In earlier lessons, students compared fractions by rewriting one fraction as an equivalent fraction with the same denominator as the second fraction. In this activity, students see that—although it’s still possible to compare the fractions—this particular strategy doesn’t work if neither of the denominators of the two fractions is a factor or multiple of each other. Students learn that in such a case, both fractions can be expressed as equivalent fractions with a common denominator that is different from either fraction’s original denominator and is a multiple of both.

• Display Priya and Lin’s reasoning for all to see. Select students to share their observations on how the two are alike and how they are different.
• Highlight the fact that both students rewrote the two fractions so that they have a common denominator.
• “Why might it be helpful to write equivalent fractions with the same denominator?” (It is easier to compare the fractions when the fractional part is the same size.)
• “Can we choose any number to be the common denominator?” (No, it must be a multiple of both of the original denominators.)
• “Does it matter if we choose a smaller or a larger common multiple?” (No, but it could work better to multiply by a smaller number.)

This activity serves two main goals: to prompt students to rewrite pairs of fractions as equivalent fractions with a common denominator, and to consider this newly developed skill as a possible way to compare fractions.

To write equivalent fractions, many students are likely to reason numerically (by multiplying or dividing the numerator and denominator by a common number). Some may, however, find equivalent fractions effectively by continuing to reason about how many of this fractional part is in that fractional part.

To compare the fractions in the second question, students may choose to write equivalent fractions with a common denominator because they were just learning to do so. The fractions, however, were chosen so that students have opportunities to choose an approach strategically, rather than writing equivalent fractions each time. For instance, students may notice that:

• In part a, one fraction is away from , and the other is from .
• In part b, one fraction is greater than 2 and the other is less than 2.
• In part c, writing an equivalent fraction for only one given fraction (rather than for both) is sufficient for comparing.
• In part d, one fraction is less than , and the other is greater than .

• See Lesson Synthesis.