CHAPTER 1 & 2a
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.
SPIRAL - patterns in a circular curving line that goes around a central point while getting closer to or farther away from it.
SYMMETRY - balanced proportions
MOSAICS - something made up of different things that together form a pattern.
STRIPES - a line or long narrow section differing in color or texture from parts adjoining.
TESSELLATIONS - the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps.
RADIAL - arranged or having parts arranged in straight lines coming out from the center of a circle.
-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.
-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
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.
Reflection in A Plane Mirror
An image formed by an object in a plane mirror can be explained mathematically by the law of reflection.
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.
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.
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.
Modelling and Formulating - creating appropriate representations and relationships to mathematize the original problem.
Transforming and Manipulating - changing the mathematical form in which a problem is originally expressed to equivalent form that represents solution.
Inferring - applying derived results to the original problem situation and interpreting and generalizing the result.
Communicating - reporting what has been learned about a problem to a specified audience.
Mathematical language is non-temporal
Mathematical language is devoid of emotional content
Mathematical language is precise
Operational Terms and Symbols
Addition
Subtraction
Multiplication
Division
Translating algebraic expression into verbal phrases/sentences:
By, of, to - susunod
From, than - may sinundan
Multivariate Mathematical Expressions - have more than one variable
Example: 5xy + 9x – 12
Constant- a symbol that assumes one specific value
Example: 3, 5, 1000, , ½, 0.4
Variable - a symbol that assumes many values
Example: m, n, x, y, z,
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 termsNumerical Coefficient - Constant factor of the term
Example: 5xy + 9x – 12 (1st term: 5 2nd term: 9 3rd term: -12)
Literal Coefficient - A factor representing variable of a term
Example: 5xy + 9x – 12 (1st term: xy 2nd term: x 3rd term: none)
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 - 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
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 ()
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.
SPIRAL - patterns in a circular curving line that goes around a central point while getting closer to or farther away from it.
SYMMETRY - balanced proportions
MOSAICS - something made up of different things that together form a pattern.
STRIPES - a line or long narrow section differing in color or texture from parts adjoining.
TESSELLATIONS - the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps.
RADIAL - arranged or having parts arranged in straight lines coming out from the center of a circle.
-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.
-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
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.
Reflection in A Plane Mirror
An image formed by an object in a plane mirror can be explained mathematically by the law of reflection.
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.
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.
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.
Modelling and Formulating - creating appropriate representations and relationships to mathematize the original problem.
Transforming and Manipulating - changing the mathematical form in which a problem is originally expressed to equivalent form that represents solution.
Inferring - applying derived results to the original problem situation and interpreting and generalizing the result.
Communicating - reporting what has been learned about a problem to a specified audience.
Mathematical language is non-temporal
Mathematical language is devoid of emotional content
Mathematical language is precise
Operational Terms and Symbols
Addition
Subtraction
Multiplication
Division
Translating algebraic expression into verbal phrases/sentences:
By, of, to - susunod
From, than - may sinundan
Multivariate Mathematical Expressions - have more than one variable
Example: 5xy + 9x – 12
Constant- a symbol that assumes one specific value
Example: 3, 5, 1000, , ½, 0.4
Variable - a symbol that assumes many values
Example: m, n, x, y, z,
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 termsNumerical Coefficient - Constant factor of the term
Example: 5xy + 9x – 12 (1st term: 5 2nd term: 9 3rd term: -12)
Literal Coefficient - A factor representing variable of a term
Example: 5xy + 9x – 12 (1st term: xy 2nd term: x 3rd term: none)
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 - 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
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 ()