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# Expanding brackets

Algebra Foundations · Brackets

<!-- Speaker note: Students build k(ax + b) as rows of algebra tiles, find that the number outside multiplies every term inside (signs included), expand, then collect like terms, writing their working one step per line (AC9M8A01, node 1). 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** the number outside a bracket multiplies every term inside it. 3(2x + 4) is 3 rows of 2x + 4, so 6x + 12. Then collect any like terms.

- I can expand a bracket by multiplying every term inside by the number outside.
- I can expand when there are negative numbers inside or outside the bracket.
- I can expand, then collect like terms, and show each step.

<!-- Speaker note: Before class: Do the Run board and the working check yourself first. Students need Algebra Foundations' brackets and expressions, and adding and subtracting negatives (Number Foundations). Multiplying two negatives (−2 × −5 = 10) is Year 8 Number: the tiles show it (a negative row flips every tile), but teach or recap it in class if it is new. The one idea: the number outside multiplies every term inside. The common mistake is multiplying only the first term (4(x + 3) = 4x + 3); it is planted after the method, in Sam's working. Answers are marked by form: an answer that is equal but still has brackets, or still has like terms to collect, is told what to do next rather than marked right. The working check marks each line against the one before, so students see which step went wrong. -->

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## Why it matters

> Three friends each buy a drink and a $4 snack. Three drinks and three snacks: the 3 goes to both. That is all expanding a bracket is.

<!-- Speaker note: In Algebra Foundations you wrote expressions with brackets: $3(x + 2)$ means "add 2 to $x$, then multiply by 3". **Expanding** writes the same expression without the bracket. In this lesson you find out what the number outside the bracket does to the terms inside, using algebra tiles: a long tile for each $x$ and a small square for each 1. -->

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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 expression is the same as $3(x + 2)$?**

- $x + 6$
- $3x + 2$
- $3x + 6$
- $3x + 5$

<!-- Speaker note: What happens: 3x + 6. 3(x + 2) is 3 lots of x + 2: three x-tiles and three lots of 2 ones, which is 6. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Students who chose 3x + 2 are thinking the 3 only reaches the x. Do not correct it yet. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Fexpanding%2Fstep-2&code=widget%3A+tiles%0Aslider+k%3A+1%2C+5%2C+2%0Aslider+a%3A+1%2C+3%2C+1%0Aslider+b%3A+1%2C+6%2C+1%0A&fresh=1}}

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

<!-- Speaker note: $k$ is the number outside the bracket. $a$ is the number of $x$ inside, and $b$ the number on its own. The board draws $k$ rows of the bracket, then counts every tile. -->

<!-- Speaker note: Teacher: ask students to read the count under the tiles aloud: '3 rows of x + 2: 3 x-tiles and 6 ones'. -->

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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?

* 3x + 6. 3(x + 2) is 3 lots of x + 2: three x-tiles and three lots of 2 ones, which is 6.

<!-- 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>

## Expand, then collect

Here is how to expand a bracket and then collect like terms.

Try one: expand $2(5x - 3)$.

$$3(2x + 4) - x$$

1) $3 \times 2x = 6x$ — The 3 multiplies the first term.
2) $3 \times 4 = 12$ — And the second term: every term in the bracket.
3) $6x + 12 - x$ — The bracket is gone. The $-x$ was outside it, so it stays as it is.
4) $5x + 12$ — Collect like terms: $6x - x = 5x$. The 12 has no like term.

<!-- 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 3(2x + 4) - x and say "Expand the bracket first." Line 1: write 3 × 2x = 6x and say "The 3 multiplies the first term." Line 2: write 3 × 4 = 12 and say "And the second term: every term in the bracket." Line 3: write 6x + 12 - x and say "The bracket is gone. The -x was outside it, so it stays as it is." Line 4: write 5x + 12 and say "Collect like terms: 6x - x = 5x. The 12 has no like term." -->

<!-- Speaker note: Teacher: model the worked example live. Draw 3 rows of tiles for 2x + 4, write 3 × 2x and 3 × 4 at writing pace, then the − x, then collect. Reveal the prepared lines afterwards to compare. -->

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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>

## Expand, then collect

**Expand $2(5x - 3)$.**

* Answer: 10x - 6

<!-- Speaker note: 2 × 5x = 10x and 2 × (−3) = −6, so 10x − 6. -->

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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

**Sam expands $4(x + 3)$. Click the line where his working goes wrong.**

```
 1  Expand 4(x + 3).
 2  The 4 multiplies the x: 4 × x = 4x.
 3  The 3 stays as it is.
 4  So 4(x + 3) = 4x + 3.
```

* Answer: line 3

<!-- Speaker note: Line 3: the 4 multiplies every term in the bracket, so the 3 becomes 4 × 3 = 12. 4(x + 3) = 4x + 12. -->

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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>

## Your working

**Expand and simplify $3(2x + 4) - x$. Write one step per line.**

* Answer: 3(2x + 4) - x → 6x + 12 - x → 5x + 12

<!-- Speaker note: First expand: 6x + 12 − x. Then collect the like terms 6x and −x: 5x + 12. -->

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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



- **Bigger numbers:** Each task changes one thing. The board shows the tiles; type the expansion.
- **Subtraction inside:** 
- **Quick check:** Multiplying two negatives, from Number Foundations.
- **A negative outside:** $k$ can be negative now: watch every tile change sign.
- **Collect afterwards:** 
- **Two brackets:** No board for this one: expand each bracket, then collect.

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

<!-- Speaker note: What this step is for: One change at a time: bigger numbers, subtraction in the bracket, a negative outside (after a quick check of multiplying negatives), extra terms to collect, then the edge case of two brackets. Hints, in order: Set k, a and b, then count: k × a x-tiles and k × b ones. / b is negative here: each row has 2 negative ones. / Set k to −2. Every tile flips sign. / −2 × x = −2x and −2 × (−5) = +10. / Expand the bracket first: 2(x + 3) = 2x + 6. / Then collect: 2x + 3x, and 6 − 1. For students who need more support: Use the tiles board for every expansion: set k, a and b, read the count, then write it. Physical algebra tiles help if you have them. For early finishers: Expand and simplify 3(2x − 1) − 2(x − 4). (4x + 5.) Then: find the bracket with the biggest number outside that expands to 12x + 18. When students get stuck: A student multiplies only the first term (Sam's mistake). Ask: How many rows are there? What is in each row? Fix: Build it on the board: every row has the ones as well as the x. / A student gets the sign wrong with a negative inside (4(3x − 2) = 12x + 8). Ask: What is in each row: add 2 or take away 2? Fix: 4 rows each take away 2, so 8 taken away: −8. / A student expands correctly but stops before collecting. Ask: Are there any terms that are the same kind? Can they be added? Fix: Underline the x terms and the number terms in different colours, then add each kind. -->

<!-- Speaker note: Teacher: before the negative-outside task, ask: what happens to a tile if you take it away twice? Then let the board show −2(x − 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>

## Positive

**Expand $6(x + 2)$.**

* Answer: 6x + 12

<!-- Speaker note: 6 × x = 6x and 6 × 2 = 12. -->

<!-- Speaker note: Teacher: the working check marks the first wrong line. Ask students who get a cross to say what that step did. -->

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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>

## Negative outside

**Expand $-3(2x - 4)$.**

* Answer: -6x + 12

<!-- Speaker note: −3 × 2x = −6x and −3 × (−4) = +12. -->

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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

**Expand and simplify $4(2x - 3) + 5$. Write one step per line.**

* Answer: 4(2x - 3) + 5 → 8x - 12 + 5 → 8x - 7

<!-- Speaker note: 8x − 12 + 5 = 8x − 7. -->

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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

**Review: which expression means "add 3 to $n$, then multiply by 5"?**

- $5n + 3$
- $n + 3 \times 5$
- $5(n + 3)$

* Answer: $5(n + 3)$

<!-- Speaker note: Brackets group the part done first: 5(n + 3). -->

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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: brackets backwards

Find a bracket that expands to $6x + 12$. Then find a different one.

Stretch: which bracket with the biggest possible $k$ expands to $12x + 18$?

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

<!-- Speaker note: What this step is for: Finding a bracket that expands to a given expression reverses the lesson, and shows more than one bracket can give the same result. Hints, in order: Which number goes into both 6 and 12? / Try k = 3: then 3 × a = 6 and 3 × b = 12. -->

<!-- Speaker note: Teacher: collect the different brackets for 6x + 12 and ask which uses the biggest number outside. That is the lead into factorising. -->

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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

**Expand and simplify 5(x − 2) + 3x.**

* 5x − 10 + 3x = 8x − 10.

<!-- 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: factorising, which is expanding backwards (the Make task), then rearranging formulas (nodes 2 and 3). Where it leads: binomial products (x + 2)(x + 3) in Year 9, which the same area model shows as a grid. -->