Finding the constant of proportionality for directly proportional relationships

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Lesson details

Learning outcome

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

Key learning points

y can be directly proportional to x^n
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'.

Teacher tip

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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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

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Prior knowledge 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:$$y={\sqrt4{x}}$$,$${\sqrt4}$$

$${\sqrt4}$$

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

$$3\over4$$

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

$${\sqrt3}$$

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

$$1\over4$$

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

$$5\over6$$

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

Assessment 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

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

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

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

$$\sqrt{3}$$

Q3.
Match the following:

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

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

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

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

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

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

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

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

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

$$a$$ 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

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Finding the constant of proportionality for direct proportion

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Finding the constant of proportionality for directly proportional relationships

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.What letter is used to represent the constant of proportionality?

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

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

Q4.$$y$$ is directly proportional to $$x$$. When $$x$$ = 70, $$y$$ = 245. Write an equation connecting $$x$$ and $$y$$.

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.

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.

Assessment exit quiz

6 Questions

Q1.Which of the following equations represent $$y$$ being directly proportional to $$x$$ or a power of $$x$$ in some way?

Q2.Match each equation to its correct value of $$k$$.

Q3.Match the following:

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.

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.

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.

How to plan a lesson using our resources

Q1.
What letter is used to represent the constant of proportionality?

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

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

Q4.
$$y$$ is directly proportional to $$x$$. When $$x$$ = 70, $$y$$ = 245. Write an equation connecting $$x$$ and $$y$$.

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.

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.

Q1.
Which of the following equations represent $$y$$ being directly proportional to $$x$$ or a power of $$x$$ in some way?

Q2.
Match each equation to its correct value of $$k$$.

Q3.
Match the following:

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.

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.

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.