1.1.1B Numbers, Representation & Counting

Grade: 
1
Subject:
Math
Strand:
Number & Operation
Standard 1.1.1

Count, compare and represent whole numbers up to 120, with an emphasis on groups of tens and ones.

Benchmark: 1.1.1.2 Number Representation

Read, write and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

Benchmark: 1.1.1.3 Count Forward & Backward

Count, with and without objects, forward and backward from any given number up to 120.

Benchmark: 1.1.1.5 Compare & Order Numbers

Find a number that is 10 more or 10 less than a given number.

For example: Using a hundred grid, find the number that is 10 more than 27.

Benchmark: 1.1.1.6 Relative Size

Find a number that is 10 more or 10 less than a given number.

For example: Using a hundred grid, find the number that is 10 more than 27.

Overview

Big Ideas and Essential Understandings 

tens ones

Standard 1.1.1 Essential Understandings

First graders read, write and represent whole numbers up to 120. Representations for numbers include numerals, pictures, tally marks, number lines, addition and subtraction and manipulatives (including base ten blocks). Formal work with place value begins as students describe two-digit numbers in terms of tens and ones; i.e. 37 can be represented as 37 ones or as 3 tens and 7 ones or as 2 tens and 17 ones. First graders use place value knowledge to compare and order numbers up to 120 and to find a number that is 10 more or 10 less than a given two-digit number.  They describe the relative magnitude of numbers using words such as equal to, more than, less than, fewer than, or about the same as.

All Standard Benchmarks

1.1.1.1  Use place value to describe whole numbers between 10 and 100 in terms of groups of tens and ones.  For example: Recognize the numbers 11 to 19 as one group of ten and a particular number of one

1.1.1.2  Read, write and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks

1.1.1.3  Count, with and without objects, forward and backward from any given number up to 120.

1.1.1.4  Find a number that is 10 more or 10 less than a given number.  For example: Using a hundred grid, find the number that is 10 more than 27.

1.1.1.5  Compare and order whole numbers up to 100.

1.1.1.6  Use words to describe the relative size of numbers. For example: Use the words equal to, not equal to, more than, less than, fewer than, is about, and is nearly to describe numbers.

1.1.1.7  Use counting and comparison skills to create and analyze bar graphs and tally charts.For example: Make a bar graph of students' birthday months and count to compare the number in each month.

Benchmark Cluster 

Benchmark Group B

1.1.1.2  Read, write and represent whole numbers up to 120.  Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.

1.1.1.3  Count, with and without objects, forward and backward from any given number up to 120.

1.1.1.5  Compare and order whole numbers up to 100.

1.1.1.6  Use words to describe the relative size of numbers.

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

  • read, write and represent numbers up to 120;
  • count forward and backward from any given number to 120;
  • represent numbers up to 120 using place value models of tens and ones;
  • compare and order numbers up to 100;
  • describe the relative magnitude of a number using words such as equal to, not equal to, more than, less than, fewer than, etc.

Work from previous grades that supports this new learning includes:

  • read and write numbers 0-31;
  • represent numbers 0-31 using numerals, objects, pictures and drawings;
  • compare and order numbers 0-20 with and without objects;
  • understand one-to-one correspondence and apply it when counting objects.
Correlations 

NCTM Standards

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

Pre-K - 2 Expectations:

  • count with understanding and recognize "how many" in sets of objects;
  • use multiple models to develop initial understandings of place value and the base-ten number system;
  • develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;
  • develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;
  • connect number words and numerals to the quantities they represent, using various physical models and representations.

Common Core State Standards

Extend the counting sequence.

1.NBT.1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

Understand place value.

1.NBT.2.  Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

1.NBT.2a. 10 can be thought of as a bundle of ten ones - called a "ten."

1.NBT.2b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

1.NBT.2c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

1.NBT.3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

Represent and interpret data.

1.MD.4.  Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

Misconceptions

Student Misconceptions 

Student Misconceptions and Common Errors

Students may think ...

  • they only have to use the digit in the ones place when comparing and ordering numbers.
  • the tens and ones places can be used interchangeably when representing numbers. 
  • base ten blocks are the only way to represent numbers.
  • the number 43 can only be represented as 4 tens and 3 ones.

 

 

Vignette

In the Classroom

My Number is Greater!

In this activity, first graders use numeral cards (four each of 0-9) and base ten blocks to compare two-digit numbers in order to determine who has the largest number. Number cards are shuffled and placed face down between two players. 

Play begins as each player draws two cards and then decides which two-digit number to represent. If a 2 and an 8 are drawn the player has a choice of using 28 or 82. Since the object of the game is to have the number that is the largest, most will choose to continue playing with the larger of the two numbers.

                                    Player A                                              Player B

 Comparing numbers

  Each player represents his/her chosen number with base ten blocks.

 

                                    Player A                                              Player  B

 

 

 10 blue blocks10 blue blocks1010 blue blocks10 blue blocks10 blue blocks10 blue blocks10 blue blocksgreen blockgreen block                                    10 blue blocks10 blue blocks1010 blue blocks green block

Players determine which of the two numbers is the greatest.  The player with the greatest number captures the number cards.  If the two numbers are equal, each player takes his/her own cards. 

Play continues with each player drawing two number cards, deciding which  number to use, building the chosen number with base ten blocks, and determining which of the two numbers is the greatest.

As first graders play, My Number Is Greater, the teacher is observing and asking questions such as:

  • How did you decide which number to use?
  • Why did you choose to use the number ____ instead of ____?
  • Which number is greatest?
  • Which number is the least?
  • I see you have ________ tens and _______ ones. What number does that represent?
  • If_______ is greater than ________ which of your two numbers is the least?

Extending the Activity

  • The game can be played to find the number that is the least.
  • The game can be played in groups of three or four.
  • Once first graders understand the place value of two-digit numbers they can play the game using only number cards.
  • First graders can draw three cards and make the greatest or least two-digit number using two of the three cards.

Resources

Instructional Notes 

Teacher Notes

  • Students may need support in further development of previously studied concepts and skills
  • First graders need to see multiple representations for any two-digit number.  For example, 43 can be represented as 4 tens and 3 ones, 43 ones, 2 tens and 23      ones, etc.
  • It is easier for first graders to order two-digit numbers using number cards. Students can make changes as they think about why they are placing numerals in a specific order. This is very different than ordering numbers on paper.
  • First graders may be using the words "greater than" and "less than" when comparing numbers but they are not using the symbols  <  and  >.  Teachers may be using these symbols when they record student thinking.
  • 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, where is 40 on each of these number lines?

number lines: 10-50 and 20-60

  • 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 

Instructional Activities

Hundred Chart Puzzle

First graders use their knowledge of hundred chart patterns and number relationships to assemble the Hundred Chart Puzzle. Copy a hundreds chart on construction paper or card stock. Cut the chart apart along the grid lines creating irregular shapes that can be reassembled into the original hundreds chart. More irregular pieces will make the puzzle harder.

Get to 100

In pairs, first graders use two number dice, place value materials and place value mats as they try to get to one hundred.  In turn each player rolls the dice, adds the numbers, and  then adds the corresponding tens and ones to his/her place value mat. The exchange of ten ones (units) for a ten (rod) are made throughout the game.  The game ends when one of the players reaches one hundred.  (Bamberger, Oberdorf, and Schultz-Ferrell, 2010)

Get To Zero

In pairs, first graders use two number dice, place value materials, and place value mats as they go from nine groups of ten on each place value mat to zero on the mats. In turn, each player rolls the number dice, adds the numbers, and the removes  that amount from the place value mat.   Since players started with nine tens (rods) on their respective place value mats, it is likely an exchange of a ten for ten ones will be made prior to removing anything from the mat.  The game ends when one of the players reaches zero.  (Bamberger, Oberdorf, and Schulz-Ferrell, 2010)


Ordering Numbers Game
Choose the range of numbers to be ordered and if you want to order in ascending or descending order.  

Get to zero

Comparing and Ordering Numbers
Students need a set of number cards (5 - 35, 35-69, 70-100, or a set of thirty random numbers from 0-100).  It is easy to make a number card master and reproduce on construction paper.  Students cut the cards apart for a personal set.   For example, this set of cards (from 35-69) would be cut apart and used by a student to complete a variety of tasks.

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

Using number cards,

  • students draw three, four, or five cards and find the number that is the greatest.
  • students draw three, four or five cards and find the number that is the least.
  • students draw three, four, or five cards and arrange them from least to greatest.
  • students draw three, four, or five cards and arrange them from greatest to least.


Count Around
Choose a number range, say 35 - 45. Children stand in a circle and count around, each child saying the next number in the sequence. Start the count at 35, the smallest number in the range. The child who says 45, the largest number in the range, sits down. The next child begins the count again at 35, the smallest number in the range. Counting backwards also works well in Count Around.

Additional Instructional Resources

Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.

Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA:  Math Solutions Press.

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

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

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.

New Vocabulary 

least The smallest in a group.

greatest  The most in a group.

less than Shows relationships between numbers - not as many as

greater than Shows relationships between numbers  - more than

equal  Having the same amount or value

compare  Examine for similarities or differences

  "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

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

word

definition

illustration

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

Additional Resources for Vocabulary Development

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

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

Professional Learning Communities 

Reflection - Critical questions regarding the teaching and learning of these benchmarks

  • What are the key ideas related to place value at the first grade level? How do student misconceptions interfere with mastery of these ideas?
  • What models should a student be able to make when representing 84, using tens and ones?
  • When checking for student understanding of place value at the first grade level, what should teachers...

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

  • How can teachers assess student learning related to these benchmarks?
  • Examine student work related to a task involving place value at the first grade level. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
  • How are these benchmarks related to other benchmarks at the first grade level?

Professional Learning Community Resources

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

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

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

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

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

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 1: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

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.

Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. London: Paul Chapman Publishing.

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

References 

References    

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

Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.

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 k-2. 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. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

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

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

Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA: Math Solutions Press.

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

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

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

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

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

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

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

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

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

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.

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

Wright, R., Martland, J., Stafford, A., & Stanger, G. (2006). Teaching number (2nd end). London: Sage Publishing.

Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006).  Teaching number in the classroom with 4-8 year-olds. London: Paul Chapman Publishing.

 

 

Assessment

Performance Assessments

  • Counting Forward in Sequences

Start from 28 and count up to 34.
Now start from 44 and count up to 52.
Continue with other short sequences to 120.

Solution:  Count forward correctly.
Benchmark:  1.1.13

  • Counting Backward in Sequences

Start from 33 and count back to 26. 
Now, start from 48 and count back to 35.
Similarly, 52 back to 47,
85 back to 77,
94 back to 86, and so on.
Continue with other short sequences from 120.

Solution:  Count backward correctly.
Benchmark:  1.1.1.3

  • Comparing and Ordering Numbers  (adapted from Wright, Stanger, Stafford, & Martland, 2006)

Using ten sequenced number cards in random order. For example, 43 - 52, 67 - 75, etc.

Students will

  • read the numbers
  • order the numbers from least to greatest or greatest to least
  • read the cards in the order they were placed
  • answer the following questions:
    •  How did you know which was the largest? Smallest?
    • What did you look at or what helped you place the numbers in order?

Solution:  Correctly names, compares and orders numbers. Explains thinking.
Benchmark:  1.1.1.5  & 1.1.1.6

  • Circle the boxes that show 24.

Test question: circle 24.

Solution: Boxes that show 24 are circled.
Benchmark:  1.1.1.2

  • Write how many.

Write how many

Solution: 14, 18
Benchmark:   1.1.1.2

Differentiation

Struggling Learners 

Struggling learners need to use models to represent numbers when comparing and ordering.  Base ten blocks provide a model that can also be used when combining and separating two-digit numbers in second grade.

Number lines can also help students see the relationship of one number to another.

Using the counting numbers, the number to the immediate right on the number line is one more and the number to the immediate left is one less.

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 k-2. Sausalito, CA: Math Solutions.

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

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

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

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

English Language Learners 
  • Using base ten blocks and other place value materials to represent numbers provides a common experience and helps English Language Learners understand the language of the base ten system. It is easier for students to explain their thinking when they have used models.
  • 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:

Ten more than _________ is ___________.

Ten less than ___________ is ______________.

The number _____________ has ___________ tens and __________ ones.

The number __________ is less/more  than _______________.

  • When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding.

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 can explore the relationship of one more/less and ten more/less on the hundreds chart using the following activity:

Additional Resources

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

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

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

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

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

Students are ...         

Teachers are ...        

comparing and ordering numbers up to 120.  Finding one/more less and ten more/less than a given two digit number.

 

asking students to justify the ordering of numbers.

 

making multiple representations of numbers to 120.  These representations include base ten blocks, words and numerals

modeling multiple representations of a given number and connecting place value vocabulary to those representations

 

 

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

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

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

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

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

Parents 

Parent Resources

Mathematics handbooks to be used as home references:

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

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

Helping your child learn 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