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Year 11
Higher

Problem solving with direct and inverse proportion

I can use my knowledge of direct and inverse proportion to solve problems.

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New
New
Year 11
Higher

Problem solving with direct and inverse proportion

I can use my knowledge of direct and inverse proportion to solve problems.

Download all resources
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Share activities with pupils
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Lesson details

Key learning points

  1. Algebraic manipulation is needed to solve proportion problems.
  2. Proportional relationships can be modelled graphically and algebraically.

Keywords

  • Inversely proportional - Two variables are inversely proportional if there is a constant multiplicative relationship between one variable and the reciprocal of the other.

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

Common misconception

Directly proportional graphs all start from (0,0) and are above the y axis.

Directly proportional means there is a multiplier between y and x. This graph is always y=kx and is linear passing through (0,0). A proportionality graphs show the multiplicative relationship between y and x^n and can be linear and non linear.

A Desmos lesson is an excellent opportunity for pupils to create their own graphs and identify if their graph is directly proportional or inversely proportion. They will also see if their graph has the required number of roots and turning points to complete Q3 in Task C.
Teacher tip

Licence

This content is © Oak National Academy Limited (2024), 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.
Two variables are in proportion if they have a constant multiplicative relationship.
Correct answer: direct
inverse
opposite
exact
proper
Q2.
Which graph shows no proportional relationship?
An image in a quiz
Correct Answer: F, f
Q3.
$$y \propto x^2$$, when $$y=192$$, $$x=4$$. Work out the value of $$y$$ when $$x=10$$.
Correct Answer: 1200, 1 200
Q4.
Which of these graphs show $$y \propto \frac{1}{x^n}$$?
An image in a quiz
Correct answer: A and B
B and C
C and D
D and E
E and F
Q5.
$$y \propto \sqrt{x}$$, when $$y=40$$, $$x=4$$. Work out the value of $$y$$ when $$x=100$$.
Correct Answer: 200
Q6.
$$y \propto \frac{1}{x^2}$$, when $$y=9.6$$, $$x=5$$. Work out the value of $$y$$ when $$x=10$$. Give your answer as a decimal.
Correct Answer: 2.4

Exit quiz

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6 Questions

Q1.
Two variables are inversely proportional if there is a constant relationship between one variable and the reciprocal of the other.
Correct Answer: multiplicative
Q2.
When $$y\propto \sqrt{x}$$, when $$x$$, increases by 96%, what proportion does $$y$$ increase by?
96%
Correct answer: 40%
80%
12%
Q3.
Given the following proportions, select which are the correct proportional relationships when : $$a \propto c$$, $$c \propto b$$, $$b \propto \frac{1}{e}$$ and $$a \propto \frac{1}{f}$$.
Correct answer: $$a \propto b$$
Correct answer: $$a \propto \frac{1}{e}$$
$$b \propto f$$
$$c \propto f$$
Q4.
When $$y \propto x$$ and $$x \propto z^2$$, when $$y=75$$, $$x=3$$ and when $$x=128$$, $$z=4$$. Write $$y$$ in terms of $$z$$.
Correct answer: $$y=200z^2$$
$$y=8z^2$$
$$y=25z^2$$
$$y=2z^2$$
Q5.
When $$y\propto x^2$$, when $$x$$ increases by 20%, what proportion does $$y$$ increase by?
20%
40%
Correct answer: 44%
12%
Q6.
Match the graph with the equations.
An image in a quiz
Correct Answer:A,$$y=x(x-5)(x+5)$$

$$y=x(x-5)(x+5)$$

Correct Answer:B,$$y=4x^3$$

$$y=4x^3$$

Correct Answer:C,$$y=x^2-10x$$

$$y=x^2-10x$$

Correct Answer:D,$$y=9x^2$$

$$y=9x^2$$

Correct Answer:E,$$y=4^x$$

$$y=4^x$$

Correct Answer:F,$$y=\frac{2}{x}$$

$$y=\frac{2}{x}$$