3.1.1A Number Representation & Place Value

Grade: 
3
Subject:
Math
Strand:
Number & Operation
Standard 3.1.1

Compare and represent whole numbers up to 100,000 with an emphasis on place value and equality.

Benchmark: 3.1.1.1 Number Representation

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.

Benchmark: 3.1.1.2 Place Value

Use place value to describe whole numbers between 1000 and 100,000 in terms of ten thousands, thousands, hundreds, tens and ones.

For example: Writing 54,873 is a shorter way of writing the following sums:

5 ten thousands + 4 thousands + 8 hundreds + 7 tens + 3 ones

54 thousands + 8 hundreds + 7 tens + 3 ones

Overview

Big Ideas and Essential Understandings 

Standard 3.1.1 Essential Understanding

Third graders expand their work with place value to include numbers from 1,000-100,000. 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 sixty-seven thousand four-hundred sixty-five.   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 five-digit number. Find 1000 more or 1000 less than a given four- or five-digit number. Find 100 more or 100 less than a given four- or five-digit 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.

Benchmark Cluster 

Benchmark Group A

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.

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

  • Read and write whole numbers up to 100,000.
  • Represent whole numbers up to 100,000 using words, pictures, numerals, expressions with operations, and number lines.
  • Using base ten materials, represent numbers from 1000 to 100,000 using ten thousands, thousands, hundreds, tens and ones.
  • Represent numbers from 1000 to 100,000 in written form. For example, 4,756 can be represented as 4000 + 700 + 50 + 6 or as 4700 + 56, or as 47 hundreds and 56 ones, or 45 hundreds and 25 tens and 6 ones, or four thousand seven hundred fifty six.
  • Identify value and place in large numbers, for example, in the number 4,756, 7 is in the hundreds place and has a value of 700.

Work from previous grades that supports this new learning includes:   

  • Read and write numbers up to 1000.
  • Represent whole numbers up to 1000 using words, pictures, and numerals.
  • Represent whole numbers up to 1000 using hundreds, tens, and ones. For example, 844 is the same as 800 + 40 + 4, or 8 hundreds, 4 tens and 4 ones, or 6 hundreds and 24 tens and 4 ones, or 844 ones, or 84 tens and 4 ones).
Correlations 

NCTM Standards 

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

Grade 3 - 5 Expectations:

  • understand the place-value structure of the base-ten 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 multi-digit arithmetic.

3.NBT.1. Use place value understanding to round whole numbers to the nearest 10 or 100.

Misconceptions

Student Misconceptions 

Student Misconceptions and Common Errors

Students may think...

  • after reading or hearing the words seven hundred fifty thousand fifty-eight they write 7005000058, a literal interpretation.
  • there is only one representation for a number when using place value. They do not recognize there are 46 hundreds or 467 tens in 4,678.
  • they represent 2,324 as 2,000 + 300 + 20 + 4, but are unable to see 2,324 as 1,000 + 1,200 + 120 + 4 (one thousand, twelve hundreds, twelve tens and four ones).

Vignette

In the Classroom

Vignette 1

Ms. Anderson writes the following number on the board:  76,823

Ms. Anderson:  Would you please read this number?

Students:  Seventy six thousand eight hundred twenty-three.

Ms. Anderson:  What do you know about this number?

Student:  It has 6 thousands.

Student:  It has 8 hundreds

Student:  There are five digits in 76, 823.

Ms. Anderson:    What can you tell me about the seven?

Student:  Seven is in the ten thousands place.

Ms. Anderson:    Yes, the digit seven is in the ten thousands place. 

Ms. Anderson sees the students can identify the place but is not sure if they understand the value of the place.

Ms. Anderson:  What is the value of the 7 in that place?

Student:  Ten thousand.

Ms. Anderson:  Ten thousand is the place.  How many ten thousands are there in 76,823

Student:  There are seven.

Student:  Seven ten thousands.

Ms. Anderson:  Lets count all seven ten thousands.

Ms. Anderson records as students count 10,000's.

10,000

20,000

30,000

.

.

.

70,000

80,000

90,000

Annya:  Hey! We don't have that many.

There is only seven ten thousands. Just erase 80,000 and 90,000. 

Ms. Anderson does not erase 80,000 or 90,000.

Ms. Anderson:  Annya, how do you know there are only seven ten thousands in 76,823?

Annya:  Look at the number.  The seven is in the ten thousands place. That means seven ten thousands.

She points to the digit 7 in 76,823.

Ms. Anderson writes seven ten thousands on the board.

Ms. Anderson:  Lets count seven ten thousands.

Using the numbers previously recorded while students count by ten thousands, Ms. Anderson points to 10,000.

Ms. Anderson:  How many ten thousands in 10,000?  

Students:  One.

Ms. Anderson:  How many ten thousands in 20,000?

Students:  Two

This continues until students verify that there are seven 10,000 in 70,000.

Student:  Stop! That's all we have.

Annya:  I said there are 7 ten thousands. We do not need to count any farther. Erase 80,000 and 90,000.

Ms. Anderson:  Do you agree that we counted seven 10,000's when we get to

70,000?

The class agrees and Ms. Anderson writes the following on the board.

7 ten thousands is equal to 70,000

7 ten thousands = 70,000

Ms. Anderson knows many of her students understand and can identify the place of a digit. For example, the 7 in 76,823 is in the ten thousands place. However, many still struggle with understanding the value of the place. For example, the seven in 76,823 means 70,000.  Understanding the place and the value of the place is the foundation of place value understanding. Students who lack this understanding struggle with arithmetic operations using multi-digit numbers.

Vignette 2

Number Sense in the Classroom (1997 MN Framework)

Students apply their place value knowledge as they relate large numbers to the real-world.

Ms. H. wanted her students to develop an understanding of number magnitude and the relative sizes of different numbers. She also wanted them to develop and use "benchmark" numbers for navigating around the base 10 place value system when comparing and estimating. With these goals in mind, she assigned the following homework:

How Many?

Fill in as many of the blanks in the chart below as you can. In the first column find populations of groups or places that fall into the indicated ranges (e.g., number of people in your family or in your city). In the second column, find units of time or activities that are measured in seconds (e.g., seconds in a minute or in an hour; seconds it takes to bus to school).

You may use any resources you want - brains, books, friends, calculators - just be sure to record what resource you used for each number you find.

population table

Population Seconds

The next day, a large grid was attached to the board and students posted their answers and their references in the appropriate slots with Post-it® notes.

(Although this activity can be used from elementary school through high school, here are some of the responses of Ms. H's class.)

Ms. H:  I see some similarities and differences in your population figures. I'd like to hear some of your answers. Remember to tell us how you found the number. Let's start by looking at the answers in the 1's category.

STUDENT: Five people live in my house. I know because I counted them.

STUDENT: Four people live in my house. I just know.

MS. H:  Do all the answers to "How many people live in my house" fit in the range 1-9?

STUDENT: Not if someone had 15 people living in their house. Then the answer would be in the 10's category.

Note:  When numbers were small, students said they "just knew" answers. (One child "just knew" there were 5 million people in Minnesota!) As the numbers increased, they began to use a variety of resources.

The degree of accuracy is important. If you are off by 1 or 2 in a 4 person family that is similar to being off 1 or 2 million in a state of 5 million. Being off by 10,000 out of 4 million is usually a relatively small error.

Ms. H:  Years ago, when Americans in general had larger families, that might have been a more typical answer. It still is in some communities and some countries. What question could someone from a larger family use to still get an answer in the 1's category?

Student: How many brothers do I have? or sisters?

Student: Or maybe they could count their pets.

Student:  For 10's I have, "How many students are in the class." I know because the attendance totals are written right there on the board. Everyone in the class would have the same number.

Student: But not everyone in the other classes in the school!

Ms. H:  How do you know their answers would be different?

Student: Well, I know my brother's class has only 20 kids. We could ask other teachers. I don't think any class has 100. Maybe that's how many people are in the school.

Ms. H:  Did anyone else put "How many students are in the school" in the 100's category?

Student: I did. I figured there are 20 classes and each class has about 25 students so there are 500 students in the school.

Ms. H:  So you estimated. So far we have been able to fill categories from our experience: count- ing, estimating, just knowing. Did anyone need to use reference materials for 1000's?

Student: I used the telephone book and there are about 500 people on one page, so I said there are about 1000 people on 2 pages in the phone book.

Student: I went from the number of people on my block to the number of people on twenty blocks and finally got a little over 1000. But I know that really is an estimate, because not every block is like mine.

Student: I looked in the Atlas and found a lot of places in Minnesota that have a population between 1000 and 10,000 - like Ada (1707), Adrian (1126), and Afton (2891). It said Minneapolis has 386,691 people and St. Paul had 285,068.

Student: The population of a small town would fill only a few blocks in St. Paul.

Student: Some even have less. Round Lake has a population of 376. That would only go in the 100's category.

Different resources may give different populations for the same place. These students found that an atlas, an almanac and the census did not agree on the exact population for Minneapolis.

Ms. H: Why is the population of Minneapolis different in these three sources?

Student: Maybe people were just born or just died and they didn't get to count them yet.

Student: Maybe they didn't count every person - they just rounded or estimated. And they all did it a little differently.

Note:  One of the mathematical goals of this lesson is to help each individual student review and/or develop meaningful benchmarks.

The students see how their peers figured their answers. Some used mental arithmetic, some used paper and pencil, some used a calculator, and some used all of them.

The teacher and students have to listen carefully to each other's reports in order to check for misunderstandings.

Student: Maybe they didn't measure the exact same area. My dad says that sometimes they count people inside the city limits and sometimes they include some of the suburbs, which would make the numbers bigger.

Ms. H:  How close are the different figures? Are they close enough to each other for our use?

Student: It looks a little bit like some of our estimates - where close is good enough.

Student: That could be true for my million category. According to the Almanac, 3 million people play Little League baseball. They should say "about" 3 million.

Ms. H:  What types of figures did you find for the "seconds" column?

Student: For 10's, I have, "How many seconds in a minute" - that's 60. But there's no name for something between a second and a minute.

Student: Well, I say 1 second is how long it takes me to spin around.

Student: I can run across the gym in 5 seconds. I timed it on a stopwatch.

Ms. H:  Blinking your eyes and sneezing - I see you do these things in under 10 seconds. How about writing your name?

Student: Someone with a really long name might take longer to write it than someone with a really short name. And if you had to write your first, middle and last name, that might take a long time - maybe more than ten seconds. We could time it sometime and see.

Ms. H:  These figures are like the number of people in a family. There are a number of right answers, but each type of answer might cover a range of number categories.

Student: For 100's I used "How many seconds it takes me to brush my teeth." I know because I timed it with a stopwatch. Other people would get different answers.

Student: I timed my mother getting money from the bank machine.

Student: I tried to figure out how old I was in seconds, thinking it would be in the 1000's category. I'm 10 years old, so I used a calculator to multiply 60 seconds in a minute times 60 minutes in an hour times 24 hours in a day. That was 86,400. So how many seconds in a day is in the 10,000's and how many seconds in an hour is 3600, so that's in the 1000's.

Student: I did something like that to figure out how many seconds were in a school year. It's in the 10,000,000's!

Student: No wonder it feels like we spend so much time in school.

Ms. H: Did you use 24 hours in a day to figure out how many seconds were in a school year?

Student: Uh-oh. I guess we're only in school 6 hours a day.  So I have to start again.

Student: No, you have four 6's in 24 hours, so divide your answer by 4.

Student: That's in the millions now.

Note:  Comparing the magnitude of numbers and investigating their relationships develops an ability to negotiate around the number system.

Student: It still feels like we spend a lot of time in school.

Student:  It takes 1 second to scoop an ice cream cone. I decided to figure out how long it would take to give everyone in the class an ice cream cone. The students found this problem intriguing and decided to go further.

Student:  I see in the chart that there are 308 million people in the United States. If they stood in line for ice cream, how long would it take to give everyone an ice cream cone?

Working together, Ms. H and her students arrived at a little more than 9 years.

Student:  But people don't work 24 hours a day. Since the store would be open only 8 hours a day, we have to multiply by 3. That will be 27 years.

Student: But at the end of 24 years, more people will be born and there will be more people in line. Will we feed all the ones who are born or will we say they have to be born by 1997 to get a free cone? If we kept scooping I wonder if we'd ever finish feeding them all.

The homework assignment and the classroom discussion allowed students to explore number benchmarks and some relationships. That exploration work is now leading to a level of inquiry, with students using the numbers to explore their own questions. Ms. H used these questions as examples of how to write math problems associated with the data the students had collected.

Students worked in groups to write and solve problems. A few examples were:

1. If there are 1 - 5 people in a household, how many households are there in the Twin Cities? in Minnesota? in the United States?

2. How much time does this class spend brushing their teeth each week?

3. If there are about 250,000,000 people in the United States, how many cars do you think there are in the United States? How many televisions? Explain your thinking.

4. If an average family (4 - 5 people) ate at McDonalds once a week, about how much would this cost in a year? Explain your thinking.

Resources

Instructional Notes 

Teacher Notes

  • Students may need support in further development of previously studied concepts and skills.
  • 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 need to see various ways of representing numbers using place value, such as 46 hundreds being the same as 4,600 ones.
  • Students need to see and be able to translate between multiple representations of a number; e.g., written words, expanded notation, and representing a number using base-ten blocks in combination with the written form.
  • Students need to see symbolic representations connected to physical models and/or pictorial representations. Abstract representations for large numbers can be difficult for students.
  • 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 

Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). Navigating through number and operations in grades 3-5. Reston, VA: National Council of Teachers of 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., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

Wickett, M., & Burns, M. (2005). Teaching arithmetic: Lessons for extending place value, grade 3. Sausalito, CA: Math Solutions.

New Vocabulary 

thousands ????????

value: relative worth; magnitude; numerical quantity that is assigned or determined by calculation or measurement; number represented by a figure, symbol

"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.

Frayer Model

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.

Example /Non Example Chart

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 

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 third grade level?  How do student misconceptions interfere with mastery of these ideas?

What is the difference between the place and the value of the place?

How would you know a student understands the place value system when using numbers from 1,000 to 100,000?

What representations should a student be able to make for the number 34,724 if they understand place value?

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

When checking for student understanding of place value, 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 work, determine what understanding is observed by the work.

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 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.

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.

References 

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

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades 3-5. 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 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., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. 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, K-6. 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 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. 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, J., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

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

Assessment

  • Show 4, 678 in at least two ways.

Solution:  Student correctly represents 4,678 in at least two ways.
Benchmark:  3.1.1.1 and 3.1.1.2

  • Write 39,536 in expanded form.

Solution:  30,000 + 9,000 + 500 + 30 + 6
Benchmark:  3.1.1.1 and 3.1.1.2  What is another way to show 14,608?

A. 146+8
B. 14,000+60+8
C. 14,000+600+8
D. 14,000+600+80

Solution:  C.
Benchmark:  3.1.1.1 and 3.1.1.2

  • An elephant weighs 3,000 + 200 + 30 +4 pounds.  Write this number in standard form and word form.

Solution: 3,234 and three thousand, two hundred thirty-four
Benchmark: 3.1.1.1 & 3.1.1.2  MCAIII Item Sampler

  • Write the number 20,000 + 4,000 + 200 + 4 in standard form.

Solution: 24,204
Benchmark: 3.1.1.1 & 3.1.1.2.  MCA III Item Sampler

  • 7,000+600+35 is the same as:

A. 7,600,350
B. 7,635
C. 76,350
D. 7,065
 
Solution: B. 7,635
Benchmark: 3.1.1.1 & 3.1.1.2.  MCA III Item Sampler

  • What is the value of the 6 in 469,752?

Solution: 60,000
Benchmark: 3.1.1.2

  • Which number has a 5 in the ten thousands place?

A. 104,352
B. 365,971
C. 582,607
D. 951,480

Solution: D. 951,480
Benchmark: 3.1.1.5.  MCA III Item Sampler

Differentiation

Struggling Learners 

Students need to understand and represent place value of four-digit numbers before working with larger numbers. This understanding includes knowing 300 is the same as 30 tens or 300 ones and 4300 is the same as 43 hundreds or 430 tens or 4300 ones.

Students need help in the transition from pictorial and physical models to the symbolic representation of numbers. Initially, teachers need to make this connection for students.

Concrete - Representational - Abstract Instructional Approach

(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. 

Concrete Triangle

Additional Resources

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 3-5.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., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.

English Language Learners 

Vocabulary is an area sensitive to ELLs. Be careful when introducing words with mathematical meaning that have a different meaning in everyday usage such as place, value, base, etc. 

Small group settings allow students more opportunities to construct numbers using base ten materials and to incorporate the language needed to describe the base ten place value system.

  • 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.

Frayer Model

  • 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:

In the number ____________ the ______ is in the ____________place.

The value of the _________ in the number _____________ is ________.

  • 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 k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 
  • Students represent multi-digit numbers in more than two ways, composing and decomposing using place values other than the face values. For example, 46,832 can be represented as 43,000 + 3,600 + 30 + 2 or 40,000 + 6,500 + 330 + 2.
  • Given a number, ask students the total number of tens or hundreds or thousands when not using the face value. For example, how many thousands in 14, 751? 14 thousands.

How many tens in 14, 751?   1,475 tens.

Additional Resources

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 3-5. 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.

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

Students are:

Teachers are:

constructing numbers using base-ten blocks and other manipulatives in order to make multiple equivalent representations.

making manipulatives available, such as base-ten blocks, to facilitate constructing multiple representations of large numbers.

describing numbers from 1,000 to 100,000 using place value vocabulary.

using place value vocabulary in describing student representations and translating between equivalent representations.

constructing and explaining representations for numbers from 1000 to 100,000.

using place value materials to make relationships between and among numbers visible.

Finding and describing patterns in numbers.  These patterns are based upon equivalent place value representations.

leading discussions that facilitate finding patterns in our base-ten number system. Ask students to justify their thinking, etc.

identifying the place and the value of the place in multi-digit numbers.  For example, in 14, 872, the 4 is in the thousands place and has a value of four thousand.

using the terms "place" and "value of the place" in order to make place value concepts visible during instruction. 

 

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.
Additional Resources

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.

For Administrators

Burns, M. (Edt).(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 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 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