Course Outline: 10 Modules
Module 1: Properties of Triangles and Introductory Reasoning
Core Concept: Relationships between sides and angles in triangles; the framework of geometric proof.
Key Content:
Triangle side-angle relationships (Triangle Inequality Theorem, sum of interior angles = 180°).
Formulas for the sum of interior and exterior angles of polygons.
Classification of triangles (by sides and by angles).
Introduction to geometric proof: understanding propositions, conditions, conclusions, and writing formal proof steps.
Module 2: Congruent Triangles: The Cornerstone of Geometric Proof
Core Concept: Congruence as a rigorous definition of “perfect coincidence,” serving as a key tool for proving segment or angle equality.
Key Content:
Definition and properties of congruent triangles.
Triangle Congruence Theorems (SSS, SAS, ASA, AAS). Introduction to the HL theorem, with deeper exploration reserved for Grade 9.
Applying triangle congruence to prove geometric conclusions.
Analysis of classic fundamental geometric configurations (e.g., vertical angles, common sides, angle bisector models).
Module 3: Axial Symmetry and Isosceles Triangles
Core Concept: Axial symmetry as a geometric transformation and its integration with properties of special shapes.
Key Content:
Definition and properties of axially symmetric figures.
Theorems and converses regarding perpendicular bisectors and angle bisectors.
Properties and criteria of isosceles and equilateral triangles.
Coordinate axis symmetry (coordinate changes for points reflected across the x-axis and y-axis).
Module 4: Multiplication, Division, and Factoring of Polynomials
Core Concept: Structured operations on algebraic expressions and their inverse process—decomposition.
Key Content:
Laws of exponents (multiplication/division with same base, power of a power, power of a product).
Multiplication of polynomials (monomial × monomial/polynomial, polynomial × polynomial).
Special Product Formulas: Difference of Squares, Perfect Square Trinomials.
Factoring: Common Monomial Factor, using special product formulas.
Module 5: Rational Expressions and Their Operations
Core Concept: Extending the concepts and rules of arithmetic fractions to algebraic expressions.
Key Content:
Definition of rational expressions, basic properties (reduction to lowest terms, finding common denominators).
The four fundamental operations with rational expressions (addition, subtraction, multiplication, division, and powers).
Integer exponents (including zero and negative exponents).
Simple rational equations and their applications.
Module 6: Radical Expressions (Square Roots)
Core Concept: Extending the number system within the real numbers and learning their algebraic representation and manipulation.
Key Content:
Definition of square root expressions (the dual property of non-negativity).
Multiplication and division of square root expressions.
Simplest radical form and like radicals.
Addition and subtraction of square root expressions.
Rationalizing the denominator.
Module 7: The Pythagorean Theorem and Its Converse
Core Concept: The fundamental theorem relating the sides of a right triangle, serving as a prime example of the integration of algebra and geometry.
Key Content:
Exploration and proof of the Pythagorean Theorem.
Applying the Pythagorean Theorem for calculations (finding side lengths in right triangles).
The Converse of the Pythagorean Theorem and its application (determining if a triangle is right-angled).
Simple real-world applications of the theorem (e.g., distance problems).
Module 8: Quadrilaterals (Part 1) – Parallelograms
Core Concept: Progressing from triangles to polygons, investigating the property and classification system of special quadrilaterals.
Key Content:
Definition, properties (opposite sides/angles, diagonals), and criteria for parallelograms.
Distance between two parallel lines.
Definitions, properties, and criteria for rectangles, rhombuses, and squares; understanding their hierarchical relationships.
Triangle Midsegment Theorem.
Module 9: Introduction to Functions – Linear Functions
Core Concept: Formally establishing a mathematical model for correspondence between variables, representing a leap in algebraic thinking.
Key Content:
Variables, constants, and the concept of a function (domain, range).
Representations of functions (equation, table, graph).
Definitions, graphs, and properties of linear functions and direct variation (meaning of k and b, increasing/decreasing behavior).
Determining the equation of a line using the method of undetermined coefficients.
Relationships between a linear function and linear equations/inequalities in one variable.
Module 10: Data Analysis
Core Concept: Progressing from data description to data analysis, learning the basis for data-informed decision making.
Key Content:
Measures of Central Tendency: mean (including weighted mean), median, mode.
Measures of Dispersion: range, variance, standard deviation (concept and calculation).
Characteristics and application scenarios of different statistical measures (When to use mean vs. median? What does variance signify?).
Preliminary data analysis activities.
Course Features & Instructional Suggestions:
Spiral Progression: Geometric proof progresses from initial exposure (Module 1) to systematic training (Module 2), and then to complex application (Modules 3 & 8), with gradually increasing difficulty.
Algebraic Structure: Polynomials (Module 4) → Rational Expressions (Module 5) → Radicals (Module 6), demonstrating the expansion and unification of the algebraic expression system.
Apex of Integration: Module 7 (Pythagorean Theorem) and Module 9 (Linear Functions) represent the highest level of integrating algebraic and geometric thinking in this academic year.
Assessment Methods: A combination of formative (module quizzes, inquiry reports, proof-writing exercises) and summative (mid-term, final exams) assessment is recommended. Special emphasis should be placed on the logical process in geometric proofs and conceptual understanding of functions.
