Math // Algebra

Simple formulae

Understand and use standard mathematical formulae; rearrange formulae to change the subject, performing inverse operations to isolate a different variable

Ages
12 to 14
Type
Method
Builds on
3 topics
Leads to
4 topics
Standards
1 code

Quick check

If a formula gives speed as distance ÷ time, can the learner rearrange it to find distance or time when they're the unknowns instead?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Identify the subject of a given formula
  • Use inverse operations to rearrange a formula to make a different variable the subject
  • Rearrange formulae involving multiple steps including fractions and powers

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

3 topics
  • Substituting into FormulaeRequired

    Algebra // Ages 11 to 12

    Rearranging formulae requires fluent substitution to verify results

  • Using inverse operationsRequired

    Multiplication & Division // Ages 11 to 12

    Rearranging formulae to change the subject is an application of inverse operations to algebraic expressions

  • Solving Linear EquationsRequired

    Algebra // Ages 11 to 14

    Rearranging formulae uses the same inverse-operation techniques as solving equations

You are here

Simple formulae

Ages 12 to 14 // Method

Leads to

4 topics
  • Trigonometry basicsRequired

    Geometry // Ages 13 to 15

    Using trig ratios to find unknowns requires formula rearrangement

  • Compound UnitsHelpful

    Ratio & Proportion // Ages 11 to 14

    Rearranging compound unit formulae (speed = distance/time) uses formula rearrangement skills

  • Plotting Linear GraphsHelpful

    Algebra // Ages 12 to 14

    Rearranging to y = mx + c form uses formula rearrangement skills

  • Applying Pythagoras' theoremHelpful

    Geometry // Ages 13 to 14

    Rearranging a² + b² = c² to find a shorter side uses formula rearrangement

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Rearranging a formula ('changing the subject') isolates a chosen variable by applying inverse operations to BOTH sides — the same technique as solving a linear equation, except the 'answer' is an expression in the other letters. The core discipline is doing each inverse operation to the WHOLE side (every term), undoing operations in the reverse order from how they were built up, and keeping the formula balanced. It can be taught by first solving the numeric equation, then generalising to letters with a 'same to both sides' layout, and checking by substituting numbers into both forms (a check for errors, not a proof of equivalence), while noting any values that cannot be divided by, such as zero.

Common misconceptions

  • Applies the inverse to only part of the side: y = mx + c → x = y − c/m (divides only c by m, not the whole (y − c)).
  • 'Swap side, swap sign' over-generalised to multiplication/division (y = mx → x = y − m).
  • Sign error when moving a term across (subtracts where it should add or vice versa).
  • Mishandles the subject when it is in a denominator, under a root, or appears twice.

Useful representations

  • 'Same to both sides' balance layout
  • Function-machine / inverse-operation chain run backwards
  • Original vs rearranged formula checked by substitution
  • y = mx + c rearranged to x = (y − c) / m

Teaching practices

  • Bridge from solving equations: solve 3x + 1 = 10 numerically, then rearrange y = 3x + 1 for x with identical steps — the letters just replace the number.
  • Apply each inverse operation to the WHOLE side, not one term, and undo operations in reverse order.
  • Use a balanced 'do the same to both sides' layout so no term is left behind.
  • Verify a rearrangement by substituting numbers into both the original and the rearranged form.
  • Handle harder cases explicitly: subject in a denominator, under a root, or appearing twice (multiply up, or collect and factor the subject out).

Activities to try

  1. 1

    From number to letter: rearrange with identical steps

    Why it works: Changing the subject is exactly the same balancing technique as solving an equation; the letters simply stand in for the numbers, so the inverse operations and their reverse order are unchanged.

    1. Solve the numeric equation 3x + 1 = 10 step by step, writing each inverse operation you apply to both sides (subtract 1, then divide by 3) to reach x = 3.
    2. Now take y = 3x + 1 and rearrange it to make x the subject using the SAME two steps: subtract 1 from both sides, then divide the whole side by 3, reaching x = (y − 1)/3.
    3. Write out, side by side, how each line of the numeric solution matches each line of the algebraic rearrangement.
    4. Finally rearrange the general form y = mx + c for x, undoing +c first, then ×m, to get x = (y − c)/m, provided m ≠ 0.
  2. 2

    Divide the WHOLE side, not one term

    Why it works: An inverse operation must be applied to the entire side (every term), not a single term; the numeric check exposes the flawed version because the y-term was left undivided.

    1. Attempt to make x the subject of y = mx + c. First try the tempting shortcut x = y − c/m and notice it divides only the c-term by m.
    2. Test this shortcut with numbers: let m = 2, c = 3, x = 4. Compute y from the original (y = 2×4 + 3 = 11), then put y = 11 into your shortcut and see what x you get back.
    3. Now bracket the whole side before dividing: x = (y − c)/m. Substitute again (y = 11): x = (11 − 3)/2 = 8/2 = 4, recovering the starting value.
    4. State in your own words why the division has to act on every term of the side at once.
  3. 3

    Undo multiplication by dividing, not by sign-swapping

    Why it works: 'Swap side, swap sign' applies only to addition/subtraction; a multiplied factor is undone by dividing, not by changing its sign. Keeping sign direction correct for added terms is a separate discipline.

    1. Take y = mx and make x the subject. Name the operation attaching m to x — it is multiplication.
    2. Undo multiplication with its true inverse, division, to get x = y/m (provided m ≠ 0). Do NOT write x = y − m.
    3. Check numerically: if m = 2 and x = 4, then y = 8, and y/m = 8/2 = 4 ✓, whereas y − m = 8 − 2 = 6 ✗.
    4. Also practise a sign move: rearrange y = mx + c and confirm you subtract c (not add it) when moving it across, verifying with the same substitution.

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Alg.5

Math // Algebra

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