# Lesson 6

¿Cuál es el cociente?

## Warm-up: Conversación numérica: Dividamos entre 3 (10 minutes)

### Narrative

This Number Talk encourages students to look for and use the structure of base-ten numbers and properties of operations to mentally find the value of division expression (MP7). The reasoning elicited here will be helpful later in the lesson when students find quotients of multi-digit numbers.

### 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.”

### Activity

• 1 minute: quiet think time
• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Encuentra mentalmente el valor de cada expresión.

• $$48 \div 3$$
• $$480 \div 3$$
• $$528 \div 3$$
• $$5,\!280 \div 3$$

### Activity Synthesis

• “¿Cómo se relaciona cada expresión con la anterior a ella?” // “How is each expression related to the one before it?”
• “¿Alguien puede expresar el razonamiento de _____ de otra forma?” // “Who can restate _____’s reasoning in a different way?”
• “¿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?”
• “¿Alguien quiere agregar algo a la estrategia de _____?” // “Does anyone want to add on to _____’s strategy?”

## Activity 1: Divisiones sin terminar (15 minutes)

### Narrative

In a previous lessons, students saw that there are many ways to find products of multi-digit numbers. In this activity, students analyze and connect different ways to divide a multi-digit whole number by a single-digit whole number, and complete calculations to find the value of the quotient. In the synthesis, students compare the different methods and explain their preference.

Engagement: Provide Access by Recruiting Interest. Provide choice. Tell students they will be finding the value of $$7,\!465 \div 5$$, and that there are four unfinished strategies to look at. Invite students to choose whether they want to solve it in their own way or look at the unfinished strategies first.
Supports accessibility for: Organization, Attention, Social-Emotional Functioning

### Launch

• Groups of 2–4
• “Escojan por lo menos dos cálculos para terminar. Asegúrense de que su grupo complete todos los cálculos” // “Choose at least two calculations to finish. Make sure each calculation is completed by someone in your group.”

### Activity

• 3–4 minutes: independent work time
• 2 minutes: small-group discussion

### Student Facing

Estos son cuatro cálculos para encontrar el valor de $$7,\!465 \div 5$$, pero cada uno está sin terminar.

Completa por lo menos dos de los cálculos que están sin terminar. Prepárate para explicar cómo sabes qué hacer para completar el trabajo.

\displaystyle \begin{align} 5,\!000 \div 5 &= 1,\!000\\ 60 \div 5 &=\phantom{0,\!0}12\\ 5 \div 5 &=\phantom{0,\!00}1 \end{align}

7,465 es un poco menos que 7,500.

\begin{align}7,\!500 \div 5 &= 1,\!500\\ 35 \div 5 &=\phantom{0,\!00}7 \end{align}

### Student Response

Students may determine quotients other than 1,493. Consider asking:

• “¿Por qué tiene sentido este método? ¿Cómo explicarías los números que hay en él?” // “How did you make sense of this method? How would you explain the numbers in it?”
• “¿En qué se parece este método a los demás? ¿En qué es diferente?” // “How is this method like the others? How is it different?”
• “¿Cómo podrías usar la multiplicación para comprobar el valor del cociente que encontraste?” // “How could you use multiplication to check the value of the quotient you found?”

### Activity Synthesis

• “¿En qué se parecen las cuatro estrategias? ¿En qué son diferentes?” // “How are the four strategies the same? How are they different?” (The first three are the same because they involve partial quotients. Each one records the partial quotients in different ways. The last one involves estimation.)
• “¿Cuáles métodos les parecen más fáciles de entender? ¿Cuáles les parecen complicados?” // “Which method or methods do you find easy to follow? Which did you find hard to follow?”
• “¿En cuál método se usan más pasos? ¿En cuál método se usan menos pasos?” // “Which method uses the most steps? Which uses the fewest steps?”
• “¿Cuáles métodos sirven para encontrar el valor de cualquier cociente? ¿Cuál puede servir para esta expresión, pero puede ser menos útil para otras?” // “Which methods would work to find the value of any quotient? Which might work for this expression, but might be less useful for others?”

## Activity 2: ¿Por dónde empezamos? (20 minutes)

### Narrative

This activity serves two goals. First, it prompts students to consider whether the order in which parts of the dividend are divided makes a difference in the process or in the result. Second, it deepens students’ understanding of the structure of algorithms that use partial quotients.

Students first explain why different initial steps could be equally productive for starting a division process. Next, they analyze and complete some partial-quotients calculations with missing numbers. The missing numbers could be partial quotients, parts of the dividend being removed, or results of subtraction. To find the unknown numbers, students need to recognize and make use of the structure of the algorithm (MP7). Lastly, students use the algorithm to find a quotient, being mindful of their starting move and of the efficiency of their process.

MLR2 Collect and Display. Collect the language students use to explain how they found the quotient. Display words and phrases such as: “cociente” // “quotient,” “cociente parcial” // “partial quotient,” and “dividendo” // “dividend.” During the synthesis, invite students to suggest ways to update the display: “¿Qué otras palabras o frases deberíamos incluir?” // “What are some other words or phrases we should include?” Invite students to borrow language from the display as needed.

• Groups of 2

### Activity

• 6–8 minutes: independent work time on the first two sets of questions
• 2–3 minutes: partner discussion
• Monitor for students who:
• can clearly explain why Jada and Noah’s initial steps are both effective
• recognize the structure of the partial quotients method and can articulate how it helps to find the missing numbers
• Pause for a discussion before the last question. Select students to share responses and reasoning.
• When discussing the second set of questions, ask: “¿Cómo encontraron los números que faltaban?” // “How do you determine what the missing numbers were?” Display the incomplete calculations to facilitate students’ explanations.
• Consider annotating the calculations to clarify the structure (for instance, by drawing arrows between partial quotients and the corresponding parts of the dividend being subtracted, labeling the parts, and so on).
• 3–4 minutes: independent work time on the last question
• Monitor for students who take different first steps to divide 5,016 by 8.

### Student Facing

1. Jada y Noah encuentran el valor de $$3,\!681 \div 9$$. Jada dice que empiecen dividiendo 81 entre 9. Noah dice que empiecen dividiendo 3,600 entre 9.

1. Explica por qué ambas sugerencias son útiles para encontrar el cociente.
2. Encuentra el valor de $$3,\!681 \div 9$$. Muestra cómo razonaste.

2. Encuentra los números que faltan para que cada forma de calcular muestre un cálculo correcto de división.

3. Considera la expresión $$5,\!016 \div 8$$.

1. ¿Qué harías para empezar a encontrar el valor del cociente?
2. Muestra cómo encontrarías el valor usando el menor número posible de pasos.

### Student Response

Students may find some, but not all of the missing numbers in the algorithms. Consider asking:

• “¿Cuáles números que faltan estás seguro de que son correctos? ¿Cómo lo sabes?” // “Which missing numbers are you sure are accurate? How do you know?”
• “¿Podrías usar la multiplicación para encontrar los números que faltan? ¿Cómo lo harías?” // “Could you use multiplication to find the missing numbers? How might that work?”
• “¿Podrías trabajar devolviéndote para encontrar los números que faltan? ¿Cómo te ayudaría eso?” // “Could you work backwards to find the missing numbers? How would that help?”

### Activity Synthesis

• See lesson synthesis.

## Lesson Synthesis

### Lesson Synthesis

“Hoy estudiamos distintas formas de dividir números de varios dígitos entre divisores de un dígito” // “Today we studied different ways to divide multi-digit numbers and single-digit divisors.“

Select students who took different initial steps to find $$5,\!016 \div 8$$ to share their calculations. Discuss:

“¿Por qué decidieron empezar con ese número?” // “Why did you decide to start with that number?”

“¿Cómo decidieron cuál era la siguiente parte que iban a dividir y quitar?” // “How did you determine the next chunk to divide and remove?”

“¿Pueden pensar en una forma de encontrar el cociente en menos pasos?” // “Can you think of a way to find the quotient with fewer steps?”