# Composing and Decomposing Whole Numbers

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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 whole-number 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 problem-solving 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 C-1

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 C-3

4 Grade 2 Mathematics, Quarter 1, Unit 1.1 Composing and Decomposing Whole Numbers (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C-4

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 two-digit 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 C-5

6 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from (10 days) Written Curriculum Grade-Level 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) 2-3 Conceptual understanding of math operations involving addition and subtraction of whole numbers by solving problems involving joining actions, separating actions, part-part whole relationships, and comparison situations; and addition of multiple one-digit 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 two-digit multiples of ten (e.g., 60+80, 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 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 C-6

8 Grade 2 Mathematics, Quarter 1, Unit 1.2 Understanding Whole Numbers from (10 days) Cumberland, Lincoln, and Woonsocket Public Schools C-8

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 C-11

12 Grade 2 Mathematics, Quarter 1, Unit 1.3 Understanding Properties of Addition and Equality (8 days) Cumberland, Lincoln, and Woonsocket Public Schools C-12

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 C-13

14 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations (7 days) Grade-Level Expectations Written Curriculum 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) 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 one-to-one 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 C-14

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 mathematics-content 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 C-15

16 Grade 2 Mathematics, Quarter 1, Unit 1.4 Solving Problems Involving Combinations (7 days) Cumberland, Lincoln, and Woonsocket Public Schools C-16

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 non-numeric 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 C-17

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 non-linear patterns. Fourth- and fifth-grade 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 C-19

20 Grade 2 Mathematics, Quarter 1, Unit 1.5 Identifying and Extending Patterns (5 days) Cumberland, Lincoln, and Woonsocket Public Schools C-20

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