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Who Says There Is No Relation Between Algebra And Daily Life?
During my tutoring sessions many students ask themselves that where do we use all these variables (x, y, z, n, etc.) in our daily life? My students are spot on as they don’t see any use of these variables or algebraic expressions in their lives directly. But, I tell you that all algebra and its concepts are invented to help us in our daily life and algebra is our best companion. They wonder and ask me for more explanations. Then I systematically explain how algebra is embedded in our everyday life using concrete examples. One such example that I want to share with my dear readers is presented below;
The basics, algebra begins.
The basic concepts in algebra are
- The variables
- Constants and
- Algebraic expressions.
Let’s take the following example from a daily life situation to understand all the previous terms in algebra;
Consider every weekend, Arthur; a 9th grade student begins helping his brother in his landscaping business. Every time Arthur goes with his brother to work, Arthur pays him $60 to stay with him all day to do a little cleaning work at the workplace.
Sometimes there are two customers right next to each other where Arthur can work on the lawnmower and so his brother pays him $25 more for every lawn Arthur cuts.
Consider the first Saturday, Arthur didn’t get a chance to work on the lawnmower.
Can you guess how much he won for the day?
Easy! Your answer might be $60, because he gets his basic cleaning services and there’s no money to work on the lawnmower.
The next day is Sunday and Arthur had the opportunity to work on the machine for two customers and cut two lawns.
Can you say how much money Arthur won this Sunday?
Next weekend, which is the second Saturday, Arthur cut five lawns, what are his earnings for the day?
The next day is Sunday and Arthur cut a lawn. What are your winnings for this Sunday? You probably know the answers to all of the above questions.
But, I want to stop here for explanations to show clearly that, how this activity of everyday life is algebra. For this we are realizing a very important concept of algebra in this example.
I want to show you the work you have done in your brain to find the answers to all the questions above. So, below are all the explanations;
Arthur’s earnings have two parts.
The first part is a fixed part, which is $60 per day, he worked for his brother to do cleaning work.
The second part is not fixed and depends on the number of lawns you mow, if you have the opportunity to use the mower.
Your thought process is as follows:
Arthur’s earnings = (fixed part) ADDED TO (25 times the number of lawns cut by Arthur)
Arthur’s earnings = 60 + 25 x 0 = 60 + 0 = $60
25 is multiplied by zero, since he did not mow the lawn this Saturday.
Arthur’s earnings = 60 + 25 x 2 = 60 + 50 = $110
25 is multiplied by 2, since he cut two lawns this Sunday.
Arthur’s earnings = 60 + 25 x 5 = 60 + 125 = $185
25 is multiplied by 5 because Arthur cut 5 lawns this Saturday.
Arthur’s earnings = 60 + 25 x 1 = 60 + 25 = $85
25 is multiplied by 1 since only one lawn is cut this Sunday.
Is Arthur’s income the same for each day?
The answer is no. The earnings are not the same; they are different for different days. As you already know that something that changes in mathematics is called a “variable”. Also the dictionary meaning of variable is changing. Therefore, Arthur’s earnings can be represented by a variable.
Now the mathematicians have their options, they can say, “Arthur’s earnings are changing.”
Isn’t that a very long sentence to use in math problems?
Yes, this is a long sentence to represent a variable which is, Arthur’s earnings.
So the world’s mathematicians agreed on a standard. This standard is to represent variable quantities or variable activities with letters of the alphabet. Lowercase letters are most often used to represent variables.
In our example we can represent Arthur’s earnings with the letter “e”. It’s very important to remember that Arthur’s earnings for a particular day are always a dollar figure, but that figure keeps changing every day that Arthur works. So we need a common representative for the earnings every weekend, which is a variable.
Also, Arthur’s earnings for the next few weekends are unknown until he actually works the next few days. So we need to represent this unknown amount of money by a variable. It can be said that this variable is; “Arthur’s earnings are unknown until he has finished his work for a particular day.”
On the other hand, we can choose a lowercase letter to represent the entire sentence above. Therefore, the mathematicians opted for the second option. Therefore, we choose the letter “e” to represent Arthur’s earnings for any day of a weekend.
Also, as you know Arthur’s income (e) depends on the number of lawns he mows, which again is not fixed for the day. In other words, the number of lawns cut by Arthur is another variable in our example. And we can represent it with any letter other than “e” (since two different variables need different symbols), of the alphabet. Consider that the number of lawns cut in one day by Arthur is represented by the letter “n”
Finally, we write the two variables;
Arthur’s earnings for one day = e
Number of lawns he mows = n
That’s all; there are two variables (changing activities) in our example. Now, I want to go back to your thought process. There is something common (in terms of the mathematical operations of plus, minus, multiply or divide) in all these earnings calculations for the first and second Saturday and Sunday.
To find Arthur’s earnings (e), add 60 to 25 times the number of lawns Arthur cut. Isn’t this process common for all days to calculate Arthur’s income? Yes, it is. This common relationship between the pattern of earnings is actually algebra, and understanding and representing these kinds of relationships is understanding and representing algebra.
Mathematically, we can write the above thought process as follows:
Earnings for the day = 60 + 25 x Number of lawns cut
Above is an example of an algebraic relationship between two variables.
As you already know that the earnings of the day is not fixed and is denoted by the letter “e”, also the number of lawns cut is not fixed and is denoted by the letter “n”. Therefore, the above algebraic relationship can be rewritten using symbols to simplify, as shown below:
e = 60 + 25 xn
Remember that there is no need to show the multiplication sign between the number and its variable as it is understood for mathematical purposes. Therefore, “25 xn” and “25n” represent the same number. Thus, our relationship comes to;
e = 60 + 25n
We have achieved a simple algebraic expression between two variables by taking a situation from everyday life.
Note that our variables (changing activities or unknown activities) are as follows;
1. Arthur wins the day. Since his income is not the same every day he works, we can say that his daily income is changing or unknown until he finishes his day’s work. Any unknown or changing activity is called a variable in mathematical language, so Arthur’s daily income is a variable and we have used the letter “e” to represent it.
2. Number of lawns Arthur cut in one day. Because the number of lawns he cut is not the same for each day he works. This is the second variable and we have used the letter “n” to represent it.
Notice that Arthur’s earnings (e) depend on the number of lawns he cut (n), so we have one variable dependent on the other.
“n” is an independent variable as it does not rise or fall with earnings, it actually drifts earnings up or down. Therefore e is the dependent variable.
Rewrite our algebraic expression as follows:
e = 60 + 25n
“e” and “n” are the variables and now you know what a variable is.
The fixed value 60 is called the constant term; remember that, constant terms are numbers without any variables.
25 the multiplier of “n” and is called the coefficient of “n”.
Note that e is alone. In algebraic relations if a variable is written alone, it gets the coefficient U. Yes, “e” means “1e” and “-e” means “-1e”.
Algebra is a branch of mathematics that deals with changing or unknown activities (variables) in our daily lives.
A variable is represented by a letter (most often a lowercase letter) of the alphabet.
Variables represent numbers, but those numbers are unknown until the right time or certain conditions are met. This is why variables are replaced by numbers in algebraic expressions or their values must be found out.
A constant term in an algebraic relation is a fixed number. As in the example given, Arthur knows that he will be paid $60 for each day he does cleaning work with his brother.
A coefficient is a number that is multiplied by the variable. In the example given, 25 is multiplied by n, which is the number of lawns cut by Arthur. Therefore, 25 is the coefficient of n.
If there is only one variable (no number in front) in an algebraic expression, that means it has the coefficient “ONE” which is not shown and understood in the math.
I hope this helps you make algebra your best friend like many of my students do.
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