Mastering Algebra with Ease: An Introduction to the Basics
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By Naveen Devinda (Contributing Author)
Algebra is widely regarded as one of the three pillars of mathematics along with Geometry and Analysis. The origin of Algebra dates back to 2000 B.C. from ancient Babylon and Egypt. The first use cases of Algebra were in construction and finance where it helped find the surface areas of objects and calculate the interest for money being lent.
Algebra has continuously evolved over the years and is now being utilized in a broad range of applications. From generating online shopping bills to the navigation of a spaceship in outer space, algebra plays an important role. It can be thought of as an attempt to generalize a solution to accommodate different conditions of a problem. Students can find Algebra challenging as it is the course where they are first exposed to the idea of abstractness in math.
A great start to your Algebra course
One of the best ways to kickstart Algebra with more confidence is to understand what practical problems it helps solve. This is true with all STEM subjects and it’s certainly worth it if both teachers and students can put in that extra effort to go beyond listing applications to obtain more insight.
Below are a few real-world applications of Algebra:
● A shopping mall trying to come up with Terms and Conditions for a purchasing limit to restrict how much it gives away on discounts. (For example, at malls sometimes they give you 50% discounts, but on their Terms and Conditions they might say there is a limit for $150 because they don’t want to give away too much as discounts).
● The pilot wants to make sure the plane has got enough fuel to carry the passengers and the cargo.
● YouTube uses algebra to rank its recommendations when you search for “Funny fails”.
● City planners need to decide how wide a city road needs to be to handle 50,000 people per hour per direction
● Convert Fahrenheit to Celsius and vice versa. Siri and Alexa do this for you, but it’s algebra that’s being applied on the programming level.
Could you think of the variables associated with each of these?
Underlying principles of Algebra
At the roots of algebra, we see that it is a study of calculating through ‘completion’ and ‘balancing’. If students have a strong grasp of the fundamentals of math, they will find that the concepts of Algebra are intuitive. If not, students will need to exert themselves to memorize mechanisms of solving problems. Early learners of Algebra can benefit if they gather a deep understanding of “Balancing” and “Inverse”.
-Balancing
Teachers incorporate the terms “balanced equation”, “balance the right and left-hand side”, etc. during their sessions but only a few students grasp the meaning of it. An effective way to deliver the notion of balancing is the “merchant’s weighing scale” scenario. This will allow learners to reinforce three principles:
1. Equality is established when both sides are balanced.
2. Any change to one side makes the scale unbalanced.
3. Making the same change to both sides will maintain equality (balanced).
An activity that can promote this conceptual knowledge is to solve elementary algebraic equations with the weighing scale scenario before delving into symbols and other algebraic notation. For example: Consider a balanced weighing scale.
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- On the right scale, we have 15 bananas and a 500g weight.
- On the left scale, we have four 200g weights and 7 bananas.
- How could we use the scale to find the weight of a banana?
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Here, you want the students will realize that they need to get bananas onto one side and the weights onto the other (the concept of isolation or isolating the variable). Then, they could divide the combined total weight of the weights on one side by the number of bananas on the other. This would yield the weight per banana.
This method is powerful because it is visual, practical, and promotes conceptual understanding.
-Inverse
An inverse operation is basically one that is the reverse of the other. The basic operations of addition, subtraction, division, and multiplication are related to each other. Relationships like addition being the inverse of subtraction, division being the inverse of multiplication, etc. These are conceptual knowledge that students will build on when learning Algebra. Exponentiation and square roots are other operations that many of us find challenging during the first stages of algebra. For instance, learners may not be able to see how the operations, raising a number to the power of 2 and obtaining the square root are related, but they too are the inverse of each other.
A great exercise to develop knowledge on inverse operations is the “construction and deconstruction” of functions.
Let’s start with the term “x”. How do we construct the expression 5x – 15?
● First, we can multiply x by 5 to get 5x
● Next, we can subtract 15 to get 5x – 15
Here, students can be encouraged to think whether they can get to the expression using various other steps. We can also practice deconstructing 5x – 15 and get back to x.
● Association of terms – If we take the example discussed above, a might students might use the following steps to try to deconstruct the expression 5x – 15
1. Divide by 5 and then get x – 15
2. Add 15 to get “x”
The first step is a common mistake and is due to a lack of conceptual understanding that every operation being done on the expression has to be done to the entire expression, not individual terms.
5x – 15 divided by 5, would give us x – 3 not x – 15
Common mistakes and misconceptions
As mentioned earlier, the abstract nature of algebra makes students more susceptible to misconceptions. Some of the most common mistakes and misconceptions include:
● Operations with letters – Students find it counterintuitive to apply the basic rules of math to letters.
Adding “x” and “2x” or subtracting “p” by “p” can be confusing. Embedding the idea that the letters just stand for an unknown and don’t have to be alphabetical letters but could be words or even pictures is effective: Banana + 2 banana = 3 bananas.
● Variables and constants – Quite often students consider the solutions for the unknown to be constants, where they believe algebra is about finding the missing value. Like we discussed at the start of this article, allowing the student to see the bigger picture and the context where algebra is applied is a way to address this misconception.
C = (F − 32) × 5/9
For example, if the equation above is the algebraic equation to convert Fahrenheit to Celsius, “F” and “C” are variables, and can vary with time.
Final message
As with other areas of math, visualization is an effective method for deep learning Algebra. It is a subject that should be approached in a practical manner. A good understanding of the context will help students reason their way through problems and also arrive at “sensible” answers.
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