Changing one thing at a time

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.

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

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

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

Doubling h

Keep l and w the same.

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?12141624

TryTry it on the board and compare the result with your prediction.

Change only h

Move one slider at a time.

Try it yourselfYour edits are saved in this browser.

Keep $l = 2$ and $w = 3$ . Move $h$ up from 1 to 4 and watch $V$ . Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

A fair test

Look at the V row.

Read Tom's working. What did he change?

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 $h$ ?

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

To find the effect of

w

V = lwh

, what should you do?Change w and h togetherChange w only, and keep l and h the sameChange l, w and h togetherChange l and h only

ModifyChange the problem on purpose and see what happens.

Other formulas

Hold the time at 3.

Need a hint?

d = s × t, so 150 = s × 3.

Another hint

Try speeds of 30, 40, 50.

Try it yourselfYour edits are saved in this browser.

Set the time to 3 hours. Choose the speed so the distance is 150 km.

Use the board. This ticks when you get there.

d = st

. Speed and time both double. What happens to the distance?It becomes 3 times as bigIt stays the sameIt doublesIt becomes 4 times as big

A square has area $A = s \times s$ . Make the side 6 cm. What does the area become?

Use the board. This ticks when you get there.

V = lwh

with

l = 2

w = 3

h = 4

(V = 24). Adding 1 to just one variable, which gives the biggest V?Adding 1 to w (V = 32)They all give the same VAdding 1 to l (V = 36)Adding 1 to h (V = 30)

PractiseA few on your own, to check you have it.

Vary and describe

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

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

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

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

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

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.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Find three numbers up to 6 that multiply to 60.

Another hint

Try l = 6: then w × h must be 10.

Make a box with volume 60 where $l > w > h$ (all different).

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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).

Running the lesson

Predict: ask what is staying the same.
Run: ask students to say the pattern in V before moving the next slider.
Investigate: ask Tom what he should change.
Modify: ask whether the distance question works for any speed.
Make: ask for the table, then the sentence.

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.

Stretch and support

Stretch: 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?

Support: Give a printed table with l and w filled in and a blank row for V.

Where it leads

This completes AC9M7A06.

Where it leads: formulas in a spreadsheet, and changing the subject of a formula (Algebra, Year 8).

What each step is for

doubling h: Committing to what doubling h does to V shows whether they see volume as a product, where doubling one factor doubles the output.
change only h: Moving only h and reading V, with l and w held still, shows the pattern for one variable at a time.
a fair test: A fair test changes one variable (Tom changes two); a table shows the pattern; some variables matter more than others.
other formulas: 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.
vary and describe: Reading tables, single changes and a substitution review, mixed; forgetting to hold the others is the planted slip; the substitution review keeps formulas fresh.
a test of your own: Choosing numbers to meet a target then describing the effect of each variable puts the method to use.

Now you can

Change one variable and hold the others still.
Describe in words what happens to the output.
Say which variable has the biggest effect, and why.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.You reached the endBack to Linear relations

Source: content/maths/linear-relations/vary-formulas/lesson.md