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# Prime factors

Number Foundations · Squares, powers and primes

<!-- Speaker note: Students grow factor trees on the board until every branch is prime, write the prime factorisation in exponent notation, check it by expanding, and see that every tree for a number ends with the same primes (AC9M7N02, node 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** split a number into factor pairs until every branch is a prime: 36 = 2 × 2 × 3 × 3 = 2² × 3². Every tree for 36 ends with the same primes.

- I can use a factor tree to write a number as a product of primes.
- I can write the prime factorisation with exponents and check it by expanding.
- I can explain why different trees for a number give the same primes.

<!-- Speaker note: Before class: Do the Run board yourself first, twice, with different first splits. Students need factor pairs, primes (Year 6) and exponent notation (previous lesson). The one idea: split until every branch is prime; the primes are the same whichever way you split (uniqueness, informally). The common mistake is stopping at composite branches (Leo's mistake). This is the hinge for HCF and LCM in the next lesson: the method depends on there being one factorisation to compare. -->

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## Why it matters

> Every whole number above 1 is built from primes in exactly one way, like a code. 60 is 2² × 3 × 5, and no other number has that code.

<!-- Speaker note: A **prime** has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, … (1 is not prime.) Every whole number above 1 that is not prime can be split into primes. Writing it that way, with exponents, is its **prime factorisation**: $$36 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2$$ A **factor tree** finds it: split the number into any factor pair, then split each branch that is not prime, until every branch ends in a prime. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**Sam starts a factor tree for 36 with $6 \times 6$. Ali starts with $4 \times 9$. When both finish, will they have the same primes?**

- Yes, the same primes
- Only if they both start with 2
- No: different starts give different primes

<!-- Speaker note: What happens: Yes. 6 × 6 → 2 × 3 × 2 × 3, and 4 × 9 → 2 × 2 × 3 × 3: both are 2² × 3². A number has only one prime factorisation. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands; then have half the class start 36 with 6 × 6 and half with 4 × 9. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fnumber%2Fprime-factors%2Fstep-2&code=widget%3A+factortree%0Anumber%3A+36%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/number/prime-factors/?full=step-2#step-2)

<!-- Speaker note: Prime branches are circled. The board writes the product with exponents when the tree is finished. -->

<!-- Speaker note: Teacher: collect the final products. Are they the same? -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* Yes. 6 × 6 → 2 × 3 × 2 × 3, and 4 × 9 → 2 × 2 × 3 × 3: both are 2² × 3². A number has only one prime factorisation.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Split, then write with exponents

Here is a tree for 60, written as working.

Now try one: which is the prime factorisation of 60?

$$60 = 6 \times 10$$

1) $6 = 2 \times 3, \quad 10 = 2 \times 5$ — Split every branch that is not prime.
2) $60 = 2 \times 2 \times 3 \times 5$ — Every branch is prime: collect them, smallest first.
3) $60 = 2^2 \times 3 \times 5$ — Write repeated primes with exponents. Check: 4 × 3 × 5 = 60.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write 60 = 6 × 10 and say "Start with any factor pair." Line 1: write 6 = 2 × 3, 10 = 2 × 5 and say "Split every branch that is not prime." Line 2: write 60 = 2 × 2 × 3 × 5 and say "Every branch is prime: collect them, smallest first." Line 3: write 60 = 2^2 × 3 × 5 and say "Write repeated primes with exponents. Check: 4 × 3 × 5 = 60." -->

<!-- Speaker note: Teacher: insist on checking by expanding every time. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Split, then write with exponents

**What is the prime factorisation of 60?**

- $2 \times 3 \times 10$
- $2 \times 30$
- $4 \times 15$
- $2^2 \times 3 \times 5$

* Answer: $2^2 \times 3 \times 5$

<!-- Speaker note: 60 = 6 × 10 = 2 × 3 × 2 × 5 = 2² × 3 × 5. Check: 4 × 3 × 5 = 60. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Leo writes 36 as a product of prime factors. Click the line where his reasoning goes wrong.**

```
 1  Write 36 as a product of prime factors.
 2  36 = 4 × 9
 3  4 and 9 are factors of 36, so the tree is finished.
 4  36 = 4 × 9
```

* Answer: line 3

<!-- Speaker note: Line 3: 4 and 9 are not prime, so the tree is not finished. 4 = 2 × 2 and 9 = 3 × 3, so 36 = 2² × 3². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Primes only

**Why is $2 \times 2 \times 9$ not the prime factorisation of 36?**

- The 2s should be written first
- 9 is not prime: it splits into 3 × 3
- It does not equal 36

* Answer: 9 is not prime: it splits into 3 × 3

<!-- Speaker note: Every factor must be prime: 9 = 3 × 3, so 36 = 2² × 3². -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **Seventy-two:** Each task changes one thing.
- **Back to the number:** 
- **A prime:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/prime-factors/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a bigger tree, expanding a factorisation back to its number, then a prime as the edge case (and why 1 is left out). Hints, in order: Start with 8 × 9, or 2 × 36. For students who need more support: Use the board for every factorisation, and keep a list of primes to 30 visible. For early finishers: Which numbers below 100 have only one prime in their factorisation (like 32 = 2⁵)? What do their trees look like? The hard part, on purpose: Knowing when a tree is finished. Ask students to check each end of a branch for factors other than 1 and itself, rather than circling the primes for them. When students get stuck: A student stops at 4 or 9 (Leo's mistake). Ask: Is every circled number prime? Can 4 be split? Fix: Keep splitting until each branch has only 1 and itself as factors. / A student writes 1 in the factorisation. Ask: Is 1 prime? Does × 1 change anything? Fix: Leave 1 out: it is not prime. / A student writes 2 × 2 × 3 × 3 but not 2² × 3². Ask: How many 2s? How many 3s? Fix: Count each prime and write the count as its exponent. -->

<!-- Speaker note: Teacher: for 13, ask what a factor tree for a prime looks like. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Ninety

**What is the prime factorisation of 90?**

- $2 \times 3 \times 15$
- $2 \times 5 \times 9$
- $2 \times 3^2 \times 5$
- $9 \times 10$

* Answer: $2 \times 3^2 \times 5$

<!-- Speaker note: 90 = 9 × 10 = 3 × 3 × 2 × 5 = 2 × 3² × 5. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Back to the number

**Which number has the prime factorisation $2^3 \times 5$?**

* Answer: 40

<!-- Speaker note: 2³ = 8, and 8 × 5 = 40. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## One hundred

**What is the prime factorisation of 100?**

- $2 \times 5^2$
- $4 \times 25$
- $2^2 \times 5^2$
- $10^2$

* Answer: $2^2 \times 5^2$

<!-- Speaker note: 100 = 10 × 10 = 2 × 5 × 2 × 5 = 2² × 5². -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: powers

**Review: work out $3^4$.**

* Answer: 81

<!-- Speaker note: 3 × 3 × 3 × 3 = 81. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a tree for 360

Then compare with someone who started differently. Check by expanding: $8 \times 9 \times 5$.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/number/prime-factors/?full=step-6#step-6)

<!-- Speaker note: What this step is for: A larger number, with a first split of the student's choosing, shows the factorisation is the same whatever the route. Hints, in order: Start with 36 × 10, or 6 × 60, or 2 × 180. / Keep going until every branch is circled. -->

<!-- Speaker note: Teacher: compare the 360 trees across the room: different shapes, same primes. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**Write 84 as a product of prime factors, with exponents.**

* 84 = 2² × 3 × 7.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: HCF and LCM (node 3), comparing two factorisations. Where it leads: simplifying fractions and finding common denominators by primes, and simplifying surds (Year 10). -->