INSTRUMENTATION IN

MATHEMATICSTNESC

HISTORY OF MATH

Before the Ancient Greeks: Egyptians and Babylonians (c. 2000 BC):Knowledge comes from “papyri”Rhind Papyrus

A brief history of Mathematics

•Main source: Plimpton 322•Sexagesimal (base-sixty) originated with ancient Sumerians (2000s BC), transmitted to Babylonians … still used —for measuring time, angles, and geographic coordinates

Babylonian Math

• Thales (624-548 BC)• Pythagoras of Samos (ca. 580 - 500 BC)• Zeno: paradoxes of the infinite • 410- 355 BC- Eudoxus of Cnidus (theory of proportion)• Appolonius (262-190): conics/astronomy• Archimedes (c. 287-212 BC)

Greek Mathematicians

Archimedes, Syracuse

Euclid (c 300 BC), Alexandria

Ptolemy (AD 83–c.168),

Roman Egypt

• Almagest: comprehensive treatise on geocentric astronomy

• Link from Greek to Islamic to European science

Al-Khwārizmī (780-850), Persia

• Algebra, (c. 820): first book on the systematic solution of linear and quadratic equations.

• He is considered as the father of algebra:

• Algorithm: westernized version of his name.

Leonardo of Pisa (c. 1170 – c. 1250) aka Fibonacci

• Brought Hindu-Arabic numeral system to Europe through the publication of his Book of Calculation, the Liber Abaci.

• Fibonacci numbers, constructed as an example in the Liber Abaci.

• illegitimate child of Fazio Cardano, a friend of Leonardo da Vinci.

• He published the solutions to the cubic and quartic equations in his 1545 book Ars Magna.

• The solution to one particular case of the cubic, x3 + ax = b (in modern notation), was communicated to him by Niccolò Fontana Tartaglia (who later claimed that Cardano had sworn not to reveal it, and engaged Cardano in a decade-long fight), and the quartic was solved by Cardano's student Lodovico Ferrari.

Cardano, 1501 —1576)

• Popularized use of the (Stevin’s) decimal point.• Logarithms: opposite of powers• made calculations by hand much easier and quicker, opened the way to

many later scientific advances. • “MirificiLogarithmorumCanonisDescriptio,” contained 57 pages of

explanatory matter and 90 of tables,• facilitated advances in astronomy and physics

John Napier (1550 –1617)

• “Father of Modern Science”

• Proposed a falling body in a vacuum would fall with uniform acceleration

•Was found "vehemently suspect of heresy", in supporting Copernican heliocentric theory … and that one may hold and defend an opinion as probable after it has been declared contrary to Holy Scripture.

Galileo Galilei (1564-1642)

Developed “Cartesian geometry” :

uses algebra to describe geometry.

Invented the notation using superscripts to show the powers or exponents, for example the 2 used in x2 to indicate squaring. René Descartes

(1596 –1650)

important contributions to the construction of mechanical calculators, the study of fluids, clarified concepts of pressure and.

wrote in defense of the scientific method. Helped create two new areas of mathematical research:

projective geometry (at 16) and probability theory

Blaise Pascal (1623 –1662)

• conservation of momentum • built the first "practical" reflecting telescope

• developed a theory of color based on observation that a prism decomposes white light into a visible spectrum.

• In mathematics:• development of the calculus. • demonstrated the generalised binomial theorem, developed the so-called "Newton's method" for approximating the zeroes of a function....

Sir Isaac Newton (1643 – 1727)

•important discoveries in calculus…graph theory.

•introduced much of modern mathematical terminology and notation, particularly for mathematical analysis,

•renowned for his work in mechanics, optics, and astronomy.

Euler (1707 –1783)

Invented or developed a broad range of fundamental ideas, in invariant theory, the axiomatization of geometry, and with the notion of Hilbert space

David Hilbert (1862 –1943)

• famous for having founded “information theory” in 1948.

•digital computer and digital circuit design theory in 1937

•Demonstrated that electrical application of Boolean algebra could construct and resolve any logical, numerical relationship.

• It has been claimed that this was the most important master's thesis of all time.

Claude Shannon (1916 –2001)]

Women

Mathematicians

Theano

Theano was the wife of Pythagoras. She and her two daughters carried on the Pythagorean School after the death of Pythagoras. She wrote treatises on mathematics, physics, medicine, and child psychology. Her most important work was the principle of the “Golden Mean.”

Hypatia

Hypatia was the daughter of Theon, who was considered one of the most educated men in Alexandria, Egypt.

Hypatia was known more for the work she did in mathematics than in astronomy, primarily for her work on the ideas of conic sections introduced by Apollonius.

Home

Caroline Herschel

Her first experience in mathematics was her catalogue of nebulae.

She calculated the positions of her brother's and her own discoveries and amassed them into a publication.

One interesting fact is that Caroline never learned her multiplication tables.

Sophie Germain

She is best known for her work in number theory.

Her work in the theory of elasticity is also very important to mathematics.

Emilie du Chatelet

Among her greatest achievements were her “Institutions du physique” and the translation of Newton's “Principia”, which was published after her death along with a “Preface historique” by Voltaire.

Emilie du Châtelet was one of many women whose contributions have helped shape the course of mathematics

Methods of Teaching

Mathematics

Model Method• is a visual way of picturing a situation. Instead of

forming simultaneous equations and solving for the variables, model building involves using blocks or boxes to solve the problem. The power of using models can be best illustrated by problems, often involving fractions, ratios or percentages, which appear difficult but if models can be drawn to show the situation, the solution becomes clearer, sometimes even obvious.

Socratic Method

• Teaching by asking instead of telling.• Involves asking a series of questions until a contradiction

emerges invalidating the initial assumption. • Socratic irony is the position that the inquisitor takes that

he knows nothing while leading the questioning.

Advantages involves discussion students can actively engage with their knowledge instead of simply memorizing or retaining it students can exchange opinions and ideas, develop excellent speaking and communication skills The Socratic Method is a fun yet educational way to teach your

students how to make use of their knowledge. The Socratic Method also teaches students how to think critically,

accept others' opinions or viewpoints, and apply their knowledge to the real world and to other forms of knowledge.

Lecture Method

• The teacher has a great responsibility to guide the thinking of the students and so he must make himself intelligible to them. Unlike other methods where motivations can come from subsequent activities, in the lecture, students interest depends largely on the teacher.

• Getting the attention. • Comprehension by the class is the measure of success

Deductive Method• The teacher tells or shows directly what

he/she wants to teach. This is also referred to as direct instruction.

Example 1:Find a2 X a10 = ?Solution:General : am X an = am+n

Particular: a2 X a10 = a2+10 = a12

Inductive method

• method of solving a problem from particular to general• in this we first take a few examples and then generalize• it is a method of constructing formula with the help of

sufficient number of concrete examples

Critical Thinking

Creative Thinking

Collaborating

Communicating

Learning Skills

Project Method

• This methods aims to bring practically designed experience into the classroom. Often conducted over a period of three to six months, the projects give students an opportunity to work in a team environment and apply theory learned in the classroom. There are some parts of the curriculum in which students are necessarily dependent on the teacher and others in which they can work more independently.

Presenting…

The 21st Century Learners Skills

The term "21st-century skills" is generally used to refer to certain core competencies such as collaboration, digital literacy, critical thinking, and problem-solving that advocates believe schools need to teach to help students thrive in today's world.

The 21st Century Learners Skills

COLLABORATION the action of working with someone to produce or create something.TEAMWORK

The process of working collaboratively with a group of people in order to achieve a goal.

CREATIVITY

Mental characteristic that allows a person to think outside of the box, which results in innovative or different approaches to a particular task.

Imagination, also called the faculty of imagining, is the ability to form new images and sensations in the mind that are not perceived through senses such as sight, hearing, or other senses.

CRITICAL THINKING the objective analysis and evaluation of an issue in order

to form a judgmentPROBLEM SOLVING

process of working through details of a problem to reach a solution.

The 21st century skills are a set of abilities that students need to develop in order to succeed in the information age.

The Partnership for 21st Century Skills lists three types:

What are 21st Century Skills?

Information Literacy

Media Literacy

Technology Literacy

Literacy Skills

Flexibility

Initiative

Social Skills

Productivity

Leadership

Life Skills

PRINCPLES AND STANDARD IN TEACHING

MATHEMATICS

PRINCIPLES1. Equity Principle

 Excellence in mathematics education requires equity—high expectations and strong support for all students.2. Curriculum Principle

 A curriculum is more than a collection of activities: it must be coherent, focused on important mathematics, and well articulated across the grades.

3. Teaching PrincipleEffective mathematics teaching

requires understanding what students know and need to learn and then challenging and supporting them to learn it well.4. Learning Principle

Students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge.

5. Assessment Principle Assessment should support the

learning of important mathematics and furnish useful information to both teachers and students.6. Technology Principle

Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students' learning.

STANDARDS

• Content StandardThe five Content Standards each

encompass specific expectations, organized by grade bands:1. Number and Operations Instructional programs from prekindergarten through grade 12 should enable all students to: Understand numbers, ways of

representing numbers, relationships among numbers, and number systems.

Understand meanings of operations and how they relate to one another.

Compute fluently and make reasonable estimates.

2. Algebra Instructional programs from

prekindergarten through grade 12 should enable all students to: Understand patterns, relations, and

functions Represent and analyze mathematical

situations and structures using algebraic symbols

Use mathematical models to represent and understand quantitative relationships

Analyze change in various contexts

3. Geometry Instructional programs from

prekindergarten through grade 12 should enable all students to: Analyze characteristics and properties  of

two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Specify locations  and describe  spatial relationships using coordinate geometry and other representational systems

Apply transformations  and use symmetry to analyze mathematical situations

Use visualization , spatial reasoning, and geometric modeling to solve problems

4. MeasurementInstructional programs from

prekindergarten through grade 12 should enable all students to:Understand measurable

attributes of objects and the units, systems, and processes of measurement

Apply appropriate techniques, tools, and formulas to determine measurements.

5. Data Analysis and ProbabilityInstructional programs from

prekindergarten through grade 12 should enable all students to: Formulate questions  that can be

addressed with data and collect, organize, and display relevant data to answer them

Select and use  appropriate statistical methods to analyze data

Develop and evaluate  inferences and predictions that are based on dataUnderstand and apply  basic concepts of probability

Process StandardThe five Process Standards are described through

examples that demonstrate what each standard looks like and what the teacher's role is in achieving it: 1. Problem Solving

Instructional programs from prekindergarten through grade 12 should enable all students to:build new mathematical knowledge through problem solving;

solve problems that arise in mathematics and in other contexts;

apply and adapt a variety of appropriate strategies to solve problems;

monitor and reflect on the process of mathematical problem solving.

2. Reasoning and Proof.Instructional programs from

prekindergarten through grade 12 should enable all students to: recognize reasoning and proof as

fundamen- tal aspects of mathematics make and investigate mathematical

conjec- tures; develop and evaluate mathematical argu-

ments and proofs; select and use various types of reasoning

and methods of proof.

3. CommunicationInstructional programs from

prekindergarten through grade 12 should enable all students to: organize and consolidate their

mathematical thinking through communication;

communicate their mathematical thinking coherently and clearly to peers, teachers, and others;

use the language of mathematics to express mathematical ideas precisely

4. ConnectionsInstructional programs from

prekindergarten through grade 12 should enable all students to: recognize and use connections among

mathematical ideas; understand how mathematical ideas

interconnect and build on one another to produce a coherent whole;

recognize and apply mathematics in con- texts outside of mathematics.

5. RepresentationInstructional programs from

prekindergarten through grade 12 should enable all students to: create and use representations to

organize, record, and communicate mathematical ideas;

select, apply, and translate among mathe- matical representations to solve problems;

use representations to model and interpret physical, social, and mathematical phenomena.

Transfer

of

learning

Transfer An act of moving something or someone to another place.

Is the study of the dependency of human conduct, learning, or performance on prior experience.

Learning An act of gaining knowledge or skill by experience, study, being

taught, or creative thought.

Occurs when learning in one context or with one set of materials impacts on performance in another context or with other related materials.

Teaching of learning: Importance

If there were no transfer, students would need to be taught every act that they would ever perform in any situation. Because the learning situation often differs from the context of application, the goal of training is not accomplished unless transfer occurs All new learning involves transfer based on previous learning (Bransford, 41).

If we did not transfer some of our prior knowledge, then each new learning situation would start from scratch. Assumption of education: what is taught in a course will be used in relevant situations in other courses, in the workplace and out of school.

It is the very essence of understanding, interacting and creating. Furthermore, it is the ultimate aim of teaching and learning.

LEVELS AND TYPES OF TRANSFER

Positive Transfer Transfer is said to be positive when learning in one context improves learning or performance in another context.

Negative Transfer Negative transfer occurs when previous learning or experience inhibits or interferes with lear.ning or

performance in a new context

LEVELS AND TYPES OF TRANSFER

Positive Transfer Transfer is said to be positive when learning in one context improves learning or performance in another context.

Negative Transfer Negative transfer occurs when previous learning or

experience inhibits or interferes with lear.ning or performance in a new context

Near and Far Transfer Another distinction used is between near and far transfer. Usually these terms distinguish the closeness or distance

between the original learning and the transfer task.

Near transfer Has also been seen as the transfer of learning within the school context, or between a school task and a very similar task.

Far transfer is used to refer to the transfer of learning from the school context to a non-school context.

High Road and Low Road Transfer Low road transfer

Happens when stimulus conditions in the transfer context are sufficiently similar to those in a prior context of learning to trigger well-developed semi-automatic responses.

High road transfer in contrast, depends on mindful abstraction from the context of

learning or application and a deliberate search for connections. 

STRATEGIES FOR PROMOTING TRANSFER Teach subject matter in meaningful contexts.

Employ informed instruction.Students should learn not only how to explain a concept, but also

to understand when and why the concept is useful.

Teach subject matter in circumstances as similar as possible to those in which it will be employed  

Provide chances to practice using the subject matter in situations that embody the full range of practical applications that the learner is likely to come across.

Present opportunities for allocating practice after the information has been originally learned.

Practice should be spread out over a period of time (not combined into a single study session) Encourage positive attitudes toward subject matter

Students will be less likely to avoid topics when they are encountered somewhere else.

General

of

learning

Bloom's Taxonomy was created in 1956 under the leadership of educational psychologist Dr Benjamin Bloom in order to promote higher forms of thinking in education, such as analyzing and evaluating concepts, processes, procedures, and principles, rather than just remembering facts (rote learning). It is most often used when designing educational, training, and learning processes.

1. Remembering:Recognizing or recalling knowledge from memory. Remembering is when memory

is used to produce or retrieve definitions, facts, or lists, or to recite previously learned information.

2. Understanding: Constructing meaning from different types of functions be they written or graphic

messages or activities like interpreting, exemplifying, classifying, summarizing, inferring, comparing, or explaining.

3. Applying: Carrying out or using a procedure through executing, or

implementing. Applying relates to or refers to situations where learned material is used through products like models, presentations, interviews or simulations.

4. Analyzing: Breaking materials or concepts into parts, determining how the parts relate to one another or how they interrelate, or how the parts relate to an overall structure or

purpose. Mental actions included in this function are differentiating, organizing, and attributing, as well as being able to distinguish between the components or parts.

When one is analyzing, he/she can illustrate this mental function by creating spreadsheets, surveys, charts, or diagrams, or graphic representations.

5. Evaluating: Making judgments based on criteria and standards through checking and critiquing.

Critiques, recommendations, and reports are some of the products that can be created to demonstrate the processes of evaluation. In the newer

taxonomy, evaluating comes before creating as it is often a necessary part of the precursory behavior before one creates something.

6. Creating:Putting elements together to form a coherent or functional whole; reorganizing

elements into a new pattern or structure through generating, planning, or producing. Creating requires users to put parts together in a new way, or synthesize

parts into something new and different creating a new form or product. This process is the most difficult mental function in the new taxonomy.

Situated

Learning

Situated learning is an instructional approach developed by Jean Lave and Etienne

Wenger in the early 1990s, and follows the work of Dewey, Vygotsky, and others (Clancey, 1995) who claim that students are more inclined to learn by actively participating in the learning experience. Situated learning essentially is a matter of creating meaning from the real activities of daily living (Stein, 1998, para. 2) where learning occurs relative to the teaching environment.

Constructivism

What is constructivism?Is a theory or a philosophy about teaching and learning that supports the notion that:Learners must be independent thinkers (cognitive)

Learners create their own knowledgeLearners work in teamsLearning is active & student-centered

What does it means ?Constructivism is the idea that learning doesn’t just happen by the traditional methods of teachers standing in front of the class and lecturing. Traditionally, teachers present knowledge to passive students who absorb it. No wonder students are often bored

Jerome Brunner

Very influential psychologistHis concern with cognitive psychology “led to a particular interest in the cognitive development of the children and just what the appropriate form of education might be”

Jean Piaget

Develop the cognitive learning theory.

Felt children were “active learners” who constructed new knowledge

“as they moved through different cognitive stages, building on what they already know”.

Lev Vygotsky

Developed the social cognitive theory which “assert that culture is the prime determinant of individual development” because humans are the only creatures to have created cultures and therefore it effects our learning development

John Dewey

Believed that learning should be engaging to the students. They will learn better if they are interested.

Cooperative

Learning

Cooperative learning - is a generic term that is used to describe an instructional arrangement for teaching academic and collaborative skills to small, heterogeneous groups of students.

According to the National Council of Teachers of Mathematics (NCTM), learning environment

should be created that promote active learning and teaching.

Cooperative learning can be used to promote classroom discourse and oral language development.

Wiig and Semel describe mathematics as "conceptually dense".

CooperativeLearning

Components

Lesson preparation Teachers should: a. select the mathematics and collaborative objectives to target for instruction and cooperative learning groups, b. plan the math activity, c. identify ways to promote the elements of cooperative learning, d. identify roles, e. establish groups.

Lesson instructions

This refers to the time in which cooperative

learning activity occur.

This is to assess students' mastery of the objectives and the group's ability to work cooperatively.

Lesson evaluation

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Euclid was a Greek mathematician, often referred to as the "Father of Geometry". Euclid was born in 365 B.C. He went to school at Plato's academy in Athens, Greece. He founded the university in Alexandria, Egypt. He taught there for the rest of his life. One of his students was Archimedes.Euclid was kind, fair, and patient. Once, when a boy asked what the point of learning math was, Euclid gave him a coin and said, "He must make gain out of what he learns." Another time, he was teaching a king. When the king asked if there was an easier way to learn geometry Euclid said, "There is no royal road to geometry." Then he sent the king to study.In his time he was thought of as being too thorough. Now, in our time, we think he wasn't thorough enough. Euclid died in 275 B.C.Euclid's most famous work was the Elements. This series of books was used as a center for teaching geometry for 2,000 years. It has been translated into Latin and Arabic.The Elements were divided into thirteen books, which subjects are as follows: Books 1-6= plane geometry, books 7-9= number theory, book 10= Eudoxus's theory of irrational numbers, and books 11-13= solid geometry. More than 1,000 editions of Elements have been published since 1482. Elements were popular until the 20th century.

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Euclid’s Biography

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Born: c. 365 BCBirthplace: Alexandria, EgyptDied: c. 275 BCLocation of death: Alexandria, EgyptCause of death: unspecifiedOccupation: Mathematician, EducatorNationality: Ancient GreeceExecutive summary: Father of geometryUniversity: Plato's Academy, Athens, GreeceTeacher: Library of Alexandria, Alexandria, Egypt Asteroid Namesake 4354 Euclides

Lunar Crater Euclid (7.4S, 29.5W, 11km dia, 700m height)

Eponyms Euclidean geometry Slave-owners

Author of books: Elements (13 volumes) Data (plane geometry) On Divisions (geometry) Optics (applied mathematics) Phenomena (astronomy)

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EUCLID’S DEFINITONS

Some of the definitions made by Euclid in volume I of ‘The Elements’ that we take for granted today are as follows :-

A point is that which has no part.

A line is breadth less length. The ends of a line are points. A straight line is that which has length only.

Euclid's construction of a regular dodecahedron.

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A surface is that which has length and breadth only.

The edges of a surface are lines A plane surface is a surface which lies evenly with the straight lines on itself oAxioms or postulates are the assumptions which are obvious universal truths. They are not proved. oTheorems are statements which are proved, using definitions, axioms, previously proved statements and deductive reasoning .

EUCLID’S AXIOMs

SOME OF EUCLID’S AXIOMS WERE :-

Things which are equal to the same thing are equal to one another.

i.e. if a=c and b=c then a=b.

Here a, b and c are same kind of things.

If equals are added to equals, the wholes are equal.

i.e. if a=b and c=d, then a+c = b+d

Also a=b then this implies that a+c = b+c .

If equals are subtracted, the remainders are equal.

Things which coincide with one another are equal to one another.

Things which are double of the same things are equal to one another

The whole is greater than the part. That is if a > b then there exists c such that a =b + c. Here, b is a part of a and therefore, a is greater than b.

Things which are halves of the same things are equal to one another.

EUCLID’S FIVE

POSTULATES

EUCLID’S POSTULATES WERE :-

POSTULATE 1 :- A straight line may be drawn from any one point to any other point

Axiom :- Given two distinct points, there is a unique line that passes through them

POSTULATE 2 :- A terminated line can be produced infinitely

POSTULATE 3 :- A circle can be drawn with any centre and any radius

POSTULATE 4 :- All right angles are equal to one another

POSTULATE 5 :- If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles.

THEOREMS WITH PROOFTHEOREM :-Two distinct lines cannot have more than one point in common

PROOF :- Two lines ‘l’ and ‘m’ are given. We need to prove that they have only one point in common

Let us suppose that the two lines intersects in two distinct points, say P and Q

That is two line passes through two distinct points P and Q

But this assumptions clashes with the axiom that only one line can pass through two distinct points

Therefore the assumption that two lines intersect in two distinct points is wrong

Therefore we conclude that two distinct lines cannot have more than one point in common

Euclid Division lemmaTHEOREM :-Given positive integers a and b, there exist unique integers q and r satisfying a = bq + r, ≤ r < b.

Tatyana Alexeyevna Afanasyeva

Dutch/Russian mathematician who advocated the use of visual aids and example for introductory courses in geometry for high school students

Robert Lee Moore(November 14, 1882 – October 4, 1974)

Born: November 14, 1882Dallas, Texas

Died October 4, 1974 (aged 91)Austin, Texas

Nationality AmericanField MathematicsInstitutions University of Texas at AugustinAlma Mater University of Chicago ((Ph.D.,1905)Thesis Sets of Metrical Hypothesis

Robert Lee Moore

Robert Lee Moore

Robert Lee Moore (November 14, 1882 – October 4, 1974) was an American mathematician, known for his work in general topology and the Moore method of teaching university mathematics.

The Moore method

George Polya1887 - 1985

Father of Problem Solving

George Polya1887 - 1985

Father of Problem Solving

George Polya1887 - 1985

Father of Problem Solving

George Polya1887 - 1985

Father of Problem Solving

George Polya1887 - 1985

Born December 13, 1887Budapest, Austria-Hungary

Died September 7, 1985(aged 97)Palo Alto, California

Nationality Hungarian (–1918)Swiss (1918–1947)American (1947–his death)

Fields Mathematics

Institutions ETH ZürichStanford University

Alma mater Eötvös Loránd University

Doctoral advisor Lipót Fejér

Biography

His first job was to tutor Gregor the young son of a baron. Gregor struggled due to his lack of problem solving skills. Polya (Reimer, 1995) spent hours and developed a method of problem solving that would work for Gregor as well as others in the same situation. Polya (Long, 1996) maintained that the skill of problem was not an inborn quality but, something that could be taught.

In 1945 he published the book How to Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages. In this text he identifies four basic principles

POLYA’S FIRST PRINCIPLE:UNDERSTAND THE

PROBLEM

Can you state the problems in you own words?

What are you trying to find or do?What are the unknowns?What information do you obtain from the problem?

What information, if any is missing and not needed?

Devise a Plan

POLYA’S SECOND PRINCIPLE:

Look for a patternRemember related problemsBreak the problem down into different parts

Make a table Make a diagramWrite an equationUse a guess and checkWork backwardIdentify a subgoal

POLYA’S THIRD PRINCIPLE:

Carry Out the Plan

Check the result in the original problem. Does your answer make sense? Is it reasonable?

Determine whether there is another method of finding the solution..

If possible, determine problems for which techniques

Polya’s Fourth Principle:

Look Back

Implement the strategy in Step 2 and perform the necessary math computations.

Check each step of the plan as you do it.Keep an accurate record of your work.Organize your work into easy to

understand visuals.Double check your math work.

About Toru

KumonHistory

Kumon Method

Toru Kumon (1914-1995)

About

Toru Kumon

Toru Kumon ( 公文 公 Kumon Tōru , March 26, 1914 – July 25, 1995)

Japanese mathematics educatorBorn in Kōchi Prefecture, JapanHe graduated from the College of Science at Osaka University with a degree in mathematics and taught high school mathematics in his home town of OsakaAsteroid 3569 Kumon is named after him

During his 33-year career, he taught math at his alma mater Tosa Junior/Senior High School, and later at Sakuranomiya High School in Osaka City, as well as at other schools

In 1958, he established the Osaka Institute of Mathematics, which later became the Kumon Institute of Education Co., Ltd in 1983

Toru Kumon died in Osaka on July 25, 1995 at the age of 81, from pneumonia

He devoted the rest of his life to improving the Kumon method and making it available to more and more people around the world.

During his 33-year career, he taught math at his alma mater Tosa Junior/Senior High School, and later at Sakuranomiya High School in Osaka City, as well as at other schools

In 1958, he established the Osaka Institute of Mathematics, which later became the Kumon Institute of Education Co., Ltd in 1983

Toru Kumon died in Osaka on July 25, 1995 at the age of 81, from pneumonia

He devoted the rest of his life to improving the Kumon method and making it available to more and more people around the world.

In 1954 in Japan, a grade 2 student by the name of Takeshi scored poorly in a math test. His mother, Teiko, asked her husband, Toru Kumon, to take a closer look at their son’s school textbooks. Being a high school math teacher, Toru Kumon thought Takeshi's textbooks did not give children enough practice to be confident in a topic. Mr. Kumon decided to help his son by handwriting worksheets for him to practice. This was the start of the Kumon Method

By the time Takeshi was in Year 6, he was able to solve differential and integral calculus usually seen in the final years of high school. This was the beginning of the Kumon Method of Learning

As a result of Takeshi's progress, other parents became interested in Kumon's ideas, and in 1956, the first Kumon Center was opened in Osaka, Japan

In 1958 the Kumon Institute of Education was established in Osaka, Japan.

The Origins of the Kumon MethodHalf a century ago, the Kumon Method was born out of a father' s love for his son

The late Chairman Toru Kumon developed the prototype of the Kumon Method in 1954 while he was a high school mathematics teacher. At the time, Toru Kumon's wife, Teiko, asked him to take a look at his eldest son Takeshi's second-grade arithmetic studies because she was not satisfied with the results of a test he had taken.

Toru Kumon wrote out calculation problems on loose-leaf paper for Takeshi, and the materials that he created from 1955 became the prototype for today’s Kumon worksheets

Based on his experience as a high school teacher, Toru Kumon knew that many senior high school students had problems with their math studies because of insufficient calculation skills

Therefore, he focused on developing Takeshi’s calculation skills, and created materials that made it possible for his son to learn independently

Kumon has two core programs, the Kumon Math and Kumon Native Language Program (language varies by country)

How Kumon works is that each student is given an initial assessment of his or her abilities

Based on the results and the student's study skills, a Kumon Instructor will create an individualized study plan.

Kumon method of learning

Mathematics programAs a high school mathematics teacher, Mr. Kumon understood that an understanding of calculus was essential for Japanese university entrance exams so in writing worksheets for his son, Mr. Kumon focused on all the topics needed for a strong understanding of calculus starting from the basics of counting.

Level 6A: Counting numbers to 10, reading numbers.Level 5A: Reading numbers to 50, sequence of numbers.Level 4A: Reading numbers, writing numbers to 120.Level 3A: Numbers up to 120, adding up to 3.Level 2A: Adding up to 10.Level A: Horizontal addition, Subtraction from numbers up to 20.Level B: Vertical addition and subtraction.Level C: Basic multiplication, division.Level D: Long multiplication, long division, introduction to fractions.

Level E: FractionsLevel F: Four operations of fractions, decimals.Level G: Positive/negative numbers, exponents, Algebraic expressions, Single-Variable Equations with 1-4 steps.Level H: Transforming Equations, Linear/simultaneous equations, inequalities, algebraic functions and graphs, adding and subtracting Monomials and Polynomials.Level I: Factorization, square roots, quadratic equations, Pythagorean theorem.Level J: Algebra II.Level K: Functions: Quadratic, fractional, irrational, exponential.

Level L: Logarithms, basic limits, derivatives, integrals, and its applications.Level M: Trigonometry, straight lines, equation of circles.Level N: Loci, limits of functions, sequences, differentiation.Level O: Advanced differentiation, integration, applications of calculus, differential equations.Level X (elective level): Triangles, vectors, matrices, probability, statistics.

Reading ProgramThe Kumon Native Language Programs are designed to expose students to a broad range of texts and develop the skill of reading comprehension. A number of Kumon Centres also use audio CDs to help students with pronunciation. (Note: Levels vary slightly by country

Level 7A: Look, Listen, Repeat.Level 6A: Reciting Words with Pictures.Level 5A: Letter Sounds.Level 4A: Consonant Combinations and Vowel Sounds.Level 3A: Advanced Vowel Sounds & Advanced Sounding Out.Level 2A: Functions of Words (nouns, verbs, adjectives), Reading Aloud.Level AI: Structure of Simple Sentences.Level AII: Sentence Structure, Sentence Topics, Thought Sequence.Level BI: Subject and Predicate.Level BII: Comparing and Contrasting.

Level CI: Constructing Sentences.Level CII: Organizing Information.Level DI: Combining Sentences.Level DII: Main Idea, Understanding Paragraphs.Level EI: Clauses.Level EII: Reason and Result.Level FI: Referring Words, Interpreting Text.Level FII: Concision, Analysis of & Recounting Events from Paragraphs.Level G: Point Making, Theme, Story Elements, Summary.Level H: Summation.Level I: Persuasion.Level J: Critical Reading.

Level K: Elements of Literature.Level L: Interpretation.

Level K: Elements of Literature.Level L: Interpretation.

Therefore, the Kumon Method has been welcomed into communities around the world with widely differing cultures, values, and educational systems*total enrollments for all subjects (as of March 2016)

1958: Kumon is established1974: First steps of overseas expansion1985: Increasing numbers of Kumon students around the world2009: Kumon as a global brand as we move toward the next 50 years of our history2014: Instructors around the world learn from children

Born on 23 January 1935 in New York City

Moses, Robert Parris

 Moses grew up in a housing project in Harlem. He attended Stuyvesant High School, an elite public school, and won a scholarship to Hamilton College in Clinton, New York.

•  He earned a master’s degree in philosophy in 1957 from Harvard University, and was working toward his doctorate when he was forced to leave because of the death of his mother and the hospitalization of his father. 

• Moses returned to New York and became a mathematics teacher at Horace Mann School a prestigious private high school.

In the 1990s, the Algebra Project students learned to think and speak mathematically through tackling problems that arose in their daily lives.

• In 12 years the program helped more than 10,000 students master fundamental algebraic skills in cities across the country.

• In 1992, Moses returned to Mississippi to start the Delta Algebra Project. Moses would later tell the New York Times. ” But this time, we’re organizing around literacy-not just reading and writing, but mathematical literacy… Now math literacy holds the key.”

His first involvement came with the Southern Christian Leadership Conference (SCLC) where he organized a youth march in Atlanta to promote integrated education.

• In 1960 Moses joined the Student Nonviolent Coordinating Committee (SNCC) and two years later became strategic coordinator and project director with the newly formed Council of Federated Organizations (COFO) which worked in Mississippi.

• In 1963 Moses led the voter registration campaign in the Freedom Summer movement. The following year he helped form the Mississippi Freedom Democratic Party which tried to replace the segregationist-dominated Mississippi Democratic Party delegation at the 1964 Democratic National Convention.

In regards to entertainment, Moses served as an influential figure in the construction of two World’s Fairs. Shea Stadium and Lincoln Center.

During his decades, Robert Moses brought:

Pierre Van Hiele andDianna Van Hiele

Biography of Pierre Van Hiele

Van Hiele was famous for his theory that describes how students learn geometry, he was born in 1909 and died November 1,2010. This theory came about in 1957 when he got his doctoral at Utrecht University in Netherlands. He was also a publisher, he published a book titled Structure and Insight in 1986 which further describe his theory. The theory came about by two Dutch educators, Diana Van Hiele-Gelof and Pierre Van Hiele (wife and husband).

Critical examination of how students learn based on Van Hiele’s theory

Based on Diana Van Hiele-Gelof and Pierre Van Hiele theory there are five levels to describe how students learn or understand geometry. These are:-Level 0: Visual-Level 1: Description-Level 2: Relational-Level 3: Deductive-Level 4: Rigor

Example of how to demonstrate the thinking process that children use in

learning math base on his theory

Van Hiele strongly believed that using his theory in Geometry it would improve the student learning. For example in geometry at the visual level the teacher could draw some triangles on the board, so the students would know what a triangle looks like.

How Hiele ‘s theory contribute to Mathematics education and it application

to the Jamaican classroom?This theory contributes greatly to Mathematics Education since it is a Geometry theory and most students find Geometry difficult. This theory can be applied through five phases.

Phase 1- (Information/ Inquiry): At this stage teacher introduce a new idea and allow student to work with it. This new idea is normally easier to understand than the original but it means the same. So students get a better understanding. Example: Alternate angles are equal but she could say ‘Z’ angles are equal. So students will understand easier.

Phase 2-(Guided or Direct Orientation): At this stage teacher give lots of work to students for practice so they get aquatinted with the concepts and learn it well.

Phase 3-(Explication): At this stage teacher told students to, in their own word describe what they learn using mathematical terms. Example: Reflection writing.

Phase 4-(Free Orientation): This is where teacher allow students to apply relationships they learn to solve harder problems. Example: They learn from a cxc and allow using the same principles learn to solve questions from a cape book.

Phase 5-(Integration): This is where students reflect on what they learn and find easier way to do what they learn.

These phases when perform will build geometry students understanding therefore build better students including Jamaica.

Robert and Ellen Kaplan

Robert kaplan

Born on 1933He is an author

Ellen Kaplan

A wife of Robert kaplan

She is also an author

Books are originally published

The nothing that is: A natural history of zeroOriginally published- January 1,1999The ‘invention’ of zero made arithmetic

infinitely easer- try to doing division in roman numerals- and it now forms part of the binary code which powers all our computers…

Hidden harmonies :the lives and time of the pythagorean theorem

Originally published- January 4, 2011 A squared equals c squared. It sounds

simple, doesn’t it? Yet this familiar expression opens a gateway into the riotous garden of mathematics, and send us on a journey of exploration in…

Raymundo Favila

A Filipino mathematician. He has his Ph D. from the University of California, Berkeley from 1939 under the supervision of Pauline Sperry, and had his career at the University of the Philippines in Manila.[3] Dr. Raymundo Favila was elected as Academician of the National Academy of Science and Technology in 1979

He was one of those who initiated mathematics in the Philippines. He contributed extensively to the progression of mathematics and the mathematics learning in the country. He has made fundamental studies such as on stratifiable congruences and geometric inequalities. Dr. Favila has also co-authored textbooks in algebra and trigonometry.

AMADOR C. MURIELwas born on November 24, 1939

The Filipino Legend

Dr. Amado Muriel was recognized because of his important works and marvelous contributions to the field of theoretical physics, in particular, his advancement of theoretical apparatus to clarify turbulence. His new kinetic equation is valuable for discovering essential problems of non-equilibrium statistical procedure.

Dr. Muriel discovered the accurate and estimated solutions for the performance of a two-level system, which were considered by his peers as a revolutionary contribution to a quantum Turing machine, now a rising field in quantum computing. Also, in his studies on stellar dynamics, he has recognized realistically that self-gravitation alone is adequate to create a hierarchy of structures in one dimension.

Dr. Melecio Magno was born on May 24, 1920.

Dr. Melecio S. Magno was Chairman of NSDB from1976 to 1981. He obtained a BS in MiningEngineering, and MS in Physics from the Universityof the Philippines.

His interests spannedseismology, metrology, meteorology, physics andheld administrative positions in these fields in UP.He was appointed Minister of NSDB from 1978 to1981. He later became President of the ScienceFoundation of the Philippines, Vice President of theNational Academy of Science and Technology, andPresident of the National Research Council of thePhilippines.

He pushed for the establishment of the National Academy of Science and Technology.Under Dr. Magno, the NSDB pursued mission-oriented R&D programs in different sectors.

Dr. Magno has researched on the assimilation and fluorescence spectroscopy of crystals, particularly rare-earth crystals; effects of typhoons on the allotment of gravitation; and the idea of science. He also a co-author of a Textbook in physics at the University of the Philippines named University Physics.

Dr. Apolinario Nazarea made significant role to the theories on biophysics and recombinant biotechnology including his own conceptual framework on the structure of RNA/DNA investigation. Born on October 11, 1940.

On the turning point of the country's technical course, his part in the expansion of biophysics and biotechnology is both essential and well timed. With his worldwide-cited systematic work and hypothetical expertise, he has laid the groundwork for the design of artificial vaccines on a sounder molecular beginning.

Gregrio Zara

A native of Lipa, Batangas, Zara finished primary schooling at Lipa Elementary School, where he graduated as valedictorian in 1918. In 1922, he again graduated valedictorian in Batangas High School, an accolade which warranted him a grant to study abroad

In the middle of his first semester, he finally got the scholarship when his rival got sick and died abroad. Dr. Zara then enrolled at the Massachusetts Institute of Technology (MIT) in the United States, and graduated with a degree of BS in Mechanical Engineering in 1926.

he obtained a Master of Science in Engineering (Aeronautical Engineering) at the University of Michigan, USA, graduating summa cum laude.

Zara then sailed to France to take up advanced studies in physics at the Sorbonne University in Paris. In 1930 he again graduated summa cum laude with a degree of Doctor of Science in Physics, with "Tres Honorable," the highest honor conferred to graduate students. Zara was the first Filipino given that honor. Madam Marie Curie was given the same accolade for her discovery of radiu

became chief of the aeronautical division of the DPWC.

1936, he was assistant director and chief aeronautical engineer in the Bureau of Aeronautics of the Department of National Defense.

For 21 years, he was director of aeronautical board, a position he held and confirmed by the Congress of the Philippines up to 1952. Considered expert in the Field, he was chosen to be the technical editor of Aviation Monthly and at various times, he worked as vice chairman and acting chairman of the National Science Development Board, where a number of Science projects were impetus.

Inventions, re-designs and theoriesThe Zara Effect – He discovered the physical law of electrical

kinetic resistance called the Zara effect (around 1930)[3]He improved methods of producing solar energy including

creating new designs for a solar water heater (SolarSorber);A sun stove, and a solar battery (1960s); Invented a propeller-cutting machine (1952);He designed a microscope with a collapsible stage;helped design the robot Marex X-10; Invented the two-way television telephone or videophone

(1955) patented as a "photo phone signal separator network"; Invented an airplane engine that ran on plain alcohol as fuel

(1952);

picture phone

Name: Padlan, Eduardo A.Specialization: Ph.D. BiophysicsDivision: Chemical, Mathematical and Physical Sciences

Dr. Padlan was elected as Academician on 2003 and born on August 31, 1940. His latest work on humanized antibodies has possible applications in the healing of different diseases as well as cancer. He has fourteen approved and awaiting patents on diverse part and uses of antibodies.

Honors and award received-Concepcion Dadufalza Award for Distinguished Achievement, University of the Philippines, 2007-Severino & Paz Koh Lectureship in Science, Philippine-American Academy of Science and Engineering, 2008

Research involvement-Research Interests-Molecular Immunology, Protein Structure & Function

Aside from doing research in geometry, Dr. Marasigan served as President of the Mathematical Society of the Philippines for two terms. His many achievements include pushing for the formulation of Policies and Standards for Mathematics, approved by the government in 1989. For ten years, he served as chair of the Ateneo Math Department and headed the Math and Operations Research Division of the National Research Council of the Philippines.

Mathematics andLiterature

According to a study Douglas Clements and Julia Saramel (2006), “there’s an overlap

between language and math” and success in

one area reinforces the other.

Math requires “precision in

language.”, so talking about math helps kids increase their vocabularies.

What may be less obvious is that many

mathematical concepts are embedded in

children’s stories…

Literature is an effective tool for mathematics instruction because it:

• incorporates stories into the teaching and learning of mathematics

• introduces math concepts and contexts in a motivating manner

• acts as a source for generating problems and building problem solving skills

• helps build a conceptual understanding of math skills through illustrations

The study uses the example of Goldilocks and the Three Bears.

By Joseph LaCoste and Mikala Smith

Goldilocks and the Three Bears is

simply one of many modern

interpretations of Robert Southey's

 original. 

Robert Southey 

August 12, 1774 in Bristol – March 21, 1843 in London) was an English poet of the Romantic school, one of the so-called "Lake Poets", and Poet Laureate for 30 years from 1813 to his death in 1843. Although his fame has long been eclipsed by that of his contemporaries and friendsWilliam Wordsworth and Samuel Taylor Coleridge, Southey's verse still enjoys some popularity.

the mathematical principal of ordering

correspondences between ordered sets

patterningclassifications and cause and effect thinking

“Such concepts can be applied

to simultaneous comprehension

of math and literature.”

“Using math related children's literature can help children realize the variety of situations in which people use mathematics for real purposes.” Professor David Whitin, Wayne State University

Language Concepts and Skills that can be embedded into

Mathematics activities:

-Increasing vocabulary -Cause & Effect -Matching to sample -Same & different -Articulation skills -Answering “Wh” questions -Calendar concepts -Labeling objects/pictures/symbols - Answering yes/no questions

-Sorting by color, shape, size -Classifying/Categorizing -Using 1-1 correspondence -Counting - Sequencing -Number recognition -Identifying functions of objects/pictures/symbols -Increasing Mean Length of Utterance - Comparisons (more/less; large/small; long/short)

-Ordinal numbers - Recognizing shapes -Following a pattern - Problem solving -Money skills -Time concepts -Utilizing AAC device -Making predictions -Following a task analysis - Comparing and Contrasting

Making predictions….like making estimates before solving math problems

Writing things down in graphic organizers can help reinforce reading comprehension, while writing things down in a similar way in math can help them “develop, cement and extend understanding.”

When responding to literature children’s writing is never identical, there are several ways to come up with a “right answer”.

The same idea holds true for Mathematics. Teachers should “encourage different methods for reasoning, solving problems and presenting solutions.”

In conclusion, the significance of developing children’s English through math skills is very obvious. It is in the early grades that lifelong foundations are formed for skills these important subject areas.

Confidence and interest play a large role in learning as well. Finally, as Clement and Sarama (2006) state about children’s literature and math, “by connecting the two areas children also build a far deeper understanding of each.”

Stage 1: getting organized

Mapping the current process

LEARNING STYLES

4 PARTS1. Visual Learners- Students

prefer the use of images, maps and graphic organizers to access and understand new information.

Neil Fleming, 2006

2. Auditory Learners- best understand new content through listening and speaking.

3. Read and Write Learners- students learn best through words.

4. Kinesthetic Learners- Students that are hands-on learners and learn best through figuring things out by hand.

(Howard Gardner)1. Verbal Linguistic- People who possess this learning style learn best through reading, writing, listening, and speaking.

2. Logical/Mathematical- Those who exhibits this kind of intelligence learn by classifying, and thinking abstractly about patterns.3. Visual-These people learn by drawing looking at the picture.

4. Auditory- Students who are music smart learn using rhythm or melody.5.Bodily Kinesthetic- Body smart individuals learn best through touch and movement.6.Interpersonal- Those who are people smart relating to others.

7. Intrapersonal- Learn best by working alone.8.Naturalistic- Working with nature

Teaching and Learning

Mathematics using Research

Need and Importance of Research in Mathematics

Education

What you mean by Mathematics education?

In contemporary education, Mathematics Education is the practice of teaching and learning mathematics, along with the associated scholarly research.

What is the meaning of Research in Mathematics Education?

Researches in Mathematics education primarily concerned with the tools, methods and approaches that facilitate practice or the study of practice.

Pedagogy of Mathematics has developed into an extensive field of study, with its own concepts, theories, methods, national and international organizations, conferences and literature.

ObjectivesThe teaching and learning of basic numeracy skills to all pupils.The teaching of practical mathematics (arithmetic, elementary

algebra, plane and solid geometry, trigonometry) to most pupils, to equip them a trade or craft.

The teaching of abstract mathematical concepts (such as set and function) at an early age.

The teaching of selected areas of mathematics (such as Euclidean Geometry) as an example of an axiomatic system and a model of deductive reasoning.

The teaching of heuristic and other problem solving strategies to solve non-routine problems.

Need and its Importance

1. Continuing math research is important because incredibly useful concepts like cryptography, calculus, image and signal processing to continue to come from mathematics and are helping people solve real-world problems. - This “math as tool” is absolutely true and probably the easiest way to go about making the case for math research. - It’s a long-term project.- We don’t know exactly what will come out next, or when, but if we follow the trend of “useful tools”.- We trust that math will continue to produce for society.- Mathematics is omnipresent in the exact science.- Mathematics is basic stuff that has been known for decades or centuries.

2. Continuing math research is important because it is beautiful. It is an art form, and more than that, an ancient and collaborative art form, performed by an entire community. Seen in this light it is one of the crowning achievements of our civilization.

- compare mathematics research directly with some other fields like philosophy or even writing or music.

- Our existence informs us on the most basic questions surrounding what it means to be human.

3.Continuing math research is important because it trains people to think abstractly and to have a skeptical mindset.

- Mathematicians properly trained are psyched to hear a mistake pointed out their argument because it signifies process. There is no shame in being wrong.- It is an inevitable part of the process of learning.

The following results are examples of some of the current findings in the field of mathematics education:

Important resultsConceptual

UnderstandingFormative AssessmentHome workStudents with

difficultiesAlgebraic reasoning

ConclusionAt the current stage of research in mathematics education,

its main contribution to practice may be to raise teacher awareness and deepen teacher understanding of the complicated nature of mathematical knowledge, knowing, and learning.

Reading and discussing research articles, contributed to the teachers learning, in general, that students construct their knowledge.

The mini-study made this theoretical idea more specific, concrete and relevant for the teachers. They learned what the constructivist view might mean in a practical context.

FRACTION WALL

This instructional material is made for students to :• easily review on the basic concepts on

fractions• identify the basic skills in using fractions • solve algebraic operations with fractions and for mastery of any problems involving fractions.

PEDAGOGICAL USE

• Basic operation on fractions

• Solving algebraic equations involving fractions

• Solving word problems involving fractions

NUMBER LINE WALL

This instructional material is made for students to master :

the rules in solving basic operations on integers (the laws of signed numbers)

Solving problems on integers.

ADDITION

To add a positive on the number line, move to the right, towards the larger numbers. To add a negative on a number line you move to the left.

Simple ruleRule for adding integers with different signs:Subtract the absolute values of the numbers and the use the sign of the larger absolute value.

SUBTRACTION

To subtract a positive number, move to the left on the number line. This is the same thing that happens when we add a negative number.

Subtract a negative number we need to move to the right.

Simple Rule:KEEP the first number the same. CHANGE the subtracting to adding. Then CHANGE the sign of the second number

MULTIPLICATION AND DIVISION

Multiplying is really just showing repeated adding. To add 2 three times. 2 + 2 + 2 = 6

SIMPLER RULES

Rule #1:If the signs are the same, the answer is positive.

Examples:

Rule #2:If the signs are different, the answer is negative.

Dividing integers are the same as the rules for multiplying integers.

Remember that dividing is the opposite of multiplying. So we can use the same rules to solve.

Rule #1:If the signs are the same, the answer is positive.

Rule #2:If the signs are different, the answer is negative.

PEDAGOGICAL USE• Introduction of integers

• Basic operations on integers

• Solving algebraic equations

ALGEBRA TILES

This instructional material is made for the learners to:

better understand ways of algebraic thinking and the concepts of

Algebra.

Each tiles represents to a certain variable/ constant

x2

x

1

2X2 + 2x+ 3

PEDAGOGICAL USE• Concepts on Algebra

basic operations on signed numbersSimple substitutionSolving equationsDistributive propertyRepresenting polynomialsBasic operations on polynomialsFactoring polynomialsCompleting the square

• Geometric figures on square and parallelogram

GRID LINE WALL

This instructional material will help the learners :

be introduced with the concepts of plane figuresto master the skill in solving areas and perimeter of

plane figures.

Pedagogical use• Concepts of plane figures; area and

perimeter

• Characteristics of polygons

GEOBOARDThis instructional material will help the learners :be introduced with the concepts of plane figures

and its characteristicsto use concrete material on finding the area and

perimeter of plane figuresto master the skill in solving areas and perimeter

of plane figures

PEDAGOGICAL USE

Concepts of plane figures; area and

perimeter

Characteristics of polygons

PIE/ CIRCLE

OBJECTIVESThis instructional material is made for the students to:

solve for the area and circumference of a circle

identify the relationship between a circle and a parallelogram.

PEDAGOGICAL USE• Concept of a circle; area and perimeter

• Relationship of a parallelogram and a circle

• Fraction

• Division of numbers

Define Perimeter and Area. Illustrate the formulas on finding the perimeter and area of plane figures. Find the perimeter and area

of common plane figures.

PERIMETERThe perimeter of any polygon is the sum of the measures of the line segments that form its sides. OR SIMPLY, the measurement of the distance around any plane figure.

Perimeter is measured in linear units.

TriangleThe perimeter P of a triangle with sides of lengths a, b,

and c is given by the formula

P = a + b + ca

b

c

SQUARE

The perimeter P of a square with all sides of length s is given by the formula

P = 4s

s

s

s

s

RECTANGLE

The perimeter P of a rectangle with length l and width w is given by the formula

P = 2L + 2W

W

L

W

L

AREA

The amount of plane surface covered by a polygon is called its area. Area is measured in square units.

RECTANGLE

The area of a rectangle is the length of its base times the length of its height.

A = bh

HEIGHT

BASE

PARALLELOGRAM

The area of a parallelogram is the length of its base times the length of its height.

A = bhWhy?

Any parallelogram can be redrawn as a rectangle without losing area.

BASE

HEIGHT

TRIANGLEThe area of a triangle is one-half of the length of its base times the

length of its height.

A = ½bh

Why?

Any triangle can be doubled to make a parallelogram.

HEIGHT

BASE

TRAPEZOIDRemember for a trapezoid, there are two parallel sides, and they are

both bases.The area of a trapezoid is the length of its height times one-half of the

sum of the lengths of the bases.

A = ½(b1 + b2)hWhy?

Red Triangle = ½ b1hBlue Triangle = ½ b2hAny trapezoid can be

divided into 2 triangles.

HEIGHT

BASE 2

BASE 1

Kite/RhombusThe area of a kite is related to its diagonals.Every kite can be divided into two congruent triangles.

The base of each triangleis one of the diagonals.The height is half of theother one.

A = 2(½•½d1d2)A = ½D1D2

d1

d2

Diameterd=2r

CircumferenceC=2πr

AreaA=πr2

radius

diameter

Rectangle P = 2l + 2w A = bh

Square P = 2l + 2w A = bh

Triangle P = side + side + side

A = ½ bh

Parallelogram P = 2l + 2w A = bh

Trapezoid P = 2l + 2w

Circles C = 2∏r A = ∏r²

1 21 ( )2

A b b h

SURFACE AREA AND VOLUME OF SOLID FIGURES

SURFACE AREAthe amount of paper you’ll

need to wrap the shape

VOLUMEthe number of

cubic units contained in the

solid.

SURFACE AREA

Total surface area:

6 (side) or 6(s) ² ²

Lateral surface area:

4(side) or 4 (s) ² ²

CUBE

VOLUME

CUBE/SQUARE PRISM

V = s²H The product of its height H and the area of its base s². S

SH

SURFACE AREA

Total surface area:

2(lb+bh +lh)

Lateral surface area:

2(l+b)h bl

h

RECTANGULAR PRISM

VOLUME

V = lwhThe product of

its length , width/base and

heightw

l

h

SURFACE AREA

Curved surface area 2 π rh

+area of the circle

2 π r2 0r

Total surface area: πrh +2 π r2

=2 π r(h+r)

CYLINDER

VOLUME

V = BhV= πr²h

The product of its base (πr²) and

height (h)

h

b

SURFACE AREA

SA = ½ lp + BWhere l is the Slant

Height andp is the perimeter and

B is the area of the Base

TRIANGULAR PRISM

VOLUME

(1/3) Area of the Base x height

Or

(1/3) BhOr

1/3 x Volume of a Prism

b

h

SURFACE AREA

Total surface area of cone:

π r(s+r)

Lateral surface area of cone-

π rs

CONE

VOLUME

V = ⅓Bh V= ⅓ πr²h

where B is the area of the base and h is the height of the cone.

(1/3 the area of a cylinder)

THE

END