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      Finding the constant of proportionality for direct proportion

      Lesson details

      Learning outcome

      I can use the general form for a directly proportional relationship to find k.

      Key learning points

      1. Direct proportion equations are of the form y = kx
      2. k is the constant of proportionality and it is also the gradient of the graph.
      3. To find the gradient, k, you can use two sets of coordinates (pairs of values).

      Keywords

      • Direct proportion - Two variables are in direct proportion if they have a constant multiplicative relationship

      Common misconception

      Failing to recognise that x/2 is the same as 1/2 × x and therefore shows a directly proportional relationship where the value of k is 1/2.

      It is useful to go back to looking at unit fractions. E.g. 2/3 = 1/3 × 2 and 7/9 = 1/9 × 7, etc. And, then look at algebraic terms leading up to for example, 2x/3. 'What is the the multiplier?'

      Teacher tip

      Using mini-whiteboards ask students to come up with their own examples and non-examples of equations representing direct proportion.

      Licence

      This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0
      except where otherwise stated. See Oak's terms & conditions
      (Collection 2).

      Lesson video

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      Prior knowledge starter quiz

      6 Questions

      Q1.
      Any non-zero number multiplied by its is equal to 1.

      additive inverse
      Correct answer: reciprocal
      difference

      Q2.
      Which of the following tables show direct proportion?

      An image in a quiz
      Correct answer: a
      Correct answer: b
      c
      Correct answer: d

      Q3.
      Match the numbers to their reciprocals.

      Correct Answer:$$1\over8$$,8
      Correct Answer:$$8\over3$$,$$3\over8$$
      Correct Answer:$$1\over3$$,3
      Correct Answer:6,$$1\over6$$
      Correct Answer:$$6\over5$$,$$5\over6$$

      Q4.
      Decide which of the following would share a directly proportional relationship.

      The time taken to make 10 000 items and the number of machines
      Correct answer: The height of a staircase and the number of stairs
      Correct answer: The cost to wallpaper a room and the size of the room
      The number of sweets in a bag and the number of people sharing it

      Q5.
      $$h$$ is directly proportional to $$g$$. When $$g$$ = 17, $$h$$ = 93.5. Find $$h$$ when $$g$$ = 25.

      Correct Answer: 137.5

      Q6.
      $$h$$ is directly proportional to $$g$$. When $$h$$ = 17, $$g$$ = 93.5. Find $$g$$ when $$h$$ = 60.5.

      Correct Answer: 11, eleven

      6 Questions

      Q1.
      When dealing with equations of direct proportion we use the equation $$y = kx$$. What does $$k$$ represent?

      the additive relationship between $$x$$ and $$y$$
      the difference between $$x$$ and $$y$$
      Correct answer: the constant of proportionality

      Q2.
      Which of the following equations represent $$y$$ being directly proportional to some form of $$x$$?

      Correct answer: $$y=3x$$
      Correct answer: $$y=\sqrt{3}x$$
      $$y=3x+1$$
      $$y=3\sqrt{x}$$
      Correct answer: $$y={x\over3}$$

      Q3.
      Match each equation to its correct value of $$k$$.

      Correct Answer:$$y={x\over6}$$,$$1\over6$$
      Correct Answer:$$y={\sqrt6{x}}$$,$${\sqrt6}$$
      Correct Answer:$$y={5x\over6}$$,$$5\over6$$
      Correct Answer:$$y={\sqrt5{x}}$$,$${\sqrt5}$$
      Correct Answer:$$y={6x\over5}$$,$$6\over5$$

      Q4.
      $$y$$ is directly proportional to $$x$$. When $$x$$ = 13, $$y$$ = 52. Write an equation connecting $$x$$ and $$y$$.

      $$y=x+4$$
      Correct answer: $$y=4x$$
      $$y=x-4$$
      $$y=4x+4$$

      Q5.
      $$y$$ is directly proportional to $$x$$. When $$x$$ = 6, $$y$$ = 13.2. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$y$$ when $$x$$ = 14.

      Correct Answer: 30.8

      Q6.
      $$y$$ is directly proportional to $$x$$. When $$x$$ = 6, $$y$$ = 13.2. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$x$$ when $$y$$ = 123.2.

      Correct Answer: 56

      To help you plan your 11 maths lesson on: Finding the constant of proportionality for direct proportion, download all teaching resources for free and adapt to suit your pupils' needs...