The purpose of this activity is to elicit strategies for finding equivalent fractions when the fractions are represented by tape diagrams or points on the number line. Students may reason in various ways, but here are two likely approaches:

• Partition given fractional parts into smaller equal-size parts and count the new parts. (For instance, partition each 1 fourth into 3 parts and then count the twelfths.)
• Bundle given fractional parts into larger equal-size groups and count the new groups. (For instance, bundle every 2 tenths to make 5 fifths in 1 whole and then count the fifths).

During the Activity Syntheses, help students recognize regularity in their moves to find equivalent fractions. In future lessons, students will connect more explicitly how diagrams of equivalent fractions relate to a numerical process for generating them. They will relate the subdividing or grouping of fractional parts to the idea of using multiples and factors to find equivalent fractions.

• “Where is 1 on the number line?”
• “If we are moving from 0 toward 1, what does this tick mark between 0 and 1 mean?”

• Select previously identified students to share how they found multiple equivalent fractions on the two kinds of representations. Display their work, or display the diagrams in the activity for students to annotate as they explain.
• “How is the process of finding equivalent fractions using diagrams like the process of using number lines?” (They both involve partitioning given parts into smaller parts, or bundling the given parts into larger parts.)
• “How are they different?” (The length of a diagram usually is 1 whole or another whole number. A number line doesn’t always show 1 whole, so we may have to figure out where it is first.)
• If time permits: “Can you write other equivalent fractions for diagram _____?” (Sample response for the last number line diagram: , )
• “How many fractions do you think you could write for that diagram?” (This prompts students to begin to realize that there are infinite equivalent fractions as the whole is partitioned into smaller parts.)

In this activity, students find equivalent fractions for fractions given numerically. They also work to clearly convey their thinking to a partner, which involves choosing and using words, numbers, or other representations with care. In doing so, students practice attending to precision (MP6) as they communicate about mathematics.

• “Create a visual display that shows how you found two equivalent fractions for the second fraction on your list: for Partner A, and for Partner B.”
• “Include diagrams, notes, and any descriptions that might help others understand your thinking.”
• See Lesson Synthesis.