Mathematics - Senior Secondary 2 - Surds I

Surds I

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TERM: 3RD TERM

WEEK: 9

Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes per period (5 periods)
Subject: Mathematics
Topic: Surds I
Focus: Rational and irrational numbers revision, simplification of surds, operations on surds (addition, subtraction, multiplication, division), rationalization of surds.

SPECIFIC OBJECTIVES:

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

  1. Differentiate between rational and irrational numbers, leading to the understanding of surds.
  2. Simplify surds (e.g., √8 = 2√2).
  3. Add and subtract surds, applying the rule for similar surds.
  4. Multiply and divide surds, including rationalizing denominators of surds.

INSTRUCTIONAL TECHNIQUES:

  • Question and answer
  • Guided demonstration
  • Practice exercises
  • Discussions
  • Use of real-life examples to connect concepts to daily life
  • Group work for collaborative learning

INSTRUCTIONAL MATERIALS:

  • Whiteboard and markers
  • Charts illustrating surd operations (addition, subtraction, multiplication, division, and conjugates)
  • Flashcards with examples of surds
  • Worksheets for simplifying and performing operations on surds

 

PERIOD 1: Introduction to Rational and Irrational Numbers and Surds

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1

Introduces rational and irrational numbers. Explains that rational numbers can be written as fractions, while irrational numbers cannot.

Students listen attentively and respond to questions.

Step 2

Defines surds as irrational numbers that cannot be simplified to remove roots (e.g., √2, √3). Explains the difference between rational and irrational numbers.

Students note down the definitions and ask questions.

Step 3

Gives examples of surds (e.g., √2, √5, 3√7) and explains why these are irrational numbers.

Students observe the examples and write them down.

Step 4

Discusses real-life examples of surds, such as the measurement of diagonal lengths in geometry.

Students engage with the teacher’s questions on real-life connections.

NOTE ON BOARD:

  • Rational Numbers: Can be expressed as a fraction.
  • Irrational Numbers: Cannot be expressed as a fraction, e.g., √2, π.
  • Surds: A specific type of irrational number that involves roots.

EVALUATION (5 exercises):

  1. Define a rational number.
  2. Define an irrational number.
  3. Provide an example of a surd.
  4. Why is √5 considered an irrational number?
  5. Explain why π is not a surd.

CLASSWORK (5 questions):

  1. Identify whether the number 3/4 is rational or irrational.
  2. Is √3 a surd? Explain.
  3. Give an example of an irrational number.
  4. Simplify √18.
  5. Simplify √12.

ASSIGNMENT (5 tasks):

  1. Identify the rational and irrational numbers from a given list.
  2. Write down any two surds you know and explain why they are surds.
  3. Simplify √8.
  4. Simplify √50.
  5. Research an everyday application of surds in technology.

 

PERIOD 2 & 3: Simplification of Surds

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1

Introduces the method of simplifying surds. Shows how to express a surd in its simplest form by extracting square factors (e.g., √18 = √(9×2) = 3√2).

Students observe the examples and take notes.

Step 2

Guides the class through several examples of simplifying surds. Demonstrates extracting factors of squares to simplify surds.

Students follow along and ask questions as needed.

Step 3

Assigns practice exercises, simplifying more complex surds. Encourages independent work.

Students practice independently and in pairs for support.

Step 4

Solves examples like √72, √128, and √200 step by step, showing the factorization and simplification.

Students copy the steps and perform similar problems.

NOTE ON BOARD:

  • Simplify √a × b: √(a × b) = √a × √b.
  • Example: √50 = √(25×2) = 5√2.

EVALUATION (5 exercises):

  1. Simplify √32.
  2. Simplify √45.
  3. Simplify √200.
  4. Simplify √24.
  5. Simplify √72.

CLASSWORK (5 questions):

  1. Simplify √36.
  2. Simplify √28.
  3. Simplify √50.
  4. Simplify √72.
  5. Simplify √98.

ASSIGNMENT (5 tasks):

  1. Simplify √8.
  2. Simplify √18.
  3. Simplify √200.
  4. Simplify √45.
  5. Create 5 surds and simplify them.

 

PERIOD 4: Addition and Subtraction of Surds

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1

Explains the rule for adding and subtracting surds: only like surds (with the same radicand) can be added or subtracted.

Students listen attentively and take notes.

Step 2

Demonstrates addition and subtraction with like surds. E.g., √2 + √2 = 2√2, 3√5 - √5 = 2√5.

Students follow the examples and solve additional problems.

Step 3

Gives practice exercises with different coefficients and surd radicands for addition and subtraction.

Students work on problems in pairs.

Step 4

Recaps the key points about adding and subtracting surds, ensuring that only surds with the same index can be combined.

Students ask questions and complete exercises.

NOTE ON BOARD:

  • Like surds can be added or subtracted: √a + √a = 2√a.
  • Unlike surds cannot be directly added or subtracted: √2 + √3 ≠ √5.

EVALUATION (5 exercises):

  1. Simplify √2 + √2.
  2. Simplify 3√7 - √7.
  3. Simplify 5√3 + 2√3.
  4. Simplify 4√5 - 3√5.
  5. Simplify √6 + √6.

CLASSWORK (5 questions):

  1. Simplify √3 + √3.
  2. Simplify 2√5 - √5.
  3. Simplify 5√2 + √2.
  4. Simplify √8 + √8.
  5. Simplify 3√6 - √6.

ASSIGNMENT (5 tasks):

  1. Simplify √12 + √12.
  2. Simplify √18 - √8.
  3. Simplify 2√3 + 4√3.
  4. Simplify √2 + √5.
  5. Explain the rule for adding surds.

 

PERIOD 5: Multiplication and Division of Surds (Including Rationalization)

PRESENTATION:

Step

Teacher’s Activity

Student’s Activity

Step 1

Introduces multiplication and division of surds. Demonstrates how to multiply surds (e.g., √2 × √3 = √6).

Students follow the teacher’s demonstration.

Step 2

Explains rationalization of the denominator. Shows how to rationalize surds in a denominator (e.g., 1/√2 becomes √2/2).

Students listen carefully and take notes.

Step 3

Performs practice exercises on multiplying and dividing surds, including rationalizing denominators.

Students work independently and in pairs for additional practice.

Step 4

Reviews the steps and provides additional examples of rationalizing surds.

Students practice and ask for clarification if needed.

NOTE ON BOARD:

  • Multiplying surds: √a × √b = √(a×b).
  • Rationalization: To rationalize √2/√3, multiply both numerator and denominator by √3: (√2/√3) × (√3/√3) = √6/3.

EVALUATION (5 exercises):

  1. Multiply √2 × √3.
  2. Multiply √5 × √5.
  3. Divide √18 by √2.
  4. Rationalize 1/√2.
  5. Multiply 3√5 × √7.

CLASSWORK (5 questions):

  1. Multiply √3 × √2.
  2. Divide √32 by √2.
  3. Rationalize 1/√5.
  4. Multiply 2√3 × √2.
  5. Rationalize 1/√7.

ASSIGNMENT (5 tasks):

  1. Rationalize 1/√6.
  2. Multiply √3 × √3.
  3. Divide √27 by √3.
  4. Simplify √5 × √4.
  5. Explain why rationalization of denominators is necessary in mathematics.