Exploring Prime and Composite Numbers

Active learning helps students grasp prime and composite numbers because it turns abstract properties into concrete investigations. By physically manipulating numbers through sieves, factor pairs, and discussions, students move beyond memorisation to see patterns and relationships among them.

• Year 6
• Mathematics
• 3 activities
• 15 min–45 min

Learning Objectives

1. Classify whole numbers up to 100 as either prime or composite, providing justification based on factor pairs.
2. Calculate the prime factorization of composite numbers using factor trees.
3. Explain the unique properties of the number 1 and the number 2 in relation to prime and composite classifications.
4. Compare and contrast the characteristics of prime and composite numbers using divisibility rules.
5. Demonstrate the concept of prime numbers as building blocks for other whole numbers through examples.

Inquiry Circle

⏱ 40 min·Small Groups

Inquiry Circle: The Sieve of Eratosthenes

In small groups, students use a large 1-100 grid to systematically cross out multiples of prime numbers. They discuss why certain numbers remain and identify the patterns that emerge, such as the 'diagonal' nature of multiples of three.

Prepare & details

Why is the number 1 neither prime nor composite?

Facilitation Tip: During The Sieve of Eratosthenes, move between groups to prompt students to explain why they are crossing out certain numbers, reinforcing divisibility language.

Setup: Groups at tables with access to source materials

Materials: Source material collection, Inquiry cycle worksheet, Question generation protocol, Findings presentation template

• Analyze
• Evaluate
• Create
• Self-Management
• Self-Awareness

Think-Pair-Share

⏱ 15 min·Pairs

Think-Pair-Share: The Case of Number One

Students individually reflect on why the number 1 does not fit the definition of prime or composite. They then pair up to refine their argument before sharing their logical reasoning with the whole class.

Prepare & details

How can prime factorization help us find the greatest common factor of two numbers?

Facilitation Tip: In The Case of Number One, listen carefully to students’ arguments before revealing the definition, allowing their misconceptions to surface naturally.

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

⏱ 45 min·Small Groups

Stations Rotation: Factor Pair Challenge

Students rotate through stations using different manipulatives like MAB blocks or tiles to create all possible rectangular arrays for a given number. They record which numbers only have one possible array (primes) and which have multiple (composites).

Prepare & details

In what ways do prime numbers act as the building blocks for all other whole numbers?

Facilitation Tip: For the Factor Pair Challenge, provide mini whiteboards so students can quickly test and revise their pairs without erasing mistakes.

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

Teaching This Topic

Teachers approach this topic by balancing exploration with structure. Start with hands-on tools like sieves and arrays to build visual understanding, then formalise definitions with clear examples and counterexamples. Avoid rushing to the textbook—instead, let students articulate rules in their own words before introducing the precise terminology.

What to Expect

By the end of these activities, students confidently identify prime and composite numbers, justify their choices with factor pairs, and explain why 1 is neither. They use mathematical language precisely and support their reasoning in both written and spoken forms.

Watch Out for These Misconceptions

Common Misconception

During Station Rotation: Factor Pair Challenge, watch for students labeling numbers like 9 and 15 as prime because they are odd.

What to Teach Instead

Have students list all factor pairs for these numbers on their whiteboards and circle the pairs. Ask them to compare with a partner whether these numbers fit the rule of having exactly two distinct factors.

Common Misconception

During Think-Pair-Share: The Case of Number One, listen for students arguing that 1 is prime because it has only one factor.

What to Teach Instead

Ask them to model factor pairs for 1 and compare it to the definition of a prime number. Use a T-chart to show that 1 has only one factor pair (1,1), which does not meet the requirement of two distinct factors.

Assessment Ideas

Quick Check

After Station Rotation: Factor Pair Challenge, present students with a list of numbers (e.g., 18, 29, 45, 53). Ask them to write 'P' or 'C' and for two numbers, list their factor pairs to justify their choice.

Exit Ticket

During Think-Pair-Share: The Case of Number One, give each student a card with a composite number (e.g., 30). Ask them to create a factor tree and on the back, write one sentence explaining why 1 is not prime or composite.

Discussion Prompt

After Collaborative Investigation: The Sieve of Eratosthenes, pose the question: 'If prime numbers are the building blocks of all whole numbers, what does that mean for composite numbers?' Facilitate a class discussion where students use examples of prime factorization to support their reasoning.

Extensions & Scaffolding

• Challenge: Give students a prime number larger than 100 and ask them to prove it is prime using a method of their choice (e.g., trial division, Sieve extension).
• Scaffolding: Provide a number line with multiples highlighted to help students identify composite numbers during the Factor Pair Challenge.
• Deeper: Ask students to research and present on one real-world application of prime numbers (e.g., cryptography, nature).

Key Vocabulary

Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7.
Composite Number	A whole number greater than 1 that has more than two factors. For example, 12 is composite because its factors are 1, 2, 3, 4, 6, and 12.
Factor Pair	Two whole numbers that multiply together to equal a given number. For example, the factor pairs of 10 are (1, 10) and (2, 5).
Prime Factorization	Expressing a composite number as a product of its prime factors. For example, the prime factorization of 12 is 2 x 2 x 3.
Divisibility Rule	A shortcut to determine if a number can be divided evenly by another number without performing the division. For example, a number is divisible by 2 if its last digit is even.