A link between Math and Life
By Leo Lam
(Test Preparation (ACT/SAT/SSAT), Math and Physics tutor at The Edge Learning Center)
Not many students would call Mathematics their favorite subject. After all, it encompasses various fields of studies, and students are exposed to a tremendous amount of knowledge in a restricted amount of time. So it does not surprise me when my students, from both Test Prep courses and one-on-one lessons, ask, “what is the point of learning (insert math topic)?”
Mathematics plays a very important role in our academic life: there are two sections of Math in the new SAT, it’s the longest – and arguably the hardest – section in the ACT, and it’s a mandatory subject in high school, whether you are going through the U.S. system or IB. Yet it is hard to pinpoint what benefit the knowledge will bring us later on after graduation.
Therefore, it makes sense that the biggest roadblock for students to really get into the subject is this: why should I bother when I won’t need it? It’s a tough question, something that even the most caring teachers have a hard time answering. Mathematics is a very interesting and unique subject. Unlike most science subjects that are based on physical observations and experiments, many mathematics concepts are, well, just concepts. They are abstract ideas that do not manifest in reality. For example, let us take a look at the building blocks of solid geometry. Before we study objects in space, we learn about points, lines, and planes. We think of objects as three-dimensional with length, width and height, while planes and shapes are two-dimensional, consisting of length and width only. Drop down to a line, and we are in the realm of one-dimension: only length. Looking at a single point, we have reached zero-dimension. Theoretically, zero-dimension has no length, width, nor height to speak of. Yet when students learn about geometry, there it is, a point is drawn on the board. We have made something that doesn’t physically exist materialized. So students have a valid argument when they ask, “what’s the point (no pun intended)?”
This makes it very hard for teachers to encourage students to “care” about Math, when we are dealing with topics that mostly exist in our imagination and not around our surroundings. Yet mathematics was originally built around our world. We began with the idea of quantity: attaching “names” for different amounts to solidify the idea of counting. Using these numbers, we developed rules to put them together, or arithmetic. Eventually, we realized we would need to work backwards to find some missing information, and algebra was born. Working with our environment, we found the need to measure objects around us, and the concepts of geometry were cemented. As more and more ideas were being created and polished, various theorems, equations, and formulae emerged. It becomes quite customary for us to “memorize” instead of fully discover the meaning of these ideas. After all, why should we have to reinvent the wheel when it is readily available? However, I believe it is this disconnect between “how to solve a question” and “why do I solve the question this way” that is discouraging students from enjoying Mathematics.
So today I would like to share a few of my tips on how to “care” about the topics one would be encountering during their school years. While many of the modern, advanced topics are purely theoretical (geometry in the nth-dimension, analysis with complex variables, to name a couple), concepts that are placed in school curricula are more “down to earth.” It may not answer the question “what’s the point”, but I hope that after reading this, you can find some comfort in thinking that, “it’s not as unrelated as I originally thought.”
Domain and Range of a Function
The concept of function is discussed in a more elaborate manner to prepare students for specific ideas like graphs, trigonometry, and calculus. A common analogy for function is a machine that takes some kind of input to produce some kind of output. Many students find this concept challenging due to the numerous forms it can take, and the many terms that are involved (domain, range, variable, exception, inverse, to name a few).
We are taught that domain is the set of all possible values for the independent (or input) variable, and that range is the set of all possible values for the dependent (or output) variable. Common way to remember this is to say domain is , while range is . Simple enough. But how do they work? It turns out we learn about domain and range at a much earlier time in our life.
One of the definitions for domain is “a territory over which dominion is exercised.” We encountered this when we were toddlers. As we learned to walk, we started exploring around the house, our domain (people living with children know it is the toddlers who exercise the dominion, not the adults). But rarely were we allowed to roam free; more likely than not, there would be barriers blocking us from entering certain areas (like the kitchen). Our domain was restricted. Should we cross these walls, there would be dire consequences (not speaking from personal experiences). Just like in life, domain in a function acts as a boundary between what is allowed and what is not. The barriers are the asymptotes (which can be more than just vertical and horizontal if we look beyond the domain and range). Try to cross these invisible walls and we end up with undefined results.
The range is the consequence. Based on our behaviors, we receive different reactions; some acts would result in praises, while others would reciprocate in punishments. You wouldn’t actually know the outcome unless you try to behave a certain way (though you should have a good idea which way the result would lean toward). A function behaves similarly, which makes finding the range more complicated than finding the domain; you actually have to input each value from the domain to clearly understand all the possible values in the range. However, if we are able to visualize the input-output pairs and identify a trend, it becomes easier to find the range. Thus, a visual representation of these pairs, called a graph, becomes a tremendously helpful tool to analyze functions.
Many students have been exposed to this concept without even realizing it. When a question asks you to show a certain idea, it is actually asking you for a proof.
Despite the fact that a proof does not find an “answer”, it is one of the most important tools in the development of mathematics. Anyone can make a claim, and likely provide a crude example to support that claim. However, it doesn’t make it true for all cases. A proof is used to show that the claim works at all instances, not just some specific ones. This makes proving an invaluable asset in mathematics.
We all have to prove ourselves during different stages of our lives. You would like more freedom from your parents? It’s unlikely that you would get it magically. Neither can you ask them to give you that freedom first so you can show them how well you deserve it. Instead, you would be asked to “prove” you are responsible enough before such demand is granted. And how does one prove responsibility? Definitely not through one or two acts. You provide numerous examples, argue logically how these consistent acts should lead to less worry from your parents, thus they can delegate less time monitoring your and more time enjoying themselves. With enough conviction, you would get your freedom. (This is just an example. Every parent is different; your results may vary.)
Mathematical proofs inherently work the same way. They don’t magically happen, and you can’t start with assuming your statement is true. You have to start with something else. Work with preexisting facts and knowledge, and apply logic to lead to your next step. As long as you have done each step correctly, you will reach your desired result.
If you have gone through your trigonometry lessons without forcing yourself to memorize SOHCAHTOA, I commemorate you. The rest of us rely on this mnemonic phrase to get through the maze around the right-angle triangles (here is a song to help you remember). We train ourselves to remember the differences between sine, cosine, and tangent, and learn to identify when to use these functions. Turns out that trigonometry didn’t even begin as studying triangles. Instead, it started off as calculating the position of the various objects in our heavenly bodies (i.e. the sky).
The key about trigonometry is the idea of similar triangles. When two triangles have the same internal structure (their corresponding angles have the same measurements), their corresponding pairs of lengths will have the same ratio. This becomes very helpful when we want to look at the structure of a ginormous triangle, like an imaginary one between the earth, a star, and another planet. Instead of actually measuring the distance in between, we find the angles that are formed, and create a smaller triangle with the same internal angles. Miniaturizing the scale makes it much easier to comprehend the structure without losing the accuracy.
But what’s more important is the fact that we are specifically looking at right-angle triangles. Their rigidity ensures a structure that we can examine easily. For starter, the three sides are bounded by a simple relationship: the Pythagorean Theorem (Christopher, one of our awesome Math teachers here at The Edge, wrote a blog that gives us a quick glimpse on how to prove the theorem. You can find it here). Hence, with only two sides, we would have a unique triangle (two triangles having the exact same lengths are congruent. You can play with the widget here to see how congruent triangles work, and find out when are they not guaranteed to be congruent). This way, looking at any pair of sides will give you a unique ratio for that particular triangle. And that is basically the purpose of Sine, Cosine, and Tangent; they are ratios between two sides.
Once we understand this core mechanic behind the trigonometric functions, the application becomes straight forward; it’s a conversion between the lengths of the right-angle triangle to its angle, and vice versa. Think of it as taking the ACT, getting 60 questions correct on the Math section, which would convert to a score of 36 in the section. While the conversion between the raw score and the section score can vary between each test on the ACT, the conversion for the trigonometric functions are fixed: a sine ratio of a half is converted to an angle of 30° and vice versa.
There are many more ideas in Mathematics that can be hard to understand when you encounter them for the first time. But don’t worry, there is always a way to make some sense out of it. Come join me so I can help you link these ideas with our daily life, so they are no longer a beast but a cuddly puppy that you can easily handle.