45-minute lesson
Secondary 1 Mathematics: Prime factors and index notation
Lesson objective
- Recognise prime and composite numbers.
- Write a whole number as a product of its prime factors using a factor tree.
- Use index notation to write and evaluate repeated multiplication.
Success criteria:
- I can explain the difference between a prime number and a composite number.
- I can draw a factor tree and write a number as a product of primes, such as 84 = 2² × 3 × 7.
- I can write 2 × 2 × 2 as 2³ and work out its value.
Materials
- Paper and pencil
- A list of the prime numbers up to 50 (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47)
- A calculator for checking (optional)
Introduction
5 minIntroduce today's words:
- Factor: A whole number that divides exactly into another number. The factors of 12 are 1, 2, 3, 4, 6 and 12.
- Prime number: A whole number greater than 1 with exactly two factors: 1 and itself, such as 2, 3, 5, 7 and 11.
- Composite number: A whole number greater than 1 with more than two factors, such as 4, 9 or 15.
- Prime factorisation: Writing a number as a product of prime numbers only, such as 12 = 2 × 2 × 3.
- Index (power): The small raised number that shows how many times the base is multiplied by itself. In 2³ the index is 3.
- Base: The number being multiplied repeatedly. In 2³ the base is 2.
Ask your child what they already know about prime factors and index notation.
Explanation
9 minA prime number has exactly two factors: 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17 and 19. A composite number has more than two factors. The number 1 is special: it has only one factor, so it is neither prime nor composite. The number 2 is the only even prime.
Every composite number can be built by multiplying primes together, and there is only one way to do it (apart from the order). This is why primes are sometimes called the “building blocks” of whole numbers.
A factor tree is a simple way to find the prime factors. Write the number at the top, split it into any two factors, and keep splitting each branch until every branch ends in a prime. Circle the primes as you reach them. It does not matter which pair of factors you start with — you always end with the same primes.
Quick divisibility checks help you find a first factor:
- Divisible by 2 if the last digit is even.
- Divisible by 3 if the digits add to a multiple of 3 (for 51: 5 + 1 = 6, so 51 = 3 × 17).
- Divisible by 5 if the last digit is 0 or 5.
Index notation is a short way to write repeated multiplication. Instead of 2 × 2 × 2 we write 2³, read as “2 to the power of 3” or “2 cubed”. The 2 is the base and the 3 is the index. So 3⁴ = 3 × 3 × 3 × 3 = 81. Take care: 2³ means 2 × 2 × 2 = 8, not 2 × 3 = 6.
When we write a prime factorisation, we list the primes from smallest to largest and use index notation for any repeated primes. For example, 2 × 2 × 3 × 7 is written 2² × 3 × 7.
Worked examples
7 minFactor tree for 84
Use a factor tree to write 84 as a product of prime factors in index notation.
Step 1: 84 is even, so split it into 2 × 42. Circle the 2 because it is prime.
Step 2: 42 is even, so split it into 2 × 21. Circle the 2.
Step 3: 21 = 3 × 7. Both are prime, so circle them.
Step 4: The circled primes are 2, 2, 3 and 7, so 84 = 2 × 2 × 3 × 7.
Step 5: Two 2s can be written as 2².
Answer: 84 = 2² × 3 × 7
Evaluating a power
Write 3 × 3 × 3 × 3 in index notation and find its value.
Step 1: There are four 3s multiplied together, so the base is 3 and the index is 4: 3⁴.
Step 2: Multiply step by step: 3 × 3 = 9, 9 × 3 = 27, 27 × 3 = 81.
Answer: 3⁴ = 81
A larger number
Write 180 as a product of prime factors in index notation.
Step 1: 180 = 2 × 90. Circle the 2.
Step 2: 90 = 2 × 45. Circle the 2.
Step 3: 45 = 3 × 15 (4 + 5 = 9, a multiple of 3). Circle the 3.
Step 4: 15 = 3 × 5. Circle the 3 and the 5.
Step 5: The primes are 2, 2, 3, 3 and 5. Check: 4 × 9 × 5 = 180.
Answer: 180 = 2² × 3² × 5
Guided practice (do together)
9 min1. Which of these numbers is prime?
(a) 21 (b) 27 (c) 29 (d) 33
2. How is 3 × 3 × 3 × 3 written in index notation?
(a) 3⁴ (b) 4³ (c) 12 (d) 3 × 4
3. Which is the prime factorisation of 36?
(a) 2² × 3² (b) 4 × 9 (c) 2 × 3 × 6 (d) 2³ × 3
4. What is the value of 2⁵?
(a) 10 (b) 25 (c) 32 (d) 7
Independent practice
11 min5. Write 60 as a product of prime factors using index notation.
6. Is 51 a prime number or a composite number?
7. Which number has the prime factorisation 2³ × 5?
8. Write 72 as a product of prime factors using index notation.
Questions to check understanding
- Can you explain the difference between a prime number and a composite number?
- Can you draw a factor tree and write a number as a product of primes, such as 84 = 2² × 3 × 7?
- Can you write 2 × 2 × 2 as 2³ and work out its value?
- What was the trickiest part today?
Answer guide
1. 29 — 29 has only the factors 1 and 29. The others are composite: 21 = 3 × 7, 27 = 3 × 9 and 33 = 3 × 11.
2. 3⁴ — The base 3 is multiplied by itself 4 times, so it is 3⁴.
3. 2² × 3² — 36 = 2 × 2 × 3 × 3 = 2² × 3². The options 4 × 9 and 2 × 3 × 6 contain composite numbers, and 2³ × 3 = 24.
4. 32 — 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. It is not 2 × 5 = 10.
5. 2² × 3 × 5 — 60 = 2 × 30 = 2 × 2 × 15 = 2 × 2 × 3 × 5, which is 2² × 3 × 5.
6. Composite — The digits add to 5 + 1 = 6, a multiple of 3, so 51 = 3 × 17. It has more than two factors, so it is composite.
7. 40 — 2³ = 2 × 2 × 2 = 8, and 8 × 5 = 40.
8. 2³ × 3² — 72 = 2 × 36 = 2 × 2 × 18 = 2 × 2 × 2 × 9 = 2 × 2 × 2 × 3 × 3, which is 2³ × 3².
Review
4 minRecap the success criteria together. Watch for these common misconceptions:
- Thinking 2³ means 2 × 3 = 6. Have the learner write out the repeated multiplication (2 × 2 × 2) every time until it becomes automatic.
- Thinking 1 is a prime number. A prime must have exactly two different factors, and 1 has only one.
- Stopping a factor tree too early, for example writing 36 = 4 × 9. Ask “Is every number at the end of a branch prime?” and keep splitting until it is.
Extension activities
- Use a 1–100 grid to sieve out the primes: cross out multiples of 2, 3, 5 and 7 (but not the numbers themselves) and see what is left.
- Use prime factorisations to find the highest common factor of 84 and 180 (2² × 3 = 12) by choosing the primes they share.
- Investigate square numbers: why does every prime appear an even number of times in the prime factorisation of a perfect square such as 36 = 2² × 3²?
Suggested follow-up
- Revisit prime factors and index notation 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
- Ask the learner to check every factorisation by multiplying the primes back together — it should give the original number.
- Let them start factor trees with different pairs (for 36, try 4 × 9 and 6 × 6) to see that the final primes are always the same.
- Practise the divisibility tests for 2, 3 and 5 as quick mental games while out and about, using numbers on letterboxes or price tags.