---
title: "Expanding brackets"
subtitle: "Algebra Foundations · Brackets · Lesson 10"
---

# Expanding brackets

**Name:** {{field-24}} **Date:** {{field-15}} **Class:** {{field-15}}

> **Key question:** 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.


:::teacher-note

**In one sentence:** 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).

**Students should be able to say:** 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.

**Width:** the unmarked parts are the floor every student does (predict, run, investigate and make). Typical adds the modify practice; wider adds the real-world contexts. They add contexts, not difficulty. Students work in the online tutorial alongside this sheet.

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## Predict

Think of $x + 2$ as one row of tiles: one $x$ and two ones. What do 3 rows hold?

:::keep-together

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

☐ $x + 6$

☐ $3x + 2$

☐ $3x + 6$

☐ $3x + 5$

My prediction, and why:

{{lines-3}}

:::


:::teacher-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.

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## Run

Try it on the board in the tutorial. Was your prediction right? What was different?

{{lines-2}}

## Multiply every term


:::keep-together

### Expand, then collect

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

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

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

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 10x - 6

2 × 5x = 10x and 2 × (−3) = −6, so 10x − 6.

:::


:::keep-together

### 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.
```

Line: {{field-15}}

Why is it wrong?

{{lines-2}}

:::


:::teacher-note

**Answer:** line 3

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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:::keep-together

### Your working

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

Answer: {{field-15}}

:::


:::teacher-note

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

First expand: 6x + 12 − x. Then collect the like terms 6x and −x: 5x + 12.

:::


:::typical

## 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.

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

:::


## Expand and simplify


:::keep-together

### Positive

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** 6x + 12

6 × x = 6x and 6 × 2 = 12.

:::


:::keep-together

### Negative outside

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

Answer: {{field-15}}

:::


:::teacher-note

**Answer:** -6x + 12

−3 × 2x = −6x and −3 × (−4) = +12.

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:::keep-together

### Full working

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

Answer: {{field-15}}

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:::teacher-note

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

8x − 12 + 5 = 8x − 7.

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:::keep-together

### Review

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

☐ $5n + 3$

☐ $n + 3 \times 5$

☐ $5(n + 3)$

:::


:::teacher-note

**Answer:** $5(n + 3)$

Brackets group the part done first: 5(n + 3).

:::


## 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 $12x + 18$?

Work in the tutorial. Plan first: what will you change, and how will you test it?

{{box-4}}

## Exit ticket

:::keep-together

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

{{lines-3}}

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:::teacher-note

**Answer:** 5x − 10 + 3x = 8x − 10.

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