At a glance. One lesson of about 60 minutes. Students write numbers as products of primes by factor tree and ladder, use index form, and explain why the result does not depend on the first split.
Calculator: no calculator. Nothing needs one. Splitting a number by hand is the skill being built.
What students will learn
Know
- Every whole number greater than 1 is a product of primes.
- Index form writes repeated primes as powers.
- Two methods: factor tree and repeated division (ladder).
Understand
- The prime factorization of a number is unique, apart from the order.
- A prime factorization shows the factors of a number at a glance.
- Different first splits give different trees but the same primes.
Do
- Write a number as a product of primes in index form.
- Find the number from its prime factorization.
- Use a prime factorization to answer questions about factors and squares.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: writing numbers in prime factor form in familiar and new cases (questions 1 to 6).
- Criterion B. Investigating patterns: the second figure. Ask students to choose three different first splits for one number and say what they notice.
- Criterion C. Communicating: explaining why 2 × 3 × 4 is not a prime factorization and why the answer does not depend on the tree (question 9).
- Criterion D. Applying mathematics in real-life contexts: using factorization to decide about groupings, squares and divisibility (extra questions).
The one thing to do with the figure
Ask for the prediction before opening the second figure. “Two students split 60 differently, as 6 × 10 and as 4 × 15. Will they find the same primes?” Take the vote, then show it.
Then ask what is the same (the collection of primes, with repeats) and what is different (the shape of the tree). That sameness is the reason we can speak of the prime factorization of a number.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook, prediction and the first figure. |
| 10 min | Key idea; the second figure. Students choose their own number and first split. |
| 10 min | Worked example with the ladder; students finish the faded one in pairs. |
| 5 min | Question 1, the check. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9. |
The answers
| Warm-up | 2³ × 5 = 40; 51 = 3 × 17, so not prime; 1 × 20, 2 × 10, 4 × 5. |
|---|---|
| Worked example | 84 = 2² × 3 × 7. |
| Faded example | 180 = 2² × 3² × 5. |
| 1. Prime factorization of 36 | C, 2² × 3². A, B and D are products equal to 36 but contain a composite number. |
| 2. Counting prime factors | 4: 2, 2, 3, 5. |
| 3. Evaluating index form | 120. 8 × 15. |
| 4. From index form to a number | 180. 4 × 9 × 5. |
| 5. Reading a prime factorization | 3. 72 = 8 × 9. |
| 6. Prime factorization of 84 | B. A, C and D have one composite factor each. |
| 7. Making a square | 3. 12 × 3 = 36 = 2² × 3². |
| 8. Counting factors from primes | 12. (3 + 1) × (2 + 1). |
| 9. Is it fully factorized? | B. A and D accept 4; C does not know every number above 1 has a prime factorization. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Stops with a composite factor, such as 6 × 6. | Circle each factor and ask: is this prime? If not, split it. |
| 3 or 10 in question 2 | Counted different primes or added them. | Write the product in full with every repeated factor, then count. |
| 30 or 90 in question 3 | Ignored the exponent, or multiplied base by exponent. | Write 2³ as 2 × 2 × 2 first. |
| 30, 60 or 120 in question 4 | Ignored or misread an exponent. | Evaluate each power on its own line before multiplying. |
| 5 or 6 in question 8 | Added the exponents, or multiplied them without adding 1. | List the factors in a table of twos and threes. |
| A or D in question 9 | Accepts 4 as a prime factor. | Ask for the factors of 4. |
Questions worth asking
- Does it matter which factor pair you start with? No: the primes at the bottom are always the same. Listen for “it ends up the same” and ask why.
- Why do we write index form? It is shorter and makes comparison easy: 2³ × 3 and 2² × 3² can be compared at a glance.
- How can you tell from 2² × 3² that the number is a perfect square? Every exponent is even, so each prime is in pairs.
- Why is 1 not used in a prime factorization? Because 1 is not prime, and multiplying by 1 adds nothing; it would make the factorization not unique.
Differentiation
A way in for anyone stuck
- Use a ladder (repeated division) rather than a tree for students who find branches confusing.
- Give a list of the primes to 30 on the desk.
- Print the tree outline for the first few numbers.
Stretch
- Question 8, then: how many factors does 360 have? (360 = 2³ × 3² × 5, so 4 × 3 × 2 = 24.)
- Find the smallest number with exactly 12 factors. (60.)
Students learning in English as an additional language
“Product” means the result of multiplying. “Split” and “break down” are used interchangeably in classrooms. Point to the symbols each time.
| Word | Plain meaning | Example |
|---|---|---|
| prime factor | a factor that is a prime number | the prime factors of 12 are 2 and 3 |
| factor tree | a diagram that splits a number into factors | 60 splits into 6 and 10 |
| product | the result of multiplying | the product of 3 and 5 is 15 |
| index form | a number written with powers of primes | 60 = 2² × 3 × 5 |
| unique | only one way, apart from the order | the prime factorization is unique |
| perfect square | a number that is a whole number squared | 36 = 6² |
Sentence starters for explaining aloud or in writing:
- … is prime, so I stop. … is not prime, so I split it into … × …
- The prime factorization of … is … because …
- I started with … × … and got the same primes because …
Two things not to say
“Just keep going until you run out of numbers.” Say what stops the splitting: a number that is prime. For the ladder, divide by 2 while you can, then 3, then 5, and so on.
“Different trees give different primes.” They do not: every tree gives one answer, written in a different order.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- Index formWrite 45 as a product of primes in index form.
3² × 5. - Earlier workList the factor pairs of 28.
1 × 28, 2 × 14 and 4 × 7.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ReverseWhat number is 2⁴ × 3 × 5?
240. - TeamsA coach has 100 players. Use 100 = 2² × 5² to say how many team sizes make equal teams, from 1 player up to 100.
(2 + 1) × (2 + 1) = 9 team sizes: 1, 2, 4, 5, 10, 20, 25, 50, 100.
3Challenge
Reasoning, generalising and working backwards.
- FactorsUse 36 = 2² × 3² to show that 36 has an odd number of factors.
(2 + 1) × (2 + 1) = 9, odd, because every exponent is even. - SmallestWhat is the smallest number with exactly 10 factors?
48 = 2⁴ × 3, with 5 × 2 = 10 factors.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Draw factor trees on the board and let students choose the first split.