Constant questions and debates about how useful life is at the level taught in school and required for passing exams have not subsided so far. However, the usefulness of mathematics is often outside the surface. We not only count money and prices when planning purchases while in the supermarket – for this purpose, there is a calculator and many other applications that have appeared in our time and are always at hand. Mathematics teaches you to think more comprehensively and profoundly and solve everyday problems in often non-trivial ways.
Let’s start with a simple one: mathematics, whatever one may say, is primarily an account. We learn to add and subtract, multiply, and divide (SplashLearn, n.d. a) from the first grade at school; however, arithmetic abilities are not limited in their applied application only to these operations. For example, even without much emphasis, sooner or later, we begin to notice that numbers ending in 0 or 5 are sure to be divisible by 5 without a remainder. The signs of divisibility by 3 and 9 (IXL, n.d.) are not so obvious, but they also contribute to critical thinking, which saves time using a calculator. It seems, why do we need this knowledge, if all the same when counting large numbers, we will resort to the help of computers that will cope much faster?
Critical thinking teaches you to be open and mobile. You must have been wondering why you shouldn’t divide by zero and why this action should be avoided as if it were forbidden by the laws of nature. It is rare for a teacher to give you a simple answer to this question (Cuemath, n.d.), but it is the same interest that brings out your openness to new solutions, new information, and methods! After all, when choosing a new phone, we may wonder why there are no models, for example, with cameras on both sides of more than 50 megapixels. Perhaps you will find the answer to this question on the Internet, or maybe you will become the first to create such a device. As a result, you immerse yourself in the problem, looking for an understandable solution in the same way as when solving a mathematical example.
We see that the applied meaning of mathematics is hidden from us on a conscious level; it is not just working with numbers as such. So, in mathematics, starting from a certain level of knowledge, numbers are becoming less common; they are replaced by mysterious letters, becoming either the roots of complex equations or the names of various geometric units. Finding the source of even the most straightforward equation (Byjus, n.d.), we, first of all, learn not only how to count and how to solve such problems, but we again return to critical thinking. In our world, much is unknown to every person, much cannot be predicted, and much does not depend on us. Under these conditions, actions are always associated with risks, which means that decisions can be very different. By ordering the equation, translating all unknowns on one side of the equal sign, and all known numbers on the other, we learn to solve a variety of everyday problems similarly. After all, when leaving home for an important meeting, we always leave a little early to take into account various unforeseen circumstances, such as traffic jams. The straightforward calculation of the time is supplemented by several unknowns, which we have on the same side as the known indicators, to achieve a precise result after the equal sign – to arrive on time. And next time, we will already have experience in what the given unknowns are at least approximately, approximately, to solve such equations better each time.
When moving to more complex categories from arithmetic operations, mathematics can cause rejection at this stage due to a large number of vague terms. The laws of geometry, correlation, and research of functions and variables become a dark forest for many underachieving students, and having missed the basics in learning at this step, they lose motivation further. So let’s break down these basics together with helpful resources and illustrative examples. Let’s start with the term “correlation,” (Benedict, 2014), which refers to the dependence of one variable on another. For example, the more points you score in your favorite game in one round, the higher you become in the leaderboard, or the more money you make doing something, the more likely you will afford your favorite cake for dessert today. These illustrations are an example of direct correlation: the higher the value of one variable, the other increases accordingly. However, this is only sometimes the case. Looking at the thermometer outside the window on a winter evening, the lower the degrees you see, the more clothes you will wear before going outside. Another example is that the more smartphones a company sells, the less inventory it has in its warehouses. These illustrations demonstrate an inverse correlation: as one variable rises, the other falls.
Why is this kind of analysis necessary? First of all, you will be able to see the usefulness in planning any actions, from the simple examples above to predicting the investment effectiveness of your savings. However, a simple understanding of this mechanism is not limited to its potential utility. Much more often in mathematics, illustrative examples of data visualization are used: the ability to understand, read and interpret them is vital and, among other things, is tested on the FSA exam. Let’s go through the main features of data visualization because to present any of your projects to the public, it is much more efficient to use pictures in a presentation than text, even the best formatted! The tabular view (Khan Academy, n.d.) is one of the first to be studied and involves the cumulative placement of data into cells of columns and rows. If you are fond of sports, then you probably analyze the position in the table of your favorite team and consider how many points it needs according to victories, games played, and goal difference, in the case of football, to catch up with the nearest rival. The ability to work with tables opens up significant opportunities in the future: you can master Excel as one of the best tools for working with extensive tabular data. Please do not underestimate the role of mathematics here; it is fundamental in this matter!
It is also possible to arrange a list of purchases or guests for a party using tables. However, there is also a graphical representation using charts. The most popular is the graph (Skillsyouneed, n.d.), which best reflects the correlation phenomenon. Let’s see how it works. The graph is located on the coordinate plane (SplashLearn, n.d. b), representing a horizontal ruler aligned with the thermometer vertically. The graph points are located in a certain way, each corresponding to the value on the ruler – the x coordinate, and on the thermometer – the y coordinate. The graph itself is just a line connecting these points. Thus it is possible to explore the correlation and predict it by substituting the values of any coordinate.
However, fundamental understanding requires knowing the definition of a mathematical function (Math is Fun, n.d. a). The function describes the graph; for example, the linear function y=x+1 looks like a straight line passing through each coordinate that satisfies the given condition. The function y=x2 looks like a parabola, and there are many such examples; you can read them all here. To adequately construct a function graph in the case of a straight line, it is enough to select two points: substitute the x coordinate in the equation and calculate y, after which draw a straight line through two points. However, in reality, including in the exam, you will encounter the phenomenon of correlation when a set of values does not precisely satisfy any function.
For example, due to the unstable economic situation, you decide to track the price of a new smartphone you want to buy every day. In this case, the “ruler” x is the time when the days are clearly counted. On the “thermometer” y, you record the price readings in a given period. As a result, you get a broken line that is not described by a mathematical function known to you. However, you may notice that over time, despite various deviations, prices go up, or vice versa, down, along a set of points that move in the appropriate direction. In this case, we can conclude that there is a direct relationship or correlation between the passage of time and the price of a smartphone. Here, you need to build a so-called trend line (Baby Pips, n.d.) that runs as close as possible to these points and shows the price trend to rise or fall. And now this line, which will be an ordinary straight line, can be described by a mathematical function with concrete real numbers! With it, you can predict the approximate price of a smartphone on the date of the proposed purchase and better manage the financial records of your funds.
The equations you solve gradually acquire new inputs, unique representations of numbers, or arithmetic operations. Above, you may have noticed the phrase “real numbers” what is it? We are used to working in reality with natural, negative numbers and zero, integer, and fractional values. By the way, a fraction (SplashLearn, n. d. c) is just one of the forms of the division sign, and it should be taken that way. We add that percentages (Math is Fun, n. d. b) also originate in fractions. However, all numbers in the world in mathematics are described by a set of real numbers, including irrational ones. Look at the root: the mysterious symbol of the “root,” (Mometrix, n.d.), which can sometimes be found above any familiar number. If you meet a root, in simple terms, you need to determine which two identical numbers multiplied by each other; what did you end up with a radical expression? In the case of the root of 9, you immediately give the answer: 3. However, what if we see the root of 28? 1.07? In this case, it is impossible to get not only an integer but even a number in the form of a decimal fraction with a finite number of decimal places. Therefore, for convenience of notation, the root is used, and such numbers are irrational (Math is Fun, n.d. c). You should be able to simply identify them – a calculator can handle an approximate calculation. This ability will allow real life to correlate the time spent on work with the potential degree of accuracy of the result: only some businesses are worth the time spent.
The root, among other things, results from another essential mathematical operation – the degree of a number (Math is Fun, n.d. d). In fact, the degree indicates how many times a given number needs to be multiplied by itself. This action has several relatively simple properties, allowing you to transform highly complex expressions at first glance into a straightforward solution. You can verify this by solving problems from the exam, but then you can apply this skill in real life. A seemingly complex problem can be solved with a series of small steps, each of which you are familiar with. Thus, even the most ambitious goals in life can wait until the hour of their achievement: a small step for a person is often a big step for all humankind.
Solution strategies for this VFT lie not only in the search for motivation and the acquisition of skills important for real life but also in the availability of resources for further self-learning. Distant learning can help you now save more time than you used to spend on the road and transfers! Even if it’s only a matter of minutes, they can be spent on extracurricular preparation for the exam! Moreover, face-to-face mentoring in a full-time classroom setting has its advantages, but there may not be enough teachers for everyone. Use this time to fill in the gaps in the material covered and move on if the current topic has already become crystal clear to you. There are many resources (Edgenuity, n.d.; i-Ready, n.d.; Imagine Learning, n.d.; MathCab, n.d.) that you can use on your own without the help of a teacher. At the same time, mentors can now manage their time more productively, using video conferencing and other technology opportunities to assess students’ knowledge more adequately and timely. With close and mutually effective cooperation, much more results can be achieved with proper motivation. And as we have already said, mathematics is exciting and valuable! Let’s continue to consider specific examples and tasks you can meet on the exam.
References
Baby Pips. (n.d.). Trend Line Definition. Web.
Benedict. (2014). Correlation – The Basic Idea Explained [Video]. YouTube. Web.
Byjus. (n.d.). Solution of Linear Equation. Web.
Cuemath. (n.d.). Division by Zero. Web.
Edgenuity. (n.d.) Algebra 1. Web.
i-Ready. (n.d.). Maths. Web.
Imagine Learning. (n.d.) Maths. Web.
IXL. (n.d.). Divisibility Rules. Web.
Khan Academy. (n.d.). Modeling with tables, equations, and graphs. Web.
Math is Fun. (n.d. a). What is a function?. Web.
Math is Fun. (n.d. b). Percents. Web.
Math is Fun. (n.d. c). Irrational Number. Web.
Math is Fun. (n.d. d). Degree (Algebra). Web.
MathCab. (n.d.). FSA Practice Tests. Web.
Mometrix. (n.d.). Roots. Web.
Skillsyouneed. (n.d.). Graphs and charts. Web.
SplashLearn. (n.d. a). Column Method – Definition with Examples. Web.
SplashLearn. (n.d. b). Coordinate Plane – Definition with Examples. Web.
SplashLearn. (n.d. c). Fraction – Definition with Examples. Web.