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Introduction to Algebra

Algebra is a branch of mathematics involving patterns, letter and symbols. Looking at the pattern of triangles below, do you think there is any relationship between the number of triangles and the number of sticks used to make them?

Pattern of triangles

We can tabulate it to see the pattern:

Number of triangles 1  2  3 4 5
Number of sticks 3  6  9  12  15

 

We can observe that we need 3 sticks to make every triangle. This also means that the number of sticks required is 3 times the number of triangles. Letters of the alphabet can be used as symbols to represent numbers of the pattern. Usage of letters in this way is called algebra, and the letters represented are called pronumerals

Now, let ‘T’ represent the number of triangles, and ‘N’ represent the number of sticks needed to make a triangle. The above pattern expressed in words says “the number of sticks equals 3 times the number of triangles”. Representing this in an algebraic way using pronumerals –

T  =  3 x N

 

Example 1: Let us look at a different pattern. The first triangle needs 3 sticks to make, but subsequent triangles need only 2 extra sticks. In other words, every triangle can be made with 2 sticks, but the first one needs an extra one stick.

triangle pattern

Number of triangles 1 2 3 4 5
Number of sticks 3 5 7 9 11

 

This pattern can be expressed in words as follows:  the number of sticks equals one plus number of triangles times two.

N  =  2 x T   +   1,     or  N  =  2T  + 1

This rule is called an algebraic equation.

 

Example 2: Now let us look at another pattern – 1, 4, 7, 10, 13, and so on. In this sequence, the next number goes up by 3. Since the first number is 1, we can express this sequence as the earlier/first number multiplied by 3 plus one. So if the next number is ‘T’, and the term number is ‘n’, this pattern can be represented as –

T  =  3 x n  +  1   or   T  =  3n + 1.

For n = 1, 2, 3, 4, and so on, we can get the pattern to be 1, 4, 7, 10, 13, and so on.

 

Example 3: If we are given the algebraic rule (T  =  3 + n), we can generate the pattern by using n = 1, 2, 3, and so on.

Hence we obtain the pattern – 4, 5, 6, 7, ….

 

Example 4: When T  =  5n – 5, we generate the pattern by giving values of n as n = 1, 2, 3, 4, etc.

The pattern here is: 5×1 – 5, 5×2 – 5, 5×3 – 5, …. which is 0, 5, 10, 15, ….