MATH PRELIMS

Fractals

, are infinitely self-similar, iterated mathematical constructs having fractal dimension.

Spirals,

is a curve which emanates from a point, moving farther away as it revolves around the point.

Stripes,

These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce.

Cracks,

are linear openings that form in materials to relieve stress.

Cracks,

define whether the material is elastic or not

7 Patterns in nature,

1. Symmetry 2. Fractals 3. Spirals 4. Tessellations 5. Bubbles/ Foams 6. Stripes 7. Cracks

Sequence,

Is an ordered list of numbers, called terms that may have repeated values. The arrangement of these terms is set by a definite rule.

The Fibonacci Sequence,

is the series of numbers. The next number is found by adding up the two numbers before it.

Textual or Narrative Presentation

combines text and figures in a statistical report. It aims to direct the reader's attention to some data which need particular emphasis and probably comparison.

Patterns in nature

, are visible regular forms found in the natural world.

Patterns,

are regular, repeated form or design

Aristotle,

who started the concept of nature

Patterns in nature

are visible regular forms found in the natural world.

Symmetry,

is when different sides of something are alike.

Bilateral or Mirror Symmetry

is symmetry with respect to reflection.

Rotational symmetry or radial symmetry,

is also found at different scales among non-living things including the crown-shaped splash pattern formed when a drop falls into a pond, and both the spheroidal shape and rings of a planet like Saturn.

Fivefold symmetry,

is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. The reason for the fivefold (penta- radiate) symmetry of the echinoderms is puzzling.

Six-fold symmetry,

each flake's structure forming a record of the varyingconditions during its crystallization, with nearly the same pattern of growth on each of its six arms.

Fractals

, Infinite iteration is not possible in nature so all patterns are only approximate.

Tessellations

, are patterns formed by repeating tiles all over a flat surface.

Bubbles/ Foams,

A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed.

Leonardo Fibonacci,

introduced the Fibonacci sequence

Fibonacci formula,

Fn=Fn-1+Fn=2

Arithmetic Sequence,

following a pattern of adding a fixed amount from one term to the next term

Arithmetic formula,

An=A1+(n-1)d

Geometric formula,

multiplying the proceeding term by a constant number

Geometric formula

An=A1r^n-1

How to get the ratio,

divide the 1st term with the 2nd term

Terms,

Is an ordered list of numbers, called

Precise,

(able to make very fine distinctions)

Concise,

(able to say things briefly)

Powerful,

(able to express complex thoughts with relative ease).

Expression,

correct arrangement of mathematical symbols used to represent a mathematical object of interest.

Expression,

does not state a complete thought

Sentence,

it is a correct arrangement of mathematical symbols that states a complete thought.

Sentence,

have verb "="

Sentence,

can be (always) true, (always) false, or sometimes true/sometimes false.

Set,

Is a collection of distinct well-defined objects called elements.

Set,

Are denoted by upper case letter and a curly braces

Relation,

has domain of the relation, and range

Function

, is a relation for which each value of domain is not associated with others

Binary Operation

, a set that involves two elements of the set to produce another element of the set.

Propositional logic,

is a mathematical system for reasoning about propositions and how they relate to one another.

Propositional Variables

, A variable that represents propositions

The logical connectives,

are defined by truth tables

logical negation,

~ p is true if and only if p is false.

logical conjunction

, p^q is true if both p and q are true.

logical disjunction,

pvq is true if at least one of p or q are true (inclusive OR)

Logical IMPLIED,

(IF..., THEN...): p→q

Logical EQUIVALENT,

p and q are logically equivalent if they have identical truth values under all possible situations.

Logical EQUIVALENT,

p↔︎q or p=q

Logical EQUIVALENT,

if and only if

Quantifiers,

quantifiers include "all", "none", "some", and "not all".

Inductive Reasoning,

is the process of getting a general conclusion by observing the specific examples or set.

Deductive Reasoning,

is the process of reaching a conclusion by general assumption, procedures or principle.

Proof,

The old, colloquial meaning of "prove" is: Test, try out, determine the true state of affairs.

Mathematical intuition,

is coming across a problem, glancing at it, and using your logical instinct to pull out an answer without asking further questions.

Certainty,

is something that is accurate and absolute.

George Polya,

was a Hungarian who immigrated to the United States in 1940. His major contribution is for his work in problem solving.

four-step process for problem solving

, 1. Understand the problem. 2. Devise a plan. (Translate) 3. Carry out the problem. (Solve) 4. Look back. (Check and interpret)

Second step in Polya,

Devise a plan. (Translate)

First step in Polya

Understand the problem

Third step in Polya,

Carry out the problem. (Solve)

Fourth step in Polya,

Look back. (Check and interpret)

Data,

is a collection of facts, such as numbers, words, measurements, observations or just descriptions of thing.

Qualitative Data,

Also called attribute data

Qualitative Data

the items under consideration.

attribute data,

are not easy to analyze

Quantitative Data,

Also called as numerical data

Quantitative Data,

result from a process that quantifies which includes counts of (how many) or measurements (length, weight, and so on) using numerical techniques.

Discrete numerical data

, data that represents counts

Continuous numerical data,

data that represents measurement

Nominal Level,

Is characterized by data that consist of names, labels, or categories only.

Nominal Level,

The data cannot be arranged in an ordering scheme.

Ordinal Level,.

may be arranged in some order but differences data values either cannot be determined or are meaningless

Interval Level,

It has no inherent (natural) zero starting point where the none of the quality is present.

Ratio Level,

include the inherent zero starting point where zero indicates that none of the quantity is present.

Random Sampling,

each member of the population has the same chance of being chosen, with no one member being favored over any other.

Systematic Sampling,

This is typically used when a list of the population is readily available.

Systematic Sampling,

samples are determined by choosing every nth item on the list until the desired number of samples are obtained.

Stratified Random Sampling

, the population is divided into sub-groups or strata and then a small sample is taken from each stratum. Each subsample is then drawn in equal numbers from each stratum.

Stratified Random Sampling

population/strata - stratum - subsample are equal

Cluster Method

, population is divided into strata, except that now certain strata are selected randomly, and the sample is chosen from only those strata.

Sampling,

The art of gathering data is called

Interviews and Focus Groups,

are often used to gather detailed and qualitative descriptions.

Interviews and Focus Groups,

is conducted on one-on-one basis while focus groups are conducted in small groups.

Surveys and questionnaires,

These are indirect and commonly used methods f gathering data.

Observation,

are generally unobtrusive method for gathering information about a certain concern or interest.

Registration Method,

This method of gathering data or information is enforced by certain laws.

Experiment Method

This method is used when the objective is to determine the cause-and-effect relationship of certain phenomena under controlled conditions.

Data is categorized into two,

qualitative and quantitative.

Data is classified into four level of measurement,

nominal, ordinal, interval and ratio.

Gathering data is done through sampling,

Random sampling, Systematic sampling, Stratified random sampling, and cluster method.

Collecting data is done through

, Interviews, Surveys and questionnaires, Observations, Registration method and Experiment method.

Tabular Presentation

because it provides numerical facts in a more conscience and systematics manner.

Graphical Presentation

When data are shown in terms of visually interpreted illustrations, the reader easily sees essential facts and relationships.

Line Graph

used to represent changes in data over a period of time.

Bar Graph

are consist of series of rectangular bars where the length of the bar represents the magnitude to be demonstrated.

simple bar graph

is one in which each bars represents one unit only.

multiple bar graph

is one where different units are used on the same diagram in order to emphasize comparisons.