THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This document summarizes a study on designing and delivering a continuing professional development (CPD) course on mathematics for A-level Biology. It involved biology teachers in designing the course content on exponentials/logarithms and statistics. The day was delivered to 20-30 teachers. Data was collected on the impact on teachers' math confidence and teaching practice. Preliminary findings showed the design process improved math teaching confidence and the day further increased statistics confidence while providing pedagogical support ideas. Challenges included time away from school and differing research/practice cultures.
Study on Acidity of Fruits and Vegetables JuicesVanshPatil7
Vansh Patil conducted a study to test the acidity of various fruit and vegetable juices using pH paper. The juices tested included orange, apple, pomegranate, and guava. Most juices were found to be acidic, with pH values ranging from 2.8 to 4.5. Orange juice and pomegranate juice were the most acidic. The acidity is due to the presence of citric acid and phosphoric acid in the fruits. The study concluded that fruit juices are generally acidic in nature.
This slide was presented by the Maths Department of Cochin Refineries School for the Inter-School workshop conducted as a part of World Mathematics Day celebration. "Mathematics in day to day life"
This document discusses the history of Indian mathematics through several prominent mathematicians such as Aryabhata, Bhaskaracharya, Varaha Mihira, and Srinivasa Ramanujan. It notes that while attitudes are slowly changing, Indian mathematical contributions remain neglected or attributed to other cultures. The document aims to address this neglect by discussing several influential Indian mathematicians and their achievements, as well as examining why Indian works were neglected and why this represents an injustice.
This document provides an overview of mathematics and its relationship to concepts of beauty, architecture, and human life. It discusses how mathematical patterns like the golden ratio and Fibonacci sequence are found in nature and influence concepts of beauty. It also explores how mathematics influenced ancient architecture and how geometry guides both fields. Additionally, it examines how mathematicians think and how numbers are fundamental to mathematics, similar to how words are to language. The document aims to convey the breadth of mathematics and its applications beyond numerical calculations.
This document appears to be a biology investigatory project on drug addiction completed by a student. It includes sections on the objective, classification of drugs, how addiction begins, effects of specific drugs like tobacco and alcohol, and conclusions. The project received guidance from the student's biology teacher and utilized several references in its completion.
The document discusses biodiversity at three hierarchical levels - genetic, species, and ecological diversity. It provides examples of genetic diversity within different species. Species diversity depends on the number and richness of species in a region. Ecological diversity includes different ecosystem types. Tropical regions generally have higher biodiversity than temperate or polar areas. Species richness increases with area up to a limit based on species-area relationships. The document outlines threats to biodiversity from habitat loss, overexploitation, invasive species, and co-extinctions.
Importance of mathematics in our daily lifeHarsh Rajput
The document discusses the history and origins of mathematics. It notes that mathematics originated from practical needs like measurement and counting, with early forms found on notched bones and cave walls. Over thousands of years, mathematics has developed from attempts to describe the natural world and arrive at logical truths. Today, mathematics is highly specialized but also applied in diverse fields from politics to traffic analysis. The document also provides examples of how concepts in commercial mathematics, algebra, statistics, geometry are useful in daily life.
This document summarizes a study on designing and delivering a continuing professional development (CPD) course on mathematics for A-level Biology. It involved biology teachers in designing the course content on exponentials/logarithms and statistics. The day was delivered to 20-30 teachers. Data was collected on the impact on teachers' math confidence and teaching practice. Preliminary findings showed the design process improved math teaching confidence and the day further increased statistics confidence while providing pedagogical support ideas. Challenges included time away from school and differing research/practice cultures.
Study on Acidity of Fruits and Vegetables JuicesVanshPatil7
Vansh Patil conducted a study to test the acidity of various fruit and vegetable juices using pH paper. The juices tested included orange, apple, pomegranate, and guava. Most juices were found to be acidic, with pH values ranging from 2.8 to 4.5. Orange juice and pomegranate juice were the most acidic. The acidity is due to the presence of citric acid and phosphoric acid in the fruits. The study concluded that fruit juices are generally acidic in nature.
This slide was presented by the Maths Department of Cochin Refineries School for the Inter-School workshop conducted as a part of World Mathematics Day celebration. "Mathematics in day to day life"
This document discusses the history of Indian mathematics through several prominent mathematicians such as Aryabhata, Bhaskaracharya, Varaha Mihira, and Srinivasa Ramanujan. It notes that while attitudes are slowly changing, Indian mathematical contributions remain neglected or attributed to other cultures. The document aims to address this neglect by discussing several influential Indian mathematicians and their achievements, as well as examining why Indian works were neglected and why this represents an injustice.
This document provides an overview of mathematics and its relationship to concepts of beauty, architecture, and human life. It discusses how mathematical patterns like the golden ratio and Fibonacci sequence are found in nature and influence concepts of beauty. It also explores how mathematics influenced ancient architecture and how geometry guides both fields. Additionally, it examines how mathematicians think and how numbers are fundamental to mathematics, similar to how words are to language. The document aims to convey the breadth of mathematics and its applications beyond numerical calculations.
This document appears to be a biology investigatory project on drug addiction completed by a student. It includes sections on the objective, classification of drugs, how addiction begins, effects of specific drugs like tobacco and alcohol, and conclusions. The project received guidance from the student's biology teacher and utilized several references in its completion.
The document discusses biodiversity at three hierarchical levels - genetic, species, and ecological diversity. It provides examples of genetic diversity within different species. Species diversity depends on the number and richness of species in a region. Ecological diversity includes different ecosystem types. Tropical regions generally have higher biodiversity than temperate or polar areas. Species richness increases with area up to a limit based on species-area relationships. The document outlines threats to biodiversity from habitat loss, overexploitation, invasive species, and co-extinctions.
Importance of mathematics in our daily lifeHarsh Rajput
The document discusses the history and origins of mathematics. It notes that mathematics originated from practical needs like measurement and counting, with early forms found on notched bones and cave walls. Over thousands of years, mathematics has developed from attempts to describe the natural world and arrive at logical truths. Today, mathematics is highly specialized but also applied in diverse fields from politics to traffic analysis. The document also provides examples of how concepts in commercial mathematics, algebra, statistics, geometry are useful in daily life.
Applications of mathematics in our daily lifeAbhinav Somani
The document discusses the history of mathematics. It states that the study of mathematics as its own field began in ancient Greece with Pythagoras, who coined the term "mathematics." Greek mathematics refined methods and expanded subject matter. Beginning in the 16th century Renaissance, new mathematical developments interacting with scientific discoveries occurred at an increasing pace. The document also notes that mathematics has been used since ancient times, with early uses including building the pyramids in Egypt.
The document discusses the Fibonacci sequence and its relationship to the golden ratio. It begins by introducing Leonardo of Pisa, who helped spread the use of the modern number system and knowledge of the Fibonacci sequence. The sequence is defined as a pattern where each number is the sum of the two preceding ones, starting with 1, 1, 2, 3, 5, etc. This sequence appears throughout nature and can be seen in spirals of shells, pinecones, and sunflowers. The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1.618, an irrational number important in art and architecture considered aesthetically pleasing. The golden ratio can also be observed in proportions of the human body.
Mathematics is the science of logic, quantity, and arrangement. It is used in many aspects of daily life without realizing it, whether cooking, sports, gardening, banking, navigation, or architecture. Math concepts like ratio, proportion, probability, mensuration, trigonometry, and geometry are essential for tasks like following recipes in cooking, analyzing sports performance, measuring land for gardening, calculating interest for banking, using coordinates for navigation, and constructing buildings. Mathematics is a universal language that is important everywhere.
This document summarizes the key aspects of radioactivity covered in a physics project. It discusses the three main types of radioactive decay - alpha, beta, and gamma decay. Alpha decay occurs when a nucleus has too many protons, causing it to emit a helium nucleus. Beta decay involves either a neutron converting to a proton or vice versa, emitting an electron or positron. Gamma decay occurs when a nucleus shifts between energy states by emitting a photon, without changing its composition. The rate of radioactive decay is characterized by the half-life, the time for half of a radioactive sample to decay.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
Indian mathematicians made many important contributions throughout history. Some key figures introduced the decimal number system and concept of zero, including Aryabhata. Brahmagupta was the first to treat zero as a number. Later mathematicians such as Ramanujan, Mahalanobis, Rao, and Kaprekar made advances in areas like statistics, number theory, and linear programming. Overall, Indian mathematicians have significantly influenced mathematics globally with pioneering concepts and solutions that have shaped the field.
Maths of nature and nature of maths 130513 vorAmarnath Murthy
1) Mathematics is present throughout nature in patterns like bubbles, waves, branches and more. The hexagonal shape of honeycombs and compound eyes maximizes their efficiency.
2) Fibonacci numbers appear in nature, like the spiral patterns of sunflowers and pinecones. They also describe rabbit populations.
3) Bees, flowers, galaxies and more display the golden ratio/spiral, an irrational number close to 1.618 seen in divisions of the Fibonacci sequence. This divine proportion is found throughout nature.
Mathematical activity has changed in every field. We discuss some of these trends and how they could influence the future of mathematical education. The aim of this paper is to study the recent trends in the present day mathematics and the role of mathematics in other disciplines. Dr. A. K. Yadav | Dr. Sushil Kumar | Dr. Rashmi Chaudhary "Recent Trends of Mathematics in Education" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31130.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/31130/recent-trends-of-mathematics-in-education/dr-a-k-yadav
To find the refractive indexes of (a) water,(b) oil using a plane mirror, an ...AnkitSharma1903
1. Ankit Sharma completed a physics project to determine the refractive indices of water and oil using a plane mirror, convex lens, and adjustable needle under the guidance of his teacher Mr. P.K. Sha.
2. The project involved using the lens formula to calculate the focal lengths of the convex lens alone and in combination with water or oil, then using these values and the radius of curvature of the lens to determine the refractive indices.
3. The refractive indices calculated were 1.0831 for water and 1.2886 for oil.
This document provides information about a mathematics magazine created by students of class 10-A. It summarizes the contributors and their roles, as well as the topics covered in the magazine such as stories, cartoons, mathematics lessons, puzzles, and activities. The document explains that the work was divided and assigned by the chief editor, editors, and teacher Ms. Sushma Singh. It aims to make mathematics easily understandable for readers through interesting facts and language. The students worked hard to complete the project successfully with the help of their teacher.
Mathematics is present in everyday activities like cooking, decorating, shopping, business, and more. It is used to measure quantities of ingredients in cooking, surface areas when painting rooms, calculating sales and profits in business. Geometry specifically is applied in building structures, kitchen utensils, sports equipment, traffic signals, musical instruments, and transportation. Math underlies many activities in daily life without us consciously realizing it.
Mathematics guides all sciences and social sciences by providing principles and models. During the 19th century, mathematics was seen as abstract but it is now widely applied across many fields from engineering to genetics due to developments in applied mathematics spurred by World War 2 and Sputnik. Modern technologies like CAT scanners and economic models all depend on sophisticated mathematical foundations. Engineering in particular utilizes differential equations, geometry, and other areas of mathematics.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
1) This document discusses the relationship between maths and physics. Mathematics allows physicists to calculate and equate physical events using equations that translate elements like forces, mass, and motion into numbers.
2) An example is provided of using mathematical equations to predict how an object sliding down an inclined plane will behave based on its mass, the inclination, and distance to the ground.
3) Physics relies heavily on mathematics to calculate and help understand phenomena like motion, forces, energy, and more. Mathematics provides the logical framework for physics.
class 11 Physics investigatory project(cbse)Ayan sisodiya
The document is a physics project report submitted by Pradeep Singh Rathour to his teacher, Mrs. Kalpana Tiwari, on Bernoulli's theorem. The report includes an introduction, acknowledgments, index, and sections explaining key concepts like pressure, Pascal's law, continuity equation, Bernoulli's equation, and applications such as venturi tubes. It discusses how Bernoulli's principle explains lift in airplanes by creating lower pressure above the wing. The report concludes that while Bernoulli's law is often misapplied to explain lift, the accurate explanation requires considering conservation of mass, momentum and energy simultaneously.
This document is a biology project submitted by Mikhil Chandnani of class 12 on the effects of maternal behavior on fetal development. It includes an introduction on how a fetus is affected by the mother's state of mind and nutrition during pregnancy. The project then discusses several causes in detail, including alcohol abuse, drug use, cigarette smoking, stress, and fetal injury, and their effects on the fetus such as birth defects, low birth weight, and developmental issues. It also covers changes during pregnancy and contraceptive methods. The conclusion emphasizes the need for mothers to be careful during pregnancy to support the healthy development of the fetus.
TOPIC-To investigate the relation between the ratio of :-1. Input and outpu...CHMURLIDHAR
TOPIC-To investigate the relation between the ratio of :-1. Input and output voltage.2. Number of turnings in the secondary coil and primary coil of a self made transformer.
The document discusses how patterns in nature can be modeled mathematically through concepts like the Fibonacci sequence and golden ratio. It provides several examples of how these concepts appear in structures like pine cones, sunflowers, nautilus shells, galaxies, and even the human body. The Fibonacci sequence describes the breeding patterns of rabbits introduced by Leonardo Fibonacci in the 13th century, and the ratios between its numbers approach the golden ratio - a number linked to patterns in architecture, music, and nature.
This document discusses the role of mathematics in medicine. It provides several examples:
1. Doctors and nurses use math every day to calculate dosages, interpret medical scans like CAT scans, and analyze medical data.
2. Medicines are identified based on their shape, size, and other geometric properties that can be described using basic shapes.
3. Numbers and units are crucial in medicine for tasks like counting pills, measuring weights and volumes, and statistical analysis.
4. Concepts like ratios, proportions, percentages, and statistics are applied when administering medications and analyzing medical trends.
5. Advanced math underlies technologies like MRI machines that provide crucial medical images.
Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
This document is a student paper submitted to the University of Education in Okara that discusses mathematical chemistry. It introduces mathematical chemistry and its applications in areas like chemical graph theory. It explains how mathematics is used to model chemical phenomena and explore concepts in chemistry. The paper also outlines the scope and importance of mathematical chemistry and compares it to related fields. It concludes by discussing the history of using mathematics to model chemical phenomena.
Applications of mathematics in our daily lifeAbhinav Somani
The document discusses the history of mathematics. It states that the study of mathematics as its own field began in ancient Greece with Pythagoras, who coined the term "mathematics." Greek mathematics refined methods and expanded subject matter. Beginning in the 16th century Renaissance, new mathematical developments interacting with scientific discoveries occurred at an increasing pace. The document also notes that mathematics has been used since ancient times, with early uses including building the pyramids in Egypt.
The document discusses the Fibonacci sequence and its relationship to the golden ratio. It begins by introducing Leonardo of Pisa, who helped spread the use of the modern number system and knowledge of the Fibonacci sequence. The sequence is defined as a pattern where each number is the sum of the two preceding ones, starting with 1, 1, 2, 3, 5, etc. This sequence appears throughout nature and can be seen in spirals of shells, pinecones, and sunflowers. The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1.618, an irrational number important in art and architecture considered aesthetically pleasing. The golden ratio can also be observed in proportions of the human body.
Mathematics is the science of logic, quantity, and arrangement. It is used in many aspects of daily life without realizing it, whether cooking, sports, gardening, banking, navigation, or architecture. Math concepts like ratio, proportion, probability, mensuration, trigonometry, and geometry are essential for tasks like following recipes in cooking, analyzing sports performance, measuring land for gardening, calculating interest for banking, using coordinates for navigation, and constructing buildings. Mathematics is a universal language that is important everywhere.
This document summarizes the key aspects of radioactivity covered in a physics project. It discusses the three main types of radioactive decay - alpha, beta, and gamma decay. Alpha decay occurs when a nucleus has too many protons, causing it to emit a helium nucleus. Beta decay involves either a neutron converting to a proton or vice versa, emitting an electron or positron. Gamma decay occurs when a nucleus shifts between energy states by emitting a photon, without changing its composition. The rate of radioactive decay is characterized by the half-life, the time for half of a radioactive sample to decay.
"Application of 3D and 2D geometry" explains the importance of geometry in our lives. Geometry is found everywhere from nature to human made machines. I have tried to inculcate all
its applications.
I hope it helps in providing guidance to those who are aspiring to understand geometry. I have taken help from internet and some books to acquire knowledge.
thank you for clicking my slide.
Indian mathematicians made many important contributions throughout history. Some key figures introduced the decimal number system and concept of zero, including Aryabhata. Brahmagupta was the first to treat zero as a number. Later mathematicians such as Ramanujan, Mahalanobis, Rao, and Kaprekar made advances in areas like statistics, number theory, and linear programming. Overall, Indian mathematicians have significantly influenced mathematics globally with pioneering concepts and solutions that have shaped the field.
Maths of nature and nature of maths 130513 vorAmarnath Murthy
1) Mathematics is present throughout nature in patterns like bubbles, waves, branches and more. The hexagonal shape of honeycombs and compound eyes maximizes their efficiency.
2) Fibonacci numbers appear in nature, like the spiral patterns of sunflowers and pinecones. They also describe rabbit populations.
3) Bees, flowers, galaxies and more display the golden ratio/spiral, an irrational number close to 1.618 seen in divisions of the Fibonacci sequence. This divine proportion is found throughout nature.
Mathematical activity has changed in every field. We discuss some of these trends and how they could influence the future of mathematical education. The aim of this paper is to study the recent trends in the present day mathematics and the role of mathematics in other disciplines. Dr. A. K. Yadav | Dr. Sushil Kumar | Dr. Rashmi Chaudhary "Recent Trends of Mathematics in Education" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-4 , June 2020, URL: https://www.ijtsrd.com/papers/ijtsrd31130.pdf Paper Url :https://www.ijtsrd.com/mathemetics/other/31130/recent-trends-of-mathematics-in-education/dr-a-k-yadav
To find the refractive indexes of (a) water,(b) oil using a plane mirror, an ...AnkitSharma1903
1. Ankit Sharma completed a physics project to determine the refractive indices of water and oil using a plane mirror, convex lens, and adjustable needle under the guidance of his teacher Mr. P.K. Sha.
2. The project involved using the lens formula to calculate the focal lengths of the convex lens alone and in combination with water or oil, then using these values and the radius of curvature of the lens to determine the refractive indices.
3. The refractive indices calculated were 1.0831 for water and 1.2886 for oil.
This document provides information about a mathematics magazine created by students of class 10-A. It summarizes the contributors and their roles, as well as the topics covered in the magazine such as stories, cartoons, mathematics lessons, puzzles, and activities. The document explains that the work was divided and assigned by the chief editor, editors, and teacher Ms. Sushma Singh. It aims to make mathematics easily understandable for readers through interesting facts and language. The students worked hard to complete the project successfully with the help of their teacher.
Mathematics is present in everyday activities like cooking, decorating, shopping, business, and more. It is used to measure quantities of ingredients in cooking, surface areas when painting rooms, calculating sales and profits in business. Geometry specifically is applied in building structures, kitchen utensils, sports equipment, traffic signals, musical instruments, and transportation. Math underlies many activities in daily life without us consciously realizing it.
Mathematics guides all sciences and social sciences by providing principles and models. During the 19th century, mathematics was seen as abstract but it is now widely applied across many fields from engineering to genetics due to developments in applied mathematics spurred by World War 2 and Sputnik. Modern technologies like CAT scanners and economic models all depend on sophisticated mathematical foundations. Engineering in particular utilizes differential equations, geometry, and other areas of mathematics.
The document announces a mathematics project competition open to students in forms 3 and 4 at Maria Regina College Boys' Junior Lyceum. Teams of two students can participate by creating one of the following: a statistics project, charts, or a PowerPoint presentation on a given theme related to mathematics history or concepts. The top five entries will represent the school in the national competition and prizes will be awarded to the top teams nationally. Proposals are due by November 30th and completed projects by January 18th.
1) This document discusses the relationship between maths and physics. Mathematics allows physicists to calculate and equate physical events using equations that translate elements like forces, mass, and motion into numbers.
2) An example is provided of using mathematical equations to predict how an object sliding down an inclined plane will behave based on its mass, the inclination, and distance to the ground.
3) Physics relies heavily on mathematics to calculate and help understand phenomena like motion, forces, energy, and more. Mathematics provides the logical framework for physics.
class 11 Physics investigatory project(cbse)Ayan sisodiya
The document is a physics project report submitted by Pradeep Singh Rathour to his teacher, Mrs. Kalpana Tiwari, on Bernoulli's theorem. The report includes an introduction, acknowledgments, index, and sections explaining key concepts like pressure, Pascal's law, continuity equation, Bernoulli's equation, and applications such as venturi tubes. It discusses how Bernoulli's principle explains lift in airplanes by creating lower pressure above the wing. The report concludes that while Bernoulli's law is often misapplied to explain lift, the accurate explanation requires considering conservation of mass, momentum and energy simultaneously.
This document is a biology project submitted by Mikhil Chandnani of class 12 on the effects of maternal behavior on fetal development. It includes an introduction on how a fetus is affected by the mother's state of mind and nutrition during pregnancy. The project then discusses several causes in detail, including alcohol abuse, drug use, cigarette smoking, stress, and fetal injury, and their effects on the fetus such as birth defects, low birth weight, and developmental issues. It also covers changes during pregnancy and contraceptive methods. The conclusion emphasizes the need for mothers to be careful during pregnancy to support the healthy development of the fetus.
TOPIC-To investigate the relation between the ratio of :-1. Input and outpu...CHMURLIDHAR
TOPIC-To investigate the relation between the ratio of :-1. Input and output voltage.2. Number of turnings in the secondary coil and primary coil of a self made transformer.
The document discusses how patterns in nature can be modeled mathematically through concepts like the Fibonacci sequence and golden ratio. It provides several examples of how these concepts appear in structures like pine cones, sunflowers, nautilus shells, galaxies, and even the human body. The Fibonacci sequence describes the breeding patterns of rabbits introduced by Leonardo Fibonacci in the 13th century, and the ratios between its numbers approach the golden ratio - a number linked to patterns in architecture, music, and nature.
This document discusses the role of mathematics in medicine. It provides several examples:
1. Doctors and nurses use math every day to calculate dosages, interpret medical scans like CAT scans, and analyze medical data.
2. Medicines are identified based on their shape, size, and other geometric properties that can be described using basic shapes.
3. Numbers and units are crucial in medicine for tasks like counting pills, measuring weights and volumes, and statistical analysis.
4. Concepts like ratios, proportions, percentages, and statistics are applied when administering medications and analyzing medical trends.
5. Advanced math underlies technologies like MRI machines that provide crucial medical images.
Basic algebra, trig and calculus needed for physics.
**More good stuff available at:
www.wsautter.com
and
http://www.youtube.com/results?search_query=wnsautter&aq=f
This document is a student paper submitted to the University of Education in Okara that discusses mathematical chemistry. It introduces mathematical chemistry and its applications in areas like chemical graph theory. It explains how mathematics is used to model chemical phenomena and explore concepts in chemistry. The paper also outlines the scope and importance of mathematical chemistry and compares it to related fields. It concludes by discussing the history of using mathematics to model chemical phenomena.
The document discusses various architectural structures from around the world and through history that incorporate principles of geometry, mathematics, and the golden ratio. Many famous structures like the Parthenon, Taj Mahal, and Notre Dame used proportions and dimensions based on the golden ratio or other mathematical concepts. City planning in ancient India also incorporated mandalas and geometric patterns rooted in cosmological principles. Overall, the document shows how mathematics and principles of structure have long been applied in architectural design around the world.
1) Biology is the scientific study of life. It comes from the Greek words "bios" meaning life and "ology" meaning study of.
2) There are several main branches of biology including botany (the study of plants), zoology (the study of animals), and ecology (the study of interactions between organisms and their environment).
3) Science uses evidence and experimentation to build knowledge about the natural world. The goal is to understand and explain natural phenomena through observations, hypotheses, experiments, and conclusions.
Mathematics can be used to describe the natural world. Symmetry is seen throughout nature, from the patterns in flowers and trees to the shapes of rocks. Symmetry refers to an object being unchanged after a transformation, such as reflection. It is observed from mathematical, scientific, and artistic perspectives. While symmetry implies sameness, asymmetry suggests some things are different or special. Relationships between equals strive for symmetry, while power dynamics introduce asymmetry.
The document provides an overview of the key concepts in the science of biology. It discusses what science aims to do and outlines the scientific method. It describes different types of observations, data expressions, and graphs that are used. Key experiments in biology like those conducted by Redi, Pasteur, and Needham are summarized. The characteristics of living things and levels of biological organization from subatomic to biosphere are highlighted. Different branches of biology are also listed.
Math concepts can be taught through food. Ratios are used in cooking recipes and scaling quantities for different serving sizes. Understanding ratios helps students apply math to real world activities like cooking and baking.
This module introduces the field of biology and its key concepts. It is divided into three lessons: 1) defines biology and its branches, discusses unifying ideas and life processes; 2) describes biotechnology and genetic engineering, how scientists manipulate genes and create recombinant DNA; 3) will discuss tools used in biology like microscopes and contributions of scientists. The purpose is to help students understand the nature and scope of biology and biological concepts applied in technology through interactive lessons and self-tests.
Biology is the study of living organisms. It has two main branches: the study of plants (botany) and animals (zoology). Each branch has many specialized subfields that focus on different types of organisms like mosses, trees, grasses, mammals, birds, fish, insects, and microorganisms. Biology also has branches that study specific areas like anatomy, physiology, biochemistry, ecology, genetics, and evolution. These branches examine topics such as the form, tissues, cells, development, functions, chemistry, environment, heredity, and changes of organisms over time.
This document discusses the relationship between mathematics and nature. It provides examples of symmetry, patterns, ratios and shapes found in nature that relate to mathematical concepts like the golden ratio, pi, the Fibonacci sequence, fractals and perfect shapes. These natural phenomena were inspiration for mathematical discoveries made by ancient civilizations. Symmetry is seen in structures like starfish and flowers, while spirals in shells, pinecones and sunflowers relate to the Fibonacci sequence. The human body and works of art also exhibit mathematical proportions. Overall, the document argues that mathematics is deeply embedded throughout nature.
Dance requires the use of mathematical concepts like rhythm, shapes, movements, spatial organization, and symmetry. Dancers must count steps to keep time with music, form geometric shapes with their bodies, and balance their movements to reflect symmetrical patterns. Key relationships include staying synchronized to music through counting, creating angles, lines, and shapes with arms and legs, and using spatial formations like parallel lines and circles to maximize use of the performance space.
the mathematics of chemistry stoichiometry dimensional analysis.pptjami1779
Here is a stoichiometry question I wrote for the given reaction:
If 3.25 grams of barium chloride are available to react, how many grams of barium sulfate can be produced?
This document discusses the connections between mathematics and food. It provides examples of how concepts like symmetry, ratios, pi, and proportions are seen in foods. Recipes also involve using mathematics through quantities, fractions, temperatures, and times. Creating a balanced diet and salad pyramid also relies on mathematical ratios and proportions. Overall, the document aims to show how mathematics can be found in many aspects of food.
Physics is the study of natural phenomena through observation, experimentation, and quantitative analysis. It uses mathematics to describe the relationships between matter, energy, and fundamental forces. Key areas of physics include mechanics, electromagnetism, thermodynamics, and modern physics. Accurate measurement is important in physics, requiring the appropriate instruments like rulers, callipers, and micrometers to quantify physical properties consistently and accurately.
The document discusses how mathematics is present in nature. It provides examples of symmetry, shapes, parallel lines, and the Fibonacci spiral that can be observed in the natural world. Radial and bilateral symmetry are seen in structures like flowers and the human body. Common shapes found in nature include spheres, hexagons used by bees to build hives efficiently, and cones formed by volcanoes. Parallel lines can be seen in dune formations, and the Fibonacci spiral appears in nautilus shells. The document aims to show how nature demonstrates mathematical concepts and patterns.
1) An airplane accelerates down a runway at a rate of 3.20 m/s^2 for 32.8 seconds until it lifts off.
2) Using the equation for uniformly accelerated motion, the document discusses key concepts like displacement, velocity, acceleration, and graphical representations of motion.
3) Sample problems are worked through, applying the equations of motion to situations involving linear motion, free-falling objects, and projectile motion.
Radial, bilateral, translational, and wallpaper symmetries are prevalent in nature. Examples include starfish with dihedral symmetry, trees with translational strip patterns, hexagonal structures like honeycombs and the Giant's Causeway that tessellate plane patterns, and the approximate bilateral symmetry found in human faces. Symmetry represents a balance and order that is reflected in both man-made and natural designs.
Uniformly accelerated motion (free fall) problems and solutionsSimple ABbieC
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
A lecture note on Microbial Growth and Nutrition, and Clones, Enzymes and Inf...Akram Hossain
This was an assignment of preparing “A lecture note on Microbial Growth and Nutrition, and Clones, Enzymes and Informative Hybridizations” for the course "General Microbiology"
Hope you will find it useful.
The lesson plan is for a biology class on microorganisms. It will discuss the occurrence of microorganisms, focusing on bacteria. Students will learn about the discovery of microorganisms, their structures, and different types. The class will explain bacteria's occurrence, unicellular nature, sizes and shapes. Key aspects like nutrition, growth and reproduction through binary fission will also be covered. Evaluation includes questions to test students' understanding and a homework assignment on drawing bacterial structures and their life processes.
Reproduction and growth of bacteria by TanzirTanzir Ahmed
This document provides information about various modes of bacterial reproduction and growth. It discusses binary fission as the main form of asexual reproduction in bacteria, where the parent cell divides into two identical daughter cells. The normal bacterial growth cycle is described through four phases: lag phase, log or exponential phase, stationary phase, and death phase. Methods for measuring bacterial growth include direct microscopic counting, plate counts, and indirect techniques like turbidity measurements and assessing metabolic activity. Synchronous bacterial cultures aim to study all cells at the same stage of the cell cycle for more precise analyses.
This document provides information about the course AMBE-101 Agricultural Microbiology which is worth 2 credits and includes a 1 hour lecture and 1 hour lab. It discusses various topics related to microbial growth including bacterial reproduction methods, factors affecting growth, growth curves and phases, synchronous growth, and continuous culture techniques like chemostats which are used to maintain bacterial cultures in exponential phase. Continuous culture using a chemostat works by maintaining a constant dilution rate and limiting nutrient concentration to control bacterial growth rate and keep cell density constant.
Bacteria reproduce through binary fission in which one parent cell divides to form two progeny cells, allowing for exponential growth. The bacterial growth cycle consists of four phases - lag phase, log/exponential phase, stationary phase, and death phase. During the lag phase, cells do not divide as they adapt to their environment. The log phase is marked by rapid cell division, while stationary phase occurs when resources are depleted. The death phase follows as cell numbers decline. Bacteria can be aerobic, using oxygen for respiration, facultative anaerobic, using oxygen or fermentation, or obligate anaerobic, unable to tolerate oxygen. Important bacterial groups include mycoplasmas, spirochetes, actinomy
MICROBIAL GROWTH, REPRODUCTION AND CONTROLPeterKenneth3
Microbial growth is defined as an increase in the number of cells. A microbial cell has a lifespan and a species is maintained only as a result of continued growth of its population. Growth is the ultimate process in the life of a cell – one cell becoming two and subsequently leading to an increase in the number in a population of microorganisms.
In microbiology, growth is synonymous to reproduction. This unit examines the term growth, binary fission, the mode of cell division in prokaryotic cells, stages in the growth curve and the mathematics of growth.
Definition of Growth
Growth is defined as an increase in the number of cells in a population of microorganisms. It is an increase in cellular constituents leading to arise in cell number when microorganisms reproduce by processes like binary fission or budding.
The Prokaryotic Cell Cycle
A prokaryotic cell cycle is the complete sequence of events from the formation of a new cell through the next division. Most prokaryotes reproduce by binary fission, budding or fragmentation.
Binary Fission
Binary fission is a form of asexual reproduction process. In which a single cell divides into two cells after developing a transverse septum(cross wall).Binary fission is a simple type of cell division and the processes involved are: the cell elongates, replicates its
chromosomes and separates the newly formed DNA molecules so that there is a chromosome in each half of the cell. A septum is formed at mid cell; divide the parent cell into two progeny cells and each having its own chromosome and a copy or complement of other cellular constituents.
Microbial interactions can be positive or negative. Positive interactions include mutualism, where both organisms benefit, and commensalism, where one benefits and the other is not affected. Negative interactions include competition, where resources are limited and both are harmed; amensalism where one is harmed; and parasitism where one benefits and the other is harmed. Examples of microbial interactions discussed include lichens as mutualism, gut bacteria as commensalism, and viruses as parasites. Microorganisms interact in complex ways that allow colonization of diverse environments.
Microbes-Introduction and significance.pptxyogesh301636
Microbiology is the study of microorganisms that are microscopic and includes bacteria, archaea, fungi, protozoa, algae, and viruses. The history of microbiology began in the 17th century with discoveries made using early microscopes. Key figures who advanced the field included Van Leeuwenhoek who first observed microbes, Pasteur who disproved spontaneous generation, and Koch who established the germ theory of disease. Microbiology now encompasses the study of these diverse microbes including their characteristics, habitats, and roles in nature, industries, and diseases.
1. The document discusses microbial growth curves and the different phases of bacterial growth: lag, exponential, stationary, and death phase.
2. It explains key concepts like generation time, growth rate constant, batch and continuous culture techniques, and the importance of understanding growth curves.
3. The mathematics of growth is also covered, including definitions and calculations of generation time and growth rate constant.
The document provides an overview of the contents of a biology textbook organized into 5 units covering various topics in biology. Unit 1 discusses diversity in the living world, including chapters on the living world, biological classification, plant kingdom, and animal kingdom. The summary provides the high-level structure and scope of the textbook without extensive detail on the document contents.
This document outlines the course content for a course on microbial ecology. It covers 5 units: (1) microbes and ecological theory, (2) microorganisms in ecosystems, (3) physiological adaptations of microorganisms, (4) bioconversion, and (5) microbial interactions. Unit 1 defines key terms like ecosystem, habitat, niche, and discusses colonization and succession. It provides examples of succession on cellophane films, dung, and human hair. The overall objective of the course is to study microbial community dynamics and interactions between microbes, plants, animals and their environments.
Biology First Year Complete 14 ChaptersSeetal Daas
This document provides notes on biology for a first year class. It begins with definitions of biology and classifications of living organisms. It then discusses the key kingdoms - Monera, Protoctista, Fungi, Plantae, and Animalia. Later sections cover branches of biology, biological methods, examples like malaria, concepts like cloning and hydroponics, and levels of biological organization from subatomic particles to the biosphere. It concludes by discussing biochemistry and the important properties of water for biological functions.
This document discusses microorganisms and provides information about their types and roles. It begins by defining microorganisms as organisms that are mostly microscopic in size and can be seen with a microscope. It then lists the main types of microorganisms as fungi, bacteria, protozoa, algae, and viruses. The document also discusses how some microorganisms like lactobacilli and yeast can be good, protecting the body from diseases, while others like certain bacteria can cause illnesses. It concludes by stating that bacteria can be both good and bad for humans, as some are needed for digestion while others cause pathogenic infections.
The document provides science objectives for 4th grade students related to ecology and interdependence between plants and animals. The objectives include: [1] Describing the interdependence of plants and animals through behaviors, body structures, and life cycles; [2] Tracing the flow of energy through food chains; and [3] Identifying characteristics of organisms. The objectives aim to teach students how plants and animals rely on each other to survive through relationships like predation, competition, and symbiosis.
Microbial growth involves an increase in cell number through cell division. There are four phases of bacterial growth in a batch culture: lag phase, exponential/log phase, stationary phase, and death phase. During exponential phase, the population doubles in each generation time. Environmental factors like temperature, pH, oxygen levels, and nutrients influence microbial growth rates.
Bacteria require certain environmental conditions for growth, including oxygen, pH, temperature, and light. The bacterial growth curve shows four phases of bacterial growth over time: lag phase, exponential/log phase, stationary phase, and death phase. During the lag phase, cells prepare for replication but do not divide. Exponential growth then occurs as cells divide rapidly by binary fission. As nutrients become depleted, growth slows in the stationary phase. In the death phase, cell numbers decline as nutrients are depleted and waste builds up. Optimal growth conditions vary between bacterial species and environments.
1. Microbiology is the study of microbes including bacteria, fungi, protozoa, algae, and viruses. Key figures who advanced microbiology include Antony van Leeuwenhoek who discovered microbes using microscopes, Louis Pasteur who disproved spontaneous generation through experiments with swan neck flasks, and Alexander Fleming who discovered penicillin.
2. Cells are the fundamental unit of life. Prokaryotic cells like bacteria lack a nucleus and other membrane-bound organelles, while eukaryotic cells include fungi, plants, animals and have a membrane-bound nucleus. The surface area to volume ratio of cells constrains their size, as larger cells require more membrane surface area for
The document provides guidance on using the features and tools available on the TwinSpace online platform for eTwinning projects. It explains how to set up pages, forums, and multimedia galleries to organize project content and discussions. Instructions are given for inviting students and teachers, setting permissions, and using chat and other communication features.
This document provides information about Małgorzata Garkowska, a math teacher of 25 years who has been involved with eTwinning since 2006. It discusses tools she uses for teaching like Google Maps, Google Earth, Google Tour Builder, and GeoGebra. It provides examples of student activities and projects that can be done with these tools including creating maps, virtual field trips, and interactive math constructions. Hands-on instructions are given for students to collaboratively create maps, tours, and complete math tasks using the tools.
This document provides information about Małgorzata Garkowska, a math teacher with over 20 years of experience who has been involved with eTwinning since 2006. It then discusses several free online tools that can be used for educational purposes: Google My Maps for creating customized maps; Google Earth for virtual exploration of places; Google Tour Builder for creating geographic storytelling tours; and GeoGebra for interactive math learning. Instructions are provided on features and functions of each tool. The document concludes with directions for partners to work together using the hands-on tasks of creating maps and tours with Google tools, and constructing geometric shapes and graphs with GeoGebra.
The European Commission has selected the 2012-1-ES1-COM06-52752 project "Why Maths?" as a "success story" based on its impact, contribution to policy-making, innovative results, and creative approach. As a result of this selection, the project will receive increased visibility on Commission websites and social media, and at conferences. The project coordinator may also be contacted by ECORYS, the Commission's contractor for disseminating and exploiting project results, to provide additional materials about the project. The selection recognizes the commitment, enthusiasm, and high-quality work of the project partners.
1) Se presentan ecuaciones diferenciales ordinarias que involucran funciones trigonométricas como seno, coseno y sus derivadas.
2) Se resuelven las ecuaciones aplicando técnicas como separación de variables y sustitución de funciones.
3) Se obtienen expresiones para las funciones desconocidas en términos de constantes.
1) Se presentan ecuaciones diferenciales ordinarias que involucran funciones trigonométricas como seno, coseno y sus derivadas.
2) Se resuelven las ecuaciones aplicando propiedades de las funciones trigonométricas y técnicas de resolución de ecuaciones diferenciales.
3) Se obtienen las soluciones en función de constantes arbitrarias y el intervalo de definición indicado para cada una.
THIS BROCHURE WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA
COM
3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
Maths in Art and Architecture Why Maths? Comenius projectGosia Garkowska
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WHY MATHS?
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS ( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA (ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ (POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA (PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This document contains trivia questions and answers about mathematics, art, geography, and astronomy related to the Comenius project involving schools from Portugal, Spain, Italy, Belgium, Poland, and Ireland. There are over 100 multiple choice and short answer questions covering topics like the capital cities and populations of the countries involved, famous artists and their use of mathematical concepts in works of art, properties of planets and galaxies, and principles of map making and geography.
Maths and Astronomy Why Maths Comenius projectGosia Garkowska
This document discusses various topics related to math and astronomy. It begins by introducing distance units used in space like astronomical units (AU), light years, parsecs and larger multiples. It then discusses how to calculate acceleration due to gravity on different planets using Newton's law of universal gravitation. Next, it examines weight on other planets and how gravity differs based on planetary characteristics. The document concludes by calculating the orbital speeds of planets around the sun.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
हिंदी वर्णमाला पीपीटी, hindi alphabet PPT presentation, hindi varnamala PPT, Hindi Varnamala pdf, हिंदी स्वर, हिंदी व्यंजन, sikhiye hindi varnmala, dr. mulla adam ali, hindi language and literature, hindi alphabet with drawing, hindi alphabet pdf, hindi varnamala for childrens, hindi language, hindi varnamala practice for kids, https://www.drmullaadamali.com
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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Executive Directors Chat Leveraging AI for Diversity, Equity, and InclusionTechSoup
Let’s explore the intersection of technology and equity in the final session of our DEI series. Discover how AI tools, like ChatGPT, can be used to support and enhance your nonprofit's DEI initiatives. Participants will gain insights into practical AI applications and get tips for leveraging technology to advance their DEI goals.
Physiology and chemistry of skin and pigmentation, hairs, scalp, lips and nail, Cleansing cream, Lotions, Face powders, Face packs, Lipsticks, Bath products, soaps and baby product,
Preparation and standardization of the following : Tonic, Bleaches, Dentifrices and Mouth washes & Tooth Pastes, Cosmetics for Nails.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
2. 2
THIS EBOOK WAS PREPARED
AS A PART OF THE COMENIUS PROJECT
WWHHYY MMAATTHHSS??
by the students and the teachers from:
BERKENBOOM HUMANIORA BOVENBOUW, IN SINT-NIKLAAS
( BELGIUM)
EUREKA SECONDARY SCHOOL IN KELLS (IRELAND)
LICEO CLASSICO STATALE CRISTOFORO COLOMBO IN GENOA
(ITALY)
GIMNAZJUM IM. ANNY WAZÓWNY IN GOLUB-DOBRZYŃ
(POLAND)
ESCOLA SECUNDARIA COM 3.º CICLO D. MANUEL I IN BEJA
(PORTUGAL)
IES ÁLVAREZ CUBERO IN PRIEGO DE CÓRDOBA (SPAIN)
This project has been funded with support from the European Commission.
This publication reflects the views only of the author, and the
Commission cannot be held responsible for any use which may be made of the
information contained therein.
4. 4
HHOOWW IISS NNAATTUURREE RREELLAATTEEDD TTOO MMAATTHHSS??
Mathematics might seem an ugly and irrelevant subject at school, but it's
ultimately the study of truth - and truth is beauty! You might be surprised to find that
maths is in everything in nature from rabbits to seashells. Mathematics is everywhere in
this universe, even though we may not notice it. In this chapter, we are going to explore
a few properties of mathematics that are depicted in nature, mainly in:
I. Symmetry:
• Bilateral;
• Radial.
II. Patterning:
• Bacterial population growth;
• Snowflakes;
• Diamonds.
III. Fibonacci sequence
IV. Tesselations
5. 5
II.. SSYYMMMMEETTRRYY
Symmetry is when a figure has
two sides that are mirror images of
one another. It would then be possible to
draw a line through a picture of the object
and along either side the image would look
exactly the same.
There are two kinds of symmetry:
One is bilateral symmetry, in which an
object has two sides that are mirror images of each
other. The human body would be an excellent
example of a living being that has bilateral
symmetry.
The other kind of symmetry is radial
symmetry. This is where there is a center
point and numerous lines of symmetry could
be drawn. The most obvious geometric
example would be a circle. (which can be
found, for example, on a spider web).
7. 7
IIII.. PPAATTTTEERRNNIINNGG
Patterns in nature are visible
regularities of form found in the
natural world. These patterns occur in
different contexts and can sometimes
be shaped mathematically. Patterns in
living things are explained by the
biological processes of natural and
sexual selection.
These are some good examples of
patterns found in nature:
Bacterial population growth
Under favorable conditions, a growing
bacterial population doubles at regular
intervals. Growth is by geometric progression:
1, 2, 4, 8, etc. [or 20, 21, 22, 23.........2n (where n
= the number of generations)]. This is called
exponential growth.
Bacteria are everywhere around us.
Given good growing conditions, a bacterium grows slightly in size or length.
A new cell wall grows through the center forming two daughter cells, each with the same
genetic material as the parent cell: if the environment is optimum, the two daughter
cells may divide into four in 20 minutes. So in a very short time we can have many
duplicates of the parent cell as their growth is like this:
8. 8
We can notice that the shape of the curve we obtain is the one of the exponential
function and the growth is very fast. Then why isn't the earth covered with bacteria?
The primary reason is that the conditions in which bacteria live are rarely optimum:
scientists who study bacteria try to create the optimum environment in the lab that is
culture medium with the necessary energy source, nutrients, pH, and temperature, in
which bacteria grow predictably.
Let's have a look at this short video that shows us an example of bacterial growth:
LINK
But in the real world the grow curve is the same only at the beginning of the observation,
then it looks like this:
Let's analyze what happens when bacteria are grown in a closed system (also called
batch culture) like a test tube, the population of cells almost exhibits these growth
dynamics:
9. 9
LAG PHASE: Growth is slow at first and the cells have to adapt to the new
environment and to acclimate to the food and nutrients in their
new habitat. In this phase cellular metabolism is accelerated, cells
are increasing in size, but the bacteria are not able to replicate and
therefore there is no increase in cell mass. The length of the lag
phase depends directly on the previous growth condition of the
organism: when the microorganism growing in a rich medium is
inoculated into nutritionally poor medium, the organism will take
more time to adapt with the new environment. Similarly when an
organism from a nutritionally poor medium is added to a
nutritionally rich medium, the organism can easily adapt to the
environment, it can start the cell division without any delay, and
therefore will have less lag phase it may be absent.
EXPONENTIAL
PHASE:
Once the metabolic machinery is running, they start multiplying
exponentially, doubling in number every few minutes until they
run out of space or nutrients. The growth medium is exploited at
the maximal rate, the culture reaches the maximum growth rate
and the number of bacteria increases exponentially and finally the
single cell divide into two, which replicate into four, eight, sixteen,
thirty two and so on (That is 20, 21, 22, 23.........2n, n is the
number of generations: this will result in a balanced growth. The
time taken by the bacteria to double in number during a specified
time period is known as the generation time, that tends to vary
with different organisms.
STATIONARY
PHASE:
As more and more bugs are competing for dwindling food and
nutrients, booming growth stops and the number of bacteria
stabilizes. As the bacterial population continues to grow, all the
10. 10
nutrients in the growth medium are used up by the microorganism
for their rapid multiplication: this result in the accumulation of
waste materials, toxic metabolites and inhibitory compounds such
as antibiotics in the medium. This shifts the conditions of the
medium such as pH and temperature, thereby creating an
unfavourable environment for the bacterial growth. The
reproduction rate will slow down, the cells undergoing division
tends to be equal to the number of cell death, and finally
bacterium stops its division completely. The cell number is not
increased and thus the growth rate is stabilised. If a cell taken
from the stationary phase is introduced into a fresh medium, the
cell can easily move on the exponential phase and is able to
perform its metabolic activities as usual.
DEATH PHASE: Toxic waste products build up, food is depleted and the bugs begin
to die, so the number of bacteria decreases quite quickly. The
depletion of nutrients and the subsequent accumulation of
metabolic waste products and other toxic materials in the media
will facilitates the bacterium to move on to this phase in which the
bacterium completely loses its ability to reproduce: individual
bacteria begin to die due to the unfavourable conditions and the
death is rapid and at uniform rate. The number of dead cells
exceeds the number of live cells.
LONG TERM
STATIONARY
PHASE:
A small number of bacteria can survive for long periods of time in
a non-growing state. This particular phase of growth is interesting
for research into a number of pathogens as it is thought to best
represent the state in which bacteria survive during a number of
diseases.
The formula for the growth of population can summarized in this way:
Let's see an example.
In 1950, the world's human population was 2,555,982,611. With a growth rate of
approximately 1.68%, what was the population in 1955?
First, let's figure out what everything is:
11. 11
We can ignore the decimal part since it's not a full person.
So, our guess is that the world's population in 1955 was 2,779,960,539.
The actual population was 2,780,296,616 so we were pretty close.
12. 12
Snowflakes
Snowflakes’ patterns can be
incredibly complex. Since snowflakes
can branch differently down to
individual water molecules, the
number of possibilities is extremely
large. Without a restrictive theory
that constrains snowflakes to a
limited number of shapes, it seems
probable that no two snowflakes are
alike.
13. 13
Fractals aren't just something we learn about in math class. They are also a gorgeous
part of the natural world. Here are some of the most stunning examples of these
repeating patterns.
Romanesco broccoli is a particularly symmetrical fractal.
The fern is one of many flora that are fractal; it’s an especially good example.
Each part is the roughly the same as the
whole. When we break a leaf off of the
original and it looks like the original – break
a leaf off of that leaf and that looks like the
original also.
The delicate Queen Anne’s Lace, which
is really just wild carrot, is a beautiful
example of a floral fractal. Each blossom
produces smaller iterative blooms. This
particular image was shot from
underneath to demonstrate the fractal
nature of the plant
14. 14
IIIIII.. FFIIBBOONNAACCCCII SSEEQQUUEENNCCEE
Leonardo Fibonacci was an Italian
mathematician. Fibonacci was regarded as the first
great European mathematician of the Middle Ages
and he’s responsible for the creation of the
Fibonacci Sequence:
The first two terms are F0= 0 and F1= 1. This sequence has a simple law: every element,
after the third, is obtained by adding the previous two. See: 1 +1 = 2, 2 +1 = 3, 3 +2 = 5,
and so on.
Fibonacci and a population of rabbits (example)
Fibonacci considered the growth of a population idealized (not biologically
realistic) of rabbits. The numbers describe the number of couples in the population of
rabbits after n months if we deduce that:
● on the first month only a couple is born;
● couples are only expected to be sexually mature (and reproduce) after the
second month of life;
● there are no problems in genetic inbreeding;
● every month, every fertile couple gives birth to a new couple;
● rabbits never die.
● The rabbits born in January 1. will be fertile after 2 months. Therefore, on March
1 they will have descendents
● On April 1, the initial couple are still fertile so they will have another couple of
descendents.
● If we reason similarly, we can deduce that, on June 1, there will be 8 pairs of
rabbits; on July 1, 13 couples; on August 1, 21 couples and so on.
● After a year, that is, January 1st of the following year, we’re expected to have 144
pairs of rabbits.
16. 16
Fibonacci numbers in nature
The Fibonacci numbers appear everywhere in nature, from the leaf arrangement
in plants, to the pattern of the florets of a flower, the bracts of a pinecone, or the scales
of a pineapple.
Probably most of us have never taken the time to examine very carefully the
number or arrangement of petals on a flower. If we were to do so, we would find that the
number of petals on a flower for many flowers is a Fibonacci number.
3 petals lily, iris
5 petals buttercup, wild rose, larkspur, columbine
8 petals delphiniums, clematis
13 petals ragwort, corn marigold, cineraria
21 petals aster, black-eyed susan, chicory
34 petals plantain, pyrethrum
55, 89 petals michelmas daisies, the asteraceae family
18. 18
“Logarithmic Spiral” of a common shell.
In nature, we have a lot of examples of the golden ratio
and the Fibonacci sequence. We can find them in all
natural world. The Fibonacci spiral appears not only in
the perfect nautilus shell but also in pinecone, pineapple,
in hurricanes, ram's horns. The Fibonacci numbers
increase at a ratio that is revealed in objects and spirals.
The Chambered Nautilus if cut in half reveals a series of
chambers. Each chamber increases in size as the mollusk
grows. They also grow in a spiral shape.
19. 19
This same spiral and ratio is present in a great many products of nature; the pinecone,
the pineapple.
If we look at the bottom of a pinecone. We can found same kinds of spirals. They don’t
go around and around in a circle – they go out like fireworks. Look at the pictures above,
to see what that looks like.
Another examples of spiral:
21. 21
SSEEEEDD PPAATTTTEERRNNSS OOFF
SSUUNNFFLLOOWWEERRSS
All the sunflowers in the world
show a number of spirals that
are within the Fibonacci
Sequence.
Look at the following images of a sunflower:
By observing closely the seeds configuration you will see how appears a kind of spiral
patterns. In the top left picture we have highlighted three of the spirals typologies that
could be found on almost any sunflower.
Well, if you look at one of the typologies, for example the one in green, and you go to the
illustration above right you can check that there is a certain number of spirals like this,
specifically 55 spirals.
22. 22
We have more examples in the two upper panels, cyan and orange, they are also
arranged following values that are within the sequence: 34 and 21 spirals.
A lot of people love honey made by tiny bees.
These insects use so much mathematical
strategy throughout their daily lives. Just their
hives use angles, shape, tessellation and
addition.
Wasps and bees exhibit polygons in
their nests. Hexagons create nests
that require less material and work
to build. It is an efficient way of
partitioning that also saves energy.
Hexagonal cell requires minimum
amount of wax for construction
while it stores maximum amount of
honey
23. 23
IIVV.. TTEESSSSEELLAATTIIOONNSS
In nature we can see samples of tessellations. This phenomenon is really beautiful and
incredible. Here you can see some examples :
a) The Giant's Causeway, located in Ireland, is an fascinating formation found in
nature. It is a collection of hexagons tesselating the ground - even in 3D at some
points.
24. 24
b) Rock formation in "White Pocket", Vermillion Cliffs National Monument,
c) Veins in a leaf.
d) Dragonfly
e) Cracked dried
25. 25
VV.. TTHHEE GGAAMMEE OOFF LLIIFFEE
"The mathematician’s patterns, like the
painter’s or the poet’s must be
beautiful; the ideas like the colours or
the words, must fit together in a
harmonious way. Beauty is the first
test: there is no permanent place in the
world for ugly mathematics."
GH Hardy (1877 – 1947)
Introduction
Everybody can easily understand that a population of living beings can evolve
depending on the environmental conditions around it. Under their premises, for more
than one century scientists and mathematicians in particular have been studying and
developing different models to better understand the future evolution of a given
population under a set of given environmental conditions. This research field was called
“Theoretical Biology”: traditional population
genetic models are frequently stochastic. For
instance around 1925 two mathematicians,
the Italian Vito Volterra and the American
Alfred James Lotka, developed
independently a model, later named Lotka-
Volterra prey-predator model, to describe
the dynamics of biological systems in which
two species interact, one as a predator and
the other as prey.
In recent years, mathematical modeling of developmental processes has earned new
respect. Not only mathematical models have been used to validate hypotheses made
from experimental data, but designing and testing these models has led to testable
26. 26
experimental predictions. There are now many different impressive cases in which
mathematical models have given a substantial new contribution to a better knowledge of
biological systems by suggesting how connections between local interactions among
system components relate to their wider biological effects.
In evolutionary game theory, biologists work together with mathematicians to obtain a
deterministic mathematical form, with selection acting directly on inherited genetic
types.
The game
The Game of Life is a simulation game that reproduces the real-life processes of the
living organisms: it is not a game in the conventional sense as there are no players and
nobody wins or loses; once the "pieces" are placed in the starting position, the rules
determine everything that happens later. Nevertheless, the game is full of surprises: in
most cases, it is impossible to look at a starting position (pattern) and see what will
happen in the future. The only way to find out is to follow the rules of the game.
It was invented by the British mathematician
John Conway (Liverpool 1937) who was teaching
at Cambridge University.
In 1970 “Scientific American” published an article
in which is mentioned for the first time this game.
To play the game is necessary to have a
chessboard extended infinitely in all directions:
the basic idea is to start with a simple
configuration of organisms, one to a cell, then
observe how it changes as we apply
Conway's 'genetic laws' for births, deaths
and survivals.
Conway's genetic laws are easy:
• each cell can have eight neighbors.
• each cell can evolve according to one of these 3 rules:
• Survival: every cell with 2 or 3 neighbors cells survives for the next
generation
• Death:
• each cell with 4 or more neighbors dies for overpopulation
• every cell with one neighbor or none dies for isolation.
27. 27
• Births: each empty cell adjacent to exactly 3 neighbors is a birth cell.
Conway choose these rules quite carefully after trying many other possibilities: some
caused the cells to die too fast while others caused too many cells to be born. The rules
he eventually choose balance these tendencies, making it hard to tell whether a pattern
will die out completely, form a stable population, or grow forever.
Conway conjectures that no pattern can grow without limit. He has offered a prize of
$50 to the first person who can prove or disprove this conjecture.
We can observe that:
• there should be no initial pattern for which there is a simple proof that the
population can grow without limit;
• there should be simple initial patterns that apparently grow without limit;
• there should be simple initial patterns that grow and change for a considerable
period of time before going to end in three possible ways: fading away completely
(from overcrowding or becoming too sparse), setting into a stable configuration that
remains unchanged, or entering in oscillating phase in which they repeat an endless
cycle of two or more periods.
From a random initial pattern of living cells on the grid, observers will find the
population constantly changing as the generations tick by.
The patterns that emerge from the simple rules may be considered a form of beauty.
Small isolated subpatterns with no initial symmetry tend to become symmetrical.
Once this happens, the symmetry may increase in richness, but it cannot be lost unless a
nearby subpattern comes close enough to disturb it. In a very few cases the society
eventually dies out, with all living cells vanishing, though this may not happen for a
great many generations. Most initial patterns eventually "burn out", producing either
stable figures or patterns that oscillate forever between two or more states; many also
produce one or more gliders or spaceships that travel indefinitely away from the initial
location.
The game and reality
Life is just one example of a cellular automaton, which is any system in which rules are
applied to cells and their neighbors in a regular grid.
There has been much recent interest in cellular automata, a field of mathematical
research. Life is one of the simplest cellular automata to have been studied, but many
others have been invented, often to simulate systems in the real world.
The Game of Life is one of the simplest examples of "emergent complexity" or "self-
organizing systems" : this topic captured the attention of scientists and mathematicians
in diverse fields who are studying of how elaborate patterns and behaviors can emerge
from very simple rules. For instance, it helps us understand how the petals on a rose or
the stripes on a zebra can arise from a tissue of living cells growing together or it how
28. 28
the diversity of life that evolved on earth along the ages.
The game and computer programming
Early patterns with unknown futures led computer programmers across the world to
write programs to track the evolution of Life patterns. Most of the early algorithms were
similar; they represented Life patterns as two-dimensional arrays in computer memory.
Typically two arrays are used, one to hold the current generation, and one in which to
calculate its successor. Often 0 and 1 represent dead and live cells respectively. A nested
for-loop considers each element of the current array in turn, counting the live
neighbours of each cell to decide whether the corresponding element of the successor
array should be 0 or 1. The successor array is displayed. For the next iteration the arrays
swap roles so that the successor array in the last iteration becomes the current array in
the next iteration.
The rules can be rearranged from an egocentric approach of the inner field regarding its
neighbors to a scientific observers point: if the sum of all nine fields is 3, the inner field
state for the next generation will be life if the all-field sum is 4, the inner field retains its
current state and every other sum sets the inner field to death.
The game and its inventor
This is an interview to Conway talking about the game:
http://a.parsons.edu/~joseph/k2/gameoflife/
Let's play the game... using this link:
http://www.kongregate.com/games/shaman4d/conways-game-of-life
To see more - here is the LINK to the presentation prepared by Italian students.
29. 29
VVII.. CCHHAAOOSS TTHHEEOORRYY AANNDD MMAATTHHSS
“Chaos was the law of nature: Order was the dream of man”
-Henry Adams-
INTRODUCTION: When we talk about chaos we normally mean a sense of disorder and
randomness. But It’s very important to understand that, in the new science conception,
chaos is seen, on the contrary, as a very complex order, so complex to escape to the
human’s perception and comprehension. According to this, chaos is the real essence of
order.
THE HISTORY OF CHAOS: Until the 1960’ the world of science was relatively simple:
everything could be explained with simple formulas and everything behaved in a
predictable way. The theory of chaos was worked out when the
classical science could explain anymore the irregular and
inconstant aspects of nature. In 1961 Edward Lorenz, an
American meteorologist and one of the pioneer of the theory of
chaos, was working on his weather forecasting machine and he
decided he wanted to examine a previous day sequence in more
detail. He typed the numbers from the previous day’s print out
into the computers and went to get a coffee. When he returned
he couldn’t believe at his eyes. The new weather was nothing like
the original. Then he realized what happened: he had made a
minimal error that normally would have been considered
insignificant. Repeating the simulation with slightly different values the results was
completely different. Lorenz realized that weather is a chaotic system, that is to say an
extreme sensible system dependent upon initial conditions.
BUTTERFLY EFFECT: This simply means that even the most minute and almost
imperceptible change in the starting conditions could well generate unpredictable
results in the final outcome. Lorenz explained this theory with the simple example of the
Butterfly Effects: a flap of a butterfly’s wings in China has the potential to cause a
hurricane in North America. In conclusion it will never be possible to predict the
behavior of systems that shows extreme sensitivity. That’ s why we can’ t predict
whether for more than a week in advance.
FRACTALS: Fractals are strictly connected with chaos. Gaston Julia is considered the
father of fractals but the study of this structures really evolved with the use of
computers. The French professor Benoit Mandelbrot discovered them studying nature:
he noted that nature has a peculiar tendency to repeat itself, often in strange and
unpredictable patterns. Fractals describe some chaotic behavior and they can be
expressed by complex equations. Fractals are for example the never-ending spirals on a
head of broccoli or the rhythmic scales of snakes.
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INTRODUCTION OF BIOLOGY: Thanks to Mandelbrot scientist started to think that
fractals can be applied in the field of biology. In fact he proposed that lungs show signs
of fractal geometry.
Thanks to fractals and chaos we can accurately describe
and understand various parts of the human body. Some
scientists support that such inefficiency exists in the world
of medicine because the human body is a chaotic system In
Biology, Chaotic systems can be used to show the rhythms
of heartbeats, walking strides, and even the biological
changes of aging. Fractals can be used to model the
structures of nerve networks, circulatory systems, lungs,
and even DNA.
You could find it strange but, it was proved that the absence
of chaos is a sign of disease. According to this theory a group of scientists found that
some subjects with congestive heart failure actually experienced periods of time where
there was no chaos in their heartbeat. When humans walk with variation in their step
they are normal. With the onset of a disease, such as Parkinson’s disease, the human
stride is more constant. An absence of chaos can also be a sign of aging, as our bodies
and the systems within them lose their chaotic characteristics as we grow older.
Brain is another example of chaotic system that can only be modeled using fractal
geometry. In addition to being able to model electrical signals of nerves and the brain,
chaos theory may help solve neurological diseases and progress the invention of
artificial intelligence.
The most significant contribution that chaos theory could make in the field of medicine
would have to do with the creation of chaotic models that would be able to predict
progression of aging or diseases within the body. Applying chaotic models to other
systems of the body such as the immune system could help us in understanding how it
ages. Some scientists support that the knowledge we have gained in fractals and chaos
theory could be helpful to cure diseases such as cancer and cystic fibrosis instead stem
cell, radiation treatment or any other uncertain cures. In conclusion we can affirm that
a human body is healthy when there is a balance between order and chaos.
To see more - here is the LINK to the presentation prepared by Italian students.