IB Math Visualized
(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.
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.
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)
Next is infinite sum, which appears in the topic “Geometric Sequence and Series.” Given the first term and the common ratio of a geometric sequence, computing its infinite sum is nothing more difficult than plugging numbers into the formula listed in the formula booklet, namely
provided that the common ratio satisfies the condition
Take the geometric sequence
as an example. Its first term is and its common ratio is . According to the formula, its infinite sum is , 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.
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
with initial values
Here is one of the well-known Fibonacci identities:
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 by ? To make this concept more concrete, let’s consider the first 8 Fibonacci numbers.
The next term is given by + = 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 . 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?
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.