1. Number and Operations in Base Ten
In this module we cover the place value understanding portions of the progression document for the Number and Operations in Base Ten domain. Our objective is that teachers will understand the way students develop place value understanding and how that understanding supports arithmetic in all four operations.
Place Value

2. Place Value Task
Find a number greater than 0 and less than 1,000 that:
is closer to 500 than 0,
and
Is closer to 200 than 500.
There are many correct answers to this problem. Describe all of the numbers that are correct.
Have the teachers do the task and share strategies and answers. Then ask, “How do we get from beginning counting to, three years later, being able to do this kind of work?” Discuss the commentary.
Illustrativemathematics.org (finish)

3. What is the Big Deal about Place Value?
The base-ten place value system is the way we communicate and represent anything that we do with whole numbers and later with decimals. (Van De Walle, 2013)
Read and discuss. How does the base ten place value system do what Van de Walle claims?

4. Place Value Understanding
A Progression
Cluster 2.NBT.A Cluster 2.NBT.B Cluster 3.NBT.A (varied algorithms) Cluster 4.NBT.A (standard algorithm)
Strategies  
Strategies
Place Value Understanding
Algorithms
Note that in the core there is a progression of ideas and operations that leads from place value understanding to using strategies for operations to using algorithms for those operations. Thus, place value understanding becomes extremely important. Have participants look at the clusters under each arrow and discuss in small groups how that cluster helps students to move to the next point in the progression. Note that these are just examples. There are many other standards in the core that illustrate the same progression.

5. What Does Place Value Mean?
Video at LearnZillion – Replace – David is doing this one.
using-pictures
Watch the video and discuss.

6. Counting and Cardinality
Talk to your neighbor.
What do you know about the following terms: cardinality, one to one correspondence, subitizing?
Read the Counting and Cardinality progression in your group.
Share out – Can anyone tell us what one to one correspondence is and demonstrate it? How about cardinality? Subitizing?
Ask the teachers to turn to their neighbor and talk about cardinality, one to one correspondence and subitizing. Share what they know with each other. Then hand out the Counting and Cardinality progression sheets from the CC and OA Progression document (included in the folder). Have them read and discuss the various points. Summarize the main points on poster paper.

7. Counting and Cardinality
Two Tasks:
K.CC.B.4 – Goody Bags
K.CC.B.4 – Counting Mats
How do these tasks help students prepare for place value study?
Tasks from
Hand out the Goody Bag Task and the Counting Mat Task (in the folder). Ask the teachers to read them over and discuss them briefly in their groups. Ask, “How do these tasks help prepare students for place value study?”

8. K.NBT.A.1 Work with numbers 11-19 to gain foundations for place value.
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = ); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
Have teachers read the kindergarten section in the NBT progression and discuss it in small groups. What does this standard require of students?

9. Turn and Talk
What would this standard look like in a kindergarten classroom?
Facilitate conversation
Collect/chart answers

10. Find a good Example Replace – Stacie is working on this one.
Watch the video and move on to the next slide.

11. Video
Discuss the teaching moves the teacher made in this video to lead to the conceptual understanding the standard requires.
Have the teachers discuss in small groups the moves the teacher made to help the students understand the concept of 10 ones and some more ones.

12. How Can We Use These Tools?
Small Group Discussions – How can tools such as these representations be used to reinforce conceptual understanding of place value?

13. The “Teen Numbers”
What special difficulties do the “teen” numbers present for both counting and place value? 17, for example, doesn’t sound like 1 ten (ten ones) and 7 ones.
Discuss at your table how to help children deal with the teen numbers in counting and in place value. Then click to the layered place value card idea.

14. Big Idea 1
Sets of ten (and tens of tens) can be perceived as single entities or units. For example, three sets of ten and two singles is a base-ten method of describing 32 single objects. This is the major principle of base- ten numeration.
In the NBT progression, page 2, this idea is listed under Base-ten Units.
Introduce the big ideas of place value on the next 5 slides. Don’t turn to the progression document until you have presented all five big ideas. Make certain that teachers have ample opportunities to comment and ask questions as you go through the big ideas. Pause often and ask “Why is this so?” and “Does this really matter?” and the like.

15. 1.NBT.A.2 Understand place value.
Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
a. 10 can be thought of as a bundle of ten ones — called a “ten.”
b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
Have teachers read the first grade section in the NBT progression and discuss it in small groups. What does this standard require of students?

16. Discussion
Is there a difference between the kindergarten standard and the first grade standard?
If so, what is the difference?
How does the teacher’s language help students with understanding the content of this standard?
In Kindergarten we refer to ten ones and some other ones. In first grade we talk about 1 ten and some ones. The idea of ten as a unit is introduced.

17. Big Idea 2
The positions of digits in numbers determine what they represent and which size group they count. This is the major organizing principle of place value numeration and is central for developing number sense.
In the NBT progression, page 2, this idea is listed under Position.

18. Layered Flash Cards Visual Supports 8 6 80 6 layered separated
Drawings and place value cards can be used to connect number words and numerals to their base-ten meanings. Eighty-six is shown as “eight tens and six ones” and written as 86 and not as 806, a common Grade 1 error of writing what is heard.
layered
separated
86 = 80 +6

19. Big Idea 3
There are patterns in the way that numbers are formed. For example, each decade has a symbolic pattern reflective of the sequence (e.g., 20, 21, ).
This idea is discussed in the NBT progression in first grade, page 6.

20. Very Hungry Caterpillar Task
How does this task develop place value understanding?
Use the hungry caterpillar task in the folder. Be sure to have a copy of the book available!

21. 2.NBT.A.1 Understand place value.
Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
a. 100 can be thought of as a bundle of ten tens — called a “hundred.”
b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
Have teachers read the second grade section in the NBT progression and discuss it in small groups. What does this standard require of students?

23. An Example Stacie will do this one.
Use the Protocol for Discussion of a Learn Zillion video to guide the discussion of this video.

24. Here is an example of a place value worksheet that you get when you search for “place value worksheets” online. It is also a non-example of work that would elicit conceptual understanding. As you can see, it would not be possible to assess whether your students had a conceptual understanding of place value by them completing this worksheet. It would be fairly obvious to a student who does not understand place value that the first number goes with hundreds, the 2nd number with tens and so on. Even on problem letter h, where it could have asked for deeper understanding, the worksheet places a 0 for tens to eliminate any need for thinking.

25. Why is this a better way to show conceptual understanding. Discuss
Why is this a better way to show conceptual understanding? Discuss. Be sure the following ideas come out.Here is a snapshot of a worksheet practicing place value understanding. You can see how a teacher would be able to assess a student’s conceptual understanding of place value more clearly with the results of this worksheet. In problems 6-8, the base ten units in 106 are bundled in different ways. This is helpful when learning how to subtract in a problem like In #9, we see that if the order is always given “correctly,” then all we do is teach students rote strategies without thinking about the size of the units or how to encode them in positional notation.

26. Big Idea 4
The groupings of ones, tens, and hundreds can be taken apart in different but equivalent ways. For example, beyond the typical way to decompose 256 of 2 hundreds, 5 tens, and 6 ones, it can be represented as 1 hundred, 14 tens, and 16 ones but also as 250 and 6. Decomposing and composing multi-digit numbers in flexible ways is a necessary foundation for computational estimation and exact computation.
This idea is discussed in the NBT progression in Grades 1 and 2 in the place value clusters.

27. Base Ten Riddles I have 23 ones and 4 tens. Who am I?
I have 4 hundreds, 12 tens and 6 ones. Who am I?
I have 30 ones and 3 hundreds. Who am I?
I am 45. I have 25 ones. How many tens do I have?
I am 341. I have 22 tens. How many hundreds do I have?
I have 13 tens, 2 hundreds, and 21 ones. Who am I?
If you put 3 more tens with me, I would be 115. Who am I?
I have 17 ones. I am between 40 and 50. Who am I? How many tens do I have.
Van de Walle, J. A., Karp, K.S., & Bay-Williams, J. M. (2013). Elementary and Middle School Mathematics: Teaching Developmentally, Pearson Education. P. 200
Using base ten riddles like this one from Van de Walle’s book “Elementary and Middle School Mathematics: Teaching Developmentally” helps students think conceptually and flexibly in terms of place value. Exercises like this one can be used as warm-ups, time fillers, or instructionally.

28. Teaching Channel Video
math-lesson
This video shows a teacher engaging students in reinforcing place value concepts in a game format. As teachers watch the video ask them to consider these questions:
How does the repetition in a game allow for practice without redundancy?
Notice the different ways in which the game requires students to compare quantities using place value.
How could the teacher have reinforced the idea that a digit has a different value depending on its position?
Hand out the Trash Can Game sheet and the 101 and Out sheet.

29. 3.NBT.A.1
Use place value understanding and properties of operations to perform multi- digit arithmetic.
Use place value understanding to round whole numbers to the nearest 10 or 100.
Have teachers read the third grade section in the NBT progression and discuss it in small groups. What does this standard require of students?

30. 3.NBT.A.1
Building upon the understanding that 10 represents a bundle of ten ones, and represents a bundle of ten tens, students should be able to locate a given number on a number line and round to the nearest ten or hundred.

31. Background Knowledge for Rounding to Ten and Hundred
David will get a background video

32. Rounding to Tens
Which of the following numbers will round to 40? How do you know? Use two different strategies.
37, 32, 43, 46, 50
Have the teachers work on this task using at least two different strategies. Ask them to share their strategies and to justify their answers.

33. Adapted from http://www.illustrativemathematics.org/illustrations/745
Rounding to Tens
What is the smallest whole number that will round to 40? How do you know?
What is the largest whole number that will round to 40? How do you know?
How many different whole numbers will round to 40? How do you know?
Adapted from
Again, have the teachers work these tasks.

34. Rounding to Hundreds
Which of the following numbers round to 600? How do you know?
550, 575, 620, 645, 680

35. Adapted from http://www.illustrativemathematics.org/illustrations/745
Rounding to Hundreds
What is the smallest whole number that will round to 600?
What is the largest whole number that will round to 600?
How many different whole numbers will round to 600?
Adapted from

36. 4.NBT.1
Generalize place value understanding for multi-digit whole numbers.
Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.
Have teachers read the fourth grade section in the NBT progression and discuss it in small groups. What does this standard require of students?

37. Rounding to 100
Watch this Learn Zillion video

38. This visual shows that when 30 tens are multiplied by 10 the three in the tens place is shifted one place to the left to represent 3 hundreds.

39. 4NBT Place Value Task
Sometimes when we subtract one number from another number we decompose a number, and sometimes we don't. For example, if we subtract 38 from 375, we can decompose a ten into ten ones:
Find a 3-digit number to subtract from 375 so that:
You don't have to decompose any numbers.
You would naturally decompose a ten.
You would naturally decompose a hundred.
You would naturally decompose both a hundred and a ten.
In each case, explain how you chose your numbers and complete the problem.
From
Have the teachers work the task in groups and then share their strategies and their explanations. Why is composing/decomposing/regrouping a place value strategy?

40. 5.NBT.A.1, 5.NBT.A.2 Understand the place value system.
5.NBT.A.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
5.NBT.A.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of Use whole-number exponents to denote powers of 10.
Have teachers read the fifth grade section in the NBT progression and discuss it in small groups. What does this standard require of students?

41. Understand the Place Value System
Stacie is working on a fifth grade video

42. When digits are divided by ten their value shifts one place to the right. So tens become ones, ones become tenths, tenths become hundredths and so forth. The bottom graphic illustrates that the place names are related when reflected across one in the number line or in a place value graph.

43. 5th Grade NBT Task
Kipton has a digital scale. He puts a marshmallow on the scale and it reads 7.2 grams. How much would you expect 10 marshmallows to weigh? Why?
Kipton takes the marshmallows off the scale. He then puts on 10 jellybeans and then scale reads 12.0 grams. How much would you expect 1 jellybean to weigh? Why?
Kipton then takes off the jellybeans and puts on 10 brand-new pink erasers. The scale reads grams. How much would you expect 1,000 pink erasers to weigh? Why?
Hand out the 5th Grade NBT task and discussion after the teachers do the task and share strategies.

44. Big Idea 5
“Really big” numbers are best understood in terms of familiar real-world referents. It is difficult to conceptualize quantities as large as 1000 or more. However, the number of people who will fill the local sports arena is, for example, a meaningful referent for those who have experienced that crowd.
(Big Ideas are from Van de Walle, 2013, pg. 192)

45. Very Large Numbers How many people do you think are in this stadium?
Ask the participants to think about how they could use pictures like this to help students visualize large numbers.
Ask the participants to estimate how many people are in this crowd at the Rose Bowl in Pasadena, California. Have them discuss an answer and write it down. Share strategies, tell them Its capacity is 95,542, and, as it is full, the crowd is probably right around that number.

46. Just Don’t Do This http://www.youtube.com/watch?v=yOz8Fq0nNqg
There just isn’t any understanding. Just don’t do it!

47. Notes-Future modules Place Value progression Comparing progression
Operations using place value strategies (algorithm article)
Addition
subtraction
Multiplication
division