HCF of whole numbers and differences between LCM and HCF for JSS 1

Class: JSS 1
Term: First Term
Week: 6
Age: 12 years
Duration: 45 minutes
Subject: Mathematics
Curriculum Theme: NUMBER NUMERATION
Focal competence: Calculating the HCF of two or more whole numbers
Key competencies/values: Collaboration; ICT and Digital Competencies
Skills:

• Calculating of LCM and HCF

Previous Lesson: Meaning of a factor and a common factor
Topic: Highest Common Factor (HCF): HCF of Whole Numbers
Subject Matter: HCF of whole numbers, differences between LCM and HCF, Quantitative Aptitude involving HCF

Specific Objectives

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

Cognitive Domain

• Differentiate between LCM and HCF.
• Find the HCF of whole numbers.
• Solve problems on quantitative aptitude tasks involving HCF of whole numbers.
• Identify the differences between LCM and HCF.

Affective Domain

• Collaborate effectively in groups to identify common factors.

Psychomotor Domain

• Apply the common factor method to find HCF.
• Apply the prime factorization method to find HCF.

Reference Materials

The following resources were used in planning this lesson:

• 2025 Revised 9 Years Basic Education Curriculum
• Relevant State Unified Scheme of Work
• The HeadTeacher Scheme of work

Instructional Materials

The teacher will teach this lesson with the aid of:

• Mathematics Open references
• Interactive HCF calculator
• Maths Is Fun
• Coolmath HCF lesson and practices
• LCM and HCF by CK-12
• Whiteboard and markers
• Charts showing factors and multiples
• Worksheets with HCF problems

Rationale for the Lesson

Understanding HCF is fundamental in mathematics, as it helps in simplifying fractions and solving real-world problems involving equal distribution or grouping. This lesson builds on previous knowledge of factors and multiples, providing students with essential problem-solving skills applicable in various mathematical contexts and everyday situations.

Prerequisite/Previous Knowledge

Students should have prior knowledge of factors, multiples, prime numbers, and basic division of whole numbers.

Lesson Content/Board Summary

Highest Common Factor (HCF): HCF of Whole Numbers

Meaning of a Factor

A factor of a whole number is a number that divides it exactly, without leaving any remainder.

For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 because each of these numbers divides 12 exactly.

Exact Division and Factor Pairs

Exact division occurs when one number divides another number completely, resulting in a whole number quotient and zero remainder. Factors often come in pairs. For instance, if (a times b = c), then (a) and (b) are factors of (c).

Example: For 24, (1 times 24 = 24), (2 times 12 = 24), (3 times 8 = 24), (4 times 6 = 24). The factor pairs are (1, 24), (2, 12), (3, 8), (4, 6).

Listing Factors of Whole Numbers

To list the factors of a number, find all the numbers that divide it exactly, starting from 1 up to the number itself.

Example:

1. Factors of 18: 1, 2, 3, 6, 9, 18
2. Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Common Factors

Common factors are factors that two or more numbers share. They are the numbers that divide all the given numbers exactly.

Example: Find the common factors of 12 and 18.

1. Factors of 12: 1, 2, 3, 4, 6, 12
2. Factors of 18: 1, 2, 3, 6, 9, 18
3. Common factors of 12 and 18 are 1, 2, 3, 6.

Highest Common Factor (HCF)

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), is the largest number that divides two or more numbers exactly without leaving a remainder.

Methods for Finding HCF

Listing Common Factors Method

This method involves listing all factors of each number, identifying the common factors, and then selecting the largest among them.

Example 1

Question: Find the HCF of 12 and 18 using the listing method.

Solution:

Step 1: List the factors of each number.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: Identify the common factors.

Common factors: 1, 2, 3, 6

Step 3: Select the highest common factor.

The highest common factor is 6.

Answer: HCF(12, 18) = 6

Prime Factorization Method

This method involves expressing each number as a product of its prime factors. The HCF is then found by multiplying the common prime factors raised to their lowest powers.

Example 2

Question: Find the HCF of 24 and 36 using the prime factorization method.

Solution:

Step 1: Express each number as a product of its prime factors.

(24 = 2 times 2 times 2 times 3 = 2^3 times 3^1)

(36 = 2 times 2 times 3 times 3 = 2^2 times 3^2)

Step 2: Identify the common prime factors and their lowest powers.

Common prime factors are 2 and 3.

Lowest power of 2 is (2^2).

Lowest power of 3 is (3^1).

Step 3: Multiply these common prime factors with their lowest powers.

(HCF = 2^2 times 3^1 = 4 times 3 = 12)

Answer: HCF(24, 36) = 12

Differences Between LCM and HCF

The Least Common Multiple (LCM) and Highest Common Factor (HCF) are distinct concepts in number theory.

1. Definition: HCF is the largest number that divides two or more numbers exactly. LCM is the smallest number that is a multiple of two or more numbers.
2. Value: HCF is always less than or equal to the smallest of the given numbers. LCM is always greater than or equal to the largest of the given numbers.
3. Application: HCF is used in problems involving dividing items into equal groups or simplifying fractions. LCM is used in problems involving events repeating at regular intervals or finding a common time/length.

Quantitative Aptitude Involving HCF

Quantitative aptitude problems involving HCF often require finding the largest possible size or quantity that can divide given numbers exactly.

Example 3

Question: A shopkeeper has 48 apples and 60 oranges. He wants to pack them into baskets such that each basket contains an equal number of apples and an equal number of oranges, with no fruit left over. What is the greatest number of baskets he can make?

Solution:

Step 1: Understand the problem. We need to find the largest number that divides both 48 and 60 exactly, which is the HCF.

Step 2: Find the prime factors of 48 and 60.

(48 = 2 times 2 times 2 times 2 times 3 = 2^4 times 3^1)

(60 = 2 times 2 times 3 times 5 = 2^2 times 3^1 times 5^1)

Step 3: Identify common prime factors and their lowest powers.

Common prime factors are 2 and 3.

Lowest power of 2 is (2^2).

Lowest power of 3 is (3^1).

Step 4: Multiply these common prime factors.

(HCF = 2^2 times 3^1 = 4 times 3 = 12)

Answer: The greatest number of baskets he can make is 12.

Teaching Methods/Instructional Techniques

Discussion, Demonstration, Guided Practice, Question and Answer, Explanation, Problem Solving, Pair Work, Group Work, Individual Practice.

Instructional Procedures

Step 1: Introduction

Time: 5 minutes

Teaching Skill: Activating prior knowledge

Teacher’s Activity: The teacher greets the students and reviews the previous lesson on factors and multiples. The teacher asks students to list factors of simple numbers like 6 and 10.

Pupils’ Activity: Students respond to questions and list factors of given numbers.

Learning Point: Recall of factors

Step 2: Explanation of Factors and Exact Division

Time: 7 minutes

Teaching Skill: Explaining concepts

Teacher’s Activity: The teacher explains the meaning of a factor and exact division with examples, emphasizing that factors divide a number without a remainder. The teacher demonstrates finding factor pairs.

Pupils’ Activity: Students listen, ask questions, and identify factor pairs for given numbers.

Learning Point: Understanding factors

Step 3: Listing Factors of Whole Numbers

Time: 6 minutes

Teaching Skill: Guided practice

Teacher’s Activity: The teacher guides students to list all factors of numbers like 20 and 32, ensuring they understand the systematic approach.

Pupils’ Activity: Students actively participate in listing factors and practice with new numbers.

Learning Point: Listing factors accurately

Step 4: Introduction to HCF and Common Factors

Time: 6 minutes

Teaching Skill: Concept introduction

Teacher’s Activity: The teacher introduces the concept of common factors and HCF. The teacher asks students to find common factors of two numbers (e.g., 12 and 18) and then identify the highest among them.

Pupils’ Activity: Students identify common factors and the HCF through guided examples.

Learning Point: Common factors and HCF

Step 5: Methods for Finding HCF

Time: 8 minutes

Teaching Skill: Demonstration and problem-solving

Teacher’s Activity: The teacher demonstrates finding HCF using both the listing common factors method and the prime factorization method with clear examples (e.g., HCF of 24 and 36). The teacher then presents a quantitative aptitude problem involving HCF and guides students to solve it.

Pupils’ Activity: Students observe, ask questions, and practice solving HCF problems in pairs or small groups using the demonstrated methods.

Learning Point: HCF calculation methods

Step 6: Differences between LCM and HCF

Time: 5 minutes

Teaching Skill: Comparison and differentiation

Teacher’s Activity: The teacher facilitates a discussion on the key differences between LCM and HCF, using examples to illustrate the distinct characteristics and applications of each.

Pupils’ Activity: Students contribute to the discussion, identify differences, and clarify their understanding of LCM vs. HCF.

Learning Point: Distinguishing LCM and HCF

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 a factor of a number?
2. List the factors of 28.
3. Find the HCF of 16 and 24 using any method.
4. State two differences between HCF and LCM.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Assessment of HCF understanding

Step 8: Note-Taking

Time: 4 minutes

Teaching Skill: Guided Writing

Teacher’s Activity: The teacher guides pupils/students to copy the essential Board Summary notes on HCF, its methods, and differences from LCM into their notebooks.

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

Learning Point: Recording HCF notes

Step 9: Conclusion

Time: 3 minutes

Teaching Skill: Reinforcement

Teacher’s Activity: The teacher summarises the key points of the lesson, reiterating the importance of HCF in solving real-life problems and its distinction from LCM. The teacher encourages students to practice more problems.

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

Learning Point: HCF concept consolidation

Continuous Assessment/Further Study

Type: Homework
Instruction: Solve the following problems in your notebook:

1. Find the HCF of 45 and 75 using the prime factorization method.
2. A farmer has 72 chickens and 96 ducks. He wants to put them into the largest possible number of pens such that each pen has the same number of chickens and the same number of ducks. How many pens can he make?
3. Explain in your own words the main difference between HCF and LCM.
4. List all the factors of 56.

Lesson Keywords

• Factor – A number that divides another number exactly without a remainder.
• Exact Division – Division that results in a whole number quotient and zero remainder.
• Common Factors – Factors shared by two or more numbers.
• Highest Common Factor (HCF) – The largest number that divides two or more numbers exactly.
• Prime Factorization – Expressing a number as a product of its prime factors.
• Quantitative Aptitude – Problems involving numerical ability and problem-solving.

Differentiation

For weaker learners, provide additional practice with listing factors of small numbers and guide them step-by-step through the listing method for HCF. Faster learners can be challenged with problems involving three numbers for HCF or more complex quantitative aptitude tasks, encouraging them to explore alternative methods or real-world applications.

Suggested Lesson Videos

For further understanding, search on YouTube for: HCF of whole numbers JSS1 Mathematics

Teacher Guide for Using This Lesson Plan

Before the lesson, ensure you have charts or visual aids prepared for listing factors and demonstrating prime factorization. Begin by revisiting factors and multiples to ensure a strong foundation. When introducing HCF, use concrete examples relevant to students’ experiences. Encourage group work for identifying common factors and applying the prime factorization method, fostering collaboration. Clearly explain the distinction between HCF and LCM, perhaps using a comparison table. For quantitative aptitude problems, guide students to identify keywords that indicate the need for HCF. During the note-taking stage (Step 8), ensure students copy the Board Summary accurately. Pay close attention to common errors, such as confusing HCF with LCM or making mistakes in prime factorization, and address them promptly. Provide individual support to students who struggle and offer extension activities for those who grasp the concepts quickly.