Properties of Integers: Factors, Multiples, Primes

Active learning helps students grasp factors, multiples, and primes by letting them manipulate objects and see patterns firsthand. These topics are abstract when taught through rules alone, so games and hands-on tasks make relationships visible and memorable for all learners.

• 4th Class
• Mastering Mathematical Thinking: 4th Class
• 4 activities
• 25 min–45 min

Learning Objectives

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

Simulation Game

⏱ 30 min·Small Groups

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.

Prepare & details

Differentiate between prime and composite numbers, providing examples of each.

Facilitation Tip: During Factor Bingo, circulate to listen for students who confuse factors with multiples and redirect them with a quick example like 'Is 15 a factor of 5 or a multiple of 5?' to clarify.

Setup: Flexible space for group stations

Materials: Role cards with goals/resources, Game currency or tokens, Round tracker

• Apply
• Analyze
• Evaluate
• Create
• Social Awareness
• Decision-Making

Think-Pair-Share

⏱ 45 min·Pairs

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.

Prepare & details

Explain how prime factorisation can be used to find the greatest common divisor (GCD) and least common multiple (LCM).

Facilitation Tip: When running the Sieve of Eratosthenes, model how to cross out multiples starting from 2, then 3, and ask students to explain why 4 is already crossed out to reinforce the process.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Stations Rotation

⏱ 40 min·Small Groups

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.

Prepare & details

Construct a method for determining if a large number is prime.

Facilitation Tip: In Factor Tree Station Rotation, provide colored pencils so students can trace each branch of the tree and spot missed factors more easily.

Setup: Tables/desks arranged in 4-6 distinct stations around room

Materials: Station instruction cards, Different materials per station, Rotation timer

• Remember
• Understand
• Apply
• Analyze
• Self-Management
• Relationship Skills

Think-Pair-Share

⏱ 25 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.

Prepare & details

Differentiate between prime and composite numbers, providing examples of each.

Facilitation Tip: For the Multiple Chains Relay, set a timer and call on pairs to explain their chain before moving to the next number to encourage precision over speed.

Setup: Standard classroom seating; students turn to a neighbor

Materials: Discussion prompt (projected or printed), Optional: recording sheet for pairs

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach these topics by starting concrete and moving toward abstract. Use physical objects like counters or tiles to build arrays for factors, then transition to skip-counting for multiples. Avoid rushing to formal definitions; let students discover rules through repeated exposure in varied contexts. Research shows that students who construct their own understanding of primes through systematic elimination (like the Sieve) retain concepts longer than those who memorize lists.

What to Expect

Students will confidently identify factors, multiples, primes, and composites, and explain their reasoning using correct terminology. They will use prime factorisation to break down numbers systematically and justify their classifications with evidence from their work.

Watch Out for These Misconceptions

Common Misconception

During Factor Bingo, watch for students who mark 1 as prime.

What to Teach Instead

Pause the game and ask the group to list the factors of 1 using their bingo cards, then count aloud to show it has only one distinct factor, so it belongs in a separate category.

Common Misconception

During the Sieve of Eratosthenes, watch for students who skip odd numbers beyond 3.

What to Teach Instead

Have them test 9 and 15 with counters, counting groups of 3 and 5 to reveal their composite nature, then resume the sieve with odd primes.

Common Misconception

During Factor Trees Station Rotation, watch for students who stop after dividing by 2 repeatedly.

What to Teach Instead

Prompt pairs to check if the result is prime; if not, they must branch again, using their colored pencils to track each new prime factor until only primes remain.

Assessment Ideas

Quick Check

After Factor Bingo, present students with a list of numbers (e.g., 15, 17, 21, 23). Ask them to circle primes and underline composites, then write the factors for one composite number using their bingo cards as reference.

Exit Ticket

During Factor Trees Station Rotation, give each student a number (e.g., 36). Ask them to complete two tasks: 1. Write the prime factorisation of the number on their exit ticket. 2. Explain in one sentence whether the number is prime or composite and why, using evidence from their tree.

Discussion Prompt

After Multiple Chains Relay, 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 (from Factor Bingo) and the prime factorisation method (from Factor Trees).

Extensions & Scaffolding

• Challenge early finishers to create a number riddle using only prime factors, such as 'I am a 3-digit number with prime factors 2, 3, and 5. Who am I?'
• For students who struggle, provide a partially completed factor tree on their sheet with one branch missing and ask them to identify the missing prime factor.
• Allow extra time for students to design their own composite number and write a step-by-step guide for a peer to find its prime factorisation.

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.