L.C.M and H.C.F of Whole Numbers, Multiples, Factors and Reasoning for Primary 6

Class: Primary 6
Term: 1st Term
Week: 4
Age: 11 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Numbers and Numeration
Previous Lesson: Types, Comparing, Ordering and Decimal Conversion of Fractions.
Topic: L.C.M AND H.C.F OF WHOLE NUMBERS
Subject Matter: Multiples of numbers up to first 20 multiples and common multiples, LCM of 2-digit whole numbers, factors and common factors of 2-digit whole numbers, HCF of 2-digit whole numbers, quantitative reasoning involving LCM and HCF.

Specific Objectives

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

Cognitive Domain:

• Define multiples, factors, Least Common Multiple (LCM), and Highest Common Factor (HCF).
• List multiples and factors of 2-digit numbers.
• Find common multiples and common factors of 2-digit numbers.
• Calculate the LCM of 2-digit numbers using different methods.
• Calculate the HCF of 2-digit numbers using different methods.
• Solve quantitative reasoning problems involving LCM and HCF.

Affective Domain:

• Appreciate the real-life applications of LCM and HCF.
• Develop a positive attitude towards solving mathematical problems.
• Participate actively in classroom discussions.

Psychomotor Domain:

• Accurately apply the steps for finding LCM and HCF.
• Solve word problems by correctly identifying whether to use LCM or HCF.

Social Domain:

• Work collaboratively with peers to solve problems.
• Communicate mathematical ideas clearly.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum
• State Unified Scheme of Work
• Primary Mathematics for Nigerian Schools, Book 6
• https://www.education.gov.ng/
• https://www.nerdc.gov.ng/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Textbook
• Charts showing multiples and factors
• Posters with examples of LCM and HCF calculations
• Workbook
• Cardboard
• Marker

Rationale for the Lesson

This lesson helps pupils understand how numbers relate to each other through their multiples and factors. Learning about LCM and HCF enables pupils to solve practical problems encountered in everyday life, such as scheduling events or dividing items equally. It also builds a strong foundation for more advanced mathematical concepts.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of multiplication, division, prime numbers, and prime factorization.

Lesson Content/Board Summary

L.C.M. AND H.C.F. OF WHOLE NUMBERS

Multiples of Numbers

A multiple of a number is the result of multiplying that number by an integer. Multiples are essentially the numbers that appear in the multiplication table of a given number.

To find the multiples of a number, multiply the number by 1, 2, 3, 4, and so on.

Example 1: List the first 5 multiples of 8.

Solution:

Multiples of 8:

• 8 x 1 = 8
• 8 x 2 = 16
• 8 x 3 = 24
• 8 x 4 = 32
• 8 x 5 = 40

The first 5 multiples of 8 are 8, 16, 24, 32, 40.

Example 2: List the first 10 multiples of 12.

Solution:

Multiples of 12:

• 12 x 1 = 12
• 12 x 2 = 24
• 12 x 3 = 36
• 12 x 4 = 48
• 12 x 5 = 60
• 12 x 6 = 72
• 12 x 7 = 84
• 12 x 8 = 96
• 12 x 9 = 108
• 12 x 10 = 120

The first 10 multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120.

Common Multiples

Common multiples are the numbers that are multiples of two or more given numbers.

To find common multiples, list the multiples of each number and identify the numbers that appear in all lists.

Example: Find the common multiples of 6 and 9 up to 40.

Solution:

• Step 1: List multiples of 6: 6, 12, 18, 24, 30, 36, 42…
• Step 2: List multiples of 9: 9, 18, 27, 36, 45…
• Step 3: Identify common numbers in both lists.

The common multiples of 6 and 9 up to 40 are 18 and 36.

Least Common Multiple (LCM)

The Least Common Multiple (LCM) of two or more numbers is the smallest non-zero number that is a multiple of all the given numbers.

There are two common methods to find the LCM:

Method 1: Listing Multiples

• Step 1: List the multiples of each number.
• Step 2: Identify the common multiples.
• Step 3: The smallest common multiple is the LCM.

Example 1: Find the LCM of 10 and 15.

Solution:

• Step 1: List multiples of 10: 10, 20, 30, 40, 50, 60…
• Step 2: List multiples of 15: 15, 30, 45, 60, 75…
• Step 3: The common multiples are 30, 60…
• Step 4: The smallest common multiple is 30.

Therefore, the LCM of 10 and 15 is 30.

Method 2: Prime Factorization Method

• Step 1: Find the prime factorization of each number.
• Step 2: List all prime factors that appear in any of the factorizations.
• Step 3: For each prime factor, use the highest power (exponent) that appears in any of the factorizations.
• Step 4: Multiply these highest powers together to get the LCM.

Example 2: Find the LCM of 12 and 18 using the prime factorization method.

Solution:

• Step 1: Find prime factors:
  • 12 = 2 x 2 x 3 = 2² x 3¹
  • 18 = 2 x 3 x 3 = 2¹ x 3²
• Step 2: List all prime factors: 2, 3.
• Step 3: Take the highest power of each prime factor:
  • Highest power of 2 is 2²
  • Highest power of 3 is 3²
• Step 4: Multiply the highest powers: LCM = 2² x 3² = 4 x 9 = 36.

Therefore, the LCM of 12 and 18 is 36.

NOTE: The LCM is always greater than or equal to the largest of the given numbers.

Factors of Numbers

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

To find the factors of a number, identify all the numbers that can divide the given number evenly.

Example 1: List all factors of 24.

Solution:

Factors of 24 are the numbers that divide 24 exactly:

• 1 x 24 = 24
• 2 x 12 = 24
• 3 x 8 = 24
• 4 x 6 = 24

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

Example 2: List all factors of 36.

Solution:

Factors of 36 are:

• 1 x 36 = 36
• 2 x 18 = 36
• 3 x 12 = 36
• 4 x 9 = 36
• 6 x 6 = 36

The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.

Common Factors

Common factors are the numbers that are factors of two or more given numbers.

To find common factors, list the factors of each number and identify the numbers that appear in all lists.

Example: Find the common factors of 12 and 18.

Solution:

• Step 1: List factors of 12: 1, 2, 3, 4, 6, 12.
• Step 2: List factors of 18: 1, 2, 3, 6, 9, 18.
• Step 3: Identify common numbers in both lists.

The common factors of 12 and 18 are 1, 2, 3, 6.

Highest Common Factor (HCF)

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides all the given numbers exactly.

There are two common methods to find the HCF:

Method 1: Listing Factors

• Step 1: List all the factors of each number.
• Step 2: Identify the common factors.
• Step 3: The largest common factor is the HCF.

Example 1: Find the HCF of 20 and 30.

Solution:

• Step 1: List factors of 20: 1, 2, 4, 5, 10, 20.
• Step 2: List factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.
• Step 3: The common factors are 1, 2, 5, 10.
• Step 4: The largest common factor is 10.

Therefore, the HCF of 20 and 30 is 10.

Method 2: Prime Factorization Method

• Step 1: Find the prime factorization of each number.
• Step 2: List only the common prime factors.
• Step 3: For each common prime factor, use the lowest power (exponent) that appears in any of the factorizations.
• Step 4: Multiply these lowest powers together to get the HCF.

Example 2: Find the HCF of 24 and 36 using the prime factorization method.

Solution:

• Step 1: Find prime factors:
  • 24 = 2 x 2 x 2 x 3 = 2³ x 3¹
  • 36 = 2 x 2 x 3 x 3 = 2² x 3²
• Step 2: List common prime factors: 2, 3.
• Step 3: Take the lowest power of each common prime factor:
  • Lowest power of 2 is 2²
  • Lowest power of 3 is 3¹
• Step 4: Multiply the lowest powers: HCF = 2² x 3¹ = 4 x 3 = 12.

Therefore, the HCF of 24 and 36 is 12.

NOTE: The HCF is always less than or equal to the smallest of the given numbers.

Quantitative Reasoning Involving LCM and HCF

Quantitative reasoning problems often involve word problems where you need to decide whether to use LCM or HCF to find the solution. Generally:

• Use LCM when the problem asks for something that happens at the *same time again* or when finding the *smallest group* or *least amount* that can be divided by multiple numbers. Keywords: “least”, “smallest”, “first time again”, “together”.
• Use HCF when the problem asks to *divide things into equal groups*, find the *largest possible size* of something, or *distribute items evenly*. Keywords: “greatest”, “largest”, “maximum”, “divide equally”, “share”.

Example 1 (LCM): Two bells ring at intervals of 6 minutes and 8 minutes respectively. If they both ring together at 9:00 AM, when will they next ring together?

Solution:

• Step 1: Understand the problem: We need to find when they will ring together again, which implies finding a common multiple. Since we want the *next* time, it’s the Least Common Multiple.
• Step 2: Find the LCM of 6 and 8.
  • Multiples of 6: 6, 12, 18, 24, 30…
  • Multiples of 8: 8, 16, 24, 32…
  • The LCM of 6 and 8 is 24.
• Step 3: Add the LCM to the starting time.
  • 24 minutes after 9:00 AM is 9:24 AM.

The bells will next ring together at 9:24 AM.

Example 2 (HCF): A farmer has 48 oranges and 60 apples. He wants to pack them into baskets so that each basket contains the same number of oranges and the same number of apples, without mixing fruits. What is the greatest number of baskets he can use?

Solution:

• Step 1: Understand the problem: We need to find the largest number of baskets that can divide both 48 oranges and 60 apples evenly. This means finding the Highest Common Factor.
• Step 2: Find the HCF of 48 and 60.
  • Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • The common factors are 1, 2, 3, 4, 6, 12.
  • The greatest common factor is 12.

The greatest number of baskets the farmer can use is 12.

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 asks them to recall what they know about multiplication and division. The teacher then introduces the topic by explaining that today’s lesson will build on their knowledge of multiplication and division to understand multiples, factors, LCM, and HCF, which are important for solving everyday problems.

Pupils’ Activity: Pupils respond to the teacher’s questions and listen attentively to the introduction of the new topic.

Learning Point: Pupils are prepared for the lesson and recall prior knowledge of multiplication and division.

Step 2: Multiples and Common Multiples

Time: 7 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher defines multiples and demonstrates how to find multiples of 2-digit numbers using examples. The teacher then explains common multiples and guides pupils to find common multiples of given numbers using charts or board examples.

Pupils’ Activity: Pupils define multiples, list multiples of numbers, and identify common multiples as guided by the teacher.

Learning Point: Pupils understand and can list multiples and common multiples of numbers.

Step 3: Least Common Multiple (LCM)

Time: 8 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher defines LCM and explains two methods for finding it: listing multiples and prime factorization. The teacher solves examples of finding the LCM of 2-digit numbers using both methods, ensuring clear steps and explanations.

Pupils’ Activity: Pupils listen to explanations, ask questions, and practice finding LCM using the demonstrated methods.

Learning Point: Pupils can find the LCM of 2-digit numbers using listing and prime factorization methods.

Step 4: Factors and Common Factors

Time: 7 minutes

Teaching Skill: Explanation/Demonstration

Teacher’s Activity: The teacher defines factors and demonstrates how to find factors of 2-digit numbers. The teacher then explains common factors and guides pupils to find common factors of given numbers.

Pupils’ Activity: Pupils define factors, list factors of numbers, and identify common factors as guided by the teacher.

Learning Point: Pupils understand and can list factors and common factors of numbers.

Step 5: Highest Common Factor (HCF)

Time: 8 minutes

Teaching Skill: Explanation/Guided Practice

Teacher’s Activity: The teacher defines HCF and explains two methods for finding it: listing factors and prime factorization. The teacher solves examples of finding the HCF of 2-digit numbers using both methods, ensuring clear steps and explanations.

Pupils’ Activity: Pupils listen to explanations, ask questions, and practice finding HCF using the demonstrated methods.

Learning Point: Pupils can find the HCF of 2-digit numbers using listing and prime factorization methods.

Step 6: Quantitative Reasoning Involving LCM and HCF

Time: 5 minutes

Teaching Skill: Problem-Solving/Application

Teacher’s Activity: The teacher explains how to identify whether a word problem requires LCM or HCF. The teacher presents simple word problems and guides pupils through the steps to solve them, emphasizing the reasoning behind choosing LCM or HCF.

Pupils’ Activity: Pupils listen to explanations, identify keywords, and participate in solving quantitative reasoning problems.

Learning Point: Pupils can apply LCM and HCF concepts to solve simple quantitative reasoning 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:

1. Define a multiple and a factor of a number.
2. List the first five multiples of 14.
3. Find the LCM of 15 and 20.
4. Find the HCF of 28 and 42.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes

Teaching Skill: Summarization

Teacher’s Activity: The teacher summarizes the key points of the lesson, reiterating the definitions of multiples, factors, LCM, and HCF, and their importance. The teacher assigns homework for pupils to practice finding LCM and HCF of other 2-digit numbers and solving simple word problems.

Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.

Learning Point: Pupils consolidate their understanding and are given further practice.

Home Task: Solve three new exercises on L.C.M AND H.C.F OF WHOLE NUMBERS. Show each step and check each answer.

Lesson Keywords

• Multiple – The result of multiplying a number by an integer.
• Factor – A number that divides another number exactly without a remainder.
• Common Multiple – A number that is a multiple of two or more given numbers.
• Common Factor – A number that is a factor of two or more given numbers.
• Least Common Multiple (LCM) – The smallest non-zero common multiple of two or more numbers.
• Highest Common Factor (HCF) – The largest common factor of two or more numbers.

Differentiation

For pupils who grasp concepts quickly, the teacher can provide more complex 3-digit number problems for LCM and HCF or more challenging quantitative reasoning questions. For pupils who need more support, the teacher can provide additional guided practice with smaller numbers, use physical manipulatives (e.g., counters for groups), or work in small groups for peer support.

Note for teachers using this lesson plan

Ensure pupils have a solid understanding of prime numbers and prime factorization before introducing the prime factorization methods for LCM and HCF. Encourage pupils to explain their reasoning when solving problems, not just provide answers. Use visual aids like multiplication charts and factor trees to support learning. Encourage group work for problem-solving activities.