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# Factorising

Algebra Foundations · Brackets

<!-- Speaker note: Students share algebra tiles into equal rows to factorise sx + t, learn to take out the highest common factor (numbers, then letters) and check by expanding (AC9M8A01, node 2). About 45 minutes (Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min). -->

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## Today

**We are learning to** factorising is expanding backwards. Share the terms into equal rows: 6x + 9 is 3 rows of 2x + 3, so 3(2x + 3). Take out the highest common factor, then check by expanding.

- I can factorise an expression by taking out a common factor.
- I can take out the highest common factor, including a letter, so the expression is fully factorised.
- I can check a factorisation by expanding it.

<!-- Speaker note: Before class: Do the Run board and the Make task yourself first. Students need Expanding brackets (the lesson before) and highest common factors from Year 7 Number. The one idea: factorising divides every term by a common factor, and the factor goes outside the bracket. The common mistake is taking out a common factor that is not the highest (12x − 8 = 2(6x − 4)); it is planted after the method, in Ava's working. Her answer is equal to 12x − 8, so it is not wrong, just not finished: that is the conversation to have. Answers are marked by form: an equal answer that could still be factorised further is told so ("could take out more") rather than marked right. -->

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## Why it matters

> 6 x-tiles and 9 ones. Can you share them into equal rows, with nothing left over? The number of rows goes outside the bracket.

<!-- Speaker note: In Expanding brackets you turned $3(2x + 4)$ into $6x + 12$. **Factorising** goes the other way: from $6x + 12$ back to a bracket. In this lesson you share algebra tiles into equal rows. The number of rows goes outside the bracket, and one row goes inside. -->

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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

**Which is $6x + 9$ written as a bracket, with the biggest possible number outside?**

- $3(2x + 3)$
- $3(2x + 9)$
- $2(3x + 4.5)$
- $6(x + 9)$

<!-- Speaker note: What happens: 3(2x + 3). 3 goes into 6 and into 9: 6x ÷ 3 = 2x and 9 ÷ 3 = 3. Check by expanding: 3 × 2x = 6x and 3 × 3 = 9. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 6(x + 9) divided only the first term; those who chose 2(3x + 4.5) found a factor of 6 but not of 9. Do not correct it yet. -->

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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%2Falgebra%2Ffactorising%2Fstep-2&code=widget%3A+tiles%0As%3A+6%0At%3A+9%0Aslider+k%3A+1%2C+6%2C+1%0A&fresh=1}}

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

<!-- Speaker note: $k$ is the number of rows. When the tiles share equally, the board writes the bracket: $k$ outside, one row inside. -->

<!-- Speaker note: Teacher: ask why 2 rows do not work (9 ones do not share into 2) and why 3 does. -->

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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?

* 3(2x + 3). 3 goes into 6 and into 9: 6x ÷ 3 = 2x and 9 ÷ 3 = 3. Check by expanding: 3 × 2x = 6x and 3 × 3 = 9.

<!-- 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>

## Take out the HCF

Here is how to factorise, then check.

Try one: factorise $10x + 15$.

$$12x - 8$$

1) $\text{HCF of } 12 \text{ and } 8 = 4$ — Take out the biggest: the highest common factor.
2) $12x \div 4 = 3x$ — Divide every term by 4.
3) $-8 \div 4 = -2$ — The sign comes too.
4) $4(3x - 2)$ — The HCF goes outside; what is left goes inside.
5) $4 \times 3x - 4 \times 2 = 12x - 8$ — Check by expanding.

<!-- 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 12x - 8 and say "Find a number that goes into every term." Line 1: write HCF of 12 and 8 = 4 and say "Take out the biggest: the highest common factor." Line 2: write 12x ÷ 4 = 3x and say "Divide every term by 4." Line 3: write -8 ÷ 4 = -2 and say "The sign comes too." Line 4: write 4(3x - 2) and say "The HCF goes outside; what is left goes inside." Line 5: write 4 × 3x - 4 × 2 = 12x - 8 and say "Check by expanding." -->

<!-- Speaker note: Teacher: model the worked example live, including the check by expanding. Then ask: is Ava's answer wrong? -->

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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>

## Take out the HCF

**Factorise $10x + 15$ fully.**

* Answer: 5(2x + 3)

<!-- Speaker note: The HCF of 10 and 15 is 5: 10x ÷ 5 = 2x and 15 ÷ 5 = 3. Check: 5 × 2x + 5 × 3 = 10x + 15. -->

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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

**Ava factorises $12x - 8$. Click the line where her working goes wrong.**

```
 1  Factorise 12x − 8.
 2  The biggest number that goes into 12 and 8 is 2.
 3  12 ÷ 2 = 6 and 8 ÷ 2 = 4.
 4  So 12x − 8 = 2(6x − 4).
```

* Answer: line 2

<!-- Speaker note: Line 2: 2 goes into both, but 4 is the biggest. 6x − 4 still shares a 2, so it is not fully factorised: 12x − 8 = 4(3x − 2). -->

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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>

## Check it

**How can you check that $4(3x - 2)$ is right for $12x - 8$?**

- Expand it and compare
- Check that 3 and 2 are both prime
- Add 4 to each term

* Answer: Expand it and compare

<!-- Speaker note: Expand: 4 × 3x = 12x and 4 × (−2) = −8. You are back to 12x − 8. -->

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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



- **On the board:** Each task changes one thing.
- **A minus sign:** No board from here: divide each term, then check by expanding.
- **Three terms:** 
- **A letter in common:** 
- **Nothing to take out:** 

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

<!-- Speaker note: What this step is for: One change at a time: different numbers on the board, a minus sign, three terms, a letter as the common factor, then the edge case of nothing to take out. Hints, in order: Which numbers go into both 10 and 15? Try the biggest. For students who need more support: Use the tiles board for every two-term question with positive numbers: try each k and read off the bracket. For early finishers: Factorise 6x²y − 9xy² fully. (3xy(2x − 3y).) Then: write three different expressions whose full factorisation has 6 outside. When students get stuck: A student divides only the first term (6x + 9 = 3(2x + 9)). Ask: Expand your answer. Do you get back to where you started? Fix: Expanding is the check: 3 × 9 = 27, not 9. / A student stops at a common factor that is not the highest (Ava's mistake). Ask: Look inside the bracket. Do those terms share anything? Fix: Take out what they share as well: 2(6x − 4) = 2 × 2(3x − 2) = 4(3x − 2). / A student cannot find a common factor. Ask: List the factors of each number. Which appear in both lists? Fix: Use the board: try k = 2, 3, 4… and see which numbers of rows work. -->

<!-- Speaker note: Teacher: before the letter task, ask what goes into both 2x² and 4x. Students often find the 2 and miss the x. -->

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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>

## Positive

**Factorise $15x + 20$ fully.**

* Answer: 5(3x + 4)

<!-- Speaker note: HCF 5: 5(3x + 4). -->

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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>

## A minus sign

**Factorise $8x - 12$ fully.**

* Answer: 4(2x - 3)

<!-- Speaker note: HCF 4: 4(2x − 3). -->

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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>

## Full working

**Factorise $18x + 12$ fully. You may take out a factor in steps. Write one step per line.**

* Answer: 18x + 12 → 2(9x + 6) → 6(3x + 2)

<!-- Speaker note: 2(9x + 6) still shares a 3, so take it out too: 6(3x + 2). Or take out the HCF 6 straight away. -->

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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: expanding

**Review: expand $3(2x - 5)$.**

* Answer: 6x - 15

<!-- Speaker note: 3 × 2x = 6x and 3 × (−5) = −15. -->

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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: four rows

Find an expression $sx + t$ that factorises fully as $4(\ldots)$. Then find another.

Stretch: why can't $s$ and $t$ both be multiples of 8?

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

<!-- Speaker note: What this step is for: Choosing tiles that factorise fully with exactly 4 outside reverses the lesson: from the factor back to the expression. Hints, in order: Both numbers must be in the 4 times table. / But the rows must not share further: 8 and 16 have HCF 8, not 4. -->

<!-- Speaker note: Teacher: collect students' expressions on the board and ask which ones are fully factorised with 4 outside. -->

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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

**Factorise 14x − 21 fully, then check by expanding.**

* 7(2x − 3). Check: 7 × 2x = 14x and 7 × (−3) = −21.

<!-- 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: rearranging formulas (node 3), which divides every term by a number in the same way. Where it leads: factorising quadratics in Years 9 and 10, which the same tiles show as a rectangle. -->