Adding and Subtracting Decimals Overview Number of instruction days: 12 14 (1 day = 90 minutes) Content to Be Learned Add and subtract decimals to the hundredths. Use concrete models, drawings, and strategies based on place value and the properties of operations to calculate solutions. Relate strategies used to add and subtract decimals to a written method. Reason and explain how models, pictures, and strategies are used to solve problems involving addition and subtraction of decimals to the hundredths. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Students solve problems involving decimals by applying their understanding of operations on whole numbers and fractions. Students seek the meaning of their answer and check their thinking by asking themselves, Does this make sense? 2 Reason abstractly and quantitatively. Recognize that a number represents a specific quantity. Derive a solution to a problem without performing the computation. Extend understanding from whole numbers to decimals. Extend understanding of addition and subtraction of fractions to add and subtract decimals. Estimate the sums and differences of whole numbers and decimals to the hundredths to reason abstractly and quantitatively. Essential Questions How can you add and subtract decimals to the hundredths? How can you use concrete models, drawings, and strategies based on place value and the properties of operations to calculate solutions? How can you justify the strategies you used to add and subtract decimals? How can you use models and pictures to solve problems involving addition and subtraction of decimals to the hundredths? How can understanding the relationship between addition and subtraction help you add and subtract decimals? Providence Public Schools D-53
Adding and Subtracting Decimals (12 14 days) Standards Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT Perform operations with multi-digit whole numbers and with decimals to hundredths. 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, 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 and explain the reasoning used. Common Core State Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. D-54 Providence Public Schools
Adding and Subtracting Decimals (12 14 days) Grade 5 Mathematics, Quarter 3, Unit 3.1 Clarifying the Standards Prior Learning In Grade 4, students were introduced to decimals to the hundredths place, and they compared these decimals. They used decimal notation for fractions with denominators of 10 or 100. They became fluent in addition and subtraction of whole numbers. Additionally, they used place value understanding to round multidigit whole numbers to any place. Current Learning The content in this unit is a major emphasis in Grade 5. They add and subtract decimals to the hundredths. They use place value strategies, properties of operations, and relationships between addition and subtraction in their solution paths. This work should include heavy use of concrete models and pictorial representations rather than relying solely on use of the algorithm. The standards covered in this unit include students ability to reason and explain how they use models, pictures, and strategies. Students can apply the reasoning they have developed to understand addition and subtraction of fractions to make sense of addition and subtraction of decimals. Future Learning Students will extend their understanding of numbers and the ordering of numbers to the full system of rational numbers in Grade 6. Grade 6 work will include negative rational numbers. Sixth-grade students will become fluent in adding, subtracting, multiplying, and dividing multidigit decimals using the standard algorithm for each operation. They will use decimal operations as they continue to solve using problems using numerical and algebraic expressions and equations. Additional Findings Research on symbolic learning suggests that manipulatives or other physical models used in teaching must, in order to be helpful, be represented by a learner both as the objects that they are and as symbols that stand for something else. The physical characteristics of these materials can be initially distracting to children, and it takes time for them to develop mathematical meaning for any physical model and to use it effectively. (Adding It Up, p. 198) An understanding of the base-ten number system should be extended through continued work with larger numbers as well as with decimals. (Principals and Standards for School Mathematics, p. 149) Providence Public Schools D-55
Adding and Subtracting Decimals (12 14 days) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Apply the use of concrete models, drawings, strategies based on place value and the properties of operations to add and subtract decimals to the hundredths. Apply the strategies used to add and subtract decimals to a written method. Reason and explain how models, pictures and strategies are used to solve problems involving addition and subtraction of decimals to the hundredths. Learning Objectives Instruction Students will be able to: Use concrete models, drawings, strategies based on place value and the properties of operations to calculate addition solutions. Use concrete models, drawings, strategies based on place value and the properties of operations to calculate subtraction solutions. Relate strategies used to add and subtract decimals to a written method. Explore with models, pictures and strategies to solve addition and subtraction of decimals. Reason and explain how models, pictures and strategies are used to solve problems involving addition and subtraction of decimals to the hundredths. Demonstrate understanding of the concepts and skills in this unit. D-56 Providence Public Schools
Adding and Subtracting Decimals (12 14 days) Grade 5 Mathematics, Quarter 3, Unit 3.1 Resources envision Math Grade 5, Pearson Education, Inc., 2009 Topic 2, Adding and Subtracting Whole Numbers and Decimals, Teacher Edition Section I, Supplemental Resources: Lesson 2-6A, Modeling Addition and Subtraction of Decimals Teacher Resource Masters Student Edition Investigations in Numbers, Data, and Space, Grade 5, Pearson Education, Inc., 2008 Implementing Investigations in Grade 5- Implementation Guide Unit 6, Decimals on Grids and Number Lines, Teacher Edition Investigation 2: Adding Decimals Section I, Supplemental Resources: Lesson 2.5A, Decimal Subtraction Problems Teacher Resources Binder Connected Mathematics 2, Pearson/Prentice Hall/2008, Bits and Pieces III Section I, Supplemental Resources: Investigation 1- Decimals More or Less! Pearson Online Success Net: www.pearsonsuccessnet.com/snpapp/login/login.jsp Implementing Investigations Site: http://investigations.terc.edu Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials Base ten blocks, money Instructional Considerations Key Vocabulary rounding compatible numbers compensation Planning for Effective Instructional Design and Delivery In this unit, students work with concrete models and pictorial representations to add and subtract of decimals. They will also use the algorithm, but standard (5.NBT.7), includes students reasoning and explanation of how they used models, pictures and strategies. Providence Public Schools D-57
Adding and Subtracting Decimals (12 14 days) During Unit planning, refer to Investigations, Unit 6, Teacher Note, Adding Decimals, p. 132-137 and the Dialog Box on p. 142-146. These pages offer you examples and clarification to support instruction throughout this Unit. When working with decimals, students should estimate answers for reasonableness based on their understanding of operations and the value of the numbers before calculating the exact answer. Students have rounded decimals in Unit 1.2 and now apply this knowledge to the work in this Unit. Students could use a number line as a visual model to round decimals. An explanation of this is found on p.22b in the envisions, Topic 2 TE called Rounding Decimals. To support students conceptual understanding of adding decimals, use the activity in Investigations, Unit 6, p. 87, called Fill Two. Then Close to 1 on p. 109 can be used to strengthen their knowledge of decimals. Just as with whole numbers, students should be able to express that when adding decimals, they add tenths to tenths and hundredths to hundredths. When adding or subtracting with decimals, it is important that they align the corresponding places correctly. This understanding can be reinforced by connecting addition of decimals to their understanding of addition of fractions with denominators of 10 and 100 taught in the fourth grade. Incorporate Ten Minute Math Activities, the Problem of the Day, and the Daily Spiral Review that are aligned to The Common Core State Standards for Mathematics. EnVision Center Activities and Investigations Activities offer additional practice for student learning and support small group differentiated instruction. Use teacher created common tasks as formative assessments to monitor student progress and understanding of critical content and essential questions. Use data from formal and informal assessments to guide your instruction and planning. For planning considerations, read through the teacher editions for suggestions about scaffolding techniques, using additional examples, and differentiated instruction as suggested by the envision and Investigations resources, particularly the Algebra Connections and Teacher Notes section. D-58 Providence Public Schools
Adding and Subtracting Decimals (12 14 days) Grade 5 Mathematics, Quarter 3, Unit 3.1 Notes Providence Public Schools D-59
Adding and Subtracting Decimals (12 14 days) D-60 Providence Public Schools