ordering whole numbers. They will solve 2-step word problems using the four operations and identify arithmetic patterns. Students will also learn to fluently add and subtract within 1000 using strategies based on place value.
Srinivasa Ramanujan Date Of Birth 22.12.1887Padma Lalitha
In last slide I have mentioned Srinivasa Ramanujan D.O.B. as
22.12.1987. I am extremely sorry for that. Please read it as 22.12.1887. Thanks to my friend Smt. Indira, who brought it to my notice.
This document introduces students to using ratio tables and double number line diagrams to solve ratio problems. It provides an example of using these tools to determine how long it will take Katie to make 8 bracelets if it takes her 20 minutes to make 5 bracelets. The document guides students through setting up a ratio table, stretching it out, creating a double number line diagram, using the diagram to determine the time, and checking that the answer is reasonable. It concludes with an exit ticket problem for students to solve on their own using a double number line diagram.
Srinivasa Ramanujan was one of India's greatest mathematical geniuses. He was born in 1887 in a small village in India and showed a strong aptitude for mathematics from a young age, teaching himself advanced mathematical concepts from books. Despite facing health and financial issues that prevented him from attending university, he gained recognition for his brilliant work on mathematical theories and continued to make significant contributions on his own.
Srinivasa Ramanujan was a renowned Indian mathematician born in 1887 in Tamil Nadu, India. He made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Some of his key achievements include formulating the Ramanujan prime, the Ramanujan theta function, and discovering highly composite numbers. The Indian government celebrates December 22nd as National Mathematics Day in honor of his birth anniversary. Ramanujan worked closely with mathematician G.H. Hardy at Trinity College, Cambridge and together they published many papers on advanced mathematical topics. Despite his short life and self-taught background, Ramanujan produced groundbreaking mathematical innovations and inspired many to pursue mathematics.
This lesson plan is about teaching students about quadrilaterals. It involves several hands-on activities and discussions to help students understand the different types of quadrilaterals, including squares, rectangles, parallelograms, rhombi, trapezoids, and isosceles trapezoids. The students are divided into groups to describe shapes and identify properties. The teacher also tells a story about "King Quadrilateral" and his family to reinforce the names and relationships between the different quadrilaterals. The lesson aims to help students comprehend the key concepts and properties of quadrilaterals.
This document contains a collection of mathematical problems that were historically used to discriminate against Jewish applicants during oral entrance exams for the mathematics department at Moscow State University in the Soviet Union. The problems were designed to have simple solutions but be very difficult to find. The document includes 21 such problems, along with hints and full solutions. It aims to preserve these problems and their solutions for historical and mathematical value.
K TO 12 GRADE 7 LEARNING MODULE IN MATHEMATICS (Quarter 3)LiGhT ArOhL
This document provides a lesson plan on solving linear equations and inequalities in one variable algebraically. It begins with reviewing translating between verbal and mathematical phrases and evaluating expressions. The main focus is on introducing and applying the properties of equality, including reflexive, symmetric, transitive, and substitution properties, to solve equations algebraically. Word problems involving equations in one variable are also discussed. The objectives are to identify and apply the properties of equality to find solutions to equations and solve word problems involving one variable equations.
The document provides a detailed lesson plan for a mathematics class on evaluating algebraic expressions. It includes objectives, subject matter, instructional procedures, evaluation, and assignment. The lesson involves identifying steps to evaluate expressions, applying those steps through group activities, and showing appreciation for group work. Students work in groups to evaluate expressions when given values for variables. They are evaluated based on accuracy, timeliness, and cooperation. For an assignment, students evaluate expressions when given different values for two variables.
1) Srinivasa Ramanujan was one of India's greatest mathematical geniuses who made substantial contributions to analytical number theory, elliptic functions, and infinite series.
2) He was mostly self-taught and showed extraordinary talent from a young age, mastering advanced mathematical concepts from books he received.
3) Ramanujan struggled for recognition in India but eventually his work was brought to the attention of the English mathematician G.H. Hardy, who helped arrange for Ramanujan to travel to Cambridge University in 1914 where he spent five productive years collaborating before falling ill and returning to India, where he passed away in 1920.
The document provides a detailed lesson plan for a mathematics class on evaluating algebraic expressions. It includes objectives, subject matter, instructional procedures, evaluation, and assignment. The lesson involves identifying steps to evaluate expressions, applying those steps through group activities, and showing appreciation for group work. Students work in groups to evaluate expressions when given values for variables. They are evaluated on accuracy, timeliness, and cooperation. For an assignment, students evaluate expressions when given different values for two variables.
Srinivasa ramanujan a great indian mathematicianKavyaBhatia4
Srinivasa Ramanujan was an Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. He developed his mathematical abilities largely through self-study and had a natural genius for mathematics. Ramanujan received recognition during his lifetime, including being elected as a Fellow of the Royal Society, but he died young at age 32. Even on his deathbed, he was working on theorems. His notebooks contained thousands of results without proofs that mathematicians have since worked to prove. Ramanujan's life and accomplishments have inspired biographies and films that highlight his brilliance in mathematics that was largely self-taught.
(1) Srinivasa Ramanujan was a renowned Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions despite having little formal training in pure mathematics.
(2) He was born in 1887 in India and showed an extraordinary aptitude for mathematics from a young age, mastering advanced mathematical concepts including trigonometry at age 13.
(3) Ramanujan received recognition for his genius and was invited to study at Trinity College, Cambridge in England. However, he struggled with the climate and culture in England and his health declined, and he ultimately returned to India where he passed away in 1920 at the young age of 32.
The document discusses various math problems and their solutions using algebraic methods. It introduces concepts like writing word problems algebraically by denoting the unknown as a variable like x. It explains how to solve problems by doing the inverse operations in reverse order, and discusses how this approach can be written algebraically. The document also discusses the history of algebra and its origins from ancient Egyptian and Arab mathematicians.
The document provides step-by-step solutions to 4 math problems involving rational functions, polynomials, and perimeter/area calculations. The first problem asks whether a farmer has enough fence to enclose a field given the perimeter needed and available fence. The next problems involve simplifying a rational expression, graphing a rational function, and finding the domain and range of a polynomial function. The reflection at the end discusses choosing these problems because they cover a variety of math concepts and allowed the author to feel confident explaining the material to others.
The document discusses techniques from Vedic mathematics for performing calculations more easily and quickly in one's head. It provides examples of using vertical and crosswise multiplication to multiply two-digit numbers in a single line. This technique can be adapted for division, addition, subtraction and other operations. It also presents "tricks" for mentally multiplying or squaring numbers near multiples of 10, multiplying by 9 or 11, and squaring two-digit numbers ending in 5. The goal is to make calculations faster and more intuitive through Vedic mathematical formulas.
The document discusses circles and related geometric terms. It begins by asking students to bring various circular objects to class and lists examples like coins, bottle caps, and food container lids. It then covers identifying parts of a circle like the diameter, radius, chord, and center. Students are instructed to fold paper circles to mark these points and parts. The document provides examples of using circles in daily life and has students make origami shapes demonstrating circle measurements. It concludes by having students draw a labeled circle diagram.
This document discusses solving linear equations in one unknown. It begins by listing 9 objectives related to understanding linear equations and using properties of equalities to solve them. Examples are then provided of solving linear equations by using addition, subtraction, multiplication, division and multiple properties. Techniques for expressing consecutive integers in terms of a variable are described. The document concludes by discussing George Polya's four steps for problem solving and providing example problems and solutions.
This document provides 30 algebra tricks to help students master the subject more easily. Some key tricks discussed include:
- Understanding basic rules like how signs change when terms are transferred across the equal sign in addition, subtraction, multiplication and division.
- Simplifying expressions by turning all negative signs positive or using cross-multiplication to solve fractional equations more quickly.
- Using techniques for squaring numbers like recognizing numbers are a certain amount above or below a multiple of 10.
- Memorizing tricks for multiplying or dividing specific numbers like 11 or numbers closer to bases like 10 or 100.
- Learning indicators for divisibility like a number being divisible by 3 if the sum of its digits is divisible by 3.
The document defines and describes different types of real numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It provides examples of each type of number. Real numbers consist of all rational and irrational numbers. A Venn diagram shows the relationships between the different subsets of real numbers. Euclid's division algorithm and its application to find the highest common factor of two numbers is also explained in the document.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding highest common factors and lowest common multiples. Examples of proving the irrationality of square roots like √5 are given.
The document discusses different types of real numbers including rational and irrational numbers. It provides examples and definitions of natural numbers, whole numbers, integers, rational numbers, and irrational numbers. It also includes information on Euclid's division algorithm and its application in finding the highest common factor of two numbers. Examples are provided to illustrate the algorithm.
This document provides a 3-sentence summary of a mathematics textbook chapter on polygons:
The chapter discusses different types of polygons based on their number of sides, such as triangles having three sides and pentagons having five sides. It also presents formulas for calculating the sum of interior angles in polygons by dividing them into simpler shapes. Regular polygons are defined as those with all sides of equal length and all interior angles equal.
500 most asked apti ques in tcs, wipro, infos(105pgs)PRIYANKKATIYAR2
This document provides 100 numerical aptitude questions and solutions that are commonly asked in campus recruitment drives by companies like Infosys, TCS, CTS, Wipro and Accenture. The questions cover topics such as number systems, permutations, combinations, time and work problems, percentages, profit and loss, and geometry. Shortcuts and tips are provided to solve problems more quickly. The questions are divided into parts for each company and an index provides the topic distribution of questions for each company.
This document contains a summary of cubes and cube roots. It begins with an introduction discussing how the mathematician Ramanujan recognized a number's interesting property. It then defines cubes and cube numbers, provides examples of cubes of numbers from 1 to 10, and discusses patterns with cubes. Finally, it covers finding cube roots through prime factorization and estimation from digit groups of cube numbers. The document contains examples and exercises related to cubes and cube roots.
This document is about Fibonacci and Lucas numbers. It begins with an introduction to the Fibonacci sequence and how it relates to a rabbit breeding problem. It then defines the Fibonacci sequence recursively and provides Binet's formula for calculating individual Fibonacci numbers. It also introduces the Lucas sequence, which is similar but begins with different starting values. The bulk of the document covers several important identities involving Fibonacci and Lucas numbers, including Lucas' identity relating sums of early Fibonacci numbers to later ones, Cassini's formula involving differences of Fibonacci numbers, and others. It proves these identities through recurrence relations and mathematical induction.
The document provides instructions and examples for various math concepts:
1) It explains how to round numbers to a given place value or significant figure, and provides examples of rounding 89,475 to the hundredth place and to one significant figure.
2) It demonstrates how to translate English phrases into math equations, such as "Max scored 2 times more goals than bob" becoming M=2B.
3) It defines index notations, square numbers, cube numbers, and provides examples of each.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times the other's denominator equals the other's numerator times the denominator.
This document provides an introduction to rational numbers. It defines rational numbers as numbers that can be written as fractions where the numerator and denominator are integers. This includes integers. It explains that rational numbers can be written in different equivalent forms by multiplying or dividing the numerator and denominator by the same integer. The document then discusses how to add, subtract, multiply and divide rational numbers. It presents the algebraic rules for doing each operation and provides examples. It also explains how to check if two fractions are equal by seeing if their numerators times denominators are the same.
Tutorial linear equations and linear inequalitieskhyps13
This document discusses linear equations and inequalities in one variable. It begins by defining open sentences, variables, and solutions. It then covers topics like solving linear equations using addition, subtraction, multiplication, and division. It also discusses solving multi-step equations. Graphing solutions to equations is explained. The document also covers understanding and solving linear inequalities in one variable as well as graphing inequalities. It provides examples of how equations and inequalities can be applied to everyday situations.
SQUARES AND SQUARE ROOTS.pptx powerpoint presentation square and square roots...8802952585rani
The document discusses squares and square roots. It defines a square number as a number multiplied by itself. It provides examples of perfect square numbers and their properties, such as being even or odd and having an even number of zeros at the end. The document also discusses finding the square root as the inverse of squaring a number using prime factorization or long division. Pythagorean triplets are introduced as sets of numbers that satisfy the Pythagorean theorem for a right triangle. In conclusion, the key properties of square numbers are that they end in 0, 1, 4, 5, 6, or 9 and have an even number of zeros at the end.
Beyond Degrees - Empowering the Workforce in the Context of Skills-First.pptxEduSkills OECD
Iván Bornacelly, Policy Analyst at the OECD Centre for Skills, OECD, presents at the webinar 'Tackling job market gaps with a skills-first approach' on 12 June 2024
Communicating effectively and consistently with students can help them feel at ease during their learning experience and provide the instructor with a communication trail to track the course's progress. This workshop will take you through constructing an engaging course container to facilitate effective communication.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
তাই একজন নাগরিক হিসাবে এই তথ্য গুলো আপনার জানা প্রয়োজন ...।
বিসিএস ও ব্যাংক এর লিখিত পরীক্ষা ...+এছাড়া মাধ্যমিক ও উচ্চমাধ্যমিকের স্টুডেন্টদের জন্য অনেক কাজে আসবে ...
3. 3
index
content
Page number
1. Introduction 4
2. Addition and subtraction 4
3. algebra 7
4. multiplication and division 9
5. solutions of equations 11
4. 4
EQUATIONS
Teacher who has to handle the 5th period is on leave.
So the students of std VIIIA invites their maths teacher.
‘Our teacher is on leave would you please, come?’
‘Why not?’, the teacher agreed. Students are happy. For they knew that maths
teacher would discuss even problems outside the course book. Teacher would present
puzzles, games etc,. maths club very interestingly.
Teacher: Today, why don’t we start with a puzzle?
Student: Yes, teacher
‘Please take a piece of paper and pen’ said the teacher.
The students did the same ‘Write a number you like on the
paper and keep it’ Don’t show it to anybody’.’We have written said the class.’
Add 2 to the number’.’Yes.teacher’’Ok.now multiply it by 3’Yes,we
multipiled’’Yes,we mulitip lied’’Subtract5’’Ok,we did’, Subtract the original number,
multiply by 2 and then subtract1’’It ‘s OK, teacher ‘Now it’s my turn. You say the
final number and I’II say the original number.
Well begin with kripa.’61’Kripa said. “your number is 15.Is that
correct,Kripa?”Yes,teacher,’Now haritha say the number’ “65’’Is it
16,Haritha?’Yes,HarithaTeacher ‘You’re absolutely correct teacher. How do you
make it?’Well learn the trick in our new lesson ‘EQUATIONS’
ACTIVITY
ADDITION AND SUBTRACTION:
Appu came back from the market with a bag of vegetables and other things.
Mother asked him to keep the change. It keep the change. It was 5 rupees.’Now my
saving have reached 50’.Appu said how much did he have before getting this 5 rupees?
5. 5
His savings became 50,when he got 5 rupees more. So he must have 50-5=45rupees.
Ammu bought a pen for 10 rupees from her ‘vishukaineettam’Now she
has 40 rupees remaining .It became 40 rupees,when it was reduced by 10
rupees.So it must have been 10 more than 40.
That is 40+10=50
Can’t you similarly fine the answer to the questions below?
1. Gopalan bought a bunch of bananas for his shop.7 of them had slightly turned
bad. After removing them, he had 46 left. How many were there in the bunch at
first?
The number of bananas is the bunch at first=46+7=53
2. A number subtracted from 500gave 234.what is the number subtracted?
Say the number=x
Subtracted from 500=500-x
That is 500-x=234
X=500-234=266
The number=266
6. 6
ACTIVITY
In a certain savings scheme money invested doubles in 5 years. To get
10 thousand rupees after 5 years ,how much should be invested now?
Joseph got 1500 as his share a profit from a sale. This is one-third the
total profit. What is the total profit?
The perimeter of a pentagon with equal sides is 65cms.What is the
length of each sides?
A number divided 12 gives 25.What is the number?
LOOK AT THE PROBLEM
Thrice a number and 2 together make 50.what is
the number?
Here whet were the operations done to the
number to get 50?
First multiplication by 3, then addition of 2.It
became 50,when the last 2 was added. So 50-
2=48
This means the original number multiplied by 3
gives 48.
The number =48/3=16
Thus 16 multiplied by 3 gives 48 and 2 added to
this makes 50.
What if we change the question like this. From
thrice a number ,2 is subtracted and this gives
40.What is the number?
Here what was the number before 2 was
subtracted?
40+2=42 And this is got on multiplication by 3
The first sect ion Līlāvatī (also known as
pāṭīgaṇita or aṅkagaṇita) consists of 277
verses.[6] It covers calculations, progressions,
mensurat ion, permutations, and other
topicsThe second sect ion Bījagaṇita has 213
verses.[6] It discusses zero, infinity, posit ive
and negat ive numbers, and indeterminate
equat ions including (the now called) Pell's
equat ion, solving it using a kuṭṭaka method.[6]
In part icular, he also solved the
case that was to
elude Fermat and his European
contemporaries centuries later.[6]In the third
sect ion Grahagaṇita, while t reating the
mot ion of planets, he considered their
instantaneous speeds.[6] He arrived at the
approximation:[10]
fo r clo se to , or in modern notation:[10]
.
7. 7
14 multiplied by3 gives 42and 2 subtracted from
this gives 40
ACTIVITY
1. Anitha and her friends bought some pens. For a packet of 5 pens, they got 2
rupees reduction in price. They had to pay only 18 rupees. Had they bought the
pens separately, how much would have been the price for each pen?
2. Three added to half a number gives 23. What is the number?
3. 2 Subtracted from one-third of a number gives 40. What is the number?
ACTIVITY
ALGEBRA
We are used given a number got by doing some operations on another number.
We must find the number we started with. What was the general method used?
LOOK AT THE PROBLEM ALGEBRA
1. 8 added to one-third of a number gives 15.what is
2. the number?
Let’s first write the problem in algebra
. If x/3+8=15. What is x?
Next method let’s look at the method
of solution
x/3+8=15
8. 8
x/3=15-8=7
x=7*3=21=21
Thus the original number =21
3. From the point on a line another
Line is to be drawn such in way
That, the angle on one side should
Be 500 more than the angle on the
Other. What should be the smalls
Angle?
4. A hundred rupees note was changed
Into 100 rupees notes. There were
7 notes in all .How many of each
Demonization were their?
ACTIVITY
To any number if another number is
added and then be added number
subtracted we get the original back.
This can be written using algebra like
this
(x+a)-a=x
This same fact can be put in the
different form.
If x+a=b then x= b-a
This is the algebra form of the rule for
Finding a number if the some of the
Written evidence of the use of mathematics dates
back to at least 3000 BC with the ivory labels
found in Tomb U-j at Abydos. These labels
appear to have been used as tags for grave goods
and some are inscribed with numbers.[1] Further
evidence of the use of the base 10 number system
can be found on the Narmer Macehead which
depicts offerings of 400,000 oxen, 1,422,000
goats and 120,000 prisoners.[2]
The evidence of the use of mathematics in the Old
Kingdom (ca 2690–2180 BC) is scarce, but can be
deduced from inscriptions on a wall near a
mastaba in Meidum which gives guidelines for
the slope of the mastaba.[3] The lines in the
diagram are spaced at a distance of one cubit and
show the use of that unit of measurement.[1]
The earliest true mathematical documents date to
the 12th dynasty (ca 1990–1800 BC). The
Moscow Mathematical Papyrus, the Egyptian
Mathematical Leather Roll, the Lahun
Mathematical Papyri which are a part of the much
larger collection of Kahun Papyri and the Berlin
Papyrus 6619 all date to this period. The Rhind
Mathematical Papyrus which dates to the Second
Intermediate Period (ca 1650 BC) is said to be
based on an older mathematical text from the 12th
dynasty.[4]
9. 9
number with another number and the
number added known .
Similarly, we have the following rules also
If x-a=b then x= b+a
This is the algebra form of the rule for Finding a number when the result of subtracting
another number form. This number and the number subtracted are known.
MULTIPLICATION AND DIVISION
To get a number from its product with another number, we must divide the product
by the number with which the original number was multiplied. Similarly to get a
number its quotient by another number, we must multiply the quotient by the number
by which the original was divided. Using algebra we can write this as :
If ax=b and (a±0) then x=b/a and if x/a = b then x= ab
LOOK AT THE PROBLEM ALGEBRA
1. If there a number for which its double and triple are equal?
If there a number unchanged by multiplication?
Yes! Zero
That is, if x=0 then 2x = 3x
2. Is there a number such that one added to its double gives its triple?
In the language of algebra, the question becomes, is there a number x, such that
2x+1 = 3x(there x is not equal to 0)
It can be one only.
If x = 1
Then 2x1 +1=3
If x=1 then 3x1 =3
Thus if the x= 1
Then the number 2+1 and 3x are both equal to 3
3. When we added to 10 to 2 times a number, we get four times that number.
What’s number its it?
10. 10
Lets write x for the number and translate the problem to algebra.
If 2x +10= 4x, then what is x?
2x – 4x = -10
-2x =-10
X=5
4. Ajayan is 10 years olde than
Vijayan. Next year, ajayans age
Would be twice vijayans age.
How old are they now?
Lets vijayan be x
Then ajayans age = x+10
After 1 year, vijayan’s age would
Be(x+1)and ajayans age would we
(X+10)+1 = x+11 algebra form of
the problem being
x+11= x(x+1)=2x+2
how do we find x from this.
If from the sum x+11
These subtract x then we get 11
At this statement x+11= 2x+2
Tells that the numbers x+11 and
2x+2 are the same
So the as in the first example
(2x+2)- x= (x+11)-x=11
This means x+2=11
X= 11-2=9
So vijayan’s age is 9 and ajayan’s
age is 19
Algebra (from Arabic al-jebr meaning "reunion of
broken parts"[1]) is one of the broad parts of
mathematics, together with number theory, geometry
and analysis. In its most general form algebra is the
study of symbols and the rules for manipulating
symbols[2] and is a unifying thread of all of
mathematics.[3] As such, it includes everything from
elementary equation solving to the study of abstractions
such as groups, rings, and fields. The more basic parts of
algebra are called elementary algebra, the more abstract
parts are called abstract algebra or modern algebra.
Elementary algebra is essential for any study of
mathematics, science, or engineering, as well as such
applications as medicine and economics. Abstract
algebra is a major area in advanced mathematics, studied
primarily by professional mathematicians. Much early
work in algebra, as the Arabic origin of its name
suggests, was done in the Near East, by such
mathematicians as Omar Khayyam (1050-1123).
Elementary algebra differs from arithmetic in the use of
abstractions, such as using letters to stand for numbers
that are either unknown or allowed to take on many
values.[4] For example, in the letter is
unknown, but the law of inverses can be used to discover
its value: . In , the letters and
are variables, and the letter is a constant. Algebra gives
methods for solving equations and expressing formulas
that are much easier (for those who know how to use
them) than the older method of writing everything out in
words
.
11. 11
SOLUTIONS OF EQUATIONS:
We have seen many examples in the lesson of how we can translate mathematical
problems to algebraic equations and the numbers for which these are true, are the
answers to the problem.
The numbers for which an algebraic equation is true are called solutions of the
equation and the process of finding the solutions is called solving the equation.
Example : solving the equation 2x=10 means finding the number whose double is 10
and the solution is x=5
FORMATIVE EVALUATION:
1. Cash prize is distributed among the first three places in a science exhibition. The
second place is 5/6 part of the money of the first place. The third prize is 4/5 part
of the second. If the cash distributed is 1500 rupees. How much is each prize?
2. One angle of triangle is 1/3 of another angle. The third angle is 260 more than
that angle. Find the three angles?
3. Find the three consecutive negative numbers whose sum is -54?
4. The perimeter of a triangle is 49cm.One side is 7cm more than the second side
and 5cm less than the third side. Find the length of the three sides?
5. Ramesan framed the equation 4(2x-3)+5(3x-4)=14 to find the number in a verbal
problem. What is the number?