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Chapter 1: The Nature of Mathematics Lesson 1: Mathematics in Our World Lesson 1.1: Patterns and Numbers in Nature and The World I. What is a pattern? A pattern is anything that has repetition (with recurring characteristics); It is a series of regular or cons...

Chapter 1: The Nature of Mathematics Lesson 1: Mathematics in Our World Lesson 1.1: Patterns and Numbers in Nature and The World I. What is a pattern? A pattern is anything that has repetition (with recurring characteristics); It is a series of regular or consistent arrangement according to a specific rule or sequence. II. Types of Natural Patterns 1. Symmetry There is symmetry if an imaginary line (line of symmetry) is drawn across an object, the resulting parts are mirrors of each other. 2. Spiral It is a curved pattern that focuses on a center point and a series of circular shapes that revolve around it. This is common in plants and some animals. 3. Stripes Stripes are lines or bands that differ in color or tone from an adjacent area. This pattern may be seen in various living things, especially animals. 4. Spots and Dots Spots and dots in nature are distinctive, often circular patterns that appear on the surfaces of animals, plants, fungi, and even geological formations. They vary in size, color, and distribution, creating unique designs. 5. Meander It is a series of regular sinuous curves, bends, loops, turns, or windings in the channel of a river, stream, or other watercourses. It is produced by a stream or river swinging from side to side as it flows across its floodplain or shifts its channel within a valley. 6. Cracks Cracks are linear openings that form in materials to relieve stress. The pattern of cracks indicates whether the material is elastic or not. III. Main Types of Patterns 1. Logical Patterns To identify logic patterns, you must look out four (4) things: (1) rotating shapes (2) increase and decrease in numbers/ size of shapes or patterns (3) alternating patterns, colors, and shapes (4) mirror images or reflections 2. Geometric Patterns Geometric patterns are a collection of shapes, repeating, or altered to create a cohesive design. These patterns are observed regularly. (1) Tessellations- repeating patterns of polygons that cover a plane with no gaps or overlaps. Another word for a tessellation is tiling. (2) Fractals- Fractals are mathematical constructions characterized by self-similarity. Two objects are self-similar if they can be turned into the same shape by stretching or shrinking (and sometimes rotating). Examples: Sierpinski Triangle, Pascal’s Triangle, Fractal Tree, Koch Snowflake 3. Word Patterns (1) Analogy- compares two different things, but they do it by breaking them into parts to see how they are related (2) Rhyme Scheme- rhymes' pattern at the line of a poem or song (3) Haiku- a Japanese poem with 17 syllables divided into three lines of 5, 7, and 5 syllables  4. Number Pattern A number pattern is a list of numbers that follow a particular sequence or order. Examples: Even numbers- integers that can be divided exactly by 2 …, -10, -8, -6, -4, -2, 0, 2, 4, 6, 8, 10, … Odd numbers- integers that cannot be divided into two parts equally …, -9, -7, -5, -3, -1, 1, 3, 5, 7, 9, … Prime numbers- whole numbers above 1 that cannot be made by multiplying other whole numbers 2, 3, 5, 7, 11, 13, 17, … Composite numbers- numbers with more than two factors 4, 6, 8, 9, 10, 12, 14, 15, … Arithmetic sequence- number pattern where the difference between two consecutive terms is called the common difference Geometric sequence- sequence where a term is multiplied by a constant, called the common ratio, to get the next term Triangular numbers- related to the number of dots needed to create a triangle 1, 3, 6, 10, 15, … Square numbers- the square of consecutive numbers 1, 4, 9, 16, 25, 36, … Cube numbers- the cube of consecutive numbers 1, 8, 27, 64, 125, … Lesson 1.2: The Fibonacci Sequence and the Golden Ratio I. The Fibonacci Sequence The Fibonacci Sequence is a number series in which each number is obtained by adding its two preceding numbers and was named after Leonardo Pisano Bogollo (Leonardo of Pisa), better known as Fibonacci. The breeding of rabbits led to his discovery of the numbers in the Fibonacci sequence. The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, typically starting with 0 and 1, or 1 and 1. Interesting Patterns about the Fibonacci Sequence (1) Every nth Fibonacci number is divisible by F(n). (2) The sum of squares of two consecutive Fibonacci numbers is also a Fibonacci number. (3) The sum of the squares consecutive Fibonacci numbers is a product of two consecutive Fibonacci numbers. (4) When arranged in a certain way, the Fibonacci sequence creates a special spiral pattern known as the Fibonacci spiral. (5) As the sequence progresses, the ratio of consecutive Fibonacci numbers approximates to 1.618 (golden ratio). II. The Golden Ratio The Golden ratio is also known as the golden section, golden mean or divine proportion. It is an irrational number approximately equal to 1.618 and is named after the Greek sculptor Phidias. The symbol of the golden ratio is the Greek letter "phi" The Fibonacci spiral is also known as the Golden Spiral. Values of the Golden Ratio 1.61803398874989484820… or approximately 1.618 2sin(54°)