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# An expression for any case

Algebra Foundations · Expressions

<!-- Speaker note: Students say what a letter stands for, write an expression for a situation with a fee and a rate, test it with a case they can check, and use it for any number (AC9M7A02, node 3). 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** say what the letter stands for, write an expression for the situation, then test it with a number you can check. One expression then gives the answer for every case.

- 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.

<!-- Speaker note: 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. -->

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## Why it matters

> A party for 5 people, 50 people or 500: one expression gives the cost for all of them, if you say clearly what the letter stands for.

<!-- Speaker note: So far each expression was written from one sentence about a number. A real situation has parts: a fee, a price for each one, a number of people. To write one expression for all of it, first say **what the letter stands for**, then build the expression, then **test** it with a case you can check in words. After that one expression gives the answer for any number. -->

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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

**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 bus
- The total cost
- The 12 dollars
- The number of students

<!-- Speaker note: What happens: 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. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: take a show of hands. Ask a student who chose 'the total cost' what n would be for a trip of 30 students. -->

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## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Fgeneral-rules%2Fstep-2&code=widget%3A+formula%0Aformula%3A+C+%3D+12n+%2B+50%0Aslider+n%3A+0%2C+40%2C+5%0Aunits%3A+dollars%0A&fresh=1}}

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

<!-- Speaker note: The rule is $C = 12n + 50$. Set $n$ to 0: what does the 50 mean? What does the 12 mean? Was your prediction right? -->

<!-- Speaker note: Teacher: before moving n, ask what the cost is for 0 students. What does the 50 mean? -->

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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?

* 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.

<!-- 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>

## A hall

Here is how to build an expression for a situation.

$$\text{Hall: 60 dollars, plus 5 dollars per guest}$$

1) $\text{Let } g = \text{number of guests}$ — Say what the letter stands for. The number of guests changes, so $g$ stands for that.
2) $5g$ — Each guest costs 5 dollars, so $g$ guests cost $5 \times g$.
3) $5g + 60$ — Add the hire, which is paid once.
4) $g = 10: \; 5(10) + 60 = 110$ — Test with a case you can check in words: 10 guests cost 50 + 60 = 110 dollars.
5) $g = 40: \; 5(40) + 60 = 260$ — Now the same expression gives any case, with no counting.

<!-- 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 Hall: 60 dollars, plus 5 dollars per guest and say "A hall costs 60 dollars to hire, and then 5 dollars for each guest." Line 1: write Let g = number of guests and say "Say what the letter stands for. The number of guests changes, so g stands for that." Line 2: write 5g and say "Each guest costs 5 dollars, so g guests cost 5 × g." Line 3: write 5g + 60 and say "Add the hire, which is paid once." Line 4: write g = 10: 5(10) + 60 = 110 and say "Test with a case you can check in words: 10 guests cost 50 + 60 = 110 dollars." Line 5: write g = 40: 5(40) + 60 = 260 and say "Now the same expression gives any case, with no counting." -->

<!-- Speaker note: Teacher: model the worked example live. Write 'Let g be...' first, every time. -->

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## 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.**

* Answer: 4g + 15

<!-- Speaker note: 4g + 15: 4 dollars for each of g GB, plus 15 dollars once. -->

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## 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?**

- 9 should be the letter
- The cost changes with the number of tickets, so the letter should stand for the number of tickets
- A letter can never stand for a cost

* Answer: The cost changes with the number of tickets, so the letter should stand for the number of tickets

<!-- Speaker note: The cost depends on how many tickets there are. Let n be the number of tickets, then the cost is 9n. -->

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## Your turn

**Pia writes the cost of $a$ apples at 3 dollars each. Click the line where her thinking goes wrong.**

```
 1  Apples cost 3 dollars each. Write the cost of a apples.
 2  Cost = 3a. I read 3a as "3 apples".
 3  So the cost of a apples is 3a.
```

* Answer: line 2

<!-- Speaker note: 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. -->

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## New situations



- **A taxi:** The board follows the situation and shows the result. Write your expression, then check it with a number.
- **A tank:** 
- **Two groups:** 
- **A big case:** 

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

<!-- Speaker note: What this step is for: 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. For students who need more support: Give a sentence frame: Let ___ be the number of ___. Paid once: ___. For each: ___. Expression: ___. For early finishers: 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. When students get stuck: A student reads a letter as an object (Pia's mistake: 3a means 3 apples). Ask: If a is 5, what is 3a? Is it 3 apples or something else? Fix: a stands for a number, how many apples. 3a is 3 dollars times that number. / A student multiplies the once-only amount (60h + 40 for a 60 dollar call-out). Ask: Does the call-out change if the job is 1 hour or 10 hours? Fix: An amount paid once is added on. Only the price for each hour is multiplied by h. -->

<!-- Speaker note: Teacher: ask for the sentence 'Let ___ be the number of ___' before each expression. -->

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## 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.**

* Answer: 40h + 60

<!-- Speaker note: 40h + 60: 40 dollars for each hour, plus 60 dollars once. -->

<!-- Speaker note: Teacher: for the plumber questions, ask which amount is paid once. -->

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## Using it

**Using $40h + 60$, what does a job of 3.5 hours cost, in dollars?**

* Answer: 200

<!-- Speaker note: 40 × 3.5 + 60 = 140 + 60 = 200 dollars. -->

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## The letter

**Cupcakes cost 3 dollars each. What should the letter $c$ stand for in the expression $3c$ for the total cost?**

- The cost
- The number of cupcakes
- Cupcakes

* Answer: The number of cupcakes

<!-- Speaker note: c is a number: how many cupcakes. 3c is then 3 dollars times that number. -->

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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 one

**Review: find $3(n + 4)$ when $n = 5$.**

* Answer: 27

<!-- Speaker note: 3 × (5 + 4) = 3 × 9 = 27. -->

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## Review two

**Review: write an expression for "add 5 to $n$, then multiply the result by 2".**

* Answer: 2(n + 5)

<!-- Speaker note: 2(n + 5): add 5 first, then double the whole result. -->

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## Make: a situation of your own

Make up a situation with a fee paid once and a price for each one: a club, a game, a delivery. Write "Let ___ be the number of ___." Then write an expression for the total, and test it with one case.

Stretch: swap with a partner and see if they can write your expression from the story alone.

<!-- Speaker note: What this step is for: Inventing a situation and its expression reverses the lesson: from an expression back to a story. -->

<!-- Speaker note: Teacher: have students swap situations and write each other's expressions. -->

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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

**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.

<!-- 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: 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. -->