3.1.1B Place Value to Compare and Order
Find 10,000 more or 10,000 less than a given fivedigit number. Find 1000 more or 1000 less than a given four or fivedigit. Find 100 more or 100 less than a given four or fivedigit number.
Compare and order whole numbers up to 100,000.
Overview
Standard 3.1.1 Essential Understanding
Third graders expand their work with place value to include numbers from 1,000100,000. Numbers are represented in terms of groups of ten thousands, thousands, hundreds, tens, and ones. For example, 67,465 is six 10,000s, seven 1000s, four 100s, six 10s and five 1s, as well as (67 x 1000) + (4 x 100) + (65 x 1), as well as 67,000 + 465, as well as sixtyseven thousand fourhundred sixtyfive. Students are now able to see the structure of ones, tens and hundreds repeated in thousands. This structure is repeated with work into the millions in later grades. Recognizing that there is an orderliness in numbers, and that there is regularity in our number system, leads to a deeper understanding of the base ten system. Students use place value as the basis for comparing numbers up to 100,000 as well as rounding numbers to the nearest 10,000, 1,000, 100 and 10.
All Standard Benchmarks
3.1.1.1:
Read, write and represent whole numbers up to 100,000. Representations may include numerals, expressions with operations, words, pictures, number lines, and manipulatives such as bundles of sticks and base 10 blocks.
3.1.1.2:
Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thousands, hundreds, tens and ones.
3.1.1.3:
Find 10,000 more or 10,000 less than a given fivedigit number. Find 1000 more or 1000 less than a given four or fivedigit number. Find 100 more or 100 less than a given four or fivedigit number.
3.1.1.4:
Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.
3.1.1.5: Compare and order whole numbers up to 100,000.
Third graders expand their work with place value to include numbers from 1,000100,000. Numbers are represented in terms of groups of ten thousands, thousands, hundreds, tens, and ones. For example, 67,465 is six 10,000s, seven 1000s, four 100s, six 10s and five 1s, as well as (67 x 1000) + (4 x 100) + (65 x 1), as well as 67,000 + 465, as well as sixtyseven thousand fourhundred sixtyfive. Students are now able to see the structure of ones, tens and hundreds repeated in thousands. This structure is repeated with work into the millions in later grades. Recognizing that there is an orderliness in numbers, and that there is regularity in our number system, leads to a deeper understanding of the base ten system. Students use place value as the basis for comparing numbers up to 100,000 as well as rounding numbers to the nearest 10,000, 1,000, 100 and 10.
All Standard Benchmarks
3.1.1.1:
Read, write and represent whole numbers up to 100,000. Representations may include numerals, expressions with operations, words, pictures, number lines, and manipulatives such as bundles of sticks and base 10 blocks.
3.1.1.2:
Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thousands, hundreds, tens and ones.
3.1.1.3:
Find 10,000 more or 10,000 less than a given fivedigit number. Find 1000 more or 1000 less than a given four or fivedigit number. Find 100 more or 100 less than a given four or fivedigit number.
3.1.1.4:
Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.
3.1.1.5:
Compare and order whole numbers up to 100,000.
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Grade 3  5 Expectations:
 understand the placevalue structure of the baseten number system and be able to represent and compare whole numbers and decimals;
 recognize equivalent representations for the same number and generate them by decomposing and composing numbers;
 develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers;
 use models, benchmarks, and equivalent forms to judge the size of fractions;
 recognize and generate equivalent forms of commonly used fractions, decimals, and percents;
 explore numbers less than 0 by extending the number line and through familiar applications;
 describe classes of numbers according to characteristics such as the nature of their factors.
Common Core State Standards
Use place value understanding and properties of operations to perform multidigit arithmetic.
3.NBT.1. Use place value understanding to round whole numbers to the nearest 10 or 100.
Misconceptions
Student Misconceptions and Common Errors
Students may think:
 they only need to look at the first or the last digit in a multidigit number when comparing and ordering numbers.
 no patterns exist in multidigit numbers.
Vignette
The students in this third grade classroom are comparing and ordering multidigit numbers. They are also finding numbers that are 10,000 more/less, 1,000 more/less, and 100 more/less than given numbers.
Teacher: Using the digits 0 through 9, write a fivedigit number on your card.
Teacher: At your table, read each number and then place your numbers in order from least to greatest.
The teacher checks in with groups as they order the numbers.
Table 1
The students at table 1 are working with the following numbers
The teacher notices that the number 12,504 is placed in the center of the table.
Teacher: It looks as though you have decided to start with the number 12,504. Can you say more about that?
Student A: Since we are going from least to greatest we started with the smallest number.
Teacher: How do you know 12,504 is the smallest number?
Student B: It has one group of ten thousand. The other numbers have more than one group of ten thousand.
Student C: 12,504 has twelve thousands which is fewer than ninetyfour thousands in 94,970 or seventytwo thousands in 72,368 or ninetyeight thousands in 98,765.
Noting that this group has a strategy and an understanding of the value of the digits and their place value the teacher moves to the next group.
Table 2
The students at table 1 are working with the following numbers
and have them placed on the table as follows
96,341
58,098
58,576
42,120
Teacher: Would you each read your number out loud?
Students correctly read each number.
Teacher: Can you tell me how you ordered your numbers?
Student A: We found the biggest and the smallest. We put the big number up high and the small number at the bottom. Then the other numbers go in the middle.
Teacher: Which number is the smallest?
Student B: 42,120.
Student C: The largest is 96,341.
Teacher: How do you know?
Student B: We looked at the first number and knew 9 was biggest and 4 was the smallest.
Teacher (Pointing to 96,341): What does the nine mean?
Student D: 9 ten thousands
Teacher (Pointing to 42,120): What does the four mean?
Student C: 4 ten thousands.
Teacher: Which is larger 9 ten thousands or 4 ten thousands?
Student A: 9 ten thousands
Teacher: Why is 96,341 more than 42,120?
Student D: We know that 96,341 has the most ten thousands.
Teacher (pointing to 58,098 and 58,576):Tell me about these numbers. How did you decide how to order them.
Student B: We saw they both began with 58 so we had to look at something else.
Student A: So we looked at the other side and we know that 98 is more than 76.
Student C: That means that 58,098 is more than 58,576.
Teacher: Let's write these number another way. What does the 5 in 58,098 mean?
Student D: The number has 5 ten thousands.
The teacher records on the board
5 ten thousands
Teacher: What does this 8 in 58,098 (pointing to the 8 in the thousands place) mean?
Student B: The number has 8 thousands.
The teacher adds to the previous recording
5 ten thousands + 8 thousands
Teacher: What does the 0 in 58,098 mean?
Student A: The number has 0 hundreds.
The teacher adds to the previous recording
5 ten thousands + 8 thousands + 0 hundreds
Teacher: What does the 9 mean in 58,098?
Student A: The number has 9 tens.
Student D: And the last 8 means the number has 8 ones.
The teacher adds to the previous recording
58,098 = 5 ten thousands + 8 thousands + 0 hundreds + 9 tens + 8 ones
The teacher repeats for the number 58,576 and records the following
58,576 = 5 ten thousands + 8 thousands + 5 hundreds + 7 tens + 6 ones
Teacher: What do you notice about the numbers when they are written this way?
Student B: They both have the same number of ten thousands and thousands.
The teacher notes this observation as follows
58,098 = 5 ten thousands + 8 thousands + 0 hundreds + 9 tens + 8 ones
58,576 = 5 ten thousands + 8 thousands + 5 hundreds + 7 tens + 6 ones
Teacher: Is there anything else about the place values for these two numbers that is the same?
Student D: They have hundreds, tens and ones but they are not the same amounts.
Student B: Oh, we skipped the hundreds before. Five hundreds is more than 0 hundreds so 58,576 is more than 58,098 because it has more hundreds.
Teacher: Do you agree that we have to look at the hundreds before deciding which of these two numbers is the largest?
Student A: I get it.....I just skipped the zero but the other number has some hundreds. I have to check out both numbers.
The other students in the group agree and they correctly reorder their numbers from least to greatest as the teacher moves to the other groups.
After verifying that the other groups correctly ordered their numbers the teacher collects all number cards and asks the students to come to the front of the room with their white boards and markers.
The teacher places six number cards on the board and asks the students to read each number. The following numbers are on the board
58,576 13,742 98,760
24,612 70,972 40,973
The teacher these numbers in order to provide opportunities for students to work across place value groupings when find 10,000 more/less, 1,000 more/less or 100 more/less.
Teacher: On your white board, write the number that is 10,000 more than 88,760?
Teacher writes 88,760 on the board.
Teacher: Show me your number.
Teacher: Student K, please read your number.
Student K: 98,760
Teacher records 98,760 on the board.
Teacher: Raise your hand if you agree. All hands are raised.
Teacher: How do you know 98,760 is ten thousand more than 88,760?
Student B: 88,760 has 8 groups of ten thousand and another group of ten thousand would mean 9 ten thousands so 98,760.
The teacher writes 98,460 on the board.
Teacher: Is 98,460 ten thousand more than 88,760? Talk to your neighbor and be ready to explain your thinking.
Teacher: What do you think...Is 98,460 ten thousand more than 88,760?
Student M: We think no, because there are different amounts of hundreds, tens, and ones. The only thing that should change would be the number of ten thousands if it was just ten thousand more.
Teacher: That thinking works for these numbers. Let's try another one.
Write the number that is one thousand more than 69,972 on your white board.
Teacher writes 69,972 on the board.
Teacher: Share with your neighbor. Do you agree?
The class agrees that the only number that works is 70,972.
Teacher: In our last example, Student M said that only one digit changes. Why isn't that true in this case?
Student G: That doesn't work for this because there were 9 thousands and one more thousand would be ten thousands which is one group of ten thousand. For this one, the ten thousands change and the thousands change. It is a new place value grouping.
Student E: It is just like when we used the place value blocks and traded ten tens for a hundred except the groups are bigger.
Students are now asked to independently find numbers that are either 10,000 more/less, 1,000 more/less, or 100 more/less for a given set of numbers and record their responses in their math notebooks.
The teacher notes some students are not fully understanding the meaning of 0 in a multidigit number and will provide additional experiences to strengthen student understanding.
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Understanding the place value concepts of 10,000 more/less, 1,000 more/less, and 100 more or less is essential for the development of efficient procedures for adding and subtracting multidigit numbers. Students need many experiences to strengthen their understanding of 0 as a placeholder in a multidigit number. They also need experiences that will help them understand what happens when another grouping is added to 9 existing groups. For example, understanding 1000 more than 19,872 requires thinking beyond following rules, i.e., changing digits. In this case, the change of 1000 more results in a change in the number of ten thousands.
 Provide physical or pictorial representations for large numbers when developing the concept of 1,000 more/less, and 100 more/less.
 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, etc.
 The hundreds, tens and ones structure exists in the hundreds period, the thousands period, the millions period, etc. Students need to recognize and identify the repeated pattern of hundreds, tens and ones in each period.
 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 14,750 on the number line?
Place a point on the number line to show the location of 14,750.
 Third grade students are not required to use the < and > symbols when comparing numbers.
 Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence 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 Activities
Each student needs digit cards: 0 to 9.
The same activity can be completed many different times and at different levels of difficulty by drawing 3, 4, or 5 digit cards. Have students place their cards face down and draw a certain number. Extra cards remain in a pile to the side. Using only those cards drawn, ask students to:
 create the largest possible number
 create the smallest possible number
 write the number words for the number created above
 tell what 10 more than their number would be; 100 more; 10 or100 less
 compare number with partners (determining whose is larger, smaller, middle number)
 build number closest to 500 (5,000 or 50,000)
 make an odd number
 make a number that is a multiple of 5
 create a number between 100 and 400 (1,000 and 4,000 or 10,000 and 40,000)
Additional Instructional Resources
Van de Walle, J., Karp, K., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
Wickett, M., & Burns, M. (2005). Teaching arithmetic: Lessons for extending place value, grade 3. Sausalito, CA: Math Solutions.
value: relative worth, magnitude; numerical quantity that is assigned or determined by calculation or measurement; number represented by a figure, symbol.
number line diagram: a diagram of the number line used to represent numbers and support reasoning about them. In a number line diagram for measurement quantities, the interval from 0 to 1 on the diagram represents the unit of measure for the quantity.
"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 nonexamples.
Example/Nonexample 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 comparing and ordering numbers at the third 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 1000100,000?
What representations will help students determine 1,000 and 10,000 more than a given number?
When checking for student understanding of comparing numbers up to 100,000, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a comparing and ordering of numbers to 100,000. What evidence is needed to say a student is proficient? Using three pieces of work, determine what understanding is observed by the work.
Look at student work related to the ordering of multidigit numbers. What reasoning led to the errors you see?
How can teachers assess student learning related to these benchmarks?
How are these benchmarks related to other benchmarks at the third grade level?
Professional Learning Community Resources
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K8, (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 K6). Sausalito, CA: Math Solutions.
Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K6. 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.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA.: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades 35. Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed.). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Cavanagh, M. (2004). Math to know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K8. (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 K6). Sausalito, CA: Math Solutions.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 35. 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.
Hyde, A. (2006). Comprehending math adapting reading strategies to teach mathematics, K6. Portsmouth, NH: Heinemann.
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 elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, Miki. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. 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. 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 smarterMessages 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., & BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
Performance Assessment
 Write the number 20,562. Have students read the number. Have students write the number that is 10,000 more, 10,000 less. Next, have students write the number that is 1000 more, 1000 less, 100 more, 100 less.
Solution: Correctly reads 20,562, finds and writes 10,000 more/less, and 1,000 more/less.
Benchmark:
 Connie lists her scores from a video game. 14,087 13,345 14,613 14,301
Which list shows the scores listed from greatest to least?
A 14,613 13,345 14,301 14,087
B 14,613 14,301 14,087 13,345
C 14,087 14,613 14,301 13,345
D 13,345 14,087 14,301 14,613
Solution: B
Benchmark: 3.1.1.2 MCAIII Item Sampler
 Some of the animals at the zoo were weighed. Which list shows the animals from the heaviest to the lightest?
Animal 
Weight in Pounds 
Elephant 
7243 
Rhinoceros 
3869 
Hippopotamus 
5319 
Walrus 
3209 
a. Elephant, hippopotamus, walrus, rhinoceros
b. Walrus, rhinoceros, hippopotamus, elephant
c. Hippopotamus, elephant, rhinoceros, walrus
d. Elephant, hippopotamus, rhinoceros, walrus
(Notes: requires the ability to compare and order numbers. Solution: d)
NCTM. (2005). Mathematics Assessment Sampler Grades 35. NCTM.
 There are 23,650 people in a stadium. The stadium can hold 1,000 more people. How many people can the stadium hold?
A 22,650
B 23,750
C 24,650
D 33,650
Solution: C 24,650
Benchmark: 3.1.1.3 MCA III Item Sampler
 Circle the point represents the location of 14,750 on each number line.
A.
B.
C.
Solution:
A.
B.
C.
Benchmark: 3.1.1.1
Differentiation
Use pictorial representations to assist in ordering and comparing numbers.
Use base ten block representations of numbers to determine 100 more/less, 1000 more/less, 10,000 more/less than a given number.
Draw a number line with a small range of numbers, then have students place an identified number between the two endpoints and justify placement.
Using number cards 09, ask students to draw four cards and make the largest or smallest possible number. Represent that number with base ten materials. When students are proficient using four cards, move to five and then six cards.
Concrete  Representational  Abstract Instructional Approach
The ConcreteRepresentationalAbstract Instructional Approach (CRA) is a researchbased 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 ontask. 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.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 35.Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. 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., & BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
Wickett, M., & Burns, M. (2005). Teaching arithmetic: Lessons for extending place value, Grade 3. Sausalito, CA: Math Solutions.
Using base ten block representations of numbers to determine 100 more/less, 1000 more/less, and 10,000 more/less than a given number allows the connection to appropriate vocabulary.
Number lines provide a visual representation when comparing and ordering numbers.
 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:
I know _______ is one thousand more than ________ because ________. 
I know _______ is one thousand less than _________ because ________. 
I know __________ is one hundred more than ____________ because ______________. 
I know __________ is one hundred less than _____________ because ______________. 
 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.
Additional ELL Resources:
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k2. Sausalito, CA: Math Solutions Publications.
Extending the Learning
Draw a number line with a wide range of numbers, have students place an identified number or numbers between the two endpoints and justify placement.
Read "How to Make a Million," then have students generate other examples that can be a million.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 35. Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Parents/Admin
Administrative/Peer Classroom Observation
Students are: 
Teachers are: 
comparing numbers in terms of place value, understanding that 25,752 is between 25,000 and 26,000. 
focused on the place value descriptions of numbers and guiding students to describe numbers using the place value structure and appropriate vocabulary.

ordering four and fivedigit numbers from least to greatest and greatest to least. 
questioning students based on the place value structure of numbers.

finding a number that is 100 more/less, 1000 more/less and 10,000more/ less than a given four or fivedigit number. 
using number lines to guide student thinking while they represent, compare and order numbers. 
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 realworld 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.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k8, 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.
For Administrators
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.
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 mathematics
Provides 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
Help Your Children Make Sense of Math
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 selfconfidence 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