Lesson 13

This warm-up elicits observations about the different ways whole numbers can be expressed as fractions. Students have previously seen number lines where 1, 2, and 3 were labeled with fractions in halves, thirds, fourths, sixths, and eighths. They understand that a denominator of 2 corresponds to 2 equal parts in the length representing 1 whole. The number line marked with \(\frac{1}{1}\), \(\frac{2}{1}\), and \(\frac{3}{1}\) is shown together with those marked with halves, thirds, and fourths to highlight that a denominator of 1 means each whole has 1 part.

In the synthesis, students learn that fractions with 1 as a denominator can be used to represent whole numbers (\(\frac{2}{1} = 2\)).

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

• “¿Qué puede significar tener a 1 como denominador?” // “What could it mean to have a denominator of 1?” (The whole hasn’t been partitioned. The whole has been partitioned into 1 part.)
• Have students label the locations of \(\frac{1}{1}\), \(\frac{2}{1}\), and \(\frac{3}{1}\) on the first number line with 1, 2, and 3.

• “La longitud de 0 a 1 no se ha partido, así que cada parte tiene una longitud de 1. Esto es lo que significa tener a 1 como denominador. Si tenemos 1 parte de 1, el numerador es 1. Si tenemos 2 partes de 1, el numerador es 2 y así sucesivamente” // “The length from 0 to 1 hasn’t been partitioned, so each part has a length of 1. This is what a denominator of 1 means. If we have 1 part of 1, the numerator is 1. If we have 2 parts of 1, the numerator is 2, and so on.”
• “¿Qué otras fracciones en estas rectas numéricas podrían ser equivalentes a 1?” // “What other fractions on these number lines might be equivalent to 1?” (\(\frac{2}{2}\), \(\frac{3}{3}\), \(\frac{4}{4}\))

In an earlier lesson, students saw that whole numbers could be written as fractions. The purpose of this activity is for students to recognize fractions that are equivalent to whole numbers, using patterns in number lines to support their reasoning. To identify fractions that are equivalent to whole numbers on number lines, students may:

• Use what they know about 2 halves, 3 thirds, and 4 fourths to identify 1, and then circle fractions at the same intervals down each number line.
• Use what they know about 2 halves, 4 halves, and 6 halves to identify 1, 2, and 3 on the first number line, and then circle fractions in the same locations on other number lines.
• Use the relationship between parts and wholes (for instance, 3 thirds make 1, 6 thirds make 2, and 9 thirds make 3).

Students then record equations that show fractions that are equivalent to whole numbers. Finally, given a list of fractions, students determine which ones are equivalent to whole numbers.

When students use patterns to identify fractions that are equivalent to whole numbers, they look for and express regularity in repeated reasoning (MP8).

• Groups of 2
• “Con su compañero, marquen todas las fracciones que son equivalentes a números enteros. Asegúrense de explicar su razonamiento” // “Work with your partner to circle all the fractions that are equivalent to whole numbers. Be sure to explain your reasoning.”
• 2–3 minutes: partner work time
• “Observen las fracciones que marcaron. ¿Cómo supieron cuáles marcar?” // “Look at the fractions you circled. How did you know which ones to circle?”
• 1–2 minutes: partner discussion
• Share responses.

• “Podemos usar ecuaciones como \(\frac{4}{2}=2\) para mostrar que \(\frac{4}{2}\) y 2 están en la misma ubicación en la recta numérica, así que son equivalentes” // “We can use equations like \(\frac{4}{2}=2\) to show that \(\frac{4}{2}\) and 2 are at the same location on the number line, so they are equivalent.”
• “Trabajen individualmente para completar el segundo y el tercer problema” // “Work independently to complete the second and third problem.”
• 5–7 minutes: independent work time
• “Compartan con su compañero sus soluciones del segundo y tercer problema. Asegúrense de compartir cómo razonaron” // “Share your solutions to the second and third problems with your partner. Be sure to share your reasoning.”
• 2–3 minutes: partner discussion

If students don’t locate whole numbers on the number lines, consider asking:

• “Cuéntame sobre las fracciones en la recta numérica” // “Tell me about the fractions on the number line.”
• “¿Cómo puedes usar las particiones de 1 entero para encontrar las fracciones que están en la misma ubicación que 2?, ¿que 3?” // “How could you use the partitions in 1 whole to find the fractions at 2? At 3?”

• Invite students to share their responses and reasoning for the last set of problems.
• As students share, record fractions that are equivalent to whole numbers as equations and highlight that equations can be written starting with the whole number or fraction (for example, \(\frac{12}{6}=2\) or \(2=\frac{12}{6}\)).
• “¿Cómo supieron si cada fracción era equivalente a un número entero?” // “How did you know if each fraction was equivalent to a whole number?”
• “¿Qué patrones ven que pueden ayudar a encontrar más fracciones que son números enteros?” // “What patterns do you see that could be helpful for finding more fractions that are whole numbers?”
• Highlight strategies that are based on understanding of the number of equal parts that are in 1 whole and on observed patterns. (For instance, the thirds that are equivalent to whole numbers have a numerator that is a number we get when counting by 3.)

The purpose of this activity is for students to write whole numbers as fractions. Students may reason in any way that makes sense to them, including using patterns they noticed previously. When students observe patterns as they write whole numbers as fractions, they look for and make use of structure (MP7).

This activity uses a “carousel” structure in which students complete a rotation of tasks. Consider demonstrating the steps before students begin.

• Groups of 3
• “Tómense un momento para examinar la tabla y los números que hay en ella. ¿Cómo piensan que funciona la tabla?” // “Take a moment to look at the table and the numbers in the table. How do you think the table works?”
• 1 minute: quiet think time
• Share responses.
• “En su grupo, por turnos, expliquen por qué \(\frac{4}{1}\) es equivalente a 4, \(\frac{30}{6}\) es equivalente a 5 y \(\frac{48}{8}\) es equivalente a 6” // “In your group, take turns explaining why \(\frac{4}{1}\) is equivalent to 4, \(\frac{30}{6}\) is equivalent to 5, and \(\frac{48}{8}\) is equivalent to 6.”
• 2 minutes: group discussion

• “Trabajen en grupo para completar la tabla. Comiencen por escribir dos fracciones que sean equivalentes a cada número entero: 4, 5 y 6” // “Work with your group to complete the table. Start by writing two fractions that are equivalent to each whole number: 4, 5, and 6.”
• 2 minutes: independent work time
• “Pasen su hoja a su derecha. En la hoja que reciben, escriban una fracción nueva que sea equivalente a uno de los número enteros, el que elijan” // “Pass your paper to your right. On the paper your receive, write one new fraction that is equivalent to a whole number of your choice.”
• “Sigan pasando y escribiendo una fracción adicional que sea equivalente a uno de los números enteros hasta que la tabla esté completa” // “Keep passing and writing one additional fraction for a whole number until the table is complete.”
• Encourage students to ask clarifying questions before they begin.
• 7–8 minutes: group work time
• “Prepárense para explicar cómo saben qué fracciones escribir para cada número entero” // “Be prepared to explain how you know what fractions to write for each whole number.”
• Monitor for students who:
  • use a number line (either draw a new one or extending those from an earlier activity)
  • extend a pattern they noticed in fractions that are equivalent to whole numbers
  • use multiplication facts

If students don’t write a whole number as a fraction, consider asking:

• “Dime cómo escribirías 1 como una fracción” // “Tell me about how you would write 1 as a fraction.”
• “¿Cómo puedes usar la fracción que es equivalente a 1 para escribir este número como una fracción?” // “How could you use the fraction that's equivalent to 1 to write this number as a fraction?”

• Invite a group of students to display their completed table.
• Select previously identified students to share how they knew what fractions to write for each whole number.
• Invite the class to share other strategies for generating equivalent fractions.
• “¿Qué patrones nuevos observan en la tabla que ya completaron?” // “What new patterns do you notice in the completed table?” (Sample responses:
  • In fractions that are equivalent to 4, you can get the numerator by skip-counting by the denominator 4 times.
  • In the row with thirds, the numerator increases by 3. In the row for fourths, it increases by 4.)