In this activity, students are given several sets of fractions and some clues about the size of a particular fraction in each set. To identify a fraction that meets certain size requirements or falls within a specified range, students need to use multiple comparison strategies they have learned. For example, they can use comparisons to benchmarks, such as and 1, to eliminate some fractions, and then use equivalent fractions to compare the remaining ones.

• “Which clues helped you eliminate fractions the fastest?” (clues about size relative to 1)
• “What strategies did you use to compare fractions?” (Compare the fractions to , 1, or another benchmark. Write equivalent fractions. Reason about fractions with the same numerator or denominator.)
• “Did you ever have to use more than one strategy to compare fractions?” (Yes, two or three were often needed to find the mystery fraction.)

This activity has two purposes: to give students an opportunity to solve fraction comparison problems in context, and to reinforce the idea that two fractions can be compared only if they refer to the same whole. To serve the former, students compare fractional distance measurements. To serve the latter, they investigate fractional measurements in two different units of distance: Chinese “li” and kilometer. 

When comparing the distances in the first question, students can rely on a number of familiar strategies. Two of the fractional values are close to 2. Some students are likely to use that benchmark for efficient comparison. For example, they may note that the school and the market are both a little over 1 li from home, the library is more than 2 li, and the badminton club is a little less than 2 li.

Focus the Activity Synthesis on the last two questions about interpreting fractional measurements in two different units.

• Ask previously selected students to share their responses to the last two questions. Display their number lines, or display the number lines from the activity for them to annotate while they explain.
• Emphasize that just as 1 km is not the same distance as 1 li, km is not the same distance as li. We can’t compare two fractions that refer to different wholes.