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Comparing Fractions
Same denominator, same numerator, and benchmarks
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Comparing fractions means deciding which fraction is greater, which is less, or whether they are equal. This skill matters because fractions show up in food, measurement, money, time, and data. A fraction comparison toolbox helps students choose a smart strategy instead of guessing. Pictures like fraction bars, circles, and number lines make the size of each fraction easier to see.
Some comparisons are quick when the fractions have the same denominator or the same numerator. Other comparisons are easier when you use a benchmark, such as 1/2, to see which fraction is closer to a familiar amount. For harder pairs, cross multiplication can compare fractions without drawing a model. The goal is to understand the size of the parts, not just follow steps.
Key Facts
- Same denominator: compare the numerators, so 3/8 > 2/8.
- Same numerator: the fraction with the smaller denominator is larger, so 3/4 > 3/8.
- Benchmark with one half: compare each fraction to 1/2 to help decide which is larger.
- A fraction is greater than 1/2 when the numerator is more than half of the denominator, such as 5/8 > 1/2.
- Cross multiplication: for a/b and c/d, compare a x d and c x b.
- Comparison symbols: > means greater than, < means less than, and = means equal to.
Vocabulary
- Fraction
- A fraction is a number that names part of a whole or part of a group.
- Numerator
- The numerator is the top number in a fraction and tells how many parts are being counted.
- Denominator
- The denominator is the bottom number in a fraction and tells how many equal parts make the whole.
- Benchmark fraction
- A benchmark fraction is a familiar fraction, such as 1/2, used to estimate and compare other fractions.
- Cross multiplication
- Cross multiplication is a method for comparing two fractions by multiplying each numerator by the other fraction's denominator.
Common Mistakes to Avoid
- Comparing only the numerators is wrong because the denominators tell the size of the parts. For example, 3/10 is less than 2/3 even though 3 is greater than 2.
- Thinking a larger denominator always means a larger fraction is wrong because bigger denominators make smaller equal parts when the numerator stays the same. For example, 1/8 is less than 1/4.
- Using cross multiplication but mixing up the products is wrong because each product must match the correct fraction. For 2/3 and 3/5, compare 2 x 5 with 3 x 3.
- Forgetting that the wholes must be the same size is wrong because fractions can only be compared fairly when they refer to equal-size wholes. One half of a large pizza can be more food than three fourths of a small pizza.
Practice Questions
- 1 Compare 5/8 and 3/8 using >, <, or =.
- 2 Compare 4/6 and 5/9 using cross multiplication.
- 3 Which is greater, 3/7 or 5/12? Explain which strategy you would use and why.