Teaching unit · UK Year 10 (Key Stage 4 / GCSE Foundation, ages 14 to 15)

Changing the subject of a formula

Rearranging a formula to make a different letter the subject, including when it is squared

About three lessons of 45 to 60 minutes

Start here · hook

One formula, four different 'subjects'

The formula v = u + at connects a final speed (v) to a starting speed (u), an acceleration (a) and a time (t). Written this way, it is ready to find v. But what if you already know v, u and a, and need to find t instead? You do not need a brand new formula: you REARRANGE the one you already have, so that t becomes the subject.

'Changing the subject' means rewriting a formula so a DIFFERENT letter is alone on one side, using exactly the same balancing moves you already use to solve an equation (whatever you do to one side, you must do to the other). Once a formula is rearranged for the value you actually need, you can substitute numbers in directly, instead of solving a fresh equation from scratch every single time.

v = u + at, rearranged to make t the subjectso a time can be found directly from a speed, starting speed and acceleration
P = 2l + 2w, rearranged to make w the subjectso a missing width can be found directly from a perimeter and a length
A = pi $r^{2}$ , rearranged to make r the subjectneeds a square root, since r itself is squared in the original formula
A temperature conversion formula rearranged the other waythe same 'undo it in reverse order' idea, whatever the context

Learning objective

What students will be able to do

Students will rearrange a linear formula to make a given letter the subject by undoing each operation in reverse order, and rearrange a formula where the subject is squared by additionally taking a square root as the final step.

Success criteria

I can identify, in order, which operations were applied to build the current subject of a formula.
I can undo those operations in REVERSE order, applying the same move to both sides each time, to make a different letter the subject.
I can rearrange a two-step linear formula (such as v = u + at or P = 2l + 2w) to make either letter the subject.
I can rearrange a formula where the subject is squared (such as A = pi $r^{2}$ ) by taking a square root as the final step.
I can check a rearrangement by substituting numbers into both the original and the rearranged formula and confirming they agree.

Curriculum anchor

Standards this unit teaches

GCSE Algebra #5UK GCSE Mathematics (DfE, England)
Rearranging formulae to change the subject
Subject content statement (Department for Education, "GCSE mathematics: subject content and assessment objectives", published 1 November 2013, reference DFE-00233-2013, "Algebra" section, "Notation, vocabulary and manipulation", item 5, standard type, https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives): students should "understand and use standard mathematical formulae; rearrange formulae to change the subject". Standard type means ALL GCSE students, Foundation and Higher tier, are taught and assessed on it.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Subject of a formula: the letter that is alone on one side of the formula, such as v in v = u + at
Rearrange: to reorganise a formula using balancing moves, without changing what it means, usually to change its subject
Inverse operation: the operation that undoes another: subtraction undoes addition, division undoes multiplication, a square root undoes squaring
Formula: an equation connecting two or more quantities by name, such as P = 2l + 2w for the perimeter of a rectangle

Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Undoing operations in reverse order

Concrete

Look at how the CURRENT subject was built. In v = u + at, to get v you take u, then add at (which itself is a times t). To make t the subject instead, undo those steps in reverse: subtract u first, THEN divide by a.

v = u + at. Subtract u from both sides: v - u = at. Divide both sides by a: (v - u) / a = t. So t = (v - u) / a.

The same idea rearranges P = 2l + 2w to make l the subject: subtract 2w from both sides (P - 2w = 2l), then divide both sides by 2 (l = (P - 2w) / 2).

Worked example

Make t the subject of v = u + at.

Subtract u from both sides: v - u = at.
Divide both sides by a: (v - u) / a = t.

Answer: t = (v - u) / a

Check for understanding, ask

Why is subtracting u before dividing by a the most direct route to t?
If you validly divide every term by a first, what extra fraction must you then subtract before t is isolated?

2. Formulae with a bracket or a fraction

Pictorial

When the new subject is inside a bracket, undo whatever is OUTSIDE the bracket first (usually a multiplication), then deal with what is inside. When the new subject is being divided, multiply both sides first to clear the fraction.

A = n(x + b): divide both sides by n first (A / n = x + b), THEN subtract b (A / n - b = x). y = x / n + b: subtract b first (y - b = x / n), THEN multiply both sides by n (n(y - b) = x).

Worked example

Make x the subject of A = 4(x + 3).

Divide both sides by 4: A / 4 = x + 3.
Subtract 3 from both sides: A / 4 - 3 = x.

Answer: x = A / 4 - 3

Check for understanding, ask

Why divide by the number OUTSIDE the bracket first, rather than expanding the bracket first?
For y = x/n + b, why is the LAST step multiplying by n, not the first?

3. When the subject is squared: using a square root

Abstract

If the letter you want as the new subject is currently SQUARED, undo every other operation first, leaving just 'something squared' on one side, then take the square root of both sides as the very last step. Since a length is always positive, take the positive square root.

A = pi $r^{2}$ : divide both sides by pi first (A / pi = $r^{2}$ ), THEN take the square root (r = square-root of (A / pi)). Squaring and taking a square root are inverse operations, exactly like multiplying and dividing.

x^{2}

= 9 has two mathematical solutions, x = 3 and x = -3, but a formula's subject that represents a LENGTH only ever uses the positive root.

Worked example

Make x the subject of y = 3x^2 + 5, where x is a positive length.

Subtract 5 from both sides: y - 5 = 3x^2.
Divide both sides by 3: (y - 5) / 3 = $x^{2}$ .
Take the (positive) square root of both sides: x = square-root of ((y - 5) / 3).

Answer: x = $\sqrt{(y - 5) / 3}$

Check for understanding, ask

Why is the square root taken LAST, after every other operation has been undone?
Why does this unit only ever give the POSITIVE square root as the answer?

Watch for

Common misconceptions and how to address them

MisconceptionOperations can be undone in the same order they were originally applied.

Why it happens: Students copy the order they SEE the formula written in, rather than reversing it.

How to address it: List the steps that built the current subject, in order, then undo them starting from the LAST one applied. This is the exact reverse of solving an equation, which many students already do correctly without noticing the connection.

MisconceptionFor A = n(x + b), you should expand the bracket first before doing anything else.

Why it happens: Expanding is the first thing taught for brackets in general, so students default to it even when it creates unnecessary extra terms.

How to address it: When the bracket contains the new subject and nothing else complicated, it is faster and safer to divide by the number outside the bracket FIRST (undoing the multiplication), leaving a simple subtraction to finish, rather than expanding.

MisconceptionOnce you take a square root, you should give both the positive AND negative answer, even when the subject is a real-world length.

Why it happens: Students correctly remember that $x^{2}$ = 9 has two solutions in pure algebra, and apply that everywhere without checking the context.

How to address it: Mathematically there are two square roots, but when the subject represents a physical length, area side, or similar positive quantity, only the POSITIVE root makes sense as an answer. Always check what the letter represents before deciding.

Do it together

Guided practice (with answers)

1. Make a the subject of v = u + at.
Answer: a = (v - u) / t, because subtracting u then dividing by t undoes the formula in reverse order.
2. Make w the subject of P = 2l + 2w.
Answer: w = (P - 2l) / 2, because subtracting 2l then dividing by 2 undoes the formula in reverse order.
3. Make x the subject of y = 5x + c.
Answer: x = (y - c) / 5, because subtracting c then dividing by 5 undoes the formula in reverse order.
4. Make x the subject of y = x/3 + 7.
Answer: x = 3(y - 7), because subtracting 7 then multiplying by 3 undoes the formula in reverse order.
5. Make x the subject of A = 2x^2, where x is a positive length.
Answer: x = $\sqrt{A / 2}$ , because dividing by 2 then taking the positive square root undoes the formula in reverse order.
6. Make x the subject of y = $x^{2}$ - 6, where x is a positive length.
Answer: x = $\sqrt{y + 6}$ , because adding 6 then taking the positive square root undoes the formula in reverse order.

On their own

Independent practice worksheets

Practise rearranging linear formulae, and formulae where the subject is squared, with computed, never-wrong answer keys.

Changing the Subject: Linear Formulae (Year 10)rearrange formulae like v = u + at or P = 2l + 2w Changing the Subject: Linear Formulae (Year 11)the same skill, freshly generated, for review or a Year 11 class Changing the Subject: Squares & Roots (Year 10)rearrange formulae where the subject is squared, using a square root Grade 8 Linear Equations & Expressions worksheetextra practice on the underlying balancing moves, without a formula's second letter

Reach every student

Differentiation

Support

Before touching the algebra, describe in WORDS what was done to build the current subject, in order (e.g. 'x was added to b, then multiplied by n').
Write out each balancing step on its own line, performing exactly one operation per line, rather than combining steps.
Practise linear (non-squared) rearrangements to full fluency first before introducing any square-root example.
Check every rearrangement by substituting the same small numbers into both the original and the rearranged formula.

Extension

Rearrange a formula where the NEW subject appears on both sides of the original equation (e.g. collecting subject terms before isolating it).
Introduce a formula where the subject is inside a square root already (undoing it needs squaring instead), as the reverse case of section 3.
Ask students to rearrange the SAME formula for each of its different letters in turn, and discuss which rearrangement was hardest and why.
Explore rearranging a formula that also needs the answer's units checked (e.g. rearranging a compound-unit formula), connecting back to the compound units prior knowledge.

Check it stuck

Assessment: exit ticket

A three-question exit ticket sampling a linear rearrangement, a fraction-and-bracket rearrangement, and a squared rearrangement.

1. Make u the subject of v = u + at.
Answer: u = v - at, because subtracting at from both sides isolates u.
2. Make x the subject of y = 6(x - 2).
Answer: x = y/6 + 2, because dividing both sides by 6 then adding 2 undoes the formula in reverse order.
3. Make x the subject of A = 5x^2, where x is a positive length.
Answer: x = $\sqrt{A / 5}$ , because dividing by 5 then taking the positive square root undoes the formula in reverse order.

For the teacher

Teacher notes and timings

Rough timing across three lessons: Lesson 1 reverse-order rearranging (section 1), Lesson 2 brackets and fractions (section 2), Lesson 3 squares and roots plus the exit ticket (section 3 and assessment).
This unit assumes comfort solving a linear equation for x. Revisit the Grade 8 Linear Equations & Expressions worksheet first if the balancing moves themselves, not just applying them to a formula, are shaky.
Curriculum note: DfE GCSE Algebra item 5 ('rearrange formulae to change the subject') is standard type, so Foundation tier. This unit deliberately keeps every template to at most two undoing steps (plus, for section 3, a final square root); formulae needing the new subject collected from BOTH sides of the equation are a natural Higher-tier extension, not built here.
Every rearrangement in this unit's worksheets is generated from a small, hand-verified set of formula templates (never a general symbolic solver), and independently round-trip-checked by substituting numbers into both the original and rearranged formula in tests/ukgcsemath3.test.ts.
Present and print both work: use the Print button for a clean handout of the templates, or project the worked examples and build each rearrangement's balancing moves live, one line at a time, with the class.

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