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# Finding an unknown

Algebra Foundations · Expressions

<!-- Speaker note: Students substitute the known values into a formula, solve the equation that is left for the unknown, and check by substituting back (AC9M7A01, 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** when a formula's answer is known but one letter is not, substitute what you know. That leaves an equation; solve it by undoing, then check.

- I can substitute the values I know into a formula to make an equation.
- I can solve the equation by undoing each operation, last first.
- I can check my answer by substituting it back into the formula.

<!-- Speaker note: Before class: Do the Run and Modify boards yourself first. The board lets students hunt for the answer by moving a slider: that is fine at Run, but in Investigate and Modify they should write the equation and solve it first, then use the board to check. The new move is putting together substitution (last lesson) and backtracking (Equations unit). The common error is repeating the operation instead of undoing it (Mia's mistake: 2w = 8, so w = 16). The check catches it every time, so make checking non-negotiable. One practise question needs only substitution, on purpose: students must decide which job each question asks for. -->

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## Why it matters

> You know how much fence you have and how long the garden bed must be. How wide can it be? A formula answers that too, worked backwards.

<!-- Speaker note: So far you knew the letters and worked out the answer. Sometimes you know the answer, and one letter is the **unknown**. Substitute what you know: what is left is an equation, and you can solve it by undoing, as in the Equations unit. -->

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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 rectangle has perimeter $P = 20$ cm and length $l = 6$ cm. Using $P = 2l + 2w$, what is its width $w$?**

- 4
- 7
- 8
- 14

<!-- Speaker note: What happens: 4. 20 = 2 × 6 + 2w, so 20 = 12 + 2w. Then 2w = 8 and w = 4. (8 forgets to halve; 14 takes the length away only once.) Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask students who chose 14 or 8 to check their width in P = 2l + 2w. -->

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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%2Ffinding-unknowns%2Fstep-2&code=widget%3A+formula%0Aformula%3A+P+%3D+2l+%2B+2w%0Aslider+l%3A+1%2C+10%2C+6%0Aslider+w%3A+1%2C+8%2C+1%0Ashape%3A+rectangle%0Aunits%3A+cm%0A&fresh=1}}

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

<!-- Speaker note: Change $w$ one step at a time. How much does $P$ go up for each 1 cm of width? Why? -->

<!-- Speaker note: Teacher: ask why P goes up 2 for each 1 cm of width. -->

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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?

* 4. 20 = 2 × 6 + 2w, so 20 = 12 + 2w. Then 2w = 8 and w = 4. (8 forgets to halve; 14 takes the length away only once.)

<!-- 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>

## Area

Here is how to find an unknown in a formula.

$$P = 2l + 2w$$

1) $P = 20, \; l = 6$ — Write what you know. The unknown is $w$.
2) $20 = 2 \times 6 + 2w$ — Substitute. Now it is an equation with one unknown.
3) $20 = 12 + 2w$ — Work out what you can.
4) $8 = 2w$ — Undo the + 12: take 12 from both sides.
5) $w = 4$ — Undo the × 2: divide both sides by 2.
6) $2 \times 6 + 2 \times 4 = 20 \;\checkmark$ — Check by substituting back into the formula.

<!-- 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 "Write the formula." Line 1: write P = 20, l = 6 and say "Write what you know. The unknown is w." Line 2: write 20 = 2 × 6 + 2w and say "Substitute. Now it is an equation with one unknown." Line 3: write 20 = 12 + 2w and say "Work out what you can." Line 4: write 8 = 2w and say "Undo the + 12: take 12 from both sides." Line 5: write w = 4 and say "Undo the × 2: divide both sides by 2." Line 6: write 2 × 6 + 2 × 4 = 20 ✓ and say "Check by substituting back into the formula." -->

<!-- Speaker note: Teacher: model the worked example live. Ask after each line: what did we undo? -->

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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>

## Area

**A rectangle has area $A = 24$ cm² and length $l = 6$ cm. Using $A = lw$, find its width $w$.**

* Answer: 4

<!-- Speaker note: 24 = 6w, so w = 24 ÷ 6 = 4. Check: 6 × 4 = 24. -->

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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>

## A plumber

**A plumber charges $C = 5h + 20$ dollars. A job cost 45 dollars. How many hours, $h$?**

* Answer: 5

<!-- Speaker note: 45 = 5h + 20, so 5h = 25 and h = 5. Check: 5 × 5 + 20 = 45. -->

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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

**Mia finds the width of a rectangle. Click the line where she goes wrong.**

```
 1  P = 2l + 2w, with P = 20 and l = 6.
 2  20 = 2 × 6 + 2w
 3  20 = 12 + 2w
 4  8 = 2w
 5  w = 16
```

* Answer: line 5

<!-- Speaker note: Line 5: 2w = 8 means 2 × w = 8. To undo × 2, divide by 2: w = 4. Mia multiplied again. Check: 2 × 6 + 2 × 16 = 44, not 20. -->

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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 unknowns



- **A car trip:** Each task loads a new formula. Write the equation first, solve it, then set the board to check.
- **A different unknown:** 
- **Going down:** 
- **A box:** 

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

<!-- Speaker note: What this step is for: One change at a time: a decimal answer, a different letter unknown, a formula that subtracts, and three letters with one unknown. Hints, in order: 150 = 60t. Undo × 60. / 30 = l × 5. Undo × 5. / 60 = 100 − 4m. How much charge has been used? / 40 has been used, at 4 a minute. For students who need more support: Give a template: formula; what I know; substitute; undo; check. For early finishers: A rectangle's length is 3 cm more than its width, and its perimeter is 30 cm. Find both sides. When students get stuck: A student repeats the operation instead of undoing it (Mia's mistake). Ask: Does your answer check? Put it back into the formula. Fix: Undo × with ÷, and + with −. Check every answer by substituting back. / A student puts the known answer in for the wrong letter (45 as h). Ask: Which letter is the cost? Which is the hours? Fix: Write each known value next to its letter before substituting. -->

<!-- Speaker note: Teacher: for the battery, ask what the minus means before anyone calculates. -->

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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>

## Tickets

**Concert tickets cost $T = 12n + 6$ dollars. An order cost 66 dollars. How many tickets?**

* Answer: 5

<!-- Speaker note: 66 = 12n + 6, so 12n = 60 and n = 5. -->

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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>

## Substitute or solve?

**Using $T = 12n + 6$, what do 3 tickets cost?**

* Answer: 42

<!-- Speaker note: T = 12 × 3 + 6 = 42 dollars. Here you know n, so you just substitute. -->

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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>

## A bike ride

**Average speed is $s = d / t$. A cyclist rides at 20 km/h for $t = 3$ hours. How far, $d$, in km?**

* Answer: 60

<!-- Speaker note: 20 = d ÷ 3, so d = 20 × 3 = 60 km. -->

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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

**Review: solve $4x - 3 = 17$.**

* Answer: 5

<!-- Speaker note: 4x = 20, so x = 5. Check: 4 × 5 − 3 = 17. -->

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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 garden bed

The bed must use exactly 24 m of edging, and be twice as long as it is wide.

Stretch: with the same 24 m, which rectangle gives the biggest area?

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/algebra/finding-unknowns/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Meeting two conditions at once (a fixed perimeter and a shape rule) reverses the lesson: the student chooses the unknowns. Hints, in order: The length is twice the width: P = 2(2w) + 2w = 6w. / 6w = 24. What is w? -->

<!-- Speaker note: Teacher: ask how they found it: guessing with the slider, or writing 6w = 24? -->

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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 T = 15w + 30 dollars for w weeks. Someone paid 120 dollars. How many weeks? Check your answer.**

* 120 = 15w + 30, so 15w = 90 and w = 6. Check: 15 × 6 + 30 = 120.

<!-- 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: the Expressions unit continues with writing expressions from words. Solving for an unknown is the Equations unit, and in Linear relations it becomes finding x for a given y on a graph. -->