3.1.1C Rounding Numbers

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: 3.1.1.4 Rounding Numbers

Round numbers to the nearest 10,000, 1000, 100 and 10. Round up and round down to estimate sums and differences.

For example: 8726 rounded to the nearest 1000 is 9000, rounded to the nearest 100 is 8700, and rounded to the nearest 10 is 8730.

Another example: 473 - 291 is between 400 - 300 and 500 - 200, or between 100 and 300.

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 C

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.

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

  • round four and five-digit numbers to the nearest 10,000, 1,000, 100 and 10.
  • use rounding to estimate sums and differences.

Work from previous grades that supports this new learning includes: 

  • round three digit numbers up and down to the nearest 10 and 100.
  • compare and order numbers to 1000.
  • estimate sums and differences up to 100.
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

  • they should focus only on the digit in the place they are asked to round to. For example, when asked to round 273 to the nearest ten, students will say 280 because 7 is closer to 8.

Resources

Instructional Notes 

Teacher Notes

  • Teaching rules for rounding does not teach conceptual understanding.
  • When rounding to the nearest ten thousand, students need to identify the two consecutive multiples of ten thousand that the given number lies between. Students can then determine which of these two numbers is closest to the given number. Likewise for rounding to nearest thousand, hundred or ten.
  • The number line provides the basis for conceptual understanding when rounding numbers. When rounding numbers, the scale of the number line should represent the position being rounded to. For example, if rounding 4,824 to the nearest hundred, a scale of a hundred would be used on a number line from 4000 to 5000. Students first determine that 4,824 is between 4,800 and 4,900 and then focus on the distance from 4,824 to 4,800 and to 4,900 in order to make a decision.
  • There are ten possible digits for any place in the base ten number system.
  • When making the decision to round up or down, the first five digits (0,1,2,3,4) round down and the last five digits (5,6,7,8,9) round up.
  • Students may need support in further development of previously studied concepts and skills.
  • In the real-world, the decision to round up or down depends on the situation.
  • Asking students to decide if a sum or difference is more or less than a given number encourages the use of rounding. For example, is 7,465 + 347 more or less than 7,900? Is 7,465 + 347 closer to7,800 or 7,810 or 7,820?
  • 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 

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 

Round: changing a number to a nearby number that ends in zero

Nearest: being relatively closer

Closest: near or near together in kind or relationship

 "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 

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

What are the key ideas related to rounding at the third grade level? How do student misconceptions interfere with mastery of these ideas?

How would you know a student understands rounding five-digit numbers to the nearest ten, hundred, thousand or ten thousand?

Look at student work related to rounding. What reasoning contributed to the errors you see?

When checking for student understanding of rounding, what should teachers

●     listen for in student conversations?

●     look for in student work?

●     ask during classroom discussions?

Examine student work related to a rounding 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 

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

  • Is 7,465 + 347 more or less than 7,900?

Solution:  Less than 7,900     
Benchmark: 3.1.1.4

  • Is 7,465 + 347 closer to
    A.        7,800
    B.        7,810
    C.        7,820

Solution:
Benchmark: 3.1.1.4

  • What is 153,924 rounded to the nearest thousand?

A 150,000
B 153,000
C 153,900
D 154,000

Solution: D
Benchmark: 3.1.1.4

Differentiation

Struggling Learners 

Struggling Learners

Number lines help make the mathematics of rounding visible.  Struggling learners need to work with a smaller range of numbers before rounding larger numbers to ten thousand or one thousand.

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 

English Language Learners

Number lines make the mathematics of rounding visible.  Language can then be connected to this representation.

  • 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:

I rounded ___________ to ____________ 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 k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 
  • Use rounding to estimate sums and differences of larger numbers.
  • Look up and provide latest census information from Kids Count Data Center. Have students use different data about Minnesota to round and estimate.  For example, below is the 2008 MN population for children 0 to 9 years old.

Approximately how many 8 and 9 year olds were in MN in 2008?

Minnesota

 

<1

73,019

 

1

73,418

 

2

74,281

 

3

72,336

 

4

70,921

 

5

71,448

 

6

69,515

 

7

67,236

 

8

68,792

 

9

67,103

 

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:

using rounding as an estimation strategy.

providing situations that require an estimate involving rounding.

 

developing front-end addition and subtraction rounding strategies.

 

identifying and labeling strategies students use in the classroom.

identifying the two consecutive multiples of a given place value that a given number lies between.

using number lines that highlight the consecutive multiples of a given place value.


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