The-Nature-of-Mathematics-for-stud.pdf

Patterns and Numbers in Nature and the World Prepared by: Marylyn U. Inocencio 1.1 Patterns and Numbers in Nature and the World 1.1.1 Patterns in Nature Anything that REPEATS with recurring characteristics or a SERIES of a regular or consistent arrangement according...

Patterns and Numbers in Nature and the World Prepared by: Marylyn U. Inocencio 1.1 Patterns and Numbers in Nature and the World 1.1.1 Patterns in Nature Anything that REPEATS with recurring characteristics or a SERIES of a regular or consistent arrangement according to a specific rule or SEQUENCE is considered a PATTERN. Patterns in nature are visible regularities of form found in the natural world. These patterns recur in different contexts and can sometimes be modeled mathematically.  Types of Natural Patterns 1. Symmetry. There is symmetry if an imaginary line is drawn across an object, the resulting parts are mirrors of each other, like the following figures. 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. Aloe polyphylla Tendrils Navy red flower Millipede Chameleon’s tail Ram’s horns 3. Meander 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. 4.Cracks are linear openings that form in materials to relieve stress. The pattern of cracks indicates whether the material is elastic or not. Stripes is a line or band that differs in color or tone from an adjacent area. This may be seen in various living things, especially animals. emperor angelfish zebra heliconius charithonia tiger 1.1.2 Other Types of Patterns Aside from the natural patterns, you can also come across four (4) main types of patterns in which you are also familiar with or if not, now is the time to know them. 1.1.2.1 Logical Patterns Logic reasoning and pattern observing are the first two math standards, which are the most important measurement of IQ and the core component of many careers. Logical patterns are usually the first to be observed since making categories or classification comes before numeration. For children, logical patterns include studying shapes and colors. For older ones, logic tests can be seen on aptitude tests wherein takers are shown a sequence of pictures and asked to select which figure comes next among several choices. To identify logic patterns, you have to look out four (4) things, namely: (1) rotating shapes (2)increase and decrease in numbers of shapes or patterns (3)alternating patterns, colors, and shapes (4)mirror images or reflections In solving problems in logical reasoning, you have to look for patterns or rules and identify which object does follow those patterns or rules.  Here is a list of examples of Logical Patterns and how they are to be identified. 1.) Identify the odd one out. 2.) Identify the missing square. ► In this item, you have to look for the following: 1. Relative Positional Rule: This is how the black square is positioned inside each box. 2. Movement Rule: This pertains to how the square moves in each box, in the clockwise direction. 3. The arrows in the first and third columns are reflections of one another. 3.) Which figure completes the grid? 1.Notice the following from the given: ⮚ The first and the fourth columns and the first and second rows contain one shape only. ►The diamond and crescent moon shapes are dark; the circles and the squares are white. 2. Divide the grid by columns, by rows, or by four groups of four squares and look for relationships. Therefore, to complete the grid, the answer is A. Which Figure is the Odd One Out. 1.1.2.2 Geometric Patterns Geometric patterns are a collection of shapes, repeating, or altered to create a cohesive design. These patterns are observed regularly. They appear in paintings, drawings, tapestries, wallpapers, tiling, and carpets. ►Tessellations Tessellations are repeating patterns of polygons that cover a plane with no gaps or overlaps. Some examples showing tessellations are the honeycombs made by honey bees and scales of fish. It may be a regular tessellation (composed of regular polygons symmetrically tiling the plane), semi- regular tessellation (made of two or more regular polygons), or semi- regular tessellation (or polymorph). ►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). They are some of the most beautiful and most bizarre objects in all of mathematics. This means as one examines finer and finer details of the object, the magnified area is similar to the original but is not identical to it. Some famous fractals are the Sierpinski Triangle, Pascal's Triangle, Koch Snowflake, and Fractal Tree. ►Sierpinski Triangle The Sierpinski triangle begins as an equilateral triangle. The recursive procedure is to replace the triangle with three smaller congruent equilateral triangles such that each smaller triangle shares a vertex with the large triangle. To draw the Sierpinski triangle easier, start with an equilateral triangle. Then mark the midpoint of each side and connect these points. Repeat the procedures to create the triangle. ► Pascal's Triangle The Pascal's triangle contains the numerical coefficients of binomial expansions. The triangle below shows the coefficients of up to. In Pascal's triangle, the Sierpinski triangle can also be drawn by connecting or shading the odd numbers. Fractal Tree In making a fractal tree, start at some point and move a certain distance in a particular direction. At that point, make a branch (two branches in this example). Turn some angle to the right (and left) and then repeat the previous step using a shorter distance. Then, do the same in making the succeeding branches. ►Koch Snowflake In drawing a Koch Snowflake, one needs to start by drawing an equilateral triangle. Then, divide each side into three equal parts. After that, draw an equilateral triangle on each middle part.  Then, divide each outer side into thirds and again, draw an equilateral triangle, but draw on each middle part. Repeat until you're satisfied with the number of iterations, like the example below. Word Pattern Number Patterns Definition The Fibonacci Sequence Who is Fibonacci? Here is a statement of Fibonacci’s rabbit problem At the beginning of a month, you are given a pair of new rabbits. After a month the rabbits have produced no offstring ; however, every month thereafter, the pair of rabbits produces another pair of rabbits. The offstring in exactly the same manner. If none of the rabbits dies, how many rabbits will there at the start of each succeeding month? By definition ►The first two numbers of Fibonacci sequence are 1,1, and each subsequent number is the sum of the previous two , the recurrence ►Relation: F1 =1 and F2=1 The Golden Ratio The Indispensability of Mathematics