Sieve of Eratosthenes

mathsbot.com/activities/sieveOfEratosthenes

Using the MathsBot Sieve of Eratosthenes

The MathsBot Sieve of Eratosthenes is an, interactive classroom tool that displays a 1–100 grid. Pupils systematically eliminate the multiples of each prime, leaving only the prime numbers highlighted. It brings a 2,000-year-old algorithm to life in seconds, and makes the pattern of primes genuinely visible.

A little background: who was Eratosthenes?

Eratosthenes was a Greek mathematician who lived around 276–194 BC. Among his many achievements, he accurately estimated the circumference of the Earth. He devised this beautifully simple method for finding prime numbers. Rather than testing each number individually, his "sieve" works by removing what isn't prime, leaving only what is.

How the sieve works

Start with the numbers 1 to 100 in a grid.

1. Cross out 1. It is not prime — a prime must have exactly two factors, and 1 has only one.
2. Circle 2, then cross out all its multiples (4, 6, 8…). These can't be prime — they all have 2 as a factor.
3. Circle 3, then cross out all remaining multiples of 3 (6 is already gone, so 9, 15, 21…).
4. 4 is already crossed out — move on.
5. Circle 5, cross out its remaining multiples (25, 35, 55…).
6. Circle 7, cross out its remaining multiples (49, 77, 91…).
7. You're done. Because √100 = 10, any composite number up to 100 must have a factor of 7 or below. Every number still standing is prime.

What remains are the 25 prime numbers up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Ideas for using the tool in the classroom

Whole-class introduction (5–10 minutes)
Display the MathsBot tool. Walk through the sieve together as a clasd. Ask pupils to predict which numbers will be eliminated before you click. Pause at key moments: "Why has 9 already been crossed out by the time we get to it?" or "Why do we stop checking after 7?"

Paired or independent activity
Give pupils a printed 100-grid alongside the digital tool. Ask them to complete the sieve on paper first, then use MathsBot to check their work. The act of doing it by hand first means the digital version becomes a confirmation and discussion tool, not just a shortcut.

Discussion questions to prompt deeper thinking

• Why is 1 not a prime number? (link to the definition of needing exactly two factors)
• Why is 2 the only even prime?
• Can you spot any patterns in where the primes appear? (Column patterns, gaps between primes, twin primes like 11 & 13 or 41 & 43)
• Why don't we need to check multiples of 11 or higher when working up to 100?
• Which number was eliminated by the most different primes? What does that tell you about it?

Helping pupils remember the primes to 100

Knowing the sieve process is not the same as knowing the primes. Pupils benefit from being able to recall them fluently. Here are some strategies to bridge the gap:

Chunk and anchor
Group the primes into manageable sets and give pupils an anchor fact for each:

• Under 10: 2, 3, 5, 7 - just four to learn
• Teens: 11, 13, 17, 19 - all four odd, none ending in 5
• Twenties and thirties: 23, 29, 31, 37 - notice 25 and 35 are not prime (common errors)
• Forties to seventies: 41, 43, 47, 53, 59, 61, 67 - a dense stretch worth practising
• Seventies to 100: 71, 73, 79, 83, 89, 97

Common misconceptions to address directly

• "51 must be prime — it's odd and doesn't end in 5." No: 51 = 3 × 17. The sieve catches this clearly.
• "All numbers ending in 1, 3, 7 or 9 are prime." Show counterexamples: 21, 27, 33, 49, 51, 57, 63, 77, 91.
• "The bigger the number, the less likely it is to be prime." True in general, but not a reliable test. 97 is prime; 91 is not.

Retrieval practice
After pupils have used the sieve, use low-stakes quizzes to build fluency:

• "Write down all the primes between 50 and 100."
• "I'll give you a number, you tell me if it's prime and explain how you know."
• "Which prime is closest to 60? To 80?"

Returning to these questions across multiple lessons, rather than only at the point of teaching, helps primes move from working memory into long-term recall.

Why this tool is worth using

The sieve does something a list of primes cannot: it shows pupils why these numbers survive. The visual process of elimination makes the definition of a prime feel earned rather than arbitrary. Pupils who work through it often remember not just which numbers are prime, but why, and that conceptual hook is far more durable than rote memorisation alone.

Want to extend the activity? Ask pupils to try extending the sieve to 200 and work out which new primes they need to check — a lovely problem that naturally introduces the idea of √200 ≈ 14, meaning they only need to check multiples of 2, 3, 5, 7, 11 and 13.