The assessment criteria

What each of the five exploration criteria rewards, what each level looks like, what earns the top band, why marks are commonly lost, and how criterion E differs between SL and HL.

IB Mathematics: Analysis and Approaches
Internal assessment
SL and HL

17 min readLast reviewed 14 Oct 2026

The exploration is marked against five criteria, A to E, for a total of 20 marks. Each criterion has a short scale of levels, and each level is described by a few key words. The descriptions are brief, so the meaning of each key word matters: the difference between "some" and "significant" personal engagement, or between "meaningful" and "critical" reflection, is the difference between levels.

The level descriptions here are written in plain language to explain what the IB's descriptors mean in practice. They are not the official wording; teachers mark using the IB's own descriptors and the clarifications that come with them.

Criterion	Focus	Marks
A Presentation	Organisation, coherence, conciseness	4
B Mathematical communication	Notation, terminology, representations	4
C Personal engagement	Ownership, independent and creative thinking	3
D Reflection	Critical evaluation of results and methods	3
E Use of mathematics	Relevance, level, correctness, understanding	6

How the criteria are applied

Each criterion is marked on its own. The teacher reads the whole exploration and, for each criterion, chooses the level whose description fits the work best. Where the work sits between two levels, the teacher decides which description it matches more closely, so an exploration does not have to meet every word of a level to earn it. A level of 0 is given when the work does not reach the lowest description.

Criterion A: Presentation

Criterion A judges how the exploration is organised as a piece of writing, not the mathematics itself. It has three key words: coherent, organised and concise.

Level	What the work looks like
0	Does not reach level 1
1	Shows either some coherence or some organisation, but not both
2	Shows some coherence and some organisation
3	Coherent and well organised
4	Coherent, well organised and concise

What the key words mean

Coherent: the exploration develops logically and is easy to follow. The aim is stated, each section follows from the one before, and the conclusion answers the aim.
Well organised: the exploration has a clear introduction (with the motivation and the aim), a body in a sensible order, and a conclusion. Graphs, tables and diagrams appear where they are discussed, not gathered at the end.
Concise: nothing is irrelevant or repeated. There are no pages of identical calculations, no long background that the mathematics never uses, and no graphs that are not discussed.

What earns 4

A reader can follow the exploration from start to finish without having to go back and look for missing information. Each section has a clear purpose that is linked to the aim. Repetitive work is shown once in full and then summarised in a table, with the details in an appendix if needed. The length is within the guidance of 12 to 20 pages.

Why marks are lost

No clear aim, or an aim that changes part-way through.
A long introduction on history or background that the rest of the work does not need.
The same calculation repeated many times in full for different data values.
Graphs or tables placed in an appendix when the argument depends on them, so the reader has to keep turning pages.
A conclusion that summarises what was done instead of answering the aim.

Criterion B: Mathematical communication

Criterion B judges how the mathematics is written: the notation, symbols and terminology, and the use of representations such as formulas, graphs, tables and diagrams. Its key words are relevant, appropriate and consistent.

Level	What the work looks like
0	Does not reach level 1
1	Some relevant mathematical communication, only partly appropriate
2	Some relevant mathematical communication, mostly appropriate
3	Relevant and appropriate mathematical communication, mostly consistent
4	Relevant and appropriate mathematical communication, consistent throughout

What the key words mean

Relevant: the representations used are the ones that help, such as a graph where a trend needs to be seen and a table where values need to be compared.
Appropriate: notation and terminology are correct mathematics, as in a textbook. Variables and key terms are defined. Graphs have labelled axes with units and scales. Results are given to a sensible accuracy.
Consistent: the same symbol means the same thing everywhere, accuracy is handled the same way throughout, and the quality of presentation does not drop in the later sections.

What earns 4

Every variable is defined when it first appears, with units. The mathematics is typed with an equation editor, never in calculator or programming notation. Every graph and table has a number, a title or caption, labelled axes or headings, and is referred to in the text. Rounded values say how they were rounded, and unrounded values are used in later calculations.

Why marks are lost

Calculator or spreadsheet notation, such as "x^2", "2*x", "sqrt(x)", "3.2E-4" or "y=ax+b, a=1.84".
Variables used without being defined, or a symbol that changes meaning (for example $t$ meaning time in one section and temperature in another).
Graphs pasted from software without labels, units or a caption, or with no comment in the text.
Inconsistent accuracy: values given to 2 decimal places on one page and 6 significant figures on the next.
"=" used between things that are not equal, such as " $= 3.2$ m/s $=$ fast".

A student writes the following under a graph of population data.

"Exponential regression on GDC gives y=2.31*e^(0.042x), r^2=.987 so it fits good."

Rewrite this sentence so that it communicates the mathematics appropriately.

Show the solution

The original uses calculator notation ("*", "^" and ".987"), does not define $x$ or $y$ , gives no units and uses an informal judgement ("fits good").

An improved version:

"Using exponential regression on a graphic display calculator (GDC), the model

P = 2.31 e^{0.042 t}

was obtained, where $P$ is the population in thousands and $t$ is the number of years after 1950. The coefficient of determination is $R^{2} = 0.987$ (3 s.f.), which means that about 98.7% of the variation in the logarithm of the population is explained by the model. Figure 4 shows the model with the data."

Each symbol is defined, the notation is typed correctly, the accuracy is stated and the graph is referred to.

Criterion C: Personal engagement

Criterion C judges how far the exploration is the student's own work in the sense of thinking: whether the student has asked their own questions, made their own decisions and presented the mathematics in their own way. Its key words describe how much evidence there is: some, significant and outstanding.

Level	What the work looks like
0	Does not reach level 1
1	Some evidence of personal engagement
2	Significant evidence of personal engagement
3	Outstanding evidence of personal engagement

What personal engagement looks like

Personal engagement is not the same as personal interest. A paragraph explaining that the student loves football shows interest in the topic; it does not show engagement with the mathematics. Evidence of engagement is found in the work itself.

Asking questions and following them up: "Would the model still work if the ball were thrown from a height?"
Making a conjecture, testing it and changing it when it fails.
Collecting original data, or designing an experiment or simulation.
Choosing between methods and explaining the choice.
Presenting a known idea in a new way, such as a new diagram, a new example or a different proof.
Applying a method to a new case that no source has worked through.

What earns 3

The student's thinking is visible throughout, not only in the introduction. Decisions are explained ("A logistic model was tried next, because the residuals of the exponential model became increasingly negative after 1990"), and the work goes beyond what a textbook or website contains. A reader can tell that the student understands the question well enough to take it in their own direction.

Why marks are lost

Engagement claimed in the introduction but absent in the mathematics.
Work that follows a well-known online example closely, with only the numbers changed.
A topic far beyond the course, where the student can only repeat what a source says.
Mechanical work, such as fitting a model with technology and reporting the output without questioning it.

Here are two extracts from explorations on basketball free throws.

Extract 1: "I chose this topic because I have played basketball since I was seven and it is my passion. Basketball is the most popular sport in my school."

Extract 2: "My success rate over 300 shots was 72%, but I had the impression that I scored more often straight after a hit. To test this, I recorded each shot in order and split the shots by the result of the previous one."

Explain which extract shows more personal engagement, and why.

Show the solution

Extract 1 shows personal interest in the topic but contains no mathematics. It could be removed without changing the exploration at all.

Extract 2 shows personal engagement with the mathematics: the student has their own data, has noticed something in it, has turned the impression into a question that can be tested ("does the probability of a hit depend on the previous shot?") and has designed a way to test it (recording the order of shots). The question comes from the student's own experience, and it leads directly to conditional probability.

Criterion D: Reflection

Criterion D judges how the student reviews and evaluates the work: the results, the methods and their limitations. Its key words are limited, meaningful and substantial critical reflection.

Level	What the work looks like
0	Does not reach level 1
1	Limited reflection, often only a description of results
2	Meaningful reflection, linked to the aim and the results
3	Substantial evidence of critical reflection

What the key words mean

Limited (superficial) reflection describes what was found or repeats a result in words, for example "the model fits well". It appears, if at all, only in the conclusion.
Meaningful reflection explains what the results mean in the context of the aim, links them to each other and identifies limitations of the method or data.
Critical reflection goes further. It asks what the results imply, how reliable they are, what the strengths and weaknesses of the chosen approach were compared with others, and what could be explored next. Substantial means this happens throughout the exploration, not only at the end.

What earns 3

Reflection appears after each significant result, not just in a final section. The student explains what each result means in context, checks whether it is reasonable, discusses the effect of assumptions and measurement error, compares the approach with alternatives and suggests specific extensions that follow from what was found.

Why marks are lost

Reflection only in the conclusion.
Generic limitations such as "more data would make it more accurate" or "human error", without saying how or why.
Restating results instead of interpreting them.
No comparison of methods, and no comment on how far the conclusion can be trusted.

A student has found that, over 300 free throws, 160 of the 215 shots following a hit went in, and 55 of the 84 shots following a miss went in. The student's reflection reads:

"After a hit my success rate was 74.4% and after a miss it was 65.5%. This shows that the hot hand exists. My results could be more accurate with more shots."

Explain why this reflection is limited, and write a more critical version.

Show the solution

The reflection restates the two percentages, draws a strong conclusion without testing it, and ends with a generic limitation.

A more critical version tests whether a difference this large could arise by chance. If the shots were independent with success probability $0.72$ , the number of hits $X$ among the 84 shots after a miss would follow $X \sim B (84, 0.72)$ , with $E (X) = 60.48$ . Then

P (X \leq 55) \approx 0.114

So about 11% of the time, independent shooting would produce a result after a miss at least as low as the one observed.

"The difference between 74.4% and 65.5% supports my impression, but the binomial model shows that a gap this size would occur by chance in about one session in nine if the shots were independent. The evidence for a hot hand is therefore weak. The test also assumes that my success probability stayed constant over the session; tiredness towards the end would lower it and could create a similar pattern. A further 300 shots, split into sessions of 50 so that tiredness can be compared, would separate these two explanations."

This version interprets the result, tests it, questions an assumption and proposes a specific extension that follows from the findings.

Criterion E: Use of mathematics

Criterion E is worth 6 of the 20 marks. It judges the mathematics itself: whether it is relevant to the aim, commensurate with the level of the course, correct, and understood. At HL it also asks for sophistication, rigour and precision. This is the only criterion that differs between SL and HL.

Key words

Relevant: the mathematics helps answer the aim. Mathematics added to make the work look harder, but not used, is not relevant.
Commensurate with the level of the course: the mathematics is at the level of the course (SL or HL) or beyond. Mathematics from before the course (simple percentages, the mean, straight-line graphs on their own) is below this level.
Correct: there are no significant errors in the mathematics. At SL, an occasional minor slip does not prevent the work counting as correct.
Knowledge and understanding: the student explains why methods are chosen and why they work, applies them properly to the situation, and interprets the output. Copying a formula and substituting into it shows knowledge; deriving or justifying it and explaining its conditions shows understanding.

Criterion E at SL

Level	What the work looks like
0	Does not reach level 1
1	Some relevant mathematics is used
2	Some relevant mathematics is used, with limited understanding
3	Mathematics at the level of the course is used, with limited understanding
4	Mathematics at the level of the course is used; it is partly correct, with some knowledge and understanding
5	Mathematics at the level of the course is used; it is mostly correct, with good knowledge and understanding
6	Mathematics at the level of the course is used; it is correct, with thorough knowledge and understanding

Levels 1 and 2 are the ceiling for an exploration whose mathematics is below the level of the course, however correct it is. From level 3 upwards, the marks depend on how correct the work is and how well it is understood.

Criterion E at HL

Level	What the work looks like
0	Does not reach level 1
1	Some relevant mathematics is used, with limited understanding
2	Some relevant mathematics is used; it is partly correct, with some knowledge and understanding
3	Mathematics at the level of the course is used; it is correct and shows some of the sophistication expected, with some knowledge and understanding
4	Mathematics at the level of the course is used; it is correct and shows the sophistication expected, with good knowledge and understanding
5	Mathematics at the level of the course is used; it is correct and shows sophistication and rigour, with thorough knowledge and understanding
6	As level 5, and the mathematics is precise

How HL differs from SL

At HL, "the level of the course" means HL. An HL exploration using only SL techniques, however well, is not commensurate with HL, so its criterion E mark is lower than the same work would earn at SL. HL also adds three ideas to the top levels.

Sophistication means the mathematics is challenging for an HL student, or that familiar mathematics is used in an unusual way, from more than one perspective, or to link different areas of the course. Fitting a model is not sophisticated in itself; deriving the model from assumptions, comparing it with a second model and investigating how its parameters affect the result can be.
Rigour means the logic and the language are clear and exact. Claims are justified or proved rather than checked on a few cases, conditions (such as the domain of a function or the assumptions of a distribution) are stated, and each step follows from the one before.
Precision means the mathematics is free of errors and every value is given to an appropriate accuracy for its context. A single real error prevents the top level.

An HL student investigates the sum of the first $n$ odd cubes. The draft contains:

"I calculated the sum for $n = 1$ to $10$ and found that it always equals $n^{2} (2 n^{2} - 1)$ , so this formula is true for all $n$ ."

Explain why this does not show the rigour expected at HL, and show how it can be improved.

Show the solution

Checking ten cases makes the formula a conjecture, not a result. Many patterns that hold for small $n$ fail later, so a general statement needs a proof. At HL, proof by induction is part of the course, so the student can and should prove it.

Let $S_{n} = \sum_{k = 1}^{n} (2 k - 1)^{3}$ . The claim is $S_{n} = n^{2} (2 n^{2} - 1)$ for all $n \in Z^{+}$ .

For $n = 1$ : $S_{1} = 1^{3} = 1$ and $1^{2} (2 \times 1^{2} - 1) = 1$ , so the claim holds.

Assume the claim holds for $n = m$ , that is $S_{m} = m^{2} (2 m^{2} - 1)$ . Then

\begin{aligned} S_{m + 1} & = m^{2} (2 m^{2} - 1) + (2 m + 1)^{3} \\ = 2 m^{4} - m^{2} + 8 m^{3} + 12 m^{2} + 6 m + 1 \\ = 2 m^{4} + 8 m^{3} + 11 m^{2} + 6 m + 1 \end{aligned}

Expanding the claimed form for $n = m + 1$ :

(m + 1)^{2} (2 (m + 1)^{2} - 1) = (m^{2} + 2 m + 1) (2 m^{2} + 4 m + 1) = 2 m^{4} + 8 m^{3} + 11 m^{2} + 6 m + 1

The two expressions are equal, so the claim holds for $n = m + 1$ whenever it holds for $n = m$ . Since it holds for $n = 1$ , by the principle of mathematical induction it holds for all $n \in Z^{+}$ .

The exploration then has a proved result to build on: $S_{n}$ is a perfect square exactly when $2 n^{2} - 1$ is a perfect square (since $n^{2}$ already is), which is true for $n = 1$ and $n = 5$ ( $S_{5} = 1225 = 35^{2}$ ) and leads to a further question about which other $n$ work.

What earns 6

At both levels, the top band needs mathematics that is relevant and at the right level, correct throughout, and clearly understood. In practice this means each method is introduced with a reason, formulas taken from sources are explained or derived, technology output is interpreted rather than just reported, and results are checked (for example, by substituting back or comparing with a graph). At HL, the work also has to be sophisticated, rigorous and error-free.

Why marks are lost

Mathematics below the level of the course, or only a small amount at that level.
Mathematics that is too advanced to be understood, presented as a series of quoted formulas.
Errors in calculations, in algebra or in the use of technology.
Results from technology given without explanation, so understanding cannot be seen.
Mathematics that does not serve the aim, included only to look impressive.
At HL: claims checked on examples instead of being proved, and conditions left unstated.

Summary of the criteria

Criterion	Top band in brief	Most common reason for lost marks
A Presentation (4)	Coherent, well organised and concise	Unfocused aim; padding and repetition
B Mathematical communication (4)	Correct notation and representations, consistent throughout	Calculator notation; undefined variables; unlabelled graphs
C Personal engagement (3)	Outstanding evidence of the student's own thinking throughout	Interest stated but not shown in the mathematics
D Reflection (3)	Substantial critical reflection throughout	Reflection only at the end; generic limitations
E Use of mathematics (6)	Correct mathematics at course level, thoroughly understood (HL: also sophisticated, rigorous and precise)	Mathematics too simple or not understood; errors

Self-check

To do:
I can name the five criteria and the maximum mark for each.
To do:
I can explain what "coherent", "well organised" and "concise" mean for criterion A.
To do:
I can find and correct calculator notation, undefined variables and unlabelled graphs in my own work.
To do:
I can tell the difference between personal interest and personal engagement.
To do:
I can write reflection that interprets, tests and evaluates a result, not one that only describes it.
To do:
I can explain what "commensurate with the level of the course" means for my level.
To do:
I can explain sophistication, rigour and precision for HL criterion E.

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