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Mathematics - Grade 2

Math_Curriculum Guide Gr 2_07.28.11_v1 Page | 1

Mathematics Curriculum Guide Grade 2

Introduction: Mathematical Content Standards

MILWAUKEE

PUBLIC SCHOOLS



In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes. 1. Students extend their understanding of the base-ten system. This includes ideas of counting in fives, tens, and multiples of hundreds, tens, and ones, as well as number relationships involving these units, including comparing. Students understand multi-digit numbers (up to 1000) written in base-ten notation, recognizing that the digits in each place represent amounts of thousands, hundreds, tens, or ones (e.g., 853 is 8 hundreds + 5 tens + 3 ones). 2. Students use their understanding of addition to develop fluency with addition and subtraction within 100. They solve problems within 1000 by applying their understanding of models for addition and subtraction, and they develop, discuss, and use efficient, accurate, and generalizable methods to compute sums and differences of whole numbers in base-ten notation, using their understanding of place value and the properties of operations. They select and accurately apply methods that are appropriate for the context and the numbers involved to mentally calculate sums and differences for numbers with only tens or only hundreds. 3. Students recognize the need for standard units of measure (centimeter and inch) and they use rulers and other measurement tools with the understanding that linear measure involves an iteration of units. They recognize that the smaller the unit, the more iterations they need to cover a given length. 4. Students describe and analyze shapes by examining their sides and angles. Students investigate, describe, and reason about decomposing and combining shapes to make other shapes. Through building, drawing, and analyzing two- and three-dimensional shapes, students develop a foundation for understanding area, volume, congruence, similarity, and symmetry in later grades.

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning



The Assessment and Literacy Connections on this page are a general listing of research-based strategies which good teachers employ. The lists are not meant to be an “end all”. These are repeated for every grade level curriculum guide with the intent that our students experience some of the same strategies year to year, teacher to teacher. On the following pages, you will find specific usage of some of these strategies.

Classroom Assessment for Learning Literacy Connections

Assessment for learning provides students with insight to improve achievement and helps teachers diagnose and respond to student needs. (Stiggins, Rick, Judith Arter, Jan Chappuis, Steve Chappuis. Classroom Assessment for Student Learning. Educational Testing Service, 2006.) Descriptive Feedback (oral/written) Effective use of questioning Exit Slips Milwaukee Math Partnership (MMP) CABS Portfolio items Student journals Student self-assessment Students analyze strong and weak work samples Use of Learning Intentions Use of rubrics with students Use of student-to-student, student-to-teacher and teacher-to-

student discourse Use of Success Criteria Formative Assessments

Measure of Academic Progress

Summative Assessments Unit Tests WKCE/WAA

Comprehensive literacy is the ability to use reading, writing, speaking, listening, viewing and technological skills and strategies to access and communicate information effectively inside and outside of the classroom and across content areas. (CLP, p.11) Literacy strategies can help students learn mathematics: Concept mapping Graphic organizers Journaling K-W-L Literature Partner talk RAFT Reciprocal teaching Talk moves Think Aloud strategy Think, Pair, Share strategy Three-minute pause Two column note taking Vocabulary strategies (e.g. Marzano’s 6 steps; Frayer model)



Operations & Algebraic Thinking

2.OA

Represent and solve problems involving addition and subtraction.

Add and subtract within 20.

Work with equal groups of objects to gain foundations for multiplication.

1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

2. Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

1 See Glossary, Table 1. 2See standard 1.OA.6 for a list of mental strategies.



Operations & Algebraic Thinking 2.OA

Essential/Enduring Understandings Assessment

Represent and solve problems involving addition and subtraction

within 100 using the structure of the problem (part-part whole) to guide their thinking.

Fluently add and subtract within 20 using mental strategies. Work with equal groups of objects to gain foundations for

multiplication.

Given a number equation or given a part-part whole situation, students will create and solve story problems using the structure of the operation to guide their work.

Students apply number based strategies (make a ten, bridge to

ten, doubles, near doubles, etc.) to mentally compute. Given a number (up to the number 25), students demonstrate

understanding of the quantity of the number by making equal groups.

Classroom Assessment Based on Standards (CABS)

2.OA Standard 1 Grade 3 NOR 5

2.OA Standard 2 Grade 2 AR 1, 4, 9, 14 Grade 2 NOR 6, 7, 8, 9, 10, 12

2.OA Standard 3 Grade 3 NOR 1, 4

Common Misconceptions/Challenges 2.OA Cluster: Represent and solve problems involving addition and subtraction within 100 using the structure of the problem (part-part whole) to guide their thinking.

Students do not move beyond an add-to result unknown or a separate result unknown problem format. Students should become familiar with the different story problem formats (table 1 on page 9).

2.OA Cluster: Fluently add and subtract within 20 using mental strategies. Students count by ones instead of applying a strategy based on number relationships. Use dot patterns or tens frames to help students practice “seeing” smaller numbers inside larger numbers. Then help students record their thinking in an equation form so they can make connections from the concrete representation to the abstract notation.

2.OA Cluster: Work with equal groups of objects to gain foundations for multiplication.



Students count by ones instead of counting by groups. Students have limited cardinality and have not developed a unitized system for counting by a group. Students need repeated exposure to and practice with counting sequences in order to become fluent with counting. Counting routines provide opportunities for students to create the anchors they need for solving problems efficiently.

Instructional Practices 2.OA Cluster: Represent and solve problems involving addition and subtraction. 2.OA Standard: 1 Solve various types of problem with unknowns in all positions Launch addition and subtraction word problems that are connected to students’ lives. Develop fluency with addition and subtraction through exploring four different addition and subtraction situations and their relationship to

the position of the unknown.

Examples of Add-to, Take-from, and Compare problems are listed below: Add-to: David had $37. His grandpa gave him some money for his birthday. Now he has $63. How much money did David’s

grandpa give him? $37 + = $63 Compare: David has 63 stickers. Susan has 37 stickers. How many more stickers does David have than Susan? 63 – 37 = Even though the modeling of the two problems above is different, the equation, 63 - 37 = ? can represent both situations (How many more do I need to make 63?) Take- from: David had 63 stickers. He gave 37 to Susan. How many stickers does David have now? 63 – 37 = Take- from (Start Unknown) David had some stickers. He gave 37 to Susan. Now he has 26 stickers. How many stickers did

David have before? - 37 = 26

It is important to attend to the difficulty level of the problem situations in relation to the position of the unknown. Result Unknown problems are the least complex for students followed by Total Unknown and Difference Unknown. The next level of difficulty includes Change Unknown, Addend Unknown, followed by Bigger Unknown. The most difficult are Start Unknown, Both Addends Unknown, and Smaller Unknown.

Summarize problem types by discussing the action in the problem and how students represent their thinking regardless of the level of difficulty.

2.OA Cluster: Add and subtract within 20. 2.OA Standard: 2 Mental Strategies Launch use objects, diagrams, or interactive whiteboards, and various strategies to help students develop fluency to add and subtract within

20 using mental strategies. The use of mental strategies will help students during explorations and make sense of number relationships as they add and subtract within

20. The ability to calculate mentally with efficiency is very important for all students. Mental strategies may include the following: Counting on Making tens (9 + 7 = 10 + 6)



Decomposing a number leading to a ten ( 14 – 6 = 14 – 4 – 2 = 10 – 2 = 8) Fact families (8 + 5 = 13 is the same as 13 - 8 = 5) Doubles Doubles plus one (7 + 8 = 7 + 7 + 1)

Summarize the learning experience by allowing students to engage in discourse about their use of mental strategies for adding and

subtracting within 20. 2.OA Cluster: Work with equal groups of objects to gain foundations for multiplication. 2.OA Standards: 3-4 Equal Groups Launch the opportunity for students to write equations representing sums of two equal addends, such as: 2 + 2 = 4, 3 + 3 = 6, 5 + 5 = 10, 6 +

6 = 12, or 8 + 8 =16. This understanding will lay the foundation for multiplication and is closely connected to 2.OA.4. Students explore odd and even numbers in a variety of ways including the following: students may investigate if a number is odd or even by

determining if the number of objects can be divided into two equal sets, arranged into pairs or counted by twos. Example: The use of objects and/or interactive whiteboards will help students develop and demonstrate various strategies to determine even and odd numbers.

To lock-in and summarize the concept of working with equal groups, allow students to write equations that represent the total as the sum of

equal addends as shown below.

4 + 4 + 4 = 12 5 + 5 + 5 + 5 = 20

The equation when counting by rows: 20 = 5 + 5 + 5 +5

The equation when counting by columns: 20 = 4 + 4 + 4 + 4

Differentiation 2.OA Cluster: Represent and solve problems involving addition and subtraction within 100 using the structure of the problem (part-part



whole) to guide their thinking. Students use manipulatives to model and verbalize their strategies in order to solve one- and two-step addition and subtraction word

problem in all formats (part-part whole and comparison). Students start with add-to-result unknown and take-from result unknown and practice them for several weeks and then move onto other formats.

2.OA Cluster: Fluently add and subtract within 20 using mental strategies. Record students’ mental strategies using number lines, equations, and illustrations. Students could use colored pencils to identify jumps of

ten from jumps of one to make sense of mental strategies used to solve the task. 2.OA Cluster: Work with equal groups of objects to gain foundations for multiplication. Build on knowledge of composing and decomposing numbers to investigate arrays up to 5 rows and up to 5 columns in different

orientations. Form an array with 3 rows and 4 objects in each row - represent the total number of objects with equations showing a sum of equal addends two different ways – count by rows, 12 = 4 + 4 + 4; count by columns, 12 = 3 + 3 + 3 + 3. Rotate the array 90° to form 4 rows with 3 objects in each row – write two different equations to represent 12 as a sum of equal addends – count by rows, 12 = 3 + 3 + 3 + 3; count by columns, 12 = 4 + 4 + 4 - discuss this statement: The two arrays are different and yet the same.

Literacy Connections Academic Vocabulary Literacy Strategies: Use the Think-Aloud strategy to help students make sense of the

context of the story and allow the action in the story to solve it. Listen to students reasoning and keep track of the strategies they are using to solving two one-digit numbers within 20.

Math journal- Allow students to draw equal groups of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns and write a story about the drawing, solve, and write an equation to express the total.

add array equal equation even odd strategy subtract unknown number word problem

Literature Suggestions: Cuyler, Margery and Howard, Arthur. 100 Day Worries (Simon &

Schuster Children’s Publishing, 2000).

Cristaldi, Kathryn and Morehouse, Henry. Even Steven and Odd Todd (Cartwheel, 1996).



Resources

Common Core State Standards for Mathematics: Common addition and subtraction situations http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf Table 1 on page 88 in the Common Core State Standards (CCSS) for School for Mathematics illustrates twelve addition and subtraction problem situations. ORC # 4243 From the National Council of Teachers of Mathematics: Get the Picture—Get the Story http://illuminations.nctm.org/LessonDetail.aspx?ID=L212 In this lesson, students act as reporters at the Super Bowl. Students study four pictures of things that they would typically find at a football game then create problem situations that correspond to their interpretation of each of the pictures ORC # 4308 From the National Council of Teachers of Mathematics: Looking back and moving forward http://illuminations.nctm.org/LessonDetail.aspx?ID=L43 In the game Race to Zero at the bottom of the page, students take turns rolling a number cube and subtracting the number they rolled each time from 20. The first person to reach 0 wins the round. ORC # 4314 From the National Council of Teachers of Mathematics: Finding fact families http://illuminations.nctm.org/LessonDetail.aspx?ID=L57

In this lesson, the relationship of subtraction to addition is introduced with a book and with dominoes.



Table 1. Common addition and subtraction situations.6 Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown1

Put Together / Take Apart

2

Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 – 3 = ?, ? + 3 = 5



Number & Operations in Base Ten 2.NBT

Understand place value.

Use place value understanding and properties of operations to add

and subtract.

1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:

a. 100 can be thought of as a bundle of ten tens — called a “hundred.” b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to

one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).

2. Count within 1000; skip-count by 5s, 10s, and 100s. 3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form. 4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.

5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. 6. Add up to four two-digit numbers using strategies based on place value and properties of operations. 7. Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds. 8. Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900. 9. Explain why addition and subtraction strategies work, using place value and the properties of operations.

1

1 Explanations may be supported by drawings or objects.





Number & Operations Base Ten 2.NBT


Understand place value of the base ten system

Use place value understandings and properties of operations to add and subtract

Given any number under 1000, students will be able to build it using a variety of representations (pictures, models, words and numbers).

Given a number under 1000 students will be able use base-ten place value concepts to compose a new number. (i.e., 159 add 10 is the same as 169 or 562 subtract 100 is the same as 462).

Classroom Assessment Based on Standards (CABS)

2.NOR Standard 1 - 2 Grade 2 NOR 1, 2, 3, 4, 5 ,6, 10, 15, 16, 17

Common Misconceptions/Challenges 2.NBT Cluster: Understand place value of the base ten system.

Students struggle to count forward and backward in units of ten from any given digit. Use a 100’s chart to make the 10-ness of our number system explicit. Students use mini-ten frames to make a number and find out what is 10 more and 10 less of that number. Students are unable to unitize to 10 (see 10 ones as 1 ten). i.e., 358 is 300 ones plus 50 ones plus 8 ones. Students see the numbers as individual digits instead of a quantity, i.e., 4 in 46 represents 4, not 40.

2.NBT Cluster: Use place value understandings and properties of operations to add and subtract

Students are confused about bridging to 10 and then adding some more. Use tens frames to model place value and properties of operations as children work to bridge to 10 and beyond.

Instructional Practices 2.NBT Understand place value. 2.NBT Standards: 1-4 Launch the lesson by using models such as: base ten blocks, number lines, bundles, place value mat, and other manipulatives to support

students’ discovery of place value patterns.

Summarize students’ explorations by sharing representations of various amounts. It is important that emphasis is placed on the language associated with the quantity. For example, 243 can be expressed in multiple ways such as 2 groups of hundred, 4 groups of ten and 3 ones, as well as 24 tens with 3 ones. When students read numbers, they should read in standard form as well as using place value concepts.



Allow students opportunities to explore counting, up to 1000, from different starting points. They should also have many experiences skip counting by 5s, 10s, and 100s to develop the concept of place value.

Examples: The use of the 100s chart may be helpful for students to identify the counting patterns. The use of money (nickels, dimes, dollars) or base ten blocks may be helpful visual cues. The use of an interactive whiteboard may also be used to develop counting skills.

Students explore reading and writing numerals in multiple ways.

Examples: Base-ten numerals 637 (standard form) Number names six hundred thirty seven (written form) Expanded form 600 + 30 + 7 (expanded notation)

Summarize discussions highlighting models, number lines, base ten blocks, interactive whiteboards, document cameras, written words, and/or spoken words that represent two three-digit numbers.

Students apply their understanding of place value as they explore to compare numbers. They first attend to the numeral in the hundreds

place, then the numeral in tens place, then, if necessary, to the numeral in the ones place. Summarize using comparative language including but not limited to: more than, less than, one more, one less, most, greatest, least, same

as, equal to and not equal to. Students use the appropriate symbols to record the comparisons. 2. NBT Use place value understanding and properties of operations to add and subtract. 2.NBT Standards: 5-9 Properties of Operations Exploring adding and subtracting fluently, students need to share how they found an accurate answer. Fluency means accuracy (correct

answer), efficiency (basic facts computed within 4-5 seconds), and flexibility (using strategies such as making 10 or breaking numbers apart). When summarizing lessons focus fluency discussion on how students communicate their thinking and are able to justify their strategies both

verbally and with paper and pencil.

Examples of strategies:

Addition strategies based on place value for 48 + 37 may include:



Adding by place value: 40 + 30 = 70 and 8 + 7 = 15 and 70 + 15 = 85. Incremental adding (breaking one number into tens and ones); 48 + 10 = 58, 58 + 10 = 68, 68 + 10 = 78, 78 + 7 = 85 Compensation (making a friendly number): 48 + 2 = 50, 37 – 2 = 35, 50 + 35 = 85

Subtraction strategies based on place value for 81 - 37 may include: Adding up (from smaller number to larger number): 37 + 3 = 40, 40 + 40 = 80, 80 + 1 = 81, and 3 + 40 + 1 = 44. Incremental subtracting: 81 -10 = 71, 71 – 10 = 61, 61 – 10 = 51, 51 – 7 = 44 Subtracting by place value: 81 – 30 = 51, 51 – 7 = 44

Properties that students should know and use are: Commutative property of addition (Example: 3 + 5 = 5 + 3) Associative property of addition (Example: (2 + 7) + 3 = 2 + (7+3) ) Identity property of 0 (Example: 8 + 0 = 8)

Students in second grade need to communicate their understanding of why some properties work for some operations and not for others.

Examples of properties: Commutative Property: In first grade, students investigated whether the commutative property works with subtraction. The intent was for students to recognize that taking 5 from 8 is not the same as taking 8 from 5. Students should also understand that they will be working with numbers in later grades that will allow them to subtract larger numbers from smaller numbers. This exploration of the commutative property continues in second grade. Associative Property: Recognizing that the associative property does not work for subtraction is difficult for students to consider at this grade level as it is challenging to determine all the possibilities.

2.NBT Use place value understanding and properties of operations to add and subtract. 2.NBT Standard 6-7 Adding and Subtracting two-digit and three-digit numbers Students explore addition strategies with up to four two-digit numbers either with or without regrouping. Problems may be written in a story problem format to help develop a stronger understanding of larger numbers and their values. Interactive whiteboards and document cameras may also be used to model and justify student thinking. Summarize the strong connection of using place value understanding with addition and subtraction of smaller numbers. Students may use concrete models or drawings to support their addition or subtraction of larger numbers. Strategies are similar to those stated in 2.NBT Use place value understanding and properties of operations to add and subtract. 2.NBT Standard 8 Mentally Adding and Subtracting 10 or 100 to a Given Number Students need many opportunities to explore and summarize mental math by adding and subtracting multiples of 10 and 100 up to 900 using



different starting points. They can practice this by counting and thinking aloud, finding missing numbers in a sequence, and finding missing numbers on a number line or hundreds chart. Explorations should include looking for relevant patterns.

Explore mental math strategies:

counting on; 300, 400, 500, etc. 100 more than 653 is _____ (753) Start at 248. Count up by 10s until I tell you to stop.”

counting back; 550, 450, 350, etc.

10 less than 87 is ______ (77) 2.NBT Use place value understanding and properties of operations to add and subtract. 2.NBT Standard 9 Addition and Subtraction Strategies Using Place Value Properties of Operations Students need multiple opportunities to summarize their explanations of their addition and subtraction thinking. Operations embedded within a meaningful context promote development of reasoning and justification.

Example: Mason read 473 pages in June. He read 227 pages in July. How many pages did Mason read altogether? Karla’s explanation: 473 + 227 = _____. I added the ones together (3 + 7) and got 10. Then I added the tens together (70 + 20) and got 90. I knew that 400 + 200 was 600. So I added 10 + 90 for 100 and added 100 + 600 and found out that Mason had read 700 pages altogether. Debbie’s explanation: 473 + 227 = ______. I started by adding 200 to 473 and got 673. Then I added 20 to 673 and I got 693 and finally I added 7 to 693 and I knew that Mason had read 700 pages altogether. Becky’s explanation: I used base ten blocks on a base ten mat to help me solve this problem. I added 3 ones (units) plus 7 ones and got 10 ones which made one ten. I moved the 1 ten to the tens place. I then added 7 tens rods plus 2 tens rods plus 1 tens rod and got 10 tens or 100. I moved the 1 hundred to the hundreds place. Then I added 4 hundreds plus 2 hundreds plus 1 hundred and got 7 hundreds or 700. So Mason read 700 books.

Summarize lessons by having students’ connect different representations and explain the connections. Representations can include numbers, words (including mathematical language), pictures, number lines, and/or physical objects. Students should be able to use any/all of these representations as needed.

Differentiation 2.NBT Cluster: Understand place value of the base ten system. Use concrete blocks (place value blocks) to help students model and visualize the magnitude of the numbers represented. This concrete



representation of place value will lead to the development of multiple models using abstract models. (i.e., number lines, drawings, 100’s chart) Example: 236 modeled in multiple ways i.e., 236 ones, 23 tens and 6 ones 236 can be 2 hundreds, 3 tens 6 ones or 20 tens and 36 ones, 236 ones, 23 tens and 6 ones.

Use a circle counting game to help students with the more difficult numbers of eleven, twelve, and those in the teens. Arrange students in a

circle and designate a number such as twelve. Students begin to count by ones, and the student who says twelve sits down. The next student begins to count from one again; again, the student who says twelve sits down; and so on (Wright et al. 2006).

2.NBT Cluster: Use place value understanding and properties of operations to add and subtract. Allow students to share in conversations about how others compose or decompose numbers using hundreds, tens, and ones, in order to

facilitate flexibility in problem solving.

Literacy Connections Academic Vocabulary Literacy Strategies: Reciprocal teaching- Student-invented strategies will likely be based

on place value concepts, the commutative and associative properties, and the relationship between addition and subtraction. Invented strategies can also be done mentally. It is vital that student-invented strategies be shared, explored, recorded and tried by others.

addition compare count count-on digit group hundreds ones /unit place value skip count subtraction sum tens

Literature Suggestions: Boynton, Sandra. Hippos Go Berserk!(Little Simon, 2000). Lanczak Williams, Rozanne. The Coin Counting Book (Charlesbridge Publishing, 2001).



Resources 10 x 10 square grid, strips (ten connected squares) and squares base ten blocks Base-ten blocks Beans and a small cup for 10 beans Beans and beans sticks (10 beans glued on a craft stick – 10 sticks can be bundled for 100) Graph paper with numbers from 1 to 120 in rows Hundreds chart Linking cubes nickels and dimes number charts (100 and 1000 starting with a zero and a 1) number lines Place value mat or chart Plastic chain links Pregrouped materials Ten-frame Online resource for base ten blocks: http://nlvm.usu.edu/en/nav/frames_asid_152_g_1_t_1.html?from=category_g_1_t_1.html Online resource for 100s chart, use for counting by 5s and 10s: http://nlvm.usu.edu/en/nav/frames_asid_337_g_1_t_1.html?from=category_g_1_t_1.html Online place value number line: http://nlvm.usu.edu/en/nav/frames_asid_334_g_1_t_1.html?from=category_g_1_t_1.html



Measurement & Data 2.MD

Measure and estimate lengths in standard units.

Relate addition and subtraction to length.

Work with time and money.

Represent and interpret data.

1. Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes. 2. Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen. 3. Estimate lengths using units of inches, feet, centimeters, and meters. 4. Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

5. Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem. 6. Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

7. Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m. 8. Solve word problems involving dollar bills, quarters, dimes, nickels, and pennies, using $ and ¢ symbols appropriately. Example: If you have 2 dimes and 3 pennies, how many cents do you have?

9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. 10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems1 using information presented in a bar graph.

1 See Glossary, Table 1.


Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning



Measure and estimate lengths in standard units.

Relate addition and subtraction strategies when applied

measurement of length.

Tell and write time and solve word problems involving money.

Represent and interpret data.

Students measure the length of their desk and finds that it is 3

feet and 36 inches. Students should explore the idea that the length of the desk is larger in inches than in feet, since inches are smaller units than feet. This concept is referred to as the compensatory principle. Note: this standard does not specify whether the units have to be within the same system.

In gym class Kate jumped 14 inches. Mary jumped 23 inches. How

much farther did Mary jump than Kate. Write an equation and then solve the problem.

Classroom Assessment Based on Standards (CABS) 2.MD Standard 1

Grade 2 M 2 2.MD Standard 3

Grade 2 M 5, 6, 7, 8 2.MD Standard 4

Grade 1 – Looking at Graphs Grade 2 SP 5,6, 8, 11

Common Misconceptions/Challenges

2.MD Cluster: Measure and estimate lengths in standard units.

When some students see standard rulers with the numbers on the hash marks, they believe that the numbers are counting the marks instead of the units or spaces between the marks. Have students use informal units to make their own rulers. They will see that the ruler is a representation of a row of units and focus on the spaces.

2.MD Cluster: Relate addition and subtraction strategies when applied measurement of length.

Students use the ruler incorrectly. Bridge nonstandard units to standard units by providing manipulatives that are a “standard” size (such as one-inch color tiles, one-centimeter cubes, Cuisenaire Rods, base-ten blocks). For example, if you ask students to measure the length of a pencil by lining up color tiles, they will need to approximate the object’s length.

2.MD Cluster: Tell and write time and solve word problems involving money.

Students fail to make the transition from counting dimes to counting nickels. To help students understand 5, 10, and 25, tape a penny



onto a connecting cube, a nickel onto 5 connected cubes, a dime onto 2 lengths of 5 connected cubes placed side by side, and a quarter onto 5 lengths of 5 connected cubes placed side by side. This technique will help students make sense of the values of coins and how to use these values to count and compare sets of coin

Students show confusion over the hour and minute hands, the language associated with time (quarter past, quarter of, half past), and the fractions associated with periods of time create all sorts of problems. By the intermediate grades elapsed time can be more than a challenge for many students. To help clear up the confusion over the hour and minute hands, teachers can instruct students to focus on the hour hand only. Teachers should talk about things that happen at certain hours of the day, and have students observe where the hour hand is pointing. Still using the hour-hand only, show “a quarter after” and “three quarters after.” Use a circle partitioned into fourths to show the meaning of these phrases and the relationship between them.

2.MD Cluster: Represent and interpret data.

Students make inaccurate assumptions when comparing data displays based on visual differences such as a change in scale or a truncated axis. Students sometimes do not see how such attributes influence the appearance of the data, which leads to faulty interpretations. To make data collection and analysis process meaningful for students, teachers can gather a collection of graph templates to which students can refer to. These visual prompts can help students compare the potential final product and help connect the data to a potential display.

Instructional Practices

2.MD Cluster: Measure and estimate lengths in standard units. 2.MD Standards: 1-2 Measure the Length of an Object Launch lessons by asking students to identify attributes of objects and identifying items that could measure them.

The measure of an attribute is a count of how many units are needed to fill, cover or match the attribute of the object being measured. Students need to understand what a unit of measure is and how it is used to find a measurement. It is important to measure the same attribute on the same object with different-sized units. Measurement with standard units (inches, feet, centimeters, and meters) should come after many experiences using informal or non-standard length units in Grade 1.

Students explore different manipulative materials when measuring e.g., students explore with yardsticks and rulers using inch markings. 2. MD Cluster: Measure and estimate lengths in standard units. 2.MD Standards: 3 - 4 Estimation of length Launch using estimation experiences to develop familiarity with the specific unit of measure being used. To measure the length of a shoe,

knowledge of an inch or a centimeter is important so that one can approximate the length in inches or centimeters. Students should begin practicing estimation with items which are familiar to them (length of desk, pencil, favorite book, etc.).

Examples of some useful benchmarks for measurement are:



First joint to the tip of a thumb is about an inch Length from your elbow to your wrist is about a foot If your arm is held out perpendicular to your body, the length from your nose to the tip of your fingers is about a yard

Explore measurement with inches, feet, yards, centimeters, and meters to be able to compare the differences in lengths of two objects. They explore by making direct comparisons by measuring the difference in length between two objects by laying them side by side and selecting an appropriate standard length unit of measure.

Summarize students’ learning by using comparative phrases such as “It is longer by 2 inches” or “It is shorter by 5 centimeters” to describe the difference between two objects. An interactive whiteboard or document camera may be used to help students develop and demonstrate their thinking.

2.MD Cluster: Relate addition and subtraction strategies when applied measurement of length. 2.MD Standards 5 -6 Addition Launch by reviewing addition and subtraction strategies. Students need to explore connections between addition and subtraction to solve word problems which include measures of length. It is important that word problems stay within the same unit of measure. Counting on and/or counting back on a number line will help tie this concept to previous knowledge. Summarize discussions by sharing representations including drawings, rulers, pictures, and/or physical objects. Estimation equations for measurement can support the work connecting to unknown parts of an equation.

Example: A word problem for 5 – n = 2 could be: Mary is making a dress. She has 5 yards of fabric. She uses some of the fabric and has 2 yards left. How many yards did Mary use? Other examples include:

20 + 35 = c

c - 20 = 35

c – 35 = 20

20 + b = 55

35 + a = 55

55 = a + 35



55 = 20 + b Subtraction

Explore connections between this standard and demonstrating fluency of addition and subtraction facts. Addition facts through 10 + 10 and the related subtraction facts should be included.

Summarize by helping students represent their thinking when adding and subtracting within 100 by using a number line.

Example: 10 – 6 = 4

2.MD Cluster: Tell and write time and solve word problems involving money. 2. MD Standard 7 Time

Launch explorations of time by holding discussions with students about things that happen in the morning, afternoon, evening, and night and summarize discussions by helping students realize this is a repeating pattern.

Launch telling time experiences by providing examples to the nearest hour and half-hour. Explore by applying knowledge of skip-counting by 5 to recognize 5-minute intervals on the clock. They need exposure to both digital

and analog clocks. Summarize discussions recognize time in both formats and communicate their understanding of time using both numbers and language.

Common time phrases include the following: quarter till ___, quarter after ___, ten till ___, ten after ___, and half past ___.

Students explore that there are 2 cycles of 12 hours in a day - a.m. and p.m. Record their daily actions in a journal would be helpful for making real-world connections and understanding the difference between these two cycles.

2.MD Cluster: Tell and write time and solve word problems involving money. 2. MD Standard 8 Money

Since money is not specifically addressed in kindergarten, first grade, or third grade, students should have multiple opportunities to identify, count, recognize, and use coins and bills in and out of context. They should also experience making equivalent amounts using both coins and bills. “Dollar bills” should include denominations up to one hundred ($1.00, $5.00, $10.00, $20.00, $100.00).

Example: Sandra went to the store and received $ 0.76 in change. What are three different sets of coins she could have received?




2.MD Standard 9 Line plots

Explore line plots are for collecting data since they show the number of things along a numeric scale. They are made by simply drawing a number line then placing an X above the corresponding value on the line that represents each piece of data. Line plots are essentially bar graphs with a potential bar for each value on the number line. Line plots are first introduced in this grade level. A line plot can be thought of as plotting data on a number line. An interactive whiteboard may be used to create and/or model line plots.

Students should draw both picture and bar graphs representing data that can be sorted up to four categories using single unit scales (e.g., scales should count by ones). The data should be used to summarize solutions that connect to understanding from earlier standards of put together, take-apart, and compare problems as listed in Table 1.

2.MD Cluster: Represent and interpret data. 2.MD Standard 10 Picture graphs

Launch picture graphs by using graphs that show data displayed in single units. Discussions of pictographs should include a title, categories, category label, key, and data.

Second graders should explore both horizontal and vertical bar graphs. Summarize bar graph discussions by highlighting features including a title, scale, scale label, categories, category label, and data.



Differentiation 2.MD Cluster: Measure and estimate lengths in standard units.

Students pose a question about the lengths of several objects. They make a line plot using concrete objects. Then measure the objects to the nearest whole inch, foot, centimeter or meter. Create a line plot with whole-number units (0, 1, 2, ...) on the number line to represent the measurements. Connect the two experiences relating the labels on the axis and the markings on the line plot.

2.MD Cluster: Relate addition and subtraction strategies when applied measurement of length.

Give students one non-standard item (e.g., crayon, marker, etc) to measure an object. Students place the measuring tool down and places it end to end with no gaps or overlaps. Students count the number of times it takes to move the measuring device across the object. (unit iteration).


Give students different representations of the same data and have them describe the similarities and differences noted from each display.

Students create real object and picture graphs so each row or bar consists of countable parts. These graphs show items in a category and do not have a numerical scale. For example, a real object graph could show the students’ shoes (one shoe per student) lined end to end in horizontal or vertical rows by their color. Students would simply count to find how many shoes are in each row or bar. The graphs should be limited to 2 to 4 rows or bars.



Literacy Connections Academic Vocabulary Literacy Strategies: Use a Venn diagram to sort a variety of objects into length

groupings based on a designated marker. For example, students sort objects into three groups: objects shorter than the marker, longer than the marker, and about the same size as the marker. Once groups are sorted, ask students to order objects in each group from shortest to longest.

Journaling- Record weekly in their journal different times during the day that they are involved in an activity (e.g., 4:00 p.m. soccer practice, 11:30 a.m. – checked book from the library on Saturday).

Small groups- Students can work in groups to collect, organize, display and interpret relevant data to answer questions. Encourage them to create graphs that make sense to them to communicate data. Compare the student-created graphs to the type of line plot and graphs studied in this grade.

analog clock cent centimeter clock digital clock dime dollar bill graph horizontal hour hand graph key length line plot line segment measure minute hand nickel penny pictograph quarter ruler vertical

Literature Suggestions: Adler, David A.(How Tall, How Short, How Far Away? (Holiday

House, 2000). Hutchins, Pat. Clocks and More Clocks (Aladdin Paperbacks, Simon

& Schuster, 1970). McCaughrean, Geraldine. My Grandmother’s Clock ( Clarion Books,

2002). Murphy, Stuart and Reed, Mike. Race Around (Harper Trophy,

2001).



Resources ORC # 3991 From the National Council of Teachers of Mathematics: Hopping Backward to Solve Problems http://illuminations.nctm.org/LessonDetail.aspx?ID=L76 In this lesson, students determine differences using the number line to compare lengths. ORC # 3979 From the National Council of Teachers of Mathematics: Where Will I Land? http://illuminations.nctm.org/LessonDetail.aspx?ID=L118 In this lesson, the students find differences using the number line, a continuous model for subtraction.

From the National Library of Virtual Manipulatives, Utah State University: Time – Match Clocks http://nlvm.usu.edu/en/nav/frames_asid_317_g_1_t_4.html?from=category_g_1_t_4.html Students manipulate a digital clock to show the time given on a face clock. They can also manipulate the hands on a face clock to show the time given on a digital clock. Times are given to the nearest five minutes. ORC # 1133 From the National Council of Teachers of Mathematics: Number Cents http://illuminations.nctm.org/LessonDetail.aspx?ID=U67 In this unit, students explore the relationship between pennies, nickels, dimes, and quarters. They count sets of mixed coins, write story problems that involve money, and use coins to make patterns. National Library of Virtual Manipulatives, Utah State University: Bar Chart http://nlvm.usu.edu/en/nav/frames_asid_190_g_1_t_1.html?from=category_g_1_t_1.html This manipulative can be used to make a bar chart with 5 to 20 rows and 3 to 12 columns. The colors for the bars are predetermined however users can type in their own title for the graph and labels for the bars.

Instructional Resources/Tools Centimeter rules and tapes Inch rulers and tapes Yardstick Meter stick



Geometry

Reason with shapes and their attributes. 2.G

1. Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. 3. Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

1 Sizes are compared directly or visually, not compared by measuring.





Essential /Enduring Understandings Assessment Recognize and draw shapes having specified attributes what makes

shapes alike and different can be determined by geometric properties.

Partitioning rectangles develops an understanding area. Students

apply their knowledge of equal groups as they count rows and columns to consider that area is covering with no gaps or overlaps.

Partitioning shapes into equal piece lays the foundation for

fractional thinking.

Given a shape and a specific number of pieces (2, 3, or 4 equal pieces) students have to partition the shape into 2, 3 or 4 equal pieces. Given a rectangular shape, students partition it into rows and columns of same shaped squares and count to find the total number of them. Classroom Assessment Based on Standards (CABS)

o 2.G Standard 1 Grade 2 G 3, 4, 5, 6, 7

o 2.G Standard 3 Grade 1 Fill the shape Grade 2 G 14, 15, 16 Grade 3 G 5

Common Misconceptions/Challenges 2.G Cluster: Recognize and draw shapes having specified attributes what makes shapes alike and different can be determined by geometric properties.

Some students may think that a shape is changed by its orientation. They may see a rectangle with the longer side as the base, but claim that the same rectangle with the shorter side as the base is a different shape. This is why is it so important to have young students handle shapes and physically feel that the shape does not change regardless of the orientation, as illustrated below.

2.G Cluster: Partitioning shapes into equal piece lays the foundation for fractional thinking.



Students also may believe that a region model represents one out of two, three or four fractional parts without regard to the fact that the parts have to be equal shares, e.g., a circle divided by two equally spaced horizontal lines represents three thirds.

Instructional Practices 2.G. Cluster: Recognize and draw shapes 2.G. Standard: 1 Identifying shapes

Launch lessons by asking students to identify shapes of objects in their world. Students explore shapes to identify, describe, and draw triangles, quadrilaterals, pentagons, and hexagons. Pentagons,

triangles, and hexagons should appear as both regular (equal sides and equal angles) and irregular. Explore shapes in a variety of orientations and configurations. Summarize lessons by helping students to understand all four sided shapes are quadrilaterals. Summarize lessons by helping students use the vocabulary word “angle” in place of “corner” but they do not need to name

angle types.

2.G. Cluster: Partition a rectangle into rows and columns 2.G. Standard: 2 Partitioning rectangles

Launch by having student discuss what it means to cover a 2 – shape (e.g., a floor with carpet or table with a cloth). This standard is a precursor to learning about the area of a rectangle and using arrays for multiplication. Students explore the idea of equal groups as they extend their thinking to explain that rows are horizontal and columns are

vertical.



2.G. Cluster: Partitioning circles and rectangles into two, three, and four equal shares. 2.G. Standard: 3 Partitioning into Equal Shares

Launch discussions with Students explore fractions as an area model with different sizes of circles, and rectangles. Summarize student learning by focusing on the conversation that students should recognize that when they cut a circle into

three equal pieces, each piece will equal one third of its original whole. In this case, students should describe the whole as three thirds. If a circle is cut into four equal pieces, each piece will equal one fourth of its original whole and the whole is described as four fourths.

Students explore circles and rectangles partitioned in multiple ways so they learn to recognize that equal shares can be different shapes within the same whole.

Differentiation

2.G. Cluster: Partition a rectangle into rows and columns Modeling multiplication with partitioned rectangles promotes students’ understanding of multiplication. Tell students that they will be



drawing a square on grid paper. The length of each side is equal to 2 units. Ask them to guess how many 1 unit by 1 unit squares will be inside this 2 unit by 2 unit square. Students now draw this square and count the 1 by 1 unit squares inside it. They compare this number to their guess. Next, students draw a 2 unit by 3 unit rectangle and count how many 1 unit by 1 unit squares are inside. Now they choose the two dimensions for a rectangle, predict the number of 1 unit by 1 unit squares inside, draw the rectangle, count the number of 1 unit by 1 unit squares inside and compare this number to their guess. Students repeat this process for different-size rectangles. Finally, ask them to what they observed as they worked on the task.

2.G. Cluster: Partitioning shapes into equal piece lays the foundation for fractional thinking.

It is vital that students understand different representations of fair shares. Provide a collection of different-size circles and rectangles cut from paper. Ask students to fold some shapes into halves, some into thirds, and some into fourths. They compare the locations of the folds in their shapes as a class and discuss the different representations for the fractional parts. To fold rectangles into thirds, ask students if they have ever seen how letters are folded to be placed in envelopes. Have them fold the paper very carefully to make sure the three parts are the same size. Ask them to discuss why the same process does not work to fold a circle into thirds.

Literacy Connections Academic Vocabulary

Literacy Strategies Use a K-W-L to introduce students to the study of shapes. As students study the attributes of shapes, continue adding to the KWL chart. Use small groups and partner talk to clarify understanding of equal parts when partitioning an object.

columns halves hexagon pentagon partition quadrilateral rectangle rows square unit thirds triangle unit

Literature connections Burns, M. The Greedy Triangle. Math Solutions, (1995) Rogers, P. The Shapes Game. Holt, NY (1989)



Resources

Instructional Resources/Tools

ORC # 3991 From the National Council of Teachers of Mathematics: Hopping Backward to Solve Problems http://illuminations.nctm.org/LessonDetail.aspx?ID=L76 In this lesson, students determine differences using the number line to compare lengths.

ORC # 3979 From the National Council of Teachers of Mathematics: Where Will I Land? http://illuminations.nctm.org/LessonDetail.aspx?ID=L118

Grid paper ORC # 1481 From the Math Forum: Introduction to fractions for primary students http://mathforum.org/varnelle/knum1.html http://mathforum.org/varnelle/knum2.html http://mathforum.org/varnelle/knum5.html

This four-lesson unit introduces young children to fractions. Students learn to recognize equal parts of a whole as halves, thirds and fourths.

From the National Library of Virtual Manipulatives, Utah State University: Time – Match Clocks http://nlvm.usu.edu/en/nav/framesasid317g1t4.html?from=categoryg1t4.html

Students manipulate a digital clock to show the time given on a face clock. They can also manipulate the hands on a face clock to show the time given on a digital clock. Times are given to the nearest five minutes.

ORC # 1133 From the National Council of Teachers of Mathematics: Number Cents http://illuminations.nctm.org/LessonDetail.aspx?ID=U67

http://illuminations.nctm.org/LessonDetail.aspx?ID=L76

http://illuminations.nctm.org/LessonDetail.aspx?ID=L118

http://nlvm.usu.edu/en/nav/framesasid317g1t4.html?from=categoryg1t4.html

http://illuminations.nctm.org/LessonDetail.aspx?ID=U67



Table 1. Common addition and subtraction situations.6

Result Unknown Change Unknown Start Unknown

Add to

Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ?

Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5

Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5

Take from

Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ?

Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3

Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3

Total Unknown Addend Unknown Both Addends Unknown1

Put Together / Take Apart2

Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ?

Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ?

Grandma has five flowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2

Difference Unknown Bigger Unknown Smaller Unknown

Compare3

(“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ?

(Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 – 3 = ?, ? + 3 = 5

6Adapted from Box 2-4 of Mathematics Learning in Early Childhood, National Research Council (2009, pp. 32, 33).



1These take apart situations can be used to show all the decompositions of a given number. The associated equations, which have the total on the left of the equal sign, help children understand that the = sign does not always mean makes or results in but always does mean is the same number as. 2Either addend can be unknown, so there are three variations of these problem situations. Both Addends Unknown is a productive extension of this basic situation, especially for small numbers less than or equal to 10. 3For the Bigger Unknown or Smaller Unknown situations, one version directs the correct operation (the version using more for the bigger unknown and using less for the smaller

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