Mathematics · Year 7

Active learning ideas

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.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

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

Activity 01

• 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.

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.

What to look for

Provide students with the number 36. Ask them to: 1. List all its factors. 2. List its first five multiples. 3. Determine if 36 is prime or composite and explain why. 4. Write its prime factorization.

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

Activity 02

• 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.

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.

What to look for

Display two numbers, for example, 18 and 24. Ask students to find the HCF and LCM using two different methods (e.g., listing and prime factorization). Have them write down their chosen methods and the results.

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

Activity 03

• 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.

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.

What to look for

Pose the question: 'Why are prime numbers considered the building blocks of all whole numbers?' Facilitate a class discussion where students share their ideas, referencing the concept of unique prime factorization.

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

Activity 04

• 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.

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.

What to look for

Provide students with the number 36. Ask them to: 1. List all its factors. 2. List its first five multiples. 3. Determine if 36 is prime or composite and explain why. 4. Write its prime factorization.

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

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

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Methods used in this brief

• Problem-Based Learning