Algebra 1: Evaluating Numerical Expressions with Rational Numbers

Overview

• Focus on evaluating numerical expressions involving rational numbers.
• Key concept: Order of operations with fractions and mixed numbers.

Key Concepts and Steps

Order of Operations

• Follow the order of operations: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Multiplication and division are at the same level; perform these operations as they appear from left to right.

Working with Fractions and Mixed Numbers

• Multiplication and Division with Fractions:
  • Use the reciprocal for division (e.g., divide by 5/1 is the same as multiply by 1/5).
  • Simplify fractions when possible by reducing.
• Addition and Subtraction with Fractions:
  • Convert all fractions to a common denominator before adding or subtracting.
  • Be careful with signs, especially negative signs.

Examples Discussed

1. Example 1:

   • Expression: 1 + (-2/3) * 3/4
   • Simplified: Do multiplication first, then addition.
   • Result: 1/2

2. Example 2:

   • Expression: Division changed to multiplication using reciprocal.
   • Result: Proper use of order gives 13/4.

3. Example 3:

   • Multiplication and division with decimals.
   • Conversion of operations for clarity.

4. Example 4:

   • Complex operations involving negatives and mixed numbers.
   • Result: Proper order yields 1/4.

5. Example 5:

   • Challenge with common denominators.
   • Resulting fraction: -3/20 or -0.15 as a decimal.

6. Example 6:

   • Negative operations with fractions.
   • Result: Simplified to 3.

7. Example 7:

   • Handling parentheses and squaring negatives.
   • Result: 15/16.

8. Example 8:

   • Mixed numbers converted to fractions.
   • Result: 39/25.

9. Example 9:

   • Squaring and cubing operations.
   • Result: -33/64.

10. Example 10:

    • Working with decimals and fractions.
    • Result: -0.747 as a decimal.

11. Example 11:

    • Negative fractions and careful subtraction.
    • Result: -179/40.*

Tips for Success

• Always simplify fractions when possible.
• Be cautious with negatives and double negatives.
• Use common denominators for addition and subtraction.
• Consider doing calculations by hand for better understanding, but check complex calculations with a calculator.

Conclusion

• Key takeaway is the importance of following the correct order of operations and being meticulous with sign changes, especially when dealing with fractions and mixed numbers.