Mastering Mathematical Thinking: 4th Class · 4th Class · Number Systems and Place Value · Autumn Term

Properties of Integers: Factors, Multiples, Primes

Investigating properties of integers including factors, multiples, prime numbers, composite numbers, and prime factorisation.

• NCCA Curriculum Specifications
• NCCA: Junior Cycle - Number - N.3
• NCCA: Junior Cycle - Number - N.4

About This Topic

Properties of integers focus on factors, multiples, prime numbers, composite numbers, and prime factorisation. Students explore factors as pairs of numbers that multiply to give the original integer, such as 1 and 12 or 2 and 6 for 12. Multiples arise from skip-counting, like 12, 24, 36 for multiples of 12. Prime numbers have exactly two distinct factors, 1 and themselves, while composites have more. Prime factorisation breaks numbers into prime building blocks, for example, 60 = 2 × 2 × 3 × 5.

This topic aligns with NCCA Junior Cycle Number standards N.3 and N.4, building skills in number decomposition essential for greatest common divisor (GCD) and least common multiple (LCM). Students learn to use factor trees for GCD, like GCD(12,18)=6, and listing multiples for LCM(12,18)=36. Key questions guide differentiation of primes and composites, prime factorisation applications, and primality tests for larger numbers through divisibility rules up to square roots.

Active learning suits this topic because students manipulate concrete tools like number tiles or arrays to visualise factors and multiples. Collaborative games reveal patterns in primes, turning abstract rules into shared discoveries that strengthen retention and problem-solving confidence.

Key Questions

1. Differentiate between prime and composite numbers, providing examples of each.
2. Explain how prime factorisation can be used to find the greatest common divisor (GCD) and least common multiple (LCM).
3. Construct a method for determining if a large number is prime.

Learning Objectives

• Classify integers as prime or composite, providing justification for each classification.
• Calculate the prime factorisation of any given integer up to 100.
• Compare and contrast the methods for finding the greatest common divisor (GCD) and least common multiple (LCM) of two integers.
• Develop and apply a systematic method to determine if a given integer is prime.

Before You Start

Introduction to Multiplication and Division

Why: Students need a solid understanding of multiplication and division to grasp the concepts of factors and multiples.

Number Recognition and Counting

Why: Basic number sense is required to identify and work with integers, particularly to understand the definition of prime numbers (greater than 1).

Key Vocabulary

Factor	A number that divides exactly into another number without a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple	A number that can be divided by another number without a remainder. Multiples of a number are found by skip-counting or multiplying it by integers.
Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. Examples include 2, 3, 5, and 7.
Composite Number	A whole number greater than 1 that has more than two factors. Examples include 4, 6, 8, 9, and 10.
Prime Factorisation	Breaking down a composite number into a product of its prime factors. For example, the prime factorisation of 12 is 2 × 2 × 3.

Watch Out for These Misconceptions

Common Misconception

1 is a prime number.

What to Teach Instead

1 has only one factor, itself, so it fits neither prime nor composite definitions. Sorting activities with factor tiles help students count distinct factors visually, clarifying through group tallying that primes need exactly two.

Common Misconception

All odd numbers greater than 2 are prime.

What to Teach Instead

Odds like 9 or 15 have factors beyond 1 and themselves. Divisibility games with counters reveal these composites, as students test multiples of 3 or 5 collaboratively, adjusting mental lists.

Common Misconception

Prime factorisation is just repeated division by 2.

What to Teach Instead

It requires all primes in order. Factor tree races encourage systematic branching with multiple primes, where peers check completeness, building accurate decomposition habits.

Active Learning Ideas

Simulation Game: Factor Bingo

Prepare bingo cards with numbers 1-100. Call out factors, and students mark multiples on their cards. First to complete a line shouts 'Factors!'. Discuss winning cards to identify prime and composite numbers. Extend by having students create their own cards.

30 min·Small Groups

Hands-On: Sieve of Eratosthenes

Print a 1-100 number grid. Students circle multiples of 2, then 3, crossing out composites. Remaining primes spark discussion on patterns. Pairs test larger numbers using divisibility checks.

45 min·Pairs

Stations Rotation: Factor Trees

Set up stations with dice to generate numbers, tree templates, and coloured markers for primes. Students build factor trees, then swap to find GCD or LCM of pairs. Rotate every 10 minutes with peer feedback.

40 min·Small Groups

Relay: Multiple Chains

Teams line up. First student writes a number's first multiple, next adds the next, racing to 100. Correct chains earn points. Debrief on LCM connections with factorisation.

25 min·Small Groups

Real-World Connections

• Cryptographers use prime numbers to create secure encryption keys for online transactions and digital communication. The difficulty in factoring large prime numbers is the basis of many security systems.
• In music theory, understanding factors and multiples can help composers analyze rhythmic patterns and harmonic structures. For instance, finding common multiples can reveal how different melodic lines might align.

Assessment Ideas

Quick Check

Present students with a list of numbers (e.g., 15, 17, 21, 23). Ask them to circle the prime numbers and underline the composite numbers. Then, have them write the factors for one composite number from the list.

Exit Ticket

Give each student a number (e.g., 36). Ask them to complete two tasks: 1. Write the prime factorisation of the number. 2. Explain in one sentence whether the number is prime or composite and why.

Discussion Prompt

Pose the question: 'If you wanted to find the greatest common divisor of 24 and 30, what are two different methods you could use?' Facilitate a class discussion comparing the factor listing method and the prime factorisation method.

Frequently Asked Questions

How do you explain prime factorisation to 4th class?

Use factor trees: start with the number at the top, split into factors, continue until primes remain. Model with 36: 36=6×6, then 6=2×3. Students practise with everyday numbers like 48 or 72, drawing trees on mini-whiteboards. Link to real use in simplifying fractions later.

What activities teach GCD and LCM using factors?

Venn diagrams for factors: overlap shows GCD, union for LCM precursors. Students list factors of pairs like 12 and 18, shade common ones. Games with cards reinforce prime factorisation as the shortcut, comparing methods for efficiency.

How can active learning help students master primes and factors?

Active tasks like factor hunts with manipulatives or prime sieves make abstract properties tangible. Small group discussions during rotations correct errors in real time, while relays build speed in recognition. These approaches boost engagement, reveal patterns through play, and improve retention over rote memorisation.

How to test if a large number is prime in 4th class?

Teach divisibility rules: check 2,3,5 first, then odds up to the square root. For 49, test up to 7 (√49=7), finding 7×7. Partner challenges with stopwatches gamify checks, emphasising no divisors other than 1 and itself confirms primality.