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Who Says There Is No Relation Between Algebra And Daily Life?
During my tutoring sessions many of students ask that where do we use all those variables (x, y, z, n etc.) in our daily life? My students are right at their place as they don’t see any use of those variables or algebraic expressions in their lives directly. But, I tell them that all the algebra and its concepts are invented to help us in our daily life and algebra is our best buddy. They wonder and ask me for more explanations. Then I explain them systematically, how algebra is embedded in our daily life by using concrete examples. One of those examples I want to share with my valued readers is given below;
The Basic concepts, algebra starts with.
The basic concepts in algebra are
- Variables
- Coefficients
- Constants and
- Algebraic expressions.
Let’s do the following example from a daily life situation to understand all of the above terms in algebra;
Consider every weekend, Arthur; a grade 9 student starts to help his brother in his landscaping business. Every time Arthur goes with his brother for work, brother pays him $60 to staying with him all the day to do little clean up work at the job site.
Some times, there are two customers side by side, where Arthur can work on the lawn mower machine and for that, his brother pays him another $25 for each lawn, Arthur mows.
Consider first Saturday, Arthur didn’t get a chance to work on the lawn mower.
Can you guess how much he made for the day?
Easy! Your answer might be $60, because he gets paid for his basic cleaning services and no money for working on the lawn mower.
Next day is Sunday and Arthur got a chance to work on the machine for two customers and he mowed two lawns.
Can you tell how much money Arthur made this Sunday?
On next weekend, i.e. second Saturday, Arthur mowed five lawns, what are his earnings for the day?
Next day is Sunday and Arthur mowed one lawn. What are his earnings for this Sunday? Probably, you know the answers to all of the above questions.
But, I want to stop here for explanations to show clearly that, how this daily life activity is algebra. For this we are accomplishing a very important concept of algebra in this example.
I want to show you the work you did in your brain to come up with the answers for all the above questions. So, below are all the explanations;
Arthur’s earnings have two parts.
Part one is fixed part, which is $60 for the day, he worked for his brother to do cleanup job.
Part two is not fixed and depends upon the number of lawns he mows, if he gets a chance to operate the mowing machine.
Your thinking process is as follows:
Arthur’s earnings = (Fixed Part) ADDED TO (25 times Number of lawns mowed by Arthur)
First Saturday
Arthur’s earnings = 60 + 25 x 0 = 60 + 0 = $60
25 is multiplied by zero as he mowed no lawn this Saturday.
First Sunday
Arthur’s earnings = 60 + 25 x 2 = 60 + 50 = $110
25 is multiplied by 2 as he mowed two lawn on this Sunday.
Second Saturday
Arthur’s earnings = 60 + 25 x 5 = 60 + 125 = $185
25 is multiplied by 5 as Arthur mowed 5 lawns this Saturday.
Second Sunday
Arthur’s earnings = 60 + 25 x 1 = 60 + 25 = $85
25 is multiplied by 1 as only one lawn is mowed this Sunday.
Are Arthur’s earnings same for each day?
The answer is no. The earnings are not same; they are different for different days. As you already know that something changing in mathematics is called a “variable”. Also the dictionary meaning of variable is changing. Hence, Arthur’s earnings can be represented by a variable.
Now, mathematicians have their options, they can say, “Arthur’s Earnings are changing.”
Isn’t this a very long sentence to use in math problems?
Yes, this is a long sentence to represent a variable that is, Arthur’s earnings.
So, the mathematicians of the world agreed upon a standard. That standard is, to represent the variable quantities or variable activities by letters from the alphabet. Most often letters in the lower case are used to represent variables.
In our example we can represent the Arthur’s earnings by letter “e”. This is very very important to remember that Arthur’s earnings for a particular day is always a number in dollars, but that number keeps on changing every day Arthur works. Therefore we need a common representative for earnings on all the weekends, which is a variable.
Also, Arthur’s earnings for the coming weekends are unknown until he actually works in those coming days. Therefore, we need to represent that unknown amount of money by a variable. One can say that this variable is; “Arthur’s earnings are unknown until he finished his work for a particular day.”
On the other hand we can pick a small letter to represent the entire previous sentence, instead. So, mathematicians went for the second choice. Therefore, we pick the letter “e” to represent Arthur’s earning for any day on a weekend.
Further, as you already know that Arthur’s earnings (e) depend upon the number of lawns he mowed, which is again 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” (as two different variables need different symbols), from the alphabet. Consider the number of lawns mowed in a day by Arthur is represented by letter “n”
Finally, let’s write both the variables;
Arthur’s earnings for a 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 thinking process. There is something common (in terms of math operations of plus, minus, multiply or divide) in all of those calculations of earnings 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 mowed. Isn’t this process is common for all the days to calculate Arthur’s earnings? Yes, it is. This common relation between the earnings pattern is actually algebra, and understanding and representing this type of relations is, understanding and representing algebra.
Mathematically, we can write the above thinking process as follows:
Earnings for the day = 60 + 25 x Number of lawns mowed
Above is an example of algebraic relation between two variables.
As you already know that earnings for the day is not fixed and is denoted by letter “e”, also the number of lawns mowed is not fixed and denoted by letter “n”. So the above algebraic relation can be rewritten using symbols for simplicity as shown below:
e = 60 + 25 x n
Remember that there is no need to show multiply sign between the number and its variable as it is understood for math purposes. Therefore “25 x n” and “25n” represent the same number. So, our relation comes to;
e = 60 + 25n
We have accomplished a simple algebraic expression between two variables taking a daily life situation.
Keep in mind that our variables (changing activities or unknown activities) are as given below;
1. Arthur’s earning for the day. As his earnings are not same everyday he works, we can say that his daily earnings are changing or unknown until he finishes his work for the day. Any unknown or changing activity is called a variable in math language, so Arthur’s daily earnings is a variable and we used letter “e” to represent it.
2. Number of lawns Arthur mowed in a day. As the number of lawns he mowed is not same for every day he works. This is the second variable and we used letter “n” to represent it.
Notice that Arthur’s earnings (e) depends upon the number of lawns he mowed (n), therefore we have one variable depending on the other.
“n” is independent variable as it does not go up or down with earnings, actually it derives the earnings up or down. Therefore 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 is alone. In algebraic relations if a variable is written alone, it got the coefficient ONE. Yes, “e” means “1e” and “-e” means “-1e.”
Summary
Algebra is branch of mathematics which deals with changing or unknown activities (variables) in our daily life.
A variable is represented by a letter (most often a lower case letter) from the alphabet.
Variables represent numbers, but these numbers are unknown till the right time or certain conditions are met. That’s why variables are replaced by numbers in algebraic expressions or their values are needed to be found out.
A constant term in an algebraic relation is a fixed number. As in the given example, Arthur knows that he will get $60 for each day for performing cleanup work with his brother.
A coefficient is a number multiplying to the variable. In the given example 25 is getting multiplied by n which is number of lawns mowed by Arthur. Hence, 25 is coefficient of n.
If there is only variable (without a number at front) in an algebraic expression, this means it got coefficient “ONE” which is not shown and is understood in mathematics.
Hope that this will help you to make algebra your best friend as many of my students do.
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