Objective:

This topic gives an overview of; 

• Finding the Value of an Expression
• Perimeter formulas
• Area formulas
• Rules for number patterns
• Pattern in geometry

Finding the Value of an Expression

We know that the value of an algebraic expression depends on the values of the variables forming the expression. There are a number of situations in which we need to find the value of an expression, such as when we wish to check whether a particular value of a variable satisfies a given equation or not.

 We find values of expressions, also, when we use formulas from geometry and from everyday mathematics. For example, the area of a square is l², where l is the length of a side of the square. If l = 5 cm., the area is 5² cm² or 25 cm² ; if the side is 10 cm, the area is 10² cm² or 100 cm² and so on. 

Using Algebraic Expressions Formulas and Rules

We have seen earlier also that formulas and rules in mathematics can be written in a concise and general form using algebraic expressions. We see below several examples.

• Perimeter formulas

1. The perimeter of an equilateral triangle = 3 × the length of its side. If we denote the length of the side of the equilateral triangle by l, then the perimeter of the equilateral triangle = 3l.
2. Similarly, the perimeter of a square = 4l where l = the length of the side of the square.
3. Perimeter of a regular pentagon = 5l where l = the length of the side of the pentagon and so on.

• Area formulas

1. If we denote the length of a square by l, then the area of the square = l²
2. If we denote the length of a rectangle by l and its breadth by b, then the area of the rectangle 
3. Similarly, if b stands for the base and h for the height of a triangle, then the area of the triangle

Once a formula, that is, the algebraic expression for a given quantity is known, the value of the quantity can be computed as required.

For example, for a square of length 3 cm, the perimeter is obtained by putting the value l = 3 cm in the expression of the perimeter of a square, i.e., 4l. The perimeter of the given square = (4 × 3) cm = 12 cm.

Similarly, the area of the square is obtained by putting in the value of l(= 3 cm) in the expression for the area of a square, that is, l² ; Area of the given square = (3)² cm² = 9 cm² .

• Rules for number patterns

Study the following statements:

1. If a natural number is denoted by n, its successor is (n + 1). We can check this for any natural number. For example, if n = 10, its successor is n + 1=11, which is known.
2.  If a natural number is denoted by n, 2n is an even number and (2n + 1) an odd number. Let us check it for any number, say, 15; 2n = 2 × n = 2 × 15 = 30 is indeed an even number and 2n + 1 = 2 × 15 + 1 = 30 + 1 = 31 is indeed an odd number.

• Some more number patterns

Let us now look at another pattern of numbers, this time without any drawing to help us 3, 6, 9, 12, ..., 3n, ...

The numbers are such that they are multiples of 3 arranged in an increasing order, beginning with 3. The term which occurs at the nth position is given by the expression 3n. You can easily find the term which occurs in the 10th position (which is 3 × 10 = 30); 100th position (which is 3 × 100 = 300) and so on.

• Pattern in geometry

What is the number of diagonals we can draw from one vertex of a quadrilateral? Check it, it is one From one vertex of a pentagon? Check it, it is 2.From one vertex of a hexagon? It is 3.

The number of diagonals we can draw from one vertex of a polygon of n sides is (n – 3). Check it for a heptagon (7 sides) and octagon (8 sides) by drawing figures. What is the number for a triangle (3 sides)? Observe that the diagonals drawn from any one vertex divide the polygon in as many non-overlapping triangles as the number of diagonals that can be drawn from the vertex plus one. 

Summary

• Expressions are made up of terms. Terms are added to make an expression. For example, the addition of the terms 4xy and 7 gives the expression 4xy + 7.
• A term is a product of factors. The term 4xy in the expression 4xy + 7 is a product of factors x, y and 4. Factors containing variables are said to be algebraic factors.
• The coefficient is the numerical factor in the term. Sometimes anyone factor in a term is called the coefficient of the remaining part of the term.
• In situations such as solving an equation and using a formula, we have to find the value of an expression. The value of the expression depends on the value of the variable from which the expression is formed. Thus, the value of 7x – 3 for x = 5 is 32, since 7(5) – 3 = 35 – 3 = 32.
• Rules and formulas in mathematics are written in a concise and general form using algebraic expressions:
  • Thus, the area of rectangle = lb, where l is the length and b is the breadth of the rectangle. 
  • The general (nth) term of a number pattern (or a sequence) is an expression in n. 
  • Thus, the nth term of the number pattern 11, 21, 31, 41, . . . is (10n + 1).