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# Letters that vary

Algebra Foundations · Expressions

<!-- Speaker note: Students tell the variables in a formula from the constants, and see that in an equation a letter can be one unknown number (AC9M7A01, node 1). 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** in a formula, letters stand for quantities that can change (variables) and numbers stay the same (constants). In an equation, a letter can stand for one unknown number.

- I can say which letters in a formula are variables and which numbers are constants.
- I can say what a constant means in a real formula, such as a call-out fee.
- I can tell a variable (can take many values) from an unknown (one value makes the equation true).

<!-- Speaker note: Before class: Do the Run board yourself first: change l and w and watch the working. The new move is reading a letter as a quantity that changes. Students often think each letter has one fixed value, or link a letter to its place in the alphabet (Kim's mistake). Keep saying the quantity aloud: "l, the length", not just "l". This lesson does not ask students to calculate much; the next lesson is substitution. The board does the arithmetic so attention stays on what changes. -->

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## Why it matters

> Recipes, phone bills and the perimeter of a garden bed all follow a formula. The letters are what changes; the numbers are what stays.

<!-- Speaker note: In maths a letter can stand for a number. A **formula** such as $P = 2l + 2w$ uses letters for quantities, here the perimeter $P$, length $l$ and width $w$ of a rectangle. This lesson is about what the letters and numbers in a formula mean. -->

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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

**$P = 2l + 2w$ gives the perimeter of a rectangle. When you draw a different rectangle, which parts of the formula change?**

- Only $P$ changes
- $l$, $w$ and $P$ change; the 2s stay the same
- The 2s change as well
- Nothing: each letter stands for one fixed number

<!-- Speaker note: What happens: l, w and P change, because different rectangles have different lengths, widths and perimeters. The 2s stay the same for every rectangle: there are always two lengths and two widths. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask students who chose 'Nothing changes' to describe two different rectangles. Do they have the same length? -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Falgebra%2Fvariables%2Fstep-2&code=widget%3A+formula%0Aformula%3A+P+%3D+2l+%2B+2w%0Aslider+l%3A+1%2C+10%2C+4%0Aslider+w%3A+1%2C+6%2C+3%0Ashape%3A+rectangle%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note: Change $l$ and $w$. Watch the rectangle, the working and $P$. Which numbers in the formula never change? -->

<!-- Speaker note: Teacher: before they reach l = 7, w = 2, ask: which numbers in the working never changed? -->

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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?

* l, w and P change, because different rectangles have different lengths, widths and perimeters. The 2s stay the same for every rectangle: there are always two lengths and two widths.

<!-- 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>

## Reading a formula

Here is how to read the parts of a formula.

$$P = 2l + 2w$$

1) $P, \; l, \; w$ — These letters are **variables**: quantities that change from one rectangle to the next.
2) $2$ — The 2s are **constants**: they are the same for every rectangle.
3) $l = 5, \; w = 3$ — For one particular rectangle, each variable has a value.
4) $3x + 2 = 14$ — In an equation, a letter can be an **unknown**: one particular number.
5) $x = 4$ — Only one value makes this equation true.

<!-- 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 P = 2l + 2w and say "The perimeter of a rectangle: twice the length plus twice the width." Line 1: write P, l, w and say "These letters are variables: quantities that change from one rectangle to the next." Line 2: write 2 and say "The 2s are constants: they are the same for every rectangle." Line 3: write l = 5, w = 3 and say "For one particular rectangle, each variable has a value." Line 4: write 3x + 2 = 14 and say "In an equation, a letter can be an unknown: one particular number." Line 5: write x = 4 and say "Only one value makes this equation true." -->

<!-- Speaker note: Teacher: model the worked example live. Draw the contrast on the board: formula (variables, many values) and equation (unknown, one value). -->

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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>

## Reading a formula

**A plumber charges $C = 5h + 20$ dollars for a job of $h$ hours. Which are the variables?**

- 5 and 20
- $C$ and $h$
- Only $h$

* Answer: $C$ and $h$

<!-- Speaker note: C and h change from job to job. 5 (dollars an hour) and 20 (the call-out fee) are constants: the same for every job. -->

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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>

## Find the mistake

**Kim works out the perimeter. Click the line where her thinking goes wrong.**

```
 1  P = 2l + 2w
 2  l is the 12th letter of the alphabet, so l = 12.
 3  w is the 23rd letter, so w = 23.
 4  P = 2 × 12 + 2 × 23 = 70.
```

* Answer: line 2

<!-- Speaker note: Line 2: a letter stands for a quantity, here the length, not its place in the alphabet. l can be any length: you need to measure or be told it. -->

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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>

## Unknown or variable?

**In the equation $3x + 2 = 14$, what does $x$ stand for?**

- An unknown: one number makes it true
- The letter x, not a number
- A variable that can be any number

* Answer: An unknown: one number makes it true

<!-- Speaker note: In an equation, the letter is an unknown: only x = 4 makes 3x + 2 = 14 true. In a formula, a letter is a variable that can take many values. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Other formulas



- **A plumber:** Each task loads a new formula. Change the variables and watch what stays the same.
- **A car trip:** 
- **Area:** 
- **A letter that never changes:** 

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

<!-- Speaker note: What this step is for: One change at a time: a formula with a constant added, one with a single variable, a product of two variables, and a letter that is a constant. For students who need more support: Give a two-column table, 'changes' and 'stays the same', to sort the parts of each formula into before answering. For early finishers: Write a formula for the cost of n pizzas at 15 dollars each with a 5 dollar delivery fee. Name every variable and constant, and say what each constant means. When students get stuck: A student gives a letter its alphabet value (Kim's mistake). Ask: What does l stand for in this formula? How would you find its value? Fix: A letter stands for a quantity. Its value is measured or given, and can change from one rectangle to the next. / A student says the 2 in 2l is a variable because it is next to a letter. Ask: Draw a different rectangle. Did the 2 change? Fix: Constants stay the same for every case. 2l means 2 × l: two lengths. -->

<!-- Speaker note: Teacher: for π, point out that a letter is not automatically a variable. -->

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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 the constant means

**Water in a tank: $V = 3t + 10$ litres after $t$ minutes. What does the 10 mean?**

- The tank fills in 10 minutes
- 10 litres in the tank at the start
- 10 litres added every minute

* Answer: 10 litres in the tank at the start

<!-- Speaker note: When t = 0, V = 10: the 10 litres were there at the start. 3 litres are added each minute. -->

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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>

## Variables

**Which letters are variables in $T = 12n + 6$ (the cost of $n$ concert tickets)?**

- $T$ and $n$
- 12 and 6
- Only $T$

* Answer: $T$ and $n$

<!-- Speaker note: T and n change with the order. 12 (dollars a ticket) and 6 (the booking fee) stay the same. -->

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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>

## Unknown

**In $2y + 1 = 9$, what is $y$?**

- An unknown: $y = 4$ makes it true
- A variable: any value works
- A constant

* Answer: An unknown: $y = 4$ makes it true

<!-- Speaker note: It is an equation, so y is an unknown: 2 × 4 + 1 = 9, and no other value works. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a formula of your own

Name the variables and the constant in your formula.

Stretch: write a formula for the perimeter of a triangle with sides $a$, $b$ and $c$.

<!-- Speaker note: What this step is for: Writing a formula for a familiar shape reverses the lesson: from the quantities to the letters and constants. Hints, in order: A square has four equal sides, each s long. / How many lots of s make the perimeter? -->

<!-- Speaker note: Teacher: have two students compare 4s with s + s + s + s. Are they the same formula? -->

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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 taxi charges T = 2d + 4 dollars for d km. Name the variables and the constants, and say what the 4 means.**

* Variables: T and d. Constants: 2 and 4. The 4 is the fee to get in, paid even for 0 km.

<!-- 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: Next: substituting into formulas, then finding an unknown in a formula. Later, linear relations use the same letters: in y = mx + c, x and y vary and m and c are constants for one line. -->