Every student has had the evening where a formula sheet is read ten times, each formula written out twenty times, and three days later the quadratic formula has a minus sign in the wrong place. The method is the problem, not the memory. Formulas written out are stored like a phone number; formulas used are stored like a route you have walked. This page is the method that works, in five steps, why rote fails by the exam, which formulas are worth memorising and which should be derived on the spot, class by class from Class 8 to Class 12 and for GCSE and the SAT, and a two-week plan for a sheet of thirty formulas.
Why writing a formula out fifty times fails
Copying is recognition, not recall. When the formula is on the page in front of you, your brain checks it against what it sees; nothing is retrieved from memory, so nothing is strengthened. Three days later, on blank paper, there is nothing to retrieve. The research on memory has shown this for decades: the act of pulling something out of memory is what makes it stay, and re-reading or copying barely registers. Students who re-read feel more confident and score lower than students who test themselves, every time it is measured.
The second problem is that a formula copied is a string of symbols, and strings of symbols are the hardest things for a human memory to hold. The same formula understood is a short story: "the area of a trapezium is the average of the two parallel sides times the height, because you can cut it and rearrange it into a rectangle of that width." Stories are what memory is built for.
The five-step method
1. Derive it once, or see where it comes from. Not a proof for every formula; a picture or a two-line reason. The area of a circle is πr² because it can be cut into thin wedges that lay out into a rectangle π r long and r high. The quadratic formula is completing the square done once in general. The sum of an AP is the first plus the last, times half the number of terms, because pairing the first with the last gives the same total every time. A formula with a reason behind it is one the brain can rebuild if it slips.
2. Use it twenty times, over two weeks, not in one evening. Twenty problems of different kinds, where the formula is applied to a question rather than copied. The number is not magic, but it is about where a formula moves from "I know it" to "I cannot forget it". Ten on day one, five on day four, five on day ten.
3. Test from blank paper. Close the book, write the formula sheet from memory, open the book, mark it. Rewrite only the ones you got wrong. This is the single most effective step and the one almost nobody does, because it is uncomfortable: it shows you exactly what you do not know.
4. Space the revision. Day 1, day 3, day 7, day 14, then once a week. Each gap should be just long enough that the formula is slightly hard to recall; that difficulty is what makes the memory stronger. Cramming the same formulas five times in one night gives the feeling of knowing and almost none of the keeping.
5. Group formulas by the shape they come from, not by the chapter. All the "half times something times something" formulas together (area of a triangle, area of a rhombus from diagonals, the kinetic energy formula in physics); all the "square root of a sum of squares" formulas together (Pythagoras, distance between two points, the modulus of a vector, the magnitude of a complex number). A group of six formulas that share a shape is one memory, not six.
What to memorise and what to derive, class by class
The rule: memorise what you cannot rebuild in under a minute and use often; derive what comes from a picture. The lists below are the formulas worth the memory in each year, with the ones to derive in brackets.
| Class or exam | Memorise | Derive on the spot |
|---|---|---|
| Class 8 | the laws of exponents; the three algebraic identities, (a+b)², (a−b)², (a+b)(a−b); area of a trapezium and rhombus; volume of a cuboid and cylinder; simple and compound interest | surface area of a cuboid (six faces); percentage change (the fraction) |
| Class 9 | the expansions of (a+b)³, (a−b)³, a³+b³, a³−b³; the Heron's formula; surface area and volume of a cone and sphere; the distance of a point from the axes; the mean | surface area of a cylinder (unroll it); the mid-point |
| Class 10 | the quadratic formula; nth term and sum of an AP; the distance and section formulas; the trigonometric ratios, the standard values (0°, 30°, 45°, 60°, 90°) and the three identities; area of a sector and arc length; the mode, median and mean of grouped data | the discriminant cases (from the formula); the sum of an AP's first n terms from first and last; tan from sin and cos |
| Class 11 and 12 | the compound and double-angle formulas for sin, cos, tan; the binomial theorem; the derivatives and integrals of xⁿ, sin, cos, tan, eˣ, ln x; the chain, product and quotient rules; the standard limits; the equation of a line in its forms; the sum formulas for n, n², n³ | sin 3x from sin(2x + x); the half-angle formulas; integration by parts (from the product rule); the distance between parallel lines |
| GCSE (UK) | the quadratic formula; the sine rule, cosine rule and ½ab sin C; the laws of indices; the circle theorems as statements; speed, density and pressure; the compound interest multiplier | the area of a trapezium (given on no sheet since 2017); Pythagoras in 3D (twice in 2D) |
| SAT (US) | the slope and the forms of a line; the quadratic formula and the vertex form; the exponent rules; the percent change multiplier; the special right-triangle ratios (the test gives the area and volume formulas and the basic triangle facts) | the distance formula (Pythagoras); the midpoint; the circle equation from the centre and radius |
Our Class 10 maths formula sheet and SAT maths formula sheet are the lists to apply the method to; both are printable.
The mnemonics that actually help, and the ones that do not
Four earn their place: SOH CAH TOA for the three trigonometric ratios; BODMAS (or PEMDAS) for the order of operations; "All Students Take Coffee" for which ratios are positive in the four quadrants (all, sine, tangent, cosine); and the hand trick for the standard values of sin from 0° to 90° (√0/2, √1/2, √2/2, √3/2, √4/2, read off the fingers). These work because the things they encode are arbitrary lists, and mnemonics are for arbitrary lists.
The ones that do not help: a rhyme for the quadratic formula (the rhyme is longer than the formula and does not tell you where the 2a goes under a stress), a mnemonic for the area of a sphere (4πr² is one memory, a sentence about it is eight), and anything for the identities, which are better remembered from the one picture of the unit circle. If a formula has a reason, learn the reason; use a mnemonic only when there is none.
A two-week plan for a sheet of thirty formulas
| Day | What to do | Time |
|---|---|---|
| 1 | Read the sheet once. For each formula, write the one-line reason or draw the picture. Group them by shape. | 45 min |
| 2 | Ten problems using the first ten formulas. Blank-paper test of those ten at the end. | 40 min |
| 3 | Ten problems on the next ten. Blank-paper test of all twenty. Rewrite the ones missed. | 40 min |
| 4 | Ten problems on the last ten. Blank-paper test of all thirty. | 40 min |
| 5 | Rest, or a mixed set of five problems. | 15 min |
| 7 | Blank-paper test of all thirty, timed. Five problems on whichever formulas were missed. | 30 min |
| 10 | Blank-paper test. Five mixed problems. | 25 min |
| 14 | Blank-paper test, timed, standing in for the exam. Anything missed gets five more problems. | 25 min |
| then | Once a week, blank-paper test, until the exam. | 10 min |
About five hours in total, against the ten or fifteen hours that copying out formulas costs for a worse result. The plan works for a chapter's worth of formulas or for the whole year's; for the whole year, run it chapter by chapter and keep the weekly test cumulative.
When forgetting formulas is not a memory problem
A child who forgets the quadratic formula in Class 10 has usually not been taught completing the square, so the formula has nothing to hang on. A child who muddles the trigonometric ratios has usually not drawn enough triangles. In our classes the method above is built into the teaching: every formula is derived once on the whiteboard, used in the class and in the homework, and tested from blank paper at the start of the next class. The child who "cannot remember formulas" usually remembers them within a month of that, because the problem was never the memory. We wrote about the larger version of this in why is maths so hard.
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Questions parents ask
1What is the easiest way to memorize maths formulas?
2How do I remember maths formulas quickly before an exam?
3Which maths formulas should I memorise and which should I derive?
4Do mnemonics work for maths formulas?
5Why do I forget formulas in the exam even though I knew them the night before?
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Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.