# Lesson 17

Fractions as Partial Quotients (optional)

## Warm-up: What Do You Know About $\frac{60}{6} + \frac{6}{6}$? (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.
• “What do you know about $$\frac{60}{6} + \frac{6}{6}$$?”
• 1 minute: quiet think time

### Activity

• Record responses.

### Student Facing

What do you know about $$\frac{60}{6} + \frac{6}{6}$$?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “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: Select Expressions (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. Select all the expressions that are equivalent to $$\frac {78}{6}$$. Explain or show your reasoning.

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. What is the value of $$78 \div 6$$? Explain or show your thinking.

### Student Response

For access, consult one of our IM Certified Partners.

If students do not identify all of the expressions whose value is equal to $$\frac{78}{6}$$, ask, “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: $$78 \div 6 =\frac {78}{6}$$
• “How do we know this is true?” (A fraction shows you are dividing the numerator by the denominator.)
• Display: $$78 \div 6 = \frac {60}{6} + \frac {18}{6}$$
• “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.)
• “In the next activity we will use expressions with fractions to find values of other quotients.”

## Activity 2: Choose One Expression (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: “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:
$$\frac{60}{6} + \frac{18}{3}$$
$$\frac{55}{6} + \frac{13}{6}$$
• “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
• “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. Use each expression to find the value of $$165 \div 15$$. Explain or show your thinking.

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. Choose one expression and use it to find the value of $$540 \div 18$$. Explain or show your thinking.

1. $$\frac {180}{18} + \frac {180}{18} + \frac {180}{18}$$
2. $$\frac {500}{18} + \frac {40}{18}$$
3. $$\frac {360}{18} + \frac {180}{18}$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

MLR1 Stronger and Clearer Each Time
• “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.
• “Can you give an example to help show . . . ?”
• “Can you use the word _____ in your explanation?”
• “The part I understood best was . . .”
• “Revise your initial draft based on the feedback you got from your partners.”
• 2–3 minutes: independent work time

## Lesson Synthesis

### Lesson Synthesis

Display:

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

“How do we know this expression is equal to $$\frac{540}{18}$$?” ($$180 + 180 + 180 = 540$$ and they are 18ths.)

“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:

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

“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:

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

“How do we know this expression is equal to $$540 \div 18$$?” ($$500 + 40 = 540$$ and they're 18ths)

“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.)

## Cool-down: Choose One Expression (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.