Using Place Value Understanding and Properties of Operations to Solve Addition Word Problems; Building Foundations for Multiplication

Transcription

1 Grade 2 Mathematics, Quarter 2, Unit 2.1 Using Place Value Understanding and Properties of Operations to Solve Addition Word Problems; Building Foundations for Multiplication Overview Number of Instructional Days: 15 (1 day = minutes) Content to be Learned Add within 100 to solve one- and two-step word problems with unknowns in all positions. Solve word problems involving adding to and putting together by using drawings and equations. Use a symbol for the unknown number represented in the problem. Determine whether a group of objects has an odd or even number of members. Write an equation to express an even number as a sum of two equal addends (doubles facts). Skip count by 5s, 10s, and 100s Use place value strategies and properties of operations to add up to four two-digit numbers. Essential Questions How can you show your thinking to prove if a group of objects has an even or an odd number? What strategies would you use to solve this problem? What information from this problem do you need to find the answer? Mathematical Practices to Be Integrated Look for and make use of structure. Recognize if there is a pattern or a structure. Recognize that some problems can be complex. Look for and express regularity in repeated reasoning. Recognize if calculations are repeated. Recognize general methods and shortcuts. Maintain oversight of the process while attending to the details. Evaluate the reasonableness of their results. How can you use a number line to demonstrate skip counting by 5? By 10? By 100? What is the sum of =? (two-digit numbers)? How do you know you are accurate? How does place value help you solve this problem? What other strategies could you use to solve the problem? 13

2 Grade 2 Mathematics, Quarter 2, Unit 2.1 Using Place Value Understanding and Properties of Operations to Solve Addition Word Problems; Building Foundations for Multiplication (15 days) Written Curriculum Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 2.OA Represent and solve problems involving addition and subtraction. 2.OA.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 1 See Glossary, Table 1. Work with equal groups of objects to gain foundations for multiplication. 2.OA.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. Number and Operations in Base Ten 2.NBT Understand place value. 2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s. Use place value understanding and properties of operations to add and subtract. 2.NBT.6 Add up to four two-digit numbers using strategies based on place value and properties of operations. Common Core Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 14

3 Grade 2 Mathematics, Quarter 2, Unit 2.1 Using Place Value Understanding and Properties of Operations to Solve Addition Word Problems; Building Foundations for Multiplication (15 days) 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In grade 1, students added and subtracted within 20 with fluency to 10 to solve one-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions. Students also added three whole numbers whose sum was less than or equal to 20. Students used objects, drawings, and equations with a symbol for the unknown number. Also in grade 1, students used place value understanding and properties of operations to add within 100, a two-digit and one-digit number or a two-digit number and a multiple of ten. Students understood that in adding twodigit numbers, one adds tens and tens, ones and ones and sometimes it s necessary to compose a ten. Current Learning Early in the year, students add within 100 to solve one-step word problems involving adding to and putting together with unknowns in all positions using drawings and equations with a symbol to represent the unknown. Students also add two two-digit numbers based on place value. In this unit, students determine whether a group of objects have an odd or even number of members. Since this is the first time students are working with two-step word problems and odd/even differentiation, these concepts should be taught at the developmental level of instruction. This is also the first time students are exposed to the mathematical language and concept of skip counting. Students are also adding up to four two-digit numbers using strategies based on place value and properties of operations. In this unit, students learn to write equations to express an even number as the sum of two equal addends. Teachers may find it challenging to not teach the algorithm but rather lay the solid foundation for place value. Students may find it challenging to recognize that the answer to the first step in the word problem is necessary to complete the second step. To support students in maintaining an oversight of the process, it may be beneficial to model the use of multiple strategies, such as using successive drawings, or some combination of a diagram and an equation for the second step. Later in the year, students subtract within 100 to solve one and two-step word problems involving situations of taking from, taking apart, and comparing with unknowns in all positions. Students also skip count within 1,000 by fives, tens, and hundreds. Teachers should refer to Table 1 of CCSS (p. 88) for a variety of rich word problem examples. 15

4 Grade 2 Mathematics, Quarter 2, Unit 2.1 Using Place Value Understanding and Properties of Operations to Solve Addition Word Problems; Building Foundations for Multiplication (15 days) Future Learning The following year, students will use place value understanding and properties of operations to perform multidigit arithmetic. They will fluently add and subtract within 1,000 using strategies and algorithms. Students will also use their place value understanding to round whole numbers to the nearest 10 or 100. In grade 3, students will build on their understanding of addition and subtraction to solve two-step word problems using all four operations. In addition, students will apply their understanding of skip counting to identify and explain patterns, and relate skip counting to multiplication strategies. Additional Findings According to Principles and Standards for School Mathematics, when students solve word problems, explaining their pictorial and written solutions helped them articulate their thinking and make it precise (p. 119). The book also states, In developing the meaning of addition and subtraction with whole numbers, students should also encounter the properties of operations, such as the commutativity and the associativity of addition (p. 83). Recognizing the inverse relationship between addition and subtraction can allow students to be flexible in using strategies to solve problems (p. 83). Students need many instructional experiences to develop their understanding of the system, including how numbers are written. They should understand, for example, that multiples of ten provide bridges when counting (e.g., 38, 39, 40, 41) and that ten is a special unit within a base-ten system (p. 81). Most work with two-step word problems should involve single-digit addends as grade 2 students are still developing proficiency. (K 5 Operations and Algebraic Thinking, Learning Progressions, p. 18) 16

5 Grade 2 Mathematics, Quarter 2, Unit 2.2 Develop Understanding of Place Value and Properties of Operations Through Comparison and Written Representation of Numbers Within 1,000 Overview Number of Instructional Days: 15 (1 day = minutes) Content to be Learned Skip-count by 5s, 10s, and 100s within 1,000. Compare two three-digit numbers by using the symbols <, =, > to record the comparisons. Add within 1,000 using concrete models or drawings and strategies based on place value, properties of operations and relate the strategy to a written method. Compose and decompose 10s and 100s to add three-digit numbers. Explain why addition strategies work, using place value and the properties of operations. Mathematical Practices to Be Integrated Look for and make use of structure. Recognize if there is a pattern or a structure. Use the structure of the place value system to make sense of skip counting, comparing, and adding whole numbers. Look for and express regularity in repeated reasoning. Recognize if calculations are repeated. Recognize general methods and shortcuts. Maintain oversight of the process while attending to the details. Evaluate the reasonableness of their results. Essential Questions How can you use a number line to show skip counting by 5s? 10s? 100s? What symbol would you use to compare these two numbers? How do you know that these two numbers are equal? Not equal? Which number is greater? How do you know? How does place value help you compare numbers? What is your strategy for adding these numbers? How is adding two digit-numbers like adding three-digit numbers? How is it different? How do your strategies for adding two-digit numbers work for three-digit numbers? 17

6 Grade 2 Mathematics, Quarter 2, Unit 2.2 Develop Understanding of Place Value and Properties of Operations Through Comparison and Written Representation of Numbers Within 1,000 (15 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations in Base Ten 2.NBT Understand place value. 2.NBT.2 Count within 1000; skip-count by 5s, 10s, and 100s. 2.NBT.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. Use place value understanding and properties of operations to add and subtract. 2.NBT.7 2.NBT.9 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. Explain why addition and subtraction strategies work, using place value and the properties of operations. 3 3 Explanations may be supported by drawings or objects. Common Core Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 18

7 Grade 2 Mathematics, Quarter 2, Unit 2.2 Develop Understanding of Place Value and Properties of Operations Through Comparison and Written Representation of Numbers Within 1,000 (15 days) 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In grade 1, students compared two-digit numbers and recorded their results using <, =, >. They added within 100 using concrete models or drawings and strategies based on place value and the properties of operations. They related the strategy to a written method and explained the strategy used. They understood that when adding two-digit numbers it s sometimes necessary to compose a ten. Also in grade 1, students extended the counting sequence to 120 by starting at any given number less than 120. Current Learning Early in the year, students learn the concept of skip counting by 5s, 10s, and 100s. They add within 100 using two two-digit numbers and strategies of place value. They explain why addition strategies work using place value. Students also compose and decompose tens and hundreds when they add three-digit numbers. In this unit, students skip count by 5s, 10s, and 100s within 1,000. They also compare two three-digit numbers and record their results using <, =, >. They add within 1,000 using concrete models or drawings and strategies of properties of operations. Also in this unit, students explain why addition strategies work, using the properties of operations. Later in the year, students subtract with 1,000 using strategies based on the relationship of addition and subtraction. They subtract three-digit numbers and explain why the subtraction strategies work using place value and properties of operations. Future Learning In grade 3, students will fluently add and subtract within 1,000 using strategies and algorithms. Students in grade 3 round whole numbers to the nearest 10 or 100. They will also use this knowledge to identify patterns and explain them using properties of operations. 19

8 Grade 2 Mathematics, Quarter 2, Unit 2.2 Develop Understanding of Place Value and Properties of Operations Through Comparison and Written Representation of Numbers Within 1,000 (15 days) Additional Findings Principles and Standards for School Mathematics states, Students need many instructional experiences to develop their understanding of the system, including how numbers are written. They should understand, for example, that multiples of ten provide bridges when counting (e.g., 38, 39, 40, 41) and that ten is a special unit within a base-ten system (p. 81). According to Benchmarks for Science Literacy, Although children at this level are not yet comfortable enough with numbers to succeed much in comparing magnitudes, they have knowledge of place value using hundreds, tens, and ones to conceptualize an understanding of which number would be greater than, less than, or equal to the other (p. 277). 20

9 Grade 2 Mathematics, Quarter 2, Unit 2.3 Recognize, Draw, and Partition Shapes Having Specified Attributes Overview Number of Instructional Days: 10 (1 day = minutes) Content to be Learned Recognize that shapes have specified attributes. Draw shapes with specified attributes. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Partition a rectangle into rows and columns of same-sized squares. Count to find the total of same-sized squares in the partitioned shape. Essential Questions What are the defining attributes of a cube? Which shape has only (3) angles? Mathematical Practices to Be Integrated Look for and make use of structure. Recognize the significant lines in geometric figures. Recognize complicated things, such as single objects as being composed of several objects. How do your drawings show shapes with (4) sides? What are their names? How is it possible to partition this rectangle into rows and columns of equal-sized squares? What is the total number of squares? 21

10 Grade 2 Mathematics, Quarter 2, Unit 2.3 Recognize, Draw, and Partition Shapes Having Specified Attributes (10 days) Written Curriculum Common Core State Standards for Mathematical Content Geometry 2.G 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. 5 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes. 5 Sizes are compared directly or visually, not compared by measuring. 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them. Common Core Standards for Mathematical Practice 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning In kindergarten and first grade, students have had multiple experiences with 2-D and 3-D geometry. In first grade, students distinguished between defining attributes and non-defining attributes. They built and drew shapes to possess defining attributes. They partitioned circles and rectangles into two and four equal shares. They understood that decomposing shapes into more equal shares created smaller shares. They have had experience with two-dimensional and three-dimensional shapes. Current Learning In grade 2, students recognize and draw shapes having specified attributes at the reinforcement level of instruction. They identify triangles, quadrilaterals, pentagons, hexagons, and cubes. They compare sizes directly or visually. They do not compare by measuring. At a developmental level, students in grade 2 partition a rectangle into rows and columns of same-sized squares, and they count to find the total amount in each row and column. The standards in this unit are addressed in their entirety. It is applied in future units, but not directly taught. 22

11 Grade 2 Mathematics, Quarter 2, Unit 2.3 Recognize, Draw, and Partition Shapes Having Specified Attributes (10 days) Future Learning In third grade, students will understand that shapes in different categories will share attributes. Students will use their knowledge of partitioning rectangles as a foundation to developing conceptual understanding of area, fractions, and the use of arrays in multiplication. Additional Findings According to Principles and Standards for School Mathematics, students should learn to represent 2- and 3-dimensional shapes with drawings, block constructions, dramatizations, and words. Teachers must help students gradually incorporate conventional terminology into their descriptions of 2- and 3- dimensional shapes. However, terminology itself should not be the focus of the PreK 2 geometry program (p. 97). 23

12 Grade 2 Mathematics, Quarter 2, Unit 2.3 Recognize, Draw, and Partition Shapes Having Specified Attributes (10 days) 24