Composing and Decomposing Whole Numbers


 Margery Francis
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1 Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Demonstrate understanding of mathematical operations involving addition and subtraction of whole numbers. Solve addition and subtraction problems. Mentally add and subtract wholenumber facts to a sum of 20. Add and subtract mentally. Use doubles, near doubles, and combinations of tens to solve problems. Essential questions How do you know when to use addition strategies to solve a problem? How do you know when to use subtraction strategies to solve a problem? How can you use a hundreds chart to solve this problem? How did you solve this (addition/subtraction) problem? What is your strategy for mental addition/subtraction? How can you use doubles, near doubles, and combinations of ten to solve problems? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Self monitor problemsolving steps, and evaluate the solution. Ask, does this make sense? Plan a solution pathway. Use appropriate tools strategically. Be familiar with available tools. Make sound decisions when using and selecting tools. Routines No routines during this unit other than establishing classroom expectations regarding calendar and accessing and using tools and manipulatives. Cumberland, Lincoln, and Woonsocket Public Schools C1
2 Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers (10 days) Written Curriculum GradeLevel Expectations M(N&O) 2 3 Demonstrates conceptual understanding of mathematical operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, partpart whole relationships, and comparison situations; and addition of multiple onedigit whole numbers. (State) (See Appendix A.) M(N&O) 2 6 Mentally adds and subtracts whole number facts to a sum of 20; names the number that is 10 more or less than the original number, and mentally adds and subtracts twodigit multiplies of ten (e.g., , 90 30). (Local) (IMPORTANT: The intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning In kindergarten and first grade, students demonstrated an understanding of mathematical operations involving addition and subtraction of whole numbers from 0 to 30. Students solved problems using joining actions, separating actions, partpartwhole relationships, comparison situations, and addition of multiple onedigit whole numbers. Students used calendar patterns to determine elapsed and accrued time. Current Learning Calendar work introduced and practiced in kindergarten and grade 1 should be continued in second grade in the context of using a calendar to solve problems and identify patterns. By the end of second grade, students demonstrate a conceptual understanding of the mathematical operations of addition and subtraction by using techniques including joining and separating actions, partpartwhole relationships, and comparison techniques. Students add multiple onedigit numbers and are introduced to a variety of strategies to be used throughout the year, enabling them to mentally add and subtract whole numbers to a sum of 20. According to Bloom s Taxonomy, student responses should be at the comprehension, application, and analysis levels. Future Learning In grades 3 5, students will use models, number lines, and explanations to demonstrate understanding of inverse relationships between addition and subtractions, repeated addition and multiplication, repeated subtraction and division, and multiplication and division of whole numbers. They will also add and subtract positive fractional numbers using like denominators and will add and subtract decimals and positive fractions with unlike denominators. They will describe or illustrate the meaning of the remainder with respect to division of whole numbers, using models and explanations and also through solving problems. Cumberland, Lincoln, and Woonsocket Public Schools C2
3 Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers (10 days) Additional Research Findings According to A Research Companion to Principles and Standards for School Mathematics, computational fluency is one vital component of developing mathematical power; other components include understanding the uses and methods of computation (p.71). Table 6.1 (p. 70 in A Research Companion) demonstrates the types of addition and subtraction situations given as word problems and the variety of methods for solving these problems. According to Principles and Standards for School Mathematics, students should encounter a variety of meanings for addition and subtraction of whole numbers. And it states that as children develop an understanding of whole numbers and the operations of addition and subtraction, instructional attention should focus on strategies for computing with whole numbers so that students develop flexibility and computational fluency. Additionally, Principles states that part of being able to compute fluently means making smart choices about which tools to use and when (pp ). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C3
4 Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C4
5 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Represent quantities from 0 to 199 in multiple ways. Make equal quantities using models, explanations, or other representations. Write numbers in expanded notation. Order whole numbers using a hundreds chart, number line, and manipulatives. Estimate groups of objects up to 50. Essential questions What are some ways you can show a twodigit number? How can you find 10 more or 10 less? What method could be used to estimate how many objects are in a set? About how many are in this container? Mathematical practices to be integrated Model with mathematics. Interpret results using tools. Analyze relationships of numbers. Make sense of problems and persevere in solving them. Explain correspondence between descriptions, written representations and models. Explain meaning of problems. Routines Mentally add and subtract whole number facts to a sum of 20. Solve addition and subtraction problems. Cumberland, Lincoln, and Woonsocket Public Schools C5
6 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from (10 days) Written Curriculum GradeLevel Expectations M(N&O) 2 1 Demonstrates conceptual understanding of rational numbers with respect to: whole numbers from 0 to 199 using place value, by applying the concepts of equivalency in composing or decomposing numbers (e.g., 34 = ; 34 = ); and in expanded notation (e.g., 141 = 1 hundred + 4 tens + 1 one or 141 = ) using models, explanations, or other representations; and positive fractional numbers (benchmark fractions: a/2, a/3, or a/4, where a is a whole number greater than 0 and less than or equal to the denominator) as a part to whole relationship in area and set models where the denominator is equal to the number of parts in the whole using models, explanations, or other representations. (State) M(N&O) 2 2 Demonstrates understanding of the relative magnitude of numbers from 0 to 199 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (10, 25, 50, 75, 100, 125, 150, or 175); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using 1 more, 1 less, 10 more, 10 less, 100 more, or 100 less ; or by connecting number words and numerals to the quantities they represent using models, number lines, or explanations. (State) M(N&O) 2 7 Makes estimates of the number of objects in a set (up to 50) by selecting an appropriate method of estimation. (Local) (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) Routines M(N&O) 23 Conceptual understanding of math operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, partpart whole relationships, and comparison situations; and addition of multiple onedigit whole numbers (state) (see Appendix A) M(N&O) 26 Mentally adds and subtracts whole number facts to a sum of 20; names the number that is 10 more or less than the original number, and mentally adds and subtracts twodigit multiples of ten (e.g., 60+80, 9030). (local) (IMPORTANT: the intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning In grades K 1, students learned to identify coins (penny, nickel, dime, quarter) and to count groups of like coins. They read, wrote, and ordered numbers up to 100 and compared whole numbers by using 1 more/1 less and 10 more/10 less. They demonstrated understanding of rational numbers up to 100 using place value. Students have had experience estimating up to 30 objects in a set. Cumberland, Lincoln, and Woonsocket Public Schools C6
7 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from (10 days) Current Learning Secondgraders are expected to know numbers through 199, to understand the concept of equivalency, and to connect a variety of models of the same quantity. They order and compare numbers using 10 more/10 less. Students have opportunities to make estimates up to 50 and to select an appropriate method of estimation. Routines and procedures are being introduced and will eventually become reinforcement activities for previously learned skills. This unit allows students to use models, known facts, properties, and relationships to explain their thinking. Students also use patterns and relationships to analyze mathematical situations (e.g., counting by 1s and 10s). According to Bloom s Taxonomy, students level of response should be in the areas of application and analysis. Future Learning In third grade, students will demonstrate conceptual understanding of rational numbers with respect to whole numbers In fourth grade, that will extend to 0 999,999, and in fifth grade, to 0 9,999,999. Students in grades 3 5 will demonstrate an understanding of relative magnitude of numbers up to 9,999,999. They will make estimates using appropriate methods, determining the level of accuracy given the situation and analyzing the effect of an estimation method on the accuracy of results. Beyond second grade, the context of money is taught through decimals rather than through counting coins. Additional Research Findings: According to Principles and Standards for School Mathematics, by the end of second grade a key idea is that a number can be decomposed and thought about in many ways (p. 33). Additionally, Benchmarks for Science Literacy states that by the end of second grade students should know that it is useful to estimate quantities without knowing them exactly. Also, by the end of second grade, students should know that numbers can be used to count things, to place them in order, and/or to name them (p. 211). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C7
8 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C8
9 Grade 2 Mathematics, Quarter 1, Unit 1.3 Understanding Properties of Addition and Equality Overview Number of instructional days: 8 (1 day = minutes) Content to be learned Demonstrate the properties of numbers (including odd/even) in addition. Simplify computations involving whole numbers to solve problems. Demonstrate conceptual understanding of equality by finding the value that makes an open sentence true. Understand vocabulary for addition properties (commutative, identity, associative). Essential questions What combinations can you make when adding a string of numbers to solve a problem? How does using combinations of 10 help you solve problems with 3 or four addends? How can you tell if an answer is incorrect? How do you know if a number is even or odd? Why does the order you add the addends not change the sum? What patterns do you see when adding numbers to zero? Mathematical practices to be integrated Look for and make use of structure. Identify patterns when developing understanding of numbers and their relationships. Use number structure to determine number magnitudes, and relationships. Look for and express regularity in repeated reasoning. Look for patterns when developing strategies. Evaluate results and reflect about applied strategies. Routines Estimate how many are in a given container. Explain the estimation strategy used. Answer questions about estimation (i.e., is the total more or less than 30? How do you know?) Cumberland, Lincoln, and Woonsocket Public Schools C9
10 Grade 2 Mathematics, Quarter 1, Unit 1.3 Understanding Properties of Addition and Equality (8 days) Written Curriculum GradeLevel Expectations M(N&O) 2 8 Applies properties of numbers (odd and even) and field properties (commutative for addition, identity for addition, and associative for addition) to solve problems and to simplify computations involving whole numbers. (Local) M(F&A) 2 4 Demonstrates conceptual understanding of equality by finding the value that will make an open sentence true (e.g., 2 + o = 7). (limited to one operation and limited to use addition or subtraction) (State) Routine M(N&O) 27 Makes estimations of the number of objects in a set (up to 50) by selecting an appropriate method of estimation (local) (IMPORTANT: The intent of this GLE is to embed estimation throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning By the end of first grade, students applied properties (odd, even, composition, and decomposition) and field properties (commutative and identity for addition) to solve problems and simplify computations for whole numbers. Students demonstrated conceptual understanding of equality by finding the value that would complete an open sentence. They also used models, verbal explanations, and written equations to demonstrate understanding. Current Learning By the end of second grade, students apply properties of numbers (odd and even) and field properties, such as the commutative, identity, and associative properties for addition, to solve problems and simplify computations with whole numbers. Students demonstrate an understanding of equality by finding the value to make an open sentence true, limited to a oneoperation sentence, either addition or subtraction. According to Bloom s Taxonomy, students level of response should be application and comprehension. Future Learning In grades 3 5, students will apply properties of numbers, including odd, even, and the multiplicative property of zero for singledigit whole numbers. They will also apply field properties (commutative and associative properties for addition; identity and commutative properties for multiplication). Students will demonstrate conceptual understanding of equality by showing equivalence between two expressions using models or different representations of the expressions. Students will find the value that will make an open sentence true (limited to one operation: addition, subtraction, or multiplication). They will also apply properties of numbers (odd, even, multiplicative property of zero, and divisibility) and Cumberland, Lincoln, and Woonsocket Public Schools C10
11 Grade 2 Mathematics, Quarter 1, Unit 1.3 Understanding Properties of Addition and Equality (8 days) field properties (commutative, associative, identity, and distributive) to solve problems and to simplify computations. Additional Research Findings According to Principles and Standards for School Mathematics, important learning can result when students are given opportunities to develop, record, and explain strategies for solving computation problems (p. 35). Research suggests that it is important to teach for understanding, rather than rote memory, according to A Research Companion to Principles and Standards for School Mathematics (pp ; p. 70, table 6.1). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C11
12 Grade 2 Mathematics, Quarter 1, Unit 1.3 Understanding Properties of Addition and Equality (8 days) Cumberland, Lincoln, and Woonsocket Public Schools C12
13 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations Overview Number of instructional days: 7 (1 day = minutes) Content to be learned Use counting techniques and a variety of strategies to solve problems involving combinations. Add coins to solve problems. Demonstrate understanding of strategies used to solve combining problems. Essential questions How many ways can you make 50 cents using nickels, dimes, and quarters? If given (for example, 43) cards in a group, how many packets of 10 can you make and how many are left over? How can you show the strategy you used to solve the problem? Mathematical practices to be integrated Make sense of problems and persevere in solving them. Understand and identify correspondences between different approaches and strategies. Analyze mathematical situations. Reason abstractly and quantitatively. Represent problem situations with numbers. Create a problem situation from a given number sentence. Routines Find 1 more, 1 less, 10 more, 10 less than a given number. Order numbers by using benchmark numbers. Connect number words and numerals to the quantities they represent. Cumberland, Lincoln, and Woonsocket Public Schools C13
14 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations (7 days) GradeLevel Expectations Written Curriculum M(N&O) 25 Demonstrates understanding of monetary value by adding coins together to a value no greater than $1.99 and representing the result in dollar notation; making change from $1.00 or less, or recognizing equivalent coin representations of the same value (values up to $1.99). (State) M(DSP) 2 4 Uses counting techniques to solve problems involving combinations using a variety of strategies (e.g., student diagrams, organized lists, tables, tree diagrams, or sc others); (e.g., How many ways can you make 50 cents using nickels, dimes, and quarters?) (State) Routines M(N&O) 2 2 Demonstrates understanding of the relative magnitude of numbers from 0 to 199 by ordering whole numbers; by comparing whole numbers to each other or to benchmark whole numbers (10, 25, 50, 75, 100, 125, 150, or 175); by demonstrating an understanding of the relation of inequality when comparing whole numbers by using 1 more, 1 less, 10 more, 10 less, 100 more, or 100 less ; or by connecting number words and numerals to the quantities they represent using models, number lines, or explanations. (State) Clarifying the Standards Prior Learning Students in kindergarten and grade 1 interpreted given representations created by the class (e.g., models, tally charts, pictographs with onetoone correspondence, and tables) to answer questions related to the data or to analyze the data to formulate conclusions using words, diagrams, and verbal/scripted responses. They also analyzed patterns, trends, and distributions in data by determining or using the terms more, less, or equal. Current Learning By the end of second grade, students use counting techniques to solve problems with combinations using a variety of strategies (e.g., diagrams, organized lists, tables, and tree diagrams). According to Bloom s Taxonomy, student responses should be at the knowledge, comprehension, application, and analysis levels. Future Learning In grades 3 5, students will interpret a given representation, including bar graphs, pictographs, circle graphs, and line graphs, to answer questions related to data, to formulate and justify conclusions, to make predictions, and to solve problems. They will analyze patterns, trends, or distributions of data in a variety of contexts by determining the most frequent (mode), least frequent, largest, and smallest item. They will also justify conclusions, solve problems, and make predictions. Cumberland, Lincoln, and Woonsocket Public Schools C14
15 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations (7 days) Additional Research Findings This unit of study is supported by research in Principles and Standards for School Mathematics, which states, students working together often begin to solve problems one way and, before reaching a solution, change their strategies. In addition, as they create and modify their strategies, students often recognize the need to learn more mathematics (p. 118). Additionally, the book states that instead of teaching problem solving separately, teachers should embed problems in the mathematicscontent curriculum. When teachers integrate problem solving into the context of mathematical situations, students recognize the usefulness of strategies. Teachers should choose specific problems because they are likely to prompt particular strategies and allow for the development of certain mathematical ideas (Principles, p. 119). The graphic representation map on page 115 of The Atlas of Science Literacy shows the cognitive sequence for developing data analysis and graphic representation skills. It also states: The graphic display of numbers and relationships can be a powerful aid in discovering and communicating patterns not easily seen in tables or equations (p. 114). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C15
16 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations (7 days) Cumberland, Lincoln, and Woonsocket Public Schools C16
17 Grade 2 Mathematics, Quarter 1, Unit 1.5 Identifying and Extending Patterns Overview Number of instructional days: 5 (1 day = minutes) Content to be learned Identify and extend numeric and nonnumeric patterns. Find missing elements in numeric patterns. Count combinations of coins (penny, nickel, dime, quarter). Recognize an equivalent sets of coins. Demonstrate understanding of value by adding coins. Essential questions What is the next object in this pattern? What is the fifth (shape, color, letter, number) in this pattern? What is the missing number in this pattern? What is the value of these coins (given any two coins)? How can you decide if two sets of coins have the same value? What is your strategy for adding coins? Mathematical practices to integrated Attend to precision. Communicate clearly with others. Express answers clearly verbally and/or in written form. Look for and make use of structure. Identify a pattern or structure. Routines Mentally add and subtract whole number facts to the sum of 20. Solve a variety of addition and subtraction problems. Cumberland, Lincoln, and Woonsocket Public Schools C17
18 Grade 2 Mathematics, Quarter 1, Unit 1.5 Identifying and Extending Patterns (5 days) Written Curriculum GradeLevel Expectations M(F&A) 2 1 Identifies and extends to specific cases a variety of patterns (linear and nonnumeric) represented in models, tables, or sequences by extending the pattern to the next element, or finding a missing element (e.g., 2, 4, 6,, 10). (State) M(N&O) 2 5 Demonstrates understanding of monetary value by adding coins together to a value no greater than $1.99 and representing the result in dollar notation; making change from $1.00 or less, or recognizing equivalent coin representations of the same value (values up to $1.99). (State) Routines M(N&O) 23 Conceptual understanding of math operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, partpart whole relationships, and comparison situations; and addition of multiple onedigit whole numbers (state) (see Appendix A) M(N&O) 26 Mentally adds and subtracts whole number facts to a sum of 20; names the number that is 10 more or less than the original number, and mentally adds and subtracts twodigit multiples of ten (e.g., 60+80, 9030). (local) (IMPORTANT: the intent of this GLE is to embed mental arithmetic throughout the instructional program, not to teach it as a separate unit.) Clarifying the Standards Prior Learning In grades K 1, students were exposed to a variety of nonnumeric sequences (shape, color, movement, letters, etc.). They extended these patterns or translated them across formats. Numeric patterns were introduced in first grade. Also in kindergarten and first grade, students used positional words to locate and describe where an object was in their environment and in reference to another location on a map or diagram. Current Learning Secondgrade students are familiar with skip counting by 2, 5, 10, and 25. They recognize other patterns though problem solving. In grade 2, students count groups of like coins and demonstrate understanding of spatial relationships using location and position through use of positional language in two and threedimensional situations. According to Bloom s Taxonomy, secondgrade responses should be in the categories of knowledge, comprehension, application, analysis, and synthesis. Cumberland, Lincoln, and Woonsocket Public Schools C18
19 Grade 2 Mathematics, Quarter 1, Unit 1.5 Identifying and Extending Patterns (5 days) Future Learning In grades 3 5, students will identify and extend to specific cases a variety of patterns (linear and nonnumeric). In grades 4 5, they will extend learning to nonlinear patterns. Fourth and fifthgrade students will also write rules or symbols for finding specific cases of a linear relationship. In grades 3 5, students will give directions from one location to another, identifying locations on a map or coordinate grid. Additional Research Findings According to Benchmarks For Science Literacy, students should realize that mathematics is the study of many different types of patterns (pp ). Principles and Standards for School Mathematics describes how recognition of patterns is the basis for algebra (pp ). And A Research Companion to Principles and Standards for School Mathematics talks about integrating patterns, functions, and algebra throughout the K 12 curriculum (pp ). Notes About Resources and Materials Cumberland, Lincoln, and Woonsocket Public Schools C19
20 Grade 2 Mathematics, Quarter 1, Unit 1.5 Identifying and Extending Patterns (5 days) Cumberland, Lincoln, and Woonsocket Public Schools C20
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