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Year 11
Higher

Finding the constant of proportionality for directly proportional relationships

I can use the general form for the directly proportional relationship y = kx^n to find k.

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Year 11
Higher

Finding the constant of proportionality for directly proportional relationships

I can use the general form for the directly proportional relationship y = kx^n to find k.

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These resources will be removed by end of Summer Term 2025.

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Lesson details

Key learning points

  1. y can be directly proportional to x^n
  2. This is because x^n could be written as a different variable.

Keywords

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

Common misconception

Setting up the initial proportion statement and hence the general form of the equation the wrong way around.

Pay particularly attention to the language used. Give pupils the opportunity to verbalise what different proportion statements mean. E.g. y ∝ x^2, 'read this aloud using the correct mathematical vocabulary'.


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

In pairs using mini-whiteboards one pupil writes down a proportional statement using the ∝ symbol and the other verbalises how the statement is read aloud.
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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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Starter quiz

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6 Questions

Q1.
What letter is used to represent the constant of proportionality?
Correct Answer: k
Q2.
Which of the following equations represent $$y$$ being directly proportional to some form of $$x$$?
$$y=5x+1$$
Correct answer: $$y={x\over5}$$
Correct answer: $$y=\sqrt{5}x$$
Correct answer: $$y=5x$$
$$y=5\sqrt{x}$$
Q3.
Match each equation to its correct value of $$k$$.
Correct Answer:$$y={\sqrt4{x}}$$,$${\sqrt4}$$
tick

$${\sqrt4}$$

Correct Answer:$$y={3x\over4}$$,$$3\over4$$
tick

$$3\over4$$

Correct Answer:$$y={\sqrt3{x}}$$,$${\sqrt3}$$
tick

$${\sqrt3}$$

Correct Answer:$$y={x\over4}$$,$$1\over4$$
tick

$$1\over4$$

Correct Answer:$$y={5x\over6}$$,$$5\over6$$
tick

$$5\over6$$

Q4.
$$y$$ is directly proportional to $$x$$. When $$x$$ = 70, $$y$$ = 245. Write an equation connecting $$x$$ and $$y$$.
$$y=3.5x+1$$
$$y=3.5x-1$$
Correct answer: $$y=3.5x$$
$$y=x+3.5$$
Q5.
$$y$$ is directly proportional to $$x$$. When $$x$$ = 8, $$y$$ = 13.2. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$y$$ when $$x$$ = 35.
Correct Answer: 57.75
Q6.
$$y$$ is directly proportional to $$x$$. When $$x$$ = 8, $$y$$ = 13.2. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$x$$ when $$y$$ = 66.
Correct Answer: 40

Exit quiz

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6 Questions

Q1.
Which of the following equations represent $$y$$ being directly proportional to $$x$$ or a power of $$x$$ in some way?
Correct answer: $$y=3x^2$$
Correct answer: $$y=3\sqrt{x}$$
$$y=3^x$$
Correct answer: $$y=2x^3$$
Correct answer: $$y={x^2\over3}$$
Q2.
Match each equation to its correct value of $$k$$.
Correct Answer:$$a=5{b}^2$$,5
tick

5

Correct Answer:$$a=2{b}^3$$,2
tick

2

Correct Answer:$$a=3{\sqrt[3]{b}}$$,3
tick

3

Correct Answer:$$a=\sqrt{3}{\sqrt{b}}$$,$$\sqrt{3}$$
tick

$$\sqrt{3}$$

Q3.
Match the following:
Correct Answer:$$a\propto{b}$$,$$a$$ is directly proportional to $$b$$
tick

$$a$$ is directly proportional to $$b$$

Correct Answer:$$a\propto{b}^3$$,$$a$$ is directly proportional to $${b}^3$$
tick

$$a$$ is directly proportional to $${b}^3$$

Correct Answer:$$a\propto{\sqrt[3]{b}}$$,$$a$$ is directly proportional to $${\sqrt[3]{b}}$$
tick

$$a$$ is directly proportional to $${\sqrt[3]{b}}$$

Correct Answer:$$a\propto{\sqrt{b}}$$,$$a$$ is directly proportional to $${\sqrt{b}}$$
tick

$$a$$ is directly proportional to $${\sqrt{b}}$$

Correct Answer:$$a\propto{b}^2$$,$$a$$ is directly proportional to $${b}^2$$
tick

$$a$$ is directly proportional to $${b}^2$$

Q4.
$$y$$ is directly proportional to $$x^2$$. When $$x$$ = 6, $$y$$ = 108. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$y$$ when $$x$$ = 11.
Correct Answer: 363
Q5.
$$y$$ is directly proportional to $${\sqrt[3]{x}}$$. When $$x$$ = 64, $$y$$ = 20. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$y$$ when $$x$$ = 343.
Correct Answer: 35
Q6.
$$y$$ is directly proportional to $${\sqrt[3]{x}}$$. When $$x$$ = 64, $$y$$ = 20. Write an equation connecting $$x$$ and $$y$$ and use this to find the value of $$x$$ when $$y$$ = 25.
Correct Answer: 125