SLC S22W1 : Variables and Expressions

Most people are scared of studying mathematics because they feel they won't be good enough or that failure may be a backbone in their journey. Why? This is probably because of how the mathematics is being taught or probably from the interest of the student. The teacher plays an essential part in making students hate mathematics as a course. The complexity involved makes it a weird subject by most and they flee to something simple.

Now, this post is talking about Variables and Expressions which seems to be the basics when it comes to algebraic expression. Without accurate knowledge of these two terms, it must be difficult to navigate your way through algebraic except doing so blindly. In the course of this post, I'll be breaking things down using the tasks involved for a lay man to understand and those that have little or no interest in mathematics.

Before mentioning the two types of variables and expressions not mentioned in the tutorial post, I'll love to shed more light on these two terms. Let's take this illustration. It's more like you're baking a bread in a bakery. You'll definitely need a recipe of 2 cups of flour or more, 1 cup of sugar and a temperature. The amount of flour and sugar varies depending on the number of baked bread but the temperature is constant.

Variables are symbols or components that represent a value which can vary and are just like the measurements or amount of ingredients that can change. In terms of expressions, it's likened to the instructions in the recipe used to make the bread. It doesn't represent set of letters alone but is a combination of numbers, variables, constants, and operations that gives you an outcome just as you combine the flour and sugar with the instructions in the recipe to make the bread.

Why it's called instructions in the recipe is because you'll add, add and add or subtract, add and subtract.

X - Variable

2x - Dependent Variable

2x + 3..... Expression

We have different types of variables which can be extended to other course of study, research etc. I'll be focusing on two types of variables which are mathematically related. Constant Discrete Variable.

• Constant Variable: A constant variable is a mathematical variable and mostly used experimentally in other field. It's a variable whose value does not change under conditions that are specified. Unlike other variables that can change, constant variable as the name implies is fixed and provides stability in calculations. How constant is this variable?

Let's take these examples. Constants as we know serves as parameters used in solving equations and ensuring calculations are consistent with themselves. We have seen constant variables as such in experimentations like physics. Some constant variables are;

g= 9.8m/s², plank's constant, c= 3×10⁸m/s...etc.

Even in real life, we have constant values in terms of the price of an item in a closed market where you don't get to price it. Mathematically, we can show this constant variable in linear equations.

y = mx + c

C there serves to be the constant value most especially when plotting graphs. In quadratic equations, we have

y = ax² + bx + c.....

• A discrete variable is mathematically related and is one that has specific counting when applied and there are values in between values like decimal or fractions as the case may be. They are values that are distinct and separate from each other, mostly appearing as whole numbers.

What I mean here is that, discrete variables are usually represented when counting but the counts involves distinct whole numbers. E.g in algebraic expressions, the number of free spaces can be calculated as F = 50 - n where n can be any number or discrete variable.

Same is applied when given as expression that involves n days. We could write it as

P= p¹ +p² + ........+ Pn

The practical application of this is when you're reminded that discrete variables can only be counted with specific values. They can't be measured as it's done in continuous variables. It's more like comparing the speed of a car and the number of balls in a house. The speed of a car can only be measured while that of the balls can be counted with specific values which can be extended to what we call n values in physics.

Having considered variables and their Practical applications, let's look into Expressions carefully and how they are applied. I'll be emphasising on two types of expressions which includes Polynomial and Rational Expressions

• Polynomial Expressions are expressions that consists of variables raised to non-negative integer powers, constant and coefficients. What do I mean? This expression involves a variable being raised to the power of an integer that doesn't have a negative symbol attached to it. Let's take this example in physics where we have to use the laws of motion to explain this, practically.

Let's assume we are given an expression such as this and are asked to identify what kind of expression this is.

h(t) = 5t² + 20t + 6

In this expression, we can easily denote a physics expression which signifies projectiles and that's how this polynomial expression is linked, practically.

Mathematically, using the algebraic example, we can outrightly identify a polynomial from its power. Polynomials usually have variables raised to the power of 3 or more while quadratic is 2. .

Example: 3x³ - 6x² + 2x + 6

In this case, this is an expression but when an equality sign is introduced, it becomes an equation.

3x³ - 6x² + 2x + 6 = 0....... Equation

• Exponential Expressions: This could be more tempting as one may feel it's a dependent variable but it's not. The variable at the superscript represent power. An exponential expression involves variables or constant, raised to the power where the exponent is a variable.

How? The exponent here is the variable and this variable is the raised power of a base. Practically, we've had series of formulas with exponential expressions as

P(t) = P°Ert

We usually see it used in calculating the population of a place and mathematically, and we usually come across expressions such as

3^x . 3²x ÷ 3²x+¹

Polynomial	Exponential

Variables and Expressions are closely related and we use them in day-to-day activities. Both can do without each other and can become independent of themselves.

Evaluating an algebraic expression is just like unlocking a secret code. You need to be accurate in where you touch first otherwise you'll have to retry after 24 hours. There are rules governing how algebraic expressions behave. You don't just see these expressions and start working on them without understanding the logic and mathematical rules involved.

In my primary school, I was taught BODMAS. This was and is still used in this century to solve most mathematical problems. The rule is as simple as the structure of the house. You can't enter a house through the window without first passing through the door. You can't enter your room without going to the lobby area first, where your room is stationed.

Same applies in mathematics. You can't count 2 when you haven't counted 1. So let's understand what the letters represent first before venturing into the practical application of what I'm saying.

B - Bracket: In most algebraic expressions, you get to see brackets covering variables and you're expected to open these brackets but how? Do you carry key to open the brackets? Definitely not. This brackets are the priorities in BODMAS. It usually in the form of parentheses. These brackets usually mean multiplication in cases where you have a number in front of the bracket with a sign.

Example: (3+5) = 8

2 ( 3+5) = 16✅

You'll open the brackets by solving what's inside before opening it. This is done irrespective of the signs involved. Once you open the brackets, the integers or variables stand on its own and from there, you can be able to solve them as like terms or distinct terms as the case may be.

O - Order This means exponents or powers usually seen in some equations. These exponents as mentioned earlier under exponential expressions are powers like squares, cubes, and any other powers which may range into polynomial. These exponents are usually at the superscript of an integer or variable variable and can be numbers or letters. In some cases, they are found raised at the top of a bracket which needs to be open first before touching the parenthesis

Example: 2² + 4 = 4 + 4 = 8

(3-2)² ÷ 1 = (1)² ÷ 1 = 1✅

D - DIVISION: As the name implies, this have to do with the division of numbers, variables etc in the expression. When this division is done as the third rule after opening the brackets, multiplication would be introduced as the fourth rule that governs our operation in algebraic.

Example: 15÷5 = 3

(5+3) ÷ 2 = 8 ÷ 2 = 4✅

M - MULTIPLICATION: This is actually the next rule after the division rule. It follows a sequence and after the division aspect comes the division. It is usually represented with the symbol. X

Example: 3×5÷5 = 3 × 1 = 3

8 × 2 = 16✅

A - ADDITION: This is simply the addition of numbers and letters in an expression. The symbol used to denote this is the + sign. Mathematically, we open brackets, divide and multiply before we add and then subtract which is denoted by the symbol -

Example: 6+3-1

9-1

8 ✅

S - SUBTRACTION: This is a BODMAS RULE that involves subtracting integers and variables which could be independent and dependent.

Now let's put everything together in an application. Let's imagine we've been given an algebraic expression.

Evaluate 2 - 5(3+4) + 25 ÷ 5 - (3×6) using BODMAS rule

Evaluate 4 + (12÷3)² × 2

We outrightly know that the above is an expression and not an equation. If you notice in the expression, we have brackets, division, exponent, multiplication and addition. You should take note of what you have in the expression first before venturing to something else.

Step 1: 4 + (12÷3)² × 2

Open Brackets:

4 + (4)³ × 2

Step 2: We would work on the Order or exponent that's after solving the brackets.

4 + (4 × 4 ) × 2
4 + 16 × 2.............

After solving the exponent and getting a plain exponent, we have to still use the rule because if we want to punch our calculators, we would get a different answer.

Step 3: Multiplication comes first before addition as shown in the expression. Now let's multiply 16 by 2 in the expression.

4 + 16 × 2......

4 + 32 ........ Multiplication

Step 4: We then add to get our final answer as simple as it could be. Now, for something complex, I'll be showing you via video.

36.... We've gotten our final answer for the expression.