Expanding brackets

Classroom resources

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Students will be able to

• 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.

Timing

Predict and Run 8 min · Investigate 15 min · Modify 12 min · Practise and Make 10 min

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.

Running the lesson

1. Predict: take a show of hands. Students who chose 3x + 2 are thinking the 3 only reaches the x. Do not correct it yet.
2. Run: ask students to read the count under the tiles aloud: '3 rows of x + 2: 3 x-tiles and 6 ones'.
3. Investigate: 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.
4. Modify: 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).
5. Practise: the working check marks the first wrong line. Ask students who get a cross to say what that step did.
6. Make: collect the different brackets for 6x + 12 and ask which uses the biggest number outside. That is the lead into factorising.

When students get stuck

Answers to the checks

1. 3(x + 2): Which expression is the same as $3(x + 2)$?
   3x + 6. 3(x + 2) is 3 lots of x + 2: three x-tiles and three lots of 2 ones, which is 6.
2. build it with tiles: Set the board to $3(x + 2)$ and count the tiles. Was your prediction right?
   
3. Expand, then collect: Expand $2(5x - 3)$.
   10x - 6
   2 × 5x = 10x and 2 × (−3) = −6, so 10x − 6.
4. Find the mistake: Sam expands $4(x + 3)$. Click the line where his working goes wrong.
   line 3
   Line 3: the 4 multiplies every term in the bracket, so the 3 becomes 4 × 3 = 12. 4(x + 3) = 4x + 12.
5. Your working: Expand and simplify $3(2x + 4) - x$. Write one step per line.
   3(2x + 4) - x → 6x + 12 - x → 5x + 12
   First expand: 6x + 12 − x. Then collect the like terms 6x and −x: 5x + 12.
6. Bigger numbers: Expand $5(2x + 3)$. Build it on the board to check.
   10x + 15
   5 × 2x = 10x and 5 × 3 = 15.
7. Subtraction inside: Expand $4(3x - 2)$.
   12x - 8
   4 × 3x = 12x and 4 × (−2) = −8.
8. Quick check: Quick check before a negative outside: what is $-2 \times (-5)$?
   10
   Same signs give a positive: 10 (Number Foundations, Multiplying and dividing integers).
9. A negative outside: Expand $-2(x - 5)$.
   -2x + 10
   −2 × x = −2x and −2 × (−5) = +10, so −2x + 10.
10. Collect afterwards: Expand and simplify $2(x + 3) + 3x - 1$. Use p and q for the terms after the bracket.
    5x + 5
    2x + 6 + 3x − 1 = 5x + 5.
11. Two brackets: Edge case, two brackets: expand and simplify $2(x + 3) + 3(x - 1)$.
    5x + 3
    2x + 6 + 3x − 3 = 5x + 3. Expand each bracket on its own, then collect.
12. Positive: Expand $6(x + 2)$.
    6x + 12
    6 × x = 6x and 6 × 2 = 12.
13. Negative outside: Expand $-3(2x - 4)$.
    -6x + 12
    −3 × 2x = −6x and −3 × (−4) = +12.
14. Full working: Expand and simplify $4(2x - 3) + 5$. Write one step per line.
    4(2x - 3) + 5 → 8x - 12 + 5 → 8x - 7
    8x − 12 + 5 = 8x − 7.
15. Review: Review: which expression means "add 3 to $n$, then multiply by 5"?
    $5(n + 3)$
    Brackets group the part done first: 5(n + 3).
16. brackets backwards: Set $k$, $a$ and $b$ so the bracket expands to $6x + 12$, with $k$ bigger than 1.
    
    2(3x + 6), 3(2x + 4) and 6(x + 2) all expand to 6x + 12. Going backwards like this is called factorising.

Solutions

One possible solution for each task. Other correct programs are fine.

one change at a time › Bigger numbers

Open task: any program that does what the step asks.

one change at a time › Subtraction inside

Open task: any program that does what the step asks.

one change at a time › Quick check

Open task: any program that does what the step asks.

one change at a time › A negative outside

Open task: any program that does what the step asks.

one change at a time › Collect afterwards

Open task: any program that does what the step asks.

one change at a time › Two brackets

Open task: any program that does what the step asks.

brackets backwards

Open task: any program that does what the step asks.

Stretch and support

Stretch: 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.

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.

Coach prompt (optional)

For a student who has done the lesson and is still stuck. They paste it into an AI tool your school allows; it asks questions about this lesson's one idea and doesn't give answers. Students see it on the course recap page only through a link ending in ?coach=1: recap with coach prompts.

You are a patient maths coach for a Year 8 student in Australia. Help me understand ONE idea only, from the lesson "Expanding brackets".

The idea: Expanding and simplifying linear expressions: I can Students expand brackets in linear expressions using the distributive property and collect like terms to simplify.

What understanding looks like:
- Correctly expands brackets including those with negative coefficients.
- Collects like terms accurately.
- Does not partially distribute (e.g. only multiply the first term).

Mistakes students often make here:
- A student multiplies only the first term (Sam's mistake).
- A student gets the sign wrong with a negative inside (4(3x − 2) = 12x + 8).
- A student expands correctly but stops before collecting.

How to coach me (rules from the Waypoints Kit coach):
- Ask me one short question at a time (2 to 4 sentences) and wait for my answer. Never give me the answer before you have heard my thinking, and don't hide the answer inside your question.
- Start in Discover: ask me to explain the idea in my own words, or to predict what happens in a simple example. I can switch any time: "teach it back" (you play a beginner and I explain), "spot the flaw" (you show me an answer with one common mistake and I find it), or "check" (I explain, and you tell me what is solid and what to revisit).
- If I am stuck or say "just tell me", ask what I think first, then give a small hint or a simpler example. Explain only after I have had a real go.
- Stay in this idea: don't teach later ideas or add extra content. If I ask about something else, say it is for another lesson and bring me back.
- After you help, ask me one reflective question, such as how I know my answer is right or what I would check.
- Praise my effort and method, not my ability.
- You can be wrong: if I rely on something you said, remind me to check it against my notes or the lesson. If you don't know, say so.
- Use simple words.
- Stop when I can explain the idea in my own words and get one new example right. Then tell me to go back to the lesson "Expanding brackets".

Exit ticket

Expand and simplify 5(x − 2) + 3x.

5x − 10 + 3x = 8x − 10.

What each step is for

• 3(x + 2): Committing to an expansion of 3(x + 2) before building it tests whether they think the 3 reaches the 2.
• build it with tiles: Building k rows of the bracket and counting the tiles shows the number outside multiplies every term inside, not just the first.
• multiply every term: Expanding multiplies each term in the bracket by the number outside, signs included; collecting like terms then tidies up. Sam's mistake multiplies only the first term.
• one change at a time: 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.
• expand and simplify: Expanding with each sign, collecting, and full working, mixed; the review keeps writing brackets from words fresh.
• brackets backwards: Finding a bracket that expands to a given expression reverses the lesson, and shows more than one bracket can give the same result.

Where it 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.