# Lesson 20

Interpret Remainders in Division Situations

## Warm-up: Choral Count: 2, 3, and 5 (10 minutes)

### Narrative

The purpose of this warm-up is to elicit strategies and understandings students have for identifying multiples of small numbers, primarily by looking for and making use of structure (MP7). These understandings will be helpful later when students solve division problems that involve distinguishing quotients with and without a remainder.

### Launch

• “Count by 2 starting at 90.”
• Record as students count.
• Stop counting and recording at 112.

### Activity

• “What patterns do you notice in each of the recorded counts?”
• Repeat with 3 and 5.
• “Count by 3 starting at 90.” Stop counting and recording at 114.
• “Count by 5 starting at 90.” Stop counting and recording at 115.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Is 105 a multiple of 2, 3, or 5? How do you know?”
• “Is 105 a multiple of 15?”

## Activity 1: Muffins and Seats (15 minutes)

### Narrative

This activity encourages students to interpret the quantities in situations, represent them mathematically, use their representations to find solutions, and then interpret their solutions in context (MP2). The dividends here are limited to three-digit numbers.

Action and Expression: Develop Expression and Communication. Develop fluency with multiplication by 4 and 9. Provide a partially completed two-column table for each factor, and suggest that students record times tables that might be helpful for this activity before getting started. For example, in the left column of one table, students should complete a list of equations showing 9 times 1–10 ($$9 \times 1 = 9$$, $$9 \times 2 = 18$$. . . $$9 \times 10 = 90$$). On the right, students should complete a list of equations showing 9 times multiples of 10 ($$9 \times 10 = 90$$, $$9 \times 20 = 180$$. . . $$9 \times 100 = 900$$). Repeat for 4, and invite students to use this reference they’ve made to complete the task.
Supports accessibility for: Conceptual Processing, Memory

• Groups of 2

### Activity

• 5–6 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for students who appropriately interpret remainders based on the situation.

### Student Facing

1. Two bakers at a bakery made 378 muffins. The muffins are put in boxes of 4.

• The first baker says they will need 94 boxes for all the muffins.
• The second baker says 95 boxes are needed.

Who do you agree with? Explain or show your reasoning.
2. An auditorium seats 258 people. The seats are arranged in rows of 9, but there is one short row with fewer than 9 seats.

How many rows of 9 seats are there? How many seats are in the shorter row?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses and reasoning.
• For problem 1, highlight that both 94 and 95 boxes are plausible if they could be defended and the assumptions are made clear. (For example, students might say that the bakers could have two leftover muffins rather than trying to sell them in boxes, so 94 boxes are enough.)
• “What equation could we write to describe the relationship between the number of muffins and the number of full boxes?” (One possible equation: $$(94 \times 4) + 2 = 378$$.)
• For the last problem, ask: “What equation could we write to describe the relationship between the number of rows and the number of seats?” (One possible equation: $$(28 \times9) + 6 = 258$$.)

## Activity 2: Save for a Garden (20 minutes)

### Narrative

In this activity, students continue to solve contextual problems that involve division (MP2). Here, the dividends extend to four-digit numbers and the problems demand a greater lift.

In the second half of the activity, students are asked to reason in the opposite direction: given a division expression, they are to invent a situation that it can represent and interpret the value of the expression in context.

If time permits, consider asking students to create a visual display of the situation they invent for problem 2, so they can present the situation and their reasoning to the class.

MLR8 Discussion Supports. Synthesis: At the appropriate time, give groups 2–3 minutes to plan what they will say when they present to other groups. “Practice what you will say when you share your strategy with another group. Talk about what is important to say, and decide who will share each part.”
Advances: Speaking, Conversing, Representing

• Groups of 2

### Activity

• 3–4 minutes: independent work time on the first problem
• Invite students to share responses and reasoning.
• “What equation(s) can we write to represent the relationship between the amount saved each month, the savings, the number of months of saving, and the amount needed for the garden?” (One possible equation: $$(8 \times 158) + 6 = 1,\!270)$$
• 5–7 minutes: independent work time on problem 2

### Student Facing

1. A school needs \$1,270 to build a garden. After saving the same amount each month for 8 months, the school is still short by \$6.

How much did they save each month? Explain or show your reasoning.

2. Choose one of the following division expressions.

$$711 \div 3$$

$$3,\!128 \div 8$$

1. Write a situation to represent the expression.
2. Find the value of the quotient. Show your reasoning.
3. What does the value of the quotient represent in your situation?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Students find a partner who chose a different expression and take turns presenting their work. The person listening should consider whether the response makes sense and check if the quotient is correct.
• If time permits, students share with a different student or partnership.

## Lesson Synthesis

### Lesson Synthesis

“Today we solved problems that involved division. What strategies did you find yourself using to divide numbers? Did you:
• use partial products?
• use partial quotients?
• draw diagrams?
• divide by place value (thousands, hundreds, tens, and ones)?
• write a series of equations?
• estimate first?”

## Cool-down: Miscounting? (5 minutes)

### Cool-Down

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## Student Section Summary

### Student Facing

In this section, we solved different problems that involve dividing whole numbers.

We recalled two ways of thinking about division. For example, suppose $$274 \div 8$$ represents a situation where 274 markers are put into equal groups. The value of $$274 \div 8$$ can tell us:

• how many markers are in each group if there were 8 groups, or
• how many groups can be made if there were 8 markers in each group.

We learned that the 274 in $$274 \div 8$$ is called the dividend. We then explored different ways to find the value of a quotient (or the result of the division). For $$274 \div 8$$, we can:

• Divide by place value and think about putting 2 hundred, 7 tens, and 4 ones into 8 equal groups.
• Divide in parts and find partial quotients. For example, we can first find $$160 \div 8$$ (which is 20), and then $$80 \div 8$$ (which is 10), and then $$32 \div 8$$ (which is 4).
• Think in terms of multiplication. For example, we can think of $$8 \times 20 = 160$$, $$8 \times 10=80$$, and so on.

Here is one way to record division using partial quotients:

Sometimes a division results in a leftover that can’t be put into equal groups or is not enough to make a new group. We call the leftover a remainder. Dividing 274 by 8 gives 34 and a remainder of 2.