To present a maths problem set well, write each solution as a short argument a reader can follow line by line. Markers give method marks for visible, justified steps, even when the final answer slips.
This page covers layout, notation, step-by-step working, proof techniques and LaTeX, with worked examples, common mistakes and a submission checklist. It ends with how STEM Donkey can help.
The Rules of a Well-Presented Maths Solution
Restate what you are finding, write one logical step per line, justify each non-obvious step with the rule or theorem used, keep equals signs between equal things only, and box or clearly state the final answer.
Write in sentences where reasoning is involved. "Since f is continuous on [0, 1], the intermediate value theorem gives..." reads far better than a bare chain of symbols.
Why Presentation Matters in Maths
A correct answer with no working often earns little credit, because the marker cannot see your method. Clear presentation also helps you catch your own mistakes before submission.
| What markers look for | How to show it |
|---|---|
| Correct method | Name the technique at the start of the solution |
| Logical flow | One step per line, connected by words such as "so", "hence" and "since" |
| Justification | Cite rules, theorems and conditions as you use them |
| Correct notation | Use standard symbols and define any new ones |
| Clear answer | State the result in exact form, with units if applied |
Notation Rules to Follow
- Use = only between equal quantities. Do not chain unequal expressions such as "x = 3 = 9".
- Use ⇒ for "implies" and ⇔ for "if and only if", and only when the logic really holds in that direction.
- Define variables before using them: "Let n be a positive integer."
- Write vectors in bold or with an arrow, and matrices in capitals, consistently.
- Keep exact values such as √2, π/3 or ln 5 until the final step, then give decimals if asked.
- Include units in applied problems, for example 9.81 m/s².
How to Structure a Step-by-Step Solution
- Restate the question briefly or label it clearly, for example "Question 3(b)".
- Say which method you will use.
- Work through the steps, one per line, justifying each.
- State the answer clearly.
- Check it: substitute back, differentiate an integral, or test a special case.
Worked example: a definite integral, set out in full. Evaluate ∫₀¹ x eˣ dx.
Integrate by parts with u = x and dv = eˣ dx, so du = dx and v = eˣ.
∫₀¹ x eˣ dx = [x eˣ]₀¹ − ∫₀¹ eˣ dx
= (1·e − 0) − [eˣ]₀¹
= e − (e − 1)
= 1.
Check: an antiderivative is eˣ(x − 1), and eˣ(x − 1) evaluated from 0 to 1 gives 0 − (−1) = 1.
Writing Mathematical Proofs
A proof is a written argument, so it needs sentences, not just symbols. State what you will prove, prove it, and say clearly when you have finished.
| Technique | When to use it |
|---|---|
| Direct proof | Assume the hypothesis and derive the conclusion |
| Proof by contradiction | Assume the statement is false and reach an impossibility |
| Contrapositive | Prove "not Q implies not P" instead of "P implies Q" |
| Induction | Statements about all positive integers |
| Counterexample | Disproving a general claim with one specific case |
Worked example: proof by induction. Prove that 1 + 3 + 5 + ... + (2n − 1) = n² for every positive integer n.
Base case. For n = 1, the left side is 1 and the right side is 1² = 1, so the statement holds.
Inductive step. Assume the statement holds for n = k, so 1 + 3 + ... + (2k − 1) = k². Adding the next odd number, 2(k + 1) − 1 = 2k + 1, gives k² + 2k + 1 = (k + 1)². So the statement holds for n = k + 1.
Conclusion. By induction, the statement holds for all positive integers n. ∎
Typesetting in LaTeX
LaTeX is the standard for typed maths and many modules require or reward it. An align environment keeps multi-line working tidy.
\begin{align*}
\int_0^1 x e^x \, dx &= \Big[x e^x\Big]_0^1 - \int_0^1 e^x \, dx \\
&= e - (e - 1) \\
&= 1
\end{align*}The ampersand aligns each line at the equals sign, and \, adds a thin space before dx. Use \mathbb{R} for the real numbers and \mathbf{v} for bold vectors.
Presentation Tips by Area
- Linear algebra: show each row operation, for example R₂ → R₂ − 3R₁, and check eigenvectors by computing Av = λv.
- Calculus and analysis: state conditions such as continuity or differentiability before applying a theorem.
- Statistics and probability: define events and random variables, state distributions and their parameters, and round only at the end.
- Applied maths: list assumptions, keep units throughout and comment on whether the answer is reasonable.
Common Presentation Mistakes
| Mistake | Fix |
|---|---|
| Answer with no working | Show each step and the rule used |
| Misused equals signs | Start a new line for each new equation |
| Rounding too early | Keep exact values until the end |
| Proof that assumes the result | Start from what is given, not from what you want to show |
| Undefined symbols | Introduce every variable with "Let..." |
Problem Set Checklist
- Every question and part is clearly labeled.
- Each solution names its method.
- Every non-obvious step is justified.
- Notation is standard and consistent.
- Final answers are clearly stated, exact where possible, with units if applied.
- Each answer has been checked.
- Handwriting is legible or the work is typed, as your module requires.
How STEM Donkey Helps with Maths Problem Sets
Send the problem sheet, your module notes or textbook chapter, and any presentation rules, such as LaTeX or a required method. A writer matched to your level prepares custom, fully worked solutions with justified steps, clear proofs and checked answers.
Use them to understand each method, then attempt similar problems on your own. Free revisions within the scope of your original request are included.
Want Every Solution Set Out Clearly?
Send the problem sheet and your module notes. You get custom, fully worked solutions with justified steps, correct notation and LaTeX typesetting on request.
Order Your Problem SetFree revisions · Full refund if late · Written from scratch for your order
Frequently Asked Questions
Justify every step that is not routine algebra. Name the rule, theorem or assumption whenever a reader might ask "why?".
That is often fine. Give the exact form, check your working once more, and give a decimal approximation if the question asks.
Many modules accept neat handwritten work. Check the rules; some require typed or LaTeX submissions.
State the conclusion in words, then add a symbol such as ∎ or "QED" if your module uses one.
Yes. Short phrases such as "hence", "since" and "by the chain rule" make the logic clear and earn method marks.
The basics take an afternoon. Start with inline maths, the align environment and common symbols, then build up.