Lesson 6

The purpose of this warm-up is for students to recognize that two values of reference are needed to determine the number that a point on the number line represents. The numbers 0 and 1 are commonly used when the numbers of interest are small. With only one number shown (for example, only a 0 or a 1), we can’t tell what number a point represents, though we can tell if the number is greater or less than the given number. These understandings will be helpful later in the lesson, as students determine the size of fractions relative to \(\frac{1}{2}\) and 1.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su pareja lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿Cómo podríamos saber qué número representa el punto? ¿Qué nos hace falta poner ahí?” // “How would we know what number the point represents? What’s missing and needs to be there?” (A label for one of the tick marks so that we’d know what each interval represents.)

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 also 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.

• Groups of 2–4
• “Díganle a su compañero una fracción que sea mayor que 1 y una fracción que sea menor que 1. Explíquenle cómo lo saben” // “Tell your partner a fraction that is greater than 1 and a fraction that is less than 1. Explain how you know.”
• 1 minute: partner discussion
• Share responses and ask how they used 1 whole to choose their fractions.
• Read the task statement as a class. Make sure students understand that they are to do three things for each number line diagram.

• “Antes de discutir con su grupo, trabajen unos minutos individualmente en al menos dos diagramas” // “Take a few minutes to work independently on at least two diagrams before discussing with your group.”
• 5 minutes: independent work time
• 5–7 minutes: group work time
• Monitor for students who:
  • label one or more tick marks with unit fractions
  • locate the number 1 on the number line when it is not given

• Select 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 one interval between tick marks represents, and then count the number of intervals.) 
• “¿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 \(\frac{1}{2}\). Sorting 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 \(\frac{1}{2}\).

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.

• Create a set of fraction cards from the blackline master for each group.

• Groups of 2–4
• Give each group one set of fraction cards

• “En grupo, clasifiquen las tarjetas de fracciones en tres categorías: menores que \(\frac{1}{2}\), iguales a \(\frac{1}{2}\) y mayores que \(\frac{1}{2}\). Prepárense para explicar cómo lo saben” // “Work with your group to sort the fraction cards into three groups: less than \(\frac{1}{2}\), equal to \(\frac{1}{2}\), or greater than \(\frac{1}{2}\). Be prepared to explain how you know.”
• “Cuando terminen, comparen su clasificación con la de otro grupo” // “When you are done, compare your sorting results with another group.”
• “Si los dos grupos no están de acuerdo sobre dónde va cierta fracción, discutan sus ideas y lleguen a un acuerdo” // “If the two groups disagree about where a fraction belongs, discuss your thinking until you reach an agreement.”
• 7–8 minutes: group work time
• 3–4 minutes: Discuss results with another group.
• “Anoten los resultados de su clasificación después de que los hayan discutido” // “Record your sorting results after you have discussed them.”

• 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 \(\frac{1}{2}\)?” // “How did the numerator and denominator of each fraction tell you how a fraction relates to \(\frac{1}{2}\)?” (Sample responses:
  • We already know fractions that are equivalent to \(\frac{1}{2}\), so we could compare any fraction to one of those equivalent fractions that has the same denominator.
  • A fraction that is equal to \(\frac{1}{2}\) has a denominator that is twice the numerator.
  • If a numerator is less than half of the denominator, the fraction is less than \(\frac{1}{2}\). If it is more than half of the denominator, it is more than \(\frac{1}{2}\).
  • If a numerator is 1 or is much less than the denominator, then the fraction is small and less than \(\frac{1}{2}\).
  • If a numerator is really close to the denominator, then the fraction is close to 1, which means it is more than \(\frac{1}{2}\).)
• 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 \(\frac{1}{2}\). The purpose of this activity is to prompt students to use another benchmark value to determine the size of a fraction.

While students may be able to visually tell if a point on the number line is more or less than \(\frac{1}{2}\), finding its distance to \(\frac{1}{2}\) is less straightforward than finding its distance to 1. The former requires thinking about \(\frac{1}{2}\) in terms of equivalent fractions.

In three cases, the fraction \(\frac{1}{2}\) 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, \(\frac{1}{2}\) 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.

• Groups of 2–4
• “Identifiquemos más fracciones en rectas numéricas, pero esta vez averigüemos cómo se relacionan con \(\frac{1}{2}\)” // “Let’s identify a few more fractions on number lines, but this time, let’s find out how they relate to \(\frac{1}{2}\).”

• “Trabajen unos minutos individualmente en al menos dos diagramas. Después, discutan con su grupo” // “Work independently for a few minutes. Work through at least two diagrams before discussing with your group.”
• 5 minutes: independent work time
• 5 minutes: group work time
• Monitor for students who:
  • locate 1 and \(\frac{1}{2}\) on the number line.
  • label the point for \(\frac{1}{2}\) with an equivalent fraction whose denominator matches the number of intervals between 0 and 1. (for example, labeling the middle tick mark on the first number line with \(\frac{3}{6}\).)
  • on the last number line, subdivide the intervals of fifths into tenths in order to locate \(\frac{1}{2}\).

• “¿Cómo supieron dónde está \(\frac{1}{2}\) en la recta numérica?” // “How did you know where \(\frac{1}{2}\) is on the number line?” (Find out where 1 is, and then locate the halfway point. Use fractions that are equivalent to \(\frac{1}{2}\), such as \(\frac{3}{6}\), \(\frac{4}{8}\), and so on.)
• “¿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 \(\frac{1}{2}\) 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 \(\frac{1}{2}\) estaba la fracción?” // “What did you have to do differently to figure out how far away the fraction is from \(\frac{1}{2}\)?” (First split each fifth into tenths and then locate \(\frac{5}{10}\).)