Writing, technology and referencing

Structure an exploration, write mathematics with correct notation and defined variables, use graphs, tables and technology well, cite every source, keep to academic integrity rules and use appendices properly.

IB Mathematics: Analysis and Approaches
Internal assessment
SL and HL

13 min readLast reviewed 14 Oct 2026

An exploration is a piece of mathematical writing for a reader who has studied the same course. That reader needs to know what question is being asked, why each step is taken and what each result means. Good writing does not decorate the mathematics; it makes the mathematics easy to follow and makes the student's understanding visible.

Writing well affects every criterion. Structure and conciseness are criterion A, notation and representations are criterion B, explained decisions show personal engagement (C), comments on results are reflection (D), and clear justification of each step is how understanding is shown in criterion E.

Structure

There is no fixed template, and headings alone do not earn marks. Most strong explorations nevertheless follow a similar overall shape.

Part	Purpose	Typical length
Introduction	Motivation for the topic, the aim, and a brief outline of the approach	Half a page to one page
Background mathematics	Only the definitions and results the exploration uses, explained in the student's own words	One to two pages, or none if each idea is introduced where it is needed
Main body	The mathematics, in sections that each answer part of the aim, with comments on each result	Ten to fifteen pages
Conclusion	The answer to the aim, a critical evaluation and possible extensions	One to two pages
Bibliography	Every source used, in a consistent style	Not counted in the page limit
Appendices	Raw data, code and repetitive calculations that the reader does not need to follow the argument	Not counted in the page limit

The introduction

The introduction gives the reason for the exploration and states the aim precisely. The reason can be personal (an observation, a question from a hobby, a problem met in another subject), but it should lead straight to mathematics. One or two paragraphs of motivation are enough; a page of history or general background before any mathematics usually costs marks for conciseness.

A short outline of the approach helps the reader: "First, a quadratic model is fitted to points measured on the arch. Then a catenary is fitted, and the two models are compared by the sum of their squared residuals."

The main body

Each section of the body should have a clear job, stated in its first sentence, and should end by saying what has been found and what happens next. Headings that describe content ("Fitting a catenary") are more useful than generic ones ("Part 2"). Results are followed by a short comment: what the value means, whether it is reasonable, and what limitation affects it. Reflection placed after each result in this way supports criterion D far more than a single paragraph at the end.

The conclusion

The conclusion answers the aim directly, using the results. It then evaluates: how reliable the answer is, which assumptions matter most, what the strengths and weaknesses of the approach were, and what specific question could be explored next. A conclusion should not introduce new mathematics or simply list the steps that were taken.

A student's conclusion begins:

"In this exploration I first collected data, then I plotted it, then I tried a quadratic and a catenary, and then I compared them. I learned a lot about modelling."

Explain what is wrong with this opening and write an improved one.

Show the solution

The opening lists what was done instead of answering the aim, gives no results and ends with a general statement about learning.

An improved opening answers the aim with numbers and then begins to evaluate:

"The catenary model $y = 4.12 \cosh (\frac{x}{4.12}) - 10.31$ fits the measured arch better than the quadratic model, with a sum of squared residuals of $0.084 m^{2}$ compared with $0.311 m^{2}$ . The difference is concentrated near the base of the arch, where the two curves separate most. Because the points were measured from a photograph, each has an uncertainty of about $\pm 0.05$ m, which is small compared with the residuals of the quadratic model near the base, so the conclusion is unlikely to be caused by measurement error alone."

(The values here are illustrative: a real conclusion would quote its own results.)

Mathematical notation

The exploration should look like mathematics in a textbook. All mathematical expressions are typed with an equation editor (in a word processor, or in a typesetting system such as LaTeX), never in the notation of a calculator, spreadsheet or programming language.

Avoid	Write
x^2, 2^(x+1)	$x^{2}$ , $2^{x + 1}$
3*x, 3.x	$3 x$ , or $3 \times x$ where needed
sqrt(x), x^(1/2)	$\sqrt{x}$
4.7E-3	$4.7 \times 10^{- 3}$
ln x/2 (ambiguous)	$\frac{\ln x}{2}$ or $\ln (\frac{x}{2})$
abs(x - 2)	$∣ x - 2 ∣$
sin^-1(0.5)	$\arcsin (0.5)$ or $\sin^{- 1} (0.5)$
y1, X_bar	$y_{1}$ , $\overset{ˉ}{x}$
≈ and = mixed carelessly	$=$ for exact values, $\approx$ for rounded ones

Software output, such as a screenshot of a regression from a spreadsheet, may show its own notation. If such output is included, the student's own text around it uses correct notation and explains what the output shows.

Defining variables

Every variable is defined in words when it first appears, with its units and, where useful, its domain. Definitions can be given in the sentence before a formula or in a short list after it.

Accuracy and rounding

State the accuracy of every rounded value (for example, "to 3 significant figures") and keep unrounded values in later calculations. The accuracy should suit the context: a length measured to the nearest centimetre does not justify an answer to six decimal places, and a probability from a model is rarely meaningful beyond three significant figures. Measurement uncertainty should be stated where data are collected.

Graphs, tables and diagrams

Graphs, tables and diagrams are part of the argument. Each one is included because it shows something that words or formulas cannot show as clearly.

Graphs

A good graph has:

a number and a caption saying what it shows ("Figure 3: Height of the arch against horizontal distance, with the fitted quadratic and catenary");
labelled axes with units and a sensible scale;
a legend or labels when more than one data set or curve is shown;
data shown as points and models shown as curves, so the two can be compared;
a sentence in the text that refers to it and says what to see in it ("Figure 3 shows that the quadratic lies above the data near both ends of the arch").

A residual plot (the residuals against the independent variable) is often more informative than another graph of the data: a pattern in the residuals shows that a model is missing something, even when the curve looks close to the points.

Tables

Tables are for values that need to be compared or looked up. Give each table a number, a title, column headings with units and a consistent number of decimal places. Long tables of raw data go in an appendix, with a short summary table in the text.

Diagrams

Diagrams are essential in geometry, trigonometry and optimisation work. They should be drawn with software (such as GeoGebra), labelled with the same letters and symbols as the text, and placed where the text first needs them.

Using technology

Technology is encouraged. A graphic display calculator (GDC), graphing software such as Desmos or GeoGebra, spreadsheets and programming languages allow explorations to handle real data, large calculations and simulations that would be impossible by hand.

What to show and what to leave to technology

Technology can do the calculation, but the student must show the understanding. A useful rule is to show one example of each type of calculation in full, then use technology for the rest and present the results in a table.

A student estimates the area of a lake by measuring its width $y$ (in metres) at intervals of $h = 100$ m along its length, giving the widths $0, 120, 180, 210, 160, 90, 0$ . The student then repeats the estimate with intervals of 50 m, 25 m and 10 m. Describe how this work should be presented.

Show the solution

The first estimate is shown in full, with the method explained. With $n = 6$ strips of width $h = 100$ m, the trapezoidal rule gives

A \approx \frac{h}{2} [y_{0} + y_{6} + 2 (y_{1} + y_{2} + y_{3} + y_{4} + y_{5})]

A \approx \frac{100}{2} [0 + 0 + 2 (120 + 180 + 210 + 160 + 90)] = 50 \times 1520 = 76 000 m^{2}

The repeated estimates, with smaller intervals, are calculated in a spreadsheet and summarised in a table in the text: interval width, number of strips and estimated area. The full spreadsheet goes in an appendix. The text then comments on the trend: whether the estimates settle towards a value as the interval decreases, and how they compare with the published area of the lake.

Explaining what technology does

State which technology was used and for what ("The regression was carried out with the exponential regression function of a GDC, which fits a straight line to $\ln y$ against $x$ "). Knowing what the software actually calculates matters: for example, an exponential regression on most calculators minimises the squared residuals of $\ln y$ , not of $y$ , which affects how the fit should be compared with other models.

Output is always interpreted. A regression equation, a correlation coefficient or a simulated probability is followed by a sentence saying what it means in context and how reliable it is.

Code and simulations

Code written for a simulation or a calculation goes in an appendix. The text explains in words what the code does, why the method is valid, how many repetitions were used and why that number is enough. Results from simulations are random, so state how much they vary (for example, by repeating the simulation and comparing the estimates).

Citing sources

Every idea, formula, data set, image or piece of text that comes from somewhere else must be acknowledged at the point where it is used and listed in the bibliography. This includes material that has been paraphrased, not only direct quotations.

What to cite

Data from any source (a website, an organisation, a book, another person).
Formulas and results that are not part of the course, or that are taken from a particular source.
Methods and approaches followed from a source, even if the numbers are different.
Images, diagrams and photographs that the student did not make.
Definitions and explanations paraphrased from a source.

Results that are standard in the course, such as the quadratic formula or the derivative of $\sin x$ , do not need a citation.

How to cite

Choose one referencing style (for example APA, MLA or Harvard) and use it consistently. An in-text citation gives the author and year, or a footnote number, next to the borrowed material. The bibliography lists full details of each source in alphabetical order, including the date accessed for websites.

A student writes: "Population data for the country were taken from the internet. The logistic function is $P (t) = \frac{L}{1 + C e^{- k t}}$ ."

Rewrite this with appropriate acknowledgement, using APA style.

Show the solution

In the text:

"Population figures from 1960 to 2020 were taken from the World Bank's World Development Indicators (World Bank, 2024). The logistic model

P (t) = \frac{L}{1 + C e^{- k t}}

where $L$ is the carrying capacity, was adapted from the treatment of population growth in Stewart (2015, Section 9.4)."

In the bibliography:

Stewart, J. (2015). Calculus: Early transcendentals (8th ed.). Cengage Learning.

World Bank. (2024). Population, total [Data set]. World Development Indicators. Retrieved 3 March 2026, from https://data.worldbank.org/indicator/SP.POP.TOTL

(The access date is an example; use the actual one.)

The source of every data set should also appear in the caption of the table or graph that shows it.

Academic integrity

The exploration must be the student's own work. When it is submitted, the student confirms this, and the teacher confirms that, to the best of their knowledge, it is. Breaking these rules is academic misconduct, and it can lead to no grade being awarded for the subject.

Plagiarism is presenting someone else's words, ideas, data, images or mathematics as one's own, including by close paraphrase without acknowledgement.
Collusion is helping another student's work to be presented as theirs, for example by sharing a draft or writing part of their exploration. Discussing ideas is fine; each student's writing, calculations and decisions must be their own.
Duplication is submitting the same work for two assessments, such as the exploration and an extended essay.
Fabrication is inventing or altering data.

Generative AI tools

The IB treats text, code, images or mathematics produced by a generative AI tool in the same way as material from any other source. If such a tool is used (for example, to explain an idea or to help debug code), its use must be acknowledged in the text and in the bibliography, with details of the tool and how it was used. Content produced by the tool cannot be presented as the student's own work. In practice, an exploration whose mathematics or writing was produced by an AI tool shows no personal engagement and no understanding, so it would score poorly even if acknowledged.

Help from the teacher

The teacher can discuss topic ideas, answer questions, point to resources and give feedback on one draft. The teacher does not correct the mathematics, rewrite sections or tell the student what to change line by line. Help from anyone else, such as a tutor or a parent, must stay within the same limits.

Appendices

An appendix holds material that supports the exploration but is not needed to follow it: raw data, long tables, full spreadsheets, computer code, and repeated calculations of a type already shown in full in the text. Appendices do not count towards the page guidance.

The main text must make sense without the appendices. A teacher or moderator is not expected to read them, so anything needed for the argument, such as the key graph, the main table of results or the one worked calculation, belongs in the body.

Self-check

To do:
I can plan an exploration with a focused introduction, a body in logical sections and a conclusion that answers the aim.
To do:
I can write all mathematics with correct notation and no calculator notation.
To do:
I can define every variable, with units and domain, when it first appears.
To do:
I can label, caption and discuss every graph and table.
To do:
I can show one calculation in full and use technology for the rest, explaining what the technology does.
To do:
I can cite data, formulas, methods and images in the text and in a bibliography.
To do:
I can explain the rules on plagiarism, collusion and generative AI tools.
To do:
I can decide what belongs in an appendix and what belongs in the main text.

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