# 2.1.1A Numbers, Representation, and Place Value

2
Subject:
Math
Strand:
Number & Operation
Standard 2.1.1

Compare and represent whole numbers up to 1000 with an emphasis on place value and equality.

Benchmark: 2.1.1.1 Number Representation

Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

Benchmark: 2.1.1.2 Place Value

Use place value to describe whole numbers between 10 and 1000 in terms of hundreds, tens and ones. Know that 100 is 10 tens, and 1000 is 10 hundreds.

For example: Writing 853 is a shorter way of writing

8 hundreds + 5 tens + 3 ones

Benchmark: 2.1.1.3 100 More or 100 Less

Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.

For example: If there are 17 students in the class and granola bars come 10 to a box, you need to buy 20 bars (2 boxes) in order to have enough bars for everyone.

Benchmark: 2.1.1.5 Compare & Order Numbers

Compare and order whole numbers up to 1000.

## Overview

Big Ideas and Essential Understandings

Second graders read, write and represent whole numbers up to 1000.  Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives. They build on their previous understanding of tens and ones to include an understanding that ten tens can be grouped to make a hundred and ten hundreds can be grouped to make a thousand. They describe numbers up to 1000 in terms of hundreds, tens and ones, and can find the number that is ten more/less and a hundred more/less than a given three digit number. Comparing and ordering of whole numbers is now expanded to include numbers to 1000. Second graders develop and use an understanding of rounding by rounding up or down to the nearest 10 or 100.

All Standard Benchmarks

2.1.1.1 Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

2.1.1.2 Use place value to describe whole numbers between 10 and 1000 in terms of groups of hundreds, tens and ones. Know that 100 is ten groups of 10, and 1000 is ten groups of 100.

2.1.1.3 Find 10 more or 10 less than any given three-digit number. Find100 more or 100 less than any given three-digit number.

2.1.1.4 Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.

2.1.1.5 Compare and order whole numbers up to 1000.

Benchmark Cluster

Benchmark Group A

2.1.1.1
Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

2.1.1.2
Use place value to describe whole numbers between 10 and 1000 in terms of groups of hundreds, tens and ones. Know that 100 is ten groups of 10, and 1000 is ten groups of 100.

2.1.1.3
Find 10 more or 10 less than any given three-digit number. Find 100 more or 100 less than any given three-digit number.

2.1.1.5
Compare and order whole numbers up to 1000.

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Read, write, and represent numbers up to 1,000 in a variety of ways including:

using numerals, addition, subtraction, multiplication;
using manipulatives;
drawing pictures;
using words--oral and written.

• Use place value to describe whole numbers up to 1,000 based on hundreds, tens, and ones, for example:

742 is 7 hundreds and 4 tens and 2 ones, or
74 tens and 2 ones, or
7 hundreds and 42 ones, or
6 hundreds and 14 tens and 2 ones, or
3 hundreds and 43 tens and 12 ones, or
other place value-based descriptions.

• Find a number that is 10 more/less and 100 more/less than a given three-digit number
• Compare and order numbers to 1000

Work from previous grades that supports this new learning includes:

• read, write and represent whole numbers up to 120;
• count forward and backward from any given number up to 120;
• use place value to describe whole numbers between 10 and 100 in terms of groups of tens and ones -- 56 is 5 tens and 6 ones or 4 tens and 16 ones or 56 ones or other place value based descriptions;
• find a number that is 10 more/less than a given two-digit number;
• compare and order whole numbers up to 120;
• describe the relative size of numbers; e.g., equal to, not equal to, more than, less than, fewer than, is about, is nearly.
Correlations

NCTM Standards

Understand numbers, ways of representing numbers, relationships
among numbers, and number systems

Pre-K - 2 Expectations:

• count with understanding and recognize "how many" in sets of objects;
• use multiple models to develop initial understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;
• develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;
• connect number words and numerals to the quantities they represent, using various physical models and representations;
• understand and represent commonly used fractions, such as ¼, ⅓, and ½.

Common Core State Standards

Understand place value.

• 2.OA.1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
• 2.OA.1a. 100 can be thought of as a bundle of ten tens - called a "hundred."
• 2.OA.1b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
• 2.OA.2. Count within 1000; skip-count by 5s, 10s, and 100s.
• 2.OA.3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
• 2.OA.4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors

Students may think:

• numbers can be represented in only one way. For example, 327 can only be represented by 3 hundreds, 2 tens, and 7 ones.
• three hundred twenty-five should be written as 30025 because the number 300 has two zeros.
• they only need to look at the first or the last digit in a multi-digit number when comparing and ordering numbers.

## Vignette

In the Classroom

Second graders build their understanding of the thousands, hundreds, tens and ones structure of our number system.  The lesson engages children in creating the greatest two-digit number from a dice roll, building the number with base 10 blocks, and continuously adding additional numbers. Children practice adding ones, tens, and hundreds as they exchange ten ones for a long, ten longs for a flat, and ten flats for a cube. The actions on the place value mat are reflected in the equations students write as they play Race to a Thousand.

Race to a Thousand

Mrs. Simones excitedly announces, "Today we're going to use a place value mat and base ten blocks to play a game.  We will explore building numbers all the way up to a thousand!"

"You'll need a partner, two dice, two place value mats, and base 10 blocks. Each of you will use your own mat to show your base ten blocks.  You will also need paper and a pencil in order to record what is happening on your place value mat."

"Jackson and I will start playing 'Race to a Thousand' so you can see how the game is played," she says. All of the 2nd graders gather around a table so they can see the game.

Jackson and Mrs. Simones begin with empty place-value mats in front of them.

 thousands hundreds tens ones

"I rolled 3 and 6," Jackson says. "What now?"

Mrs. Simones poses the question, "What possible numbers could you make using the digits 3 and 6?"

Danielle volunteers, "You could make 36 or 63."

"That's right. Knowing Jackson wants to get to 1000, which number should he choose?" questions Mrs. Simones.

"I think he should pick 63," offers Mariah.  "63 is bigger than 36 and he wants to make the largest number possible."

"Great thinking, Mariah!  Can anyone give us a strategy to build the 63?" asks Mrs. Simones.

Chris shared, "63 has six tens and three ones.  We need to put six tens on the place value mat and three ones."

Sam adds, "The tens should be in the tens column and the ones in the ones column."

Jackson places the tens and ones on his place value mat.

 thousands hundreds tens ones

Mrs. Simones rolls 4 and 5 and poses the question, "What number should I make to help me get to a thousand?"

Luke says, "You should make 54! 5 tens are more than 4 tens."

"Nice thinking, Luke!  Could you place the place value blocks I need on my mat?"

Luke correctly places 5 longs in the tens place and 4 cubes in the ones place on Mrs. Simones' place value mat.

 thousands hundreds tens ones

Mrs. Simones asks, "Who is closer to making a thousand after only one turn?"

Everyone call out, "Jackson!"  She then asks, "How much closer to a thousand is he?"

Jaelyn calculates mentally and her hand goes up.  "Jackson has 9 more because 63 is 9 more than 54."

"Excellent comparing, Jaelyn!  Now it's Jackson's turn to roll."

Jackson rolls 2 and 4 and decides to use the number 42. He places 4 tens and 2 ones on his mat.

 thousands hundreds tens ones

Mrs. Simones: "What number sentence could I write to show what Jackson just did?"

"You could write 63 plus 42," Kadence says.

Mrs. Simones records on the board:  63 + 42 =

"Explain why this number sentence tells about Jackson's turn," she says.

"The 63 tells what Jackson had on his mat," Kadence explains. "Then he rolled 42 on his second turn and added that to his mat."

"What can you tell me about the blocks on Jackson's mat?" asks Mrs. Simones.

"He has 105," says Mia.

Tucker shares, "He can exchange 10 tens for a hundred because 10 tens equals one hundred."

Jackson then grabs 10 tens from his mat and sets them aside.  He then places a hundreds place value block into the hundreds place on his place value mat.

 thousands hundreds tens ones

"What number is shown on Jackson's mat now?" asks Mrs. Simones.

"He has 105 because he has one hundred and five ones," explains Cade.

Mia adds, "That's what I said before.  He had 105 when he had 10 tens and 5 ones."

Mrs. Simones records the sum on the board and Jackson writes the equation on his paper:

63 + 42 = 105

Mrs. Simones then adds:

10 tens + 5 ones = 105
1 hundred + 5 ones = 105

Mrs. Simones reviews the object of the game "Race to a Thousand" and the second graders gathers their materials and begin to play.

During the game.....

As Mrs. Simones watches Elle and Carter play, she notices Carter easily makes exchanges, while Elle sometimes misses an opportunity.  Mrs. Simones investigates her thinking by posing "Elle, I notice that you seem to know what to do when you decide how to place your digits to make the biggest number.  Can you tell me when you exchange?"

Elle hesitantly shares, "I think I can exchange when I have lots of ones or too many tens."

"Do you know how many ones you need to make an exchange?" she questions further.

"I think I need ten, "Elle hopes.

"That's right, Elle!  It may help to keep track of the ones if you line them up in rows of five. You can see when you have ten to exchange," suggests Mrs. Simones. "Let's try it."

"Oh, now I can see I have 16 ones and I can exchange!  That does help," Elle beams.

Some children choose to use the smaller of the two possible numbers after rolling the dice.  Mrs. Simones watches as Sonja rolls a 2 and a 5 and poses the question, "What numbers can you make using a 2 and a 5, Sonja?"

Sonja answers, "25 and 52."

Mrs. Simones asks, "When you are playing Race to a Thousand, do you want to use the largest possible number or the smallest possible number?"

Sonja thoughtfully states, "Oh...the largest number so I can get to a thousand faster!  I'll make 52!"

Mrs. Simones continues to question students about the numbers shown on their mats, the difference between the numbers on the players' mats, and how many more were needed for one player to win the game.

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Students need to use a variety of place value materials to represent numbers. For example, base ten blocks, bundled sticks, bean sticks, etc.
• Students should clearly grasp the value of digits according to their place.  The digit "1" in the ones place has a value of one while a "1" in the tens place has a value of ten, a "1" in the hundreds place has a value of one hundred, and a "1" in the thousands place has a value of one thousand.
• Building numbers using multiple representations is essential as students develop an understanding of place value and the base-ten number system. For example:  692 could be written as 6 hundreds 9 tens 2 ones or 200  + 90 + 2  or shown using base-ten blocks.
• Hundreds, tens and ones cards can be used to build numbers. For example, building the number 451 requires starting with the 400 card, placing the 50 card atop it, and then placing the 1 card atop both cards. Once students see the number 451 they can expand the number to see 400 and 50 and 1. Note: these cards have to be aligned in such a way as to insure only a three digit number is visible when three cards are stacked atop each other.

• Organizing numbers in different ways helps highlight patterns within the base ten number system. For example, moving left to right across each row highlights the ten more  relationship among three-digit numbers.

271                  281                  291                  301                  311

272                  282                  292                  302                  312

273                  283                  293                  303                  313

274                  284                  294                  304                  314

275                  285                  295                  305                  315

276                  286                  296                  306                  316

277                  287                  297                  307                  317

278                  288                  298                  308                  318

279                  289                  299                  309                  319

280                  290                  300                  310                  320

• Students should use number lines to demonstrate ordering and comparing of numbers. The decision as to where to place a number on a number line reflects student thinking about the location of, as well as the relative magnitude of numbers. Placing the same number on number lines with different intervals requires students to be flexible when thinking about the relative size of numbers.  For example, which point represents the location of 475 on the number lines?

• Second graders will be using the words "greater than" and "less than" but are not necessarily using the symbols <  and  >.
• Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources

NCTM Illuminations

Instructional Activities

Numbers to a Thousand  This game is effective for demonstrating the place value of three digit numbers.

Largest Number  Students use a set of three digit number cards (teacher determined). Working in pairs, each student draws a number card and then builds the number using base ten blocks. Players verify the representation by counting each other's base ten blocks and determining who had the largest number. Play continues as students draw new cards and build new representations using base ten blocks. This game can be changed to Smallest Number by asking students to determine which of the represented numbers is the smallest.

Who Has? is an activity which requires students to use their place value knowledge by connecting pictorial representations to numeric representations. Scroll to the bottom of this site and click on the PDF Base Ten Deck (tens and ones) and the PDF Place Value Deck (hundreds, tens and ones).

Ordering Numbers GameGreat practice for placing numbers in ascending or descending order.

Burns, Marilyn.(1994). Math by all means: Place value grade 2. Sausalito, CA: Math Solutions Publications.

Richardson, Kathy. (1999). Developing number concepts: Place value, multiplication, and division. Parsippany, NJ: Dale Seymour Publications.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA:  Math Solutions.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Sutton, Kim. (2003). Place value with pizazz games and activities for
meaning in place value. Arcata, CA: Creative Mathematics.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J. A., & Lovin, L. H. (2006) Teaching student-centered mathematics grades K-3. Boston: Pearson Education.

New Vocabulary

compose numbers: creating through different combinations

consecutive:  means one right after the other. Example:  A list of consecutive whole numbers:  0, 1, 2, 3, 4, 5, ...

decompose numbers: breaking apart in different ways

digit: a basic symbol used in a numeration system. The 10 digits used in our base-ten numeration system are: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

landmark (benchmark, friendly) numbers: provide a foundation for extending number sense concepts. For example, at the second grade level generally include sums of tens and getting to the next ten or counting by fives.

model: represent a mathematical situation with manipulatives (objects), pictures, numbers or symbols.

order:  put numbers in order by placing them in ascending order (from least to greatest) or in descending order  (from greatest to least).

tally marks: use to record the frequency of an item.

value: how much something is worth.

"Vocabulary literally is the

key tool for thinking."

Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful    Multiple and varied opportunities to use the words in context.  These

Use:              opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

 word definition illustration

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Professional Learning Communities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks:

What are the key ideas related to place value understanding at the second grade level? How do student misconceptions interfere with mastery of these ideas?

How would you know a student understands the place value system when using numbers from 100 - 1000?

What representations should a student be able to make for the number

472 if they understand place value? List all possible representations of 472.

What is meant by equivalent representations? How can teachers help students understand equivalent representations?

When checking for student understanding of basic facts, what should teachers

• listen for in student conversations?
• look for in student work?
• ask during classroom discussions?

Examine student work related to a place value task. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the second grade level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8, (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics

Fosnot, C. T., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press

References

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Burns, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd Edition). Sausalito, CA: Math Solutions Publications.

Burns, M. (Ed.). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class: Grades k-2. Sausalito, CA: Math Solutions Publications.

Caldera, C. (2005). Houghton Mifflin math and English language learners.  Boston, MA: Houghton Mifflin Company.

Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Chapin, S., and Johnson, A. (2006). Math mattera: Understanding the math you teach, grades k-8, (2nd ed.). Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2.  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Hyde, A. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, D.C.: National Academies Press.

Leinwand, S., (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.

Murray, M.. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.

Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S.,...& Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA:  Allyn & Bacon.

Van de Walle, John A., & Lovin, LouAnn H. (2006) Teaching student-centered mathematics grades K-3. Boston: Pearson Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

## Assessment

• Anton has 407 rocks in his collection.

ow many hundreds are in 407?
How many tens are in 407?
How many ones are in 407?

Solution:  4, either 0 or 40, either 7 or 407
Benchmark  2.1.1.2

• Write the number that represents: 800 + 50 + 2

Solution:  852
Benchmark  2.1.1.2

• What is the value of 25 in 257?

Solution:  25 tens
Benchmark:  2.1.1.2

• Write the numeral: four tens, six hundreds, and eight ones.

Solution:  648
Benchmark:  2.1.1.1 and  2.1.1.2

• Put these numbers in order from smallest to largest:

186, 952, 592, 329, 392, 681

Solution:  186, 329, 392, 592, 681, 952
Benchmark:   2.1.1.5

• Start at 336. Count up by 10s.  I'll tell you when to stop.  (Stop when the student has counted to 436.)

Solution:  336, 346, 356, 366, 376, 386, 396, 406, 416, 426, 436
Benchmark: 2.1.1.3 and 2.1.1.5

• Show the number that is 100 less than 852 using base ten blocks.

Solution:  7 hundreds, 5 tens, and 2 ones or 75 tens and 2 ones or other combinations that equal 752.

Benchmark:  2.1.1.1./2.1.1.3/2.1.1.5

• Which is the same as 638?

a. 6 tens and 38 ones
b. 6 hundreds and 3 tens and 8 ones
c. 6 hundreds and 8 tens and 3 ones
d. 8 hundreds and 3 tens and 6 ones

Solution:  B   6 hundreds and 3 tens and 8 ones
Benchmark:  2.1.1.2

Performance Assessment

• Using base ten blocks and three-digit number cards, Can you show me what this (point to card with 512 on it) means using these materials? (The child should show blocks representing 512, such as: 5 hundreds + 1 ten + 2 ones or 51 tens + 2 ones or 512 ones or 5 hundreds + 12 ones)

If successful, continue by asking:
* Can you think of another way to show 512 using these materials?
* Another way?
* How many different ways can you display 512?

If not successful with the initial question, continue by asking:

* Can you show me what this (point to card with 37 on it) means using these materials? (The child should show blocks representing 37, such as:  3 tens + 7 ones OR 2 tens + 17 ones OR 1 ten + 27 ones OR 37 ones.)

Benchmark   2.1.1.1./2.1.1.3/2.1.1.5

## Differentiation

Struggling Learners

Struggling learners need to see multiple representations of numbers. These representations include numerals, words, and place value models.

Models of numbers using base ten materials can lead to the discussion of ten more/less and 100 more/less as a ten or a hundred is added to or taken from the original representation. For example, ten more than two hundred forty-one can be represented as follows.

Base ten materials need to be used as students begin to compare and order three-digit numbers.

Use only one type of base ten manipulative until students grasp the concept of hundreds, tens and ones. Once students can successfully represent numbers using one type of material then introduce another type of manipulative to represent place value.  It will be necessary to help students see the similarities between a number represented using base ten blocks and bean sticks or bundled sticks.

(Adapted from The Access Center: Improving Access for All K-8 Students)

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J. A., & Lovin, L. H. (2006) Teaching student-centered mathematics grades K-3. Boston: Pearson Education.

English Language Learners
• English Language Learners may be able to make numbers and recognize the hundreds, tens, and ones digits, but have trouble articulating what they have done. Base ten materials will be the key in building language skills as well as a conceptual understanding of place value.
• Once students have successfully represented a number using a particular manipulative, introduce another material. It will be important to help students translate between representations using different materials.  Vocabulary plays a pivotal role in connecting representations using different materials.
• Word banks need to be part of the student learning environment in every mathematics unit of study.
• Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.

• Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

 The number  _____ has _________hundreds, __________ tens and _________ ones.
 ______________________    is 100 more than  ___________________________.
 ______________________ is 10 less than _____________________________.
• When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning

●     Base Ten Riddles

Students can solve and then write their own base ten riddles. For example:

I have 140 ones and 3 tens. Who am I?

I am between 350 and 360. I have 6 ones.  Who am I?

The Mayan numeration system is another possibility. The following link offers some information: http://www.basic-mathematics.com/mayan-numeration-system.html

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom-A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.

Classroom Observation

 Students are: Teachers are: reading, writing, & ordering numbers up to 1,000. reinforcing place value vocabulary in instruction and assessment. representing three-digit numbers in many ways using base 10 blocks and explaining the model they have constructed. using multiple representations of three-digit numbers and connecting these representations to the numbers. comparing and ordering numbers in terms of place value and justifying the ordering. guiding place value discussions to help students make connections among the many ways to represent numbers. finding a number that is ten more ore ten less than a given three-digit number. representing the concept of ten more or ten less with base ten blocks. explaining their thinking. questioning/assessing students' understanding of the place value of numbers. using calculators to look for patterns when skip counting on/back by 10 and 100. facilitating discussions about patterns formed in counting on/back by ten and one hundred. explaining and justifying their thinking. asking: Why? How do you know? Will that always be the case?

What should I look for in the mathematics classroom? (Adapted from SciMathMN,1997)

What are students doing?

• Working in groups to make conjectures and solve problems.
• Solving real-world problems, not just practicing a collection of isolated skills.
• Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
• Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
• Recognizing and connecting mathematical ideas.
• Justifying their thinking and explaining different ways to solve a problem.

What are teachers doing?

• Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
• Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
• Connecting new mathematical concepts to previously learned ideas.
• Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
• Selecting appropriate activities and materials to support the learning of every student.
• Working with other teachers to make connections between disciplines to show how math is related to other subjects.
• Using assessments to uncover student thinking in order to guide instruction and assess understanding.

For Mathematics Coaches

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach: Grades k-8, 2nd edition.  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Sammons, L., (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

Burns, M. (Ed). (1998).  Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Parents

Parent Resources

Mathematics handbooks to be used as home references:

Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Helping your child learn mathematicsProvides activities for children in preschool through grade 5

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.

Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?

Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995

Oughton, J. (1992). How the stars fell into the sky.  New York, NY: Houghton Mifflin Company.

Schwartz, D. (1999). On beyond a million:  An amazing math journey.  New York, NY:  Random House Inc.

Wells, R. (2000). Can you count to a googol? Morton Grove, IL:  Albert Whitman & Company.