IB Math Visualized

By Christopher Yu

(Math and Physics tutor at The Edge Learning Center)

Have you ever experienced difficulties in memorizing or understanding a mathematical formula or theorem? Was it because staring at it for no matter how long still gave you no clues and did not impress you? If so, how about trying to “decipher” them visually? Sounds promising? Please read on.

In this blog, I would like to show you three examples of visualizing a mathematical formula or theorem that you will come across in the IBDP Mathematics HL/SL curricula.

Pythagorean Theorem

Let’s take baby steps. Firstly, the Pythagorean theorem. It is the foundation of the topic “Trigonometry,” among many others, in both the HL and SL curricula The theorem relates the three sides of a right-angled triangle (with hypotenuse h):


It is not too hard to memorize it as it is short. However, for the curious ones or those who can make new knowledge stick only if they are convinced, you might want to ask why it is true.

Instead of a mathematically rigorous proof, which would likely overwhelm most of the students at first, the following depicts an elegant yet convincing “picture proof” of the theorem:IB math pythagorean theorem

IB math windmills theorem In words, the picture suggests that the theorem can be visualized as constructing a square of length h by putting together four identical copies of the right-angled triangles in question around a smaller square of length a – b. Other than being convinced now, even better, you probably have established a visual link between a windmill and the theorem, which might help you reinforce your memory about the picture proof.

As an aside, here is yet another fun time: Making Chinese Paper Windmill (DIY)

Infinite Sum

Next is infinite sum, which appears in the topic “Geometric Sequence and Series.” Given the first term u1 and the common ratio r of a geometric sequence, computing its infinite sum s8 is nothing more difficult than plugging numbers into the formula listed in the formula booklet, namely

S8 equation

provided that the common ratio satisfies the condition r1

Take the geometric sequence


as an example. Its first term is 1:2and its common ratio is 1:2. According to the formula, its infinite sum is 1:2:1-1:2, which equals to 1. Okay… but how does adding infinitely many positive numbers end up will a definite answer 1? It seems to be counterintuitive and difficult to “see” how such an infinite operation ends up with what the formula says.

What if now we consider a square of length 1? We first divide it into two identical halves. If we repeat the same process over and over again by dividing one of the two resulting pieces into two identical halves, we will end up with an infinite sequence of alternate rectangles and squares. The numbers in the following diagram indicate the areas of the first six rectangles/squares. It is easier to see and be convinced that the division process can go indefinitely (although we don’t want to do so). The sequence of areas is exactly the geometric sequence in our example. Whether you want to keep dividing or not, the total area of those rectangles and squares must be equal to the area of the square from which they are derived. Clearly, the area of the parent square is 1.

IB math Eye of Horus square

Fibonacci Squares

Finally, let’s take a look at a famous, classic sequence – Fibonacci numbers.

The Fibonacci numbers are a sequence of integers defined by the recurrence relation

Fn equation

with initial values


Here is one of the well-known Fibonacci identities:

E equation

In words, the sum of the squares of the first Fibonacci numbers equals the product of the n-th and the (n+1)-th Fibonacci numbers. This is merely a translation of the mathematical identity to an English description, and it probably does not help one to memorize it.

What if we think about the right hand side of the identity as a rectangle of size Fn by Fn+1? To make this concept more concrete, let’s consider the first 8 Fibonacci numbers.

F1-F8 table

The next term F9 is given by F8 + F7 = 21 + 13, which is 34.

Therefore the identity yields


Now let’s consider a rectangle of size 21 by 34:


The rectangle composes eight squares with lengths corresponding to the first eight Fibonacci numbers. For instance, the largest square on the left has length 21 and area  21^2. Pictorially, it is clear that the sum of the areas of the eight squares (the left hand side of the identity) is equal to the area of the rectangle that is nicely formed by putting the eight “Fibonacci squares” in a spiral configuration (the right hand side of the identity).

Try it out: Can you tell the sum of the squares of the first ten Fibonacci numbers by evolving the above 21 by 34 rectangle into a larger one with more squares?

Wrapping Up

To see is to believe, which is especially true for the visual learners. Even if you are not one of those, it does no harm for you to adopt it as an alternative way of learning.

IB math you have to see to believe