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.
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
- Overall directions
- When and where to find ideas
- Topic filtering
- Criteria
- Revision checklist
IA in context of your math grade
Component 1 to 7 are largely meaningless. The “boundaries” for the IA are only used to determine the overall boundary, not to determine you 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.
This means if your goal is 65% for the course, for a solid 6, then getting 13 out of 20 on the IA is on par with that goal.
Because IB marks the result and not effort, you should avoid topics that are too ambitious. You are best equipped to demonstrate knowledge of mathematics within or slightly beyond the syllabus, and this is the level of mathematics top examiners suggest you to pursue.
In SL, 1 mark in the IA is worth the same as 2 marks in Papers 1 and 2.
In HL, 1 mark in the IA is roughly 3.67 marks in Papers 1 and 2, or 2.75 marks in Paper 3.
Be pragmatic regarding the amount of time spent on the IA. 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, first published in 2019, last revised in November 2020.
Top examiners also publish comments on student work every exam session in subject reports. If your teachers refuse to tell you what’s in them, explain that not knowing expectations can place your school at a disadvantage.
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 cannot earn full marks in Use of Mathematics. Rather, you should fully explain the principles behind a formula, the conditions allowing it to be used, and its limitations. 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.
- 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. More math also means more chances to reveal that you do not fully grasp the basics or subtleties. Stay within the recommended length of 12 to 20 pages. Investigating a topic thoroughly is usually better than scratching the surface of another.
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. Personal engagement measures the extent to which your IA is different from a generic textbook explanation and from other IAs.
You do not have to collect data. AA discusses very little statistics or its theory. Surveys are difficult to design, and data analysis is difficult. Unless you are taking Biology HL, AP Statistics, or another stats-heavy course, do not do a stats IA. Statistics can more easily be misused than many other math topics.
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 a sample calculation. The IA should be well-organized with high readability. Calculations that are repetitive or below the syllabus level should be summarized rather than shown in detail.
Overall directions
It is highly recommended to look for IA topics early and over a long time (eg several months). The success of a topic depends on how much you know thoroughly. Stronger students can make a lot of topics work; weaker students can score poorly in spite of good topics.
There are many approaches. At both SL and HL, the candidate should start from a topic that they know the basics of, such as an extension of a topic in the syllabus. It is risky to learn a completely new topic for the IA. If you do not have a good understanding of the bulk of the math in your topic after two hours of research and practicing problems, the topic is too hard, too time-consuming, or too broad.
If you do decide to learn new math for the IA, 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.
The topic should allow for decision-making and evaluations by you. Look to string together several pieces of math, instead of one with a one-step setup followed by algebra or simplifications.
model a shape or object
This involves finding a mathematical model for some 2-D or 3-D shape. The curves used should be non-trivial (ie not polynomials or simple transformations in the syllabus) and allow you to discuss deeper into functions. 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, as you can easily extend the topic to include area, perimeter, volume, and surface area, which will require calculus. The biggest challenges are related to organization and clarity.
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. Math competitions and full-length YouTube videos provide an abundance of such problems. Especially at HL, this enables exploration of more sophisticated mathematics. The biggest disadvantage is that the direction is often inflexible.
risky approaches
The following pathways are harder to write, and thus can also score poorly.
“Mathematics of [some everyday topic]” tend to be broad, too basic, or too segmented. These IAs require a delicate balance between explaining the subject and the math. It is better to first identify the mathematics, prior to identifying a context or a field of applications. Furthermore, avoid spending too many paragraphs explaining something that is not mathematical.
Topics should be quantitative, not qualitative. History of mathematics would be one to avoid. That said, quantitative does not mean data. In fact, stats IAs rarely score excellently due to the lack of discussion of stats theory in the course and in the IA.
IAs with too many goals lack focus. The IA should be a single coherent work, rather than an anthology of discussions.
In spite of the name, the exploration should be focused, organized, and purposeful. Explore well-documented math at or above the syllabus with a central aim in mind. It cannot be stressed enough to strive for correct mathematics rather than original mathematics.
When and where to find ideas
As for all internal assessments (and extended essay), you should start looking for a topic early, especially when you have the time in IB Year 1. You should first pinpoint the mathematics involved in the math IA before looking at potential applications.
When you find a particular topic interesting, read more about it on your own time or find more complex problems or concepts on the topic. When this is done at a long enough timescale, you may often stumble upon a topic suitable for a 12 to 20 page discussion.
You can few math videos each month to get exposed to a wider range of mathematics. Check out “Summer of Mathematics Exposition” videos on YouTube. Another way is to explore interesting mathematics or mathematical problems you see in other subjects.
It is tempting to simply go to you generic search engine or large language model and search for “IB Math IA ideas”. While you can certainly get interesting and sometimes useful ideas, there are various shortcomings. The most significant of which is you are unable to evaluate the extent of mathematics involved. It is risky to “trust” that a topic is good, without knowing the mathematics. The strength of a topic depends very much on the writer, and what they know.
Topic filtering
A strong, safe topic has the following characteristics:
- The topic is open-ended, so you can easily adjust course to stay within the recommended length of 12 to 20 pages. A narrow topic in which there is only one way to discuss the math can be limiting hence risky. If you find aspects of the topic confusing, 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 how certain formulas work, why they were used, and their associated assumptions or simplifications. Fourier transforms, for example, are not recommended as it is considered much beyond the syllabus.
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.
In summary, think FACET: an ideal topic is Flexible and has math that is Accessible, Checkable, Established, and Tangible.
Is mY TOpiC GoOd? Well now you know what “good” means.
When selecting a topic, identify what you know and don’t know. This allows you to plan for which parts you need to learn and research. When researching, keep track of all your sources.
Criteria
There are five criteria.
| criterion | max marks |
|---|---|
| A: Presentation | 4 |
| B: Mathematical communication | 4 |
| C: Personal engagement | 3 |
| D: Reflection | 3 |
| E: Use of mathematics | 6 |
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 writing that flows nicely, and is understandable and convincing to a high-performing peer in the same course. Each section refers back to the central question or aim. The writing is cohesive and organized.
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.
The main body should not have repetitive calculations. Results from very similar calculations are summarized instead of presented in detail.
Include graphs, tables, diagrams in the body text that enhance and clarify your message.
mathematical communication (4 marks)
This criterion assesses the precision in mathematical terminology and notation.
All terminology beyond the scope of the syllabus are properly introduced. All variables are clearly defined and used consistently.
As a rule of thumb, notations should closely resemble what you see in textbooks and past papers. For example , , , , are not accepted for scientific notation, exponents, subscripts, multiplication, and division respectively.
An equation involves an equal sign; an expression does not. Equations are solved; expressions are simplified; functions are evaluated.
Instead of “plug in”, use “substitute.”
Tables, graphs, and/or diagrams are properly formatted.
Calculations are easy to follow. Proofs are set up properly.
Calculations should be reported with the appropriate precision. Keep extra decimal places in intermediate calculations
personal engagement (3 marks)
Personal engagement does not measure effort, but rather the extent to which candidates interacted with the math, or made the IA their own. Research and exploring unfamiliar mathematics almost always count towards personal engagement. Yet, 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.
Lengthy, mundane calculations by hand, does not contribute to personal engagement. Reusing standard proofs and examples are not personal engagement.
In contrast, 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. A moderate amount of creativity is rewarded. Completely original math is risky as it is harder to guarantee its correctness.
Making and testing hypotheses, and a general display of a curious mind, contribute to personal engagement. You want to demonstrate that you are taking an active role 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 investigation or problem. In other words, discuss the implications and both strengths and weaknesses of your particular investigation or results.
For example, a quick example of reflection is an intuitive explanation after a lengthy calculation, or some sanity checks that confirm your expectations.
Another example is solving the problem in multiple ways, and comparing when should each strategy be used.
Reflections at the end to holistically comment on the challenges you overcame or knowledge or appreciation for math gained, should only be a fraction of all reflections.
use of mathematics (6 marks)
This measures the extent of mathematics you are able to convince the grader that you fully understand.
It is better to demonstrate knowledge of nuances of a relatively simple topic, rather than trivial knowledge of a complex topic. The length of the IA does not allow in-depth discussions of mathematics way beyond the syllabus.
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.
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.
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.
- The overall direction is valid and correct, ie the context is correct and conditions for using the math are met.
- 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.
- Each major step is justified and appropriate. Major decisions are explained and supported. Where applicable, limitations and simplifications, especially ones in your particular problem or context, are noted.
- Reflections are discussed where they arise, as opposed to be restricted to their own section. Pieces of original math are clearly indicated as such.
- 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.
- Subheadings are used only when necessary. Topic and transitions sentences are effective.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 often unfocused or unclear.
- The language is precise, technical and unpretentious. The first person is used to show personal engagement and reflection, where appropriate.
- 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.
- 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 text is of size 10 pt or greater, with 12-pt double-spaced for body text. The body text does not contain hyperlinks or any embedded content.
- Sentences are clear. There are very few spelling or grammatical mistakes.
- References are consistently cited in-text, and listed in the bibliography. AI prompts that generated texts longer than several sentences (40 consecutive words) are included in footnotes.
Your teacher can always answer your specific questions, but can only comment on one draft.
Double check when your final due date is. IB cannot extend such internal due dates. If you need an extension, talk to your teacher in advance.