Direct and Inverse Proportion for JSS 3

Class: JSS 3
Term: First Term
Week: 3
Age: 13-14 years
Duration: 60 minutes
Subject: General Mathematics
Curriculum Theme: Number and Numeration
Previous Lesson: First Term General Mathematics Examination
Topic: DIRECT AND Inverse Proportion
Subject Matter: Direct proportion; Indirect proportion; Apply direct and inverse proportions to practical problems

Specific Objectives

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

Cognitive Domain

• Define direct proportion.
• Define inverse proportion.
• Solve problems involving direct proportion.
• Solve problems involving inverse proportion.

Psychomotor Domain

• Demonstrate the application of direct proportion to practical problems.
• Demonstrate the application of inverse proportion to practical problems.

Reference Materials

The following resources were used in planning this lesson:

• 2014 Senior Secondary Education Curriculum (SSEC)
• Relevant State Unified Scheme of Work
• A suitable General Mathematics textbook for JSS 3
• FCT ERC/NAPPS Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Whiteboard and markers
• Charts showing examples of direct and inverse proportion
• Flashcards with proportion problems
• Real-life scenarios for practical application

Rationale for the Lesson

Understanding direct and inverse proportion is fundamental in mathematics and daily life. This lesson helps students develop critical thinking and problem-solving skills, enabling them to interpret relationships between quantities and apply these concepts to real-world situations such as calculating costs, time, or resource allocation.

Prerequisite/Previous Knowledge

Students should have a basic understanding of ratios, fractions, and simple algebraic equations.

Lesson Content/Board Summary

DIRECT AND Inverse Proportion

Direct Proportion

Direct proportion describes a relationship between two quantities where an increase in one quantity leads to a proportional increase in the other, and a decrease in one quantity leads to a proportional decrease in the other. Their ratio remains constant.

If (y) is directly proportional to (x), we write (y propto x). This can be expressed as an equation:

(y = kx)

Where (k) is the constant of proportionality.

Example 1

Question: If 3 exercise books cost N450, how much will 7 exercise books cost?

Solution:

Step 1: Identify the relationship. As the number of books increases, the cost increases, so it is direct proportion.

Let (C) be the cost and (N) be the number of books. Then (C propto N), so (C = kN).

Step 2: Find the constant of proportionality, (k).

(450 = k times 3)

(k = frac{450}{3})

(k = 150)

Step 3: Use (k) to find the cost of 7 books.

(C = 150 times 7)

(C = 1050)

Answer: 7 exercise books will cost N1050.

Example 2

Question: A car travels 120 km in 2 hours. How far will it travel in 5 hours at the same speed?

Solution:

Step 1: Identify the relationship. As time increases, distance travelled increases, so it is direct proportion.

Let (D) be the distance and (T) be the time. Then (D propto T), so (D = kT).

Step 2: Find the constant of proportionality, (k).

(120 = k times 2)

(k = frac{120}{2})

(k = 60) (km/h)

Step 3: Use (k) to find the distance travelled in 5 hours.

(D = 60 times 5)

(D = 300)

Answer: The car will travel 300 km in 5 hours.

Inverse Proportion

Inverse proportion (or indirect proportion) describes a relationship between two quantities where an increase in one quantity leads to a proportional decrease in the other, and a decrease in one quantity leads to a proportional increase in the other. Their product remains constant.

If (y) is inversely proportional to (x), we write (y propto frac{1}{x}). This can be expressed as an equation:

(y = frac{k}{x}) or (xy = k)

Where (k) is the constant of proportionality.

Example 1

Question: 4 men can dig a trench in 6 hours. How long will it take 8 men to dig the same trench, assuming they work at the same rate?

Solution:

Step 1: Identify the relationship. As the number of men increases, the time taken to dig the trench decreases, so it is inverse proportion.

Let (M) be the number of men and (T) be the time. Then (T propto frac{1}{M}), so (T = frac{k}{M}) or (TM = k).

Step 2: Find the constant of proportionality, (k).

(6 times 4 = k)

(k = 24)

Step 3: Use (k) to find the time taken by 8 men.

(T times 8 = 24)

(T = frac{24}{8})

(T = 3)

Answer: It will take 8 men 3 hours to dig the trench.

Example 2

Question: A supply of food can last 10 students for 15 days. How many days will the same supply last 25 students?

Solution:

Step 1: Identify the relationship. As the number of students increases, the number of days the food lasts decreases, so it is inverse proportion.

Let (S) be the number of students and (D) be the number of days. Then (D propto frac{1}{S}), so (D = frac{k}{S}) or (DS = k).

Step 2: Find the constant of proportionality, (k).

(15 times 10 = k)

(k = 150)

Step 3: Use (k) to find the number of days for 25 students.

(D times 25 = 150)

(D = frac{150}{25})

(D = 6)

Answer: The same supply of food will last 25 students for 6 days.

Teaching Methods/Instructional Techniques

Discussion, Explanation, Guided Practice, Question and Answer, Problem Solving, Individual Practice

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Questioning/Activating prior knowledge

Teacher’s Activity: The teacher greets the students and asks questions about ratios and fractions to link to the new topic. For example, “If you buy more pens, what happens to the total cost?” or “If more people share a task, what happens to the time taken?”

Pupils’ Activity: Pupils respond to the questions and share their ideas.

Learning Point: Relating quantities

Step 2: Introduction to Direct Proportion

Time: 10 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces direct proportion, explaining its definition and giving simple real-life examples (e.g., cost of items vs. quantity). The teacher writes the definition and the formula (y = kx) on the board.

Pupils’ Activity: Pupils listen attentively, ask questions for clarification, and copy the definition.

Learning Point: Direct proportion definition

Step 3: Solving Direct Proportion Problems

Time: 10 minutes

Teaching Skill: Guided Practice/Problem Solving

Teacher’s Activity: The teacher guides students through the worked examples for direct proportion from the Board Summary, explaining each step clearly. The teacher encourages students to identify the constant of proportionality.

Pupils’ Activity: Pupils follow the examples, ask questions, and attempt to solve similar problems on their own.

Learning Point: Direct proportion problem-solving

Step 4: Introduction to Inverse Proportion

Time: 10 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher introduces inverse proportion, explaining its definition and providing contrasting real-life examples (e.g., number of workers vs. time to complete a job). The teacher writes the definition and the formula (y = frac{k}{x}) on the board.

Pupils’ Activity: Pupils listen, compare with direct proportion, and copy the definition.

Learning Point: Inverse proportion concept

Step 5: Solving Inverse Proportion Problems

Time: 10 minutes

Teaching Skill: Guided Practice/Problem Solving

Teacher’s Activity: The teacher guides students through the worked examples for inverse proportion from the Board Summary, ensuring they understand how to find the constant of proportionality and apply it.

Pupils’ Activity: Pupils actively participate in solving the examples and ask for help when needed.

Learning Point: Inverse proportion problem-solving

Step 6: Applying Proportions to Practical Problems

Time: 5 minutes

Teaching Skill: Application/Discussion

Teacher’s Activity: The teacher presents a few mixed practical problems and asks students to identify whether they are direct or inverse proportion before solving. This reinforces their understanding of when to apply each concept.

Pupils’ Activity: Pupils discuss and identify the type of proportion for each problem.

Learning Point: Identifying proportion types

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

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

1. What is direct proportion?
2. Give an example of direct proportion.
3. What is inverse proportion?
4. If 5 workers can complete a task in 10 days, how many days will it take 2 workers?

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Proportion concepts assessed

Step 8: Note-Taking

Time: 10 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on direct and inverse proportion, including definitions and worked examples, into their notebooks.

Pupils’ Activity: Pupils/students copy the notes carefully into their notebooks.

Learning Point: Recording lesson content

Step 9: Conclusion

Time: 5 minutes

Teaching Skill: Consolidation

Teacher’s Activity: The teacher summarises the main points of the lesson, reiterating the difference between direct and inverse proportion and their importance in everyday problem-solving. The teacher encourages students to look for examples in their daily lives.

Pupils’ Activity: Pupils listen and ask any final questions.

Learning Point: Lesson concepts consolidated

Continuous Assessment/Further Study

Type: Homework

Instruction: Solve the following problems in your notebook:

1. If 4 oranges cost N200, how much will 9 oranges cost?
2. A car travels at a constant speed. If it takes 3 hours to cover 180 km, how long will it take to cover 300 km?
3. 6 painters can paint a house in 8 days. How many days will it take 4 painters to paint the same house?
4. 20 students can finish a project in 15 days. If 5 more students join them, how many days will it take to finish the project?

Lesson Keywords

• Direct Proportion – A relationship where two quantities increase or decrease together at the same rate.
• Inverse Proportion – A relationship where an increase in one quantity leads to a decrease in another, and vice versa.
• Constant of Proportionality – The constant value relating two proportional quantities.

Differentiation

For students who grasp the concepts quickly, provide more complex word problems involving multiple steps or scenarios where they need to determine the type of proportion independently. For students needing additional support, provide simpler problems with clear guidance on identifying the type of proportion and setting up the equations. Use visual aids and concrete examples to reinforce understanding.

Suggested Lesson Videos

YouTube search for “direct and inverse proportion JSS3 mathematics”