Easy Techniques to Improve your Maths, Part 2: Formula Layout


Using an algebraic formula may seem obvious to most, yet many people try to shortcut the process and end up with mistakes. Use this methodical five step process to mitigate errors and maximise exam marks.

This is the second of a series of short articles for improving your mathematics by mitigating common silly mistakes, by the founder of MathU, Dr Andrew Smith.


5 Step Layout

1. Write out the Formula

Use paper - doing it all in your head will quickly lead to mistakes as questions become more complex.

Always write the formula, even during every practice question. This will reduce the chance of errors through improved memorisation of the correct formula (repeated writing helps this) and improved accuracy of substitution of values.

Furthermore, in an exam situation most teachers will assign a mark to the correct formula when written.

2. Write out the variables

List all variables and the values for each (use a question mark (?) for the unknown value sought).

Now adjust the units for individual values to ensure the units match the formula requirements.

3. Substitute

Do not calculate anything here, just substitute the variable values into the formula. This will minimise the chance of an error, which can easily occur when calculating while substituting, especially with negative or complicated algebraic values.

4. Solve

Solve the equation for the desired unknown variable.

The right amount of "working" is the amount that prevents silly calculation mistakes - don't try to do too much in your head at once, paper is cheap while mistakes are not.

Keep equals signs directly under each other (and never more than one per line), not just for etiquette but to reduce the likelihood of mistaking which side each term is on, a common source of error.

5. Answer the Question, with Units; Consider Realism

Do not skip this step. Most questions ask for an answer, not the value of a variable. You need to actually answer the original question. Often, this requires further calculation, conversion to different units, or interpretation of the calculated value.

Add units to your answer. In an exam situation, the units will have marks attached, while in a work situation a value with no units is meaningless or prone to misinterpretation.

Now consider whether your answer makes sense in the context of the question. This can reveal the presence of an error - not all errors and not where the error may be, but will alert you to review.


Example:

Question:
Using Einstein's famous equation and the speed of light as \(300\,000\,000\text{ m/s}\), how much energy is released by the annihilation of \(1\text{ mg}\) of mass in a nuclear reaction?

Answer:
\begin{align} E &= mc^2\\ \text{where: }&\\ E &=?\\ m &= 1 \text{ mg}\\ &= 0.000\,001 \text{ kg}\\ c &= 300\,000\,000 \text{ m/s}\\ \text{substituting:}&\\ E &= 0.000\,001 \times (300\,000\,000)^2\\ &= 90\,000\,000\,000 \text{ J}\\ &= 90 \text{ GJ}\\ \text{energy released is }& 90 \text{ GJ}\\ \end{align}


For further help, including maths understanding, skill, practice, and mitigations for common mistakes, check out our online maths learning tool MathU at mathu.com.au