45-minute lesson
Primary 6 Mathematics: Prime, composite and square numbers
Lesson objective
- Find all the factors of a whole number.
- Identify prime and composite numbers by looking at their factors.
- Recognise square numbers and connect them to square arrays.
Success criteria:
- I can list all the factors of a number up to 100 in pairs.
- I can explain why a number is prime or composite.
- I can recognise square numbers up to 144 and draw them as square arrays.
Materials
- 36 small counters, buttons or blocks
- Grid paper and coloured pencils
- A 100 chart (written or printed)
- Paper and pencil
Introduction
5 minIntroduce today's words:
- Factor: A whole number that divides exactly into another number. The factors of 6 are 1, 2, 3 and 6.
- Prime number: A whole number greater than 1 that has exactly two factors: 1 and itself.
- Composite number: A whole number greater than 1 that has more than two factors.
- Square number: The result of multiplying a whole number by itself, such as 5 × 5 = 25.
- Array: An arrangement of objects in equal rows and columns.
Ask your child what they already know about prime, composite and square numbers.
Explanation
9 minA factor of a number divides into it exactly, with nothing left over. Factors come in pairs that multiply to make the number. For 12 the factor pairs are 1 × 12, 2 × 6 and 3 × 4, so the factors of 12 are 1, 2, 3, 4, 6 and 12.
Counters show factors clearly. Try to arrange 12 counters in rectangular arrays: 1 row of 12, 2 rows of 6 or 3 rows of 4. Each array gives a factor pair.
Now try 7 counters. The only rectangle you can make is 1 row of 7. Numbers like this, with exactly two factors (1 and the number itself), are called prime numbers. The prime numbers less than 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
A number greater than 1 with more than two factors is called a composite number, because it can be “composed” (made) by multiplying smaller numbers. 12, 15 and 51 are composite.
Two special facts are worth remembering:
- 1 is neither prime nor composite, because it has only one factor (itself).
- 2 is the only even prime number. Every other even number has 2 as a factor, so it has more than two factors.
Some numbers of counters can be arranged into a perfect square array, with the same number of rows and columns. 9 counters make a 3 × 3 square, and 16 make a 4 × 4 square. These are square numbers. We write 4 × 4 as 4² and say “4 squared”. The square numbers up to 144 are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.
Square numbers have an odd number of factors, because one factor pair is a number multiplied by itself. For example, 36 has the factors 1, 2, 3, 4, 6, 9, 12, 18 and 36. The pair 6 × 6 only adds one new factor, so there are 9 factors.
Worked examples
7 minPrime or composite?
Is 51 a prime number or a composite number?
Step 1: Test small factors. 51 is odd, so 2 is not a factor.
Step 2: Check 3: the digits add to 5 + 1 = 6, which is a multiple of 3, so 3 might be a factor.
Step 3: 51 ÷ 3 = 17, so 3 × 17 = 51.
Step 4: 51 has factors 1, 3, 17 and 51, which is more than two factors.
Answer: 51 is composite, because 51 = 3 × 17.
Listing all the factors
List all the factors of 36.
Step 1: Work through factor pairs in order: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6.
Step 2: 5 is not a factor, because 36 ÷ 5 leaves a remainder.
Step 3: After 6 × 6 the pairs would repeat, so we can stop.
Step 4: Write the factors in order, listing 6 only once.
Answer: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors, so 36 is a square number).
Finding square numbers
Which square numbers are between 50 and 100?
Step 1: List squares in order: 7 × 7 = 49, 8 × 8 = 64, 9 × 9 = 81, 10 × 10 = 100.
Step 2: 49 is less than 50, and 100 is not between 50 and 100.
Answer: 64 and 81.
Guided practice (do together)
9 min1. Which of these numbers is prime?
(a) 21 (b) 27 (c) 29 (d) 33
2. Which of these numbers is composite?
(a) 13 (b) 17 (c) 19 (d) 51
3. Which of these is a square number?
(a) 24 (b) 36 (c) 48 (d) 50
4. Which statement about the number 1 is true?
(a) It is prime. (b) It is composite. (c) It is neither prime nor composite. (d) It is both prime and composite.
Independent practice
11 min5. How many factors does 18 have?
6. What is 7 squared (7²)?
7. What is the only even prime number?
8. How many prime numbers are there between 20 and 30?
Questions to check understanding
- Can you list all the factors of a number up to 100 in pairs?
- Can you explain why a number is prime or composite?
- Can you recognise square numbers up to 144 and draw them as square arrays?
- What was the trickiest part today?
Answer guide
1. 29 — 29 has only two factors, 1 and 29. 21 = 3 × 7, 27 = 3 × 9 and 33 = 3 × 11 are composite.
2. 51 — 51 = 3 × 17, so it has more than two factors. 13, 17 and 19 are prime.
3. 36 — 36 = 6 × 6, so 36 counters can be arranged in a 6 × 6 square array.
4. It is neither prime nor composite. — 1 has only one factor, so it does not fit the definition of prime (exactly two factors) or composite (more than two factors).
5. 6 — The factor pairs are 1 × 18, 2 × 9 and 3 × 6, so the factors are 1, 2, 3, 6, 9 and 18: six factors.
6. 49 — 7 squared means 7 × 7 = 49.
7. 2 — 2 has exactly two factors, 1 and 2. Every other even number also has 2 as a factor, so it has more than two factors.
8. 2 — The prime numbers between 20 and 30 are 23 and 29. The others (21, 22, 24, 25, 26, 27, 28) are composite.
Review
4 minRecap the success criteria together. Watch for these common misconceptions:
- Thinking all odd numbers are prime. Show counterexamples such as 9 = 3 × 3, 15 = 3 × 5 and 51 = 3 × 17.
- Believing 1 is a prime number. Revisit the definition: a prime has exactly two different factors, and 1 has only one.
- Confusing “squared” with “doubled”, for example thinking 5² = 10. Build the square array with counters: 5 rows of 5 is 25.
Extension activities
- Use the Sieve of Eratosthenes on a 100 chart: cross out 1, then cross out every multiple of 2 (except 2), of 3 (except 3), of 5 and of 7. The numbers left are the primes up to 100.
- Investigate: can every even number from 4 to 50 be written as the sum of two prime numbers? (For example, 10 = 3 + 7.)
- Add consecutive odd numbers: 1, 1 + 3, 1 + 3 + 5, … What kind of numbers do you get? Draw arrays to explain why.
Suggested follow-up
- Revisit prime, composite and square numbers tomorrow with two or three quick questions from memory.
- Try the online practice check for this topic and look at any questions that need another go.
Tips for parents
- Use counters or buttons to build arrays. Seeing that 7 counters only make one long row helps make “prime” meaningful.
- Practise factor pairs during car trips: “Name a factor pair of 24.”
- Encourage the learner to work through factor pairs in order (1, 2, 3, …) so no factor is missed.