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Which Fraction is Bigger - and Why

Comparing fractions is about more than following one procedure. Fractions can be compared in many different ways, depending on the numbers and the situation. In this topic, we explore several strategies for deciding which fraction is larger and, just as importantly, explaining why.

One common method is to use common denominators, which allows us to compare fractions that have the same-sized pieces. For example, comparing 3/4​ and 5/8​ becomes easier if both fractions are written in eighths. Another method is to use common numerators, which allows us to compare fractions with the same number of pieces. For example, 3/5​ is larger than 3/8 because fifths are larger pieces than eighths when the whole is the same size.

We will also use reasoning strategies such as comparing the size of the pieces, the number of pieces, and the distance from benchmark fractions like 0, 1/2​, and 1. Sometimes it is helpful to think about missing pieces, such as noticing that 7/8​ is only 1/8​ away from 1. Other times, it helps to think about extra pieces, especially when comparing fractions greater than 1.

As future teachers, it is important to help students see fraction comparison as reasoning about quantities, not just cross-multiplying or memorizing a rule. Different strategies make sense for different fractions, and students should learn to choose a method that fits the numbers.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Compare fractions using common denominators.

  • Compare fractions using common numerators.

  • Explain how the size of the pieces affects the size of the fraction.

  • Use benchmark fractions such as 0, 1/2​, and 1 to compare fractions.

  • Use missing pieces or extra pieces reasoning to compare fractions.

  • Choose an efficient comparison strategy based on the fractions given.

  • Explain fraction comparisons using words, pictures, and symbols.

Key Vocabulary

  • Compare - Decide whether one number is greater than, less than, or equal to another number.

  • Common Denominator - A shared denominator used to compare or combine fractions.

  • Common Numerator - A shared numerator used to compare fractions by looking at piece size.

  • Benchmark Fraction - A familiar fraction, such as 1/2​ or 1, used to estimate or compare.

  • Missing Pieces - Thinking about how far a fraction is from one whole.

  • Extra Pieces - Thinking about how far a fraction is beyond one whole.

  • Equivalent Fractions - Fractions with different names that represent the same amount.

  • Whole - The complete amount being divided or compared.

Strategies

  • Common Denominators - 3/4​ vs. 5/8 - 3/4 = 6/8​, so 3/4 is larger.

  • Common Numerators - 3/5 vs. 3/8 - Both have 3 pieces, but fifths are larger than eighths.

  • Size and Number of Pieces - 2/3​ vs. 3/8 - Two large pieces may be more than three small pieces; use a model or benchmark to check.

  • Benchmark Fractions - 4/9​ vs. 5/8 - ​4/9 is less than 1/2​, while 5/8​ is greater than 1/2​.

  • Missing Pieces - 7/8​ vs. 9/10 - ​7/8​ is missing 1/8​; 9/10 is missing 1/10​. Since 1/10​ is smaller, 9/10​ is larger.

  • Extra Pieces - 1 2/3​ vs. 1 3/5 - Both have 1 whole, so compare the extra parts: 2/3​ is larger than 3/5​.

Teacher Connection

Fraction comparison is a place where students often overuse rules without thinking about size. For example, a student might say 1/8​ is larger than 1/4​ because 8 is larger than 4. This shows that the student is thinking about the denominator as a whole number, rather than as the number of equal parts in the whole.

As a future teacher, it is important to ask students to explain their comparisons. The goal is not only for students to choose the correct symbol, but to understand why one fraction is larger, smaller, or equal to another.

Helpful teacher questions include:

  • What is the whole?

  • Are the pieces the same size?

  • Which fraction has more pieces?

  • Which fraction has larger pieces?

  • Is either fraction close to 12\frac{1}{2}21​ or 1?

  • How far is each fraction from one whole?

  • What model could help us see the comparison?

Quick Reflection Question

Why might 7/8​ and 9/10​ be easier to compare by thinking about missing pieces instead of finding common denominators?