Prime Factor Decomposition
Prime Factor Decomposition
Lesson duration is: 60 minItems in this lesson
Prime Factor Decomposition
Open with the idea that large numbers can be broken into smaller building blocks. In this lesson, those building blocks are prime numbers.
Learning goal
I can find the prime factor decomposition of a number and write it as a product of prime factors, using index form where appropriate.
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What words come to mind when you hear prime factors?
Use this to surface prior ideas such as prime, factors, division, trees, multiplication, or confusion about 1. Pick a few responses to connect to the lesson goal.
Prime and composite numbers
A prime number has exactly 2 factors: 1 and itself
A composite number has more than 2 factors
The number 1 is not prime
Prime factor decomposition uses prime numbers only
Useful examples: 2, 3, 5, 7 are prime; 4, 6, 8, 9 are composite. Stress that 1 is a special case and often causes errors.
How to check a number
To decide if a number is prime or composite, ask: Can it be made by multiplying smaller whole numbers?
If the only factors are 1 and the number itself, it is prime
If it has other factors too, it is composite
Examples: 11 is prime, 12 is composite
Keep this at KS3 level: students do not need formal tests here, just factor awareness.
What decomposition means
Prime factor decomposition means writing a number as a multiplication of prime numbers.
Examples:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
20 = 2 × 2 × 5
The final factors must all be prime.
Link the word decomposition to 'breaking down'. Emphasise that different factor tree paths can be used, but the final prime factors are the same.
Which number is prime?
1
11
12
20
Checks the key idea from slides 2 to 4, including the common mistake that 1 is prime.
Prime or composite
A prime number has exactly two factors: 1 and itself. A composite number has more than two factors. The number 1 is not prime, so prime factor decomposition uses prime numbers only.
Echoes the exact language taught in the first subtopic. Each blank has a fixed place from meaning.
Using a factor tree
Start with the whole number
Split it into two factors
Keep splitting any factor that is composite
Stop only when every branch ends in a prime number
Model one quickly on the board while students look at the slide structure. The exact branches do not matter as long as all end points are prime.
Example: factor tree for 12
One possible factor tree is:
12 → 3 × 4
4 → 2 × 2
So the prime factor decomposition is:
12 = 2 × 2 × 3
Point out that 3, 2 and 2 are all prime, so the process is complete.
A different tree, same answer
You could also start like this:
12 → 2 × 6
6 → 2 × 3
This still gives:
12 = 2 × 2 × 3
Different trees can lead to the same prime factors.
This helps prevent students thinking there is only one correct first split.
Using repeated division
Another method is to divide by prime numbers step by step.
Example for 18:
18 ÷ 2 = 9
9 ÷ 3 = 3
3 ÷ 3 = 1
So 18 = 2 × 3 × 3
Stress that division should use prime numbers. Reaching 1 shows the process is finished.
When do you stop a factor tree?
At two branches
When all factors are prime
After one split
When numbers are even
Targets the stopping rule from slides 5 to 8 and the misconception of stopping too early.
Breaking a number down
In a factor tree, start with the whole number and split it into two factors. Keep splitting any factor that is composite. Stop only when every branch ends in a prime number.
Reinforces the procedure language used on the factor tree slides.
Small-number example
Find the prime factor decomposition of 24.
One method:
24 → 6 × 4
6 → 2 × 3
4 → 2 × 2
So 24 = 2 × 2 × 2 × 3
You may also show repeated division: 24 ÷ 2 ÷ 2 ÷ 2 ÷ 3 = 1.
Slightly larger example
Find the prime factor decomposition of 60.
60 → 6 × 10
6 → 2 × 3
10 → 2 × 5
So 60 = 2 × 2 × 3 × 5
This example is large enough to show several prime factors but still manageable at this level.
Writing answers in index form
If a prime factor repeats, index form makes the answer shorter.
2 × 2 × 2 = 2³
3 × 3 = 3²
24 = 2³ × 3
36 = 2² × 3²
Remind students that the index tells how many times the factor is used.
From factors to index form
Example with 72:
72 = 2 × 2 × 2 × 3 × 3
Group equal prime factors:
three 2s → 2³
two 3s → 3²
So 72 = 2³ × 3²
Make the grouping visible when teaching, for example by circling matching factors.
Which is the prime factor decomposition of 60?
2 × 2 × 3 × 5
6 × 10
2 × 30
3 × 4 × 5
Distractors reflect unfinished or non-prime factorisations from slides 9 and 10.
Using index form
If a prime factor repeats, index form makes the answer shorter. For 24, the factors are 2 × 2 × 2 × 3, so the answer is 2³ × 3. For 72, three 2s and two 3s give 2³ × 3².
Checks understanding of repeated factors and converting to index form using examples already taught.
What does the index tell you?
Expected elements: students should say that repeated prime factors are grouped and the index shows how many times each prime is used. Accept examples such as 2 × 2 × 2 becoming 2³.
Common error: stopping too early
A factor tree is not finished if any final factor is composite.
Wrong:
20 = 4 × 5
Why wrong?
4 is composite
it must be broken down again
Correct:
20 = 2 × 2 × 5
This is one of the key misconceptions named in the prompt. Stress the check: every final factor must be prime.
Common error: missing repeated factors
Be careful to include every prime factor.
Wrong:
24 = 2² × 3
Why wrong?
2² × 3 = 12
one extra factor of 2 is missing
Correct:
24 = 2³ × 3
A quick check is to multiply the factors back together to see if they return to the original number.
Checking your answer
A correct prime factor decomposition must:
use prime numbers only
include all the factors
multiply back to the original number
use index form correctly, if factors repeat
This slide pulls together the success criteria before the glossary.
Which answer for 20 is correct?
4 × 5
2 × 10
2 × 2 × 5
2² × 10
Directly addresses the error of stopping before all factors are prime.
Check the answer
A correct prime factor decomposition must use prime numbers only and include all the factors. The factors must multiply back to the original number and any repeated factor should use index form correctly.
Summarises the success criteria and links to the common errors on slides 13 to 15.
What part of prime factors feels easiest now?
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What mistake will you watch out for next time?
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What is one thing you will remember?
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Glossary
prime number - a whole number greater than 1 with exactly two factors: 1 and itself
composite number - a whole number greater than 1 with more than two factors
factor - a number that divides exactly into another number
prime factor - a factor that is also a prime number
prime factor decomposition - writing a number as a product of prime factors
factor tree - a diagram that breaks a number down into factors
index form - a short way to show repeated multiplication of the same factor
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