Letters that vary

Classroom resources

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Students will be able to

• 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).

Timing

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

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.

Running the lesson

1. Predict: ask students who chose 'Nothing changes' to describe two different rectangles. Do they have the same length?
2. Run: before they reach l = 7, w = 2, ask: which numbers in the working never changed?
3. Investigate: model the worked example live. Draw the contrast on the board: formula (variables, many values) and equation (unknown, one value).
4. Modify: for π, point out that a letter is not automatically a variable.
5. Make: have two students compare 4s with s + s + s + s. Are they the same formula?

When students get stuck

Answers to the checks

1. what changes?: $P = 2l + 2w$ gives the perimeter of a rectangle. When you draw a different rectangle, which parts of the formula change?
   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.
2. change the rectangle: Make a rectangle 7 cm long and 2 cm wide. What is its perimeter?
   
3. Reading a formula: A plumber charges $C = 5h + 20$ dollars for a job of $h$ hours. Which are the variables?
   $C$ and $h$
   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.
4. Find the mistake: Kim works out the perimeter. Click the line where her thinking goes wrong.
   line 2
   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.
5. Unknown or variable?: In the equation $3x + 2 = 14$, what does $x$ stand for?
   An unknown: one number makes it true
   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.
6. A plumber: In $C = 5h + 20$, what is the charge for a job of 0 hours (just the call-out)?
   20
   C = 5 × 0 + 20 = 20. The constant 20 is the call-out fee.
7. A car trip: A car travels at 60 km/h: $d = 60t$. Which part stays the same for the whole trip?
   60, the speed
   The speed 60 is a constant. The time t and the distance d both change as the trip goes on.
8. Area: In $A = lw$ (the area of a rectangle), what happens to $A$ if $l$ doubles and $w$ stays the same?
   $A$ doubles
   lw means l × w. Twice the length times the same width gives twice the area. Try it on the board.
9. A letter that never changes: The circumference of a circle is $C = \pi d$. $\pi$ is a letter. Is it a variable or a constant?
   A constant: about 3.14 for every circle
   π (pi) is a Greek letter for one particular number, about 3.14, the same for every circle. C and d are the variables.
10. What the constant means: Water in a tank: $V = 3t + 10$ litres after $t$ minutes. What does the 10 mean?
    10 litres in the tank at the start
    When t = 0, V = 10: the 10 litres were there at the start. 3 litres are added each minute.
11. Variables: Which letters are variables in $T = 12n + 6$ (the cost of $n$ concert tickets)?
    $T$ and $n$
    T and n change with the order. 12 (dollars a ticket) and 6 (the booking fee) stay the same.
12. Unknown: In $2y + 1 = 9$, what is $y$?
    An unknown: $y = 4$ makes it true
    It is an equation, so y is an unknown: 2 × 4 + 1 = 9, and no other value works.
13. a formula of your own: A square has sides of length $s$. Write a formula for its perimeter $P$. Start with $P =$.
    4s
    P = 4s: four sides, each s long. 4 is a constant; P and s are variables.

Solutions

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

other formulas › A plumber

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

other formulas › A car trip

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

other formulas › Area

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

other formulas › A letter that never changes

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

Stretch and support

Stretch: 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.

Support: Give a two-column table, 'changes' and 'stays the same', to sort the parts of each formula into before answering.

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 "Letters that vary".

The idea: Variables as generalised quantities: I can Students recognise that variables represent quantities that can change, and distinguish a variable from an unknown constant.

What understanding looks like:
- Correctly identifies what each variable represents in a given formula.
- Explains the difference between a variable that can take many values and a specific unknown.
- Does not treat all variables as things to 'solve for'.

Mistakes students often make here:
- A student gives a letter its alphabet value (Kim's mistake).
- A student says the 2 in 2l is a variable because it is next to a letter.

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 "Letters that vary".

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.

What each step is for

• what changes?: Committing to which parts change exposes the idea that each letter stands for one fixed number.
• change the rectangle: Moving l and w and watching P change, while the 2s never do, shows what a variable is.
• variables, constants and unknowns: A variable is a letter for a quantity that can change; a constant stays the same; in an equation a letter can be an unknown, one particular number.
• other formulas: 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.
• read the formula: Naming variables, constants and unknowns, and saying what a constant means, in new formulas.
• a formula of your own: Writing a formula for a familiar shape reverses the lesson: from the quantities to the letters and constants.

Where it 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.