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# Changing one thing at a time

Linear relations · Rules, tables and graphs

<!-- Speaker note: Students vary one variable at a time in V = lwh, d = st and A = s², write the results as a table and describe the effect (AC9M7A06, nodes 1 and 2). About 35 minutes (Predict and Run 8 min · Investigate 8 min · Modify 10 min · Practise and Make 9 min). -->

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## Today

**We are learning to** to see what one variable does to a formula, change only that one and hold the rest still. Doubling h in V = lwh doubles V; doubling the side of a square makes its area four times as big.

- I can change one variable and hold the others still.
- I can describe in words what happens to the output.
- I can say which variable has the biggest effect, and why.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need to substitute into a formula (Substituting into formulas in Algebra). The one idea: to see what one letter does, change only that letter. Changing two at once makes it impossible to tell which did what (Tom's mistake). The wider idea is that a variable's effect depends on the formula: in V = lwh, doubling any one factor doubles V; in A = s², doubling s makes A four times as big. The second Modify question and the last are about comparing effects. The elaborations suggest spreadsheets. The Make task ends with a table: let students build it in a spreadsheet with a formula in the cells, so changing one cell updates the row. Machine-learning models are formulas with very many variables tuned in the same way (a talking point only). -->

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## Why it matters

> A scientist testing a plant changes the light, but not the water. In a formula, that is how you find what each letter does.

<!-- Speaker note: A **formula** has several variables, such as $V = l \times w \times h$. To find out what one variable does to the output, **vary it systematically**: change that one, keep every other variable the same, and write the results in a table. | $h$ | 1 | 2 | 3 | 4 | |---|---|---|---|---| | $V$ (with $l = 2$, $w = 3$) | 6 | 12 | 18 | 24 | -->

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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 box has $l = 2$, $w = 3$ and $h = 2$, so $V = 12$. What is $V$ when $h$ is doubled to 4 and the rest stay the same?**

- 12
- 14
- 16
- 24

<!-- Speaker note: What happens: 24. V = 2 × 3 × 4 = 24: doubling one factor doubles the product. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what is staying the same. -->

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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%2Flinear-relations%2Fvary-formulas%2Fstep-2&code=widget%3A+formula%0Aformula%3A+V+%3D+lwh%0Aslider+l%3A+1%2C+6%2C+2%0Aslider+w%3A+1%2C+6%2C+3%0Aslider+h%3A+1%2C+6%2C+1%0Aunits%3A+cm%C2%B3%0A&fresh=1}}

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

<!-- Speaker note: Move one slider at a time. -->

<!-- Speaker note: Teacher: ask students to say the pattern in V before moving the next slider. -->

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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?

* 24. V = 2 × 3 × 4 = 24: doubling one factor doubles the product.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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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>

## The pattern in the table

**With $l = 2$ and $w = 3$, $V$ is 6, 12, 18 when $h$ is 1, 2, 3. By how much does $V$ go up for each extra 1 in $h$?**

* Answer: 6

<!-- Speaker note: V goes up by 6 each time, which is l × w = 2 × 3. -->

<!-- Speaker note: Teacher: ask Tom what he should change. -->

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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 fair test?

**Tom wants to know what h does to V. Click the line where his test stops being fair.**

```
 1  Tom wants to know what h does to V = lwh.
 2  He sets l = 2, w = 3 and h = 1, so V = 6.
 3  Then he changes h to 2 and w to 4, so V = 16.
 4  V went from 6 to 16, so doubling h made V go up by 10.
```

* Answer: line 3

<!-- Speaker note: Line 3: he changes w as well as h, so he cannot tell which caused the change in V. Change h alone: V = 2 × 3 × 2 = 12. -->

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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 to change

**To find the effect of $w$ on $V = lwh$, what should you do?**

- Change w and h together
- Change w only, and keep l and h the same
- Change l, w and h together
- Change l and h only

* Answer: Change w only, and keep l and h the same

<!-- Speaker note: Isolate one variable at a time: that is a fair test. -->

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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



- **Find the speed:** Hold the time at 3.
- **Both double:** 
- **Side and area:** 
- **The biggest change:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/vary-formulas/?full=step-4#step-4)

<!-- Speaker note: What this step is for: The same method on distance and on a square: both variables doubling, a square growing faster than its side, and the variable with the biggest effect. Hints, in order: d = s × t, so 150 = s × 3. / Try speeds of 30, 40, 50. For students who need more support: Give a printed table with l and w filled in and a blank row for V. For early finishers: In d = st, find three different pairs of s and t that give d = 120. What pattern do you see? Which graph would show it? The hard part, on purpose: Holding the others still. Ask students to say which letters stay the same before they move a slider. When students get stuck: A student moves two sliders and describes the result. Ask: Which slider caused that change? Fix: Change one, write down V, then change the next. / A student says doubling the side doubles the area. Ask: What is 6 × 6 compared with 3 × 3? Fix: A square's area has the side twice: doubling gives 4 times. / A student reads 2l as 2 and l side by side. Ask: What does the working on the board show between 2 and l? Fix: 2l means 2 × l. -->

<!-- Speaker note: Teacher: ask whether the distance question works for any speed. -->

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## Vary and describe

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Flinear-relations%2Fvary-formulas%2Fstep-5&code=widget%3A+formula%0Aformula%3A+V+%3D+lwh%0Aslider+l%3A+1%2C+6%2C+1%0Aslider+w%3A+1%2C+6%2C+1%0Aslider+h%3A+1%2C+6%2C+1%0Aunits%3A+cm%C2%B3%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/vary-formulas/?full=step-5#step-5)

<!-- Speaker note: What this step is for: Reading tables, single changes and a substitution review, mixed; forgetting to hold the others is the planted slip; the substitution review keeps formulas fresh. -->

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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 V

**$V = lwh$ with $l = 2$, $w = 3$, $h = 4$. What is $V$?**

* Answer: 24

<!-- Speaker note: 2 × 3 × 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>

## Change w

**Now change only $w$, from 3 to 4. What is $V$?**

* Answer: 32

<!-- Speaker note: 2 × 4 × 4 = 32. -->

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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>

## Change t

**$d = st$ with $s = 60$ km/h. The time doubles from 2 to 4 hours. What is $d$ now, in km?**

* Answer: 240

<!-- Speaker note: 60 × 4 = 240 km, double the 120 km for 2 hours. -->

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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: substitution

**Review: $P = 2l + 2w$ with $l = 6$ and $w = 4$. What is $P$?**

* Answer: 20

<!-- Speaker note: 2 × 6 + 2 × 4 = 12 + 8 = 20. -->

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## Make: a test of your own

Now write a table (on paper or in a spreadsheet) that varies $h$ from 1 to 6 for your box, and describe in a sentence what happens to $V$ as $h$ goes up.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/linear-relations/vary-formulas/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Choosing numbers to meet a target then describing the effect of each variable puts the method to use. Hints, in order: Find three numbers up to 6 that multiply to 60. / Try l = 6: then w × h must be 10. -->

<!-- Speaker note: Teacher: ask for the table, then the sentence. -->

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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

**In V = lwh with l = 2, w = 3, h = 4, h goes to 8 and nothing else changes. What is V?**

* 48: 2 × 3 × 8, double the 24 it was.

<!-- 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 completes AC9M7A06. Where it leads: formulas in a spreadsheet, and changing the subject of a formula (Algebra, Year 8). -->