Lesson 7

The purpose of this warm-up is to elicit students’ prior understanding of equivalence and strategies for comparing fractions. To determine equivalence, students may rely on familiarity with benchmark fractions, use fraction strips, or think about the relative sizes of the fractional parts. They may also use their knowledge about fractions with the same numerator or denominator. In any case, students have opportunities to look for and make use of structure (MP7).

• 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

• Share and record answers and strategy.
• Repeat with each statement.

• If no students refer to a visual representation (a tape diagram or number line) to explain an equation such as \(\frac{3}{4}=\frac{6}{8}\), ask how one of these representations could help with their explanation. 
• “For the pair of fractions that you know are not equal, can you tell which fraction is greater? How?”

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, partitioning each 1 fourth into 3 parts and then counting the twelfths).
• bundle given fractional parts into larger equal-size groups and count the new groups (for instance, bundling every 2 tenths to make 5 fifths in 1 whole and then counting the fifths).

During this and upcoming 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 fractional parts to the idea of using multiples and factors to find equivalent fractions.

• Groups of 2
• “In this activity, you’ll see diagrams and number lines that represent fractions.”
• “Find at least two fractions to describe the shaded part of each diagram, and two fractions for the point on each number line.”

• “Take 5 minutes to work independently on the first problem. Afterwards, share your thinking with your partner. ”
• 5 minutes: independent work time for the first problem
• 2 minutes: partner discussion
• “Now take a few minutes to work independently on the second problem.”
• 5 minutes: independent work time for the remaining problems
• 2 minutes: partner discussion
• Monitor for students who:
  • partition each unit fraction (a single section on a tape diagram or an interval on a number line) into smaller parts
  • bundle multiple unit fractions into larger groups
• Identify 2–3 students who reason differently on tape diagrams and 2–3 who reason differently on number lines.

Students label number lines using tick marks alone. For example if 4 marks are visible (including zero) each line would be labeled as fourths instead of thirds. If this happens, consider using the idea of movement from 0 to 1. Ask: “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?” Ask students to review the labels on their number lines and decide if revisions are needed before continuing to work on the next activity.

• 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 them 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: \(\frac{15}{12}\), \(\frac{20}{16}\))
• “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.

• Groups of 2
• “Work with a partner on this activity. One person is partner A and the other is B.”
• “Your task is to find two equivalent fractions for each fraction listed under A or B, and then convince your partner that your fractions are equivalent.”

• “Take 5 to 6 quiet minutes to find equivalent fractions and to plan your explanation.”
• 5–6 minutes: independent work time
• “Take turns sharing your fractions and explanation with your partner.”
• “When your partner explains, listen carefully to their reasoning and ask them to clarify if something is unclear.”
• 5–6 minutes: partner discussion

• “Create a visual display that shows how you found two equivalent fractions for the second fraction on your list: \(\frac{10}{6}\) for Partner A, and \(\frac{14}{10}\) for Partner B.”
• “Include diagrams, notes, and any descriptions that might help others understand your thinking.”
• See lesson synthesis.