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Pandemic Control: Statistics and Module 1: Nature of probability assist in controlling Mathematics disease spread by predicting transmission and recovery rates. Lesson 1: Nature of Lesson 2: Patterns...

Pandemic Control: Statistics and Module 1: Nature of probability assist in controlling Mathematics disease spread by predicting transmission and recovery rates. Lesson 1: Nature of Lesson 2: Patterns in Nature Mathematics Patterns in Nature: Definition of Mathematics: - Patterns are repeated - Mathematics is about discovering arrangements of lines or shapes relationships and expressing them (symmetries, spirals, tessellations, symbolically (through numbers, etc.) and can often be modeled letters, diagrams, graphs). mathematically. - It deals with the logic of shape, Fibonacci Sequence and quantity, and arrangement. Golden Ratio: - Not limited to arithmetic; it's Fibonacci Sequence: Discovered about real-world applications and by Leonardo Fibonacci, it's a understanding the series of numbers where each interconnectedness of different number is the sum of the two fields (science, technology, etc.). preceding numbers. It appears in nature, such as in rabbit Mathematics in Nature: population growth or plant Predicting Natural Phenomena: patterns. Mathematics helps predict natural Fibonacci Spiral: Created from behaviors like the sun's movement Fibonacci squares, resulting in a and weather forecasts. spiral known as the Golden Population Control: Algebra Rectangle, which often appears in helps predict population growth, nature. aiding in setting policies to avoid Golden Ratio (φ = 1.618): The overpopulation. ratio between successive Fibonacci numbers approximates the Golden Ratio, which is seen in Mathematical Symbols and architecture, art, and nature as a Definitions: measure of beauty and harmony. Symbols like π, +, and = are part of the language. Module 2: Mathematics Language and Symbols Good definitions in math are clear and specific (e.g., "A rectangle is a quadrilateral with four right Lesson 1: The Language of angles"). Mathematics Translation Between English and Math: Mathematics as a Language: You can translate English Mathematics is often sentences into mathematical misunderstood as difficult, but expressions, e.g., "The sum of two when approached as a language, consecutive numbers is 31" it becomes easier. becomes x+(x+1)=31. Like any language, mathematics Lesson 2: Using Mathematics has its own vocabulary, grammar, as Language to Solve Problems and syntax. Characteristics of Mathematical Sets: Language: A collection of objects (e.g., {a, b, Precise: Exact measurements c}). and meanings (e.g., π = Different types: universal set, finite 3.1415…). set, empty set, etc. Concise: Brief and clear Operations on sets include union, expression (e.g., 1 + 2 = 3). intersection, and complement. Powerful: Able to express complex ideas succinctly. Functions: Universal quantifiers: "For all" or "For every" (denoted by ∀). A rule that assigns each element of one set (domain) to an element Existential quantifiers: "There of another set (range). exists" or "At least one" (denoted by ∃). Represented as f(x)→yf(x) \rightarrow yf(x)→y. Variables: Relations: Symbols that represent unknown values or elements in a set (e.g., A relationship between elements x, y). in sets. A relation is reflexive, symmetric, or transitive. Formality in Mathematics: Binary Operations: Statements in mathematics should be expressed formally using An operation on two elements symbols to avoid ambiguity. from a set (e.g., addition, multiplication). Module 3: Problem Solving and Satisfies closure property, Reasoning meaning the result is still within the set. Problem-Solving Heuristics Lesson 3: Basic Concepts in Logic Definition of a Problem: A situation requiring a solution, encountered in both theoretical Propositions: and real-life contexts. A declarative statement that is Polya’s Four-Step Plan (1945): either true or false. Understand the Problem: Identify Compound propositions use given information, ask questions. logical connectives like "and," "or," and "if-then." Devise a Plan: Choose a strategy or method. Quantifiers: Carry out the Plan: Implement By Pattern: Count progressively: the chosen strategy. 9 + 8 + 7 +... + 1 = 45 handshakes. Look Back: Verify if the solution is correct. By Formula: Use a combination formula for nCr. 10 Problem-Solving Strategies: ➔ Logical reasoning Inductive Reasoning ➔ Pattern recognition ➔ Working backwards Definition: Drawing general ➔ Adopting a different point of conclusions from specific view examples or observations. ➔ Considering extreme cases ➔ Solving simpler analogous Examples: problems Number Pattern: Predicting ➔ Organizing data numbers in a sequence (e.g., 2, 4, ➔ Making visual 8, 16…). representations ➔ Accounting for all Conjecture: The sum of two odd possibilities numbers is even. ➔ Intelligent guessing and testing Deductive Reasoning Definition: Using general rules to Example Problem-Solving: reach specific conclusions. Handshakes Examples: Problem: With 10 people, how many handshakes can be made if Equation Solving: Solving for x in everyone shakes hands exactly equations like 3x+15=21 once? Logic Puzzle: Deductively solving Solution Methods: age and ice cream preferences based on clues. Module 4: Mathematics as Objective: Findings are presented Statistical Tool in a way that allows others to observe the same results. Role of Science in Decision-Making: Repeatability: Scientific claims must be replicable to verify their - Many people base decisions and accuracy and uphold the integrity judgments on scientific studies. of discoveries. - Studies that do not follow the Attitudes for Scientific Inquiry: scientific method may be viewed as unreliable. - Open-mindedness: Willing to accept new ideas and findings. - Sole reliance on scientific studies for decision-making can be risky. - Skepticism: Questioning and critically evaluating claims. Definition of Science: - Fallibility: Acknowledging that - According to Meimban and mistakes can be made. Terrago (2021), science is one of many ways to acquire knowledge - Caution: Careful interpretation about the natural world. and handling of scientific data. Scientific Method: The process - Uncertainty: Recognizing that not used in science to discover hidden all answers are definitive, and knowledge about nature. further inquiry is often needed. The scientific method is governed by established rules set by the scientific community. Characteristics of Science: Empirical: Knowledge is gained through observation, perception, and experimentation. Module 4: Mathematics as Dependent variable: The variable Statistical Tool being studied or measured (e.g., satisfaction rating). Definition of Statistics Independent variable: The variable being manipulated or Collection of data: Gathering controlled (e.g., gender). data through interviews, questionnaires, experiments, and Variables can also be active observation. (manipulated) or attribute (cannot be manipulated). Presentation of data: Organizing data using tables, graphs, or Descriptive Statistics charts. Measures of Central Tendency: Analysis of data: Describing properties or correlations between Mean: The average of values. variables. Median: The middle value when Interpretation of data: Drawing data is ordered. conclusions or making predictions Mode: The most frequent value. based on data analysis. Measures of Dispersion: Variable and Data Range: Difference between Variable: A characteristic that can maximum and minimum values. vary (e.g., age, gender). Variance: Average of squared Data: Measurements of variables. deviations from the mean. Variables can be: Standard Deviation: Square root of variance. - Quantitative (numerical): Discrete (counted) or Coefficient of Variation: continuous (measured). Standard deviation relative to the - Qualitative (categorical). mean, used for comparison. Types of Variables Exploratory Data Analysis Nonprobability Sampling: Not all (EDA) members have equal chances of selection (e.g., convenience Methods to analyze patterns and sampling). relationships among variables: Normal Distribution Bar Chart: Displays categorical variable distribution. Properties: Bell-shaped, symmetric, mean = median = Pie Chart: Shows proportions. mode, extends infinitely. Line Chart: Tracks variables over Empirical Rule: 68% of values time. within 1 standard deviation, 95% Histogram: Graphs continuous within 2, and 99.7% within 3. variable distribution. Hypothesis Testing Box Plot: Shows the five-number Null Hypothesis (Ho): States no summary and identifies outliers. effect or difference (e.g., μ1=μ2). Inferential Statistics Alternative Hypothesis (Ha): Population: The entire States the effect or difference group of interest. exists (e.g., μ1≠μ2​). Sample: A subset of the Types of Tests: population. - One-tailed test: Tests a Census: Data from the specific direction. entire population. - Two-tailed test: Tests any Statistic: Measure from a difference in either direction. sample. Parameter: Measure from a Types of Errors population. - Type I Error (α): Rejecting a Sampling true null hypothesis. - Type II Error (β): Accepting Probability Sampling: Random a false null hypothesis. selection (e.g., simple random, systematic). Tests One-Sample t-Test: Tests if a sample mean differs from a known population mean. Independent t-Test: Compares means between two unrelated groups. Dependent t-Test: Compares means of two related groups. ANOVA (Analysis of Variance): Compares means across three or more groups.