An Introduction to Algebra

Yes, Algebra is math, but hey – don’t look so glum! It’s much more fun than you think. Why, just consider this – What did the Algebra 1 textbook say to the Algebra 2 text? It said (snicker, snicker), “Don’t bother me! I’ve got problems enough of my own!”

Or, how about this one: What type of upper support garment does a mermaid wear? She wears an algae-bra! Get it? (Snort, snort, tee-hee.) Okay, okay. Just hang in there. You’re going to be surprised.

Algebra truly is fun, especially if you approach it with a receptive mindset. Most people just need the right introduction to algebra. Here’s the thing to remember … if you like Sudoku, you’re going to love Algebra. Just keep an open mind – you owe yourself that much!

So, just what is algebra, other than a required school subject? What makes it special? Where did it come from and why is it important? Of course the classic, and most often asked question about algebra is this: “What am I ever going to use algebra for in real life?” (Be on the lookout for the answer, because you might just be surprised!)

A Quick History

In rudimentary form, algebra has been around for centuries. It appears to have originated in the middle east, in Mesopotamia. The word “algebra” comes from Arabic and means completion or restoration. The first known algebra book was written around 820 AD in Bagdad by a Persian mathematician by the name of Muḥammad ibn Mūsā al-Khwārizmī. Its title? The Compendious Book on Calculation by Completion and Balancing. It introduced a system of abstract techniques and representational language that is still used today to solve algebraic equations.

You Can’t Always See Algebra, but It’s There

The beauty and simplicity of algebra come from its use of abstracts. Algebra employs a lot of abstracts. Abstracts distill the idea of something which exists in reality. An abstraction can refer to something tangible, such as cars, boxes, or pounds of rice, as well as to intangible concepts, such as thoughts or emotions. They exist, but aren’t readily packaged, counted or weighed like items that are tangible. Beginning in Algebra 1 and Algebra 2 and continuing into advanced calculations, mathematical abstracts distill tangible concepts.

Once Upon a Time in a Kingdom Far, Far Away

Most Algebra 1 books begin by explaining the rules of algebra. Algebra, like the fictional kingdoms that abound in many sci-fi and fantasy book series popular today, has a unique set of rules by which it operates. Anyone wishing to interact within the walls of the Kingdom of Algebra must learn and abide by the kingdom’s rules, vocabulary and governing principles if they are to succeed in their quests.

Algebra dances with numbers. It systematically studies the way they relate to one another. It categorizes and manipulates them, and uses them and variables such as x and y to represent unknown values. It uses formulas to create equations that are employed to solve problems. To understand and employ algebra correctly, it is necessary to become well-versed in its secret language of symbols, definitions, formulas and vocabulary. In the land of language, words that relate to one another in meaningful ways create sentences. In the Kingdom of Algebra, numbers and variables provide the relationship parameters that create equations.

Math: Collect the Whole Set

Mathematics defines various sets of numbers for different purposes, and it is a wise student indeed who takes the time to memorize their differences. Examples of these sets of numbers include real numbers, natural numbers, whole numbers, integers, rational numbers and irrational numbers. Algebra works exclusively with the set of real numbers. Algebra also uses a specific set of symbols to denote various functions and concepts. For example, an algebraic expression might have one or more algebraic terms that include constants, coefficients, variables, and symbols of operation. Below is one such algebraic expression:

6x2 + 3y + 8xy + 2

There is no equal symbol on an expression such as this because it is a simple phrase, and not an equation. The terms in the above algebraic expression are 6x2, 3y, 8xy, and 2. Algebraic constants are numbers that stand alone. They are so named because their value doesn’t change. Variables, expressed as letters, represent unknown numbers. Coefficients are numbers in terms that contain a variable. Anytime there is a variable by itself, its understood coefficient is 1.

Get Smart Now

The average Algebra curriculum works to help students acquire an understanding of equations and to develop their facility to work with linear and quadratic equations as well as exponents and their functions. By learning about the elements of algebra as they also learn to solve and analyze equations and use quadratic and exponential functions, a student fine-tunes his ability to critically analyze, generalize and selectively apply algebraic concepts correctly.

The typical Algebra I course is structured to develop a student’s ability to reason algebraically, which is to say, mathematically and abstractly, and particularly as it relates to the real number system. The goal is for students to become fluent using variables to represent abstracts even as they improve their ability to formulate and solve problems in algebraic terms. For the most part, Algebra 1 textbooks give way to algebra 2 books, although there are some curricula that choose to wedge geometry between the years of algebra. Other curricula incorporate geometry directly into the algebra texts themselves, and still others follow Algebra I and Algebra II with a course in Geometry.

Algebra II extends students’ understanding of and ability to use functions. Likewise, it explores the relationship between exponential functions, their inverses, and studies trigonometric functions as they apply to the set of real numbers, their graphs, properties, and the like. The primary categories of interest in Algebra II are the constructs of functions, modeling, numbers as they relate to quantity, statistics, and probability. In Algebra I, students explored the structure of polynomials. In Algebra II, they solve polynomial equations with complex numbers.. They learn to extend the system of polynomial functions to the field of rational functions. Algebra II students become comfortable with a variety of means of data collection and deepen their understanding of the impact of randomness and design.

You Use It More Than You Realize

The Algebra I student might ask if he will ever apply what he learns in his course. The Algebra II student knows he will apply what he has learned. The average person uses learned algebra freely in the course of daily life whether they realize it, or not. Many students will use their algebra studies as stepping stones to higher math and more advanced learning and potentially careers that require an in-depth understanding of even higher level math, as is the case in many scientific and technological fields. All students should study algebra, however, as its understanding is considered a part of general education and furthermore, becomes a tool one uses to make endless numbers of necessary calculations that arise in the course of everyday life. The person who declares, “I’ve never used the algebra I learned in school,” likely does, and simply fails to realize it.

For example, if you’ve ever applied for and received a loan, you no doubt had to pay interest on the money you borrowed. Algebra was inherent in the formulas you employed to calculate the amount of that interest. The calculations that are required to determine whether one particular interest rate at a given number of years vs. a variable rate which might (or might not) change at set intervals can be surprisingly complex. Failure to calculate the least expensive method of financing is to fail to handle money responsibly.

The same principles apply when one has money to invest, and is attempting to determine whether one method or another is the safest and which one is most likely to earn the greatest amount of interest. Algebra is inherent in calculating the best rate of return no matter if one is managing a passbook savings account or an entire investment portfolio. Algebra is your friend whether attempting to determine the best car for the money or the best age to retire. It is a tool you will come to value.

The advent of YouTube has single-handedly unleashed an army of industrious DIY homeowners who, armed with the knowledge they’ve gained from watching tutorial videos, are now prepared to tile their basements, paint their walls, replace their roofs, refinish their furniture, and landscape their yards. YouTube videos frequently fail to include the instructions needed to calculate materials. How many cases of tile should you purchase? How many gallons of mortar? How much paint do you need? How much grass seed? How many plants are required to provide that desired curb appeal for the front of your home? Hopefully, you studied your algebra assignments when you had the chance, for when you run these calculations, you’ll need those skills. Check it out! There goes your algebra to work, saving you money!

Time passes so quickly. Before you know it, you’re on the PTA, planning the school’s annual fall carnival, or you’re in charge of Wednesday night suppers at church, or it’s your home where your extended family gathers each Thanksgiving. Instead of cooking for just your two, you suddenly have dozens if not hundreds of more mouths to feed! Trusty algebra is your friend when it comes to altering those recipes. You might do this as a labor of love, but in most restaurants, the chef does the same as a part of his job. He, too, is grateful for his algebra lessons!

When it comes time to go on vacation, and you’re weighing the pros and cons of a walking tour across England, or across a Greek Isle, it will be algebra that comes to your aid as you calculate mileage, airfare and currency conversions. You’ll use algebra every time you calculate the value of anything unknown. In other words, when a question starts with, “I wonder how much it will cost to ….” algebra shows up to work. The same is true when you’re trying to determine how much of something is left, which of two or three alternatives is the best offer, or even the time you should plan to leave from one particular location to reach another on time.

The doctor, the pharmacist, the personal trainer and the housewife all use algebra. So does the woman trying to determine how many skeins of yarn to purchase to make a baby afghan, the seamstress buying material for curtains, and the flooring salesman trying to help a young woman make a cost comparison between hardwoods and carpet. Practically everybody who walks into a big box building store needs two things: a tape measure and some rudimentary knowledge of algebra.

It isn’t just the NASA space scientist who needs algebra, or the electrician, the plumber, or the warehouse manager … although they all indeed do call algebra their friend. It is also the woman who hopes to lose weight on the cucumber diet that makes algebraic calculations. It is the mom trying to weigh the purchase of Christmas presents for her children against the cost of summer camp tuition.

Algebra is inherent in functions involving time, distance, weight, amounts, money, temperature, age, weather, and price per unit. Whether you are trying to determine gas mileage, batting average, the least expensive grocery store from which to buy produce, the number of calories you can afford to eat without blowing your diet, or how much worm medicine per pound to give your dog, you’ll draw upon your knowledge of algebra.

The truth? It’s everywhere, it’s everywhere! Algebra is everywhere in everyone’s’ everyday lives. We use it far more often than we realize and in ways we don’t even recognize. We might not think we use the algebra we learned in school, but we truly do, day in and day out, throughout our lives. Every time we turn around we’re bumping into it. So – here’s the real question.