# Lesson 17

Base-ten Diagrams to Represent Division

## Warm-up: Which One Doesn’t Belong: Base-ten Diagrams (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare features of base-ten diagrams, looking not only at the number and types of shapes in each diagram, but also the value each diagram represents. The activity also enables students to recall what they know about representations of numbers in base-ten and enables the teacher to hear how they talk about these representations.

The analysis here prepares students for the activities in the lesson, in which they use base-ten diagrams to find whole-number quotients.

### Launch

• Groups of 2
• Display image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Record responses.

### Student Facing

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How is it that A and C both show 111?” (If a small square represents 1, then a rectangle is 10 and a large square is 100. In A: $$100 + 10 + 1 = 111$$. In C: $$100 + 11 = 111$$.)
• “How do we know that D also shows 111?” (Each group in D represents $$(3 \times 10)$$ + 7 or 37. Three groups of 37 makes 111.)
• “Suppose we don’t know what a small square represents except that it represents the same value in all diagrams. Can we tell if C and D represent the same value? How?” (Yes. We know that 10 small squares make 1 rectangle and 10 rectangles make 1 large square. In D, we’d have 21 small squares and 9 rectangles. Trading 10 small squares for a rectangle gives 10 rectangles and 11 small squares, which is equal to 1 large square and 11 small squares.)

## Activity 1: Divide with Diagrams or Blocks (15 minutes)

### Narrative

In this activity, students use base-ten diagrams to find quotients of two-digit dividends and single-digit divisors. They think about distributing the pieces in the diagram into equal-size groups, decomposing a higher-value piece with 10 of the lower-value pieces as needed to divide.

Some students may benefit from manipulating physical blocks. Provide each group of students with access to a set of base-ten blocks.

MLR8 Discussion Supports. Use multimodal examples to show the meaning of place value. Use verbal descriptions along with gestures, drawings, or concrete objects to show how a base-ten block is equivalent to 10 one blocks and how they are interchangeable.

### Required Materials

Materials to Gather

### Launch

• Groups of 4
• Display the first diagram. Make sure students can explain why it represents 64.

### Activity

• 5 minutes: quiet think time
• 2 minutes: group discussion
• Monitor for students who see that a larger piece can be decomposed into 10 of the next smaller piece to help with distribution.

### Student Facing

1. Priya draws a base-ten diagram to find the value of $$64 \div 4$$. A rectangle represents 10. A small square represents 1.

Use the diagram (or actual blocks) to help Priya complete the division. Explain or show your reasoning.

2. Use this base-ten diagram (or actual blocks) to find the value of $$117 \div 3$$.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses and reasoning.
• Make sure students see that:
• We can think of $$64 \div 4$$ as putting 6 tens and 4 ones into 4 equal groups, and $$117 \div 3$$ as putting 1 hundred 1 ten and 7 ones into 3 equal groups.
• To divide the base-ten pieces, we can decompose a piece representing a larger place value with 10 of the next smaller place value.

## Activity 2: Help Noah Get Unstuck (20 minutes)

### Narrative

In this activity, students continue to use base-ten representations and to reason about equal-size groups to find whole-number quotients. The work reinforces the idea of decomposing a hundred into 10 tens as needed to perform division.

Students explicitly use place value understanding to decompose hundreds and tens (MP7) while making sense of a students' reason to help him complete the division problem (MP3).

Engagement: Provide Access by Recruiting Interest. Optimize meaning and value. Invite students to share examples from their own lives in which they might need to divide three-digit numbers by one-digit numbers. Invite them to imagine and share why Noah might be dividing 235 by 5.
Supports accessibility for: Attention, Social-Emotional Functioning

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the first diagram.
• “How does the diagram represent 235?” (A large square represents 100. A rectangle represents 10. A small square represents 1.)

### Activity

• 4–5 minutes: independent work time on the first question
• Monitor for students who see the 2 hundreds as 20 tens and those who see them as 200 ones.
• Pause after the first question. Make sure students see that the 2 hundreds can be decomposed into 20 tens (or 200 ones) and split into 5 equal groups, and the 3 tens can be decomposed into 30 ones and split into 5 groups. Complete the second diagram to illustrate this reasoning.
• 5 minutes: partner work time on the last question

### Student Facing

1. This diagram represents 235. To find $$235 \div 5$$, Noah draws the following diagram but then gets stuck.

He says, “There are not enough of the hundreds or the tens pieces to put into 5 groups.”

Explain or show how Noah could find $$235 \div 5$$ with his diagram.

2. Find the value of $$432 \div 6$$. Show your reasoning. Use base-ten diagrams or blocks if you find them helpful.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Select students to share their responses and reasoning for the last question. Display them for all to see.
• Highlight the idea that each unit can be decomposed into 10 units of a lower place value to make it possible to create equal-sized groups.

## Lesson Synthesis

### Lesson Synthesis

“In earlier lessons, we solved division problems such as ‘712 divided by 4’ in different ways. We used familiar multiples or multiplication facts, or divided a series of smaller numbers. We also used area diagrams to reason about division.”

“Today we used base-ten diagrams and blocks to find quotients such as $$712 \div 4$$. How is this approach like earlier ones?” (It also involves performing division in a series of steps, rather than all at once.)

“How is this approach different?” (It involves:

• using place values
• dividing the amount in each place value into equal-size groups
• thinking of a digit in a number as 10 times the value of the digit to the right of it (for example, thinking of 3 tens as 30 ones)

“Instead of drawing base-ten pieces or using blocks, suppose we represent 712 with numbers and words: 7 hundreds + 1 ten + 2 ones. Can we still find $$712 \div 4$$ by reasoning about place values?” (Yes, we can distribute the hundreds, tens, and ones into 4 equal groups.)

“Here is a student’s unfinished work for finding $$712 \div 4$$. How would you complete it?”

Display:

$$712 \div 4$$ means putting 7 hundreds + 1 ten + 2 ones into 4 equal groups.”

(178. Sample reasoning: After putting 1 hundred in each group, there are 3 hundreds, 1 ten, and 2 ones left. The hundreds can be decomposed into tens and the tens can be decomposed into ones so that there’s enough to put into 4 groups.

\begin{align}&3 \text{ hundreds} + 1 \text{ ten} + 2 \text{ ones}\\ = & \,30 \text{ tens} + 1 \text{ ten} + 2 \text{ ones} \\ = & \,28 \text{ tens} + 3 \text{ tens} + 2 \text{ ones} \\ = &\,28 \text{ tens} + 30 \text{ ones} + 2 \text{ ones} \\ = & \,28 \text{ tens} + 32 \text{ ones} \end{align}

$$28 \div 4 = 7$$, so 7 tens in each group.

$$32 \div 4 = 8$$, so 8 ones in each group.)

## Cool-down: Find the Value of a Quotient (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.