• The Think-Pair-Share activity can be used to introduce the idea of place value. Observation during this activity will give some basis for evaluating current level of student understanding.
• The discussion during Quick Response activities 1 and 2 will help to determine if further explanation or practice is necessary.

• The Partner Place Value Project may be used to determine level of comprehension of the class.
• Use the Lesson 1 Exit Ticket to determine if mastery has been achieved and if students are ready to move on to the next lesson.

W:	Students will learn about place value in multidigit numbers. Through practice and discussion, students will gain understanding of how digit location plays an important role in the value of the number.
H:	The initial activity involves students arranging place value labels onto a multidigit number. Students discuss their decision-making process with a partner using a think-pair-share event. This activity will hook students’ interest so that they become conscious of the importance of place value in large numbers.
E:	Students will examine the idea of place value more closely by using base-ten models to show the comparative size of each digit’s value. The quick response activity is another way to engage students in the learning and to make sure students are grasping the concepts thoroughly. Students experiment with writing multidigit numbers in expanded form to demonstrate an understanding of place value.
R:	Use the quick response and partner place value project as a way for students to review all of the place value concepts.
E:	Note areas of difficulty during the quick response activity. Reteach or redirect as necessary. Observation during the partner place-value project should serve as a basis for evaluating the students’ level of understanding.
T:	This lesson may be tailored to meet the needs of the students. Additional lesson ideas and modifications may be found in the Extension section for students who may benefit from additional learning opportunities or those who have accomplished mastery and wish to go beyond the requirements of the standard.
O:	The lesson introduces students to the importance of place value in multidigit numbers and goes on to investigate decimal place values. Students learn to translate numbers between expanded form and standard form. The lesson is organized so that it starts with simpler, whole-number-digit numbers and proceeds to more difficult numbers such as those with digits in several decimal places.

This lesson is organized in a discovery format with students reviewing concepts of place value for whole numbers first and applying these concepts to decimal numbers. Visual representations of place value were used to help solidify the concept of the ratio of 10 and between place values. Students practice identifying place value names and the numeric value of digits in any place value with partners, and present their findings to the class. Students are also asked to extend their understanding of written forms of numbers by writing both multidigit whole numbers and decimal numbers in several different ways. Several remediation and extension activities are suggested to help deepen student understanding of these concepts. Lessons 2 and 3 will scaffold on the ideas presented in this lesson by using similar activities and strategies to identify more patterns with ten, and compare and round decimal numbers.

Prior to students arriving, write a six-digit number on the board using six different digits from
1–9. Also cut out large paper labels with the first six whole number place values written on them (ones, tens, hundreds, thousands, ten-thousands, and hundred-thousands). If using an interactive whiteboard, create a separate rectangle for each label that can be easily moved around on the screen. Do not display the labels right away. Wait until students have had a chance to come up with them on their own during the Think-Pair-Share activity.

 

For example: 261,594

 

Think-Pair-Share Activity

“Take out a half-sheet of paper. Each of you should write down this number in the center of your paper.” Ask half of the class to write the names of the place values of each digit below the digits. Ask the other half of the class to write down the value of each digit above the digit. Allow approximately 3 minutes for students to work alone. “Now turn to your partner and share your results.” Give students 1–2 minutes each to share.

Next you will select six students randomly from the first group to share one of their place value names. Hand each of these students a paper place value label (or allow them to move a label on the interactive whiteboard) to place below the digit that represents their assigned place value on the board. The students should sit down once they place their labels. If others who worked on the place value side of the room feel a label has been placed out of order, allow them to explain why they think so. If they are correct, allow them to adjust the position of the label.

“Does everybody agree that the labels are correctly placed? Are all the digits marked with the correct place value?” Once the group agrees that all six labels are placed correctly, review the names with the whole class.

Select six students from the group who worked on the digit values to each record the value of one digit above the number on the board. Allow other students to offer suggestions on any digits that are not correctly labeled.

“Now we will look at the place values using base-ten blocks.” If base-ten blocks are available, demonstrate or explain how large each place value is by displaying blocks (or explaining how many of each figure would represent a number like 1,000 or 10,000). Place an emphasis on the fact that adjacent place values are in a ratio of 10 or .

Possible discussion: “Nice job! You have labeled our whole number places and values correctly. Now let’s review how these place values compare to each other. I am holding a unit block. How much does it represent?” (1)          

“The single blocks represent the ones place in our number. How many would we use to represent the ones place in our number?” (4) “What would happen if I had more than nine ones?” (They wouldn’t fit in the ones place. You could link 10 ones blocks together and increase the tens place by one.) “Yes, I replace ten ones with a long like this. Each long has a value of ten.”

“What if I had more than ten longs (10s)?” (They wouldn’t fit in the tens place, so we would link ten longs together to make 100. This would increase the hundreds place by one.) “Ten groups of ten (10 longs) make a flat, which has a value of 100. Ten flats (100s) could be stacked into a large cube, which would have the value of how many ones? Remember each of the 10 flats has 100 unit blocks inside of it.” (1,000)

“Our number and many other numbers are larger than 1,000. What do you think the base-ten blocks representing these numbers might look like?”

Discuss what 10 blocks of 1,000 would look like and represent (a large long with value of 10,000), and what a flat, and cube made of these larger blocks would represent (100,000 and 1,000,000 respectively).

“Notice that the values always get regrouped whenever we reach 10. This is because we are working in the base-ten number system. Each digit has a value based on its position in the number. The place values help us understand the size or magnitude of the digits.

“In our lesson today we will build on these ideas to learn about the place values of decimal numbers, and investigate a variety of methods for representing whole numbers and decimals.”

If base-ten blocks and work mats are available, allow students some time to investigate and review the whole number place value relationships before moving on. Check the Resources folder for paper versions that can be used in place of actual blocks and work mats (M-5-5-1_Paper Base-Ten Models.docx and M-5-5-1_Whole Place Value Mat.docx). Another optional tool is to display an interactive base-ten block activity on the computer or interactive whiteboard. See possible links in the resource list.

“There are several ways to display our numeric values. Writing our numbers as base-ten numerals is what we have been doing so far today and what is usually used in our number displays and calculations. Two additional representations are word form and expanded form. You may already have some experience with these forms for whole numbers. We will practice them today with whole numbers and we will expand our work to include decimals later in the lesson.”

Continue using the work that is on the board.

“We can use the labels we already set up to help us represent 261,594 in word form and expanded form. In word form you use the place values (below digits) to get started. Think about how you read this number aloud. This is how you will write it in words. For example, we have three digits (2, 6, and 1) that are in a place containing the word thousand. These will be listed together as 261 thousand followed by the hundreds, tens, and ones. Try writing this number completely in words.”

After a minute or two, check student work. Display your answer on the board. Discuss any parts the students struggled with.

Answer: Two hundred sixty-one thousand, five hundred ninety-four.

Be sure students are not separating place values with the word and.

“Now we will write this same number in expanded form. The digit values we wrote above the 261,594 are the main pieces we need. We will write this form as the sum of all the digit values. So our number would look like this:

261,594 = 200,000 + 60,000 + 1,000 + 500 + 90 + 4.”

(You may want students to state each value aloud as you write it down.)

“We can also take this a step further by rewriting each number in our sum as the product of the digit and the place value it is in. For example: 200,000 would be written as 2 × 100,000 and 500 would be written 5 × 100. Are there any questions? Take a minute to finish writing the expanded form for 261,594.”

Call on one or more students to share their answer.

 261,594 = 200,000 + 60,000 + 1,000 + 500 + 90 + 4

   = (2 × 100,000) + (6 × 10,000) + (1 × 1,000) + (5 × 100) + (9 × 10) + (4 × 1)

Give several additional numbers to practice writing in words and expanded form until students can show they are able to use these forms on their own.

“Let’s continue by thinking about another number.” Write 2,233 on the board.

“What is the value of the 3?” Some students may say three ones, some may 30, and some may want to know which three you are asking about. Take time to discuss the importance of the place value in determining which answer is correct. “The digit that is in the tens place represents 30, while the digit in the ones place represents exactly 3. How do these values compare?” Let students share their thinking.

“Since 30 ÷ 10 = 3, or 30 ×  = 3, the 3 in the ones place is  the value of the 3 in the tens place. How do the digits in the hundreds and thousands places compare?” Call on a few students to share their ideas.

“Yes, the digit in the thousands place represents 2,000 while the digit in the hundreds place represents 200. Since 2,000 ÷ 10 = 200, or 2,000 ×  = 200, the 2 in the hundreds place is  the value of the 2 in the thousands place. What is similar about these two examples?” Allow students to state the similarities they see. If no student brings up the  relationship, do so yourself. As a digit moves to the right, each place has a value  the size, so the digits value is  that of the same digit one place to its left. Likewise, as a digit moves to the left, each place value is 10 times greater. Think about 45 versus 57, in 45 the 5 represents 5, but in 57 the 5 represents 5 × 10 or 50.

“Do you think a digit is always  the value of the same digit placed one space to its left?” Many will say yes, some may say no or be unsure.  

“Make up another example on your paper to show whether you are correct.” Walk around the room to assist struggling students and correct misconceptions. After students prove this relationship again, restate that in the base-ten number system the value of a digit in a specific place is always  the value of the same digit one place to its left.

“Do you think this  relationship also works with decimal numbers? We will find out as we continue or lesson.

“Now that we have reviewed and practiced understanding place values with whole numbers, we will apply the concepts to decimal numbers. Keep in mind that decimals are also part of the base-ten number system, but each decimal place value represents a different size part of a whole that is smaller than one whole unit. Let’s take a look.”

“What do you notice about our number now?” (It has a decimal point and 5 digits after it.)

“Raise your hand if you know the name of the first place after the decimal point.” Call on students until you get the answer “tenths.” Emphasize that this is spelled like “ten” but ending with “ths.” This lets anyone reading this number know that it is not the tens place on the left of the decimal point, but a smaller fractional part of a whole located on the right side of the decimal point.

“Take a minute to discuss with the person next to you how you think the seven in this place would compare to having a seven in the ones place.”

Walk around and listen to student discussions. Ask one or more pairs of students to share their ideas. Emphasize and expand on correct responses. Use guiding questions to assist students who respond with a partial or incorrect response. Be sure students understand that the tenths place is  the size of the ones place if the same digit is used. Also the name tenths implies that we have a ratio or fractional part out of ten. Therefore, 0.7 represents a part smaller than one, which is seven-tenths . Remind students that each place as they move farther to the right continues to be 1/10 the size of the previous place value (which is to its left). Ask students to add the decimal point and new digits to the number on their paper. They should label the place value names below and numeric values above each digit.

If you have base-ten blocks and work mats available, allow students some time to investigate and review the decimal number place value relationships before moving on. Paper versions can be used in place of actual blocks and work mats (M-5-5-1_Paper Base-Ten Models.docx and M-5-5-1_Decimal Place Value Mat.docx).

Quick Response Activity Part 1

Using individual whiteboards or paper with markers, have students respond to place value names, numeric values, and the  relationship for a few more numbers such as:

• “In the number 345.67 what is the value of the 7? In what place value is the digit 6?” (The value of the 7 is 7 hundredths. The digit 6 is in the tenths place.)
• “In the number 4.88, how do the values of the 8s compare? Which 8 represents a larger amount? How much larger is it?” (The first 8 is 10 times more than the second 8. The first 8 is larger than the second 8 by 10 times.)
• “In the number 17.79, how do the values of the 7s compare? Which represents a smaller amount? How much smaller? What is the name of the place value of the digit 9? What fractional part of a whole is this?” (The first 7 is 10 times more than the second 7. The second 7 is smaller by 10 times. The digit 9 is in the hundredths place; this is  of the whole.)

Ask the questions one at a time. Have each student write down his/her answer large enough for you to see from anywhere in the classroom. When it seems most students have an answer, ask them to quickly hold up their answer for you to see. Use these responses to judge what needs further explanation and which students may need extra help.

Once students can accurately answer the Quick Response Part 1 questions, move on to writing the decimal numbers in multiple ways. Extend the same ideas used in writing whole numbers.

Use these or similar examples to discuss writing decimals in words:

• 12.9                 (twelve and nine-tenths)
• 34.09               (thirty-four and nine-hundredths)
• 3.486               (three and four hundred eighty-six thousandths)
• 4.73509           (four and seventy-three thousand five hundred nine hundred-thousandths)

“In order to write decimal numbers in words, we will begin by writing the whole number, which is to the left of the decimal, exactly as we practiced before. To include the decimal portion of the number, we use the word and to represent the decimal point. The decimal point separates the part of a whole (decimal number) from the whole. The decimal number will be written in the same manner as a whole number but it will end with the decimal place value name of the last digit. For example, for 12.9, what is the whole value?” (12)

“This will be written as twelve followed by and.”

Ask students if they can figure out how to write the decimal portion of this number.

Answer: 12.9 = twelve and nine-tenths because 9 is in the tenths place

“We can also represent this number using expanded form by showing it as the sum of its parts. This number would begin with 10 + 2, but we would also add the decimal number. If we have more than one digit in the decimal portion of the number, we add each digit’s value separately, just like with whole numbers. For this example our expanded form would begin as 10 + 2 + 0.9. We could also break this down further to (1 × 10) + (2 × 1) + (9 × 0.1). Notice that nine-tenths is represented as the product of 9 and 0.1 because 0.1 means
one-tenth. This is the same as saying 9 × , which equals , or 0.9.”

Work through another example or two from above (more if needed) and have student pairs or groups complete another while you monitor. Check for understanding by doing the Quick Response Activity Part 2, again using whiteboards or paper that students can use to quickly show responses.

Quick Response Activity Part 2

• “Write the number 35.678 in both written and expanded form.” (thirty-five and six hundred seventy-eight thousandths; (3 × 10) + (5 × 1) + (6 × 0.1) + (7 × 0.01) +
  (8 × 0.001))
• “Write the number 272.09 in both written and expanded form.” (two hundred seventy-two and nine hundredths; (2 × 100) + (7 × 10) + (2 × 1) + (0 × 0.1) +
  (9 × 0.01))
• “Write the number 4.065 in both written and expanded form.” (four and sixty-five thousandths; (4 × 1) + (0 × 0.1) + (6 × 0.01) + (5 × 0.001))
• “In the number 716.784, what represents a part smaller than one? How would you write this part as a fraction? Could you write the entire number as a mixed fraction?” (The .784 part is smaller than 1. As a fraction, this part is . As a mixed fraction, the entire number is .)

Partner Place-Value Project

Place students in pairs for this activity. Hand out the Partner Place-Value Project sheet (M-5-5-1_Partner Place Value Project and KEY.docx). Allow students to work for approximately 20 minutes. Allow additional time for students to share. While students are working, choose a portion of each pair’s project for them to present to the class. Select a variety that will represent the different aspects of the lesson and help summarize it. Students should complete the entire activity on the sheet, but prepare a poster or other display to share the portion you selected for them with the class.

While students are working on the Partner Place Value Project, walk about the room observing work and conversations. Make suggestions or ask guiding questions to assist any students who are struggling or demonstrate a misunderstanding of one or more concepts. Pay particular attention to any students you identified as needing help during the Quick Response activities. Encourage students to make appropriate adjustments during work time, or after they present, if a misconception is discovered at that time.      

Exit Ticket

Each student or pair of students should complete the exit ticket at the end of the lesson (M-5-5-1_Lesson 1 Exit Ticket and KEY.docx). These may be collected from students as they leave class and used to determine concept mastery.

Extension:

• Routine: Discuss the importance of understanding and using the correct vocabulary words to communicate mathematical ideas clearly. During this lesson the following terms should be entered into students’ vocabulary journals: base-ten blocks, base-ten number, expanded form, hundredths, place value, tenths, thousandths, and word form. Keep a supply of Vocabulary Journal pages (M-5-5-1_Vocabulary Journal.docx) on hand so students can add pages as needed. Bring up practical instances using place value as they are seen throughout the school year in math examples and other content areas such as science class or shopping at home. Ask students to bring in examples of large and/or decimal numbers that they see in magazines or newspaper stories. Encourage them to describe the meaning of the number and its place values in that particular context.
• Small Group: Place Value Stations

Use these or similar stations for students who are having difficulty with the concept of whole number or decimal place values or number forms. Students can work alone or in pairs as they move through the stations.

Work Station 1: Place the extra set of place value name cards at this station, mixed in random order. Write four (or more) multidigit numbers on a piece of chart paper. Post instructions directing students to arrange the place value labels for each digit above the digits. When the labeling is complete, students should be directed to raise their hand to have you check their label placement. If labels are placed correctly, the student(s) should read the numbers aloud and name the value of each digit.

Use numbers such as:

• 92,488                   (Ninety-two thousand, four hundred eighty-eight;

9 ten thousands, 2 thousands, 4 hundreds, 8 tens, 8 ones)

• 7,243,689              (Seven million, two hundred forty-three thousand, six hundred

eighty-nine;

7 millions, 2 hundred thousands, 4 ten thousands, 3 thousands, 6 hundreds, 8 tens, 9 ones)

• 58.97                     (Fifty-eight and ninety-seven hundredths;

5 tens, 8 ones, 9 tenths, 7 hundredths)

• 364.592                 (Three hundred sixty-four and five hundred ninety-two

thousandths;

3 hundreds, 6 tens, 4 ones, 5 tenths, 9 hundredths, 2 thousandths)

Work Station 2: Using base-ten blocks (or paper models) create four (or more) different groups of blocks in zip-top baggies or small bins. These blocks can represent whole numbers and/or decimal numbers. Post directions asking students to sort the pieces using place value mats (whole number, decimal or both) to find and write the value of each number. Students should be directed to check their answers with you or an answer envelope placed at the station. At least two of the numbers should have some place values higher than 10 so students need to regroup them. Use less regrouping if students need to work more on basic place value concepts.

Use numbers such as:

• 3 thousands, 5 hundreds, 14 tens, and 11 ones                (4,911)
• 9 thousands, 16 hundreds, 8 tens, and 17 ones                (10,697)
• 12 ones, 7 tenths, 5 hundredths, 11 thousandths             (12.7511)
• 6 ones, 6 tenths, 14 hundredths, and 14 thousandths                  (6.754)

Provide place value mats, paper or chart paper, and pencils or markers.

Work Station 3: Post directions asking students to write the numbers provided in both written form and expanded form. Place a list of four numbers (or more) at the station.

Direct students to raise their hands when finished to have you check their work, or have an answer envelope prepared at the station.

Use numbers such as:

• 37,291                   (Thirty-seven thousand, two hundred ninety-one;

(3 × 10,000) + (7 × 1,000) + (2 × 100) + (9 × 10) + (1 × 1))

• 206,354                 (Two hundred six thousand, three hundred fifty-four;

(2 × 100,000) + (6 × 1,000) + (3 × 100) + (5 × 10) + (4 × 1))

• 91.82                     (Ninety one and eighty-two hundredths;

(9 × 10) + (1 × 1) + (8 × 0.1) + (2 × 0.01))

• 304.709                 (Three hundred four and seven hundred nine thousandths;

(3 × 100) + (4 × 1) + (7 × 0.1) + (9 × 0.001))

Provide paper or chart paper and markers or pencils.

• Technology Connection:  Whole Number Place Value Practice

Use this activity for students who are having difficulty with the concept of whole number place values. Students can work alone or in pairs on the computer using one the following web resources:

1. Manny’s Rumba (computer required)

http://www.learningbox.com/Base10/BaseTen.html

Students are asked to create specific whole number values (up to three digits) by sliding the correct number of base-ten block pieces into a framed area. They will combine ones, tens, and hundreds to create three-digit numbers.

2. Math Cats Place Value Party (computer required)

http://www.mathcats.com/explore/age/placevalueparty.html

This is an interactive activity allowing students to change digits to different place values (represented by candles and layers of a birthday cake) to discover or reinforce the relationship of 1/10 between adjacent place values.

• Expansion: The Metric Connection to Base-Ten

This activity (M-5-5-1_Expansion and KEY.docx) is appropriate for students who have shown proficiency in writing numbers in multiple forms, naming place values and meanings, and understanding the relationships between place values. Students with a strong understanding of the relationship of powers of ten when moving multiple place values can be challenged with this activity on metric conversions. This activity can be used as an independent activity if the directions are posted, or a guided activity with instruction from you.