WHAT IS AN EQUATION?
An equation is used to represent the equality of two expressions that are on either side of the “equal” (=) sign. Each side of the equal to sign consists of an expression. The part of the equation which is written to the left of the “equal” sign is called the “left-hand side” or “LHS” and the part written to the right is called the “right-hand side” or “RHS”. An equation usually consists of a set of variables or unknowns whose value needs to be determined.
Equations can be categorised into multiple types. Let’s take a look at a few.
TYPES OF EQUATIONS
These are also known as polynomial equations. They are further classified based on the highest degree of the indeterminate in the equation. Linear equations (degree 1), quadratic equations (degree 2), cubic equations (degree 3) etc., are some of the equations that fall under this category. All equations are of the form: P = 0.
Linear equations: x + 5 = 0 (highest degree of the indeterminate x is 1)
Quadratic equations: x2 – 3x + 2=0 (highest degree of the indeterminate x is 2)
Cubic equations: x3+ 4×2- x – 4=0 (highest degree of the indeterminate x is 3)
The equations used to represent geometric shapes are called Geometric equations. They are represented as cartesian equations or parametric equations.
A circle with radius one is given by the cartesian equation x2 + y2=1. Similarly, it can be defined using the parametric format x=cost, y=sint.
This is an equation which holds true for all possible values of the variables. Calculus and algebra use a lot of identities while solving equations.
E.g. x2-y2 = (x-y)(x+y)
Trigonometry also consists of several identities that can be manipulated to form trigonometric equations.
E.g. sin2+cos2= 1
An equation wherein the unknown represents a function is called a functional equation. It is used to relate one value of the function at a point to another value of the function at a different point.
E.g. f(x)= f(x+1)/x
There are other types of equations, such as differential and integral equations. However, in this article, we will stick to Algebraic and Geometric equations and discuss different methods to teach the concepts to students.
As seen above, students come across different types of equations, and various operations are used to solve them. Hence, this can get quite a bit overwhelming.
Let us discuss a few methods which will aid the teaching and learning process.
METHOD 1: INTEREST
Equations are an integral part of many subjects. Hence, first of all, a student needs to develop an interest in parent subjects such as Algebra, Geometry, trigonometry, etc. While introducing a topic to students, the best way to get them interested is by slightly digressing from the classical curriculum. You can present them with interesting facts or a fun-filled history lesson on the concerned subject.
With regards to Algebraic equations, it will be interesting to share the fact that their study is probably as old as Mathematics. There have been records that in 2000BC Babylonian Mathematicians had developed a method to solve quadratic equations. Brahmagupta, the Indian mathematician, came up with the quadratic formula. Islamic Mathematicians gave a general solution for quadratic equations as they recognised the importance of the determinant.
The father of Geometry, Euclid, also conducted extensive research in the subject. Learning about the different types of equations that can be used in geometry came as a by-product of his research.
Discussing a few Historical facts helps students to get interested and also helps to expand their general knowledge.
METHOD 2: PLAY A GAME
Usually, topics that require the use of equations include several theorems, axioms and proofs. These also include identities. After explaining how these proofs and formulae are derived, it is a good practice to quiz students on the same. Students must remember these formulae by heart and should be able to apply them quickly to questions. It becomes easy for them to do so when they have clarity on the concepts; however, a memory game always aids the process with the added advantage of making the entire process fun.
You can follow a simple template to help guide you through the process as follows:
- Depending on the parent topic (e.g. Algebra), select a sub-topic (e.g. quadratic equations).
- Give students a few days to prepare.
- Ask students questions regarding the same. You can ask them to find the roots of an equation, frame questions on the determinant, ask true or false questions etc. For older students, you can even combine topics and pose simple questions such as – what would be the curve that follows a particular quadratic equation?
This activity can be done as a team too. It enables students to think fast and apply their knowledge while solving questions. It also helps them to memorise several postulates that would otherwise be a mundane task. Students can develop their reasoning and logical skills, as well.
METHOD 3: USE GRAPHS
Irrespective of the topic at hand, the best way to understand how to solve equations is by using graphical methods. You can try showing the pictorial meaning of an identity or an equation. It helps students to visualise an equation which eventually aids them in solving problem sums.
If we take the example of a system of linear equations,
3x + 2y – z = 1
4x – 4y + 8z = -4
-2x + y – 2z = 0
On solving, we get the values of x,y,z:
x= 1, y=-2, z=-2
x,y,z depict the point of intersection of the three lines. Thus, by graphically explaining what each equation represents and what the solution indicates, students can understand the concepts better. It is not only about solving a problem correctly but also about understanding the analytical meaning behind the question and the answer. This way, a student will have great clarity in the concepts.
You can also use computer simulations for the same. Coding a simple question using a programming language such as Python, R, C++, etc., helps students in understanding not only the topic at hand but can instil in them a love for computer science.
METHOD 4: REAL-LIFE EXAMPLES
While explaining a particular concept, using real-life references helps students understand the significance of the topic. As discussed already, equations make their way into algebra, geometry, trigonometry, etc. These concepts are used in a wide variety of real-world applications. Some examples are as follows:
- Finance: Algebraic equations are used in finance industries to calculate loss and profit levels. Algebra is used to represent exchange rates and interest rates while linear equations are used to describe financial graphs.
- Sports: Algebraic quantities, in soccer, are used in calculating the force and distance with which a player kicks the football into the goal. Sprinters estimate the speed with which they are required to reach the endpoint using algebra.
- Architecture: Geometric equations are used to create the aesthetic aspects and safety of structures.
- Astronomy: Geometric equations are used to calculate many problems in astronomy, one of them being the distance between celestial bodies.
By giving students an idea of real-world applications, they get interested in the subject and understand the significance of a particular topic. It also aids them in relating academic problems to real-world questions.
METHOD 5: USE AN EXTERNAL SOURCE
While learning about equations, irrespective of the topic, practice is the key to mastering them. Generating interest while keeping up with the school’s curriculum becomes a problematic task. Hence, taking help from an external source could prove to be beneficial. Cuemath is a beautiful online educational platform that combines fun with studies to learn math
Cuemath helps students by providing them with puzzles, apps, math boxes and workbooks. A teacher can use these resources to guide a student in his learning process. It keeps the process visual and interactive. Once students gain good clarity in the concepts, good grades follow.
Dispelling a student’s fear of the subject is an essential part of teaching. With an organised curriculum filled with facts, games and quality learning, a student will be able to master the topic quickly.
Most Mathematical topics use equations. They are used to represent a problem, solution or both. Hence, working with equations needs to be incorporated into a student’s curriculum. If they get comfortable with the concepts of equations at a young age, solving complex research-level questions will be a cake-walk for them.
The methods listed in this article can be tailored to suit a teacher’s or student’s requirement. You can also build more on it and come up with your creative strategies to study/teach the subject. By keeping the focus on holistic development, Mathematics as a subject will become second nature. Hopefully, this article inspires everyone to give equations a try!