Mathematics Lecture Notes: Grade 9 - Square Roots

Introduction

• Lecture focuses on the concept of square roots as part of the grade nine mathematics curriculum.
• Delivered in English.

Key Concepts

Perfect Square

• Definition: A number that can be expressed as the product of an integer by itself.
  • Example: 5 x 5 = 25, thus 25 is a perfect square.
  • Alternative definition: The second exponent of an integer (e.g. 5² = 25).
• Examples of Perfect Squares:
  • 1 (1 x 1)
  • 4 (2 x 2)
  • 9 (3 x 3)
  • 16 (4 x 4)
  • 25 (5 x 5)
  • 36 (6 x 6)
  • 49 (7 x 7)

Square Roots

• Finding Square Roots of Perfect Squares:
  • Example: √4 = 2, √49 = 7, √121 = 11, √169 = 13

Non-Perfect Squares

• Non-perfect squares do not have integer square roots (e.g., √13, √10, √105).
• Method to Find Square Roots of Non-Perfect Squares:
  • Use perfect squares to approximate.
  • Example: √20 is approximated between the square roots of 16 and 25, thus between 4 and 5.
• Process:
  • Identify two perfect squares between which the number lies.
  • Example: 20 is between 16 (4²) and 25 (5²), so √20 is between 4 and 5.

Solving Examples

1. Find the Square Root of 20:

   • 20 is not a perfect square.
   • Locate between 16 (4²) and 25 (5²), hence between 4 and 5.

2. Square Root of 110:

   • 110 is between 100 (10²) and 121 (11²), hence between 10 and 11.

3. Square Root of Negative Numbers:

   • Example: Negative √15, locate between √9 and √16, hence negative 4 and negative 5.

Practice Questions

1. Square Root Lies Between 11 and 12:

   • Numbers between 121 (11²) and 144 (12²): Answer is 131.

2. 44 Located Between Two Integers:

   • Between 36 (6²) and 49 (7²): Answer is between 6 and 7.

3. Value Between Negative 7 and Negative 8:

   • Numbers between 49 (7²) and 64 (8²): Answer is 57.

4. 72 Lie Between Which Two Integers:

   • Between 64 (8²) and 81 (9²): Correct range is between these two squares.

5. Square Root Between 14 and 15:

   • Numbers between 196 (14²) and 225 (15²): Answer is 200.

Conclusion

• Recap of the importance of understanding perfect squares and estimating non-perfect square roots.
• Emphasis on using a list of perfect squares for efficient calculations.

• End of Class: Encouragement for students to practice and apply these methods.
• Instructor Sign-off: "I hope you get benefits from this class."

Additional Resources

• Practice worksheets on perfect squares and square roots.
• Online calculators for practicing square root estimations.