Class: Senior Secondary School 1 (SS1, SS 1, SSS1, SSS 1)
Term: First Term
Week: 7
Age: 15 years
Duration: 45 minutes
Subject: Further Maths
Curriculum Theme: Further Mathematics
Previous Lesson: Surds.
Topic: COORDINATE GEOMETRY (The straight line III)
Subject Matter: transforming non linear relationships into linear form, area of triangles with given coordinates, area of quadrilaterals with given coordinates

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

Define transformation of non-linear relationships into linear form.
State the methods for finding the area of triangles and quadrilaterals using coordinates.
Calculate the area of a triangle given the coordinates of its vertices.
Calculate the area of a quadrilateral given the coordinates of its vertices.

Affective Domain:

Appreciate the importance of transforming non-linear relationships for easier analysis.
Show interest in solving problems involving coordinate geometry.

Psychomotor Domain:

Graphically represent transformed linear relationships.
Accurately apply formulas to find areas of geometric shapes on a coordinate plane.

Participate actively in class discussions and problem-solving activities.
Collaborate with peers to solve coordinate geometry problems.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Further Mathematics.
State Unified Scheme of Work for Further Mathematics SSS 1.
New Further Mathematics Project 1 for Senior Secondary Schools.

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing examples of transformed non-linear relations.
Coordinate geometry diagrams on the whiteboard.
Graph paper, rulers, and pencils.

Rationale for the Lesson

This lesson helps pupils understand how to simplify complex non-linear relationships into linear forms, which is useful for data analysis. It also enables them to calculate areas of basic geometric shapes precisely using coordinate points, strengthening their problem-solving skills in geometry.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of plotting points on a coordinate plane, calculating the gradient of a straight line, and basic algebraic manipulation.

Lesson Content/Board Summary

COORDINATE GEOMETRY (The straight line III)

1. Transforming Non-Linear Relationships into Linear Form

Some relationships between two variables, x and y, are not linear but can be transformed into a linear form Y = mX + c by making suitable substitutions. This allows for easier analysis and graphing.

Common non-linear relationships and their transformations include:

If y = axⁿ, taking logarithms gives log y = log a + n log x. Let Y = log y, X = log x, C = log a, then Y = nX + C.
If y = ab^x, taking logarithms gives log y = log a + x log b. Let Y = log y, C = log a, M = log b, then Y = Mx + C.
If y = a/x + b, let X = 1/x, then y = aX + b. Here Y=y, m=a, c=b.
If y = a + b√x, let X = √x, then y = bX + a. Here Y=y, m=b, c=a.

The steps to transform a non-linear relationship into a linear form are:

Identify the non-linear equation.
Apply suitable algebraic or logarithmic manipulations to achieve the form Y = mX + c.
Clearly define the new variables Y, X, and the constants m and c in terms of the original variables and constants.

2. Area of Triangles with Given Coordinates

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), and (x₃, y₃) can be calculated using the determinant method (also known as the shoelace formula).

The formula is given by:

Area = ½ | (x₁(y₂ – y₃) + x₂(y₃ – y₁) + x₃(y₁ – y₂)) |

Alternatively, using the shoelace method:

Area = ½ | (x₁y₂ + x₂y₃ + x₃y₁) – (y₁x₂ + y₂x₃ + y₃x₁) |

Note: The absolute value ensures the area is always positive.

3. Area of Quadrilaterals with Given Coordinates

The area of a quadrilateral with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃), and (x₄, y₄) can also be calculated using an extended shoelace formula.

The formula is given by:

Area = ½ | (x₁y₂ + x₂y₃ + x₃y₄ + x₄y₁) – (y₁x₂ + y₂x₃ + y₃x₄ + y₄x₁) |

Another method is to divide the quadrilateral into two triangles using a diagonal and sum the areas of the two triangles.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the concept of straight lines and their equations. The teacher then introduces the idea that not all relationships are linear but can sometimes be made linear for easier study. The teacher also reminds pupils about basic area calculations.
Pupils’ Activity: Pupils recall previous knowledge and listen attentively to the introduction.
Learning Point: Pupils connect previous knowledge to the new topic and understand the scope of the lesson.

Step 2: Transformation of Non-Linear Relationships

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the concept of transforming non-linear relationships into linear form (Y=mX+c). The teacher demonstrates with examples like y=axⁿ and y=ab^x, showing the appropriate substitutions and logarithmic manipulations required. The teacher writes the key steps on the board.
Pupils’ Activity: Pupils observe the examples, take notes, and ask questions for clarification.
Learning Point: Pupils understand how to convert non-linear equations into a linear form.

Step 3: Area of Triangles using Coordinates

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces the formula for calculating the area of a triangle given the coordinates of its vertices. The teacher demonstrates the application of the formula with a worked example on the board, explaining each step clearly.
Pupils’ Activity: Pupils copy the formula and the example, ensuring they understand the substitution of coordinates.
Learning Point: Pupils learn the formula and method for calculating the area of a triangle using coordinates.

Step 4: Area of Quadrilaterals using Coordinates

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher extends the concept to quadrilaterals, presenting the shoelace formula for quadrilaterals or demonstrating how to divide a quadrilateral into two triangles. The teacher works through a simple example.
Pupils’ Activity: Pupils observe the method and formula, asking questions where necessary.
Learning Point: Pupils learn how to calculate the area of a quadrilateral using coordinates.

Step 5: Practice Exercises

Time: 8 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher provides a few practice questions on the board for pupils to attempt, involving both transformation and area calculations. The teacher monitors pupils’ work and provides assistance.
Pupils’ Activity: Pupils work on the given exercises individually or in pairs, applying the learned formulas and techniques.
Learning Point: Pupils consolidate their understanding through practice.

Step 6: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Explain why some non-linear relationships are transformed into linear forms.
State the general linear form equation used for transformation.
Given vertices A(1,2), B(4,2), C(3,5), calculate the area of triangle ABC.
Mention two ways to find the area of a quadrilateral given its vertices.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 5 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the main points of the lesson, reiterating the importance of transforming non-linear equations and the methods for calculating areas using coordinates. The teacher assigns homework involving more complex problems.
Pupils’ Activity: Pupils listen to the summary and copy down the homework.
Learning Point: Pupils review the key concepts of the lesson.

Lesson Keywords

Linear Transformation – The process of converting a non-linear relationship into a linear form.
Non-linear Relationship – A relationship between variables that does not produce a straight line when graphed.
Coordinates – A set of values that show an exact position on a graph.
Area of Triangle – The measure of the two-dimensional space enclosed by a three-sided polygon.
Area of Quadrilateral – The measure of the two-dimensional space enclosed by a four-sided polygon.
Shoelace Formula – A method used to find the area of a polygon whose vertices are described by Cartesian coordinates.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more challenging non-linear equations to transform or quadrilaterals with more complex coordinates. For pupils who need more support, the teacher will provide additional guided practice, simpler examples, and step-by-step instructions, possibly using visual aids extensively.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in logarithms and basic algebraic manipulation before tackling the transformation of non-linear relationships. Emphasize the importance of writing down the formulas correctly and substituting coordinates accurately to avoid errors in area calculations. Encourage pupils to draw diagrams for area problems to visualize the shapes.