Class: Junior Secondary School 3 (JSS3, JSS 3)
Term: 1st Term
Week: 10
Age: 14 years
Duration: 45 minutes
Subject: Mathematics
Curriculum Theme: Mathematics
Previous Lesson: Graphical Solution of Simultaneous Equations
Topic: Variation, Word Problems
Subject Matter: Definition of variation, Direct variation y = kx, Inverse variation y = k/x, Partial variation y = kx + c, Joint variation y = kpq, where k is a constant. Translate word problems into numerical expression, Interpreting and solving given word problems.

Specific Objectives

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

Cognitive Domain:
(a) Identify the different types of variation.
(b) Solve mathematical problems involving direct, inverse, partial, and joint variation.
Affective Domain:
(a) Develop a positive attitude towards solving mathematical word problems.
Psychomotor Domain:
(a) Translate word problems on variation into correct algebraic expressions.
Social Domain:
(a) Work collaboratively to solve variation problems in groups.

Reference Materials

The following resources was used in planning this lesson:

9 Years Basic Education Curriculum
Lagos State Unified Scheme of Work for Junior Secondary Schools
Math Coach’s Corner – Problem-Solving vs Word Problems
Relevant Textbooks

Instructional Materials

The teacher will teach this lesson with the aid of:

Charts showing different types of variation equations
Flash cards with sample word problems
Whiteboard and markers
Graph sheets
Mathematical sets and calculators

Rationale for the Lesson

This lesson will help pupils understand the relationship between two or more variables and how to represent them mathematically to solve real-life problems.

Prerequisite/Previous Knowledge

Pupils already understand basic algebraic expressions and can simplify, substitute values, and solve for unknowns in simple equations.

Lesson Content/Board Summary

Variation

Variation is the relationship between two or more quantities where a change in one affects the other. The relationship can be direct, inverse, partial, or joint.

Direct Variation

Direct variation occurs when one quantity increases as the other increases in the same ratio.
It is written as y ∝ x or y = kx, where k is the constant of proportionality.

Examples include:

If y varies directly as x and y = 20 when x = 5, find y when x = 8.
- Solution:
  ( y = kx )
  ( 20 = k(5) Rightarrow k = 4 )
  ( y = 4x )
  When ( x = 8 ), ( y = 4(8) = 32 ).
The distance travelled varies directly as time. If a car travels 60 km in 2 hours, how far will it go in 5 hours?
- ( d = kt )
  ( 60 = k(2) Rightarrow k = 30 )
  ( d = 30(5) = 150 ) km.

Inverse Variation

Inverse variation occurs when one quantity increases as the other decreases.
It is written as y ∝ 1/x or y = k/x, where k is a constant.

Examples include:

If y varies inversely as x and y = 8 when x = 3, find y when x = 6.
- ( y = frac{k}{x} )
  ( 8 = frac{k}{3} Rightarrow k = 24 )
  When ( x = 6, y = frac{24}{6} = 4 ).
The time taken to complete a task varies inversely as the number of workers. If 5 workers take 8 days, how many days will 10 workers take?
- ( t = frac{k}{w} )
  ( 8 = frac{k}{5} Rightarrow k = 40 )
  ( t = frac{40}{10} = 4 ) days.

Partial Variation

Partial variation occurs when one quantity partly varies directly with another and partly remains constant.
It is written as y = kx + c, where k and c are constants.

Example:
If y = 20 when x = 2 and y = 50 when x = 8, find the equation of the variation.

Using ( y = kx + c ):
When ( x = 2, 20 = 2k + c )
When ( x = 8, 50 = 8k + c )
Subtract: ( 30 = 6k Rightarrow k = 5 )
Substitute: ( 20 = 2(5) + c Rightarrow c = 10 )
Hence, ( y = 5x + 10 ).

Joint Variation

Joint variation occurs when a quantity varies directly as the product of two or more quantities.
It is written as y = kpq, where k is the constant of proportionality.

Example:
If y varies jointly as p and q, and y = 60 when p = 4 and q = 3, find y when p = 5 and q = 6.

( y = kpq )
( 60 = k(4)(3) Rightarrow k = 5 )
When ( p = 5, q = 6, y = 5(5)(6) = 150 ).

Word Problems on Variation

Word problems involve interpreting real-life situations into mathematical equations of variation.
To solve such problems:

Identify the type of variation (direct, inverse, partial, or joint).
Write the general equation.
Find the constant of proportionality using given data.
Substitute the new values to find the unknown.

Teaching Methods/Instructional Techniques:

Discussion, Explanation, Demonstration, Guided Discovery, Problem-Solving, Group Work

Instructional Procedures

To deliver this lesson, the teacher will adopt the following steps:

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher asks pupils how distance, time, and speed are related and leads them to recall previous lessons on ratio and proportion.
Pupils’ Activity: Pupils respond by sharing their ideas and experiences.
Learning Point: Pupils recall that some quantities are related by constant ratios.

Step 2: Explanation of Direct and Inverse Variation

Time: 10 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher defines direct and inverse variation and illustrates each on the board with examples.
Pupils’ Activity: Pupils listen, ask questions, and copy examples.
Learning Point: Pupils understand direct and inverse variation.

Step 3: Partial and Joint Variation

Time: 10 minutes
Teaching Skill: Discussion
Teacher’s Activity: The teacher explains partial and joint variation using examples and real-life applications.
Pupils’ Activity: Pupils participate in class discussions and solve a sample problem.
Learning Point: Pupils understand how variables relate through partial and joint variation.

Step 4: Solving Word Problems

Time: 10 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher guides pupils to translate given word problems into algebraic equations and solve them.
Pupils’ Activity: Pupils work in groups to solve similar problems.
Learning Point: Pupils can interpret and solve word problems involving variation.

Step 5: Note-Taking

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: Teacher writes the board summary for pupils to copy.
Pupils’ Activity: Pupils copy the note.
Learning Point: Pupils have accurate notes on the topic.

Step 6: Evaluation/Review

Time: 3 minutes
Teaching Skill: Questioning
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

What is direct variation?
Give one example of inverse variation.
State the general equation for partial variation.
Solve: y varies jointly as p and q, y = 12 when p = 2, q = 3. Find y when p = 4, q = 5.
Pupils’ Activity: Pupils respond orally and on the board.
Learning Point: Pupils demonstrate comprehension of variation problems.

Step 7: Conclusion

Time: 2 minutes
Teaching Skill: Reinforcement
Teacher’s Activity: The teacher summarizes key points and emphasizes how variation applies to daily life.
Pupils’ Activity: Pupils listen and ask clarifying questions.
Learning Point: Pupils appreciate the usefulness of variation in mathematics and real-life situations.

Lesson Keywords

Variation – the relationship between quantities that change together
Constant (k) – a fixed number that defines the relationship
Direct variation – when one variable increases with another
Inverse variation – when one variable increases as another decreases
Proportionality – equality of ratios between related quantities

Differentiation

Support slower learners by breaking problems into smaller steps, while advanced learners solve complex word problems independently. Encourage peer assistance during practice.

Note for teachers using this lesson plan

Ensure pupils grasp each type of variation before solving mixed problems. Use relatable examples like speed-time-distance and workmen-time relationships for better understanding.