IB Mathematics

The International Baccalaureate Organisation (IBO) reviews the IB Diploma Programme (IBDP) subject content every seven years to ensure the curriculum fits the purpose. The new IBDP Maths curriculum was introduced in 2019, with the first assessment to be conducted in 2021.

The old version of the IB Maths curriculum offers four levels of Maths for students to choose from. This includes Maths at a Higher Level (HL), Maths at a standard level (SL), Math Studies and Further Maths.

Most of the IBDP schools offer IB Maths HL, Maths SL as well as Maths Studies. In the Maths HL syllabus, students have to pick one option out of the 4 options offered. The options include Statistics and Probability, Sets, Relations and Groups, Calculus and Discrete Mathematics.

In the new IBDP Maths curriculum (first assessment in 2021), instead of offering four levels of Maths, two different streams are offered. These include the Mathematics: Analysis and Approaches (Maths AA), and Mathematics: Applications and Interpretation (Maths AI), and each of them are offered at both HL and SL.

While there is a significant change of the syllabus designs in IB Maths, the new Maths AA SL is fairly similar to the previous Maths SL, while Maths AA HL is similar to the previous Maths HL. Maths AI SL is similar to the previous Math Studies. IB Maths AI HL, on the other hand, does not resemble any previous IB Maths courses. One thing to note is that in Maths AA HL, students do not have to choose the HL options now, as previously needed in the Maths HL. The old options are actually divided between AA HL and AI HL. While most of the topics covered by the old calculus options are now part of the AA HL syllabus, part of the old "discrete maths and the statistics and probability" options have made it into the new AI syllabus. (All topics in the old group and set theory options have been removed from the new syllabus).

In the new curriculum, students who are taking HL are required to complete a new Paper 3. The paper has 2 compulsory extended-response problem-solving questions, and it will be scored out of a possible 55 marks in a 1-hour assessment period. The question is usually easier in the beginning and gets more challenging, students should be cautious with the time management for this paper, especially as it constitutes 20% of the final grade. One thing to take note of is that the old Paper 3 only included questions from options topics. Therefore practicing the old syllabus Paper 3 past papers are not useful for students taking the new syllabus.

Last but not least, students in both Maths AA or AI SL or HL are required to complete the internal assessment which is an integral part of the mathematics HL. The internal assessment is worth 20% of the final grade.

There are a few considerations as to whether to choose Maths AA or AI, and HL or SL. Firstly, students should identify whether there’s any IB Maths prerequisites required by the university for the subject that they desire to read. Secondly, students should review whether they can handle the respective challenges offered at the Maths level that they wish to take.

Who should be taking Maths AA HL?

For students who would like to do Maths, Engineering, Physical Science, Computer Science related disciplines at university, they should do Maths AA HL. Most of these disciplines demand that students have done the Maths AA HL. Some might accept either Maths AA HL or Maths AI HL.

Take Cambridge University Mathematics as an example. Students are expected to have done the Maths AA HL to be considered for a place. (https://www.undergraduate.study.cam.ac.uk/courses/mathematics)

Take Imperial College London as another example. Their Electrical and Electronic Engineering course requires students to have done HL Maths in the IB, and both Maths AA or AI HL students can be considered. (http://www.imperial.ac.uk/electrical-engineering/study/undergraduate/entry-requirements/).

For students who are aiming to do Economics at top schools, IB Maths at HL is highly preferred or required. Take LSE as an example, Maths HL is a prerequisite for their competitive Economics course. (https://www.lse.ac.uk/study-at-lse/Undergraduate/Degree-programmes-2021/BSc-Economics).

Take another example - Cambridge University requires students applying for the Economics program to have taken the Maths AA HL. (https://www.undergraduate.study.cam.ac.uk/courses/economics).

Who should be taking Maths AA SL

For students who are not interested in the above disciplines mentioned, and don’t mind investing some time in doing Maths to challenge oneself.

Who should be taking Maths AI HL

For students that are very good at using a graphing calculator, and have to particular reason to do HL Maths e.g. college requirements, not having another subject to put as HL etc.

Who should be taking Maths AI SL

For students who want to minimise time spent on Maths, and are not interested in doing a quantitative university course, taking AI SL is a better choice for university admission, as students can spare time in other subjects to maximize the overall IB grade to increase their admissions chances.

Maths AA SL is a subset of the Maths AA HL syllabus. Students doing the Maths AA SL will be covering topics like Functions, Numbers and Algebra, Geometry and Trigonometry, Calculus - Differentiation, Calculus- Integration, Statistics and Probability. For Maths AA HL students, they will be doing all that is covered in the Maths AA SL syllabus, as well as a few advanced topics such as differential equations and complex numbers.

Maths AA HL students will be sitting Paper 3 in the IB exam. Investigative questions in paper 3 often contain many parts. Each part of the question is usually related to/dependent on the solutions to the previous parts. In addition, in the new paper 3, students are often asked to make generalizations and prove their claims (similar to IGCSE mathematics paper 6, but far more advanced).

Students often ask for research question ideas on their EE (Extended Essay). Realistically, a good EE research question/ topic should be appropriately designed so that you would be able to 1) gather meaningful primary and/or secondary data if needed and 2) be able to scope things like the interpretation of the data, analysis and evaluation of the data / events within the 4,000-word limit.

Here are some sample Maths AA or Maths AI EE research questions:

• Texts Classification using Naive Bayesian Filters
• Comparison of Fast Multiplication Algorithms
• Artificial Neural Networks and Function Approximations
• Economic Applications of Lagrange Multipliers
• Generating Fractals using Matrices
• How to make a world map?

Here are some sample IB Mathematics AA and AI sample IA topics (Recommend to be 12-20 page):

• Analyzing the Role of Vaccination using SVIR Models
• Estimating the Gini Coefficient
• Benford's Law and Fraud detection
• Solving equations using Origami
• Calculating the value of Pi
• The Mathematics of Plinko Boards
• Do mobile games lie about the probability of getting rare items or characters?
• Is the flight path of airplanes always the shortest between the destinations?

Choosing a research question

With IB Maths IAs, everyone knows that the most difficult part is choosing a question. Your teachers usually assign questions in class, and so it might seem strange at first to come up with your own. To figure out what you would like to research, let’s first look at what counts as a good research question.

A good research question:

• Is clear in its aim and direction
• Is specific in what it wants to find out
• Gives you a problem that you can answer within your Maths IA
• Suggests possible methodology and approaches

Here’s a bad example. If I want to do a Maths IA on football and write “The Mathematics behind Football” as my research question, then my reader has no idea of what I want to research at first glance. This research topic is too general and it isn’t even technically a question!

Here’s a better approach. I write “What position, direction, and angle should a football be kicked in order to maximise probability of it hitting the net?” This is a lot more wordy, but it is a much clearer and therefore better research question. From the research question alone, your reader can already get a sense of your methodology and what kind of approaches you’ll take in your experiments. How far should you stand from the net? What direction should you shoot? Should you kick from the right, the left, or from front and centre? What angle and trajectory will give you the highest probability of getting the ball in the net? Your reader can easily foresee that you will conduct these kinds of experiments to answer your overall question.

What is personal engagement, and how do I fulfil it?

Another requirement on the IB syllabus is the fulfilment of personal engagement in your Maths IA. Readers will not be interested in an IA if you, too, are not interested in it and just consider it as another piece of homework. Ideally, you will use the mathematics and theories that you studied in class and apply it to study the people around you, conducting experiments linked to real life. To fulfil personal engagement, we recommend that students consider their hobbies, extra-curricular activities, or sports that they enjoy. Are there any potential maths experiments that you can conduct that are relevant to your interests?

For example, if you really like cameras and taking pictures in your spare time, you might research the following in your Maths IA: “What angle and height of the camera will result in the clearest/most beautiful pictures?” In this case, you’d also consider: What is my definition of “most clear” and “most beautiful”? Does it mean that the main subject is perfectly in the middle? How high should my tripod legs be? How shall I test the angles of the camera?

All these aspects are interesting for your reader, and excellent to use as part of your topic. This topic is something you’re passionate about, it is probably not well-researched, and it truly belongs to you and your life experiences. You benefit from the experiment’s outcome because you will improve your photo-taking skills through maths! This proves that you are personally invested in the results and fulfils personal engagement.

Where to start

If you’re still feeling conflicted and directionless about your Maths IA topic, a popular format to follow is the Curve-fitting IA. In a curve-fitting IA, you will collect data and use it to coordinate points on a graph, making a scatter plot. Then, you can think back on the functions and curves you learnt in class, e.g. quadratics, exponentials, logarithms, straight lines, etc. Which curve fits your coordinated points most closely? The curve of best fit will become your model, the equation that best describes the relationship between your x and y.

Then, you can explore why the relationship between x and y is this way. Do some evaluations and write down your theories to show that you have thought deeply about your results. Lastly, remember that your mathematics model is probably an oversimplification of a complex reality. You are limited by what you have learnt so far in high school, so your model will contain some massive assumptions and limitations. Examine these issues too!

One benefit of a curve-fitting IA is that the format and structure is very straightforward. If you follow this format, you should have no problem knowing what to focus on for each section. Another great benefit is that you can use a curve-fitting IA to fit virtually every topic that allows for data collection. If it’s a topic where you can see correlations and potential relationships between different factors, you can use it as your IA.

For example, if your passion is movies, then perhaps you will want to investigate the relationship between how much money is invested in a movie, and how much money it actually makes at the box office. Is there a relationship between the two? Is it an exponential relationship, a straight line, or quadratics, etc? Or perhaps, against your expectations, you find that there is no correlation between the amount of money invested and the amount made through ticket sales at all!

Overall, a curve-fitting IA is a great place to start because of its clarity, straightforwardness, and potential variety. Even if many different students use this format, the topic and experiments they come out with will look very different. If you are at a loss for what to do, look into a curve-fitting IA first.

When you begin your first Maths IA, you might be confident with solving the question and equations, but feel unsure about how to present it well. Your reader will have certain expectations for the language and format that you will use. Here are some tips and tricks to remember as you go forth and investigate!

1. Data-collection is not an absolute requirement, but it is a good idea.

A frequently asked question from students is: If I do a Maths IA, do I absolutely have to collect data by doing experiments and surveys? The answer is: not really. Whether you must collect data or not depends on the topic and question you have set for yourself. If you really don’t want to bother with experiments and surveys, you can design your topic to exclude them.

For example, let’s say you decided to conduct a Maths IA on shooting basketballs. You aim to find out how to shoot more accurately, the ideal flight paths, etc. - this kind of IA does not require the collection of data, but would need some knowledge and use of physics theorems. Mind you, if you want to focus on the popular curve-fitting IA as mentioned in this article, then you will have to collect data.

2. Primary data is better than secondary data

In Maths IAs, the primary data that you collect yourself is preferable, as opposed to the secondary data that you found from someone else’s experiment. Primary data is when you personally designed the survey, wrote the questions, and perhaps interviewed your classmates or went out on the street to question the public. This is much better than googling previous experiments and finding your data points online, which is secondary data. Primary data is preferred because it helps to fulfil the personal engagement criteria on the syllabus. If you designed your own survey and conducted it yourself, your own involvement in the IA is naturally much greater.

Understandably, primary data collection is impossible for some topics. For example, if your IA examines the possible correlations between the amount of money invested in movies vs. their ticket sales, you cannot possibly design your own surveys. Instead, you might go online and download the budget vs. box office figures for each movie and conduct research off this secondary data. This is okay too, though not preferred!

3. Aim for more than 30 data points

This one is pretty straightforward. If you can choose how much data to collect, a good rule of thumb is 30 or more data points. If you exceed this number, you can greatly reduce sampling variability. Your results will be less affected and will be more reliable! This is not a strict requirement but try your best to aim for over 30.

4. Don’t overcomplicate the maths

Something that teachers might not tell you is exactly how difficult mathematics needs to be. Some students might understandably assume that the harder the concepts and calculations, the better and higher the grade, right? This is not true.

If we look back at the IB Mathematics syllabus, it’s pretty clear that examiners just want you to take what you’ve learned in class and apply it to real life around you. (There is actually a whole field of mathematics called Applied Mathematics.) Your IA should show your creative thinking and effective utilisation of the maths you already know.

If you choose a super difficult university-level problem and resolve it, this does not guarantee high grades. For example, let’s say you decided to examine and solve Maxwell’s Equation of electromagnetism in your IA. This is not a good use of mathematics because it is too difficult, it focuses primarily on the theoretical, and the multivariable calculus it requires is not even taught to Maths HL students.

Some Maths IA topics absolutely require out-of-syllabus maths - this is dependent on the IA’s aim. If you are determined to use out-of-syllabus maths, we recommend that you only use a little, or limit it as much as you can. Moreover, these new, unfamiliar concepts must be constructed from the beginning and explained fully in your IA. If you have not learnt it in class, you must explain precisely and in detail how you went from the very beginning to getting your final results. Simply put, a friend in your class who does not know these concepts should be able to read your IA and follow along with the explanations perfectly.

5. Presentation and communication are crucial

An often overlooked aspect of the Maths IA is the presentation and communication. However, it is very important! Not only do you have to understand your assignment, but your readers do too.

A. Use clearly defined and proper mathematical notation

Clearly define the terms that you will use. If you say x and y, specify what they represent in your study, e.g. y represents annual income and x represents the number of years receiving education. If these are not defined well, your examiner may deduct a lot of marks!

Additionally, try to use proper maths notations. Since many students use computers nowadays to take notes during maths lessons, you might get into the habit of using a slash / to indicate division, an arrow up ^ to indicate rising power, and INT to show integration. This is not standard maths notations, so strive to write your IA using the proper notation as if you had written it out by hand. For example, x to the power of 3 should really be written x³, and the integration of a function f(x) should be written as ∫ f(x) dx but not int f(x) dx.

To format this correctly on a computer, use Microsoft Word’s Equation Editor. On the toolbar on the top right you will find commonly used maths symbols and formats that will be useful to your IA. E.g. calculus, differentiation, integration, summation signs, etc. The Equation Editor is not hard to use, so just use trial and error to explore around.

B. Have a clear logic flow

Do not overlook the steps showing your process, even if they seem really easy and obvious. When you are completing a maths question for homework or class, you might be tempted to skip some steps and your teacher may give you the mark anyway. But in a proper Maths IA, you must show the ideas and logic behind each step and make it very clear for your reader to understand. Include the details!

C. Prove your understanding

Lastly, it is a good idea to show that you know what you’re doing. Before each major section of calculations, try to explain why you must complete these particular operations or use that particular maths tool. This is particularly important if you use any out-of-syllabus maths, so explain it well. It will give your examiner a chance to see that you truly understand the maths ideas within your investigation. It also proves that you are not just blundering through each step and coming out with an answer at the end without real understanding and intention.

Follow these Five Top Tips and you are already set up for a well-presented, professional looking Maths IA!