Mathematics · Year 6 · The Power of Number Systems · Term 1

Exploring Prime and Composite Numbers

Identifying and categorizing numbers based on their factor pairs and divisibility rules.

• ACARA Content Descriptions
• AC9M6N01

About This Topic

This topic focuses on the fundamental properties of whole numbers, specifically identifying prime and composite numbers. Students learn to use factor pairs and divisibility rules to categorise numbers, building a foundation for more complex algebraic concepts. In the Australian Curriculum, this aligns with AC9M6N01, where Year 6 students develop a deep understanding of the multiplicative nature of numbers. Understanding prime numbers is essential for simplifying fractions and finding common multiples later in the year.

Beyond simple identification, students explore why certain numbers, like 2, are unique (the only even prime) and why 1 is excluded from these categories. This unit encourages students to see numbers as products of their prime factors, often referred to as the 'atoms' of mathematics. This topic comes alive when students can physically model the patterns through collaborative investigations and visual factor trees.

Key Questions

1. Why is the number 1 neither prime nor composite?
2. How can prime factorization help us find the greatest common factor of two numbers?
3. In what ways do prime numbers act as the building blocks for all other whole numbers?

Learning Objectives

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

Before You Start

Multiplication and Division Facts

Why: Students need to be fluent with multiplication and division to identify factor pairs and test for divisibility.

Identifying Factors of Numbers

Why: Understanding how to find all the numbers that divide evenly into another number is the direct precursor to classifying numbers as prime or composite.

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.

Watch Out for These Misconceptions

Common Misconception

All odd numbers are prime.

What to Teach Instead

Students often confuse 'odd' with 'prime' because many early primes are odd. Use a gallery walk of numbers like 9, 15, and 21 to show they have factors other than 1 and themselves, helping students see the difference through peer discussion.

Common Misconception

The number 1 is a prime number.

What to Teach Instead

Students think 1 is prime because it only has one factor. Teach that a prime must have exactly two distinct factors; hands-on modeling of factor pairs helps students see that 1 is a unique case.

Active Learning Ideas

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.

40 min·Small Groups

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.

15 min·Pairs

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).

45 min·Small Groups

Real-World Connections

• Cryptography, the science of secure communication, relies heavily on prime numbers. The security of online transactions and encrypted messages often depends on the difficulty of factoring very large composite numbers into their prime components.
• Computer scientists use prime factorization in algorithms for data compression and error detection. Understanding how numbers are built from primes helps in designing efficient ways to store and transmit information.

Assessment Ideas

Quick Check

Present students with a list of numbers (e.g., 15, 23, 36, 41, 50). Ask them to write 'P' next to prime numbers and 'C' next to composite numbers. For two of their choices, ask them to write down the factor pairs that justify their classification.

Exit Ticket

Give each student a card with a composite number (e.g., 24). Ask them to create a factor tree to show its prime factorization. On the back, they should write one sentence explaining why the number 1 is not considered prime or composite.

Discussion Prompt

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 share their ideas, using examples of prime factorization to support their reasoning.

Frequently Asked Questions

How can active learning help students understand prime numbers?

Active learning moves students away from rote memorisation of lists. By physically building arrays or participating in a 'human number line' where they step forward based on their factors, students internalise the properties of numbers. These social and tactile experiences make the abstract definitions of 'prime' and 'composite' concrete and memorable.

Why is prime factorisation important in Year 6?

It serves as a universal tool for simplifying fractions and finding the Lowest Common Multiple. It transitions students from additive thinking to multiplicative thinking, which is vital for high school algebra.

What are the best divisibility rules to teach first?

Start with 2, 5, and 10 as they are visually obvious. Then move to 3 and 9, which require digit summing, providing a great opportunity for students to investigate why these patterns work.

How do I explain that 2 is the only even prime?

Use a visual array. Show that any other even number can be split into two equal rows (multiples of 2), meaning they must have at least three factors: 1, 2, and the number itself.