Compare fractions using benchmarks

Learning Objectives Identify and use the key benchmarks of 0, 1/2, and 1 to compare fractions. Determine if a fraction is less than, equal to, or greater than 1/2 by comparing the numerator to half of the denominator. Determine if a fraction is less than, equal to, or greater than 1 by comparing the numerator to the denominator. Compare two fractions by analyzing their relative positions to a common benchmark. Reason about the 'distance' of a fraction from a benchmark to compare two fractions on the same side of that benchmark. Apply benchmark reasoning to evaluate the magnitude of simple rational expressions for a given value of the variable. You have 7/8 of a pizza left, and your friend has 8/10 of a pizza left. Without doing complex math, who has more? 🍕 This...

TermDefinitionExample Benchmark FractionA common, easy-to-visualize fraction (like 0, 1/2, or 1) that we use as a reference point to compare other, more complex fractions.To compare 4/9 and 5/8, we can use the benchmark of 1/2. We see that 4/9 is less than 1/2, and 5/8 is greater than 1/2. NumeratorThe top number in a fraction. It represents how many parts of the whole you have.In the fraction 3/5, the numerator is 3. DenominatorThe bottom number in a fraction. It represents the total number of equal parts the whole is divided into.In the fraction 3/5, the denominator is 5. Rational ExpressionAn expression that is the ratio of two polynomials, essentially a fraction containing variables.(x + 2) / (x - 3). Comparing fractions is a foundational skill for estimating the value of these expres...

Comparing to the Benchmark 1 For a positive fraction \( \frac{a}{b} \): If \( a < b \), then \( \frac{a}{b} < 1 \). If \( a = b \), then \( \frac{a}{b} = 1 \). If \( a > b \), then \( \frac{a}{b} > 1 \). Use this rule to quickly determine if a fraction's value is less than, equal to, or greater than one whole by simply comparing the numerator and the denominator. Comparing to the Benchmark 1/2 For a positive fraction \( \frac{a}{b} \): If \( 2a < b \), then \( \frac{a}{b} < \frac{1}{2} \). If \( 2a = b \), then \( \frac{a}{b} = \frac{1}{2} \). If \( 2a > b \), then \( \frac{a}{b} > \frac{1}{2} \). Use this rule to determine if a fraction is less than, equal to, or greater than one-half. This is equivalent to checking if the numerator is less than...