Internal assessment

The internal assessment (IA) is a take-home essay on a piece of mathematics. It is worth 20% of the final grade, and is expected to be 12 to 20 pages double-spaced.

SL and HL candidates have slightly different criteria, but there are many overlaps. HL IAs are expected to have fewer mistakes and more sophistication and rigor.

The IA is reflection of your mathematical understanding. The key to doing well is to really know the math. The exploration should be focused, organized, purposeful, and insightful.

If you find this article helpful, be sure to share it with your classmates and teacher, so your IA mark is less likely to be moderated down.

Contents

• IA in context of your math grade
• Documentation
• Expectations
• Common misconceptions
• Topic selection
  • A great topic
  • modelling
  • solve a problem
  • topics to avoid
• A singular aim
• Criteria
  • presentation (4 marks)
  • mathematical communication (4 marks)
  • personal engagement (3 marks)
  • reflection (3 marks)
  • use of mathematics (6 marks)
• Writing process
• Revision checklist

IA in context of your math grade

Only the component marks (eg 13/20) are used for final grade calculation. Your final mark out of 100 is a weighted average of your component marks. If your goal is 65% for the course, ie a solid 6, then getting 13/20 on the IA is on par with that goal.

Each IA mark is equivalent to about 2 to 3.7 marks in the papers. See Exam tips for details. Balance between writing a good IA with studying for the mocks and finals.

Because IB marks the result and not effort, avoid topics that are too ambitious. You are best equipped to demonstrate knowledge of mathematics within or slightly beyond the syllabus, which is the level of mathematics top examiners suggest you to pursue.

This page is dedicated to getting at least 15/20 in an efficient manner.

Documentation

The grading criteria are available in the analysis and approaches subject guide. 

Top examiners also publish comments on student work every exam session in subject reports. Subject reports take the guessing game out of reading sample IAs. Examiners literally tell you what they want and do not want! Some snippets below are taken from subject reports. Your teacher has full access to them.

Expectations

A 15/20 IA uses math at par with the course, is nearly all correct, demonstrates a thorough understanding of the subject matter, has a bit of originality, is easy-to-read, and is introspective towards your research or exploration process.

Graders only give marks for what you understand. Correct math alone do not score well in Use of Mathematics. Rather, you should justify what makes you use certain math and specific steps to fulfill your aim and not other potentially relevant approaches. Your IA should go beyond substituting values into formulas; it should convince a skeptical fellow student that it is correct.

Common misconceptions

The following are correct statements that address common misconceptions about the IA.

1. More advanced math is not better. Subject reports recommend students to start within or very close to the syllabus. The IA is as strong as the most blatant error or miscommunication. Avoid appearing as a hallucinating LLM, merely summarizing without thinking. Investigating a topic thoroughly is usually better than scratching the surface of another.

   

2. You are not expected to invent new math. The concepts you explore are usually well established. It is important to ensure your math is correct rather than novel.

3. You do not need data. AA discusses very little statistics or its theory. Surveys are difficult to design, and data analysis is difficult. Statistical theory can use many advanced, subtle ideas that getting the correct answer is insufficient to demonstrate understanding.

4. You should use technology to your advantage. IB encourages use of technology in the IA. To demonstrate understanding, it is best to explain how the technology calculates the math, such as with an outline of a calculation. Repetitive and mundane calculations should be summarized rather than shown in detail.

Topic selection

Avoid starting from a field that is not mathematics then explore the math of that field. This often leads to less organized writing, paraphraphs wasted on not discussing the mathematics, and the math content not always at the syllabus level. It is risky to “trust” that a topic generated by an LLM or website is good; the topic should be evaluated in context of what you understand and what you can explain.

At both SL and HL, start from a math concept that you know the basics of, such as an extension or enrichment of syllabus content. Do not learn a completely new topic for the IA. If you do not have a good understanding of the math in your topic after two to three hours of research and practicing problems, the topic is too hard, too time-consuming, or too broad. For what you do learn, be sure to practice medium level problems in the topic. The more you understand, the more likely you can express your understanding to the grader.

Look for topics early and regularly. When you encounter an interesting topic read more about it and practice a few problems to test your understanding. You do not need a perfect topic; just something that plays to your strengths and can be refined. Keep in mind a handful of options. Do not spend too long on any option unless you are convinced that it will work.

You can watch a few math videos each month to get exposed to a wider range of mathematics. Check out “Summer of Mathematics Exposition” videos (using hashtags #SoME1, #SoME2, #SoME3, #SoME4) and Mathologer  on YouTube. Another way is to explore interesting mathematics you see in other subjects.

It is better to demonstrate knowledge of nuances of a relatively simple topic, rather than trivial knowledge of a complex topic.

A great topic

A great topic has the following characteristics:

• The topic is open-ended, so you can easily pivot to have depth while being concise. A narrow topic with only one way to discuss the math can be limiting hence risky. If you find aspects of the topic difficult, a flexible topic may allow you to circumvent or delay the challenging parts.

• You already understand most of the topic at an early stage. You should be able to write a third to half the IA immediately, without learning anything new. In particular, you should be able to explain in your own words the underlying theory, assumptions and simplifications.

  KISS: Keep it simple, stupid!

• It is easy to check your solution by using different methods, via simple experiments, simulations or other technology. This allows you to ensure that the math is correct.

• You can easily find textbooks, videos, articles, problems, and possibly data on the topic. The key concepts or theorems have established and universal names.

• The concepts allow for relatively easy examples, diagrams, graphs, or other ways to allow you demonstrate understanding. Topics that are too abstract or difficult to construct visuals can be harder to convince the grader that you understand. Correct math alone is not understanding.

In summary, think FACET: a great topic is Flexible and has math that is Accessible, Checkable, Established, and Tangible.

The following are directions in which you might find a great topic.

modelling

Modelling is a broad area of applied mathematics. You may be familiar with curve fitting and regression, but it could also involve random variables, differential equations, Markov chains, graphs and networks, and other representations. It is usually easy to verify your answers, and explain your thought processes for personal engagement and reflection marks. The biggest advantage is you have control over the direction of the IA. The biggest challenges are related to organization and clarity, and using the right tool for the right job.

Examiners remark that Lagrange interpolation (polynomial interpolation), while popular, does not allow the SL student to demonstrate understanding. Use a model that is appropriate for the problem, without using unnecessarily complex models or complicated formulas that you do not fully understand. Explain why the chosen models are suitable. Marks will be deducted in multiple criteria if you include clearly wrong approaches (eg use of a polynomial when there should be a horizontal asymptote.)

See the following Wikipedia links for additional ideas: List of curves , Gallery of curves , Mathematical model 

solve a problem

This is solving a problem with a known solution, and explaining it in detail. This allows you to make connections between math topics and learn new math along the way. Some care should be taken to choose a problem with an appropriate level of complexity. Especially at HL, this enables exploration of more sophisticated mathematics. The biggest disadvantage is that the direction is often inflexible. Examiners also want to see some personal insights rather than a reprint of the solution.

topics to avoid

IB named topics such as Fourier transform, partial differential equations, SIR infectious model, and golden ratios as ones that remain common but should be avoided. These mostly come down to student transcribing existing works. Rather, IB wants explorations that include your personal voice to allow demonstration of understanding.

Similarly, even using volume or surface area of revolution, to crank out answers from a formula by itself is insufficient for the highest marks.

A singular aim

When you are comfortable with the topic, pick a specific aim or goal. Periodically relate each part of the exploration back to the aim; you cannot write an anthology of disjoint theorems or properties. The aim gives the exploration direction and purpose.

The aim should be feasible given your mathematical strengths and familiarity with the topic. It can be somewhat malleable throughout your writing, but it needs to be clear and consistent in the final version. With a clear aim, you can identify different ways and steps you can use to fulfill the aim. You also need to prioritize the important aspects or steps and make those the focus of the IA. Very often, the thought processes behind the math are more important than the algebra or calculations themselves.

Break down the aim into different steps and formulate a plan to accomplish each step. Estimate how many pages to spend on each step. Ensure that you are including content that actually contribute marks.

Criteria

There are five criteria. IB recommends knowing the criteria before writing.

All criteria are same at SL and HL, except for criterion E.

The following supplements the expectations in the subject guide.

presentation (4 marks)

Examiners reward cohesive writing that flows nicely, and is clear and convincing to a high-performing peer in the same course. Each section refers back to the central question or aim. Extensive discussion that is not directly related to the mathematical aim may be penalized. Note that the aim is much more specific than the overall topic.

Assume the reader is familiar with all prior learning topics, and in most circumstances you can use them without commentary. Briefly explain material that is within the syllabus. Thoroughly discuss material that is beyond the syllabus. Easier said than done, but you need to be concise or elaborative at different times.

Nov 2024 AA SL TZ2 subject report:

Conciseness is key to avoiding unnecessary detail while still maintaining depth. It is essential to help students focus on the critical aspects of their investigation, ensuring that each part of their work contributes meaningfully to the overall exploration.

Examiners prefer if you start with subheadings in the initial draft, then replace most of them with topic and transition sentences in the submitted version. It is to encourage you to build stronger connections amongst subtopics, rather than relying on subheadings.

While you should not directly jump into the math in the first paragraph, examiners have also disliked introductions that went too long in describing the steps in the IA. No need to include what are equivalents of abstracts or planning sheets near the start of the IA.

The main body should not have repetitive calculations. Results from very similar calculations are summarized instead of presented in detail. Examiners want any long table to be in an appendix and only present the associated key ideas in the body text.

Only include graphs, tables, diagrams in the body text that enhance and clarify your message. Give enough context prior to their inclusion. If you are using LaTeX, it may chooses to reorder locations of images as to minimize white space; double check that the final order of text and images does not place extra burden on the reader to understand the IA. Do not use embedded content such as GIFs, external links, videos, QR codes; the examiner will not attempt to open them.

Reflections exclusively discussed at the end of the IA can negatively impact your presentation if the back and forth page flipping annoyed the grader enough.

mathematical communication (4 marks)

This criterion assesses the precision of mathematical terminology and notation.

All terminology beyond the scope of the syllabus are properly introduced and explained.

All variables are clearly defined and used consistently, including with careful, consistent usages of italics, subscripts and superscripts. As a rule of thumb, notations should closely resemble what you see in textbooks and past papers. For example $6.67\text{E}-11$, $a\text{\^{\,\,}}2$, $x\_0$, $l*w$, $5\diagup2$ are not accepted for scientific notation, exponents, subscripts, multiplication, and division respectively.

Use proper math terminology. Instead of “plug in”, use “substitute.” An equation involves an equal sign; an expression does not. Equations are solved; expressions are simplified; functions are evaluated.

Present the algebra with variables and equal signs, then when you put in rounded values at the end, use ”≈”. Calculations should be reported with the appropriate precision. Keep extra decimal places in intermediate calculations.

Tables, graphs, and/or diagrams are properly formatted and labeled. Ensure the label and the visual or table are together on the same page. If a table spans multiple pages (which is highly discouraged), ensure that the column headers are duplicated when it continues on the next page. Ensure that the text in all visuals and tables are also of 12-point font. Any intentional or accidental attempt to overwhelm or confuse the reader can only negatively impact your marks.

Calculations are easy to follow. Proofs are set up properly. Take care to not allow one equation in a single step to span multiple lines. The use of variables instead of numerical values help to keep formulas readable.

personal engagement (3 marks)

Personal engagement does not measure effort, but rather the extent to which you interacted with the math, or made the IA your own. Research and exploring unfamiliar mathematics almost always count towards personal engagement. Collecting data alone is personal engagement only if it expands the perspectives of the IA. Also, be sure to balance the 3 marks for personal engagement with the 9 combined marks of reflection and use of math that require a more thorough understanding of the mathematics.

Playing around with different equations in order to understand them, may be regarded as personal engagement. This can also be seen from examining details or perspectives typically left untouched in a textbook explanation. At the same time, you should not attempt methods that are clearly inappropriate. Reusing standard proofs and examples also do not show personal engagement, but nevertheless they may be essential to justifying your claims.

Making and testing hypotheses, and a general display of a curious mind, contribute to personal engagement. You want to demonstrate that you are taking ownership in the exploration.

reflection (3 marks)

Excellent reflections anticipate and answer a curious reader’s questions when they arise. While you should make note of assumptions or limitations of the formula or model in general, reflection is more about what you can say about your particular aim. Reflections also show awareness of the impacts of various decisions you have made, in terms of direction and results.

May 2024 AA HL TZ2 subject report:

Good reflection will include … questions like: “How reliable is this result? What if …? How can the process be changed to improve reliability?” The answer to such questions followed up by action will invariably result in meaningful and critical reflection … The addition of “limitations and extensions” … as an afterthought contributes minimally towards this criterion.

A quick example of reflection is an intuitive explanation or analogy after a lengthy calculation, or some sanity checks that confirm your expectations. Reflections are not purely descriptive or summarizing.

Whenever possible, incorporate reflection throughout the essay. In addition to the above comments, examiners really do not like reflection that comes after the conclusion.

use of mathematics (6 marks)

This measures the extent of mathematics you are able to convince the grader that you fully understand. You can only show this if you actually understand!

Correct math without reasoning or justification scores poorly in this criterion. You should examine the nuances of your topic which allow better chances to show understanding than the more apparent formulas and theorems. Lengthy, mundane calculations by hand do not earn more marks than if you simply explain the steps in your own words. An example is many students calculating Pearson’s correlation coefficient by hand and that’s just wasting time and space.

When you have identified a pattern, you should explain how it arise. Justify each step and any mathematical claims you make.

If possible, try to make generalizations when supported by your exploration. Evaluating the strength of your conclusion contribute to both reflection and use of math.

Using clear, precise, and effective language can contribute to both mathematical communication and use of math.

Most of the mathematics is at least at the level of the syllabus. IB does not require you to go beyond the syllabus for a 6/6. For HL IAs, SL content remains commensurate with the level of the HL course if the SL content is used in a way that is not expected of SL students.

The mathematics is focused. There is no unnecessarily convoluted or roundabout method.

Solid understanding needs to be demonstrated throughout the IA for 6/6 in SL and 5/6 in HL. Occasional, non-consequential errors are tolerated at 6/6 in SL and 5/6 in HL. 5/6 in HL requires mathematics beyond the reach of most SL students. The IA explores subtleties or implications in detail, showing a firm grasp of mathematics at large.

Writing process

The writing process is an iterative process. You should aim to as soon as possible get a draft on paper and revise afterwards. Since you probably do not have extensive experience in writing a math exploration, you want to give yourself time to make big changes.

You are allowed one opportunity to have your written work checked by your teacher prior to submission. You may ask additional verbal and written questions throughout the writing process as long as you are asking a specific question and not “what do I do?”

Revision checklist

The checklist is for obtaining at least a 15/20 on a suitably chosen topic. It goes from macroscopic to microscopic changes. The IA is at its core, a writing assignment.

This is for organization and writing. For what content to include, see explanation of the criteria above.

1. The overall direction is valid and correct, ie the context is correct and conditions for using the math are met.
2. The math has been thoroughly checked, and preferably verified using technology. Use enough precision to allow for 3 correct significant figures in your final results.
3. Each major step (including the first step) is justified and appropriate. Major decisions are explained and supported. Syllabus content are briefly explained. Where applicable, limitations and simplifications, especially ones in your particular problem or context, are noted.
4. Reflections are discussed where they arise, as opposed to be restricted to their own section. No reflection after the conclusion. Pieces of original math are clearly indicated as such.
5. The overall structure is organized. The ordering of content is the most sensible. The introduction and conclusion are effective and direct. Long tables that span multiple pages are put in an appendix, but summarized in the main body. Lengthy explanations are followed by a short summary.
6. Subheadings are used only when necessary. Topic and transitions sentences are effective. Only informative and necessary tables and visuals included, and they are introduced with sufficient context. The reader does not need to flip back and forth. The exploration is easy to follow and convincing to a top student in the same course. Explorations longer than 20 pages of body text are usually unfocused or unclear.
7. The language is precise, technical and unpretentious. The first person is used to show personal engagement and reflection, where appropriate.
8. Mathematical notations conform to standards. Long equations that split into multiple lines are kept to a minimum. Variables are properly declared and used consistently in capitalization and font style. Important equations are numbered.
9. Figures and tables are well organized and clearly labelled. Figures and images are greyscale. Charts and diagrams are clear and valuable. Axes are labelled. All words are 12-pt and double-spaces. The body text does not contain hyperlinks or any embedded content.
10. Sentences are clear. There are very few spelling or grammatical mistakes.
11. References are consistently cited in-text, and listed in the bibliography. Each source in the bibliography must have corresponding in-text citations. AI prompts that generated texts longer than several sentences (40 consecutive words) are included in footnotes.
12. Number each page. Ensure that title page, if present, contains only the title and page count, and not any personally identifiable information such as name, school, candidate session number, personal code etc.

Your teacher can always answer your specific questions, but can only comment on one draft. The difference is by asking questions, you are taking an active role and ownership in the direction of the IA.

Examiners note that many teachers do not include comments when uploading IAs to IB. Comments and annotations are the only ways for teachers to explain why you deserve the mark. Ensure that your teachers are advocating on your behalf to IB.

Double check your final due date. IB cannot extend such internal due dates. If you need an extension, talk to your teacher in advance.