Mathematics · Year 5 · Additive and Multiplicative Structures · Autumn Term

Prime Numbers and Composite Numbers

Students will identify prime numbers up to 100 and understand the concept of composite numbers.

• National Curriculum Attainment Targets
• KS2: Mathematics - Multiplication and Division

About This Topic

Year 5 students identify prime numbers up to 100 and distinguish them from composite numbers. A prime number greater than 1 has exactly two factors: 1 and itself. They justify why 2 stands as the only even prime, since all other even numbers greater than 2 are divisible by 2. Students also apply the Sieve of Eratosthenes to mark multiples and reveal primes systematically. Composite numbers, such as 4, 6, 8, and 9, possess more than two factors.

This topic fits within the additive and multiplicative structures unit, linking to multiplication and division standards. It builds factor knowledge essential for prime factorisation in later years. Through key questions, students compare primes and composites, honing justification skills and number sense.

Active learning benefits this topic greatly, as abstract divisibility rules gain clarity through manipulation. When students cross multiples on large grids or sort number cards into prime and composite piles collaboratively, they internalise patterns. Peer debates on edge cases like 1 reinforce reasoning and make concepts stick through shared discovery.

Key Questions

1. Justify why 2 is the only even prime number.
2. Analyze the Sieve of Eratosthenes method for finding prime numbers.
3. Compare prime numbers with composite numbers, providing examples of each.

Learning Objectives

• Identify all prime numbers up to 100, demonstrating understanding of the definition.
• Explain why 2 is the only even prime number, using divisibility rules.
• Compare and contrast prime and composite numbers, providing at least three examples of each with their factors.
• Apply the Sieve of Eratosthenes method to systematically identify prime numbers within a given range.

Before You Start

Multiples and Factors

Why: Students need to understand what multiples and factors are to define and identify prime and composite numbers.

Divisibility Rules

Why: Understanding basic divisibility rules (e.g., for 2, 3, 5) helps students efficiently find factors and classify numbers.

Key Vocabulary

Prime Number	A whole number greater than 1 that has only two factors: 1 and itself. For example, 7 is prime because its only factors are 1 and 7.
Composite Number	A whole number greater than 1 that has more than two factors. For example, 10 is composite because its factors are 1, 2, 5, and 10.
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.
Sieve of Eratosthenes	An ancient algorithm for finding all prime numbers up to a specified integer. It works by iteratively marking as composite the multiples of each prime, starting with the multiples of 2.

Watch Out for These Misconceptions

Common Misconception

1 is a prime number.

What to Teach Instead

1 has only one factor, itself, so it fits neither prime nor composite definitions. Listing factors on arrays during group sorts helps students see this clearly. Peer sharing corrects the idea that more than one factor defines primes.

Common Misconception

All primes are odd numbers.

What to Teach Instead

2 is even yet prime, as its only factors are 1 and 2. Visual sieves where students physically cross evens first reveal this exception. Hands-on marking builds intuition over rote memorisation.

Common Misconception

Composite numbers have no factors other than themselves.

What to Teach Instead

Composites have three or more factors, like 6 (1,2,3,6). Factor pair games with manipulatives let students match divisors, exposing extra pairs beyond primes' single pair.

Active Learning Ideas

Whole Class: Sieve of Eratosthenes Grid

Project a 1-100 number grid on the board. Call students to the front in sequence to cross out multiples of each prime starting from 2. Remaining numbers become primes. Conclude with a class discussion on the process and why 2 is unique.

35 min·Whole Class

Small Groups: Prime Factor Hunt

Provide groups with number cards from 1 to 100. Each group lists factors for 10 numbers, sorts them as prime or composite, and justifies choices on mini-whiteboards. Groups share one example with the class.

25 min·Small Groups

Pairs: Even Prime Debate

Pairs receive statements about even numbers and primes. They debate and draw factor trees or arrays to prove why only 2 qualifies as an even prime. Switch pairs to defend opposing views.

20 min·Pairs

Individual: Prime Sieve Journal

Students create personal 1-100 grids, apply the sieve independently, and note observations like patterns in primes. They write one justification for 2 being prime.

15 min·Individual

Real-World Connections

• Cryptography, the practice of secure communication, relies heavily on prime numbers. Large prime numbers are used to create encryption keys that protect sensitive data in online banking and secure websites.
• Number theory, the study of integers and their properties including primes, is fundamental to computer science algorithms. For instance, prime numbers are used in hashing functions to distribute data evenly across storage locations.

Assessment Ideas

Exit Ticket

Provide students with a list of numbers from 1 to 50. Ask them to circle all prime numbers and underline all composite numbers. Then, ask them to write one sentence explaining why 1 is neither prime nor composite.

Quick Check

Display the number 30 on the board. Ask students to write down all of its factors. Then, have them classify 30 as either prime or composite, justifying their answer with reference to its factors.

Discussion Prompt

Pose the question: 'If you are given a very large number, how can you be sure it is prime?' Facilitate a class discussion where students explain their strategies, comparing the efficiency of trial division versus methods like the Sieve of Eratosthenes.

Frequently Asked Questions

How do I teach the Sieve of Eratosthenes in Year 5?

Start with a large 1-100 grid. Students take turns eliminating multiples of 2, then 3, and so on, up to the square root of 100. Highlight remaining primes. This method teaches systematic sieving and divisibility. Follow with students recreating it on paper to reinforce the process and discuss efficiency over trial division.

What is the difference between prime and composite numbers?

Primes greater than 1 have exactly two distinct factors: 1 and the number itself. Composites have more than two factors. Examples: 7 (prime: 1,7), 12 (composite: 1,2,3,4,6,12). Use factor rainbows or trees to visualise, helping students compare and classify numbers up to 100 accurately.

How can active learning help students understand prime numbers?

Active tasks like collaborative sieving on grids or factor hunts with cards make divisibility tangible. Students physically eliminate multiples, debate classifications, and justify choices in pairs. This builds deeper number sense than worksheets alone, as movement and discussion reveal patterns and correct misconceptions through peer interaction.

Why is 2 the only even prime number?

All even numbers greater than 2 are divisible by 2, giving them at least three factors: 1, 2, and themselves. 2 has only two: 1 and 2. Arrays or counters show 2 as two units with no subgroups, while 4 splits into two 2s. Group justifications solidify this rule.