8th Grade Math - Real Numbers Lesson

Overview

This lesson covers the set of real numbers, which is divided into two subsets: rational numbers and irrational numbers. It aligns with the Texas Essential Knowledge and Skills (TEKS) standards 8.2.a and 8.2.b.

Texas Essential Knowledge and Skills (TEKS)

• 8.2.a: Extend knowledge of sets and subsets using visual representation to describe relationships between sets of real numbers.
• 8.2.b: Approximate the value of an irrational number, including π and square roots of numbers less than 225, and locate that rational number approximation on a number line.

Real Numbers

• Composed of two subsets: Rational Numbers and Irrational Numbers.

Rational Numbers

• Can be expressed as a fraction ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ).
• Include integers, fractions, and repeating or terminating decimals.
• Subsets include:
  • Whole Numbers: Rational numbers ≥ 0 without fractions or decimals {0, 1, 2, 3, ...}.
  • Integers: Whole numbers and their negatives {..., -3, -2, -1, 0, 1, 2, 3, ...}.
  • Natural Numbers: Rational numbers > 0 without fractions or decimals.

Irrational Numbers

• Cannot be expressed as a fraction ( \frac{a}{b} ), where ( a ) and ( b ) are integers and ( b \neq 0 ).
• Include non-repeating, non-terminating decimals.

Subset Relationships

• Rational Numbers and Irrational Numbers are subsets of Real Numbers.
• Integers are a subset of Rational Numbers and also Real Numbers.
• Whole Numbers are a subset of Integers, Rational Numbers, and Real Numbers.
• Natural Numbers are a subset of Whole Numbers, Integers, Rational Numbers, and Real Numbers.

Approximating Irrational Numbers

• If a number is between two perfect squares, its square root is between the square roots of those perfect squares.

Example

• Estimate the value of ( \sqrt{30} ) as between two whole numbers:
  1. Identify perfect squares around 30: 25 and 36.
  2. ( \sqrt{30} ) is between ( \sqrt{25} = 5 ) and ( \sqrt{36} = 6 ).
• Better Approximation:
  • 30 is about halfway between 25 and 36.
  • Approximate ( \sqrt{30} ) to be between 5.4 and 5.6:
    • Check:
      • ( (5.4)^2 = 29.16 )
      • ( (5.6)^2 = 31.36 )
    • Since 30 is between 29.16 and 31.36, ( \sqrt{30} ) is approximately between 5.4 and 5.6.