Being able to explore some mathematics of your own choice and take ownership of your learning should be an exciting opportunity. However, students often find that choosing a topic for their exploration rather overwhelming. The aim of this page is to give you some inspiration and provide you with ideas that will get you started thinking about what will work for you. When I talk with students about formulating an idea and writing a plan, the first consideration I have is the level of mathematics that the exploration will involve
- Not too easy - ideally it should be mathematics that you have already covered in the course (not below)
- Not too hard - you really have to demonstrate that you understand it
Of course strong students who want to stretch themselves and explore a completely new area of mathematics (to them) may enjoy the challenge of studying unfamiliar mathematics from outside the syllabus, but students score better on their internal assessment when the level of mathematics is not too hard and equivalent to work that they are doing in class. It is worth noting that you do not get better marks if the mathematics is 'harder'!
On this page, you will find tried and trusted exploration ideas. You could use them as they stand or adapt them to something that interests you better. If you have not already looked, then you should check out all the IA examples on the page Exploration Examples first.
There are lots of successful examples of this using different objects. There is an IB exemplar piece that finds the volume of a pawn chess piece (other students have found volumes of bottles, glasses, vases, pears, etc).Ideally you should find a reason why this is useful (rather than just interesting from a mathematical point of view). A key part of modelling functions to an image is to get the scale correct first. In the case below, the student carefully measured the length of the bottle in the first place and then adjust the axes on the graphing software. You can learn how to insert an image in Geogebra here and DESMOS here.
Inspired by this tutorial about finding the equation for plotting the batman logo, a student of mine produced an graph showing the Nasa logo. The key was demonstrating that he understood all the transformations (there are some more advanced transformations reflecting in diagonal lines and rotations).
Take photographs of arches and try to model the shape using different functions. There may not be enough here to get the top marks for mathematical content for HL unless a more sophisticated approach is attempted (e.g. using differential equations).
A student took a video of a horse jumping over an obstacle and used Logger Pro to analyse the frames of the video in order to find the path of the horse. There is a lot of potential to take videos of other things and use the frames of the video to create a graph of points. A basketball shot is an obvious example, or any bouncing ball.
Find mathematical models for pure sound (sine functions) then analyse what happens mathematically for consonant chords (ones that sound good together e.g. C + E) and dissonant chords (ones that sound jarring and unpleasant).
You can use differentiation to find the (local) minimum or maximum value of a function. You can use this to solve an optimisation problem. It will probably necessary to fix a (some) parameter(s) to make a problem that is possible to be solved. Be careful that this is an exploration and not just a typical textbook problem.
Dobble is a simple card game where you will always find a matching pair of objects on any pair of cards. The student explored how this was possible. She had to break the game down to a smaller number of cards to start with then build up to a full pack. She was required to find the different number of combinations of two objects.
A student used a temperature probe to find the internal temperature of a slice of brownie as it cooled. The temperature followed an exponential model, as predicted by Newton's Law of Cooling. Brownies of different thickness were compared to see the effect this had on the parameters of the model. Unfortunately the student never brought the brownies to school for the teacher to taste :(
This student looked at calories burnt during a running race based on heart rate which was modelled using data collected from a sports watch. The question was whether this could be replaced by the consumption of energy gels consumed during the race. A model was found for the absorption of glucose and the amount of glycogen would be needed to be stored in the body prior to the race.
The student produced an investigation into the optimal shape of an ice cream cone (think short and wide versus tall and thin) in order to get the most ice cream. The volume of the cone plus the the semi-ellipsoid scoop had to be found and optimisation (the maximum volume) was used to solve the problem.
This student wanted to work out what angle a player's arms would have to be at in order to make their defensive shot go vertically upwards (and therefore give their teammates the most time to play the next shot). A function was found based on receiving the ball in different positions.
This student combined his knowledge of Physics with Calculus to work out how fast a cyclist would travel starting from rest and freewheeling (not pedalling) down a slope predicting how long it would take for the cyclist to reach terminal velocity
The student modelled the trajectory of a bullet over different distances to be able to calculate the effect of gravity on bullet drop and be able to re-adjust the gun's sights, a process known as zeroing. By including tyhe effects of air resistance on the bullet, the investigation became quite complicated. Calculus was used to generate the formulae for this exploration.
The student used Newton's Law of Cooling, solving Differential Equations using Integration to be able to find a model for the temperature of coffee in a cup. Different cups that gave different surface areas were compared in order to try to predict the values of the parameters for different cups.
The aim of this exploration was to find the angle required to make a bank shot (not using the backboard) knowing the initial velocity of the ball. The student videoed himself taking a shot and used Logger Pro software to calculate the velocity of his throw. Integration was required to solve the problem.
The aim of this exploration was to simulate the path taken by a bouncing ping pong ball, and then find the distance it has reached after each bounce. The student use Integration to find the equations of motion and factored in the effect of air resistance.
Finding the longest possible length of ladder that could be moved around the corner of a staircase. The student had to use Geometry and Trigonometric Identities to solve this problem.
The investigation looked at the area covered by the ellipse of sunlight as it entered a room at various angles. As the diagram shows, the room chosen was a dome-shaped room which for simplicity was situated at sea level on the equator. The student was required to use Geometry and Trigonometry in order to solve this problem.
The student designed an experiment to memorize two lists of words, the first without any exercise, the second having done a few minutes of light exercise. The student collected date from 50 students. She did some simple handling data techniques, as well as two hypothesis tests. She was able to show that memory and gender were independent and there was some evidence to suggest that memory was improved by light exercise. This type of exploration might be better suited to a student of Mathematics Applications, since the hypothesis tests included in this come from the AI course and not the AA course. The danger of including mathematics from outside of your course is that it can be difficult for you to demonstrate that you fully understand it.
Another Handling Data exploration, this student set up an experiment to find the flexibility of dancers using a simple hamstring flexibility test. The student used some simple statistical measures to support her hypothesis that height and flexibility are inversely proportional
The student looked at different methods of finding the sums of consecutive numbers finding, amongst other things, that any number that is a power of 2 (2, 4, 8, 16, ...) could never be found by adding together consecutive integers. This exploration showed that a carefully chosen investigation can produce an excellent internal assessment.
The student looked at a section of a racing track and compared the distance travelled using different models. The distance for the final model was found using integration and the arc length formula \(s=\int^b _a \sqrt{1+(\frac{dy}{dx})^2}\ dx\)
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