Application of Surds to Trigonometry Ratio and Trigonometry Graph – SS3 Mathematics Lesson Note (First Term)

A surd is the square root of a rational number that is not a perfect square, such as √5, √3 or √7. Its value is irrational, so it cannot be written as an exact fraction. Surds are simplified using six rules, which cover products, quotients, adding like surds and rationalising the denominator.

Behavioural objectives

By the end of the lesson, students should be able to:

1. Define surd.
2. Define rational and irrational numbers.
3. Identify rational numbers and irrational numbers.
4. State the rules of surds.

Instructional resources

A chart showing rational and irrational numbers.

What are rational and irrational numbers?

Step 1: Identification of prior ideas

Mode: The entire class

Teacher’s activities: The teacher asks students to define rational and irrational numbers.

Students’ activities: Students respond to the teacher’s questions as follows.

Rational numbers are numbers that can be expressed in the form p/q, where q ≠ 0. A rational number is also a fraction that gives an exact value without approximation, e.g. 3/1, 2/5, 1/2, √100, etc.

Irrational numbers are numbers whose decimal values have no ending and do not repeat. They include the square roots of numbers that are not perfect squares, e.g. √2, √3, √7, etc.

Step 2: More examples

Mode: Entire class

Teacher’s activities: The teacher asks the students to give more examples of rational and irrational numbers.

Students’ activities: Students respond to the teacher’s questions.

What is a surd?

Mode: Entire class

Teacher’s activities: The teacher defines the term surd as follows.

Surd: the square root of a rational number that is not a perfect square, e.g. √5, √3, √7, etc.

What are the rules of surds?

Rule 1

√(a × b) = √a × √b

Example: Simplify √18.

18 = 9 × 2 = 3² × 2, since 9 is the greatest perfect square factor of 18.

Therefore, √18 = √(3² × 2)
= √(3²) × √2
= 3√2

Rule 2

√(a/b) = √a / √b

Example:

√(12/121) = √12 / √121
= √(2² × 3) / 11
= √(2²) × √3 / 11
= 2√3 / 11

Rule 3

b/√a = (b/√a) × (√a/√a) = b√a / a

You can rationalise the denominator by multiplying the numerator and the denominator by the surd in the denominator.

Example: Rationalise 5/√7.

Multiply the numerator and denominator by √7.

5/√7 = (5/√7) × (√7/√7)
= 5√7 / 7

Rule 4

a√c ± b√c = (a ± b)√c

Example: Simplify 5√6 + 4√6.

5√6 + 4√6 = (5 + 4)√6, by the rule
= 9√6

Rule 5

c / (a + b√n)

Multiply the top and bottom by a – b√n. This rule enables us to rationalise the denominator.

Example: Rationalise 3/(2 + √2).

3/(2 + √2) = 3/(2 + √2) × (2 – √2)/(2 – √2)
= (6 – 3√2) / (4 – 2)
= (6 – 3√2) / 2

Rule 6

c / (a – b√n)

Multiply the top and bottom by a + b√n. This rule enables you to rationalise the denominator.

Application: simplifying surds

Mode: Individual

Teacher’s activities: The teacher asks the students to simplify the following surds.

1. √20
2. √12
3. √18

Students’ activities: Students solve the questions as follows.

1. √20 = √(4 × 5) = 2√5
2. √12 = √(4 × 3) = 2√3
3. √18 = √(9 × 2) = 3√2

Evaluation

Mode: Entire class/Individual

Teacher’s activities: The teacher asks the students to answer the following questions.

1. Define a surd.
2. State whether each number is rational or irrational: √100, 2/5, √7.
3. Simplify √50.
4. Simplify 7√3 + 2√3.
5. Rationalise 2/√5.

Students’ activities: The students solve the questions accordingly.

Assignment

Simplify the following to a single surd:

1. √27
2. √75
3. 9√2 – 4√2

References

1. Oluwafisayo, J. Multipurpose Mathematics
2. Bakare, L. N. Mathematics Clinic for SSCE
3. Otumudia, M. A. Hidden Facts in Further Mathematics