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Who Says There Is No Relation Between Algebra And Daily Life?
During my tutorial sessions, many students ask where do we use all these variables (x, y, z, n, etc.) in our daily lives? My students are in their shoes because they don’t directly see the use of these variables or algebraic expressions in their lives. But, I tell them that all algebra and its concepts are invented to help us in our daily life and algebra is our best friend. They question themselves and ask me for more explanations. Then I systematically explain to them how algebra is rooted in our daily lives using concrete examples. One such example that I want to share with my valued readers is given below;
The basic concepts, algebra begins with.
The basic concepts in algebra are
- Constants and
- Algebraic expressions.
Let’s do the following example from a daily life situation to understand all the above terms in algebra;
Consider every weekend, Arthur; a 9th grader starts helping his brother with his landscaping business. Every time Arthur goes to work with his brother, he pays him $60 to stay with him all day to do small cleaning jobs on the job site.
Sometimes there are two customers side by side, where Arthur can work on the lawn mower and for this his brother pays him $25 more for each lawn Arthur mows.
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 paid for his basic cleaning services and no money to work on the lawn mower.
The next day is Sunday and Arthur has had the chance to work on the machine for two customers and he has mowed two lawns.
Can you tell how much money Arthur won this Sunday?
Next weekend, that is to say the second Saturday, Arthur has mowed five lawns, what are his earnings for the day?
The next day is Sunday and Arthur has mowed a lawn. What are his earnings for this Sunday? You probably know the answers to all of the questions above.
But, I want to stop here for explanations to show clearly that, how this activity of daily life is algebra. For that, we accomplish a very important algebra concept in this example.
I want to show you the work you have done in your brain to find the answers to all of the above questions. So below are all the explanations;
Arthur’s earnings have two parts.
The first part is a fixed part, which is $60 for the day he worked for his brother to do the cleaning job.
The second part is not fixed and depends on how many lawns he mows, if he has the possibility to use the mower.
Your thought process is as follows:
Arthur’s income = (Fixed part) ADDED TO (25 times the number of lawns mowed by Arthur)
Arthur’s winnings = 60 + 25 x 0 = 60 + 0 = $60
25 is multiplied by zero since he did not mow the lawn this Saturday.
Arthur’s gain = 60 + 25 x 2 = 60 + 50 = $110
25 is multiplied by 2 since he mowed two lawns this Sunday.
Arthur’s winnings = 60 + 25 x 5 = 60 + 125 = $185
25 is multiplied by 5 because Arthur mowed 5 lawns this Saturday.
Arthur’s gain = 60 + 25 x 1 = 60 + 25 = $85
25 is multiplied by 1 because only one lawn is mowed this Sunday.
Is Arthur’s income the same every day?
The answer is no. The earnings are not the same; they are different for different days. As you already know, something that changes in mathematics is called a “variable”. Also, the dictionary meaning of the variable changes. Therefore, Arthur’s earnings can be represented by a variable.
Now the mathematicians have their options, they can say, “Arthur’s income changes.
Isn’t that a very long sentence to use in math problems?
Yes, that’s a long sentence to represent a variable which is, Arthur’s earnings.
Thus, the mathematicians of the world have agreed on a standard. This standard consists of representing variable quantities or variable activities by letters of the alphabet. Most often, lowercase letters are used to represent variables.
In our example, we can represent Arthur’s winnings with the letter “e”. It’s very, very important to remember that Arthur’s income for a particular day is always a dollar number, but that number continues to change each day that Arthur works. Therefore, we need a common representative for all weekend earnings, which is a variable.
Also, Arthur’s earnings for the upcoming weekends are unknown until he actually works in the days ahead. Therefore, we need to represent this unknown amount of money by a variable. We can say that this variable is; “Arthur’s earnings are unknown until he has completed his work for any given day.”
On the other hand, we can choose a small letter to represent the whole previous sentence instead. Thus, the mathematicians opted for the second choice. Therefore, we choose the letter “e” to represent Arthur’s earnings for any day of a weekend.
Also, as you already know, Arthur’s income depends on the number of lawns he has mowed, which is also not fixed for the day. In other words, the number of lawns mowed by Arthur is another variable in our example. And we can represent it by any letter other than “e” (because two different variables require different symbols), of the alphabet. Consider that the number of lawns mowed in a day by Arthur is represented by the letter “n”
Finally, let’s write the two variables;
Arthur’s gain for one day = e
Number of lawns it 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 payout calculations for the first and second Saturdays and Sundays.
To find Arthur’s earnings (e), 60 is added to 25 times the number of lawns Arthur has mowed. Isn’t this process common every day to calculate Arthur’s winnings? Yes it is. This common relationship between the pattern of earnings is actually algebra, and to understand and represent these kinds of relationships is to understand and represent algebra.
Mathematically, we can write the above thought process as follows:
Gain for the day = 60 + 25 x Number of mowed lawns
Above is an example of an algebraic relationship between two variables.
As you already know, the gain of the day is not fixed and is denoted by the letter “e”, likewise the number of mowed lawns is not fixed and is denoted by the letter “n”. So, the algebraic relation above can be rewritten using symbols to simplify, as shown below:
e = 60 + 25 xn
Remember that it is not necessary to display a multiplication sign between the number and its variable as understood for mathematical purposes. Therefore “25 xn” and “25n” represent the same number. So our relationship comes to;
e = 60 + 25n
We have made a simple algebraic expression between two variables by taking a situation from everyday life.
Keep in mind that our variables (changing activities or unknown activities) are as shown below;
1. Arthur wins for the day. Since his income is not the same every day he works, it can be said that his daily income changes or is unknown until he completes his work for the day. Any unknown or changing activity is called a variable in mathematical language, so Arthur’s daily earnings is a variable and we have used the letter “e” to represent it.
2. Number of lawns Arthur mowed in a day. As the number of lawns he mows is not the same for each day he works. This is the second variable and we used the letter “n” to represent it.
Note that Arthur’s income (e) depends on the number of lawns he mowed (n), so we have one variable that depends on the other.
“n” is an independent variable because it neither increases nor decreases with income, in fact it drifts income up or down. So e is the dependent variable.
Rewrite our algebraic expression again as follows:
e = 60 + 25n
“e” and “n” are the variables and now you already 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 stands alone. In algebraic relations, if a variable is written alone, it has the coefficient UN. Yes, “e” means “1e” and “-e” means “-1e”.
Algebra is a branch of mathematics that deals with the changing or unknown (variable) activities of our daily lives.
A variable is represented by a letter (usually a lowercase letter) of the alphabet.
Variables represent numbers, but these numbers are unknown until the right time or certain conditions are met. This is why variables are replaced by numbers in algebraic expressions where their values must be found.
A constant term in an algebraic relation is a fixed number. As in the example given, Arthur knows that he will receive $60 a day to do some cleaning work with his brother.
A coefficient is a number multiplied by the variable. In the given example, 25 is multiplied by n, which is the number of lawns mowed by Arthur. Therefore, 25 is the coefficient of n.
If there is only one variable (without a number in front) in an algebraic expression, it means that it has the coefficient “ONE” which is not shown and is understood in mathematics.
I hope this helps you make algebra your best friend, as many of my students do.
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