How to remember maths formulas, by memorising fewer of them

There are somewhere around sixty formulas in Class 10 Maths, depending on how you count. If you are trying to hold all sixty in your head as separate facts, no wonder some of them fall out at the worst possible moment.

The students who never seem to forget them are not remembering more than you. They are remembering considerably less, because most of what you are memorising they can rebuild in about eight seconds.

Most formulas are not separate facts

A small number of ideas generate most of the list. Understand a root properly and everything hanging off it becomes something you can derive instead of recall.

Derive each one once, properly

You do not need to derive formulas every time you use them. That would be slow and pointless. You need to derive each one once, carefully, so that you know it can be rebuilt if it ever slips.

Take the distance formula. Plot two points, draw the horizontal gap and the vertical gap, notice that you have made a right triangle, and apply Pythagoras. Do that once and:

stops being a string of symbols and becomes an obvious consequence. You will never write it with the square root in the wrong place again, because you know what the square root is doing there.

Learn what each part does

Even for formulas you do memorise, knowing the role of each piece prevents the most common errors. Take:

Three things worth understanding rather than memorising:

• The $\pm$ is why you get two roots. It is not decoration.
• The $b^2-4ac$ under the root decides everything. Negative means no real roots, because you cannot square-root a negative. Zero means the $\pm$ adds nothing and the two roots collapse into one.
• The whole numerator sits over $2a$. Students routinely divide only part of it, which is the most common way this formula is misapplied.

Someone who understands those three things cannot really get the formula wrong. Someone who memorised the string of symbols gets it wrong under pressure.

The ones you do just have to know

Some things are genuinely worth committing to memory, because deriving them every time is not practical:

1. The trigonometric values at 0, 30, 45, 60 and 90 degrees.
2. $\sin^2\theta+\cos^2\theta=1$ and the two identities that follow from it.
3. Surface area and volume formulas for the standard solids.
4. Squares up to 25 and the common Pythagorean triples like 3-4-5 and 5-12-13.

For the trig table specifically, do not memorise fifteen separate values. Learn the sine row as $\sqrt{0}/2,\ \sqrt{1}/2,\ \sqrt{2}/2,\ \sqrt{3}/2,\ \sqrt{4}/2$, notice it is just the numbers 0 to 4 under a root over 2, and then note that cosine is the same row backwards. Two patterns instead of fifteen facts.

Test, do not review

The single most effective way to make a formula stick is to try to write it from memory, fail, check, and try again tomorrow. The failing part is not a sign it is going badly. It is the part that does the work.

Keep a blank sheet. Once a week, write down every formula you can remember for a chapter. Then open the book and see what is missing. That missing list is your revision, and it takes ten minutes.