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Direct and inverse proportion - Mathematics Year 8

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Introduction

If x doubles, must y double? A relationship can increase without being directly proportional, or decrease without being inversely proportional. Test the actual invariant rather than choosing a rule from the direction of change alone.

You will classify six stated relationships, solve four missing-value problems, and sketch labelled direct and inverse graphs. Each conclusion will name its model, domain and check; the contexts are invented mathematical examples.

Identify constant ratios and constant products

For direct proportion to x, y = kx with one constant k. When x is non-zero, y/x = k; multiplying x by a factor multiplies y by the same factor. In the positive-quantity examples here k is positive. Under the stated rule, x = 0 gives y = 0, so the graph is a straight line through the origin, restricted to the chosen domain.

Original direct example. If y is 15 when x is 3 and y is directly proportional to x, then k = 15/3 = 5 and $y = 5 x$ . At x = 1,2,3,4 the corresponding values are 5,10,15,20. Each quotient y/x is 5. A unitary method first finds y for x = 1, then scales; an algebraic method records the constant in the rule.

For inverse proportion, y = k/x for non-zero x, so xy = k. In our positive domain, increasing x by a factor divides y by that factor. With $x y = 24, y = \frac{24}{x}$ , x = 3 gives y = 8, while x = 6 gives y = 4. Both pairs have product 24. Their quotients are different, so this is not direct proportion to x.

State what is being held fixed. If an ideal task requires 24 worker-hours and equally productive workers can share it perfectly, x workers take y = 24/x hours. Doubling workers halves the predicted time only under those assumptions. Real coordination, equipment and indivisible tasks can break the model; a table of invented values is not evidence that a real team works at that rate.

A direct model might charge £5 per identical ticket with no fixed booking fee: total pounds y = 5x for ticket count x. Its feasible x values are non-negative integers, not arbitrary fractional tickets. A graph can show individual feasible points even though the algebraic straight line is useful for reasoning. The constant's meaning depends on the units: £5 per ticket here, rather than a unit-free magic number.

Before calculating a missing value, identify a declared proportional relationship or enough evidence to reject one. A few matching table entries are consistent with a model but do not prove an unstated law for every possible input. A known rule supports transfer beyond the listed entries within its stated domain.

Classify six relationships and sketch their graphs

Six stated relationships to classify. Use the rule as well as the listed values. In each row the x and y entries correspond in order.

Model rule	x values	y values	Classification
y = 5x	1, 2, 3, 4	5, 10, 15, 20	Direct
y = 24/x	1, 2, 3, 6	24, 12, 8, 4	Inverse
y = 5x + 2	1, 2, 3, 4	7, 12, 17, 22	Neither
y = 2x	1, 2, 3, 4	2, 4, 6, 8	Direct
y = 12/x	1, 2, 3, 4	12, 6, 4, 3	Inverse
y = x²	1, 2, 3, 4	1, 4, 9, 16	Neither

For y = 5x + 2, the graph is straight and increasing but its y-intercept is 2, not zero. The ratios 7/1 and 12/2 differ. It can model a £2 fixed charge plus £5 per item, but that total is not directly proportional to item count. For y = x², the graph passes through the origin when zero is included but is curved and its ratio y/x changes with x. Passing through the origin alone is insufficient.

For y = 24/x, plot (1,24),(2,12),(3,8),(6,4) using a scale that fits. Draw the positive curved branch as the stated model, not a sequence of straight joins claimed to be its exact graph. It never has a point at x = 0 because division by zero is undefined. For y = 5x, label (0,0),(1,5),(2,10),(3,15); a consistent scale gives a straight line through the origin.

Use Corbettmaths’ original direct-proportion explanation to identify the constant of proportionality. The video also uses models involving powers of a variable; our six-row table tests direct or inverse proportion to x itself, so do not relabel every power model as y = kx. Original: corbettmaths, Direct Proportion - Corbettmaths. The written explanations and native tasks work without sound or the external player.

When comparing visual patterns, read axis labels and origin placement rather than judging the curve's apparent steepness. A cropped graph may hide the intercept. Check the corresponding rule and invariant to distinguish a fixed fee, a quadratic pattern and a proportional model.

Solve four missing values and challenge assumptions

Four missing-value problems using the declared source rules.

Stated model	Given value	Missing value	Check
y = 5x	x = 7	y = 35	35/7 = 5
y = 5x	y = 45	x = 9	45/9 = 5
xy = 24	x = 8	y = 3	8 × 3 = 24
xy = 24	y = 6	x = 4	4 × 6 = 24

For the first row, multiply x by the unit rate 5. For the second, undo that multiplication by dividing 45 by 5. For the inverse rows, divide the fixed product 24 by the known factor. These methods are different because the invariant is different. In every row, substituting the recovered pair checks the original rule.

Second solution paths. Under y = 5x, the known pair (3,15) scales by 7/3 to reach x = 7, so y becomes 15×7/3 = 35. The unitary route avoids the intermediate fraction by finding k = 5 first. Under xy = 24, moving from x = 3 to x = 8 multiplies x by 8/3 and therefore divides y = 8 by 8/3 to give 3. The constant-product route is simpler for these numbers. Explain efficiency without claiming that only one valid representation exists.

Error clinic. A learner doubles both x and y in the inverse pair (3,8), giving (6,16). Its product is 96, not 24, so the original model has changed. Another treats y = 5x + 2 as direct merely because it is straight; its fixed intercept and varying ratios reject that claim. A third treats any decreasing list as inverse without checking the product. Locate the first unsupported assumption, then recalculate with the appropriate invariant.

For unfamiliar transfer, direct proportion gives y = 18 when x = 4, so k = 4.5 and y = 45 at x = 10. For an inverse positive model with product 30, x = 12 gives y = 2.5. Check 45/10 = 4.5 and 12×2.5 = 30. State units from any declared context; do not add worker, cost or distance units to a purely numerical table without defining them.

Keep all six classifications with reasons, four complete missing-value solutions and two labelled graphs. Include one corrected error, model assumptions and a real later retrieval attempt after a break. Online plotting and invented worker-hour values do not certify practical measurement or observed real productivity.

Interactive Tasks

Quiz: Test Your Knowledge

What is the constant in the stated direct example y=15 when x=3? (5) (!45) (!12) (!0.2)

Which invariant defines direct proportion to non-zero x? (A constant ratio y divided by x) (!A constant sum x plus y) (!A constant difference y minus x) (!A constant product xy)

Which invariant defines the positive inverse model here? (A constant product xy) (!A constant ratio y divided by x) (!A constant sum) (!A positive gradient)

What y follows x=6 in the model xy=24? (4) (!16) (!144) (!18)

How is y=5x+2 classified relative to x? (Neither direct nor inverse) (!Direct because it is increasing) (!Inverse because it has two terms) (!Direct because every straight line is proportional)

Why is y=x squared not directly proportional to x? (Its ratio y divided by x varies) (!It never passes through the origin) (!It is a decreasing function for positive x) (!Its values are negative for positive x)

What y follows x=7 in y=5x? (35) (!12) (!1.4) (!45)

What x gives y=45 in y=5x? (9) (!225) (!40) (!0.111)

What y follows x=8 in xy=24? (3) (!192) (!32) (!16)

What extra assumption supports the ideal worker-hour model? (Equally productive workers can share fixed work perfectly) (!Every real job doubles output with any added worker) (!No fixed total work is needed) (!A decreasing time automatically proves inverse proportion)

Memory Game

Direct	Relationship with y equal to a fixed multiple of x
Inverse	Relationship with a fixed non-zero product xy here
Constant	Value held unchanged in the stated invariant
Origin	Coordinate point where both coordinates are zero
Domain	Allowed input values of the model
Intercept	Coordinate where a graph crosses a specified axis

Drag and Drop

5	Constant ratio for y=5x
24	Constant product for y=24/x
35	y value at x=7 in the direct source model
9	x value at y=45 in the direct source model
3	y value at x=8 in the inverse source model

Crossword Puzzle

DIRECT	Proportion with a constant ratio y over x
INVERSE	Proportion with a constant non-zero product
CONSTANT	Value unchanged in the stated relation
ORIGIN	Point with both coordinates zero
DOMAIN	Set of permitted inputs
PRODUCT	Result of multiplying the paired values

LearningApps

Optional extra practice. Choose an appropriate activity with your teacher. If the external collection is unavailable, use the native activities and independent tasks in this course.

Cloze Text

Open Tasks

Easy

Find the source constants: Use y 15 atx 3 and product 24 to state the two original rules.
Complete the four missing values: Solve all four table rows and substitute each recovered pair.
Name the invariant: Compare quotients for y 5 x and products for y 24/x.
Read the graph reference: Explain origin, axis scale and positive domain before plotting.

Standard

Classify all six: Use the actual six rules and values; give quotient/product/intercept reasons.
Sketch two labelled graphs: Plot the stated direct line and positive inverse branch; explain the excluded zero input.
Compare methods: Solve a missing value by unitary/scaling and algebraic-constant approaches.
Repair the first assumption: Correct doubling both values in an inverse model and the straight-line fixed-fee misconception.

Advanced

Transfer to new constants: Use direct 18 atx 4 to find y 45 atx 10 and inverseproduct 30 to find y 2.5 atx 12.
Evaluate evidence limits: Explain why finitely many matching pairs do not prove an unstated rule for all inputs.
Interpret an ideal context: State the assumptions behind fixed 24 worker-hours shared by xworkers.
Retrieve later: Keep a real later classification and missing-value problem with domain, units, first error and check.

Learning Check

Six relationship classifications: Give all six with reasons.
Four missing values: Solve and check all rows.
Two graph meanings: Describe the line and inverse branch.
Methods and unfamiliar transfer: Use unitary and constant-based reasoning.
Error and context: Correct the inverse doubling and fixed-fee claims.
Complete evidence: Retain source scope and real later work.

Worked checks and assessment guidance

y 5 x: direct; y 24/x: inverse; y 5 x+2: neither; y 2 x: direct; y 12/x: inverse; yx²: neither. Constant ratios 5/2 orproducts 24/12 identify the proportional cases; fixed intercept or varying ratio/product rejects the others.
For y 5 x, x 7→y 35 and y 45→x 9, with quotient 5. For xy 24, x 8→y 3 and y 6→x 4, with product 24. Substitution checks the stated rules, not a different guessed pattern.
The direct line includes origin and points 1, 5/2, 10/3, 15; inverse positive branch includes 1, 24/2, 12/3, 8/6, 4 and excludes x 0. Chosen axis scales and domains must be labelled; blank source grid is not already either graph.
18/4=4.5, so directx 10 gives 45. Inverseproduct 30 atx 12 gives 2.5. Scaling x by a factor scales y the same way for direct and reciprocally for inverse; fixed-rule checks agree.
Inversepair 3, 8 hasproduct 24; wrong 6, 16 hasproduct 96. y 5 x+2 hasintercept 2 andvaryingratios.24 worker-hours assumes equal productivity and perfect sharing; no real workforce data are claimed.
Keep six classifications, four missing values, two labelled graphs, checks and assumptions. Finite matching values are consistent with a stated rule, not proof of an unstated universal law. Record actual later retrieval separately.

Learning evidence

Retain six reasoned classifications, four checked missing values, two labelled graphs and an unfamiliar transfer with stated domain and context assumptions. Actual later retrieval must be recorded separately.

OER and Wikipedia

Department for Education: England mathematics programme of study, Key Stage 3, ratio, algebra, fluency, reasoning and problem solving. Year 8 allocation is editorial within KS3. Tasks, numerical models, examples and feedback are original. The selected Commons sources retain their creators and licences below; videos are embedded from the original named publisher. Reviewed 12 November 2026.

Cuisenaire-Rods.jpg, Celcom at en.wikipedia, CC BY 3.0. Unchanged original; caption and learning activities are original.

2D Cartesian Coordinates.svg, H Padleckas, Public domain. Unchanged original; caption and learning activities are original.