Lesson 11

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for dividing whole numbers. These understandings help students develop fluency and will be helpful later in this lesson when students divide a fraction by a whole number.

• Display one expression.
• “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 expressions and work displayed.
• Repeat with each expression.

• “¿Qué patrones observan en los cocientes?” // “What patterns do you notice in the quotients?” (When the dividend is split in half, the quotient is also split in half.)
• “¿Por qué ocurre eso?” // “Why does that happen?” (There are half as many to start with, so there will be half as many in each group.)

The purpose of this activity is for students to solve a contextual problem about dividing a fractional amount by a whole number. Students draw a diagram to represent the situation and relate the diagram to a division expression. Because of earlier work in this unit, students may draw one of the familiar square area diagrams showing the product \(\frac{1}{3} \times \frac{1}{2}\). Other students may make a diagram resembling the macaroni and cheese pan and divide it appropriately. The focus in the synthesis is on bringing out how the diagram shows \(\frac{1}{2} \div 3\) and how it allows students to answer the question. The relationship between \(\frac{1}{3} \times \frac{1}{2}\) and \(\frac{1}{2} \div 3\) will be brought out in later lessons. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• Groups of 2
• Display and read: “Anoche, la tía de Jada cocinó una bandeja refractaria de macarrones con queso para la cena. Hoy, la tía llevó lo que sobró a la casa de Jada para que Jada y sus hermanas lo compartieran” // “Last night, Jada’s aunt baked a pan of macaroni and cheese for dinner. Today, she brought the leftovers to Jada’s home for Jada and her sisters to share.”
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (We solved problems about macaroni and cheese before. I wonder how much macaroni and cheese Jada’s aunt brought.)
• 1–2 minutes: partner discussion

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
  • draw diagrams like the ones in the student responses
  • explain the situation as \(\frac {1}{2}\) divided into 3 equal pieces
  • recognize each person will get \(\frac{1}{6}\) of the whole pan of macaroni and cheese

• “Creen una presentación visual que muestre cómo pensaron en cada uno de los problemas. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender sus ideas” // “Create a visual display that shows your thinking about the problems. You may want to include details such as notes, diagrams, drawings, etc., to help others understand your thinking.”
• 2–5 minutes: independent or group work
• 5–7 minutes: gallery walk
• “¿Cómo muestra cada representación \(\frac {1}{2}\) bandeja refractaria de macarrones con queso?” // “How does each representation show \(\frac {1}{2}\) a pan of macaroni and cheese?”
• “¿Cómo muestra cada representación 3 partes iguales?” // “How does each representation show 3 equal pieces?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “¿Cómo sabemos que cada persona recibió la misma cantidad de macarrones con queso?” // “How do we know that each person got the same amount of macaroni and cheese?” (I divided the half into 3 equal shares.)
• “¿Cuánto de la bandeja refractaria de macarrones con queso entera recibió cada persona?” // “How much of the whole pan of macaroni and cheese did each person get?” (\(\frac {1}{6}\))
• “¿Cómo se muestra \(\frac {1}{6}\) en los diagramas?” // “How do the diagrams show \(\frac {1}{6}\)?” (Each \(\frac {1}{2}\) of the pan is divided into 3 equal pieces, so each of those pieces is \(\frac {1}{6}\) of the whole pan.)

The purpose of this activity is for students to continue to solve problems about dividing a unit fraction by a whole number. The unit fraction in both problems is \(\frac{1}{2}\) so that students will consider the relationship between the number of people sharing the macaroni and cheese and the size of the serving each person gets. When students connect the quantities in the story problem to an equation and a diagram representing the story, they reason abstractly and quantitatively (MP2).

• Groups of 2

• 1–2 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who :
  • draw diagrams like the ones in the student responses
  • describe the amount each person gets as a fraction of the whole pan
  • describe a relationship between the number of people sharing and the size of the serving each person gets

If students do not identify or explain how much macaroni and cheese each person gets, consider asking, “¿Cuánto de la bandeja de macarrones con queso completa recibió cada persona?” // “How much of the whole pan of macaroni and cheese did each person get?”

• “¿En qué se parecen las situaciones? ¿En qué son diferentes?” // “How are the situations the same? How are they different?” (Both situations are about \(\frac {1}{2}\) a pan of macaroni and cheese, but this one has more people sharing it, so each person gets less macaroni and cheese. When 4 people share, each person gets \(\frac {1}{8}\) of the whole pan. When 5 people share, each person gets \(\frac {1}{10}\) of the whole pan.)