The purpose of this activity is to remind students of a key insight from grade 3—that the same point on the number line can be named with fractions that don’t look alike. Students see that those fractions are equivalent, even though their numerators and denominators may be different. 

Students have multiple opportunities to look for regularity in repeated reasoning (MP8). For instance, they are likely to notice that: 

• Fractions that have the same number for the numerator and denominator all represent 1. 
• In fractions that describe the halfway point between 0 and 1, the numerator is always half the denominator, or the denominator twice the numerator.
• In fractions that describe , the denominator is 4 times the numerator.

These observations will help students to identify and generate equivalent fractions later in the unit.

• Select students to share their responses and reasoning for the first set of questions. Highlight explanations that convey that:
  • Any fraction with the same number for the numerator and denominator has a value of 1.
  • Equivalent fractions share the same location or are the same distance from 0 on the number line.
• Invite previously selected students to share their responses for the second set of questions. 
• Consider asking:
  • “¿Alguien puede expresar el razonamiento de _______ de otra forma?” // “Who can restate _______ 's reasoning in a different way?”
  • “¿En qué se parecen las maneras de representar los puntos con fracciones? ¿En qué son diferentes?” //  “How are these ways for labeling the points the same? How are they different?“

In this activity, students reason about whether two fractions are equivalent in the context of distance. To support their reasoning, students use number lines and their understanding of fractions with related denominators (where one number is a multiple of the other). The given number lines each have only one tick mark between 0 and 1, so students need to partition each line strategically to represent two fractions with different denominators on the diagram. 

To help students intuit the distance of 1 mile, consider preparing a neighborhood map that shows the school and some points that are a mile away. Display the map during the Launch.

• Invite previously selected students to share their responses and how they knew that is equivalent to but is not.
• To facilitate their explanations, ask students to display their work, or display blank number lines for them to annotate.
• Consider asking:
  • “¿Alguien pensó de la misma forma, pero lo explicaría de otra manera?” // “Who reasoned the same way but would explain it differently?”
  • “¿Alguien lo pensó de otra forma y aun así llegó a la misma conclusión?” // “Who thought about it differently but arrived at the same conclusion?”