Direct and inverse proportion
Finding the constant k in y = kx or xy = k, and telling the two apart from a table of values
About three lessons of 45 to 60 minutes
Does buying more always cost proportionally more?
If 4 identical tickets cost £20, most people can work out that 6 tickets cost £30, without being taught a formula, because the cost is DIRECTLY proportional to the number of tickets: double the tickets, double the cost. But some relationships work in reverse: if 4 painters can paint a fence in 6 hours, 8 painters do NOT take 12 hours, they take 3 hours, because more workers means LESS time. That is INVERSE proportion.
Both relationships are controlled by a single constant, k. For direct proportion, y = kx (y/x is always the same k). For inverse proportion, xy = k (x times y is always the same k). This unit is about finding that constant from one known pair of values, then using it to solve for anything else, and telling direct and inverse proportion apart when only a table of numbers is given.
- 4 tickets cost £20, so 6 tickets cost £30DIRECT proportion: cost / tickets is always the same constant, £5 per ticket
- 4 painters take 6 hours, 8 painters take 3 hoursINVERSE proportion: workers x hours is always the same constant, 24 painter-hours of work
- Litres of fuel bought and their total costdirect: the price per litre, k, never changes
- Pipes filling a tank and the time it takesinverse: more pipes working together finish in less time
What students will be able to do
Students will find the constant of proportionality k for direct proportion (y = kx) and inverse proportion (xy = k) from one known pair of values, use it to solve for a missing value, and determine whether a table of values shows direct proportion, inverse proportion, or neither.
- I can find k = y / x from one known (x, y) pair for direct proportion, and use it to find a missing y or x.
- I can find k = x times y from one known (x, y) pair for inverse proportion, and use it to find a missing y or x.
- I can decide whether a table of values shows direct proportion by checking whether y / x is the same for every pair.
- I can decide whether a table of values shows inverse proportion by checking whether x times y is the same for every pair.
- I can solve a real-world direct or inverse proportion word problem by first identifying which type of proportion applies.
Standards this unit teaches
- GCSE Ratio, Proportion & Rates of Change #10UK GCSE Mathematics (DfE, England)Direct and inverse proportion problems
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Ratio, proportion and rates of change" section, item 10, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "solve problems involving direct and inverse proportion, including graphical and algebraic representations". This is standard-type content, so ALL GCSE students (Foundation and Higher tier) are taught and assessed on it.
- GCSE Ratio, Proportion & Rates of Change #13UK GCSE Mathematics (DfE, England)Interpreting equations for direct and inverse proportion
Subject content statement (same document, item 13): students should "understand that X is inversely proportional to Y is equivalent to X is proportional to 1/Y" and "interpret equations that describe direct and inverse proportion" (underlined type, all students assessed). Item 13 continues, in bold type (Higher tier only, deliberately not required by this unit), that students should also "construct" such equations from scratch.
- GCSE Ratio, Proportion & Rates of Change #14UK GCSE Mathematics (DfE, England)Graphs of direct and inverse proportion
Subject content statement (same document, item 14, underlined type): students should "interpret the gradient of a straight line graph as a rate of change; recognise and interpret graphs that illustrate direct and inverse proportion".
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Ratio in the glossaryproportion is ratio applied to a changing pair of quantities, rather than a single fixed comparison
- Grade 9 Ratios, Rates & Scale worksheetthe ratio and unit-rate skills this unit's constant of proportionality k builds directly on
- Grade 7 function tables worksheetreading input/output relationships from a table, the same table-reading skill this unit applies to proportion
Words to teach and display
- Direct proportion
- a relationship where y = kx for a constant k: as x increases, y increases at the same constant rate, and doubling x doubles y
- Inverse proportion
- a relationship where xy = k for a constant k: as x increases, y decreases, and doubling x halves y
- Constant of proportionality (k)
- the fixed number connecting x and y in either y = kx (direct) or xy = k (inverse), found from one known pair of values
- Proportional
- describes two quantities connected by a constant multiplier or a constant product
- Rate of change
- how much one quantity changes for a given change in another; the gradient of a direct-proportion graph
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Direct proportion: finding and using k
ConcreteFor direct proportion, y = kx, so k = y / x, and this ratio is the SAME no matter which known pair you use. Once k is known, any missing y is found with y = k x (given x), and any missing x is found with x = y / k (given y).
4 tickets cost £20: k = 20 / 4 = 5 (£5 per ticket). So 7 tickets cost 5 x 7 = £35, and if a group spent £45, they bought 45 / 5 = 9 tickets.
y is directly proportional to x. When x = 6, y = 42. Find y when x = 10.
- k = y / x = 42 / 6 = 7.
- When x = 10, y = k x 10 = 7 x 10 = 70.
Answer: y = 70
- Why does a direct proportion graph always pass through the origin?
- If k = 5, what does doubling x do to y?
2. Inverse proportion: finding and using k
ConcreteFor inverse proportion, xy = k, so k = x times y, and again this product is the SAME for every pair. Once k is known, any missing y is found with y = k / x, and any missing x is found with x = k / y.
4 painters take 6 hours: k = 4 x 6 = 24 (painter-hours of total work). So 8 painters take 24 / 8 = 3 hours, and if a job took 2 hours, it must have used 24 / 2 = 12 painters.
y is inversely proportional to x. When x = 5, y = 8. Find y when x = 4.
- k = x times y = 5 x 8 = 40.
- When x = 4, y = k / x = 40 / 4 = 10.
Answer: y = 10
- Why does an inverse proportion graph never touch the x-axis or y-axis?
- If k = 24, what does doubling x do to y?
3. Direct, inverse, or neither: reading a table
PictorialGiven only a table of x and y values (no description), test BOTH: work out y / x for every pair (if it is always the same, it is direct proportion) and x times y for every pair (if it is always the same, it is inverse proportion). If neither is constant, it is neither.
Table: x = 2, y = 10; x = 5, y = 25; x = 8, y = 40. Check y / x: 10/2 = 5, 25/5 = 5, 40/8 = 5, always 5, so this IS direct proportion, with k = 5. Table: x = 2, y = 12; x = 3, y = 8; x = 4, y = 6. Check x times y: 2x12 = 24, 3x8 = 24, 4x6 = 24, always 24, so this IS inverse proportion, with k = 24.
Is y directly proportional to x for this table? x = 3, y = 15; x = 5, y = 25; x = 6, y = 35.
- Check y / x for every pair: 15/3 = 5, 25/5 = 5, 35/6 = 5.83 (not a whole number, and not 5).
- y / x is NOT the same for every pair.
Answer: No, this is not direct proportion, because y / x is not constant (5, 5, 5.83).
- A table has y / x constant but x times y NOT constant. What kind of proportion is it?
- Could a table be BOTH direct and inverse proportion at once? Why or why not?
Common misconceptions and how to address them
MisconceptionInverse proportion means y decreases, so any table where y goes down as x goes up must be inverse proportion.
Why it happens: Students spot the general downward trend and stop checking, instead of testing that x times y is genuinely CONSTANT.
How to address it: Many decreasing relationships are NOT inverse proportion (e.g. y = 20 - x also decreases). Always check x times y is the SAME for every pair before calling it inverse proportion; a merely decreasing table is not enough.
MisconceptionThe same formula, y = kx, is used for both direct and inverse proportion, just with a different k.
Why it happens: Students remember 'there's a constant k' but not that direct and inverse proportion use two DIFFERENT equations connecting x, y and k.
How to address it: Direct proportion is y = kx (y divided by x is constant). Inverse proportion is xy = k, equivalently y = k/x (x TIMES y is constant, not divided). Write both forms side by side until the distinction is automatic.
MisconceptionOnce k is found from one pair, it needs to be recalculated for every new pair in the same problem.
Why it happens: Students do not trust that the SAME constant applies throughout a single direct or inverse proportion relationship.
How to address it: k is fixed for the whole relationship, found once from any single known pair, then reused for every other value in that same problem. Recalculating it each time (getting the same answer) wastes time but is not wrong; using a DIFFERENT k for different pairs in the same problem is the actual error to avoid.
Guided practice (with answers)
1. y is directly proportional to x. When x = 3, y = 21. Find y when x = 8.
Answer: 56, because k = 21/3 = 7, and y = 7 x 8 = 56.
2. y is directly proportional to x. When x = 4, y = 12. Find x when y = 33.
Answer: 11, because k = 12/4 = 3, and x = 33/3 = 11.
3. y is inversely proportional to x. When x = 6, y = 10. Find y when x = 4.
Answer: 15, because k = 6 x 10 = 60, and y = 60/4 = 15.
4. y is inversely proportional to x. When x = 3, y = 20. Find x when y = 5.
Answer: 12, because k = 3 x 20 = 60, and x = 60/5 = 12.
5. A table shows x = 2, y = 6; x = 3, y = 9; x = 5, y = 15. Is this direct proportion?
Answer: Yes, because y/x = 3 for every pair (6/2, 9/3, 15/5 all equal 3).
Independent practice worksheets
Practise finding k and solving direct and inverse proportion problems, and identifying which type a table of values shows, with computed, never-wrong answer keys.
Differentiation
- Always write the relevant equation (y = kx or xy = k) at the top of the working before substituting any numbers, so which operation to use for k is never guessed.
- Practise finding k in isolation first (just one line of working) before combining it with a second step to find a missing value.
- Use a simple whole-number k (2, 3, 4, 5) for every early example, saving less friendly numbers for later.
- For the identifying task, always test BOTH y/x and x times y explicitly, rather than guessing from the shape of the numbers.
- Introduce the graphical language directly: a straight line through the origin is direct proportion, a curve approaching (but never touching) both axes is inverse proportion.
- Ask students to write the specific equation (e.g. y = 5x, or y = 24/x) for a given real-world direct or inverse proportion scenario.
- Investigate what happens to y if x is tripled, for both direct proportion (y triples) and inverse proportion (y becomes a third), without recalculating k each time.
- Compare direct proportion to simple linear relationships that are NOT proportional (y = kx + c with c not 0), reinforcing why the origin-crossing property matters.
Assessment: exit ticket
A three-question exit ticket sampling direct proportion, inverse proportion, and identifying from a table.
1. y is directly proportional to x. When x = 5, y = 30. Find y when x = 9.
Answer: 54, because k = 30/5 = 6, and y = 6 x 9 = 54.
2. y is inversely proportional to x. When x = 4, y = 9. Find y when x = 6.
Answer: 6, because k = 4 x 9 = 36, and y = 36/6 = 6.
3. A table shows x = 1, y = 8; x = 2, y = 4; x = 4, y = 2. Is this direct proportion, inverse proportion, or neither?
Answer: Inverse proportion, because x times y = 8 for every pair (1x8, 2x4, 4x2 all equal 8), while y/x is NOT constant.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 direct proportion (section 1), Lesson 2 inverse proportion (section 2), Lesson 3 identifying from a table (section 3) plus the exit ticket.
- This unit assumes comfort with ratio and unit rates. Revisit those first if finding k itself, not just applying it, is the sticking point.
- Curriculum note: this unit cites the DfE 'GCSE mathematics: subject content and assessment objectives' (2013), items 10, 13 and 14 of the 'Ratio, proportion and rates of change' section. Item 13's 'construct' clause is bold/Higher-tier only; this unit and its worksheets only ever ask students to FIND k from a given pair and INTERPRET the resulting relationship, never to construct an original equation from an open-ended scenario, keeping the whole unit within Foundation tier.
- Language to repeat: direct proportion divides (k = y/x); inverse proportion multiplies (k = xy); the SAME k applies to every pair in one relationship.
- Present and print both work: use the Print button for a clean handout, or project the two proportion graphs and build the k-finding working with the class live.