Compare Fractions with the Same Numerator

10 min

Narrative

The purpose of this True or False? is to elicit insights students have about comparing unit fractions. Students’ reasoning helps to deepen their understanding of what the denominator of a fraction means. It also will be helpful later when students compare fractions with the same numerator.

In this activity, students have an opportunity to look for and make use of structure (MP7) because they notice that a larger denominator indicates that a whole is split into more parts. The more parts into which the whole is split, the smaller those parts.

Launch

  • Display one statement.
  • “Give me a signal when you know whether the statement is true and can explain how you know.”
  • 1 minute: quiet think time
Teacher Instructions
  • Share and record answers and strategies.
  • Repeat with each statement.

Student Task

Decide whether each statement is true or false. Be prepared to explain your reasoning.

  • 12>14\frac{1}{2} > \frac{1}{4}
  • 14>13\frac{1}{4} > \frac{1}{3}
  • 16>18\frac{1}{6} > \frac{1}{8}
Solution Steps (4)
  1. 1
    Evaluate 1/2 > 1/4
    →TRUE - halves are bigger pieces than fourths
  2. 2
    Evaluate 1/4 > 1/3
    →FALSE - fourths are smaller than thirds (more parts = smaller pieces)
  3. 3
    Evaluate 1/6 > 1/8
    →TRUE - sixths are bigger than eighths
  4. 4
    Recognize pattern
    →With same numerator, larger denominator = smaller fraction

Sample Response

  • True: I know that one-fourth is half the size of one-half.
  • False: If the whole is split in 3 parts, those parts are bigger than if there were 4 parts, so one-fourth is smaller than one-third.
  • True: Eighths are smaller than sixths because if we split the whole into more parts, each part is smaller.
Activity Synthesis (Teacher Notes)
  • Consider asking:
    • “Who can restate _____’s reasoning in a different way?”
    • “Does anyone want to add on to _____’s reasoning?”
Standards
Addressing
  • 3.NF.3.d·Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
  • 3.NF.A.3.d·Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols <span class="math">\(&gt;\)</span>, =, or <span class="math">\(&lt;\)</span>, and justify the conclusions, e.g., by using a visual fraction model.

20 min

15 min