# Lesson 17

Fracciones como cocientes parciales (optional)

## Warm-up: ¿Qué sabes sobre $\frac{60}{6} + \frac{6}{6}$? (10 minutes)

### Narrative

The purpose of this What Do You Know About _____? is for students to share what they know about a sum of fractions. The fractions are selected because they represent whole numbers and the whole number values are visible. Students will work with expressions like these throughout this lesson.

### Launch

• Display the number.
• “¿Qué saben sobre $$\frac{60}{6} + \frac{6}{6}$$?” // “What do you know about $$\frac{60}{6} + \frac{6}{6}$$?”
• 1 minute: quiet think time

### Activity

• Record responses.

### Student Facing

¿Qué sabes sobre $$\frac{60}{6} + \frac{6}{6}$$?

### Activity Synthesis

• “¿Qué otras expresiones son iguales a $$\frac{60}{6} + \frac{6}{6}$$?” // “What are some expressions that are equal to $$\frac{60}{6} + \frac{6}{6}$$?” ($$10 + 1$$, 11, $$(60 \div 6)+ (6 \div 6)$$)

## Activity 1: Seleccionemos expresiones (15 minutes)

### Narrative

The purpose of this activity is for students to relate their understanding of fractions as representing division to think about decomposing a quotient into partial quotients in a way that simplifies the calculation.  To find the value of $$78 \div 6$$, students may

• use their understanding of division.
• use multiplication and find how many groups of 6 there are in 78.
• use the fraction expressions from the first part of the problem.

• Groups of 2

### Activity

• 5–8 minutes: partner work time
• Monitor for students who:
• use multiplication to find the value of $$78\div6$$.
• use the expression $$\frac {60}{6} + \frac {18}{6}$$ to find the value of $$78 \div 6$$.
• use the expression $$\frac {66}{6} + \frac {12}{6}$$ to find the value of $$78 \div 6$$.

### Student Facing

1. Selecciona todas las expresiones que son equivalentes a $$\frac {78}{6}$$. Explica o muestra tu razonamiento.

1. $$78 \div 6$$
2. $$\frac {66}{6} + \frac {12}{6}$$
3. $$\frac {60}{6} + \frac {18}{6}$$
4. $$(60 \div 6) + (18 \div 6)$$
5. $$\frac {77}{6} + \frac {8}{6}$$
6. $$(60 \div 6) + 18$$
2. ¿Cuál es el valor de $$78 \div 6$$? Explica o muestra cómo pensaste.

### Student Response

If students do not identify all of the expressions whose value is equal to $$\frac{78}{6}$$, ask, “¿Cómo decidiste cuáles expresiones eran iguales a $$\frac{78}{6}$$?” // “How did you decide which expressions were equal to $$\frac{78}{6}$$?”

### Activity Synthesis

• Invite students to share the expressions that match $$78 \div 6$$.
• Display: $$78 \div 6 =\frac {78}{6}$$
• “¿Cómo sabemos que esto es verdadero?” // “How do we know this is true?” (A fraction shows you are dividing the numerator by the denominator.)
• Display: $$78 \div 6 = \frac {60}{6} + \frac {18}{6}$$
• “¿Cómo pueden usar esta ecuación para encontrar el valor de $$\frac{78}{6}$$?” // “How can you use this equation to find the value of $$\frac{78}{6}$$?” (I know $$\frac{60}{6}$$ is 10 and $$\frac{18}{6}$$ is 3 so $$\frac{78}{6}$$ is 13.)
• “En la siguiente actividad vamos a usar expresiones que tienen fracciones para encontrar los valores de otros cocientes” // “In the next activity we will use expressions with fractions to find values of other quotients.”

## Activity 2: Escojamos una expresión (20 minutes)

### Narrative

The purpose of this activity is for students to find the whole number value of quotients using sums of fractions and to think about which sums were most helpful. They may notice that it is helpful to decompose the dividend into a multiple of the divisor and multiples of 10 are particularly helpful. This is closely related to how students found quotients using partial products which requires strategically choosing the number of groups of the divisor to subtract.

This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.

Representation: Internalize Comprehension. Invite students to identify which details were most useful to solve the problem. Display the sentence frame: “La próxima vez que el dividendo no sea divisible entre el divisor, voy a buscar múltiplos de 10 o múltiplos del divisor que me ayuden a dividir de una manera más eficiente” // “The next time the dividend is not divisible by the divisor, I will look for multiples of 10 or multiples of the divisor to help me divide more efficiently.“
Supports accessibility for: Conceptual Processing, Memory, Organization

### Launch

• Groups of 2
• Display:
$$\frac{60}{6} + \frac{18}{3}$$
$$\frac{55}{6} + \frac{13}{6}$$
• “¿Cuál de estas expresiones usarían para encontrar el valor de $$\frac{78}{6}$$?” // “Which of these expressions would you use to find the value of $$\frac{78}{6}$$?” (The first one because the fractions have nice whole number values.)
• 1–2 minutes: partner discussion
• “Van a escoger expresiones como esta, que ayudan a encontrar cocientes” // “You are going to choose expressions like this one that are helpful for finding quotients.”

### Activity

• 5–8 minutes: independent work time
• 1–2 minutes: partner discussion
• Monitor for students who:
• explain that numerators that are multiples of the divisor are helpful to divide.
• explain that numerators that are not multiples of the divisor require working with fractions.

### Student Facing

1. Usa cada expresión para encontrar el valor de $$165 \div 15$$. Explica o muestra cómo pensaste.

1. $$\frac {75}{15} + \frac {80}{15} + \frac {10}{15}$$
2. $$\frac {30}{15} + \frac {30}{15} + \frac {30}{15} + \frac {60}{15} + \frac {15}{15}$$
3. $$\frac {150}{15} + \frac {15}{15}$$
2. Escoge una expresión y úsala para encontrar el valor de $$540 \div 18$$. Explica o muestra cómo pensaste.

1. $$\frac {180}{18} + \frac {180}{18} + \frac {180}{18}$$
2. $$\frac {500}{18} + \frac {40}{18}$$
3. $$\frac {360}{18} + \frac {180}{18}$$
3. ¿Cuáles expresiones ayudaron más? ¿Cuáles expresiones ayudaron menos? Explica o muestra cómo pensaste.

### Activity Synthesis

MLR1 Stronger and Clearer Each Time
• “Compartan con su compañero sus respuestas acerca de por qué algunas expresiones ayudaron y otras no. Por turnos, uno habla y el otro escucha. Si es su turno de hablar, compartan sus ideas y lo que han escrito hasta ese momento. Si es su turno de escuchar, hagan preguntas y comentarios que ayuden a su compañero a mejorar su trabajo” // “Share your response as to why some expressions were helpful and others were not with your partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
• 3–5 minutes: structured partner discussion.
• Repeat with 2–3 different partners.
• (Optional) If needed, display question starters and prompts for feedback.
• “¿Me puedes dar un ejemplo que ayude a mostrar . . . ?” // “Can you give an example to help show . . . ?”
• “¿Puedes usar la palabra _____ en tu explicación?” // “Can you use the word _____ in your explanation?”
• “La parte que mejor entendí fue . . .” // “The part I understood best was . . .”
• “Ajusten su borrador inicial basándose en los comentarios que les hicieron sus compañeros” // “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

## Lesson Synthesis

### Lesson Synthesis

Display:

$$\frac{180}{18} + \frac{180}{18} + \frac{180}{18}$$

“¿Cómo sabemos que esta expresión es igual a $$\frac{540}{18}$$?” // “How do we know this expression is equal to $$\frac{540}{18}$$?” ($$180 + 180 + 180 = 540$$ and they are 18ths.)

“¿Cómo podemos usar esta expresión para encontrar el valor de $$540 \div 18$$?” // “How can we use this expression to find the value of $$540 \div 18$$?” ($$\frac {180}{18} = 10$$ and there are three of them so the value of $$540 \div 18$$ is 30.)

Display:

$$\frac {360}{18} + \frac {180}{18}$$

“¿Cómo podemos usar esta expresión para encontrar el valor de $$540 \div 18$$?” // “How can we use this expression to help us find the value of $$540 \div 18$$?” ($$36 \div 18 = 2$$ so $$360 \div 18 = 20$$ and $$180 \div 18 = 10$$ and $$20 + 10 = 30$$.)

Display:

$$\frac {500}{18} + \frac {40}{18}$$

“¿Cómo sabemos que esta expresión es igual a $$540 \div 18$$?” // “How do we know this expression is equal to $$540 \div 18$$?” ($$500 + 40 = 540$$ and they're 18ths)

“¿Por qué esta expresión no ayuda tanto como las otras?” // “Why is this expression not as helpful as the others?” (The values of those fractions are not whole numbers so we have to calculate with fractions.)