# Lesson 14

Representemos la división de números enteros entre fracciones unitarias

## Warm-up: Conversación numérica: Cocientes que aumentan (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for dividing a whole number by a unit fraction. These understandings will be helpful later in this lesson when students match situations to equations and solve the equations.

### Launch

• 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

### Activity

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$6 \div 1$$
• $$6 \div \frac {1}{2}$$
• $$6 \div \frac {1}{3}$$
• $$6 \div \frac {1}{6}$$

### Student Response

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### Activity Synthesis

• “¿Por qué el cociente se hace más grande en cada problema?” // “Why is the quotient getting larger with each problem?” (Because we are dividing the same number into smaller sized groups so there are going to be more groups.)

## Activity 1: Observemos patrones (20 minutes)

### Narrative

The purpose of this activity is for students to find quotients of a whole number by a unit fraction and observe patterns in how the size of the numerator and denominator influence the size of the quotient. Whereas students were provided a tape diagram in the previous lesson, here they may draw a diagram but they may also reason about the size of the quotients in other ways. When students notice a pattern or repetitive action in computation, they are looking for and expressing regularity in repeated reasoning (MP8).

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

Action and Expression: Internalize Executive Functions. Invite students to plan a strategy, including the tools they will use, for finding the value of each statement. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Organization

### Launch

• Groups of 2
• “Con su pareja, decidan quién trabajará en el conjunto A y quién trabajará en el conjunto B” // “Decide with your partner who will work on set A and who will work on set B.”

### Activity

• 3–5 minutes: independent work time
• 3–5 minutes: partner discussion

### Student Facing

Conjunto A: En cada caso, encuentra el valor que hace que la ecuación sea verdadera. Si te ayuda, dibuja un diagrama. ¿Qué patrones observas?

• $$3 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$4 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$5 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$6 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$

Conjunto B: En cada caso, encuentra el valor que hace que la ecuación sea verdadera. Si te ayuda, dibuja un diagrama. ¿Qué patrones observas?

• $$3 \div \frac {1}{2} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{3} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{4} = \underline{\hspace{1 cm}}$$
• $$3 \div \frac {1}{5} = \underline{\hspace{1 cm}}$$

¿En qué se parecen los conjuntos de problemas A y B? ¿En qué son diferentes?

### Student Response

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### Advancing Student Thinking

If students don’t solve the equations correctly, prompt them to draw diagrams to represent each equation and ask: “¿Cómo se representa la ecuación en cada diagrama?” // “How does each diagram represent the equation?”

### Activity Synthesis

MLR7 Compare and Connect
• “Con su compañero, creen una presentación visual que muestre sus ideas sobre en qué se parecen y en qué son diferentes los conjuntos de problemas. Incluyan detalles, como notas, diagramas, dibujos, etc., para ayudar a los demás a entender cómo pensaron” // “Work with your partner to create a visual display that shows your thinking about how the problem sets are the same and different. 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
• “¿En qué se parecen y en qué son diferentes los dos conjuntos de problemas?” // “What is the same and what is different between the two sets of problems?”
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Additional connections could include:
• “¿Qué cambia en cada conjunto de problemas?” // “What is changing in each problem set?” (The size of the number being divided is changing in set A and the size of the piece the number is being divided into is changing in problem set B.)
• “Si los patrones continuaran, ¿cuál sería la siguiente ecuación de cada conjunto?” // “If the patterns continued, what would be the next equation in each set?”
• “¿Por qué el cociente se hace más grande en ambos conjuntos de problemas?” // “Why is the quotient getting larger in both sets of problems?” (In set A, it is getting larger because you have 4 more $$\frac {1}{4}$$ size pieces in each additional whole. In set B, it is getting larger because the size of the piece is getting smaller, so you have more of them.)

## Activity 2: Empareja la situación con la expresión (15 minutes)

### Narrative

The purpose of this activity is for students to match situations to expressions and then find the value of the expressions (MP2). Students see expressions that show both quotients of a whole number by a fraction and quotients of a fraction by a whole number. They need to think carefully about the situations to make sure the expression they choose matches the situation (MP2).

### Launch

• Groups of 2
• Display the image from student book:
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?” (They look like seeds. How many are in the bowls? Why are there 2 bowls?)
• “El tazón pequeño contiene $$\frac {1}{4}$$ de taza de granos de maíz. Eso es una porción de palomitas de maíz. ¿Qué significa ‘una porción’?” // “The small bowl is filled with $$\frac {1}{4}$$ cup of kernels. That is one serving of popcorn kernels. What does ‘one serving’ mean?” (It is the amount you are supposed to eat at one time.)
• “¿Aproximadamente cuántas porciones hay en el tazón grande?” // “About how many servings are in the big bowl?” (Answers vary. Sample responses range from 10 to 15.)

### Activity

• 1–2 minutes: independent think time
• 5–8 minutes: partner work time
• Monitor for students who:
• draw diagrams to represent the situations
• describe the relationship between the dividend and divisor using language such as “groups of”
• use the value of the expression to describe why the expression matches the situation

### Student Facing

1. Empareja cada problema con una expresión que lo represente. Algunas expresiones no tendrán pareja. Prepárate para explicar cómo razonaste.

1. Una porción de palomitas de maíz es $$\frac {1}{4}$$ de taza de granos. En el tazón hay 3 tazas de granos. ¿Cuántas porciones hay en el tazón?

2. Una porción de jugo de naranja es $$\frac {1}{4}$$ de litro. El recipiente de jugo tiene 2 litros. ¿Cuántas porciones hay en el recipiente?
3. Una porción de granola es $$\frac {1}{2}$$ taza. La bolsa de granola tiene 5 tazas. ¿Cuántas porciones hay en la bolsa?

$$\frac {1}{4} \div 3$$

$$\frac {1}{2} \div 5$$

$$3 \div \frac {1}{4}$$

$$\frac {1}{4} \div 2$$

$$5 \div \frac {1}{2}$$

$$2 \div \frac {1}{4}$$

2. Encuentra el valor de cada expresión.

### Student Response

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### Advancing Student Thinking

If students do not correctly match the situations to the expressions, refer to each situation and ask, “¿Cómo se representa la división en la situación?” // “How does the situation represent division?”

### Activity Synthesis

• Ask previously selected students to share their solutions.
• “¿Cómo decidieron cuál expresión le correspondía a cuál situación?” // “How did you decide which expression matched which situation?” (I thought about the number of things being divided and whether it was a fraction or a whole number. I thought about the size of the pieces.)
• Display: $$3 \div \frac {1}{4}= 12$$
• “Describan cómo se representa la situación del primer problema en esta ecuación” // “Describe how this equation represents the situation in the first problem.” (There are 3 cups of kernels in the bowl and a serving is $$\frac {1}{4}$$ a cup, so there are 12 servings in the bowl.)
• Display: $$\frac {1}{4} \div 3 = \frac {1}{12}$$
• “¿Cómo saben que esta ecuación no le corresponde a la situación?” // “How do you know that this equation does not match the situation?” (The answer is a fraction. That doesn’t make sense because there is more than 1 serving in 3 cups.)

## Lesson Synthesis

### Lesson Synthesis

“Hoy usamos expresiones para representar y resolver problemas en los que se dividía un número entero entre una fracción unitaria” // “Today we used expressions to represent and solve problems involving the division of a whole number by a unit fraction.”

Display: “Jada dice que cuando se divide un número entero entre una fracción unitaria, la respuesta siempre será mayor que 1” // “Jada says when you divide a whole number by a unit fraction, the answer will always be greater than 1.”

“¿Están de acuerdo con Jada? Prepárense para explicar cómo pensaron” // “Do you agree with Jada? Be prepared to explain your thinking.” (I think so, because there will always be more than 1 unit fraction in a whole number, so even if it was 1 divided by a unit fraction, there will be however many unit fractions make up 1 whole, so that will be a whole number of unit fractions.)

## Cool-down: Resuelve y empareja la expresión (5 minutes)

### Cool-Down

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