Week 1: Construction of Angles, Triangles, and Geometrical Figures
Content: In this week, students will focus on constructing various angles (30°, 45°, 60°, 90°) and geometric shapes using a protractor and a ruler. Students will also explore the concept of equivalent angles, equidistance from two points, two lines, and fixed points. This week also involves working on four-sided plane figures, given certain conditions, and understanding the locus of given points.
- Example 1: Constructing a 30° angle using a protractor.
- Example 2: Drawing an equilateral triangle based on a given side.
- Example 3: Identifying equivalent angles in different figures.
- Example 4: Constructing a square with given conditions and understanding fixed points.
- Example 5: Locus of a point equidistant from two lines.
- Example 6: Using locus principles to draw a circle based on fixed points.
Week 2: Proofs of Basic Theorems in Euclidean Geometry
Content: This week will cover the angle sum of a triangle and how it equals 180°. We will also study the properties of triangles, focusing on the angle of a triangle being equal to the sum of two interior opposite angles.
- Example 1: Proving that the sum of the interior angles of a triangle is 180°.
- Example 2: Demonstrating the property of exterior angles of a triangle.
- Example 3: Using angles to prove properties of congruent triangles.
- Example 4: Understanding complementary angles and their role in triangle properties.
- Example 5: Proving the triangle inequality theorem.
- Example 6: Solving problems on angle sum property of triangles.
Week 3: Riders and Proofs of Properties
Content: This week explores advanced proofs of properties related to angles on parallel lines, angles in polygons, and congruent angles. We will also study parallelograms and the intercept theorem.
- Example 1: Proving the angles on parallel lines are equal (corresponding angles).
- Example 2: Proving the angle sum property in polygons.
- Example 3: Demonstrating the concept of congruent angles in geometric figures.
- Example 4: Understanding properties of parallelograms, including opposite angles.
- Example 5: Proving the intercept theorem in geometry.
- Example 6: Solving practical problems involving parallel lines and angles.
Week 4: Trigonometric Ratios – Sine, Cosine, and Tangent
Content: In this lesson, students will learn about trigonometric ratios and their application in right-angled triangles. We will explore the relationships between the sides of a right triangle using sine, cosine, and tangent.
- Example 1: Defining sine, cosine, and tangent in terms of a right-angled triangle.
- Example 2: Calculating the sine, cosine, and tangent of an angle in a given triangle.
- Example 3: Understanding how to use trigonometric ratios to solve for unknown sides.
- Example 4: Using a calculator to find sine, cosine, and tangent values.
- Example 5: Solving real-life problems using trigonometric ratios.
- Example 6: Working with trigonometric ratios in non-right-angled triangles.
Week 5: Derivation of Trigonometric Ratios for Specific Angles
Content: Students will focus on the derivation of trigonometric ratios for angles of 30°, 45°, and 60°. This involves understanding how these ratios are derived geometrically and using drawn angles.
- Example 1: Deriving the sine, cosine, and tangent of 30°.
- Example 2: Deriving the sine, cosine, and tangent of 45°.
- Example 3: Deriving the sine, cosine, and tangent of 60°.
- Example 4: Using the unit circle to explain trigonometric ratios.
- Example 5: Applying these derivations to solve for unknown angles and sides in triangles.
- Example 6: Practicing the use of trigonometric tables.
Week 6: Trigonometric Ratios – Angle of Elevation and Depression
Content: This week will cover the concepts of angles of elevation and depression and their application in solving problems involving trigonometric ratios.
- Example 1: Understanding the angle of elevation and depression in real-life scenarios (e.g., a building’s height).
- Example 2: Using trigonometric ratios to calculate the height of a structure.
- Example 3: Solving problems related to the angle of depression in navigation and engineering.
- Example 4: Using the sine and cosine laws to solve angle of elevation problems.
- Example 5: Practical application in finding angles of depression from a distance.
- Example 6: Using trigonometric ratios in surveying and architecture.
Week 7: Trigonometric Ratios in Relation to the Unit Circle
Content: Students will learn how sine and cosine values relate to the unit circle and explore the trigonometric ratios of various angles.
- Example 1: Understanding the unit circle and how sine and cosine values are derived.
- Example 2: Using the unit circle to find trigonometric ratios for common angles (e.g., 0°, 30°, 45°, 60°).
- Example 3: Solving for angles using the unit circle.
- Example 4: Exploring the relationship between the unit circle and trigonometric functions.
- Example 5: Graphing trigonometric functions based on the unit circle.
- Example 6: Applying the unit circle in real-world situations such as sound waves.
Week 8: Graph of Sine and Cosine Functions
Content: This week will focus on graphing sine and cosine functions using standard angles. Students will learn to graph the sine and cosine curves and understand their properties.
- Example 1: Plotting the sine graph using 15°, 20°, 30°, and 60° angles.
- Example 2: Identifying the amplitude, period, and phase shift in a sine graph.
- Example 3: Plotting the cosine graph for various angles.
- Example 4: Analyzing the behavior of sine and cosine graphs.
- Example 5: Understanding how changes in the function’s equation affect the graph.
- Example 6: Solving real-life problems using sine and cosine graphs.
Week 9: Length of an Arc of a Circle
Content: In this lesson, students will learn how to calculate the length of an arc of a circle and solve related problems.
- Example 1: Using the formula L=rθL = r \theta to calculate the length of an arc.
- Example 2: Finding the length of an arc given the radius and central angle.
- Example 3: Solving problems involving the circumference and arcs of a circle.
- Example 4: Calculating arc lengths in different units (e.g., radians and degrees).
- Example 5: Applying arc length concepts to real-world problems (e.g., circular tracks).
- Example 6: Working with sectors and finding the perimeter.
Week 10: Revision of First Term Work
Content: This week will focus on revising key concepts from the first term, including basic algebra, geometry, and trigonometric functions. Students will refresh their understanding and prepare for more advanced topics.
Week 11: Quadratic Equations
Content: This week will cover the solution of quadratic equations using factorization, completing the square method, and the general form leading to the formula method.
- Example 1: Solving a quadratic equation by factorization.
- Example 2: Completing the square to solve quadratic equations.
- Example 3: Applying the quadratic formula to find the roots of an equation.
- Example 4: Solving quadratic equations graphically.
- Example 5: Using real-life examples to solve quadratic equations.
- Example 6: Solving simultaneous quadratic equations.
Week 12: Sets and Venn Diagrams
Content: Students will be introduced to set theory, including universal sets, finite and infinite sets, and the use of Venn diagrams to solve problems involving unions, intersections, and complements.
- Example 1: Understanding the concept of a universal set.
- Example 2: Representing sets using Venn diagrams.
- Example 3: Solving problems involving unions and intersections of sets.
- Example 4: Applying the complement of sets in problem-solving.
- Example 5: Solving real-life problems using set notation.
- Example 6: Using Venn diagrams for analyzing survey data.
Week 13: Logic and Proofs
Content: This week introduces logical statements, including simple, compound, and conditional statements. Students will also explore negation, conjunctions, and disjunctions in logical reasoning.
- Example 1: Understanding simple and compound logical statements.
- Example 2: Exploring negation and its use in logical proofs.
- Example 3: Identifying conjunctions and disjunctions in logic.
- Example 4: Working with conditional and bi-conditional statements.
- Example 5: Using logical reasoning to solve mathematical problems.
- Example 6: Proving statements using logical deduction.
Week 14: Revision and Exams
Content: This week will focus on revising all topics covered during the term and preparing for the final exams.