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IB Math AI 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.
13.) Investigation of Mathematics in Barcodes
Mathematics:
This investigation aims to find out the number of combinations to express a number in the barcode. The way of mapping information into numbers and strips of black and white seemed similar to a function that we learn.
Procedure:
Find this using EAN (European Article Number) and tree diagram and combination formula. Find out how to calculate the number of different possible barcodes using a simplified set of rules. Calculate the check digit by using all the other numbers in the barcode.
Analysis:
There are two mathematical approaches to find the number of combinations to express a number in barcodes. Tree diagrams demonstrate all the possible combinations and the combination formula gives the total number of ways.
14.) Packaging and Geometrical Shapes
Mathematics:
The aim of this investigation is to find the ideal packaging surface area for 1-litre of Water, Milk and Apple Juice.
Procedure:
First take measurements of all three packages. Then calculate their actual surface area and volume using a mathematical formula and then using calculus find the dimensions for the least surface area of all three packages. After finding actual surfaces, find the minimum surface area and find the percentage difference to see which of the packages is closest to the minimum. To validate the results and calculations, model an equation for surface area and edge lengths. Also drawing a graph for the equation of the surface area using Graphical Display Calculator and finding the minimum value from the graph.
Analysis:
After analysing all three packages with different shapes and results, it will be evident that the milk package is closest to the ideal shape for the surface area and the water is closest in terms of ideal package for volume but is the most distant in surface area. However, none of the three packages can be categorised as an ideal packaging shape.
15.) The Gambler’s Fallacy and Casino Maths
Mathematics:
Finding expected values for games of chance in a casino. The main purpose of this investigation is to find the probability of casino games to become more successful when you play. The causes of the game known as “roulette” will be covered. This is the misconception that prior outcomes will have an effect on subsequent independent events.
Procedure:
Analyse all the probabilities in the games and construct a winning strategy. Play the casino games to see if the probabilities are correct and see the chances of winning. For Roulette, find out the probabilities, by finding out how many spins there are in the game and the ratio. After finding that out, calculate the percentage of that probability. The classic example for this is the gambler who watches a run of 9 blacks on a roulette wheel with only red and black, and rushes to place all his money on red. He is sure that red must come up – after all the probability of a run of 10 blacks in a row is 1/1024. However, because the prior outcomes have no influence on the next spin actually the probability remains at 1/2.
Analysis:
Based on the calculations, the probability of winning a game of Roulette is very high, if you have a strategy. You would anticipate the ball will arrive in a dark pocket in 18/38 turns, around 47% of the time, and a green pocket in 2/38 twists, or 5% of the time.
16.) Probability of scoring a football goal
Mathematics:
Deriving a model to calculate the probability of scoring a goal from every shooting position in the football court and applying it to predict the expected goals for different matches. The aim of this investigation is to find an equation where you can insert the location of the shot and thereby find out the probability of a goal.
Procedure:
Use the data of the Premiery League Season 2021 from Wyscout. The data should consist of all the shots that took place during that season and even the location of all the shots. Also, whether the shot resulted in a goal or not. You will be able to get the coefficients of the equation that relates the position of the shot with its probability of success. Consider the position as an independent variable having x and y coordinates. X coordinate would represent distance from the right sideline and y corrdinate would represent the vertical nearness to the goal. Calcuate the absolute value of the difference between x coordinate and 50%. First perform linear regression analysis to estimate a relationship between dependent variable (probability) and independent variable (x and y coordinates). Using the data makes an equation for probability of success.
Analysis:
You will be able to find an equation that relates the probability of the goal to the location of the shot. The equation will be able to estimate the probability of scoring a goal from any location of the football court. It will not make any difference if the player is shooting the ball from the left side or right side of the pitch. The probability of scoring a goal will depend on angle and distance. All players will have equal probability of scoring a goal. Lastly, all shots will be independent from each other.
17.) Horse Jumping
Mathematics:
This investigation will explore how much space before the take-off can leave up to chance and what are the limiting distances from the jump above which the horse will not be able to complete the jump anymore.
Procedure:
Start the investigation by first finding the curve the horse makes when jumping over a jump and try to determine some of its characteristics. That will later help to determine the limit value easier. The trajectory the horse follows is parabolic. Observe the take-off, highest point and landing. Draw the coordinate axes onto the picture where the x-axis is at the level of the ground and the y-axis coincides with the jump. Plot the points, the coordinate axes and the best-fit curve. It will represent a quadratic equation.
Analysis:
The characteristics observed from the graph parabola would be concave-down, which means there is a negative sign before the x square in the quadratic function. The vertex is the height of the jump. Lastly, the y-axis is the line of symmetry which makes x-intercepts of the graphs equally apart from the jump. This indicates that the horse needs an equal amount of space to take-off and land.
18.) The Goldbach Conjecture
Mathematics:
The Goldbach Conjecture is a famous unsolved problem in number theory that states that every even number greater than 2 can be expressed as the sum of two prime numbers. The conjecture has been shown to hold for all integers less than 4 × 1018
Procedure:
To solve a problem in number theory like the Goldbach Conjecture, mathematicians use a variety of mathematical techniques and tools such as number theory, combinatorics, probability theory, and analysis. First write all of the prime numbers on two sides of a triangle as 2,3,5,7,etc. Then draw a line leaving each prime number which is parallel to the opposite side of the triangle, and at the points of intersection of these lines write the sum of the numbers.
Analysis:
The greater the even number the more likely it will have different prime sums. As the even numbers get larger, they can be written with larger combinations of primes. That would suggest that the conjecture gets ever more likely to be true as the even numbers get larger.
19.) Monty Hall Problem
Mathematics:
How does Bayesian probability work in this real-life example, and can you add a layer of complexity to it? The Monty Hall problem is a paradoxical problem in conditional probability and reasoning using Bayes theorem. This technique is widely applicable in our daily lives as we always take risks and expose ourselves to a variety of choices where decision making is crucial. This investigation will undergo probability and statistics.
Procedure:
One way to see the solution is to explicitly list out all the possible outcomes and then count. Bayes Theorem is a formula that describes how to update the probability that a hypothesis is correct. List different combinations of choices and outcomes. Start with the Bayesian Rule of Probability. The probability of win in the Classic Monty Hall game can be expressed as switch and win and stay and win using experimental and theoretical probabilities.
Analysis:
Throughout the investigation, you will understand the importance of assumptions and the effect of decision making. There is a close relationship between real life situations and theoretical probabilities of events occurring. In general, the probability of winning increases if the player chooses to switch his or her initial decision.
20.) Correlation Coefficient between Golden Ratio and Visual Aesthetics
Mathematics:
The golden ratio is a design composition tool which is non-terminating, irrational number, known as the divine proportion and can be allegedly found in natural designs and patterns such as arrangement of sunflower seeds and spirals in the galaxy.
Procedure:
This investigation aims to find the correlation coefficient between the Golden ratio and Visual Aesthetics followed by linear regression t-test. Make a logo as per golden circles. Then determine the correlation coefficient. The golden ratio can help generate a composition that is balanced aesthetically pleasing design compositions, creating symmetrically pleasing visuals.
Analysis:
Many multinational companies use golden ratio to design their company logo. The golden ratio makes the shape much more symmetrical and geometric. As the correlation coefficient comes extremely close to 1, shows that there exists a strong positive correlation and connection between the golden ratio and visual aesthetics.
21.) Voronoi Diagrams and Graph Theory
Mathematics:
The objective of this investigation is to determine the optimal university based on the number of airports nearest to each of the 9 shortlisted universities using a Voronoi diagram.
Procedure:
In order to construct Voronoi Diagram, overlay a map of the country over a squared grid using GeoGebra. Locate the 9 universities and write their coordinates in a table. The perpendicular bisectors between every site must be found. Also find the midpoint and gradient of the perpendicular bisector between two points.
Analysis:
The location of airports is extremely important to the choice of university. To determine which university has the most airports nearby, simply add the location of the airports to the grid. Whichever cell the airport is in, it will be closest to the corresponding university.
22.) The Chinese Remainder Theorem: An insight into the mathematics of number theory
Mathematics:
This topic whilst seemingly quite abstract is a good introduction to number theory – the branch of mathematics which deals with the properties of whole numbers. The Chinese remainder theorem is a theorem which gives a unique solution to simultaneous linear congruences with coprime moduli. In its basic form, the Chinese remainder theorem will determine a number p that, when divided by some given divisors, leaves given remainders.
Procedure:
The aim of the exploration is to determine a number p, that when divided by some divisors leaves given remainders. What number has a remainder of 2 when divided by 3, a remainder of 3 when divided by 5 and a remainder of 2 when divided by 7? There are a couple of methods to solve this. Firstly, it helps to understand the concept of modulus.
Analysis:
Its proof is another application of the fact that the greatest common divisor of two numbers can be written as a linear combination of the two numbers. For example 21 mod 6 means the remainder when 21 is divided by 6. In this case the remainder is 3, so we can write 21 ≡ 3 (mod 6). The ≡ sign means “equivalent to” and is often used in modulus questions.
23.) Optimizing the angle of projection of a basketball in a real-world context
Mathematics:
Inorder to observe the projectile motion of a basketball in a real-life context, the aim of this exploration will be to investigate the height of a basketball player with the distance of the player from the hoop, and how these factors affect the optimal angle of release of the basketball. This is to be investigated to optimize the angle of release of a basketball from the free-throw line and three-point line in a real-world context in comparison to conventional kinematics and projectiles.
Procedure:
Firstly, the equations to be used must be listed and defined in order to derive the final model to represent the shot of a basketball. Equations of motion would include vertical and horizontal movement of the basketball as in projectile motion. In order to optimize the equation of motion of the basketball to maximize distance we must take the first derivative of the result. Then find the initial velocity followed by using trigonometric identities. Quadratic formula can be further used to find the optimal angle of projection.
Analysis:
The equations derived would allow the optimal angle of projection of a basketball and minimum initial velocity of the ball to be calculated respectively. The projection of a basketball is at a lower height than the final height of the basketball after the shot. It was concluded that as the height of a player increases, the optimal angle of projection as well as the minimum required initial velocity of the basketball decreases linearly.
24.) Modeling the spread of COVID-19 in India 2021
Mathematics:
Investigate and analyze the rate of COVID-19 spread in India. In addition, model its spread using a logistic model. The deaths per day and cumulative deaths graph would be relevant to understand the severity of the situation, while the new cases per day graph would tell how fast it is spreading day to day. We will use functions and R square value under statistics and probability to determine the accuracy of the model.
Procedure:
Firstly, estable the two variables; f(x) representing cumulative confirmed cases of COVID-19 and x representing the number of days after 1st confirmed case. From the graph you find determine the range. Then find the midpoint of the function. After you observe the initial exponential increase in the graph, assign a different equation using derivatives.
Analysis:
The outcome of this investigation will be to determine a model for the spread of COVID-19. Limitations and strengths will further need to be discussed. This exploration connects modeling with real life application.
25.) Modular Arithmetic and its applications in Caesar’s Cipher
Mathematics:
This investigation will explore the applications of modular arithmetic and its usage in encryption, specifically Caesar’s Cipher. This exploration will show the ways in which modular arithmetic is used to encrypt and decrypt data. Hence, finding out what makes data encryption so secure and hard to hack.
Procedure:
Caesar’s Cipher is a substitution cipher where each letter is replaced by another letter located some places further in the alphabet. Modular arithmetic is the branch of mathematics related with the “mod” functionality. The shift cipher can be given by: a = (b+3) mod 26. Modular arithmetic is based on dividing two integers and obtaining a remainder. Formula of how modular arithmetic works is A/B = C reminder D. There are three steps required to conduct a basic modular arithmetic calculation:
1. Divide the integers with a result without the partial fraction
2. Multiply the result obtained from Step 1 by the divisor
3. Subtract the result in Step 2 from the dividend – giving the remainder
Analysis:
He encrypted text will be written with capital letters whereas the decrypted text will be written with small letters. The usage of modular arithmetic in Caesar’s Cipher is extensive. The addition modulo 26 shows how all the letters wrap around that valueh
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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
17.) Chinese Postman Problem
Mathematics:
This is a problem from graph theory – how can a postman deliver letters to every house on his streets in the shortest time possible? How do we calculate the shortest possible route?
Procedure:
Solving this requires using a branch of mathematics called graph theory. To find a minimum Chinese postman route we must walk along each edge at least once and in addition we must also walk along the least pairings of odd vertices on one extra occasion. Find the route through google maps. The aim is to find the least possible time while traversing every edge end returning to the starting point. While creating the graph the important thing was to draw the nodes where the postman had a choice of either going left or right or had more than 1 option to choose from.
Analysis:
If all orders are even then the graph is traversable. If there are 2 odd vertices then we can find a traversable graph by starting at one of the odd vertices and finishing at the other.
18.) Spirals in Nature
Mathematics:
While researching mathematics in nature you will find spirals that are found in shell shapes. The shell curves are logarithmic and equiangular with slightly different proportions to other spirals such as golden ratio. In this investigation find which spiral is there in ammonite, is it archimedean spiral or logarithmic spiral.
Procedure:
Plot the points and the coordinates to make the right shape. Then see the radius and angle for each point. Then find angle in radians. If the spiral follows an Archemedian spiral, r = aϴ, so plotting r against ϴ should give a straight line of gradient a intersecting the vertical axis at the origin. Then find the absolute value error.
Analysis:
Every point on the spiral has a value of theta and a value of r and each model has an approximate value of r for that theta.
19.) Handshake Problem
Mathematics:
How many handshakes are required so that everyone shakes hands with all the other people in the room?
Procedure:
To see this, enumerate the people present, and consider one person at a time. The first person may shake hands with n-1 other people. The next person may shake hands with n-2 other people, not counting the first person again. Continue counting like this and you will get a total number of handshakes.
Analysis:
Not being able to shake hands with yourself, and not counting multiple handshakes with the same person, the problem is to show that there will always be two people at the party, who have shaken hands the same number of times. The answer is (n/2)= n ( n – 1 )/2
20.) Tangrams
Mathematics:
Investigate how many different ways different size shapes can be fitted together. The Tangram is a dissection puzzle consisting of seven flat polygons, called tans which are put together to form shapes. The objective is to replicate a pattern generally found in a puzzle using all seven pieces without overlapping. The mathematics behind tangram involves the use of geometric concepts such as congruence, similarity, and spatial reasoning to solve the puzzle and create various shapes and figures.
Procedure:
The Tans are made up of two large right-angled triangles, one medium-sized right-angled triangle, two small right-angled triangles, one parallelogram and one square. The tans can be rearranged in countless ways to create various shapes and figures, making tangram a valuable tool for developing spatial and geometric thinking skills. Discover possible ways to fit different size shapes together.
Analysis:
By measuring the sides of the tans and using them to calculate the area and perimeter of different shapes, one can develop a deeper understanding of these concepts.
21.) Does the area covered by the long jumper’s arc impact the distance?
Mathematics:
As a Long Jumper that has been competing for many years, let’s research for aspects that impact the long jump distance.
Procedure:
There are many elements that affect the outcome of the long jump distance like board contact time, take-off angle, midpoint knee flexion angle, individual strength level etc. In this investigation take the top 5 long jump athletes that have competed in Olympic games. Take screenshots of the jump and plot the coordinates and function of the jump. Then calculate pearson’s correlation and determine the type of correlation. Then find the equation and calculate the area under the curve.
Analysis:
Prediction for this investigation is distance achieved does not have any common reaction with the area covered by arc.
22.) Determine the shape of the rope of a flying kite
Mathematics:
How to find a function in the form of y=f(x) to describe the shape of the rope of a flying kite that is in static equilibrium. Consider a flying kite. The string of the kite is a curve. So the aim of this exploration is to find the shape and the mathematical function for the kite string, that is in forms of y=f(x) which only contain variables x and y constants: force from hand, density of the rope and cross section area of the rope and the gravitational acceleration on the surface of the Earth.
Procedure:
In this exploration use the force equilibrium method. To establish the model, assume after launching the kite into the air and reach the desired vertical height and stay there, in other words, all the forces acting are in balance which means the system is in static equilibrium. The gravitational potential energy of a rope partially depends on the shape of the rope, therefore if the function f(x) describes the shape of the rope changes, the gravitational potential energy V will also change. Using hyperbolic functions determine hyperbola. The derivatives of hyperbolic functions are also necessary in the process of deduction. Consider a flying kite and determine the lowest point, x and y coordinates and local minimum.
Analysis:
Through the application of Physical and Mathematical knowledge, the shape of the rope turns out to be a hyperbolic function.
23.) Height vs Spiking Ability in High School Volleyball
Mathematics:
What is the relationship between height and spiking ability in High School Volleyball? Through this investigation we will be able to consider the factors that affect spiking ability to the advantage of a player’s volleyball performance.
Procedure:
To determine the relationship between height and spiking ability among volleyball players, least square regression lines and Pearson’s correlation coefficient can be performed to interpret the trends of the data and the strength of the correlation between the two variables. Then you can further validate the investigation by using Chi square test of independence. Lastly, binomial distribution calculation to calculate the probability of players spiking ability depending on their height range.
Analysis:
Through the data collection, data analysis of strong and clear correlation between two data variables, and chi square test for independence provide enough evidence to validate the fact that height affects the spiking ability of volleyball players and are directly proportional. This investigation will justify the relationship between height and spiking frequency; as height increases the spiking frequency increases.
24.) An analysis of two industrial fields (pharmaceutical and construction) before and after the occurrence of COVID-19 pandemic
Mathematics:
It is an application investigation on the correlation between the share prices of two companies, before and after the occurrence of the covid pandemic.
Procedure:
Take two companies that are listed in Indian Stock Markets; one from the pharmaceutical industry and one from the construction industry. To conduct this investigation, collect quantitative information on the daily history of stock prices in 2019 and 2020 using stratified sampling. To explore the correlation between the share prices of both companies, before and after the occurrence of the pandemic, use scatter diagrams and Pearson’s product correlation. Moreover, to determine their relationship, t-test can be used to compare the means of the share prices for both the years. Depending on which company shows more variability between the two years, use calculus to calculate and compare the company’s maximum share price for 2019 and 2020.
Analysis:
After processing the data for the respective companies you will be able to determine if the share prices increased or decreased in which year. You will also be able to find the correlation.
25.) The Influence Body Fat and Lean Mass has on one’s Vertical Jump
Mathematics:
The aim of this investigation is to understand the influence one’s body composition has on their vertical jumping ability. Aim to identify the correlation between one’s body fat percentage and their vertical jump as well as one;s lean mass and their vertical jump.
Procedure:
Choose a common gender to collect data. Choose people based on the variations in their body compositions. Draw a table showing the standing reach and jumping reach of the participants. The equation (vertical jump = jumping reach – standing reach) can be used to calculate the participant’s vertical jump. Use a mathematical formula to calculate body fat percentage. The formula should consider one’s height, waist and neck measurements in order to calculate their body fat percentage. Use the formula to find the lean body mass. In order to better understand the relationship between both variables being body fat percentage and vertical jump a trend can be expressed through a logarithmic function.
Analysis:
Based on the logarithmic equation visualizing the relationship that Body Fat and Vertical Jump share, it is in fact true that as body fat increases, vertical jump decreases. There will be a medium positive correlation between Lean Mass and Vertical Jump.
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