Lesson 5

The purpose of this warm-up is to elicit the idea that number lines can be partitioned into intervals smaller than 1, which will be useful when students see number lines partitioned into fractions in a later activity. While students may notice and wonder many things, the idea that fractions can be represented on the number line is the important discussion point. Students do not need to identify the tick mark as showing \(\frac{1}{2}\) in the warm-up, as that will be the focus later in the lesson.

This prompt gives students opportunities to look for and make use of structure (MP7). The specific structure they might notice is that each number line is partitioned in half.

• 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.

• “¿Qué saben sobre el número que cada marca representa?” // “What do you know about the number each tick mark represents?” (On the first number line, it is 5 because it is halfway between 0 and 10. On the second number line, I think it is \(\frac{1}{2}\) because it is halfway between 0 and 1.)

The purpose of this activity is for students to further develop the idea that fractional amounts can be represented on a number line. Students sort a given set of cards showing number lines. They first sort in a way of their choice, which might include number of parts or length of the number line. Monitor for different ways groups choose to categorize the number lines, but especially for categories that distinguish between number lines with whole number partitions and fractional partitions.

When students identify common properties of the number lines for their sorts, such as the numbers listed on the tick marks or the total number of tick marks, they look for and make use of structure (MP7).

• Create a set of cards from the blackline master for each group of 2. 

• Groups of 2
• Distribute one set of pre-cut cards to each group of students.

• “Con su compañero, clasifiquen las rectas numéricas en las categorías que ustedes escojan. Asegúrense de tener un nombre para cada categoría” // “Work with your partner to sort some number lines into categories that you choose. Make sure you have a name for each category.”
• 3-5 minutes: partner work time
• Select groups to share their categories and how they sorted their cards.
• Choose as many different types of categories as time allows. Be sure to highlight categories created based on whether the tick marks represent whole numbers or fractions.
• If not mentioned by students, ask, “¿Podemos clasificar las rectas numéricas según lo que representan las marcas?” // “Can we sort the number lines based on what the tick marks represent?”
• “Veamos las rectas B y E. Ambas están partidas en 4 partes. ¿Qué representan las marcas de la recta E que no tienen un número?” // “Let’s look at B and E. Both are partitioned into 4 parts. What do the unlabeled tick marks in E represent?” (1, 2, 3) “¿Qué creen que representan las de la tarjeta B?” // “What do you think those in B represent?” (\(\frac{1}{4}\), \(\frac{2}{4}\), \(\frac{3}{4}\), or amounts less than 1)
• “Tómense un minuto para clasificar sus tarjetas dependiendo de si las marcas de las rectas numéricas representan números enteros o de si las marcas representan fracciones” // “Take a minute to sort your cards by number lines where the tick marks only represent whole numbers and number lines where the tick marks represent fractions.”
• 1-2 minutes: partner work time

If students don’t identify number lines that would have fractions marked, consider asking:

• “¿Qué otros números podemos encontrar en esta recta numérica?” // “What other numbers can we find on this number line?”
• “¿Qué pueden ser las marcas que están entre los números enteros?” // “What could the marks in between the whole numbers be?”

• “¿Cómo supieron si una recta numérica tenía marcas que representan fracciones?” // “How did you know if a number line had tick marks that represent fractions?” (If there is one or more tick marks between two back-to-back whole numbers like 0 and 1, or 1 and 2, then the tick marks between them represent fractions.)
• Attend to the language that students use to describe their number lines, giving them opportunities to describe the number lines more precisely.
• Highlight the use of phrases like “partes menores que 1” // “parts less than 1” or “partir una parte en partes más pequeñas que sean menores que 1” // “partitioning one part into smaller parts less than 1.”
• Consider displaying a number line with fractions that are less obvious, such as number line I. Ask students to help identify the fractions on that number line, and label 1 and 3 so that the tick marks between the whole numbers are clear.

The purpose of this activity is to transition students from thinking about fractional lengths on fraction strips to thinking about fractions as numbers on the number line. Students build on their experience of folding fraction strips to fold number lines into halves, thirds, fourths, sixths, and eighths and then label unit fractions.

Students begin by considering how the fraction \(\frac{1}{2}\) can be labeled on the number line. They learn that each part of the number line has a length of one half, but the endpoint of the first one-half part is the location of the number \(\frac{1}{2}\) on the number line. This distinction is important for understanding fractions as numbers that can be represented as points on the number line and for using the number line precisely (MP6).

When folding the number lines, students also need to attend to the fact that it is the interval between 0 and 1 that needs to be partitioned, rather than the length of the entire strip of paper that contains each number line.

• Each student needs at least 5 number lines from 0 to 1. Each copy of the blackline master contains a few extra number lines, in case students fold incorrectly at first.
• Create a number line folded into fourths and a fraction strip that shows fourths to display in the synthesis.

• Groups of 2
• “Hemos estado pensando en qué lugar de la recta numérica están ubicadas las fracciones. Dediquemos un tiempo a pensar cómo marcar las fracciones en la recta numérica” // “We’ve been thinking about where fractions are located on the number line. Let’s take some time to think about how to label fractions on the number line.”
• “Tómense un minuto para pensar en las rectas numéricas de Andre y Clare” // “Take a minute to think about Andre and Clare’s number lines.”
• 1 minute: quiet think time
• “Hablen con su compañero sobre por qué ambas formas de marcar la recta numérica pueden tener sentido” // “Talk to your partner about how each student’s labeling could make sense.”
• 2–3 minutes: partner discussion
• Share responses.
• Display a number line with both the distance to \(\frac{1}{2}\) and the number \(\frac{1}{2}\) marked in a color, such as:

• “Andre estaba pensando en las partes que tenían longitud \(\frac{1}{2}\), por eso él marcó con \(\frac{1}{2}\) las partes que iban de 0 a \(\frac{1}{2}\) y de \(\frac{1}{2}\) a 1” // “Andre was thinking about the parts that had length \(\frac{1}{2}\), so he labeled the parts from 0 to \(\frac{1}{2}\) and \(\frac{1}{2}\) to 1 with \(\frac{1}{2}\).”
• “Para ubicar el número \(\frac{1}{2}\), encontramos el extremo derecho de la primera mitad (la parte que empieza en 0). Luego lo marcamos” // “To locate and label the number \(\frac{1}{2}\), we find the endpoint of the first one-half part from 0 and label it.”
• Give each student a copy of the blackline master and scissors.

• “Recorten y separen las rectas numéricas” // “Cut the number lines apart.”
• “Luego, doblen una en medios, otra en tercios, otra en cuartos, otra en sextos y otra en octavos” // “Then, fold one into halves, one into thirds, one into fourths, one into sixths, and one into eighths.”
• “Mientras doblan las rectas numéricas, compartan con su compañero sus estrategias para doblarlas” // “As you fold, share your folding strategies with your partner.”
• “Dibujen marcas sobre sus dobleces y marquen la fracción unitaria en cada recta numérica” // “Draw tick marks along your folding lines and label the unit fraction on each number line.”
• 3–5 minutes: partner work time
• Monitor for students who need support lining up the 0 and the 1 as they fold. Consider suggesting that they cut off the ends of the number line at 0 and 1, or marking 0 and 1 on both sides of each paper strip to make them easier to see while folding.

• Display a number line folded into fourths and a fraction strip of fourths.
• “¿En qué se pareció partir estas rectas numéricas a partir nuestras tiras de fracciones?” // “How was partitioning these number lines similar to partitioning our fraction strips?” (The number lines were folded just like the strips were, but instead of a rectangle it’s just a line. We labeled the location of the unit fractions at the end of the first part on each number line instead of the space.)