When brackets are needed

Classroom resources

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Students will be able to

• I can say which step is done first, and what is then done to its whole result.
• I can put brackets around the part that is done first, and leave them out when they are not needed.
• I can show with a number that 2(n + 3) and 2n + 3 are different.

Timing

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

Before class

Do the Run and Modify boards yourself first. They hide the formula: the board does what the words say, so students check their brackets against it.

The new move is brackets as "do this part first". The common error is leaving them out (Jo's mistake: 6x + 2), so the multiplier only reaches the x. Always check with one number: add 2 to 1 and multiply by 6 is 18, but 6 × 1 + 2 is 8.

Not every sentence needs brackets. The 'books' and 'multiply then add' questions are there on purpose: when a later step is add or subtract, no brackets are needed. Ask: "what is done to what?"

Running the lesson

1. Predict: take a show of hands. Ask students to try 4 stickers on each option.
2. Run: ask them to say which of 2 × 4 + 3 and 2(4 + 3) the board matches, and why.
3. Investigate: model the worked example live. Underline the part done first, then bracket it.
4. Modify: before the no-brackets task, ask 'is anything done to a whole result here?'
5. Make: ask students to say the steps in words before they check their expression.

When students get stuck

Answers to the checks

1. stickers: Dan has $n$ stickers. He gets 3 more, then doubles the lot. Which expression is the number he ends with?
   2(n + 3). He adds 3 first (n + 3), then doubles all of it. 2n + 3 would double only his own stickers and add 3 afterwards.
2. check the words: Set $n = 4$. Dan has 4 stickers, gets 3 more and doubles the lot. How many does the board show?
   
3. Add 3, then double: Write an expression for "add 4 to $n$, then multiply the result by 5".
   5(n + 4)
   5(n + 4): the brackets keep n + 4 together. Check with n = 1: 1 + 4 = 5, then × 5 = 25, and 5(1 + 4) = 25.
4. Find the mistake: Which expression means "subtract 3 from $n$, then divide the result by 2"?
   $\frac{n - 3}{2}$
   The whole of n − 3 is divided by 2. The fraction bar works like brackets. Check with n = 11: 11 − 3 = 8, and 8 ÷ 2 = 4.
5. Your turn: Jo writes "add 2 to $x$, then multiply by 6". Click the line where she goes wrong.
   line 3
   Line 3: 6x + 2 multiplies only x, then adds 2. To multiply the whole x + 2 by 6 she needs brackets: 6(x + 2). Check with x = 1: 1 + 2 = 3, × 6 = 18, but 6 × 1 + 2 = 8.
6. Subtract, then multiply: Write an expression for "subtract 2 from $n$, then multiply the result by 3".
   3(n - 2)
   3(n − 2). Check with n = 5: 5 − 2 = 3, then × 3 = 9.
7. No brackets: Write an expression for "multiply $n$ by 3, then add 2".
   3n + 2
   3n + 2. Brackets are only needed when a later step applies to a result that was made by an earlier add or subtract.
8. Two letters: Write an expression for "add $a$ and $b$, then multiply the result by 4".
   4(a + b)
   4(a + b). Check with a = 2 and b = 3: 2 + 3 = 5, then × 4 = 20.
9. Halving: Write an expression for "add 5 to $n$, then halve the result".
   (n + 5)/2
   (n + 5)/2. Check with n = 3: 3 + 5 = 8, then halved is 4.
10. Books: Books cost 8 dollars each. Write an expression for the cost of $n$ books after taking 5 dollars off the total.
    8n - 5
    8n − 5: the books cost 8n, then 5 dollars is taken off.
11. A group: A group of $n$ adults and 3 children each pay 6 dollars. Write an expression for the total.
    6(n + 3)
    6(n + 3): the group has n + 3 people and each pays 6 dollars.
12. Which is bigger?: When $n = 5$, which is bigger, $2(n + 3)$ or $2n + 3$?
    $2(n + 3)$, which is 16
    2(5 + 3) = 2 × 8 = 16 and 2 × 5 + 3 = 13. Brackets change the answer.
13. Review: Review: find $4(n - 1)$ when $n = 6$.
    20
    4 × (6 − 1) = 4 × 5 = 20.
14. find the rule: The mystery board follows a rule in $n$. Move $n$ and watch $T$. Write an expression for $T$, using brackets.
    5(n - 1)
    T = 5(n − 1): subtract 1 from n, then multiply the result by 5.

Solutions

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

new sentences › Subtract, then multiply

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

new sentences › No brackets

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

new sentences › Two letters

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

new sentences › Halving

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

find the rule

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

Stretch and support

Stretch: Write 'double n, then add 3 more than n' in two ways. Which words need brackets? Test both with n = 5.

Support: Give a two-step frame: Step 1 (done first): ___; Step 2 (done to the result): ___. Then write the expression.

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 7 student in Australia. Help me understand ONE idea only, from the lesson "When brackets are needed".

The idea: Expressions with brackets and multiple operations: I can Students formulate algebraic expressions involving brackets, including those where brackets are required to preserve intended meaning.

What understanding looks like:
- Correctly writes expressions requiring brackets to preserve intended meaning.
- Can explain why brackets are necessary in a given expression.
- Evaluates expressions with brackets correctly applying order of operations.

Mistakes students often make here:
- A student writes 6x + 2 for 'add 2, then multiply by 6' (Jo's mistake).
- A student puts brackets everywhere (8(n − 5) for 'take 5 off the total').

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.
- If I seem to be missing the idea that comes before this one (Translating words to algebraic expressions), check it quickly, then come back.
- 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 "When brackets are needed".

Exit ticket

Write an expression for: (a) add 6 to n, then multiply by 3; (b) multiply n by 3, then add 6. Which one gives 15 when n = 3?

(a) 3(n + 6); (b) 3n + 6. (b): 3 × 3 + 6 = 15; (a) gives 27.

What each step is for

• stickers: Committing to an expression exposes leaving the brackets out, so the doubling only applies to n.
• check the words: The board follows the words exactly, so its number for n = 4 shows which expression keeps the order of the steps.
• where do the brackets go?: Brackets go around the part that is done first, when something is then done to the whole result.
• new sentences: One change at a time: subtract first, a sentence that needs no brackets, two letters, and dividing the whole result.
• brackets or not?: Choosing when brackets are needed, mixed, with one question where they are not; the review is substituting.
• find the rule: Finding an expression with brackets from a mystery board reverses the lesson: from numbers back to the order of steps.

Where it leads

Next: an expression for a whole situation, where you say what the letter stands for. Brackets come back when expressions are expanded and factorised (Year 8) and in Equations with brackets.