Evaluating expressions with and without changing the subject

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Learning outcome

I can evaluate an expression where the subject has been changed and where it has not.

Key learning points

Changing the subject can make it easier to evaluate a formula.
It is possible to evaluate a formula without changing the subject.
Both forms of the formula give the same result.

Keywords

Subject of an equation/formula - The subject of an equation/a formula is a variable that is expressed in terms of other variables. It should have an exponent of 1 and a coefficient of 1.

Common misconception

Changing the subject of an equation or formula is only done when you are told to.

Having students practise rearranging before substituting and vice versa allows them to decide when the skill could be useful.

Teacher tip

This is a good opportunity for pupils to practise using key formulae and this could include those from other subjects. You could ask the pupils to share formulas they have seen or used in other lessons or outside of school.

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).

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Prior knowledge starter quiz

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6 Questions

Q1.
Substitute means to put in place of another. In algebra, substitution can be used to replace __________ with values.

constants

equations

solutions

Correct answer: variables

Q2.
$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. It can be rearranged to $${2A\over{h}}-a=b$$ in order to make $$b$$ the of the formula.

Correct Answer: subject, Subject

Q3.
The formula for area of a triangle is $$A={1\over2}bh$$. The area of a triangle with base 7.5 cm and height 6 cm is cm²

Correct Answer: 22.5

Q4.
The formula $$s={d\over{t}}$$ can be rearranged to $$t={d\over{s}}$$ or $$d=s\times {t}$$. If $$d=12$$ and $$s=3$$, find $$t$$.

$$36$$

Correct answer: $$4$$

$${1\over{4}}$$

Q5.
Find the area of a trapezium with $$h=7$$, $$a=3.2$$, $$b=4.8$$.

$$A={1\over2}(3.2+4.8)+7=11$$ square units

Correct answer: $$A={1\over2}(3.2+4.8)\times7=28$$ square units

$$A=2(3.2+4.8)\times7=112$$ square units

$$A={1\over2}(3.2\times4.8)\times7=53.76$$ square units

Q6.
$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. Select all the correct rearrangements of this formula.

Correct answer: $${2A\over{h}}-b=a$$

$${Ah\over{2}}-a=b$$

$${A\over{2h}}-a=b$$

$${2h\over{A}}-a=b$$

Correct answer: $${2A\over{a+b}}=h$$

Assessment exit quiz

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6 Questions

Q1.
$$a=\sqrt{{c^2}-b^2}$$ is a __________ of $$a^2+b^2=c^2$$.

expression

Correct answer: rearrangement

solution

substitution

variable

Q2.
Which arrangement of the formula for area of a triangle makes it quickest for you to find $$b$$ if you know $$A$$ and $$h$$?

$$A={1\over2}bh$$

$${2A\over{b}}=h$$

Correct answer: $${2A\over{h}}=b$$

$${h\over{2A}}=b$$

Q3.
The formula for the area of a triangle is $$A={1\over2}bh$$. The base $$b$$ of a triangle with area 24 m² and height 8 m is m.

Correct Answer: 6, six, 6m

Q4.
Rearrange the formula $$C=2{\pi}{r}$$ to find the radius of a circle with circumference 42 cm. Leave your answer in terms of $$\pi$$.

$$r=2C\pi=2\times 42\times \pi=84\pi$$

$$r={2\pi\over{{C}}}={2\pi\over{{42}}}={\pi\over{21}}$$

$$r={2C\over{{\pi}}}={2\times 42\over{{\pi}}}={84\over{\pi}}$$

Correct answer: $$r={C\over{2{\pi}}}={42\over{2{\pi}}}={21\over{\pi}}$$

Q5.
Which rearrangement makes it easiest to find length $$b$$ if you know lengths $$a$$ and $$c$$?

$$a=\sqrt{{c^2}-b^2}$$

Correct answer: $$b=\sqrt{{c^2}-a^2}$$

$$b^2=c^2-a^2$$

$$b=\sqrt{{a^2}-c^2}$$

$$b=\sqrt{{c^2}+a^2}$$

Q6.
Jun says "You have to rearrange the formula $$a^2+b^2=c^2$$ into the form $$b=\sqrt{{c^2}-a^2}$$ with $$b$$ as the subject if you want to find $$b$$ given that $$a=6$$ and $$c=10$$." Is Jun right?

Yes, $$b$$ must be the subject before you can substitute.

Yes, because $$b=\sqrt{{10^2}-6^2}$$ is the fastest way to find $$b$$.

Correct answer: No, you can substitute $$a$$ and $$c$$ into $$a^2+b^2=c^2$$ before evaluating.

To help you plan your 9 maths lesson on: Evaluating expressions with and without changing the subject, download all teaching resources for free and adapt to suit your pupils' needs...

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Evaluating expressions with and without changing the subject

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.Substitute means to put in place of another. In algebra, substitution can be used to replace __________ with values.

Q2.$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. It can be rearranged to $${2A\over{h}}-a=b$$ in order to make $$b$$ the of the formula.

Q3.The formula for area of a triangle is $$A={1\over2}bh$$. The area of a triangle with base 7.5 cm and height 6 cm is cm²

Q4.The formula $$s={d\over{t}}$$ can be rearranged to $$t={d\over{s}}$$ or $$d=s\times {t}$$. If $$d=12$$ and $$s=3$$, find $$t$$.

Q5.Find the area of a trapezium with $$h=7$$, $$a=3.2$$, $$b=4.8$$.

Q6.$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. Select all the correct rearrangements of this formula.

Assessment exit quiz

6 Questions

Q1.$$a=\sqrt{{c^2}-b^2}$$ is a __________ of $$a^2+b^2=c^2$$.

Q2.Which arrangement of the formula for area of a triangle makes it quickest for you to find $$b$$ if you know $$A$$ and $$h$$?

Q3.The formula for the area of a triangle is $$A={1\over2}bh$$. The base $$b$$ of a triangle with area 24 m² and height 8 m is m.

Q4.Rearrange the formula $$C=2{\pi}{r}$$ to find the radius of a circle with circumference 42 cm. Leave your answer in terms of $$\pi$$.

Q5.Which rearrangement makes it easiest to find length $$b$$ if you know lengths $$a$$ and $$c$$?

Q6.Jun says "You have to rearrange the formula $$a^2+b^2=c^2$$ into the form $$b=\sqrt{{c^2}-a^2}$$ with $$b$$ as the subject if you want to find $$b$$ given that $$a=6$$ and $$c=10$$." Is Jun right?

How to plan a lesson using our resources

Q1.
Substitute means to put in place of another. In algebra, substitution can be used to replace __________ with values.

Q2.
$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. It can be rearranged to $${2A\over{h}}-a=b$$ in order to make $$b$$ the of the formula.

Q3.
The formula for area of a triangle is $$A={1\over2}bh$$. The area of a triangle with base 7.5 cm and height 6 cm is cm²

Q4.
The formula $$s={d\over{t}}$$ can be rearranged to $$t={d\over{s}}$$ or $$d=s\times {t}$$. If $$d=12$$ and $$s=3$$, find $$t$$.

Q5.
Find the area of a trapezium with $$h=7$$, $$a=3.2$$, $$b=4.8$$.

Q6.
$$A={1\over2}(a+b)h$$ is the formula for area of a trapezium. Select all the correct rearrangements of this formula.

Q1.
$$a=\sqrt{{c^2}-b^2}$$ is a __________ of $$a^2+b^2=c^2$$.

Q2.
Which arrangement of the formula for area of a triangle makes it quickest for you to find $$b$$ if you know $$A$$ and $$h$$?

Q3.
The formula for the area of a triangle is $$A={1\over2}bh$$. The base $$b$$ of a triangle with area 24 m² and height 8 m is m.

Q4.
Rearrange the formula $$C=2{\pi}{r}$$ to find the radius of a circle with circumference 42 cm. Leave your answer in terms of $$\pi$$.

Q5.
Which rearrangement makes it easiest to find length $$b$$ if you know lengths $$a$$ and $$c$$?

Q6.
Jun says "You have to rearrange the formula $$a^2+b^2=c^2$$ into the form $$b=\sqrt{{c^2}-a^2}$$ with $$b$$ as the subject if you want to find $$b$$ given that $$a=6$$ and $$c=10$$." Is Jun right?