005i - Prime Factorisation with Factor Trees

Lesson 5 — Prime Factorisation Using Factor Trees

Strand: Number | Descriptor: AC9M7N02 | Duration: 45 minutes

Learning Intentions

• To understand that every natural number greater than has a unique prime factorisation.
• To find the prime factorisation of a number using a factor tree.

Success Criteria

I can:

1. Construct a factor tree for a given number and continue until every branch ends in a prime.
2. Write the prime factorisation as a product of primes.
3. Explain why different starting splits give the same final set of primes.
4. Use divisibility rules for , , and to choose sensible first splits.

Warmup

(5 minutes — divisibility rules recall, mini whiteboards)

For each number, state whether it is divisible by , , or :

Number	÷2?	÷3?	÷5?	÷9?

Quick recall of the rules:

• 2 — the number is even.
• 3 — the digit sum is divisible by .
• 5 — the number ends in or .
• 9 — the digit sum is divisible by .

Answers: — 2 ✓, 3 ✓ (digit sum 9), 5 ✗, 9 ✓. — 2 ✗, 3 ✓, 5 ✓, 9 ✓. — 2 ✓, 3 ✗, 5 ✓, 9 ✗. — 2 ✓, 3 ✓, 5 ✗, 9 ✓ (digit sum 18).

Activities

Activity 1 — Explicit Instruction: Building a Factor Tree (12 min)

I do: Model on the board, thinking aloud.

          60
         /  \
        6    10
       / \   / \
      2   3 2   5

Conventions to state explicitly:

• Circle every prime as it appears — that branch is finished.
• Keep splitting any composite until nothing composite remains.
• Write the final answer with the primes in ascending order.

The key idea — uniqueness. Rebuild starting differently:

          60                        60
         /  \                      /  \
        4    15                   2    30
       / \   / \                      /  \
      2   2 3   5                    5    6
                                         / \
                                        2   3

All three trees give . State the Fundamental Theorem of Arithmetic: every natural number greater than can be written as a product of primes in exactly one way (ignoring order).

We do: Build trees for and .

You do: Prime factorise , , , .

Activity 2 — The Ladder (division) Method (8 min)

Offer an alternative for students who find trees messy — repeatedly divide by the smallest prime that works.

Advantage to point out: the ladder produces the primes already in ascending order and never leaves a branch unfinished.

Practice: , , .

Activity 3 — Inquiry: the Mystery Number (10 min)

Pairs.

A number’s prime factorisation is .

• What is the number?
• Without listing them by trial, work out how many factors it has.
• Now find every factor. Does your count match?

Socratic scaffolding:

Prompt	Purpose
Understand: what does the factorisation tell you?	The complete “ingredient list” of primes for this number.
What is the number itself?	.
Related problem: how is any factor of built?	From some selection of those same primes.
Devise a plan: how many s could a factor contain?	Zero, one, or two — three choices.
And how many s? How many s?	Two choices each ( or ).
Carry it out	factors.
Check	— twelve. ✓
Looking back — generalise	If then has factors.

Extension: Use the rule to predict the number of factors of , then verify. (Predicted ; the factors are .)

Checks for Understanding

(5 minutes — exit ticket)

1. Draw a factor tree for and write its prime factorisation.
2. A student writes "" as their final answer. What is wrong, and what is the correct answer?
3. Kim starts her tree for with ; Sam starts with . Will they get the same answer? Explain.
4. The prime factorisation of a number is . What is the number?

Answers: 1. ; 2. and are composite, so the tree is unfinished — ; 3. Yes — by the Fundamental Theorem of Arithmetic the prime factorisation is unique, so both reach ; 4. .

Common Misconceptions

Misconception	How to pre-empt it
Stopping the tree at composite leaves (e.g. ).	Enforce the circling convention — a branch ends only when circled, and only primes get circled.
Including in the factorisation, e.g. .	Ask what contributes. Point out that is not prime, so it has no place in a prime factorisation.
Believing a different starting split gives a different answer.	Deliberately have different students start differently, then compare on the board. Make the surprise explicit.
Recording the tree but never writing the product.	Require the final line "" in ascending order every time.
Multiplication slips when checking.	Encourage checking by regrouping into easy products, e.g. first.
Confusing prime factorisation with listing all factors.	Contrast the two directly for : prime factorisation ; all factors .

Enrichment — Competition-Style Problems

E1 (AMC Junior style). What is the largest prime factor of ?

Answer

The largest prime factor is .

E2 (Kangaroo style). A number has prime factorisation . How many factors does it have, and what is the number?

Answer

The number is . Factor count: .

E3 (Challenge). The product of three different prime numbers is . What are they?

Answer

E4 (Challenge). What is the smallest natural number divisible by every one of ?

Answer

Take the highest power of each prime needed: (for ), (for and ), . So .

Homework

1. Find the prime factorisation of: (a) (b) (c) (d) (e) .
2. Write the number whose prime factorisation is: (a) (b) (c) .
3. Use the ladder method for . Show every division step.
4. Find the largest prime factor of: (a) (b) (c) .
5. Reasoning. Explain why the prime factorisation of any even number must include at least one .
6. Challenge. A number’s prime factorisation is , where is prime. The number is between and . Find and the number.

Answers: Q1 — (a) (b) (c) (d) (e) . Q2 — (a) (b) (c) . Q3 — . Q4 — (a) (b) (c) . Q6 — between and means between and , so or , giving or .