The purpose of this activity is for students to identify fractions using known benchmarks on the number line and to compare them to 1. Given a point on a number line, the location of 0, and one other benchmark value, students decide if the point represents a number greater or less than 1. They also quantify the distance of that number from 1. Students do so by relying on what they know about the number of fractional parts in 1 whole, as well as by looking for and making use of structure (MP7).

The work here develops students’ ability and flexibility in using number lines to reason about fractions. In later lessons, students will work with number lines that are increasingly more abstract to help them reason about fractions in more sophisticated ways.

• Invite previously selected students to share their responses. Display their work, or display the number lines from the task for them to annotate as they explain.
• “¿Cómo supieron qué fracción representa cada punto?” // “How did you know what fraction each point represents?” (Figure out what each space between tick marks represents, and then count the number of spaces.)  
• “¿Cómo supieron si es más que 1 o menos que 1?” // “How did you know if it’s more or less than 1?” (It is more than 1 if the point is to the right of 1, or if the numerator is greater than the denominator.)

In this optional activity, students sort a set of fractions into groups based on whether they are less than, equal to, or greater than . This sorting task enables students to estimate or to reason informally about the size of fractions relative to this benchmark before they go on to do so more precisely. In the next activity, students reason about fractions represented by unlabeled points on the number line and their distance from .

As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

This activity is optional because it asks students to reason about fractions without the support of the number line.

• Invite groups to share how they sorted the fractions. 
• “¿Cómo les ayudó el numerador y el denominador a saber cómo se relacionaba una fracción con ?” // “How did the numerator and denominator of each fraction tell you how a fraction relates to ?” (Sample responses: 
  • We already know fractions that are equivalent to , so we could compare any fraction to one of those equivalent fractions that has the same denominator.
  • A fraction that is equal to has a denominator that is twice the numerator.
  • If a numerator is less than half of the denominator, the fraction is less than . If the numerator is more than half of the denominator, it is greater than .
  • If a numerator is 1 or is much less than the denominator, then the fraction is small and less than  (except for  itself).
  • If a numerator is really close to the denominator, then the fraction is close to 1, which means it is greater than  (except for  itself).
• Give students 2–3 minutes of quiet time to complete the sentence frames in the activity. 

Previously, students located fractions on number lines and considered their distance and relative position to 1. Here, they think about fractions in relation to . The purpose of this activity is to prompt students to use another benchmark value to determine the relative size of a fraction.

While students may be able to visually tell if a point on the number line represents a number that is greater or less than , finding its distance to is less straightforward than finding its distance to 1. The former requires thinking about in terms of equivalent fractions.

In three cases, the fraction  and the point of interest are each on a tick mark on the number line. This makes it possible for students to quantify the distance without further partitioning the number line. In the last diagram, is not on a tick mark, prompting students to subdivide the given intervals, relying on their understanding of equivalence and relationships between fractions.

The work here encourages students to look for and make use of structure (MP7) and will be helpful later in the unit when students compare fractions by reasoning about their distance from benchmark values.

• “¿Cómo supieron dónde está  en cada recta numérica?” // “How did you know where is on each number line?” (Find out where 1 is, and then locate the halfway point. Use fractions that are equivalent to , such as , , and .)
• “¿En qué era diferente la última recta numérica de las otras?” // “What was different about the last number line compared to the others?” (There was no tick mark to represent on the number line. The number line had an odd number of intervals.)
• “¿Qué tuvieron que hacer de otra forma para averiguar a qué distancia de estaba la fracción en la última recta numérica?” // “What did you have to do differently to figure out how far away the fraction is from  on the last number line?” (First split each fifth into tenths and then locate .)