DSE Maths | Variation (direct, inverse, joint) Lesson-Prep Guide (for tutors)

DSE Maths | Variation (direct, inverse, joint) Lesson-Prep Guide (for tutors) | model, find k, evaluate

2026-11-02

Tutor lesson-prep guides · Secondary Maths (DSE)

Variation has exactly one routine: write k, solve for k, then use k. Students only ever fail by dropping k or reversing direct and inverse.

Objectives — Students can (1) turn a sentence into y = kx, y = k/x or y = kx squared; (2) find k from one known pair; (3) evaluate a new value and self-check by asking whether the other quantity should rise or fall.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Solve for an unknown	y = kx with x = 3 and y = 12, find k	k = 4
Fractions	k over 12 equals 4, find k	48
Squares	7 squared	49

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-5 min	Diagnostic	Three checks on substitution and squares	Answer on the spot
5-18 min	Model direct variation	From a sentence to an equation, then k, then the value	Follow and write the three steps
18-32 min	Model inverse variation	Same numbers reversed, compare directions	State the direction aloud
32-48 min	Joint and partial variation	Model z depending on x and y together	Write the relation themselves
48-60 min	Error correction and exit check	Name four errors, ask three exit questions	Answer orally and self-assess

03 Teaching script (read it out)

Scene 1: Direct variation: write k first, never a bare ratio

The sentence y varies directly as the square of x means y = kx squared. Given x = 4 and y = 48, solve 48 = k times 4 squared, so k = 3.
Then for x = 7, y = 3 times 7 squared = 147.

Stress: writing the proportionality sign and substituting is where marks leak; put k in immediately.

Scene 2: Inverse variation: one rises, the other falls

The sentence y varies inversely as x means y = k/x. With x = 8 and y = 6, k = 8 times 6 = 48.
For x = 12, y = 48 over 12 = 4. Make them say the check aloud: x rose from 8 to 12 and y fell from 6 to 4.

Remind: a rising answer here means k/x was written as kx; go back and fix it.

Scene 3: Joint and partial variation: watch which variable sits on top

z varies directly as x and inversely as y becomes z = kx/y. If z is jointly proportional to x and y, it is z = kxy. One word decides the position, so circle the word inversely.
Example: z = kx/y with x = 2 and y = 3 giving z = 8 means k = 12; then x = 4 and y = 6 still gives z = 8, which shows doubling both changes nothing.

Stress: partial variation has two terms, for example y = kx + c, never a single term.

04 Question types and worked steps

Type	Example	Steps	Answer
Direct variation value	y varies as x squared; y = 48 when x = 4; find y when x = 7	1 write y = kx squared 2 find k 3 substitute	147
Inverse variation value	y varies inversely as x; y = 6 when x = 8; find y when x = 12	1 write y = k/x 2 k = xy 3 substitute	4
Find k	one known pair given, find the constant	1 write relation 2 substitute 3 solve for k	k = 48 in the inverse example
Joint variation	z proportional to x over y, find a new value	1 write z = kx/y 2 find k 3 substitute	see act three

Sanity check (teach this): Direct variation must move the same way: x rises from 4 to 7 and y rises from 48 to 147; a fall signals an error.
In inverse variation find k by multiplying: 8 times 6 gives 48, while dividing gives an implausibly small number.
After finding k, substitute the original pair back: x = 4 with k = 3 returns y = 48.

05 Independent practice (3 questions)

Question	Answer	What to watch
y varies as x; y = 21 when x = 6; find y when x = 10	35	did they find k = 3.5
y varies inversely as x; y = 9 when x = 4; find y when x = 6	6	is the direction downwards
z varies as x squared over y; z = 3 when x = 2 and y = 4; find z when x = 4 and y = 6	8	is k equal to 3 and the square applied

06 Common mistakes and what to say

Mistake	Student writes	Say this
Dropping k	substituting straight into y = x squared	Write y = kx squared; without k you are assuming k = 1 unless the question says so.
Direct and inverse reversed	calling inverse variation y = kx	Inverse means one rises as the other falls, so y = k/x; test with numbers and you will not flip it.
Wrong operation for k	using k = y divided by x in an inverse question	Inverse fixes the product, so k = x times y = 48.
Missing term in joint variation	writing z = kxy when y is inverse	y is inverse, so it goes underneath: z = kx/y. Circle the word.

07 Homework (paper, section and question numbers)

Secondary Maths (DSE) lesson-prep index — compare this ten-section structure
School S5 variation paper — Paper 1 Q1 to Q6, direct and inverse variation
School S5 variation paper — Paper 1 Q7 to Q10, joint and partial variation

08 Exit check (last 5 minutes)

Question	Answer	Decision
First line of a direct variation answer	y = kx (or kx squared)	wrong means rewrite the relation
How is k found in inverse variation	multiply x by y	wrong means redo act two
Inverse means numerator or denominator	denominator	wrong means reread act three

09 Tutor FAQ

Do students memorise every form?

No. Two rules cover it: direct goes on top, inverse goes underneath, and repeat for each variable.

How do they spot partial variation?

Look for an added constant, as in y is the sum of a term in x and a constant, which gives y = kx + c.

Are units required?

Yes. If the context gives units, write them; DSE deducts for missing units.

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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.