Prime Factorisation

Active learning turns prime factorisation from abstract symbols into tangible, visual work. Students build factor trees and division ladders with their hands, seeing how primes multiply to recreate the original number. This hands-on process builds confidence and cements the idea that every composite number has one unique set of prime factors.

• Year 7
• Mathematics
• 4 activities
• 20 min–35 min

Learning Objectives

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

Stations Rotation

⏱ 25 min·Pairs

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.

Prepare & details

Explain why prime factorization is unique for every composite number.

Facilitation Tip: During the Factor Tree Relay, circulate and ask pairs to explain why their tree branches still produce the same primes even when starting with different factors.

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

Stations Rotation

⏱ 35 min·Small Groups

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.

Prepare & details

Construct a factor tree for a given composite number.

Facilitation Tip: In the Division Ladder Tournament, insist groups record each step visibly so rivals can verify results and spot division errors quickly.

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

Stations Rotation

⏱ 30 min·Whole Class

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.

Prepare & details

Compare different methods for finding the prime factorization of a number.

Facilitation Tip: For the Prime Factor Card Sort, set a time limit to raise urgency and prevent students from overanalyzing each card before grouping.

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

Stations Rotation

⏱ 20 min·Individual

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.

Prepare & details

Explain why prime factorization is unique for every composite number.

Facilitation Tip: In the Mystery Number Challenge, encourage students to record both the factor tree and division ladder for each clue to build dual-method fluency.

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 should model multiple paths for the same number, showing that the order of factors does not change the product. Avoid rushing to the final prime list; instead, ask students to predict the next branch and explain their reasoning. Research shows that students who verbalize their factor choices during tree building develop stronger conceptual understanding than those who only write the final list.

What to Expect

By the end of these activities, students can decompose any composite number into primes, explain why the result is unique, and spot common errors in factor trees. They will also justify their steps and correct peers’ mistakes during discussions.

Watch Out for These Misconceptions

Common Misconception

During Factor Tree Relay, watch for students who insist their tree must begin with 2 for even numbers or 3 for multiples of 3.

What to Teach Instead

During the Factor Tree Relay, remind students that any factor pair is valid. Ask them to rebuild a tree starting with a different branch and compare the final prime lists to see they match.

Common Misconception

During Factor Tree Relay, watch for students who include 1 as a prime factor on their branches.

What to Teach Instead

During the Factor Tree Relay, have students multiply their final primes to reconstruct the original number. If they include 1, the product will be too large, prompting them to remove it and rebuild correctly.

Common Misconception

During Prime Factor Card Sort, watch for groups who believe different factor trees can produce different prime factorisations for the same number.

What to Teach Instead

During the Prime Factor Card Sort, ask groups to swap their sorted trees and verify the prime sets match. If discrepancies appear, have them rebuild together until the sets align.

Assessment Ideas

Quick Check

After Factor Tree Relay and Division Ladder Tournament, present students with 84 and ask them to show two complete paths to its prime factorization. Collect work to check accuracy of primes and exponents.

Exit Ticket

During Prime Factor Card Sort, give each student a card with 100 and ask them to write its prime factorization on the back. Then ask: ‘Explain in one sentence why the prime factorization of 100 will always be the same.’

Discussion Prompt

During the Mystery Number Challenge wrap-up, pose to small groups: ‘You overheard a student say, “Prime factorisation is just random splitting.” How would you explain the rules and purpose to them using today’s activities?’ Have groups share their explanations.

Extensions & Scaffolding

• Challenge: Ask students to find a number whose prime factorization includes exactly three distinct primes and write a mini-proof that no smaller number meets this criterion.
• Scaffolding: Provide partially completed factor trees or division ladders with missing steps for students to fill in.
• Deeper exploration: Introduce the concept of abundant, deficient, and perfect numbers using prime factorisation to classify numbers up to 100.

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.