Fraction Bars are Secret Grouping Symbols
A fraction bar acts as a grouping symbol, meaning all operations in the numerator must be completed and all operations in the denominator must be completed before performing the final division. In Grade 6 Saxon Math Course 1 (Chapter 1: Number, Operations, and Algebra), students evaluate expressions like (3 + 7) / (2 x 5): first compute the numerator 3 + 7 = 10, then the denominator 2 x 5 = 10, then divide 10 / 10 = 1. This is equivalent to having invisible parentheses around numerator and denominator. Students who ignore this rule and divide before simplifying both parts obtain incorrect results.
Key Concepts
Property Before dividing, perform the operations above the bar and below the bar.
Examples $\frac{5+7}{1+2} = \frac{12}{3} = 4$.
$\frac{20 4}{2 \times 2} = \frac{16}{4} = 4$.
Common Questions
Why is the fraction bar a grouping symbol?
The fraction bar separates the numerator and denominator, signaling that each side must be fully simplified before the final division is performed. It acts like parentheses.
Evaluate (4 + 6) / (3 x 5).
Numerator: 4 + 6 = 10. Denominator: 3 x 5 = 15. Final: 10 / 15 = 2/3.
Evaluate (12 - 3) / (2 + 1).
Numerator: 12 - 3 = 9. Denominator: 2 + 1 = 3. Final: 9 / 3 = 3.
What mistake occurs when students ignore the fraction bar as a grouping symbol?
They may divide a single digit from the numerator by a single digit from the denominator before completing the numerator or denominator, getting an incorrect result.
How does the fraction bar relate to order of operations?
The fraction bar creates implied parentheses. Evaluate the numerator (top group) and denominator (bottom group) completely before dividing, just like you would evaluate expressions in parentheses first.