18 minute read
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. 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
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.
