Lesson 14

This Number Talk encourages students to use what they know about the meaning of fractions and about properties of operations to mentally relate fractions that are equivalent to whole numbers. 

• Display one fraction.
• “Hagan una señal cuando tengan una respuesta y puedan explicar cómo la obtuvieron” // “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

• Record answers and strategy. 
• Keep fractions and work displayed. 
• Repeat with each fraction.

• “¿Cómo les ayudaron las primeras fracciones a encontrar el número entero en el caso de la última fracción?” // “How did the earlier fractions help you find the whole number for the last fraction?”
• Consider asking:
  • “¿Alguien usó la misma estrategia, pero la explicaría de otra forma?” // “Did anyone have the same strategy but would explain it differently?”
  • “¿Alguien pensó en el problema de otra forma?” // “Did anyone approach the problem in a different way?”

The purpose of this activity is for students to analyze pairs of fractions to determine if they are equivalent. Students may use any representation that makes sense to them. Students will create a visual display and have a gallery walk to consider the different ways of looking for equivalence. Highlight representations such as diagrams and number lines, which will support students as they learn to compare fractions with the same numerator or denominator in this section.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

• Groups of 2
• “Decidan si las fracciones de cada pareja son equivalentes y muestren cómo pensaron en cada caso. Pueden usar cualquier representación que tenga sentido para ustedes” // “Decide if these pairs of fractions are equivalent and show your thinking for each one. You can use any representation that makes sense to you.”

• 3–5 minutes: independent work time
• “Compartan sus ideas con su compañero” // “Share your ideas with your partner.”
• 2–3 minutes: partner discussion

• “Con su compañero, creen una presentación visual que muestre cómo pensaron sobre \(\frac{4}{6}\) y \(\frac{5}{6}\). Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Work with your partner to create a visual display that shows your thinking about \(\frac{4}{6}\) and \(\frac{5}{6}\). You may want to include details such as notes, diagrams, drawings, and so on, to help others understand your thinking.”
• Give students materials for creating a visual display.
• 2–5 minutes: partner work time
• 5–7 minutes: gallery walk

If students don’t determine whether or not fractions are equivalent, consider asking:

• “¿Qué sabes sobre estas fracciones?” // “What do you know about these fractions?”
• “¿Cómo puedes usar diagramas o tus tiras de fracciones para decidir si son equivalentes?” // “How could you use your fraction strips or diagrams to decide if they are equivalent?”

• “¿Qué representaciones diferentes podemos usar para decidir si \(\frac{4}{6}\) y \(\frac{5}{6}\) son equivalentes o no?” // “What are the different representations we can use to decide if \(\frac{4}{6}\) and \(\frac{5}{6}\) are equivalent?” (diagrams, number lines)
• “¿Cómo mostró cada representación que \(\frac{4}{6}\) y \(\frac{5}{6}\) no son equivalentes?” // “How did each representation show that \(\frac{4}{6}\) and \(\frac{5}{6}\) are not equivalent?” (In the diagram, you can see that \(\frac{5}{6}\) has more space shaded. On the number line, they are not at the same location.)

The purpose of this activity is for students to recognize that fraction comparisons are only valid when they refer to the same whole. Previously, students analyzed pairs of fractions to determine if they were equivalent. In doing so, they were likely to have used comparison language, such as “larger or smaller than” or “greater or less than.” In this activity, students encounter a pair of fractions they saw earlier (\(\frac{4}{6}\) and \(\frac{5}{6}\)) and compare them more explicitly. The student work in this activity uses the number line, but this might also come up with student-drawn diagrams.

In order to interpret Lin’s argument that \(\frac{4}{6}\) is greater than \(\frac{5}{6}\) students will need to articulate the meaning of fractions and highlight the fact that the two wholes Lin is comparing are not equal (MP6).

• Groups of 2
• “Al igual que ustedes, Han y Lin comparan \(\frac{4}{6}\) y \(\frac{5}{6}\). Tómense un minuto para examinar su trabajo” // “Han and Lin are comparing \(\frac{4}{6}\) and \(\frac{5}{6}\) like you did. Take a minute to look at their work.”
• 1 minute: quiet think time

• “Hablen con su compañero sobre cómo pudieron Han y Lin obtener resultados diferentes” // “Talk with your partner about how Han and Lin could get different results.”
• 2–3 minutes: partner discussion
• Monitor for students who notice that the whole is different in Lin’s number lines, which makes her think that \(\frac{4}{6}\) is greater.

• Select previously identified students to share how Han and Lin could make different comparison statements for the same fractions.
• “Cuando comparamos fracciones, es importante recordar que esas fracciones deben hacer referencia a la misma unidad” // “It is important to remember when we are comparing fractions that those fractions need to refer to the same whole.”
• “Cuando la unidad de 0 a 1 es del mismo tamaño, podemos ver que \(\frac{4}{6}\) es menor que \(\frac{5}{6}\)” // “When the whole from 0 to 1 is the same size, we can see that \(\frac{4}{6}\) is less than \(\frac{5}{6}\).”
• “Esto es cierto independientemente de que dibujemos una recta numérica, usemos tiras de fracciones o dibujemos un diagrama” // “This is true whether we are drawing a number line, using fraction strips, or drawing a diagram.”
• Consider asking, “¿Qué entendieron Lin y Han sobre representar fracciones en una recta numérica?” // “What did both Lin and Han understand about representing fractions on a number line?” (The number lines need to be partitioned into equal parts. The denominator tells us then number of parts. The numerator tells us how many parts to count to locate a point. Points farther to the right are greater than those to the left.)