Mathematics · Year 7 · The Language of Number · Term 1

Prime Factorisation

Students will use factor trees and division to find the prime factorization of composite numbers.

• ACARA Content Descriptions
• AC9M7N01

About This Topic

Prime factorisation expresses composite numbers as products of prime numbers, a unique representation for each number greater than 1. Year 7 students learn this by drawing factor trees, starting with any factor pair and dividing until only primes remain, or using repeated division by primes. They explain the uniqueness: regardless of path, the primes and exponents stay the same, forming the number's signature.

This topic anchors the Language of Number unit under AC9M7N01, building fluency in number properties essential for highest common factors, lowest common multiples, and algebraic simplification later. Comparing factor trees and division methods sharpens both strategies and deepens insight into number structure.

Active learning suits prime factorisation perfectly. Collaborative races with timers or peer verification of trees encourage debate over choices, while manipulatives like linking cubes represent divisions visually. These approaches make the abstract process concrete, boost retention through movement and talk, and reveal misunderstandings early.

Key Questions

1. Explain why prime factorization is unique for every composite number.
2. Construct a factor tree for a given composite number.
3. Compare different methods for finding the prime factorization of a number.

Learning Objectives

• Calculate the prime factorization of composite numbers using factor trees and repeated division.
• Compare the efficiency and outcomes of different prime factorization methods for a given number.
• Explain the fundamental theorem of arithmetic in the context of unique prime factorizations.
• Identify the prime factors and their exponents within a prime factorization.
• Construct factor trees for composite numbers, demonstrating understanding of factor pairs.

Before You Start

Factors and Multiples

Why: Students need to be able to identify all factors of a number and understand the concept of multiples to begin finding factor pairs.

Identifying Prime and Composite Numbers

Why: Students must be able to distinguish between prime and composite numbers to correctly complete the prime factorization process.

Key Vocabulary

prime number	A whole number greater than 1 that has only two divisors: 1 and itself. Examples include 2, 3, 5, and 7.
composite number	A whole number greater than 1 that has more than two divisors. Examples include 4, 6, 8, and 9.
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.
factor tree	A diagram used to find the prime factorization of a composite number by repeatedly breaking it down into its factors until only prime numbers remain.

Watch Out for These Misconceptions

Common Misconception

All factor trees must start by dividing by 2.

What to Teach Instead

Trees can begin with any factor pair, such as 3x something for multiples of 3. Group discussions during tree-building activities let students test paths and see all lead to the same primes, building flexibility.

Common Misconception

1 counts as a prime factor in trees.

What to Teach Instead

1 is neither prime nor composite and does not appear in factorisation. Hands-on multiplication checks in pairs reveal including 1 disrupts the unique prime product, helping students self-correct.

Common Misconception

Prime factorisation is not unique for a number.

What to Teach Instead

The multiset of primes is always identical, though order varies. Collaborative verification races expose this, as groups compare trees and confirm products match the original.

Active Learning Ideas

Pairs: Factor Tree Relay

Pair students; one student draws a factor tree for a number between 48 and 96 while the partner times them and checks by multiplying primes back to the original. Switch roles for three numbers. Discuss why different branches yield the same primes.

25 min·Pairs

Small Groups: Division Ladder Tournament

Provide composite numbers; groups build division ladders by dividing by smallest primes first, racing against others. Compare ladders at end, verifying products. Extend by finding HCF of two numbers using ladders.

35 min·Small Groups

Whole Class: Prime Factor Card Sort

Distribute cards with numbers, primes, and factor sets. Class sorts into matches, justifying with quick trees on whiteboards. Reveal and correct as group, noting unique factor sets.

30 min·Whole Class

Individual: Mystery Number Challenge

Give clues like 'product of primes is 2x2x3x5'; students build trees or lists to guess numbers. Share and verify by multiplying.

20 min·Individual

Real-World Connections

• Cryptographers use prime factorization to secure online communications and financial transactions. The difficulty of factoring very large numbers into their primes is the basis for many encryption algorithms, like RSA.
• Computer scientists use prime factorization in algorithms for tasks such as finding the greatest common divisor (GCD) and least common multiple (LCM) of numbers, which are fundamental operations in many software applications.

Assessment Ideas

Quick Check

Present students with a composite number, such as 48. Ask them to find its prime factorization using two different methods: a factor tree and repeated division. Collect their work to check for accuracy in identifying prime factors and exponents.

Exit Ticket

On an index card, ask students to write the prime factorization of 36. Then, pose the question: 'Explain in one sentence why the prime factorization of 36 will always be the same, no matter how you draw the factor tree.'

Discussion Prompt

Pose the following to small groups: 'Imagine you are explaining prime factorization to someone who has never heard of it. What are the key steps? What is one common mistake they might make, and how can they avoid it?' Have groups share their explanations.

Frequently Asked Questions

How do you teach prime factorisation uniqueness in Year 7?

Stress that every composite number has one set of prime factors, regardless of method. Use factor trees and division side-by-side: show a number like 84 as 2x2x3x7 via tree or ladder. Students multiply back to verify, discussing why paths differ but primes do not. This procedural and conceptual link fits AC9M7N01 perfectly.

What are steps to construct a factor tree?

Select a composite number. Find two factors multiplying to it, circle primes at ends. Repeat on composites until all leaves are prime. For 72: 8x9, then 2x4 and 3x3, yielding 2x2x2x3x3. Practice with varied starts builds confidence and reveals the unique prime product.

Common mistakes in prime factorisation for Australian Curriculum Year 7?

Errors include listing 1 as prime, incomplete trees with composites left, or claiming non-unique factors. Division by non-primes or ignoring exponents also occur. Address via peer review: students swap trees, multiply to check, and explain fixes, aligning with AC9M7N01's emphasis on reasoning.

How can active learning help with prime factorisation?

Active strategies like pair relays and group tournaments make factor trees dynamic. Students physically draw, debate branches, and race divisions, turning solo practice into social proofing. Manipulatives model decompositions; whole-class sorts reinforce uniqueness. These methods engage kinesthetic learners, cut errors through talk, and embed concepts deeply for AC9M7N01 mastery.