An expression for any case

Classroom resources

Load these into Waypoints Prepare: choose Load URL and paste the copied address, or download the file and import it. In Prepare, the worksheet's Teacher edition shows the answers; the student copy hides them.

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Students will be able to

• I can say what the letter in an expression stands for, a number that changes.
• I can write an expression for a situation with a once-only amount and an amount for each one.
• I can test my expression with a case I can check in words, then use it for a big case.

Timing

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

Before class

Do the Modify boards yourself first. The mystery boards follow the situation, so students check their expression against them before they answer.

The new move is saying what the letter stands for before writing anything. Two common errors: letting the letter be the thing worked out ("Let n be the cost"), and reading the letter as the object (Pia's mistake: 3a means 3 apples). Insist on the sentence "Let ___ be the number of ___".

The once-only amount is a constant added on; the amount for each one is multiplied by the letter. This is the same structure as y = mx + c in Linear relations, so say so, but do not name it yet.

Running the lesson

1. Predict: take a show of hands. Ask a student who chose 'the total cost' what n would be for a trip of 30 students.
2. Run: before moving n, ask what the cost is for 0 students. What does the 50 mean?
3. Investigate: model the worked example live. Write 'Let g be...' first, every time.
4. Modify: ask for the sentence 'Let ___ be the number of ___' before each expression.
5. Practise: for the plumber questions, ask which amount is paid once.
6. Make: have students swap situations and write each other's expressions.

When students get stuck

Answers to the checks

1. what should the letter be?: A school trip costs 12 dollars per student plus a 50 dollar bus. You want one expression for the total cost. What should the letter stand for?
   The number of students. It is the quantity that changes from trip to trip. The cost is worked out from it (12n + 50), and 12 and 50 are constants.
2. one expression, many trips: $n$ is the number of students on the trip. How much does a trip of 20 students cost?
   
3. A hall: A phone plan costs 15 dollars a month plus 4 dollars per GB. Let $g$ be the number of GB used. Write an expression for the monthly cost.
   4g + 15
   4g + 15: 4 dollars for each of g GB, plus 15 dollars once.
4. Find the mistake: Zoe starts with "Let $n$ be the cost" for tickets that cost 9 dollars each. Why is that not a good start?
   The cost changes with the number of tickets, so the letter should stand for the number of tickets
   The cost depends on how many tickets there are. Let n be the number of tickets, then the cost is 9n.
5. Your turn: Pia writes the cost of $a$ apples at 3 dollars each. Click the line where her thinking goes wrong.
   line 2
   Line 2: 3a means 3 × a: 3 dollars times the number of apples. It does not mean "3 apples". The letter a stands for how many apples, a number.
6. A taxi: A taxi costs 3 dollars to get in plus 2 dollars per km. Let $d$ be the number of km. Write an expression for the fare.
   2d + 3
   2d + 3. Check with d = 4: 8 + 3 = 11 dollars.
7. A tank: A tank holds 100 litres and loses 4 litres each minute. Let $m$ be the minutes. Write an expression for the litres left.
   100 - 4m
   100 − 4m: it starts at 100 and 4 litres go each minute. Check with m = 5: 100 − 20 = 80.
8. Two groups: Adult tickets cost 8 dollars and child tickets cost 5 dollars. Let $a$ be the number of adults and $c$ the number of children. Write an expression for the total cost.
   8a + 5c
   8a + 5c: 8 dollars for each of a adults, plus 5 dollars for each of c children.
9. A big case: Tickets cost $T = 12n + 6$ dollars for $n$ tickets (6 dollars booking). Use the expression to find the cost of 50 tickets.
   606
   12 × 50 + 6 = 600 + 6 = 606 dollars.
10. A plumber: A plumber charges 60 dollars call-out plus 40 dollars per hour. Let $h$ be the number of hours. Write an expression for the cost.
    40h + 60
    40h + 60: 40 dollars for each hour, plus 60 dollars once.
11. Using it: Using $40h + 60$, what does a job of 3.5 hours cost, in dollars?
    200
    40 × 3.5 + 60 = 140 + 60 = 200 dollars.
12. The letter: Cupcakes cost 3 dollars each. What should the letter $c$ stand for in the expression $3c$ for the total cost?
    The number of cupcakes
    c is a number: how many cupcakes. 3c is then 3 dollars times that number.
13. Review one: Review: find $3(n + 4)$ when $n = 5$.
    27
    3 × (5 + 4) = 3 × 9 = 27.
14. Review two: Review: write an expression for "add 5 to $n$, then multiply the result by 2".
    2(n + 5)
    2(n + 5): add 5 first, then double the whole result.

Solutions

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

new situations › A taxi

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

new situations › A tank

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

new situations › Two groups

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

new situations › A big case

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

Stretch and support

Stretch: A shop sells pens at 2 dollars each and rulers at 3 dollars each, with a 1 dollar bag. Let p be pens and r rulers. Write the cost, then find it for 10 pens and 4 rulers.

Support: Give a sentence frame: Let ___ be the number of ___. Paid once: ___. For each: ___. 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 "An expression for any case".

The idea: Generalising arithmetic to algebraic expressions: I can Students recognise that algebraic expressions generalise arithmetic relationships, and can identify what an expression represents across multiple cases.

What understanding looks like:
- Writes an algebraic expression that correctly generalises a given arithmetic pattern.
- Verifies the expression by substituting specific values.
- Does not confuse the generalised expression with a specific equation to solve.

Mistakes students often make here:
- A student reads a letter as an object (Pia's mistake: 3a means 3 apples).
- A student multiplies the once-only amount (60h + 40 for a 60 dollar call-out).

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 (Expressions with brackets and multiple operations), 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 "An expression for any case".

Exit ticket

A gym costs 30 dollars to join plus 15 dollars a week. Write what your letter stands for, an expression for the total, and the cost after 8 weeks.

Let w be the number of weeks. Total = 15w + 30. After 8 weeks: 15 × 8 + 30 = 150 dollars.

What each step is for

• what should the letter be?: Committing to what the letter stands for exposes letting it stand for the thing that is being worked out, instead of the quantity that changes.
• one expression, many trips: Moving n and reading off the cost shows that one expression gives every case, and the parts mean what the story says.
• from a situation to an expression: Name the letter, write what changes, add what is paid once, then test with one case; the letter stands for a number, not for the objects.
• new situations: One change at a time: a fee plus a rate, a quantity that goes down, two letters, and using the expression for a big case.
• build and use: Naming the letter, building the expression and using it for a larger case, mixed; the review is brackets.
• a situation of your own: Inventing a situation and its expression reverses the lesson: from an expression back to a story.

Where it leads

This finishes the Expressions unit. The Equations unit starts from expressions like these: what number of students makes the trip cost 290? In Linear relations the same structure becomes y = mx + c and its graph.