⚖️ Ratio Province · Proportion graphs

y = kx

Model y = kx with a constant relationship and use it to find missing values. In this lesson, focus on direct proportion uses a constant relationship.

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Understand y = kx

Model y = kx with a constant relationship and use it to find missing values. In this lesson, focus on direct proportion uses a constant relationship.

Direct proportion uses a constant relationship. The ratio y ÷ x stays constant and the graph passes through the origin. For y = kx, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

y = kx: Direct proportion uses a constant relationship. Recalculate the constant ratio y ÷ x from the final pair. Keep the y = kx representation visible until the final line.

Picture the idea

Move one coordinate on a proportion graph and inspect the constant ratio or product at every point. Use the model to explain one change you notice while working on y = kx.

Check as you go

Recalculate the constant ratio y ÷ x from the final pair. Write that check beside the final y = kx answer.

Key vocabulary

proportionconstantscale factordirect proportioninverse proportionykx

Rules and key facts

Given information: y = kx — y is directly proportional to x. When x = 3, y = 6. Find y when x = 5. Method choice: Identify the mathematical relationship shown by the given values. Show one clear calculation and check it against the information in the question. Calculation or reasoning: k = 6 ÷ 3 = 2. Then y = 2 × 5 = 10. Final answer: 10. Check: Show one clear calculation and check it against the information in the question.

  • Identify whether the relationship is direct or inverse.
  • Use a known pair to calculate the constant.
  • Write the matching relationship.
  • Substitute the new value and check the constant remains unchanged. Record the check explicitly for y = kx.

Step-by-step method

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
  4. Substitute the new value and check the constant remains unchanged. Record the check explicitly for y = kx.

What you need first

  • Recognise the vocabulary: proportion, constant, scale factor.
  • Be able to explain the purpose of y = kx before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Unit pricing
  • Journey time and speed

Visual / interactive

See the idea, then move it around

Skip to Practice

Move one coordinate on a proportion graph and inspect the constant ratio or product at every point. Use the model to explain one change you notice while working on y = kx.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: y = kx — y is directly proportional to x. When x = 3, y = 6. Find y when x = 5. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 6 ÷ 3 = 2. Then y = 2 × 5 = 10. Final answer: 10. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
Secure example

Use the normal method

Given information: y = kx — y is directly proportional to x. When x = 3, y = 9. Find y when x = 17. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 9 ÷ 3 = 3. Then y = 3 × 17 = 51. Final answer: 51. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Check the y = kx result against the original information.

Challenge example

Stretch the idea

Given information: y = kx — y is directly proportional to x. When x = 3, y = 12. Find y when x = 16. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 12 ÷ 3 = 4. Then y = 4 × 16 = 64. Final answer: 64. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: y = kx — y is directly proportional to x. When x = 3, y = 15. Find y when x = 15. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 15 ÷ 3 = 5. Then y = 5 × 15 = 75. Final answer: 75. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Underline the units and command word. Show the key calculation and write the answer in context.

Common mistakes

  • Assuming every increasing relationship is direct proportion. This is a key trap when answering y = kx questions.
  • Using y ÷ x for an inverse relationship.

How to check your answer

Recalculate the constant ratio y ÷ x from the final pair. Write that check beside the final y = kx answer.

Extension challenge

Create a y = kx problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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Use proportion quest controls to solve three checked y = kx rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Study cards

Core idea

y = kx: Direct proportion uses a constant relationship. Recalculate the constant ratio y ÷ x from the final pair. Keep the y = kx representation visible until the final line.

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Key vocabulary

proportion · constant · scale factor · direct proportion · inverse proportion · y · kx

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Rules

Identify whether the relationship is direct or inverse. Use a known pair to calculate the constant. Write the matching relationship. Substitute the new value and check the constant remains unchanged. Record the check explicitly for y = kx.

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Formula / fact

Given information: y = kx — y is directly proportional to x. When x = 3, y = 6. Find y when x = 5. Method choice: Identify the mathematical relationship shown by the given values. Show one clear calculation and check it against the information in the question. Calculation or reasoning: k = 6 ÷ 3 = 2. Then y = 2 × 5 = 10. Final answer: 10. Check: Show one clear calculation and check it against the information in the question.

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Foundation example

Given information: y = kx — y is directly proportional to x. When x = 3, y = 6. Find y when x = 5. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 6 ÷ 3 = 2. Then y = 2 × 5 = 10. Final answer: 10. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: y = kx — y is directly proportional to x. When x = 3, y = 9. Find y when x = 17. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 9 ÷ 3 = 3. Then y = 3 × 17 = 51. Final answer: 51. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: y = kx — y is directly proportional to x. When x = 3, y = 12. Find y when x = 16. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 12 ÷ 3 = 4. Then y = 4 × 16 = 64. Final answer: 64. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: y = kx — y is directly proportional to x. When x = 3, y = 15. Find y when x = 15. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 15 ÷ 3 = 5. Then y = 5 × 15 = 75. Final answer: 75. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Common mistake

Assuming every increasing relationship is direct proportion. This is a key trap when answering y = kx questions.

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Exam tip

For y = kx, show the key representation before the final calculation. Use this final check: Recalculate the constant ratio y ÷ x from the final pair.

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Real-world use

Unit pricing, Journey time and speed

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Checklist

I can explain y = kx, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for y = kx

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Simple explanation

y = kx: Direct proportion uses a constant relationship. Recalculate the constant ratio y ÷ x from the final pair. Keep the y = kx representation visible until the final line.

Think of y = kx as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Identify whether the relationship is direct or inverse.
  2. Use a known pair to calculate the constant.
  3. Write the matching relationship.
  4. Substitute the new value and check the constant remains unchanged. Record the check explicitly for y = kx.

Hint 1

Start by naming the given information and the exact result required for y = kx.

Hint 2

Identify whether the relationship is direct or inverse.

Full worked solution

Given information: y = kx — y is directly proportional to x. When x = 3, y = 6. Find y when x = 5. Method choice: use the y = kx method and show each step with the stated values. Calculation or reasoning: k = 6 ÷ 3 = 2. Then y = 2 × 5 = 10. Final answer: 10. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Identify whether the relationship is direct or inverse. → Use a known pair to calculate the constant. → Write the matching relationship. → Substitute the new value and check the constant remains unchanged. Record the check explicitly for y = kx.

Common mistake warning

Assuming every increasing relationship is direct proportion. This is a key trap when answering y = kx questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain y = kx in my own words.
  • I can use these words accurately: proportion, constant, scale factor.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Assuming every increasing relationship is direct proportion.
  • I can apply this check: Recalculate the constant ratio y ÷ x from the final pair.
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