# 50+ IB Maths AI IA Ideas

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## IB Math AI HL IA Ideas

**1.) Employing Optimization To Minimize Amount Of Packaging Material**

**Mathematics:**

Using mathematical concepts like** differentiation and integration** to derive generalized conditions that minimize the surface area and hence the packaging of these differently shaped goods (example: conical, cuboid and irregular shaped goods).

**Procedure:**

The actual surface area will be found using the existing dimensions. Then, the volume of a standard cylindrical shape is differentiated to find out if a minimized surface area exists. Once the minimum dimension values are found, the minimum surface area will be calculated. This is repeated for conical, cuboid and irregular shaped goods.

**Analysis**:

Total wastage is calculated by subtracting the minimized surface area from the actual surface area.

**2.) Finding the optimal route using algorithms?**

**Mathematics:**

Using the **Brute Force algorithm, the Nearest Neighbor and the Deleted Vertex** to find the optimal route.

**Procedure:**

The Brute Force algorithm takes into account all of the Hamiltonian cycles. A formula is used to calculate the number of different routes you can take (Hamiltonian cycles) by inputting the number of points on the route. After calculating the distances of all possible routes, the route with the shortest distance can be found. The next method is the Nearest Neighbor. The way that that works is, you would start at a point and go to the nearest point with the shortest distance. This will be continued until you reach back to the beginning point. This will give the upper bound or the minimum value the distance should be. As for the deleted vertex, the first step would be to delete a vertex and all the edges connecting to that point. This will leave you with the “minimum spanning trees” where you have to add the values of these distances. Then, you find two edges of the discarded edges from before and add the values to the totaled distance from step 1. Repeat this procedure for each deleted vertex and see which one yields you the highest value.

**Analysis:**

The methodologies used are compared to conclude which the most efficient algorithm is to calculate the optimal route from one point to the other.

**3.) Modeling the velocity of a skydiver and calculate the height at which he opens the parachute**

**Mathematics:**

Using **Newton’s laws to model the velocity** of objects in motion using techniques like integration.

**Procedure:**

Using basic integration, a differential equation is found for the final velocity of free-fall. Then, an equation for the terminal velocity of the man while considering drag force is derived. Later, the height at which the diver pulls out the parachute is calculated. Integration is used to find the vertical displacement.

**Analysis:**

The outcome of this investigation would be to determine a height value for when the diver pulls out the parachute. Limitations and strengths will further need to be discussed.

**4.) Mathematical modeling behind a perfect 3-point shot**

**Mathematics:**

Calculate the** appropriate parabola, degree angle, and force for a variety of height**, accordingly, determine the perfect 3-point throw. Mathematical concepts such as trigonometry, quadratics, linear regression and moderate physics calculations are used.

**Procedure:**

Specific heights of people were collected and is considered to be the independent variable. Other constant variables such as arc lengths and angle degrees were plugged into the equations according to the height. The data will then be modeled in a graphing software to visualize the 3 point shots. Lastly, the parabola of the perfect shot is calculated for each condition.

**Analysis:**

These factors can be explored in relation to their impacts on the probability of scoring a 3-point shot by manipulating the height variable to investigate 3 different scenarios. The investigation will explore the impact of height differences in players on the modeled curve of a shot and its ability to enter the net perfectly.

**5.) Modeling the shape of a coconut to explore the best method to calculating the volume of irregular objects**

**Mathematics:**

Different methods of calculating the volumes of a coconut are compared to find the most appropriate method through the application of calculus, basic mathematical formulae and graphical software’s.

**Procedure:**

Using the dimensions of a coconut, the first method is using the ellipsoid formula. The values are substituted and the volume of the coconut is found. The second method is comparing the accuracy of an ellipse formula and manually marking the coordinates to form an ellipse using a graphing software. The more viable method is used to calculate the volume. The last method is to use a graphing software to mark coordinates that make two parabolas facing each other (up and down). These two parabolas need to be marked in a way where they take the shape of the coconut.

**Analysis:**

The results of all three methods are compared in terms of percentage error and the method that provided the most accurate answer is concluded.

**6.) Traffic Control using graph theory**

**Mathematics:**

Graph Theory has numerous applications in real life. It can be applied to solving systems of traffic lights at crossroads.

**Procedure:**

The controller to be developed has to minimize waiting time of the public transportation while maintaining the traffic flow. Particular traffic flows can be called compatible if two flows will not result in an accident caused by vehicles moving on multiple flows simultaneously. By using compatible graph, optimal waiting time at the crossroads can be determined.

**Analysis**:

Graph theory was developed because of the Konigsberg bridge problem. This branch of maths has real life importance and using it for traffic control gives us a better understanding and clarity about the concept.

**7.) Flight Scheduling using Dijstra’s Algorithm**

**Mathematics:**

If you’re interested in Aviation, this topic is for you. The Dijkstra’s algorithm is an algorithm that can find a short path on a graph effectively.

**Procedure:**

We are required to represent locations on world map therefore it can be modelled using graph theory. The graph can represent the map, the vertices represent the locations, and the edges are the connection between the locations. You may use software to finally model the situation.

**Analysis:**

This exploration will help understand that how complex it is to schedule flights and how easily it can be done using a simple mathematical tool, graph theory.

**8.) Markov chains and Monopoly**

**Mathematics:**

Markov chains reduce complex rules and systems into a simple long term probability, which is useful for making long term predictions.

**Procedure:**

For simplicity redefine monopoly rules and make the game smaller with less variables. Create a transition matrix with respect to each place on the board and find the long term probabilities. You can look at variations of this game and observe what effect it has on the results.

**Analysis:**

This exploration will make use of maths to decide whether monopoly is a fair game. It will give an insight on how math is involved in such games which we usually consider to be dependent all on luck.

**9.) Time-frequency analysis of musical instruments using Fourier Transform**

**Mathematics:**

Fourier transform comes from calculus. Fourier analysis can be used to identify fundamentals and over tunes of individual notes.

**Procedure:**

This method can be applied to various instruments such as, guitar, flute, piano etc. The method of digitally computing Fourier spectra is widely referred to as the FFT (short for fast Fourier transform). You can compute the Fourier spectra digitally and then proceed.

**Analysis:**

This exploration connects calculus with real life application.

**10.) Correlation between rice harvest and the previous year’s GDP of India**

**Mathematics:**

Utilising linear regression models to determine the correlation between the rice harvest and India’s GDP

**Procedure:**

Obtain the data for rice harvests and the year’s previous GDP of the nation for several instances. Try to research incidents that could have affected either the GDP or rice harvest and include the incidents in the analysis and produce the linear regression models.

**Analysis**:

Rice is a staple food item of India and also plays a major role in the GDP of the nation. By correlating the two, the relation will prove how important rice harvest is important to the nation’s economy

**11.) Mathematically modelling a landslide and determining the probability of a landslide occurring in a region that already has experienced one**

**Mathematics:**

Implement slope stability models and use the fracture criterion to model landslides and its probability of occurring along with rainfall data and other surface variables

**Procedure:**

Obtain the data of previously recorded landslides along with landslides that occurred again in that region and implement the models above to determine the probability of another landslide

**Analysis:**

Landslides are extremely dangerous and affect the environment and human life. By predicting the probability of a landslide occurring, we can prevent any disasters or harm from occurring to living organisms in that regions as well as to reduce the damage that occurs due to it.

**12.) Modelling the most ideal football kick to score a goal**

**Mathematics:**

Modelling the most ideal football kick with laws of projectile motion and equations of physics that determine the curvature and elevation along with power of the kick

**Procedure:**

Take a various pool of players and record their goals. Then, implement and derive data from their recordings in terms of power, projectile motion and other data

**Analysis:**

Football players in history have scored magnificent and physics-breaking goals. Now, by modelling the perfect kick to score any goal with relation to players’ goals, we can ideally attempt to model this kick and understand the entirety of how the goal occurs.

## IB Math AI SL IA Ideas

**1.) Is there a relationship between the height of a basketball player and their shooting ability?**

**Mathematics:**

This IA investigates the correlation between height and shooting ability in basketball using the **Pearson’s correlation coefficient and chi-squared.**

**Procedure**:

The data of height and number of basketball shots per person will be collected from 40 people. A line graph can be plotted to roughly see the relationship between the two variables. Then, using Pearson’s Correlation Coefficient formula, the magnitude of the correlation can be determined. Later, using the Chi2 value formula, it can be determined if the variables are independent of each other.

**Analysis:**

A group of people are taken with varying heights and their shooting ability. If standard deviation is high, the range of the data set is large. For Pearson’s correlation coefficient, if the value is 1 or close to 1, there is a strong positive correlation

**2.) Modeling the function and finding the surface area of a ceramic mug**

**Mathematics:**

**Lagrange Interpolation Method and the FITPOLY function** from GeoGebra are used as two different methods to find the quadratic expressions to calculate the surface area of ceramic mugs.

**Procedure:**

Firstly, the mug is placed in GeoGebra where the coordinates of the mug can be found. Using these coordinates, two methods are performed to find the equations of the curves. First, the Lagrange Interpolation formula is used where the coordinates are just substituted into the formula. Since the calculations are tedious, a coding website can be used to find the coefficient values. The other method is the FITPOLY function where GeoGebra estimates a curve depending on the degree of the polynomial for the respective portion of the mug. Using the more accurate equations, the surface area of the mug is found using calculus.

**Analysis:**

The equation results from performing both methods is compared to see which one resembled a mug more. These sets of equations would be used in the further calculations to find the surface area of the ceramic mug

**3.) Is there a relationship between a person’s height and his shoe size?**

**Mathematics:**

This IA investigates the correlation between a person’s height and his shoe size using the **Pearson’s Correlation Coefficient and Chi-squared method**. The Pearson’s Correlation Coefficient is a method or a way to measure the relationship between two continuous or discrete variables. A Chi2 test involved making a contingency table where two variables are compared to see if they are related to each other.

**Procedure:**

The height and shoe size data will be collected from 40 people. A line graph can be plotted to roughly see the relationship between the two variables. Then, using Pearson’s Correlation Coefficient formula, the magnitude of the correlation can be determined. Later, using the Chi2 value formula, it can be determined if the variables are independent of each other.

**Analysis:**

If standard deviation is high, the range of the data set is large. For Pearson’s correlation coefficient, if the value is 1 or close to 1, there is a strong positive correlation. As for the Chi2 test, comparing the critical value against the Chi2 table will tell if the variable is in fact independent.

**4.) Analyzing the best mathematical way to estimate the number of candies in a jar**

**Mathematics:**

Aim is to correctly guess the amount of candies inside a fully filled vase by using several methods. They are, GeoGebra equations (graphical software), square-root equations, and ellipse equations. It involves calculus, majorly, differentiation.

**Procedure:**

First, the volume of my hexagonal jar is calculated using measured dimensions. The volume of a single candy using GeoGebra software is found by rotating the function of the candy about the x axis (volume of revolution formula). Similarly functions are found in the other two methods as well and are rotated about the x axis using the same formula. After finding the individual volumes of the candy and the jar, the total amount of candies can be found by dividing them both.

**Analysis:**

to understand the reliability and the accuracy for each of these methods, the amount of candies in the jar can be counted and compared with the theoretical value. Whichever value is closest will be the most reliable method to estimate the number of candies in a jar.

**5.) Using Pringles to explore the formula of a hyperbolic paraboloid, determine the surface area and define the hyperbolic paraboloid parametrically**

**Mathematics:**

Using mathematical concepts like differentiation and integration to calculate the surface area of a hyperbolic paraboloid such as a pringle.

**Procedure:**

Firstly, you would have to draw the pringles chip and then plot it on a graph. Then you must split it into different parts, and derive the equation. Calculate the surface area using integration.

**Analysis:**

This analysis is most important when it comes to the firm’s point of view. For instance, when manufacturing Pringles, one must optimize space and ingredients to ensure consumer satisfaction and reduce costs for the firm.

**6.) Modelling musical chords using sine waves**

**Mathematics:**

Musical chords can be modelled using trigonometric functions which relates to amplitude and frequencies.

**Procedure**:

You can graph the notes using sine wave, you can put together notes of a particular chord in one graph and from there determine if its dissonant or consonant. Further find a mathematical pattern for consonant chords.

**Analysis:**

Math can help determine which chords sound pleasing and which combination of chords work the best. Using these mathematical concepts for music will help strengthen one’s understanding of applications of sinusoidal waves.

**7.) Is there a relation between university endowment and its ranking?**

**Mathematics:**

To find a relationship one can use bivariate statistics along with hypothesis testing i.e. chi squared test of independence.

**Procedure**:

One must collect required data and put it in a tabular format. Using the two methods above and finding the required statistics and concluding the test results will help make a conclusion for the same.

**Analysis:**

Using statistics, you can figure out relations and dependence of any two factors. This exploration will help understand how statistics and testing can be applied to the real world and help clear out our doubts about the assumptions we make for the things happening around us.

**8.) Use of Bayes’ theorem to evaluate depression test performance**

**Mathematics:**

If you’re interested in psychology, you might want to use probability theories for evaluation of depression tests.

**Procedure**:

Bayes’ theorem poses an interesting question, the possibility of false positives. Using this method, you can make estimations about the test. The population can also to distributed with a probability distribution for ease.

**Analysis:**

This exploration will give insights on how accurate the test is and how probability theories can be used to analyze results related to certain tests.

**9.) Statistical analysis of Cryptocurrencies**

**Mathematics:**

If you are interested in finance and investment, this topic is for you. Use of bivariate statistics and modelling can help determine relation between various factors affecting the exchange rate of cryptocurrencies.

**Procedure**:

First step would be to collect data and determine the properties that you will be analyzing depending on depth of your exploration. You can compare exchange rates of crypto with US dollar, inflation rate, Chinese government bonds etc.

**Analysis:**

This exploration will be useful to plan investments. We may not just rely on these mathematical results as field of economics is a social science. Investment and trading relies both on mathematical aspects and human behavior.

**10.) Shoelace Algorithm to find areas of polygons**

**Mathematics:**

The shoelace algorithm is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. The method consists of cross-multiplying corresponding coordinates of the different vertices of a polygon to find its area.

**Procedure**:

To apply the shoelace algorithm you will need to:

- List all the vertices and note the ordered pairs.
- Calculate the sum of multiplying each x coordinate with y coordinate in the row below
- Calculate the sum of multiplying each y coordinate with x coordinate in the row below
- Subtract the second sum from the first sum, get the absolute value.
- Divide the resulting value by 2 to get the actual area of the polygon.

**Analysis:**

The shoelace algorithm works only with a simple polygon. If the polygon crosses or overlaps itself the algorithm will fail.

**11.) Calculus and Zeno’s Arrow Paradox**

**Mathematics:**

This topic is about how derivatives are applied in Zeno’s arrow paradox. This paradox basically states that a moving arrow moves no distance at an instant, which lasts 0 second, as it occupies a space equal to its size, nor does it move at any instant for the same reason, concluding that the arrow, therefore, has no motion. Calculus is the mathematical study of change. For this reason, it is useful because it accepts the concept of infinity and a number approaching infinity and zero is necessary to try to find a mathematical solution to the paradox. In Zeno’s famous arrow paradox, he contends that an arrow cannot move since at every instant of time it is at rest. There are two logical problems hidden in this claim. Firstly, Zero is divided by zero. Secondly, Adding up zeros.

**Procedure:**

Use the derivative of the velocity of the arrow, and make time shrink towards zero, which is what happens at each instant during the movement of the arrow. Suppose an arrow is shot which travels from A to B. Consider any instant. Since no time elapses during the instant, the arrow does not move during the motion. But the entire time of flight consists of instances alone. Hence, the arrow must not have moved.

**Analysis:**

- If instants are infinitesimal, but not 0, then the arrow would cover an infinitely small distance each infinitely small unit of time.
- If time is not continuous and it has an extremely small, indivisible unit, the arrow moves during finite periods of time rather than during each instant.

Calculus allows us to think of an instant as an infinitely small unit of time, therefore allowing, in this case, movement to exist. By using a limit, the time in which the arrow moves will be an infinitely small number, almost zero.

**12.) Probability using Bayes Theorem**

**Mathematics:**

In probability, Bayes theorem is a mathematical formula, which is used to determine the conditional probability of the given event. Conditional probability is defined as the likelihood that an event will occur, based on the occurrence of a previous outcome.

**Procedure:**

According to the definition of conditional probability, derive Bayes Theorem formula. It can be derived for events A and B, as well as continuous random variables x and y. Bayes Theorem for Events: P(A/B) = (P(B/A)*P(A)) / P(B) where P(B) cannot equal zero.

**Analysis:**

Bayes theorem is used to determine conditional probability. When two events A and B are independent, P(A/B) = P(A) and P(B/A) = P(B). Conditional probability can be calculated using the Bayes theorem for continuous random variables.

**13.) Methods of approximating sin(x) as an algebraic function**

**Mathematics:**

The function sin (x) is a trigonometric relation defined in terms of unit circle as the vertical distance between the x-axis and the point of the unit circle that meets a line subtended by angle x (in radians). Sin (x) is a transcendental function which can be only defined in terms of other trigonometric expressions or infinite series of polynomials.

**Procedure:**

In the first method, substitute x for sin (x). Secondly, we can obtain a simpler function that approximates the parabolic shape of sin (x) between 0 and pi using quadratics.

**Analysis:**

The rationale behind this substitution is that the graph of sin (x) roughly coincides with the grade of f(x) = x for angles very close to zero. While a pure sine function has an infinite number of upward and downward curves, a quadratic function will only formulate one parabola, and will move away from the x-axis continuously for any values of x beyond that.

**14.) Birthday Paradox**

**Mathematics:**

This exploration is based on statistics and probability. The birthday paradox shows how intuitive ideas on probability can often be wrong. How many people need to be in a room for it to be at least 50% likely that two people will share the same birthday?

**Procedure:**

When you are comparing if any of the (n) people in the room share a birthday, you will simply make n comparisons (n, 2). Approximate the problem by working out the probability P(no shared birthday).

**Analysis:**

This is an interesting question to answer as it shows that probabilities are often counter-intuitive. The answer will be likely that you only need 23 people before you will have a 51% chance that two of them share the same birthday.

**15.) Correlation between the height and weight of a runner and their 100 m dash time**

**Mathematics:**

Measuring and creating a relation between a runner’s physical attributes and their times along with stride length and step frequency.

**Procedure**:

Take various people and measure the data of height, weight, age etc. along with their 100 m dash time. By recording their sprints, measure their average stride length and step frequency.

**Analysis:**

By analysis and relating this data, the timings and probability of an individual to win races can be determined to a great extent

**16.) Modelling the tectonic plate movement of India into the Asian continent**

**Mathematics:**

Implementation of continuous kinetic models along with the tectonic plate theorems

**Procedure:**

Obtain data of the movement of India’s tectonic plate inclusive of velocity it is moving into Asia, the distance that has collided, reduction of height of mountains such as Mt. Everest and more.

**Analysis:**

India is moving underneath Asia by a few millimetres every year and therefore, we can model and predict when it will completely be underneath the Asian tectonic plate

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