The purpose of this activity is for students to compare the sizes of different products when one factor stays the same, allowing students to focus on the size of the varying factor. Students should be encouraged to use whatever strategies and representations make sense to them. Monitor for students who:

• Use number lines.
• Use multiplication to compute the total distances.
• Reason about the relationship between the fraction of the trail and the total distance, without performing any computations.

This activity uses MLR2 Collect and Display. Advances: Reading, Speaking.

• “Are there any other words or phrases that are important to include in our display?”
• As students share responses, update the display by adding (or replacing) language, diagrams, or annotations.
• Remind students to borrow language from the display as needed.
• Display: miles
• “Whose distance does this expression represent?” (Kiran)
• “What multiplication expression represents the number of miles Noah ran?” ( )
• “How did you decide to order the lengths that each student ran?” (The length of the trail is always 5, so we can just compare the fraction factors.)
• Display the problem about Tyler.
• “What numbers make sense? Why?” (I can use any fraction that is greater than and less than . It has to be greater than so that Tyler runs farther than Noah. It has to be less than so that Elena runs farther than Tyler.)

The purpose of this activity is for students to compare a fractional amount of a whole number with that same whole number. Students may calculate, draw a diagram, or reason about the size of the factor. When they choose their own numerator or denominator to make equations and inequalities true, monitor for students who:

• Experiment with different numerators and multiply the fraction by the whole number to see if it makes the statement true.
• Choose a numerator of 1 and use their understanding of a unit fraction as part of a whole.
• Explain why more than one answer makes sense.

Action and Expression: Internalize Executive Functions. Invite students to verbalize their strategy for comparing fractional amounts before they begin. Students can speak quietly to themselves, or share with a partner.

• Display the inequality:
• Invite students to share their responses.
• “How do you know ?” (Because .)
• “How do you know ?” (Because it is 10 of 50 parts, there are 40 other parts.)
• Display inequality:
• Invite students to share their responses.
• “How do you know ?” (Because is less than 1.   is equal to 5.)
• Display the equation:
• “How did you find the number that makes this equation true?” (It has to be equal to 1 and  .)