EM2 TEACH Level 5 Module 1 Lesson 1 Thin Slice

5 A Story of Units® Fractions Are Numbers

TEACH ▸ Module 1 ▸ Place Value Concepts for Multiplication and Division with Whole Numbers

What does this painting have to do with math?

Color and music fascinated Wassily Kandinsky, an abstract painter and trained musician in piano and cello. Some of his paintings appear to be “composed” in a way that helps us see the art as a musical composition. In math, we compose and decompose numbers to help us become more familiar with the number system. When you look at a number, can you see the parts that make up the total?

On the cover

Thirteen Rectangles, 1930

Wassily Kandinsky, Russian, 1866–1944

Oil on cardboard

Musée des Beaux-Arts, Nantes, France

Wassily Kandinsky (1866–1944), Thirteen Rectangles, 1930. Oil on cardboard, 70 x 60 cm. Musée des Beaux-Arts, Nantes, France. © 2020 Artists Rights Society (ARS), New York. Image credit: © RMN-Grand Palais/ Art Resource, NY

Great Minds® is the creator of Eureka Math® , Wit & Wisdom® , Alexandria Plan™, and PhD Science®

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ISBN 978-1-64497-179-6

Module

Fractions

1 Place Value Concepts for Multiplication and Division with Whole Numbers

2 Addition and Subtraction with Fractions

3 Multiplication and Division with Fractions

4 Place Value Concepts for Decimal Operations

5 Addition and Multiplication with Area and Volume

6 Foundations to Geometry in the Coordinate Plane

Before This Module

Overview

Grade 4 Module 1

Students read, write, compare, and round multi-digit whole numbers to millions in standard, expanded, word, and unit forms. They describe the relationship between a digit in one place and the digit in the next larger place by using the multiplicative comparison 10 times as much as.

Grade 4 Modules 2 and 3

Students multiply and divide whole numbers of up to four digits by one-digit numbers (including expressing quotients with whole-number remainders) and multiply 2 two-digit numbers. They use methods based on the place value chart, area models, the commutative and associative properties of multiplication, and the distributive property.

Place Value Concepts for Multiplication and Division with Whole Numbers

Topic A

Place Value Understanding for Whole Numbers

Students use multiplicative comparison statements to explain that a digit in one place represents 10 times as much as what it represents in the place to the right. Students notice how digits of a number shift when they multiply or divide by a power of 10 and express a power of 10 in exponential form. Then students find products and quotients by using powers of 10 and convert metric measurements from larger to smaller units.

Topic B

Multiplication of Whole Numbers

Students build fluency with multiplying multi-digit numbers by using the standard algorithm. They use place value understanding to visualize the decomposition of factors while they multiply a single digit at a time by another single digit in the standard algorithm.

Topic C Division of Whole Numbers

Students use methods based on place value to find quotients of whole numbers with up to four-digit dividends and two-digit divisors. They estimate quotients, then use tape diagrams, area models, and vertical form to record quotients and remainders.

Topic D

Multi-Step Problems with Whole Numbers

Students move between written, pictorial, and numeric representations of mathematical statements. They use tape diagrams to determine when parentheses are needed in expressions and evaluate expressions containing grouping symbols. 3 times the sum of 15 and 25

There are 26 people at the park. 8 people go home. The rest of the people make 2 equal groups to play a game. How many people are in each group?

After This Module

Grade 5 Module 4

Students use place value knowledge and times as much as language to learn about decimal numbers. Students see how the strategies they use for whole-number operations extend to operations with decimal numbers. They convert metric measurements from smaller units to larger units.

Grade 6 Modules 2 and 4

There are 9 people in each group.

In module 2, students learn to divide whole numbers with any number of digits by using the standard algorithm. In module 4, students build upon grade 5 knowledge by writing and evaluating numerical expressions with terms that have whole-number bases and exponents.

to multiply and divide by powers of 10

Estimate products and quotients by using powers of 10 and their multiples.

Convert measurements and describe relationships between metric units.

Solve multi-step word problems by using metric measurement conversion.

Multiplication of Whole Numbers

Lesson 8

Multiply two- and three-digit numbers by two-digit numbers by using the distributive property.

Lesson 9

Multiply two- and three-digit numbers by two-digit numbers by using the standard algorithm.

Multiply three- and four-digit numbers by three-digit numbers by using the standard algorithm.

Multiply two multi-digit numbers by using the standard algorithm.

Division of Whole Numbers

Divide two-digit numbers by two-digit numbers in problems that result in one-digit quotients. Lesson 14

Divide three-digit numbers by two-digit numbers in problems that result in one-digit quotients. Lesson 15

Divide three-digit numbers by two-digit numbers in problems that result in two-digit quotients.

Divide four-digit numbers by two-digit numbers.

Multi-Step Problems with Whole Numbers

Write, interpret, and compare numerical expressions.

Create and solve real-world problems for given numerical expressions.

multi-step word problems involving multiplication and division.

Why Place Value Concepts for Multiplication and Division with Whole Numbers

Why does multiplication and division of whole numbers come first?

After much consideration of our students’ learning, teachers’ input, and a review of the research around how students learn and how mathematical concepts progress, we decided it makes the most sense to put place value concepts and operations with whole numbers first. Why?

1. The major emphasis of grade 5 standards involves understanding the place value system, performing operations with multi-digit whole numbers, and applying and extending knowledge of whole-number operations to fractions and decimals.

Beginning the year with a focus on place value and whole-number operations sets up students for success as they move into operations with fractions in modules 2 and 3, then with decimals in module 4.

2. Beginning the year with learning how to multiply multi-digit numbers provides an opportunity for students to develop fluency with using the standard algorithm throughout the year, as required by the standards.

3. Multiplying and dividing multi-digits numbers gives rise to developing estimation skills and to introducing powers of 10 in a meaningful way. Powers of 10 are not just the numbers on a place value chart, rather they are powerful tools for making estimates of products and quotients and for checking the reasonableness of answers.

When do students learn about decimals? Why?

Grade 5 module 4 addresses work with decimals and parallels the content of module 1. Students begin by relating adjacent place value units and use comparison language, such as 1 tenth is 10 times as much as 1 hundredth, just as they did with whole numbers. It makes sense mathematically to position decimals in module 4 after an in-depth study of whole numbers in module 1 and then fractions in modules 2 and 3. This move also makes sense pedagogically because students can use 1 10 , 1 100 , and 1 1,000 to describe relationships between numbers on the place value chart and to perform operations on decimals.

I notice students only convert from larger metric units to smaller metric units in this module. Why?

Metric conversions are limited to moving from larger units (such as kilometers) to smaller units (such as meters) in module 1 because conversions that move from smaller to larger units are best performed by using fractions or decimals. Students learn to multiply fractions in module 3 and they learn to multiply decimals in module 4. The remaining part of the metric conversion standard is fully met in module 4 as an application of decimals.

Achievement Descriptors: Overview

Place Value Concepts for Multiplication and Division with Whole Numbers

Achievement Descriptors (ADs) are standards-aligned descriptions that detail what students should know and be able to do based on the instruction. ADs are written by using portions of various standards to form a clear, concise description of the work covered in each module.

Each module has its own set of ADs, and the number of ADs varies by module. Taken together, the sets of module-level ADs describe what students should accomplish by the end of the year.

ADs and their proficiency indicators support teachers with interpreting student work on

• informal classroom observations,

• data from other lesson-embedded formative assessments,

• Exit Tickets,

• Topic Quizzes, and

• Module Assessments.

This module contains the 12 ADs listed.

5.Mod1.AD1

Write whole-number numerical expressions with parentheses.

5.Mod1.AD2

Evaluate whole-number numerical expressions with parentheses.

5.OA.A.1

5.Mod1.AD5

Solve real-world and mathematical problems that involve addition, subtraction, multiplication, and division of multi-digit whole numbers.

5.NBT

5.Mod1.AD9

Multiply two multi-digit whole numbers by using the standard algorithm.

5.Mod1.AD6

5.OA.A.1

Explain the relationship between digits in multi-digit whole numbers.

5.Mod1.AD3

Translate between whole-number numerical expressions and mathematical or contextual verbal descriptions.

5.OA.A.2

5.Mod1.AD7

Explain the effect of multiplying and dividing whole numbers by powers of 10.

5.Mod1.AD4

Compare the effect of each number and operation on the value of a whole-number numerical expression.

5.OA.A.2

5.Mod1.AD8

Express whole-number powers of 10 in exponential form, standard form, and as repeated multiplication.

5.NBT.B.5

5.Mod1.AD10

Solve problems that involve division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

5.Mod1.AD11

Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models.

5.NBT.B.6

5.Mod1.AD12

Convert among whole-number amounts within the metric measurement system to solve problems.

5.MD.A.1

The first page of each lesson identifies the ADs aligned with that lesson. Each AD may have up to three indicators, each aligned to a proficiency category (i.e., Partially Proficient, Proficient, Highly Proficient). While every AD has an indicator to describe Proficient performance, only select ADs have an indicator for Partially Proficient and/or Highly Proficient performance.

5.NBT.A.1

5.NBT.A.2

5.NBT.A.2

5.NBT.B.6

An example of one of these ADs, along with its proficiency indicators, is shown here for reference. The complete set of this module’s ADs with proficiency indicators can be found in the Achievement Descriptors: Proficiency Indicators resource.

ADs have the following parts:

• AD Code: The code indicates the grade level and the module number and then lists the ADs in no particular order. For example, the first AD for grade 5 module 1 is coded as 5.Mod1.AD1.

• AD Language: The language is crafted from standards and concisely describes what will be assessed.

• AD Indicators: The indicators describe the precise expectations of the AD for the given proficiency category.

• Related Standard: This identifies the standard or parts of standards from the Common Core State Standards that the AD addresses.

5.Mod1.AD11 Represent division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using models.

RELATED CCSSM

5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Partially Proficient Proficient

Determine the quotient for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits by using a provided model.

Use the model shown to help you divide.

Create models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

Use the expression to answer part A and part B.

4,102 ÷ 14

The quotient is

Part A

Draw a model for the expression.

Part B

Use your model to determine the quotient and remainder.

Highly Proficient

Interpret models for division of whole-number dividends with up to four digits and whole-number divisors with up to two digits.

What values could be represented by the letters in the model? Explain your thinking.

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Topic A Place Value Understanding for Whole Numbers

In topic A, students apply their understanding of place value to multiply and divide by powers of 10 and their multiples.

Prior to grade 5, students use place value understanding to round multi-digit whole numbers to any place. They compare quantities through multiplicative comparison and recognize that in a whole number, a digit in one place represents 10 times as much as what it represents in the place to the right.

The topic opens with students using place value charts to show that when two adjacent digits in a given number are the same, the digit to the left is 10 times as much as the digit to the right and the digit to the right is 10 times as small as the digit to the left. Students use dot models to understand what happens when they multiply or divide a number by 10 . Next, students apply what they learn from the dot models to conclude that when they multiply a number by 10 , it causes each digit of the number to shift one place value to the left, and when they divide a number by 10 , it causes each digit of the number to shift one place value to the right. Building on this understanding, they notice how the digits shift when they multiply or divide a number by 100 and by 1,000 .

Students find products and quotients of expressions composed only of powers of 10, such as 10,000 × 100, by using what they learn about how digits in a number shift. When students find products and quotients of expressions composed only of 10s, it gives rise to learning about exponents with a base of 10. They write powers of 10 in standard form, expanded form, and exponential form. Students extend their understanding of the shifts they notice when they multiply or divide by 10 to multiplying and dividing by 102 or 103.

Students estimate products and quotients of multi-digit numbers by rounding factors, divisors, and dividends to multiples of powers of 10 . By comparing estimates and analyzing estimation strategies, they understand what may cause an underestimate or an overestimate. Students then estimate products and quotients in real-world situations. The topic culminates with students using observations about how digits shift when they multiply by powers of 10 to convert metric measurements.

By combining multiplicative comparison language with their understanding of powers of 10 , students describe relative sizes of units of metric length, weight, and capacity. They convert between units and express larger units in terms of smaller units by using powers of 10 . Students solve multi-step word problems involving metric conversions and apply their estimation skills from previous lessons to determine whether answers are reasonable.

In topic B, students apply their understanding of place value to multiply multi-digit whole numbers.

Progression of Lessons

Lesson 1

Relate adjacent place value units by using place value understanding.

Lesson 2

Multiply and divide by 10, 100, and 1,000 and identify patterns in the products and quotients.

Lesson 3

Use exponents to multiply and divide by powers of 10.

I can represent multiplication and division by 10 on a place value chart. I notice when two adjacent digits are the same number, the digit to the left is 10 times as much as the digit to the right and the digit to the right is 10 times as small as the digit to the left.

When I multiply a number by 10, 100, or 1,000, the digits shift to the left. When I divide a number by 10, 100, or 1,000, the digits shift to the right. For example, if I multiply 4 tens by 1,000, the 4 shifts three units to the left, which is 4 ten thousands, or 40,000. If I divide 4 thousands by 100, the 4 shifts two units to the right, which is 4 tens, or 40.

I can write powers of 10 in standard form and exponential form. I can use what I know about how many 10s are in a number to efficiently multiply or divide by shifting digits to the left or the right.

Lesson 4

Estimate products and quotients by using powers of 10 and their multiples.

129 ÷ 4 ≈ 12 0 ÷ 4 = 30

I can estimate products and quotients by rounding numbers to multiples of 10. For example, I can estimate the product of 47 and 61 by finding 50 × 60. I can estimate the quotient of 316 and 45 by finding 300 ÷ 50.

Lesson 5

Convert measurements and describe relationships between metric units.

kilometer, meter, centimeter, millimeter longestshortest

I can convert larger metric units to smaller metric units by using multiplication. I can use prefixes to remind me of the relationship between metric units.

Lesson 6

Solve multi-step word problems by using metric measurement conversion.

6 m 40 cm or 64 0 cm

80 cm

I can use the Read–Draw–Write process to make sense of and to solve word problems. Models help me see different ways to solve a problem. I can solve word problems that have different metric units by converting larger units to smaller units.

Relate adjacent place value units by using place value understanding.

Lesson at a Glance

With a partner, students organize and count a collection of bills that requires them to use their place value understanding. Students model 10 times as much for each unit on the place value chart up to 1 million and determine that when two adjacent digits are the same, the digit to the left is 10 times as much as the digit to the right. Then students model division by 10 on the place value chart and find that when two adjacent digits are the same, the digit to the right is 10 times as small as the digit to the left. They compare the same digit in different places and describe the relationship between the numbers by using what they know about multiplication and division. This lesson introduces the academic verb consider.

Key Question

• How are place value units related to each other?

Achievement Descriptor

5.Mod1.AD6 Explain the relationship between digits in multi-digit whole numbers. (5.NBT.A.1)

Agenda Materials

Fluency 10 min

Launch 5 min

Learn 35 min

• Organize and Count Bills to Compare

• Compare and Relate the Same Digit with Different Values

• Problem Set

Land 10 min

Teacher

• Computer or device*

• Projection device*

• Teach book*

• Money Counting Collection (in the teacher edition)

• Place Value Chart to Millions (in the teacher edition)

Students

• Dry-erase marker*

• Learn book*

• Pencil*

• Personal whiteboard*

• Personal whiteboard eraser*

• Organizational tools

• Place Value Chart to Millions (in the student book)

* These materials are only listed in lesson 1. Ready these materials for every lesson in this module.

Lesson Preparation

• Print or copy Money Counting Collection and cut out the collections of paper money. Prepare one collection per student pair.

• Consider whether to remove Place Value Chart to Millions from the student books in advance or have students remove them during the lesson.

• Provide tools for students to choose from to help organize their counts. Tools may include cups, paper clips, whiteboards, bags, rubber bands, or graph paper.

Fluency

Choral Response: Rename Place Value Units

Students use unit form to identify a number modeled with place value disks, and then compose and rename to prepare for relating adjacent place value units.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display the 10 ones disks on the chart.

What value is represented on the chart? Say the answer in unit form.

10 ones

Display 10 ones = ten.

10 ones is equal to how many tens?

1 ten

10 ones = ten 1

Teacher Note

Use hand signals to introduce a procedure for answering choral response questions. For example, cup your hand around your ear for listen, lift your finger to your temple for think, and raise your own hand to remind students to raise theirs.

Teach the procedure by using the following general knowledge questions:

• What grade are you in?

• What is the name of our school?

• What is your teacher’s name?

Differentiation: Support

Consider having place value disks available during the activity for students who need additional support.

Display the answer and the disks bundled as a ten on the chart.

Continue the process with the following sequence:

Whiteboard Exchange: Place Value

Students identify a place value and the value of a digit in a multi-digit number, and then write the number in expanded form to prepare for relating adjacent place value units.

After asking each question, wait until most students raise their hands, and then signal for students to respond.

Raise your hand when you know the answer to each question. Wait for my signal to say the answer.

Display 2,518.

When I give the signal, read the number shown. Ready?

2,518

What digit is in the thousands place?

2

2,518

2,000 + 50 0 + 10 + 8

Teacher Note

Establish a signal (e.g., show me your whiteboards) to introduce a procedure for showing whiteboard exchange responses. Practice with basic computations such as the following until students are accustomed to the procedure:

• What is 10 + 8?

• What is 500 + 18?

Establish a procedure for providing feedback on whiteboard exchanges. Consider circulating and giving hand signals—thumbs-up or try again.

Display the underlined 2.

What value does the 2 represent in this number?

2,000

Write 2,518 in expanded form.

Give students time to work. When most students are ready, signal for students to show their whiteboards. Provide immediate and specific feedback. If students need to revise, briefly return to validate their corrections.

Display the number in expanded form.

Continue the process with the following sequence:

Launch

Students convert among different measurements and analyze their multiplicative relationships.

Introduce the Which One Doesn’t Belong? routine. Present four statements and invite students to study them.

A

1 foot = 12 inches

C 1 L = 1,000 mL

B

1 meter is the same length as 100 centimeters.

D

1,000 grams = 1 kilogram

Teacher Note

Consider asking students to express each number in expanded form differently. For example, ask students to use only addition for some numbers and incorporate multiplication for others as in the following examples:

Language Support

Consider using strategic, flexible grouping throughout the module.

• Pair students who have different levels of mathematical proficiency.

• Pair students who have different levels of English language proficiency.

• Join pairs to form small groups of four.

As applicable, complement any of these groupings by pairing students who speak the same native language.

Give students 2 minutes to find a category in which three of the items belong, but a fourth item does not.

When time is up, invite students to explain their chosen categories and to justify why one item does not fit.

Highlight responses that emphasize reasoning about the factors and multiples of 10 among the metric units.

Ask questions that invite students to use precise language, make connections, and ask questions of their own.

Use the following sample questions and prompts.

Which one doesn’t belong?

A doesn’t belong because it is the only choice that does not use metric units.

B doesn’t belong because it is the only one that uses words instead of an equal sign.

C doesn’t belong because it is the only one with abbreviated units.

D doesn’t belong because it is the only choice where 1 unit is to the right of the equal sign.

Complete this statement: × 1 mL = 1 L.

1,000 × 1 mL = 1 L

1 liter is 1,000 times as much as 1 milliliter.

Complete this statement: 1 meter = × 1 centimeter.

1 meter = 100 × 1 centimeter

1 meter is 100 times as much as 1 centimeter.

Complete this statement: 1 kilogram = × 1 gram.

1 kilogram = 1,000 × 1 gram

1 kilogram is 1,000 times as much as 1 gram.

We expressed each relationship by using multiplication.

Transition to the next segment by framing the work.

Today, we will use our place value understanding to describe the relationship between place value units by using multiplication and division.

Learn

Organize and Count Bills to Compare

Materials—S: Money counting collection, organizational tools

Students use self-selected strategies to organize and count a collection and record their process.

Partner students and distribute a counting collection to each pair.

Direct students to the recording page in their books. Briefly orient students to the materials and procedure for the counting collection activity:

• Partners collaborate to count a collection.

• Partners make their own recordings to show how they counted.

• Partners may use the place value chart and other organizational tools. Organizational tools may include readily available classroom items such as cups, paper clips, personal whiteboards, etc.

Before they begin, invite partners to work together to estimate how many dollars are in their collections. Have them write down their estimates. Then encourage them to talk about how they will organize their collections to count.

Promoting the Standards for Mathematical Practice

Students use appropriate tools strategically (MP5) as they discuss and self-select counting strategies and organizational tools for counting their collection.

Ask the following questions to promote MP5:

• What strategies or tools can help you count your collection?

• Which tool would be the most helpful to count your collection? Why?

• Why did you choose this strategy to count your collection? Did your strategy work well?

Then invite students to select organizational tools they would like to use, with the understanding that tools may be exchanged as plans are refined.

Ask partners to begin counting their collections. Circulate and notice how students engage in the following behaviors:

Organize: Strategies may include grouping bills of the same unit, making groups of 10 of the same unit, organizing bills on the place value chart, and writing expressions or equations. Students may also organize their collections by using attributes that do not support counting efficiently, such as mixing units to make equal groups of bills.

Count: Students may count subgroups and then add to find the total, or they may use a place value chart and write the digits that represent the number of each unit. Other students may use a combination of multiplication and addition to find the total.

Record: Recordings may include drawings, numbers, expressions, equations, and written explanations.

Use questions and prompts such as the following to assess and advance student thinking as they organize and count their collection:

• Show and tell me what you did.

• How can you organize your collection to make it easier for you to count?

• How does the way you organized your collection make it easier for you to count?

• How did you keep track of what you already counted and what you still needed to count?

• How did you name the larger units? Why?

• How did you know how to write your total?

• How close was your estimate to your actual count?

Teacher Note

The counting collections vary in levels of complexity. Partner students and strategically assign each pair a counting collection.

• Counting Collection A does not require composing units.

• Counting Collection B requires composing units in one place value.

• Counting Collection C requires composing units in two place values.

• Counting Collection D requires composing units in three place values.

UDL: Action & Expression

Consider offering sticky notes for labeling to support students in organizing their collections. For example, if students organize their bills like a place value chart, they can use the sticky notes to label each place. This allows flexibility as students organize and keep track of their count.

For this counting collection, I am partners with .

We are counting .

We think they have a value of .

This is how we organized and counted the collection:

We counted altogether.

An equation that describes how we counted is: .

Self-Reflection

Write one thing that worked well for you and your partner. Explain why it worked well. It was helpful to bundle when we had 10 of a unit because then we could rename it as the next largest unit. That helped us find the total.

Write one challenge you had. How did you work through the challenge?

We were not sure what some of the place value units were. We used the numbers on the bills to help us.

Gather the class and facilitate a brief discussion about how students chose to organize and count the bills.

How did you organize your bills?

We put like units together.

We put our bills into groups of 10.

We organized the bills like a place value chart.

How did you find the total?

We skip-counted by each unit.

We bundled to make larger units when we could. We found the total by writing how many of each unit we had.

We counted how many bills of each unit we had. Then we multiplied to find the amount for each unit. We added the amounts for each unit to find the total.

How did you decide when to compose a larger unit?

When we had 10 of a smaller unit, we composed them to make 1 of the next larger unit.

When we had a group of 10, we bundled it with a paper clip. Then we placed the bundle into the next larger unit on our chart.

When we had 10 thousands, we bundled them to make 1 ten thousand.

Invite each group to share the total amount of money in the collection they counted. Record the totals so they can be referred to later in the lesson.

Compare and Relate the Same Digit with Different Values

Materials—T/S: Place Value Chart to Millions

Students determine that the same digits in different places do not represent the same value and articulate how the digits in different place values are similar and different.

Direct students to Place Value Chart to Millions in their books. Have students remove the chart and insert it into their whiteboards.

Ask them to write 1,731,225 in standard form as you do the same.

Underline the 2 in the hundreds place and the 2 in the tens place. Point to them as you ask the following questions.

Do these 2s represent the same amount?

No, they represent different amounts.

Let’s write the number in expanded form so we can see more clearly how much each 2 represents.

Direct students to write 1,731,225 in expanded form as you do the same.

Gesture to the 2 in the hundreds place.

How much does this 2 represent?

200

Gesture to the 2 in the tens place.

How much does this 2 represent?

20

Gesture to the 2 in the hundreds place.

The first 2 represents 200.

Gesture to the 2 in the tens place.

The other 2 represents 20. Consider, or think about, how 2 hundreds is similar to or different from 2 tens.

Pause to allow students time to think, then invite students to respond.

Both show 2 of a unit.

2 hundreds is greater than 2 tens.

Invite students to turn and talk about whether they would rather have 2 hundred-dollar bills or 2 ten-dollar bills and why.

Direct students to show 2 tens on the place value chart.

Let’s think some more about the relationship between 2 tens and 2 hundreds. What do we need to multiply 2 tens by to get 2 hundreds? 10

Using Place Value Chart to Millions, draw two dots in the tens column. Draw an arrow, labeled × 10 from the 2 tens in the tens place to the hundreds place and draw 2 hundreds.

Display the comparison statement:

200 is  times as much as 20.

Complete the statement: 200 is times as much as 20.

200 is 10 times as much as 20.

Record 200 = 10 × 20 and direct students to do the same.

Direct students to erase.

Let’s show the relationship between 200 and 20 by using division.

Direct students to draw 2 hundreds.

Write 200 ÷ = 20. Gesture to the statement 200 = 10 × 20.

We know that 200 is 10 times as much as 20. Let’s use that to complete the statement:

200 ÷ = 20.

200 ÷ 10 = 20

Language Support

This segment introduces the term consider. Consider previewing the meaning of the term before students are asked to consider how the numbers are similar. Relate the term to thinking about the weather as they decide what to wear or thinking about reasons for choosing a recess activity.

Teacher Note

The digital interactive Place Value Chart helps students represent and compare the sizes of numbers.

Consider allowing students to experiment with the tool individually or demonstrate it for the whole class.

Draw two dots in the hundreds column. Draw an arrow, labeled ÷ 10 from the 2 hundreds in the hundreds place to the tens place and draw 2 tens.

Write the comparison statement: 20 is times as small as 200.

Complete the statement: 20 is times as small as 200. 20 is 10 times as small as 200.

When we have the same digit in adjacent places, or right next to each other, the digit on the left is 10 times as much as the digit on the right.

Let’s look at other relationships between digits in this number.

Circle the 1 in the millions place and the 1 in the thousands place.

Consider, or think about, how 1 million is similar to or different from 1 thousand.

Invite students to think–pair–share to compare the two digits. Both show 1 of a unit.

1 million is greater than 1 thousand. The 1s are in different places.

1 thousand is 10 hundreds. 1 million is 10 hundred thousands.

Direct students to the expanded form recording.

Gesture to the 1 in the millions place.

How much does this 1 represent? 1,000,000

Gesture to the 1 in the thousands place.

How much does this 1 represent?

1,000

Language Support

Consider reviewing the familiar term adjacent with students. Adjacent angles are angles that are next to each other and share a side. Angles that are nonadjacent do not share a side. Make connections to place value by discussing which places are next to each other and which are not. Highlight that the prefix non- means not to help students understand that nonadjacent means not adjacent. Create a visual you can use to highlight examples of adjacent and nonadjacent digits, such as in the following chart:

Is 1 million 10 times as much as 1 thousand? Why?

No, because the millions place is not adjacent to the thousands place. 10 times as much as 1 thousand is 1 ten thousand, not 1 million.

Let’s see how many times as much 1 million is as 1 thousand.

Draw 1 thousand on the place value chart and multiply by 10 (by using the arrow to show movement), until you reach 1 million. Label each arrow × 10.

How many times do we have to multiply by 10 to get from 1,000 to 1,000,000?

We have to multiply by 10 three times.

What is the value of 10 × 10 × 10?

1,000

Complete this statement: 1,000,000 is times as much as . 1,000,000 is 1,000 times as much as 1,000.

Record 1,000,000 = 1,000 × 1,000 and direct students to do the same.

Invite students to think–pair–share about how digits that are the same and in adjacent places are similar to or different from digits that are the same but not in adjacent places.

A digit that is the same as a digit in an adjacent place is 10 times as much as the same digit directly to its right.

Digits that are the same but not adjacent are a multiple of 10 times as much as the same digit in other place values to the right.

Let’s show the relationship between 1,000,000 and 1,000 by using division.

Direct students to draw 1 million on the place value chart while you do the same.

Write 1,000,000 ÷ = 1,000. Gesture to the statement 1,000,000 = 1,000 × 1,000.

We know that 1,000,000 is 1,000 times as much as 1,000. Let’s use that to complete this statement: 1,000,000 ÷ = 1,000.

1,000,000 ÷ 1,000 = 1,000

Differentiation: Support

Help students understand that 10 times as much as 1 thousand is 1 ten thousand by showing and bundling physical place value disks on the place value chart until their understanding of the pictorial representation is firm.

For students who need additional support, consider offering them calculators to confirm the relationship of 10 times as much and 10 times as small. Support students with entering 1 on the calculator and making the direct connection to the ones place before they begin to multiply by 10. Have students multiply by 10 and connect the tens place. Continue multiplying by 10 to millions, pointing out that each time students multiply by 10, the product is one place to the left in the place value chart. Repeat the process of dividing by 10 until students return to the ones place.

Repeatedly draw an arrow, labeled as ÷ 10, from the 1 million in the millions place to the thousands place and draw 1 thousand.

Write the comparison statement: 1,000 is times as small as .

Complete this statement: 1,000 is times as small as .

1,000 is 1,000 times as small as 1,000,000.

Invite students to turn and talk about whether they would rather have a $1,000 bill or a $1,000,000 bill and why.

Let’s see whether this works with other totals that we counted. Refer to the list of counting collection values and direct students to the number 2,988,396. Have them write the number in standard form as you do the same.

Underline the two 8s and circle the two 9s.

Use a similar sequence to guide students to describe the relationship between the 8 in the ten thousands place and the 8 in the thousands place and the relationship between the 9 in the hundred thousands place and the 9 in the tens place.

Consider using the following questions to guide students’ analysis:

• Are these two 8s equal? How do you know?

• How is 8 ten thousands similar to or different from 8 thousands?

• How is 9 hundred thousands similar to or different from 9 tens?

Gesture to the circled 9s in standard form.

If we divide 9 hundred thousands by 10, will we get 9 tens? Why?

No, we will not. The tens place is not adjacent to the hundred thousands place.

What will we get if we divide 9 hundred thousands by 10? Why?

We will get 9 ten thousands because the ten thousands place is adjacent to the hundred thousands place.

Teacher Note

Ask students to think about the relationship between the millions place and the ten millions place, and places beyond. Or ask students to think about the relationship between the tens place and the ones place. Note that the pattern continues, even as the place value units become greater or less.

Invite students to turn and talk about how they know 9 hundred thousands divided by 10 is 9 ten thousands.

Is 9 hundred thousands 10 times as much as 8 ten thousands? Why?

No, because the digits are not the same. 10 times as much as 8 ten thousands is 8 hundred thousands, not 9 hundred thousands.

Two digits that are not the same do not have the 10 times as much relationship. Display the equations.

Differentiation: Challenge

Present students with a number such as 2,458,136 and invite them to rearrange the digits to produce the number with the greatest possible value. Then ask students to choose any digit and describe its value before and after rearranging by using 10 times as much or 10 times as small language and by showing their thinking on a place value chart.

What is 100 ÷ 10?

What is 1,000 ÷ 10?

Invite students to turn and talk to predict the quotients for the remaining equations based on the pattern they see.

Invite students to think–pair–share to complete this statement: When we divide by 10, the quotient .

When we divide by 10, the quotient moves one place value unit to the right.

When we divide by 10, the quotient is 10 times as small as the dividend.

Problem Set

Differentiate the set by selecting problems for students to finish independently within the timeframe. Problems are organized from simple to complex.

Land

Debrief 5 min

Objective: Relate adjacent place value units by using place value understanding.

Facilitate a class discussion about relating adjacent place value units by using the following prompts. Encourage students to restate or add on to their classmates’ responses.

Display the number with the digits circled and underlined.

How does 3 hundred thousands relate to 3 ten thousands?

3 hundred thousands is 10 times as much as 3 ten thousands.

Is it correct to say that 3 ten thousands is 10 times as much as 2 thousands? How do you know?

No, it is not correct. 10 times as much as 2 thousands is 20,000, not 30,000.

The digits have to be the same to be 10 times as much.

How are place value units related to each other?

There is a 10 times as much relationship from one place value unit to the next when you start at the ones place and move left.

When the digits in a number are the same and adjacent, then the digit to the left is 10 times as much as the digit next to it.

Exit Ticket 5 min

Provide up to 5 minutes for students to complete the Exit Ticket. It is possible to gather formative data even if some students do not complete every problem.

Differentiation: Support

If students need support to complete problems 6–12 in the Problem Set, model how they can continue to use their place value charts as needed. See the following example for problem 5.

Language Support

Scaffold the questions for English learners by asking them to complete the following statements:

• When I see two of the same digit in a number, I know .

• For a digit to represent 10 times as much as the next digit to its right, it must be .

Expect to see varied solution paths. Accept accurate responses, reasonable explanations, and equivalent answers for all student work.

13. Consider the number shown. 8 7 7, 4 8 7

a. Complete the equation to represent the number in expanded form. 877,487 = ( 800,000 ) + ( 70,000 ) + ( 7,000 ) + ( 400 ) + ( 80 ) + ( 7 )

b. Draw a box around the digit that represents 10 times as much as the underlined digit.

c. Complete the equations to show the relationships between the boxed and underlined digits.

70,000 = 10 × 7,000

70,000 ÷ 10 = 7,000

d. Explain how the digit in the hundred thousands place is related to the digit in the tens place. 8 hundred thousands is 10,000 times as much as 8 tens.

14. Kayla and Blake both write a number.

9. 400,000 = 10 × 40,000

a. Kayla says, “The 3 in my number is 10 times as much as the 3 in Blake’s number.” Do you agree with Kayla? Explain.

No, I do not agree with Kayla. The 3 in Blake’s number represents 3,000. The 3 in Kayla’s number represents 300,000. So the 3 in Kayla’s number represents 100 times as much as the 3 in Blake’s number, not 10 times as much. The value of the 3 in Kayla’s number is 300,000, not 30,000.

b. Write a division equation to relate the 8 in Kayla’s number to the 8 in Blake’s number. 8,000 ÷ 1,000 = 8