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## Pascal's Triangle Collection...

Maple Learn

A new collection has been released on Maple Learn! The new Pascal’s Triangle Collection allows students of all levels to explore this simple, yet widely applicable array.

Though the binomial coefficient triangle is often referred to as Pascal’s Triangle after the 17th-century mathematician Blaise Pascal, the first drawings of the triangle are much older. This makes assigning credit for the creation of the triangle to a single mathematician all but impossible.

Persian mathematicians like Al-Karaji were familiar with the triangular array as early as the 10th century. In the 11th century, Omar Khayyam studied the triangle and popularised its use throughout the Arab world, which is why it is known as “Khayyam’s Triangle” in the region. Meanwhile in China, mathematician Jia Xian drew the triangle to 9 rows, using rod numerals. Two centuries later, in the 13th century, Yang Hui introduced the triangle to greater Chinese society as “Yang Hui’s Triangle”. In Europe, various mathematicians published representations of the triangle between the 13th and 16th centuries, one of which being Niccolo Fontana Tartaglia, who propagated the triangle in Italy, where it is known as “Tartaglia’s Triangle”.

Blaise Pascal had no association with the triangle until years after his 1662 death, when his book, Treatise on Arithmetical Triangle, which compiled various results about the triangle, was published. In fact, the triangle was not named after Pascal until several decades later, when it was dubbed so by Pierre Remond de Montmort in 1703.

The Maple Learn collection provides opportunities for students to discover the construction, properties, and applications of Pascal’s Triangle. Furthermore, students can use the triangle to detect patterns and deduce identities like Pascal’s Rule and The Binomial Symmetry Rule. For example, did you know that colour-coding the even and odd numbers in Pascal’s Triangle reveals an approximation of Sierpinski’s Fractal Triangle?

Or that taking the sum of the diagonals in Pascal's Triangle produces the Fibonacci Sequence?

Learn more about these properties and discover others with the Pascal’s Triangle Collection on Maple Learn. Once you are confident in your knowledge of Pascal’s Triangle, test your skills with the interactive Pascal’s Triangle Activity

## Goldbach's Conjecture on Maple Learn...

Almost 300 years ago, a single letter exchanged between two brilliant minds gave rise to one of the most enduring mysteries in the world of number theory.

In 1742, Christian Goldbach penned a letter to fellow mathematician Leonhard Euler proposing that every even integer greater than 2 can be written as a sum of two prime numbers. This statement is now known as Goldbach’s Conjecture (it is considered a conjecture, and not a theorem because it is unproven). While neither of these esteemed mathematicians could furnish a formal proof, they shared a conviction that this conjecture held the promise of being a "completely certain theorem." The following image demonstrates how prime numbers add to all even numbers up to 50:

From its inception, Goldbach's Conjecture has enticed generations of mathematicians to seek evidence of its legitimacy. Though weaker versions of the conjecture have been proved, the definitive proof of the original conjecture has remained elusive. There was even once a one-million dollar cash prize set to be awarded to anyone who could provide a valid proof, though the offer has now elapsed. While a heuristic argument, which relies on the probability distribution of prime numbers, offers insight into the conjecture's likelihood of validity, it falls short of providing an ironclad guarantee of its truth.

The advent of modern computing has emerged as a beacon of progress. With vast computational power at their disposal, contemporary mathematicians like Dr. Tomàs Oliveira e Silva have achieved a remarkable feat—verification of the conjecture for every even number up to an astonishing 4 quintillion, a number with 18 zeroes.

Lazar Paroski’s Goldbach Conjecture Document on Maple Learn offers an avenue for users of all skill levels to delve into one of the oldest open problems in the world of math. By simply opening this document and inputting an even number, a Maple algorithm will swiftly reveal Goldbach’s partition (the pair of primes that add to your number), or if you’re lucky it could reveal that you have found a number that disproves the conjecture once and for all

## Directional Displacement Problems...

Maple Learn

2-dimensional motion and displacement are some of the first topics that high school students learn in their physics class. In my physics classes, I loved solving 2-dimensional displacement problems because they require the use of so many different math concepts: trigonometry, coordinate conversions, and vector operations are all necessary to solve these problems. Though displacement problems can seem complicated, they are easy to visualize.
For example, below is a visualization of the displacement of someone who walked 10m in the direction 30o North of East, then walked 15m in the direction 45o South of East:

From just looking at the diagram, most people could identify that the final position is some angle Southeast of the initial position and perhaps estimate the distance between these two positions. However, finding an exact solution requires various computations, which are all outlined in the Directional Displacement Example Problem document on Maple Learn.

Solving a problem like this is a great way to practice solving triangles, adding vectors, computing vector norms, and converting points to and from polar form. If you want to practice these math skills, try out Maple Learn’s Directional Displacement Quiz; this document randomly generates displacement questions for you to solve. Have fun practicing!

## New Proof of The Pythagorean Theorem!...

Maple Learn

In March of 2023, two high school students, Calcea Johnson, and Ne’Kiya Jackson, presented a new proof of the Pythagorean Theorem at the American Mathematical Society’s Annual Spring Southeastern Sectional Meeting. These two young women are challenging the conventions of math as we know it.
The Pythagorean Theorem states that in a right angle triangle, the sum of the squares of the legs is equal to the square of the hypotenuse:

The theorem has been around for over two thousand years and has been proven hundreds of times with many different methods. So what makes the Johnson-Jackson proof special? The proof is one of the first to use trigonometry.
For years, mathematicians have been convinced that a trigonometric proof of the Pythagorean Theorem is impossible because much of trigonometry is based upon the Pythagorean Theorem itself (an example of circular reasoning).
That said, some results in trigonometry are independent of the Pythagorean Theorem, namely the law of sines, and the sine and cosine ratios; the latter is a result that 12-year-old Einstein used in his trigonometric proof of the theorem.
Though all the details of the Johnson-Jackson proof have not been made public, there was enough information for me to recreate the proof in Maple Learn. The idea of the proof is to construct a right angle triangle with an infinite series of congruent right angle triangles (the first of which has side lengths a, b, and c). Then, using the sine ratio, solve for the hypotenuse lengths of each small congruent triangle. To explore this construction see Johnson and Jackson’s Triangle Construction on Maple Learn.

Next, find the side lengths of the large triangle (A and B) by evaluating an infinite sum (composed of the hypotenuse lengths of the small congruent triangles). Finally, apply the law of sines to the isosceles triangle made from the first 2 congruent triangles. After simplifying this expression, the Pythagorean relationship (c2 = a2 + b2) emerges.

To see more details of the proof, check out Johnson and Jackson’s Proof of Pythagorean Theorem on Maple Learn.
This new proof of the Pythagorean Theorem shows that discoveries in math are still happening and that young people can play a big role in these discoveries!

## Geometric Transformations...

Maple Learn

A geometric transformation is a way of manipulating the size, position, or orientation of a geometric object. For example, a triangle can be transformed by a 180o rotation:

Learning about geometric transformations is a great way for students, teachers and anyone interested in math to get comfortable using x-y coordinates in the cartesian plane, and mapping functions from R2 to R2. Understanding geometric transformations is also an essential step to understanding higher-level concepts like the Transformations of Functions and Transformation Matrices.
Check out the Geometric Transformations collection on Maple Learn to learn about this topic. Start out by playing with the Geometric Transformations Exploration document to build intuition about how objects are affected by each of the four transformation types: Dilation, Reflection, Rotation, and Translation. Once you are confident in your skills, try using the Single Geometric Transformation Quiz to test your knowledge.
For those looking to expand their understanding of geometric transformations, the Combined Transformations Exploration document will let you explore how multiple transformations and the order of said transformations affect the final form of an object. For example, the blue polygon can be transformed into 2 different pink polygons depending on whether the reflection or rotation is performed first: