Factors, Multiples, and Primes

Active learning transforms abstract concepts like factors, multiples, and primes into concrete understanding through movement and collaboration. These hands-on activities build fluency that textbooks alone cannot, turning number theory into something students can see, manipulate, and discuss.

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

Learning Objectives

1. Calculate the prime factorization of composite numbers using factor trees.
2. Compare and contrast methods for finding the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of two or more numbers.
3. Explain the significance of prime numbers as the fundamental building blocks of integers.
4. Identify all factors and multiples for a given integer up to 100.
5. Classify numbers as prime or composite based on their divisibility.

Problem-Based Learning

⏱ 25 min·Pairs

Pairs: Factor Pairs Race

Pairs receive numbers from 12 to 60 and list all factor pairs on mini-whiteboards within 2 minutes per number. They swap boards to verify and discuss complete lists, noting square numbers where factors pair equally. Conclude with sharing patterns observed across the class.

Prepare & details

Analyze why prime numbers are considered the building blocks of the number system.

Facilitation Tip: During Factor Pairs Race, circulate and note pairs who skip numbers or miscount, providing immediate feedback with counters or grids.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 35 min·Small Groups

Small Groups: Prime Tower Challenge

Groups draw composite numbers up to 100 and collaboratively build prime factor trees using linking cubes or paper strips for each prime. They verify by multiplying back to the original number and present one tower to the class. Extend to finding HCF of paired numbers.

Prepare & details

Compare the methods for finding the highest common factor and lowest common multiple.

Facilitation Tip: For the Prime Tower Challenge, assign roles so each group member contributes to the physical construction and recording of prime towers.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 30 min·Whole Class

Whole Class: HCF and LCM Sorting Game

Distribute cards with pairs of numbers; students stand and sort them into zones for calculating HCF or LCM using prime factors. Discuss methods as a class, then vote on most efficient strategies. Reinforce with a quick quiz on selected pairs.

Prepare & details

Construct a prime factor tree for a given composite number.

Facilitation Tip: In the HCF and LCM Sorting Game, limit time per round and rotate cards so students experience varied number pairs and methods.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Problem-Based Learning

⏱ 20 min·Individual

Individual: Prime Hunt Puzzle

Students receive a grid of numbers 1-100 and circle primes individually, then pair up to justify choices and cross-check. Compile class list and test with divisibility rules. Use to introduce prime factorisation for composites.

Prepare & details

Analyze why prime numbers are considered the building blocks of the number system.

Facilitation Tip: When students complete the Prime Hunt Puzzle, ask them to explain their path through the puzzle to uncover misconceptions about prime structure.

Setup: Groups at tables with access to research materials

Materials: Problem scenario document, KWL chart or inquiry framework, Resource library, Solution presentation template

• Analyze
• Evaluate
• Create
• Decision-Making
• Self-Management
• Relationship Skills

Teaching This Topic

Start with concrete manipulatives like counters or grid arrays to build visual models of factors and multiples. Move students toward abstract reasoning by connecting these models to symbolic notation and prime factorization. Avoid rushing to algorithms; instead, encourage students to verbalize their process, as explaining often reveals gaps in understanding. Research shows that students who articulate their thinking develop stronger conceptual foundations than those who memorize steps.

What to Expect

Students will fluently identify factors, multiples, and primes, justify their reasoning using correct terminology, and apply prime factorization to solve problems. Mastery shows in clear explanations, accurate calculations, and the ability to connect these concepts to real-world contexts like scheduling or grouping.

Watch Out for These Misconceptions

Common Misconception

During Factor Pairs Race, watch for students who list only one pair or include 1 as a prime factor.

What to Teach Instead

Have students draw arrays for their numbers, using 1xN and Nx1 as the first pairs to highlight why 1 is not prime and why all factors must be paired.

Common Misconception

During HCF and LCM Sorting Game, watch for students who confuse which number is the divisor (factor) and which is the product (multiple).

What to Teach Instead

Ask students to physically place number cards on a table labeled 'divides into' and 'is divided by,' forcing them to articulate directionality with each card.

Common Misconception

During Prime Tower Challenge, watch for groups that force the smallest prime at the base of their tower, assuming order matters.

What to Teach Instead

Challenge groups to build their tower starting with a different prime, then compare results to show that prime factorization is unique regardless of order.

Assessment Ideas

Exit Ticket

After Factor Pairs Race, provide the number 48. Ask students to: 1. List all factor pairs. 2. State if 48 is prime or composite and explain. 3. Write its prime factorization using a tree or division method.

Quick Check

After HCF and LCM Sorting Game, display the numbers 20 and 30. Ask students to find the HCF and LCM using one method from the game and one alternative method, recording both processes and answers.

Discussion Prompt

During Prime Hunt Puzzle, pose the question: 'Why can’t 1 be a prime number?' Have students discuss in pairs, then share with the class, referencing their puzzle findings and factor lists.

Extensions & Scaffolding

• Challenge: Ask students to create a 3D model of prime factorization using linking cubes, showing how composite numbers break down into primes.
• Scaffolding: Provide partially completed prime factor trees or factor pair lists for students to finish, focusing on one step at a time.
• Deeper exploration: Explore twin primes or Goldbach’s conjecture, connecting prime pairs to historical mathematical problems.

Key Vocabulary

Factor	A number that divides exactly into another number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Multiple	A number that can be divided by another number without a remainder; it is the product of a given number and an integer. For example, multiples of 5 are 5, 10, 15, 20, and so on.
Prime Number	A whole number greater than 1 that has only two divisors: 1 and itself. Examples include 2, 3, 5, 7, and 11.
Composite Number	A whole number greater than 1 that has more than two divisors. For example, 4 has divisors 1, 2, and 4; 6 has divisors 1, 2, 3, and 6.
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.