CHAPTER 1 & 2a 

“CHAPTER 1”

Mathematics

• Is the study of the relationships among numbers, quantities, and shapes.
• Includes arithmetic, algebra, trigonometry, geometry, statistics, and calculus.
• Nurtures human characteristics like power of creativity, reasoning, critical thinking, spatial thinking and others.

NATURAL PATTERNS

1. SPIRAL - patterns in a circular curving line that goes around a central point while getting closer to or farther away from it.
2. SYMMETRY - balanced proportions
3. MOSAICS - something made up of different things that together form a pattern.
4. STRIPES - a line or long narrow section differing in color or texture from parts adjoining.
5. TESSELLATIONS - the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps.
6. RADIAL - arranged or having parts arranged in straight lines coming out from the center of a circle.

IN 19TH CENTURY

-Belgian physicist ^^Joseph Plateau^^ examined ^^soap films^^, leading him to formulate the concept of a minimal surface.

-German biologist and artist ^^Ernst Haeckel^^ ^^painted hundreds of marine organisms^^ to emphasize their symmetry.

-Scottish biologist ^^D’arcy Thompson^^ pioneered the ^^study of growth patterns in both plants and animals^^, showing that simple equations could explain spiral growth.

IN 20TH CENTURY

-British mathematician ^^Alan Turing^^ ^^predicted mechanism of morphogenesis^^ which give rise to patterns of spots and stripes.

-Hungarian Biologist ^^Aristid Lindenmayer^^ and French American Mathematician ^^Benoit Mandelbrot^^ showed how the ^^mathematics of fractals^^could create plant growth patterns.

^^-^^^^W. Gary Smith^^ ^^adopts eight patterns in his landscape work,^^ namely: scattered, fractured, mosaic, naturalistic drift, serpentine, spiral, radial and dendritic.

B. THE FIBONACCI SEQUENCE

Leonardo Pisano Bogollo – his nickname is Fibonacci which roughly means ^^“Son of Bonacci”^^

Fibonacci Day: ^^Nov. 23^^

The ratio of any two successive Fibonacci Numbers is very close to the Golden Ratio, referred to and represented as phi (𝝓) which is approximately equal to ^^1.618034 …^^

Golden Spiral – is a logarithmic spiral whose growth factor is phi (𝝓), the golden ratio.

C. PATTERNS AND REGULARITIES IN THE WORLD AS ORGANIZED BY MATHEMATICS

Spectacular patterns

• Example: rainbow, butterfly

Symmetrical patterns

• Example: shell, spider web, circle

REGULARITIES IN THE WORLD AS ORGANIZED BY MATHEMATICS

1. The Motion of Pendulum

   The mathematics of pendulum is quite complicated but harmonic. Its period or the time it takes to swing back to its original position is related to its length, but the relationship is ^^not linear^^.

2. Reflection in A Plane Mirror

   An image formed by an object in a plane mirror can be explained mathematically by the law of reflection.

3. The Motion of a Falling Object

   A free-falling object is an object that is falling under the sole influence of gravity. Any object that is moving and being acted upon the force of gravity is said to be in a state of free fall. Its motion obeys the equations of uniformly accelerated vertical motion.

4. The Action-reaction Pair of Forces

   In every interaction, there is a pair of forces acting on the two interacting objects. The amount of force on the first object equals the size of the force on the second object. The direction of the force on the first object is opposite to the direction of the force on the second object. Forces always come in pairs – equal and opposite action-reaction pairs.

D. PHENOMENA IN THE WORLD AS PREDICTED BY MATHEMATICS

Mathematics is an extraordinary exercise of the human in abstracting the result of observation to find similarities and differences between phenomena.

E. APPLICATION OF MATHEMATICS IN THE WORLD

Mathematics has everyday applications. It is a universal language different in places in different times, in different settings and different circumstances. The physical world seems to consist of countable things and any infinity encountered is a result of extending a counting process.

“CHAPTER 2”

Language - the system of words, signs and symbols which people use to express ideas, thoughts and feelings.

Mathematical Language - the system used to communicate mathematical ideas.

FOUR MAIN ACTIONS ATTRIBUTED TO PROBLEM SOLVING AND REASONING

1. Modelling and Formulating - creating appropriate representations and relationships to mathematize the original problem.
2. Transforming and Manipulating - changing the mathematical form in which a problem is originally expressed to equivalent form that represents solution.
3. Inferring - applying derived results to the original problem situation and interpreting and generalizing the result.
4. Communicating - reporting what has been learned about a problem to a specified audience.

Characteristic of Mathematical Language

• Mathematical language is ^^non-temporal^^
• Mathematical language is ^^devoid of emotional content^^
• Mathematical language is ^^precise^^

MATHEMATICAL EXPRESSIONS AND SENTENCES

Operational Terms and Symbols

1. Addition
2. Subtraction
3. Multiplication
4. Division

Translating algebraic expression into verbal phrases/sentences:

By, of, to - susunod

From, than - may sinundan

DEFINITION OF TERMS

1. Multivariate Mathematical Expressions - have more than one variable

   Example: 5xy + 9x – 12

2. Constant- a symbol that assumes one specific value

   Example: 3, 5, 1000, , ½, 0.4

3. Variable - a symbol that assumes many values

   Example: m, n, x, y, z,

4. Term - An expression preceded by plus or minus sign and involves two or more factors.

   Example: 5xy + 9x – 12 --- 3 terms

                 x3 – 3x2 + x + 27 -- 4 terms 

                 3x2 + x = 4 -- 3 terms

5. Numerical Coefficient - Constant factor of the term

   Example: 5xy + 9x – 12 (1st term: 5 2nd term: 9 3rd term: -12)

6. Literal Coefficient - A factor representing variable of a term

   Example: 5xy + 9x – 12 (1st term: xy 2nd term: x 3rd term: none)

Mathematical Sentence:

• Combination of two ^^mathematical expression^^ using a ^^comparison operator^^.
• The comparison operator includes equal, not equal, greater than, greater than or equal to, less than and less than or equal to.
• Relation symbols - the signs which convey equality or inequality
• Equation - A mathematical expression containing the equal sign
• Inequality - A mathematical expression containing the inequality sign

OPEN SENTENCE VS CLOSE SENTENCE

• Open Sentence - It uses variables, meaning that is ^^not known whether or not^^ the mathematical sentence is ^^true or false^^.
  • Example:
  • x + 2 = 5
  • 8ab –c = 1
  • x + y ≠ 5
  • 3 (m + n) = 100
  • 18 w > 16.5
  • 2xy < 3x
• Close Sentence - A mathematical sentence that is known to be either true or false.
  • True Closed Sentence – mathematical sentence that is known to be true.
  • Example:
    • 1 + 1 = 2
    • 3 ≠ 5
    • x + 2x = 3x
    • x-1 = 1/𝑥
    • log 100 = 2
    • 2 is an even number
  • False Closed Sentence – mathematical sentence that is known to be false.
  • Example:
    • 3 = 5
    • 2 + 5 = 5 – 2
    • The square root of 100 is 5
    • 3-2 = 9
    • 5 > 10
    • 0 is an odd number

TWO THINGS TO CONSIDER UNDERSTANDING THE MEANING OF MATH SYMBOL

• [ ] CONTEXT - It refers to the ^^particular topics^^ being studied.

  Example: 180^0

• [ ] CONVENTION - A technique used by mathematicians, engineers and scientists in which a ^^particular symbol^^ has a particular meaning.

  Ex: alpha (α), sigma (), phi ()