Mathematics · Year 7

Active learning ideas

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.

• ACARA Content Descriptions
• AC9M7N01

• 20–35 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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.

What to look for

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.

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

Activity 02

• 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.

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.

What to look for

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.'

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

Activity 03

• 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.

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.

What to look for

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.

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

Activity 04

• 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.

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.

What to look for

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.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

  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.

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Methods used in this brief

• Stations Rotation