The assessment criteria

The five criteria the exploration is marked against, what each level looks like at SL and HL, what sophistication and rigour mean, and how to move up a level on each.

IB Mathematics: Applications and Interpretation
Internal assessment
SL and HL

18 min readLast reviewed 15 Oct 2026

Introduction

Every exploration is marked against the same five criteria, and the marks for the five are added to give a total out of 20. Knowing the criteria well before you start writing saves a great deal of rewriting later, because they say exactly what the reader is looking for: a well-organised argument, clear mathematical writing, your own thinking, honest evaluation, and correct mathematics that you understand.

Criterion	Name	Maximum
A	Presentation	4
B	Mathematical communication	4
C	Personal engagement	3
D	Reflection	3
E	Use of mathematics	6
	Total	20

Criteria A to D are worded identically at SL and HL. Criterion E has two versions: the HL version expects correct mathematics at a lower level and, for its top levels, mathematics that shows sophistication and rigour. Both versions are given in full in the section on criterion E.

How the criteria are applied

Best fit, not a checklist

Your teacher marks each criterion separately by reading its level descriptors from the bottom up and choosing the one that describes the work most accurately. This is a best-fit judgement. The work does not need to meet every phrase of a descriptor to be given that level, and when it matches different parts of a criterion at different levels, the mark reflects the overall balance.

Where a level spans more than one mark, the upper mark goes to work that shows the qualities strongly (close to the level above) and the lower mark to work that shows them more weakly (close to the level below). Only whole marks are given. The top level of a criterion does not mean flawless work: it describes what a strong student can achieve, and teachers are told to use the full range when it fits.

Each criterion is judged on its own. Excellent mathematics does not raise the reflection mark, and a weak conclusion does not lower the communication mark. That is why a plan that serves all five criteria from the start pays off.

Criterion A: Presentation

What the levels look like

Level	Description (paraphrased)
0	Does not reach level 1.
1	Either some parts are logically connected or some parts are organised, but not both.
2	Some logical connection and some organisation are both present.
3	The whole report is coherent and well organised.
4	Coherent, well organised and also concise.

Three words carry the criterion. A coherent exploration develops logically, is easy to follow and meets its aim; that covers the overall structure (introduction, body, conclusion) and how well each part leads to the next. A well-organised exploration has an introduction, states its aim and has a conclusion, with relevant graphs, tables and diagrams placed where they are discussed rather than in appendices. A concise exploration contains nothing irrelevant and no unnecessary repetition of calculations, graphs or descriptions.

Technology is encouraged but not required. Choosing an analytic (by-hand) method rather than a technological one is not a lack of conciseness and is not penalised, but pages of the same calculation repeated are never acceptable, whichever method produced them.

Moving up on A

From 2 to 3: write the aim in the introduction and answer it in the conclusion, using the same words; give every section a heading and a first sentence saying what it does and why; move key graphs out of the appendix into the text.
From 3 to 4: cut. Remove background history not used later, repeated calculations (show one instance, then tabulate), screenshots that duplicate a table, and descriptions that repeat what a graph shows.

An extract from a draft reads: "Next I calculated the residual for 1965. The predicted value was 318.26 and the actual value was 320.04, so the residual is 1.78. Next I calculated the residual for 1970. The predicted value was ..." and continues in the same way for twelve years. Explain how this affects criterion A and rewrite it concisely.

Show the solution

The same calculation repeated twelve times is unnecessary repetition, so the report cannot be called concise, which holds criterion A at 3 at best. It also hides the pattern in the residuals, which is the reason for calculating them.

A concise version shows one instance and tabulates the rest: "The residual for each year is the observed value minus the value predicted by the model, $e_{i} = y_{i} - {\hat{y}}_{i}$ . For 1965, $e = 320.04 - 318.26 = 1.78$ . The residuals for all twelve years are given in Table 2 and plotted in Figure 4." The text then discusses what the plot shows.

Criterion B: Mathematical communication

What the levels look like

Level	Description (paraphrased)
0	Does not reach level 1.
1	Some relevant mathematical communication, only partly appropriate.
2	Some relevant and appropriate mathematical communication.
3	Relevant and appropriate, and consistent for most of the report.
4	Relevant and appropriate, and consistent throughout.

Criterion B looks at four things:

Mathematical language. Notation, symbols and terminology are correct. Calculator or computer notation (such as x^2, *, 2.3E4, sqrt(x)) is acceptable only inside output generated by software, such as a screenshot; everywhere else the report uses proper notation ( $x^{2}$ , $2.3 \times 10^{4}$ , $\sqrt{x}$ ).
Definitions. Key terms and every variable are defined where they are first used, with units.
Representation. Formulae, diagrams, tables, charts, graphs and models are used where they help. Level 4 is possible with only one form of representation if that form suits the topic.
Deduction. Where an argument or proof is given, its steps are set out logically.

Typical level 1 work has unlabelled graphs or uses computer notation throughout with little correct notation elsewhere. At level 4, minor slips that do not get in the way of clear communication are not penalised.

Moving up on B

The fastest gains on B are mechanical. Define every variable in a sentence when it first appears ("Let $t$ be the number of years after 1960 and $C$ the annual mean concentration of carbon dioxide in parts per million"). Label every axis with the quantity and its unit, number every figure and table, and give each a caption. Replace every piece of typed calculator syntax with real notation. Use the same symbol for the same quantity from start to finish; changing from $t$ to $x$ halfway is a common reason for "mostly consistent" rather than "consistent throughout".

Criterion C: Personal engagement

What the levels look like

Level	Description (paraphrased)
0	Does not reach level 1.
1	Some evidence of personal engagement.
2	Significant personal engagement: genuine engagement on a few occasions that moves the exploration forward and helps the reader understand the writer's intentions.
3	Outstanding personal engagement: genuine, high-quality engagement in many places, driving the exploration forward creatively and leaving the impression that the writer fully understands the context.

Personal engagement must be visible in the report; a teacher's comment that a student worked hard or seemed interested is not evidence. Textbook-style explorations, or reproductions of easily available mathematics without the student's own perspective, rarely reach the top levels.

Personal engagement is easy to confuse with personal background. A paragraph about why you like a topic is a reasonable start, but the marks come from what you then do with the mathematics. Evidence includes: collecting your own data, or questioning and cleaning a secondary data set; making a prediction before calculating and then testing it; trying a method, finding it unsatisfactory and explaining why you changed it; choosing a model's parameters for reasons drawn from the context; extending the question when a result suggests a new one; and presenting an idea in your own way (your own diagram, your own derivation, your own example).

Moving up on C

From 1 to 2: add a few genuine decisions that change the direction of the work, and say why you made them.
From 2 to 3: make such decisions throughout, not only in the introduction; let your understanding of the context guide the mathematics at each stage.

Two extracts introduce the same exploration on a cooling cup of tea. Extract X: "I love tea and drink it every day, so I chose this topic because it is personal to me." Extract Y: "Before measuring, I predicted the tea would cool at a constant rate, because the temperature seemed to drop steadily while I drank it. The first ten readings disproved this: the drop in the first minute was more than three times the drop in the tenth, which led me to test an exponential model instead." Compare the two as evidence for criterion C.

Show the solution

Extract X describes interest but shows no engagement with the mathematics; it would be the same whatever the student then did, and on its own it supports level 1 at most.

Extract Y shows a prediction made from experience, tested against the student's own data, found wrong, and used to change the model. That decision drives the exploration forward and helps the reader understand the writer's thinking, which is the kind of evidence that, repeated through the report, supports levels 2 and 3.

Criterion D: Reflection

What the levels look like

Level	Description (paraphrased)
0	Does not reach level 1.
1	Limited reflection.
2	Meaningful reflection.
3	Substantial evidence of critical reflection.

Describing results ("the model gives 424 ppm for 2025") is only limited reflection. Meaningful reflection goes further: it links results to the aim, says what was learned, considers a limitation or compares two mathematical approaches. Critical reflection is decisive or deeply insightful: it uses the results to develop the exploration, for example by asking what should be done next, what the results imply, what the strengths and weaknesses of the methods are, or how the problem looks from another perspective. Substantial means it appears throughout the report. Reflection that appears only in the conclusion can still reach level 3, but only if it is of high quality and shows how it developed the exploration.

Moving up on D

One simple habit raises D quickly: after every significant result, write one or two sentences answering "What does this mean, how reliable is it, and what does it change?". A conclusion should then return to the aim, say how far it was achieved, give the main limitations with their likely effect (not a list of generic sentences such as "more data would be better"), and suggest a specific next step.

A conclusion reads: "To conclude, the quadratic model was the best because it had the highest $R^{2}$ . A limitation is that I could have used more data." Explain why this is limited reflection and rewrite it to show critical reflection.

Show the solution

It describes a result and gives a generic limitation that would fit any exploration. It does not say what "best" means for the aim, how far the model can be trusted, or what the limitation changes.

A rewrite: "The quadratic model predicted the 2020 and 2025 values, which were not used to fit it, to within 0.8%, while the linear model under-predicted both by about 3–4%. A higher $R^{2}$ alone would not justify the choice, because adding a term always raises $R^{2}$ ; the decisive evidence is the quadratic model's accuracy on new data and the curved pattern in the linear residuals. The model describes the trend in the annual means only; it says nothing about the cause of the rise, and extrapolating it to 2100 would assume that emissions keep accelerating in the same way, which is a question about policy, not mathematics. A natural next step would be to test whether the acceleration itself is changing by fitting the model to successive 20-year windows."

Criterion E: Use of mathematics

SL levels

Level	Description (SL, paraphrased)
0	Does not reach level 1.
1	Some relevant mathematics is used.
2	Some relevant mathematics; limited understanding is shown.
3	Relevant mathematics at the level of the course; limited understanding is shown.
4	Relevant mathematics at the level of the course, partly correct; some knowledge and understanding are shown.
5	Relevant mathematics at the level of the course, mostly correct; good knowledge and understanding are shown.
6	Relevant mathematics at the level of the course, correct; thorough knowledge and understanding are shown.

HL levels

Level	Description (HL, paraphrased)
0	Does not reach level 1.
1	Some relevant mathematics; limited understanding is shown.
2	Some relevant mathematics, partly correct; some knowledge and understanding are shown.
3	Relevant mathematics at the level of the course, correct; some knowledge and understanding are shown.
4	Relevant mathematics at the level of the course, correct; good knowledge and understanding are shown.
5	Relevant mathematics at the level of the course, correct, showing sophistication or rigour; thorough knowledge and understanding are shown.
6	Relevant mathematics at the level of the course, precise, showing sophistication and rigour; thorough knowledge and understanding are shown.

The HL scale is steeper. An HL exploration needs correct mathematics already at level 3, where SL accepts partly correct mathematics at level 4. The top two HL levels add sophistication and rigour, and level 6 asks for precision rather than correctness.

The key words

Relevant: the mathematics helps the exploration reach its aim. Overly complicated mathematics where simpler mathematics would do is not relevant.
Commensurate with the level of the course: part of the syllabus or of a similar level, and not based entirely on prior learning. At HL the mathematics may go slightly beyond the syllabus, but that is not needed for the highest levels.
Demonstrated: made clear by reasoning or evidence, with examples or applications. A correct answer on its own does not demonstrate understanding, not even "some".
Thorough: understanding is demonstrated throughout, not in one section only.
Correct: occasional minor errors are allowed if they do not break the flow of the mathematics or lead to an unreasonable outcome.
Precise (HL level 6): free of errors, and using an appropriate degree of accuracy at every stage.

Technology is encouraged, but understanding has to be shown alongside it. At SL, understanding must be demonstrated for any level above 1; the HL descriptors ask for limited understanding already at level 1. Substituting numbers into a formula, or quoting a regression equation or a p-value from software, does not by itself show understanding. What does: explaining why the method suits the data, stating and checking its conditions, doing one instance by hand where that is reasonable, and interpreting each result in context.

The mathematics only needs to be what the aim requires. That could be a few elements of mathematics or a single subtopic, used well. Doing a few things thoroughly is better than touching many things superficially.

Sophistication and rigour (HL)

Sophistication comes from the depth with which mathematics is used, and difficult mathematics added for show does not provide it. HL-level techniques such as non-linear regression with a justified choice of model and $R^{2}$ compared fairly, a differential equation solved with Euler's method and checked against an analytic solution, a confidence interval, a Markov chain with a steady state, or a phase portrait used to interpret a system are natural sources. SL mathematics can also be sophisticated when it is used in a complex way: comparing an algebraic method with a numerical one for the same problem, or seeing that a finance problem and a geometric series are the same structure.

Rigour is about the argument. Each step follows from the last, symbols mean one thing, assumptions are stated before they are used, and claims are supported. "The data look exponential" is not rigorous. A rigorous version derives a testable consequence first ("if $C = a b^{t}$ , then $\ln C = \ln a + t \ln b$ , so a plot of $\ln C$ against $t$ should be linear"), then checks it against the data, including the residuals, before accepting or rejecting the model. At HL, lines of reasoning must be shown to justify the steps of the mathematical development.

An HL student models the number of fish in a lake with the differential equation $\frac{d P}{d t} = 0.3 P (1 - \frac{P}{2000})$ , with $P = 200$ when $t = 0$ ( $t$ in years), and writes "Using Euler's method with step size 0.5 gives $P (10) = 1352$ ." Suggest how the student could show sophistication and rigour in this part of the exploration.

Show the solution

Rigour: state and justify the model (why growth is proportional to $P$ when the population is small, and why the factor $1 - \frac{P}{2000}$ represents a carrying capacity of 2000); write the Euler recurrence $P_{n + 1} = P_{n} + h \times 0.3 P_{n} (1 - \frac{P_{n}}{2000})$ with $h = 0.5$ and $P_{0} = 200$ , show the first step by hand ( $P_{1} = 200 + 0.5 \times 0.3 \times 200 \times 0.9 = 227$ ), and state the accuracy used.

Sophistication: compare Euler's approximation with the exact logistic solution, $P (10) = \frac{2000}{1 + 9 e^{- 3}} \approx 1381$ , and explain why Euler's method underestimates here (the population grows faster and faster at first, so each step's gradient is taken from the slower start of the step, and the shortfall built up early is not recovered by $t = 10$ ); repeat with smaller step sizes and discuss how the error changes; interpret the equilibrium points $P = 0$ and $P = 2000$ and their stability in context; test the model against real data on the lake if available. Each of these looks at the problem from another perspective or links two areas of mathematics, which is what sophistication describes.

Moving up on E

Choose mathematics at the level of the course that the aim actually needs.
Explain every method before using it: what it does, why it suits these data, and what its conditions are.
Show one instance of each calculation fully (by hand where reasonable), then use technology for the rest and say what was entered.
Interpret every result in context, with units and a sensible accuracy.
Check results: compare a model with data not used to build it, verify a hand calculation with technology, test a conclusion for sensitivity to assumptions.
At HL, justify every claim the argument relies on, and look for a second method or a link to another area of mathematics.

How the criteria connect

The criteria overlap in what they reward, so good habits score on several at once. A clear aim helps A (the report meets its aim), D (results are linked back to it) and E (the mathematics is relevant to it). Your own data or your own decision about a model is evidence for C and, when explained, for D. Defining variables and labelling graphs helps B and makes the understanding behind E visible.

Habit	A	B	C	D	E
Precise aim, answered in the conclusion	✓			✓	✓
Every variable defined with units		✓			✓
Own data or own reasoned choice of model			✓	✓	✓
One calculation shown in full, then a table	✓	✓			✓
Interpretation after each result	✓		✓	✓	✓
Model tested on data not used to fit it			✓	✓	✓

Summary

The exploration is marked out of 20: A Presentation 4, B Mathematical communication 4, C Personal engagement 3, D Reflection 3, E Use of mathematics 6.
Marks are given best fit for each criterion separately; the top levels do not require perfection.
A rewards a coherent, well-organised and concise report with figures in the body, not in appendices.
B rewards correct notation (no calculator syntax outside software output), defined variables, suitable representations and logical setting out.
C rewards evidence of your own thinking that drives the exploration, not effort or enthusiasm.
D rewards reflection throughout: linking to the aim, limitations with their effect, comparing methods, implications and next steps.
E rewards relevant mathematics at the level of the course, correct, with understanding demonstrated, not just results quoted from technology.
HL criterion E is steeper: correct mathematics from level 3, sophistication or rigour for level 5, both plus precision for level 6.

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