Exploration Examples

Mathematical Topics: probability

Yahtzee is a dice game played with 5 dice. You roll the dice and put to one side any dice that you want to keep the score of. This is repeated 2 more times. The aim of this exploration is to find the probability of rolling a Yahtzee after three rolls, that is 5 dice with the same number on each dice. This was a good investigation because the student had to use different approaches to solve it. It is really well explained and easy to follow.

Mathematical Topics: geometry of a circle and trigonometry

This exploration aims to determine where a field hockey player who is receiving the ball during a penalty corner should stand to have the best chance of scoring. It considers the distance travelled by the ball and the angle that the player then has to shoot in order to score.

This exploration clearly came out of personal interest. Whilst the results were not conclusive for the position of the player, it allowed the student to compare a theoretical model in a real-life context. The mathematics in this exploration is not particularly demanding. It does not have to be, and the student was able to demonstrate good understanding of the mathematics.

Mathematical Topics: geometry, differentiation and optimisation

The aim of this exploration was to investigate different designs of guttering for houses and find the most optimal design that would allow the greatest water flow. Some constraints had to be placed on the design so that the different designs could be explored, and a best solution found. The differentiation used in this exploration was simple and well explained.

Mathematical Topics: trigonometric functions and modelling

The idea behind this exploration was a really good one. The length of the day varies periodically throughout the year, so a sinusoidal function provides a natural model. The model for the length of day would be related to the sine function. The student chose three places on earth: one was 1000km away from her home city, yet on the same line of longitude. The other was 1000km away from the same city and on the same line of latitude. The student was able to easily find data for this and find models for the length of the day in the 3 different places. The parameters of the models would be related to the positions of the three different places.

Mathematical Topics: modelling with the quadratic function

The aim of this exploration was to investigate the path of a world record blob jump (see the photo!). The student was able to show that the path was approximately a parabola (quadratic function). He went on to find equations of velocity and acceleration and consider Kinetic and Potential Energy. This exploration highlights how we can use technology to find models from photographic images. A page on this site will show you how you how you can do this with free computer software. Can you think of an exploration that might use this technique?

Mathematical Topics: modelling, equations of ellipses and area under graphs

The aim of this exploration was to determine whether a graphical model produced is true to the shape of the remembrance and real poppy flower, and to use these to find an approximation of the total area of the remembrance poppy leaves.

Modelling images using different equations is quite a popular type of exploration and finding the area formed is good for using Calculus in your internal assessment. I like to encourage students to find an actual reason for doing this. Could you find a real application for finding the area of the paper poppy?

Mathematical Topics: modelling with trigonometric functions

The aim of this exploration is to find models for the times of high tides and also the coefficient to show the relative swell of the tides. Given that tides are dependent on the moon which revolves around the earth, the obvious model was trigonometric. However, there is more than one high tide in a day. This, and the way that the data was presented, made the model for this more challenging than expected.

Mathematical Topics: equations of ellipses and integration to find areas under graphs

The aim of this exploration is to work out where to cut a Camembert (circular-shaped cheese) and a Caprice des Dieux (ellipse-shaped cheese) with parallel slices such that the resulting pieces of cheese are the same size. This was a delightful exploration, as it came from a genuine problem (the boxes that the cheese come in have a cutting guide so that the customer can cut 25g slices). The mathematics worked out to be quite a bit more challenging (despite good planning, this sometimes happens) and it became quite sophisticated.

Mathematical Topics: integration, solving differential equations

The aim of this exploration was to calculate the total horizontal distance travelled by a ping-pong ball after being dropped and just before reaching its second bounce. This was a very sophisticated look at an experiment that looked fairly simple to start with. This student used their knowledge of physics to set up experiments to determine the coefficient of restitution of the ball with the table. Calculations were made much more difficult by factoring in air resistance. This made the integration much more challenging (a differential equation requiring an integrating factor needed to be solved).

Mathematical Topics: probability and combinations

The aim of this exploration is to find the probability of winning a match of tennis if we know the probability of winning one point. For example, it was shown that if the probability of a player winning a point is 0.54, then it can be shown that the probability of winning the entire match is 0.9 showing that having only a slight advantage of wining one point has a huge impact on winning the whole match.

Mathematical Topics: modelling with circles and ellipses, integration to find surface area and volume of revolution

This student investigates volumes and surface areas of different domes from famous buildings. This exploration clearly comes from a personal interest in architecture. What makes this a useful thing to calculate is that the aim is to find the most heat efficient dome, which she speculates is the one with the smallest surface area to volume ratio. Mathematically this is an interesting exploration, since the student is required to carry out integration to find volumes and surface area of revolution. Notice that the student tries to explain each step, thereby demonstrating that she understands the mathematics.

Mathematical Topics: Calculus and Trigonometry

The aim of this exploration was to find the optimal angle and velocity to perform a free throw shot in order to maximize the chances of scoring. The model considered the angle at which the ball would enter the hoop. The higher the trajectory, the bigger the target. However, this needed to be balanced with the force in which a player could throw the ball. Hence a compromise between the two variables had to be found.

Mathematical Topics: probability

This exploration is about finding the expected score in a dice game involving 5 dice. From the student, the rules of the game… “The aim is to get the highest score... The player starts by throwing all of the dice. If they roll at least one 5 or 2, the player gets no points for this roll. This is because the 5s and 2s are ‘dead’ dice. Whenever there is at least one ‘dead’ dice in the current roll, the player obtains no score. The ‘dead’ dice are then removed, and the player rolls the left-over dice again. This process continues until all of the dice are ‘dead’. The player only gets points for a roll if there are no ‘dead’ dice – he gets the sum of the numbers on the dice. The points are added together to give their score.”

The student collected some experimental data by playing the game, then went on to analyse the game theoretically. A strength of the exploration was that the student had to try different approaches to solve it and thereby gained a better understanding of it because of that.