Class: Junior Secondary School 3 (JSS 3 / JS3)
Term: 3rd Term
Week: 3
Age: 14 years
Duration: 45 minutes
Subject: Mathematics
Curriculum Theme: Mathematics
Previous Lesson: Statistics (data Presentation): Frequency Table.
Topic: VARIATION
Subject Matter: Revision of problem solving involving Direct, Inverse, Joint, and Partial Variation.

Specific Objectives

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

Cognitive Domain:

Recall the definitions and formulas for direct, inverse, joint, and partial variation.
State the constant of variation for different types of variation.
Solve problems involving direct, inverse, joint, and partial variation.

Affective Domain:

Appreciate the application of variation in real-life situations.
Develop confidence in solving problems related to variation.
Show interest in mathematical problem-solving.

Psychomotor Domain:

Accurately apply the correct formulas to solve variation problems.
Correctly calculate the constant of variation from given data.
Present solutions to variation problems in a clear and organized manner.

Collaborate with peers to solve challenging variation problems.
Participate actively in class discussions about different types of variation.
Share problem-solving strategies with classmates.

Reference Materials

The following resources were used in planning this lesson:

9 Years Basic Education Curriculum for Mathematics JSS3
State Unified Scheme of Work
New General Mathematics for Junior Secondary Schools 3

Instructional Materials

The teacher will teach this lesson with the aid of:

Whiteboard and markers
Textbooks (New General Mathematics for JSS 3)
Charts showing formulas for different types of variation

Rationale for the Lesson

This lesson helps pupils review and solidify their understanding of variation, which is important for solving real-world problems involving relationships between quantities. It also prepares them for more advanced mathematical concepts in senior secondary school.

Prerequisite/Previous Knowledge

Pupils are expected to have prior knowledge of basic algebra, solving linear equations, and an introductory understanding of direct and inverse variation from JSS 2.

Lesson Content/Board Summary

VARIATION

1. Direct Variation

Direct variation describes a relationship where one quantity increases or decreases proportionally with another quantity. If ‘y’ varies directly as ‘x’, it means that as ‘x’ increases, ‘y’ increases, and as ‘x’ decreases, ‘y’ decreases, provided the ratio between them remains constant.

Mathematical Relationship: y α x or y = kx

Where ‘k’ is the constant of variation (a non-zero constant).

Example 1: If y varies directly as x, and y = 12 when x = 3, find the constant of variation and the value of y when x = 5.

Step 1: Write the relation: y = kx
Step 2: Substitute the given values to find k: 12 = k(3) => k = 12/3 = 4
Step 3: Write the equation: y = 4x
Step 4: Find y when x = 5: y = 4(5) = 20

Example 2: The cost (C) of electricity varies directly as the number of units (U) consumed. If 50 units cost ₦1500, how much will 80 units cost?

Step 1: Write the relation: C = kU
Step 2: Substitute known values: 1500 = k(50) => k = 1500/50 = 30
Step 3: Write the equation: C = 30U
Step 4: Find cost for 80 units: C = 30(80) = ₦2400

2. Inverse Variation

Inverse variation describes a relationship where one quantity increases as the other quantity decreases, or vice versa, such that their product remains constant. If ‘y’ varies inversely as ‘x’, it means that as ‘x’ increases, ‘y’ decreases proportionally.

Mathematical Relationship: y α 1/x or y = k/x (or xy = k)

Where ‘k’ is the constant of variation.

Example 1: If y varies inversely as x, and y = 6 when x = 4, find the constant of variation and the value of y when x = 8.

Step 1: Write the relation: y = k/x
Step 2: Substitute known values: 6 = k/4 => k = 6 * 4 = 24
Step 3: Write the equation: y = 24/x
Step 4: Find y when x = 8: y = 24/8 = 3

Example 2: The time (T) taken to complete a job varies inversely as the number of workers (N). If 5 workers can complete a job in 10 days, how many days will it take 2 workers?

Step 1: Write the relation: T = k/N
Step 2: Substitute known values: 10 = k/5 => k = 10 * 5 = 50
Step 3: Write the equation: T = 50/N
Step 4: Find time for 2 workers: T = 50/2 = 25 days

3. Joint Variation

Joint variation describes a relationship where one quantity varies directly as the product of two or more other quantities. For example, if ‘y’ varies jointly as ‘x’ and ‘z’, it means ‘y’ is directly proportional to both ‘x’ and ‘z’.

Mathematical Relationship: y α xz or y = kxz

Where ‘k’ is the constant of variation.

Example 1: If y varies jointly as x and z, and y = 24 when x = 2 and z = 3, find the constant of variation and the value of y when x = 4 and z = 5.

Step 1: Write the relation: y = kxz
Step 2: Substitute known values: 24 = k(2)(3) => 24 = 6k => k = 24/6 = 4
Step 3: Write the equation: y = 4xz
Step 4: Find y when x = 4 and z = 5: y = 4(4)(5) = 80

Example 2: The volume (V) of a cylinder varies jointly as its height (h) and the square of its radius (r). If V = 60π when h = 5 and r = 2, find the constant of variation and the volume when h = 7 and r = 3.

Step 1: Write the relation: V = khr²
Step 2: Substitute known values: 60π = k(5)(2²) => 60π = k(5)(4) => 60π = 20k => k = 60π/20 = 3π
Step 3: Write the equation: V = 3πhr²
Step 4: Find V when h = 7 and r = 3: V = 3π(7)(3²) = 3π(7)(9) = 189π

4. Partial Variation

Partial variation describes a relationship where one quantity is partly constant and partly varies directly or inversely as another quantity. It involves a sum of terms, where one term is constant and the other term is a direct or inverse variation.

Mathematical Relationship: y = a + bx (y partly constant, partly varies directly as x)

Where ‘a’ and ‘b’ are constants.

Example 1: y is partly constant and partly varies directly as x. When x = 2, y = 10 and when x = 5, y = 16. Find the relationship between y and x.

Step 1: Write the general relation: y = a + bx
Step 2: Substitute the first pair of values: 10 = a + 2b (Equation 1)
Step 3: Substitute the second pair of values: 16 = a + 5b (Equation 2)
Step 4: Solve the simultaneous equations. Subtract Equation 1 from Equation 2:
(16 – 10) = (a – a) + (5b – 2b)
6 = 3b => b = 2
Step 5: Substitute b = 2 into Equation 1: 10 = a + 2(2) => 10 = a + 4 => a = 6
Step 6: Write the relationship: y = 6 + 2x

Example 2: The cost (C) of hiring a taxi is partly constant and partly varies with the distance (D) covered. If a journey of 5 km costs ₦700 and a journey of 10 km costs ₦1100, find the cost of a journey of 15 km.

Step 1: Write the general relation: C = a + bD
Step 2: Substitute the first pair of values: 700 = a + 5b (Equation 1)
Step 3: Substitute the second pair of values: 1100 = a + 10b (Equation 2)
Step 4: Solve the simultaneous equations. Subtract Equation 1 from Equation 2:
(1100 – 700) = (a – a) + (10b – 5b)
400 = 5b => b = 80
Step 5: Substitute b = 80 into Equation 1: 700 = a + 5(80) => 700 = a + 400 => a = 300
Step 6: Write the relationship: C = 300 + 80D
Step 7: Find the cost for 15 km: C = 300 + 80(15) = 300 + 1200 = ₦1500

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 greets the pupils and reminds them of their previous knowledge of variation, asking them to recall simple direct and inverse variation examples.
Pupils’ Activity: Pupils respond to the teacher’s greeting and share examples of direct and inverse variation they remember.
Learning Point: Pupils recall prior knowledge related to the topic.

Step 2: Revision of Direct Variation

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains direct variation, writes the formula on the board, and solves two examples from the lesson content, emphasizing step-by-step working.
Pupils’ Activity: Pupils listen attentively, copy notes, and ask questions for clarification.
Learning Point: Pupils understand and can solve problems involving direct variation.

Step 3: Revision of Inverse Variation

Time: 8 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains inverse variation, writes the formula, and solves two examples from the lesson content, highlighting the difference from direct variation.
Pupils’ Activity: Pupils observe, take notes, and attempt to solve parts of the examples with the teacher’s guidance.
Learning Point: Pupils understand and can solve problems involving inverse variation.

Step 4: Revision of Joint Variation

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher introduces joint variation, explaining how one quantity relates to the product of two or more others, and solves two examples.
Pupils’ Activity: Pupils pay attention to the new concept and work through the examples shown on the board.
Learning Point: Pupils grasp the concept of joint variation and its application.

Step 5: Revision of Partial Variation

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains partial variation as a combination of constant and variable parts, demonstrates the method of solving simultaneous equations, and solves two examples.
Pupils’ Activity: Pupils listen to the explanation, copy the examples, and ask about the simultaneous equation solving process.
Learning Point: Pupils understand partial variation and how to solve problems using simultaneous equations.

Step 6: Class Practice

Time: 5 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher gives a quick problem that combines elements of different variation types or asks pupils to solve a similar problem in pairs.
Pupils’ Activity: Pupils attempt to solve the given problem individually or in pairs.
Learning Point: Pupils apply their revised knowledge to solve new problems.

Step 7: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

Define direct variation and write its general formula.
If P varies inversely as Q, and P=10 when Q=4, find P when Q=8.
What is joint variation? Give an example of a relationship that shows joint variation.
Given that y is partly constant and partly varies directly as x. If y = 15 when x = 3, and y = 25 when x = 5, find the relationship between y and x.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 3 minutes
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definitions and formulas for direct, inverse, joint, and partial variation. The teacher assigns homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the homework assignment.
Learning Point: Pupils consolidate their learning and identify areas for further practice.

Lesson Keywords

Variation – The way in which one quantity changes with respect to another.
Direct Variation – A relationship where one quantity increases or decreases proportionally with another (y = kx).
Inverse Variation – A relationship where one quantity increases as the other decreases proportionally (y = k/x).
Joint Variation – A relationship where one quantity varies directly as the product of two or more other quantities (y = kxz).
Partial Variation – A relationship where one quantity is partly constant and partly varies directly or inversely with another quantity (y = a + bx).
Constant of Variation – The constant ‘k’ in a variation equation that relates the quantities.

Differentiation

For pupils who grasp the concepts quickly, the teacher can provide more complex problems or ask them to create their own real-life examples of each variation type. For pupils who are struggling, the teacher can provide additional guided practice, simpler examples, and one-on-one support during the practice session.

Note for teachers using this lesson plan

Ensure that pupils understand the concept of the constant of variation (‘k’) as it is fundamental to solving all types of variation problems. Emphasize the importance of clear, step-by-step problem-solving. Encourage pupils to draw connections between abstract mathematical concepts and real-world scenarios to enhance understanding.