Mathematics · Year 7 · The Power of Number · Autumn Term

Factors, Multiples, and Primes

Exploring the concepts of factors, multiples, and prime numbers, including prime factorisation.

• National Curriculum Attainment Targets
• KS3: Mathematics - Number

About This Topic

Factors, multiples, and prime numbers anchor number work in Year 7 mathematics, aligning with KS3 National Curriculum goals for fluency in number operations. Students identify factors as divisors of a number with no remainder, multiples as products from repeated addition or multiplication, and primes as integers greater than 1 divisible only by 1 and themselves. Prime factorisation decomposes composite numbers into unique prime products, using trees or repeated division to reveal structure.

This unit addresses key questions like the role of primes as number system building blocks, methods for highest common factor (HCF) and lowest common multiple (LCM), and constructing factor trees. Comparing listing versus prime methods for HCF and LCM sharpens efficiency, while exploring patterns fosters number sense essential for fractions, ratios, and later algebra.

Active learning excels with this topic through visual and kinesthetic tasks that make abstract ideas concrete. Students sort tiles into factor arrays or race in teams to build factor trees, gaining instant feedback from peers and manipulatives. These approaches build confidence, correct errors on the spot, and turn routine practice into engaging problem-solving.

Key Questions

1. Analyze why prime numbers are considered the building blocks of the number system.
2. Compare the methods for finding the highest common factor and lowest common multiple.
3. Construct a prime factor tree for a given composite number.

Learning Objectives

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

Before You Start

Basic Division and Multiplication

Why: Students need to be fluent with division and multiplication to identify factors and calculate multiples.

Number Properties (Integers)

Why: Understanding what integers are and their basic properties is foundational for discussing divisibility and number types like primes.

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.

Watch Out for These Misconceptions

Common Misconception

1 is a prime number.

What to Teach Instead

Prime numbers have exactly two distinct factors: 1 and themselves; 1 has only one factor. Group factor-listing activities help students count factors visually on arrays, revealing 1's uniqueness through comparison with true primes like 2 or 3.

Common Misconception

Factors and multiples are interchangeable.

What to Teach Instead

Factors divide into the number; multiples result from multiplying it. Sorting tasks with number cards clarify directionality, as pairs manipulate and label examples, building distinct mental models via hands-on trial and peer explanation.

Common Misconception

Prime factor trees always start with the smallest prime.

What to Teach Instead

Any prime factor works if correct; order varies by division choice. Relay races constructing trees from different starts show flexibility, with group verification ensuring accuracy and exposing incomplete branches.

Active Learning Ideas

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.

25 min·Pairs

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.

35 min·Small Groups

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.

30 min·Whole Class

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.

20 min·Individual

Real-World Connections

• Cryptography relies heavily on prime numbers. The security of online transactions and encrypted communications often uses algorithms based on the difficulty of factoring very large numbers into their prime components.
• In music theory, prime numbers and their relationships can be found in harmonic intervals and rhythmic patterns, influencing the structure and composition of musical pieces.
• Computer science uses prime factorization in algorithms for tasks like generating random numbers or in certain data compression techniques, where efficient decomposition of numbers is beneficial.

Assessment Ideas

Exit Ticket

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.

Quick Check

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.

Discussion Prompt

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.

Frequently Asked Questions

Why are prime numbers building blocks of the number system?

Every integer greater than 1 factors uniquely into primes, per the Fundamental Theorem of Arithmetic. This underpins operations like simplifying fractions or finding LCM. In Year 7, students grasp this by decomposing numbers into primes repeatedly, seeing how primes generate all composites, much like letters form words, building deep number understanding for advanced topics.

How to teach HCF and LCM using prime factors?

List prime factors of each number, take lowest powers for HCF and highest for LCM. Demonstrate with factor trees side-by-side, then have students practise on paired numbers. Visual overlays of Venn diagrams with primes reinforce the method over exhaustive listing, saving time and reducing errors in multi-digit cases.

How can active learning help students master factors, multiples, and primes?

Active methods like tile arrays for factors or card games for multiples make concepts tangible, countering rote memorisation pitfalls. Collaborative races for prime hunts build fluency through competition and discussion, while manipulatives provide instant error correction. These engage kinesthetic learners, boost retention via peer teaching, and connect abstract rules to real patterns students discover themselves.

What are common errors in prime factorisation?

Students often stop at composites or repeat factors incorrectly. Address by modelling repeated division by smallest primes first, then verifying products match originals. Interactive tree-building in pairs catches gaps early, as partners challenge steps, fostering precision and confidence in this core skill.