# 50+ IB Maths AA IA Ideas

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## IB Math AA HL IA Ideas

**1.) The Math of Infectious Disease: Understanding and Interpreting Models in Epidemiology**

**Mathematics:**

Using mathematical concepts like **exponential functions** and **calculus** to derive **differential equations** from various epidemiological models like SARS, etc.

**Procedure**:

In order to proceed with this topic, you must derive differential equations based on the model and find the solution using various methods like the homogenous method, linear method, etc.

**Analysis**:

The derivation of such differential equations for epidemiological models are of great importance when it comes to analyzing trends and forecasting data. The information gathered can then be used to control the spread of the disease and prevent future outbreaks.

**2.) Investigating the mathematical harmony of Beethoven**

**Mathematics**:

If you’re a classical music geek, this one’s for you. There are various mathematical properties that surround music such as trigonometry.

**Procedure**:

In the case of Beethoven, you can graph the notes and chords with sine waves. The simplest model of a musical sound is a sine wave, in which the domain (x-axis) is time and the range (y-axis) is pressure.

**Analysis**:

Using such mathematical concepts helps strengthen knowledge in music as counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, tone and pitch are determined by using math.

**3.) Understanding China’s Income Distribution from a Mathematical Perspective using the Lorenz Curve and Gini Coefficient**

**Mathematics**:

Using various mathematical concepts like integration and functions.

China’s income distribution can be found using secondary methods or primary, depending on your geographical location.

**Procedure:**

Using the data, you can plot Lorenz’s curve using Gini Coefficient to show the ideal income inequality in an economy as well as the reality. The areas below the graph using integration and the difference between the two will depict the gap between the rich and the poor.

**Analysis:**

Such models are of great importance when it comes to analysing trends and forecasting data. The Lorenz curve is important because it helps in understanding economic inequality. When the lorenz curve keeps moving away from the baseline it indicates that the level of unequal distribution keeps increasing.

**4.) Investigating the safe operating speed for ground vehicles on a given road segment**

**Mathematics:**

Using mathematical concepts such as integration when calculating area under the graph and finding optimum speed. When driving a car, it is often necessary to slow down during a curve. This optimum speed can be found using mathematics.

**Procedure:**

An observer is placed at each end of the survey segment recording each vehicle’s passing time and license plate, in order to calculate the time spent traversing the segment. This data can be used to derive the formula for speed and differentiate it in order to find maximum and minimum points.

**Analysis:**

The optimum speed when travelling in a curve is found for various reasons, the most important one being safety. Determining the optimum speed to operate at a turning is absolutely crucial when it comes to safety.

**5.) Calculating orbital flight paths/intercept trajectories in space for rockets**

**Mathematics:**

Using mathematical concepts such as differentiation and integration to calculate the orbital flight path.

**Procedure:**

You will have to graph the pathway in which the rocket will move as well as model the rocket to determine the optimum speed and path.

**Analysis:**

Such analysis is crucial in order to reduce the fuel efficiency of rockets in space using gravity assist.

**6.) Using origami to design foldable furniture**

**Mathematics:**

There is plenty of math’s related to origami. Origami works on math principles and theorems. One can apply algebra, sequences and series, calculus and logical ability to design foldable furniture analogous to folding paper in origami.

**Procedure:**

You will have to pick a tessellation which you will use to design the piece of furniture and then decide the optimum number of folds (using calculus) for the size of furniture you’ve picked

**Analysis:**

Designing a foldable furniture is useful when there are small areas. For example, folding a dining table into a coffee table. Applying Origami and math to it would help to determine the most optimized way of doing the same.

**7.) Numerical model of the solar system**

**Mathematics:**

The planets orbit around the sun in elliptical paths, with sun at one of the foci. Using geometry and algebra you can find the equations of each planets and model the entire solar system.

**Procedure:**

Using data from the web you can find the equations using the concept of ellipses. As an extension you can find the areas that each planets covers as they revolve around the sun and compare their properties that are affected due to this revolution.

**Analysis:**

If you’re interested in space, this exploration will give you insights of planetary motion and how math is involved in understand the physical world.

**8.) Analysis of AC circuits by using complex numbers**

**Mathematics:**

If you’re a physics student and wish to dig deeper in math’s behind AC circuits, you can start by applying complex numbers to the previously leant concepts.

**Procedure:**

Complex numbers are utilized in calculations of current, voltage or resistance in AC circuits. One can make use of math to analyze AC circuits.

**Analysis:**

Math’s and Physics go hand in hand. Physics makes use of mathematical tools to justify and understand the theorems and concepts. This exploration will help understand the relation between math’s and physics.

**9.) Chess and Mathematics**

**Mathematics:**

Combinatory is the mathematics of counting and probabilities. You can explore chess and all the different ways a chess game can play out using statistics, probability and combinatory.

**Procedure:**

Math exists inherently in the game. Starting with the initial positions of the chess pieces, start to analyses all the possible moves and use mathematical tools to figure various ways in which the game can be played out. Instead of dealing with the whole game, you can also create situations on the board which a player might face and then using math calculate all the possible moves and find out the right one.

**Analysis:**

This exploration connects the game of chess with math. Although there will be numerous simulations of the game possible, one can still analyses the certain situations on the board using math’s and make conclusions.

**10.) Mathematics of solar panel**

**Mathematics:**

Solar energy has great importance in today’s world. There are various mathematical tools that can be used to design and install the most optimum solar panel. This exploration may involve vectors, trigonometry and calculus.

**Procedure:**

The various aspects which you can work on are-

- Optimum tilt of the solar panel
- Find the size of PV modules according to use
- Number of PV panels required

**Analysis:**

Solar panels are installed in many households and according to the power consumption, they must be customized. This exploration makes use of math to determine the most optimal way to make use of solar panels.

**11.) Relation of diversity of ecosystem and climate**

**Mathematics:**

Referring to previous techniques in mapping the relation and implementing them in sample data such as diversity indices and parameterization along with providing effective numbers

**Procedure**:

Obtain data of diversity and climatic conditions for a variety of regions with different properties and implement diversity indices and parameterization.

**Analysis**:

The data and mathematics will help determine the diversity of species that survive and require the variety of climatic conditions

**12.) Relationship between number of gear shifts and probability to win in a F1 race**

**Mathematics**:

Utilising previous statistics of F1 races and probability, we determine how drivers who have won shift the gears of their cars along with its frequency

**Procedure**:

Gather previous race statistics of different positioned racers in one or more series of races while noting the car model

**Analysis**:

F1 is an exhilarating sport and victories are determined not only by the car’s performance but also the decisiveness of the driver, which includes the number and pace of gear shifts throughout the track.

**13.) The Golden Chord Progression**

**Mathematics**:

Utilising the golden ratio to determine harmonious and discordant chords

**Procedure:**

The golden ratio can be applied to sample melodies and songs which can provide a progression series of chords.

**Analysis:**

The golden chord progression is a technique that enables music producers to determine chord progressions that fit a melody based on the melody’s properties.

## IB Math AA SL IA Ideas

**1.) Surface Area of Hyperbolic Paraboloid – Integration by substitution**

**Mathematics:**

**Differentiation** and **integration** are the most dominant mathematical concepts when it comes to calculating the surface area of a hyperbolic paraboloid.

**Procedure:**

Pick an object with a hyperbolic paraboloid and then plot it on a graph. Then you must split it into different parts. Derive the equation. Calculate the surface area using integration.

**Analysis:**

This analysis is most important when it comes to a business’s point of view. For instance, when manufacturing products of a hyperbolic paraboloid, one must optimise space to ensure consumer satisfaction.

**2.) Modelling the shape of an egg**

**Mathematics**:

The are various mathematical concepts when modelling the shape of an egg, one of the most important ones being **integration**.

**Procedure**:

Firstly, you would have to draw the egg and then plot it on a graph. Then you must derive the equation. Calculate volume and surface area using integration.

**Analysis:**

The geometrical properties of eggs – such as **volume and surface area** – have uses ranging from ecological, physiological and morphological studies in birds, to predictions of conditions in the poultry industry.

**3.) The correlation between fertiliser concentration and plant growth**

**Mathematics:**

Using mathematical concepts like **Pearson’s correlation**. This is a statistical tool that tests the strength of a linear correlation.

**Procedure:**

If you are a biology student, then this topic would be apt for you. Firstly, you must take at least 5 different pots with soil and a seed sown into it. Next, add varied concentration of fertilizer to each pot, as well as a control experiment. Measure plant growth over the span of 15 days (depending on the seed). Collect this data and use Pearson’s correlation to determine how strongly correlated the two variables are.

**Analysis:**

This research is especially important in finding out if the fertilizer has a significant impact on plant growth, as well as to find the optimum concentration

**4.) Modelling the pharmacokinetic profile of erythromycin**

**Mathematics:**

Using mathematical concepts such as **exponential functions** and integration. The pharmacokinetic profile is the way and the amount of time in which a drug moves and gets absorbed by your body.

**Procedure:**

In this case, you would have to collect secondary data. Using the data points, the half-life of erythromycin can be calculated. Depending on the result, you can figure out at which stage is the drug most active.

**Analysis:**

It is highly important to know at which stage the drug is most active from the consumer’s and producer’s perspectives in order to determine at what time to consume the medicine and how long its effects will last.

**5.) Modeling a cooling cup of tea**

**Mathematics:**

This IA is about finding the time it takes for a cup of tea to cool down using exponential functions.

**Procedure:**

To gather the data I needed to produce the graph a cup of tea cooling down needs to be measured. Using a temperature sensor and piece of graphing software a set of readings that measured the temperature of the tea can be produced. 3 exponential equations are derived with graphs and separate sets of data.

**Analysis:**

By comparing the average error difference of all 3 equations, one equation will be narrowed down. Using this equation, the time for the tea to cool down is calculated.

**6.) Is there a correlation between hours of study and exam grades?**

**Mathematics:**

Using bivariate statistics, one can find correlation between any two things that seem to be affecting one another. It would involve Pearson moment correlation coefficient and other statistical methods.

**Procedure:**

You will have to collect data for the same and make a scatter plot. Further calculate the required statistics and make the conclusions. Further exploration may include modelling of the data using various other functions.

**Analysis:**

There are various types of students, some work less and are still able to score well because of efficient work, and some are opposite. Does this mean the there is no relation between studying hours and grades? Using mathematical tools one can try to make a conclusion with the data collected from around them.

**7.) Optimization of profit of a start up**

**Mathematics**:

Calculus can be used to maximize profit in a business. Along with that Algebra can be used to find breakeven points, so that a person can work accordingly towards the profit goals.

**Procedure**:

First thing is to set up different expenses involved in a business and optimize all of them using calculus. Calculate the breakeven point. Further calculate the profit equation and optimize it.

**Analysis:**

This exploration will help you understand the various factors involved in a new business and the things that need to be taken care of. The math application to a start-up gives an insight on how you can build a business while considering every factor and planning profits for the long run.

**8.) Weaving a spider web**

**Mathematics:**

If you’ve every observed a spider web, you must have seen geometric patterns and symmetry. One can also make use of algebra and calculus to form a spider web.

**Procedure:**

Taking the center of the spider web to be at the origin, one can use Cartesian and polar coordinates to determine the equations of the entire web. Use of trigonometric functions will also be involved.

**Analysis:**

This exploration will help model real life scenarios by taking the example of a spider web. Also it will enable us to understand that how nature forms geometric and symmetric patterns around us with so much accuracy.

**9.) Photography and math**

**Mathematics:**

If you’re interested in photography, this could be your chance to explore math behind your camera. Geometric sequences, geometry and some Algebra will be used for this exploration.

**Procedure:**

The shutter speeds of camera form a geometric sequence, which in turn is related to the aperture of the lens. This affects the brightness of the pictures taken. Another aspect that can be explored is the rule of thirds which is used by photographers to take pictures, which comes from the well-known golden ratio.

**Analysis:**

This exploration will enable us to understand that how art and math are related. The working of a camera is mostly based on math.

**10.) Analysis of projectile motion on different planets (Varying gravity)**

**Mathematics:**

Projectile motion is modelled by quadratic equations. The path depends on the gravity, the angle of projectile and the initial speed.

**Procedure:**

The best way to go around this would be by using parametric equations for the projectile (Quadratics can also be used). Consider planets, the sun or even black holes for the analysis.

**Analysis:**

This exploration will give a great idea about how gravity affects life on various different kinds of bodies in space (hypothetically of course!).

**11.) Optimization of Packaging Materials**

**Mathematics:**

The theory of optimization to calculate the optimal packaging required for common products with cylindrical, cuboidal and conical shapes.

**Procedure**:

Derive the formulas for these shapes so that they have the least surface area and utilize minimum packaging materials. Using the formula, common products will be assessed on how much packaging materials are wasted in their production.

**Analysis**:

Using the calculations, evaluate different industrial products (conical in shape, cylindrically shaped, cuboidal products) to determine the optimality of their packaging methods.

**12.) Approximation of Phi**

**Mathematics**:

This investigation aims to explore the famous golden ratio known as Phi using various mathematical approaches like Trigonometry, Geometry (Line Segments), Quadratic Equation, Continued Fraction, Integration and Fibonacci Sequence.

**Procedure**:

- Line Segment – Draw a line segment and divide it into two segments. Each segment will result in a different ratio for the smaller line segment to the larger line segment, and of the larger line segment to the whole line segment.
- Quadratic Equation – Consider the two line segments and rearrange the equation which can be solved using quadratic formula.
- Fibonacci Sequence – Make a sequence by adding two previous numbers in succession starting with 0 and 1.

**Analysis**:

- Line Segment – Golden Ratio is obtained when two segments when two segment lengths have the same proportion as the proportion of their sum to the larger of the two lengths.
- Quadratic Equation – There will be two results obtained. However, the positive value will be the approximation to the phi value.
- Fibonacci Sequence – The ratio of two consecutive numbers in the sequence will approximate to Phi value.

**13.) Exploration of Euler’s Number “e”**

**Mathematics**:

This investigation aims to explore the Euler’s number “e”. Using multiple mathematical concepts to derive, calculate and approximate “e”. These include: probability, limits, sequence and series, continued fractions, and calculus.

**Procedure:**

Using the simple interest formula, the number of times the principal amount is compounded (n) derives the final amount after interest.

- Limits – Consider a positive integer, integrate the equation and then solve the integrals.
- Sequence and Series – Represent the exponent function by a series with coefficients that are unknown and then apply the property of the derivative of the exponent function. To calculate “e” substitute different values of k in the equation.

**Analysis:**

As the value of n increases the value of S (accumulated amount) comes closer to that of e. As n tends to infinity, S converges to “e”.

- Limits – Using graphs, plot the equation. From the graph it will be evident that there is a horizontal asymptote. Evidently, this asymptote is the line y = e.
- Sequence and Series – As the value of k increases the value of sum of the series will get closer to “e”.

**14.) Investigating Josephus Problem**

**Mathematics:**

In mathematics, the josephus problem is a theoretical problem associated with a specific counting-out game. RQ: To figure out which position to stand in, in order to be the last man standing by creating and applying different formulas through Josephus Problem.

**Procedure:**

To find out the Josephus problem, i.e. to figure out the position to stand in, in order to be the only man remaining in the circle by using different mathematical applications such as pattern recognition, solving problems using power of two. Also, derive a general formula for the Josephus problem and calculate the winning position for the Josephus problem.

**Analysis:**

Two patterns will be derived. In the first pattern all the winning positions were odd and in the second pattern the winning position is jumping by two numbers each time and resists at one at some point that is when a number is a power of two.

**15.) Discovering patterns within Pascal’s Triangle**

**Mathematics:**

One of the most interesting number patterns in Pascal’s Triangle.

**Procedure:**

In order to build the triangle, start with 1 at the top, then continue placing numbers below it in a triangular pattern. Each number is the numbers directly above it added together.

**Analysis:**

- First Diagonal will have just 1’s
- Second Diagonal will have the counting numbers (1,2.3. etc)
- Third Diagonal will have the triangular numbers
- Fourth Diagonal will have the tetrahedral numbers
- The triangle will be symmetrical
- Horizontal sums double each time (Power of 2)
- Each line is the exponent of 11.

**16.) Derivation of Kepler’s Law**

**Mathematics:**

Deriving Kepler’s Law utilising the principles of polar coordinates and calculus

**Procedure:**

Research the history of Kepler’s Law and implement the principle of polar coordinates and calculus to obtain the equations produced by the law in accordance to other principles and assumptions

**Analysis:**

Kepler’s law brought light on the movement and rotation of objects in the solar system and explains the positioning of the planets and moon.

**17.) Modelling the Surface Area of Jello**

**Mathematics**:

Calculating the surface area of jello using modelling techniques and tools along with calculus.

**Procedure**:

Modelling tools and techniques help provide equations that can also be correlated and compared to calculus equations that determine the surface area of the jello.

**Analysis:**

Modelling the surface area of jello is important in its manufacturing as it determines the ideal way to consume it while ensuring that customers are satisfied

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