How to write mathematical workings that actually score IB marks

Introduction: More Than Just Getting the Right Answer

For ten years in the IB Maths classroom, I’ve seen it time and again. A student works hard, they understand the concepts, they can solve the problem. But then the exam results come back, and they've dropped marks. Not because their answer was wrong, but because their workings didn't communicate their thinking effectively. The IB isn't just about finding the correct final answer; it's about demonstrating understanding, and that demonstration happens through your written workings.

This isn't just about showing off; it's about clarity, precision, and adherence to the unspoken rules of mathematical communication that the examiners expect. In my classroom, we focus on building these habits from DP1. Learning how to write effective mathematical workings is a skill as important as understanding calculus or statistics. It’s the bridge between knowing the maths and scoring the marks.

The IB Mark Scheme: Your Blueprint for Communication

The IB mark scheme is your best friend when it comes to understanding what examiners are looking for. It's often broken down into method marks (M), accuracy marks (A), and sometimes reasoning marks (R). What many students don’t realise is that you can often score method marks even if your final answer is wrong, provided your method is correct and clearly shown.

Consider a question where you need to solve a quadratic equation. If you write down the correct quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, and substitute your values correctly, you've likely earned an M mark, even if you make a calculation error afterwards. If you just write down the final answer without showing the substitution or the formula, the examiner has no way to award that M mark. My advice to my students is always: assume the examiner cannot read your mind. Every step, every formula, every substitution needs to be explicit.

What Constitutes a "Step"?

A "step" isn't necessarily a full calculation. It can be:

• Stating a relevant formula: $V = \pi r^2 h$ for cylinder volume.
• Substituting known values into a formula: $V = \pi (3)^2 (5)$.
• Performing a significant algebraic manipulation: $2x + 5 = 11 \Rightarrow 2x = 6$.
• Setting up an equation based on the problem description: If a perimeter is $P$, then $2l + 2w = P$.
• Identifying key values from a graph or table: "From the graph, when $x=2$, $y \approx 4.5$."

I frequently remind students that a single line of working can often combine a few ideas, but clarity is paramount. If you're doing something complicated, break it down. For example, when solving a complex trigonometry problem, state the identity you are using, then show its application.

Precision and Notation: Small Details, Big Impact

The IB expects a certain level of mathematical rigour in your notation. Sloppy notation can lead to ambiguity and loss of marks. This is particularly crucial in topics like calculus, vectors, and statistics.

Calculus: Don't Lose Your $dx$!

In calculus, I see many students drop the $dx$ when integrating. For example, if you're integrating $x^2$, it should be $\int x^2 \, dx$. If you write $\int x^2$, it's technically incomplete and can be penalised. Similarly, when performing differentiation, ensure you write $\frac{dy}{dx}$ or $f'(x)$ clearly. Simply changing $y$ to $y'$ isn't always sufficient.

For my Paper 1 students in SL AA, understanding precise notation for differentiation and integration is non-negotiable. The examiners are looking for this attention to detail.

Vectors: Scalars vs. Vectors

When working with vectors, always distinguish between a vector quantity and a scalar quantity. A vector should be bolded ($\mathbf{a}$) or underlined ($\underline{a}$). If you write 'a' when you mean $\mathbf{a}$, you're potentially losing an accuracy mark, especially in proofs or specific calculations. I often make my students practice writing vectors clearly from the start of the Vectors unit.

Significant Figures and Units

Unless specified otherwise, answers should generally be given to three significant figures. If a question asks for an exact value, leave it as a fraction, surd, or in terms of $\pi$. Rounding prematurely can cause errors later in the calculation, leading to cumulative inaccuracy. Always use appropriate units where applicable – cm, m/s, degrees, radians, etc. If the question gives units, your answer should reflect them.

Structuring Your Response for Clarity

Examiners have many scripts to mark. Make their job easier by structuring your responses clearly. A well-organized response is easier to follow and reduces the chance of an examiner missing a crucial step.

Here are some practices I encourage in my classroom:

• Start with the formula: If you're using a specific formula, write it down first. For example, for combinations: $^nC_r = \frac{n!}{r!(n-r)!}$.
• Show substitution: $^5C_2 = \frac{5!}{2!(5-2)!}$.
• Show intermediate steps: $^5C_2 = \frac{5!}{2!3!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1)(3 \times 2 \times 1)}$.
• State the answer clearly: $^5C_2 = 10$.

For questions with multiple parts, label your answers clearly (a), (b)(i), (b)(ii), etc. If you need to refer back to a previous answer, state "From part (a), we know that $x=3$." This logical flow is critical.

Using Technology: Calculator Workings

Many IB Maths exams involve graphics display calculators (GDCs). You need to show that you are using it appropriately. For example, if you solve an equation graphically:

• State the equations you entered into your GDC: "Let $y_1 = x^2 - 3x + 2$ and $y_2 = 0$."
• State the relevant window settings if they are critical to seeing the solution: "Using a GDC window of $[-5, 5]$ for $x$ and $[-5, 5]$ for $y$..."
• Indicate the function used: "Using the 'intersect' function on the GDC..." or "Using the 'zero' function..."
• Write down the coordinates or values obtained: "The points of intersection are $(1,0)$ and $(2,0)$, so $x=1$ or $x=2$."

For my students using the Casio fx-CG50, I dedicate specific lessons to how to show calculator steps properly. Simply writing down the answer after a complex calculator operation is almost guaranteed to lose marks.

Reviewing Your Work: The Final Check

Before submitting, always review your workings. Ask yourself:

• Is every step clear and logical?
• Have I used correct mathematical notation?
• Are my units and significant figures correct?
• Have I answered all parts of the question?
• Could someone else follow my thinking without needing to ask questions?

This habit of self-review is something I drill into my students during mock exams and practice sessions. It’s not just about finding errors in calculation, but also about improving the clarity of communication. Sometimes, just re-writing a messy step can clarify your thought process for an examiner.

Conclusion: Practice Makes Perfect Communication

Mastering the art of writing effective mathematical workings is a journey that starts in DP1 and refines throughout DP2. It requires consistent practice, attention to detail, and a deep understanding of what the IB mark scheme values.

It’s not just about getting the right answer; it’s about proving to the examiner that you understood the process. By being explicit with your formulas, precise with your notation, clear with your steps, and mindful of your calculator usage, you will not only improve your chances of scoring full marks but also solidify your own understanding of the mathematics. Start building these habits today, and watch your confidence and your scores grow. For more practice, remember to review past papers and their mark schemes, paying close attention to how marks are awarded for working out.