Dividing Whole Numbers With Remainders Overview Number of instruction days: 7 9 (1 day = 90 minutes) Content to Be Learned Solve for whole-number quotients with remainders of up to four-digit dividends and one-digit divisors. Represent and explain calculations using equations, rectangular arrays, and/or area models. Solve problems using the relationship between multiplication and division. Use strategies based on place value and the properties of operations to solve division situations. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Consider related problems and try simpler forms of the original problem in order to gain insight into the solution. Interpret the remainder and its relationship to the quotient. Check for reasonableness of the quotient and ask if it makes sense. 2 Reason abstractly and quantitatively. Make sense of the remainder and its relationship in the problem situation. Use multiple strategies to find the quotient. Essential Questions What is a remainder? Why is making sense of the remainder important? What are some strategies that can be used in division? How are multiplication and division related? How can you represent and explain the strategy you used to solve the problem? Providence Public Schools D-93
Dividing Whole Numbers With Remainders (7 9 days) Standards Common Core State Standards for Mathematical Content Number and Operations in Base Ten 2 2 Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000. 4.NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 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-94 Providence Public Schools
Dividing Whole Numbers With Remainders (7 9 days) Grade 4 Mathematics, Quarter 4, Unit 4.1 Clarifying the Standards Prior Learning In Grade 3, division is a major cluster according to the PARCC frameworks. Students learned to fluently multiply and divide within 100 using strategies such as the relationship between multiplication and division. Students interpreted whole-number quotients of whole numbers. They used multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities. They also determined an unknown whole number in a multiplication or division equation relating three whole numbers. Current Learning In Grade 4, using the four operations to solve problems is a major cluster according to the PARCC framework. In this unit, students extend their understanding of division to four-digit dividends and onedigit divisors. Remainders are introduced for the first time. Students recognize and apply the relationship between multiplication and division to solve problems. They also use strategies based on place value and the properties of operations to solve problems. They illustrate and explain their calculations using equations, rectangular arrays, and/or area models. Future Learning Division is a major cluster in Grade 5 according to the PARCC framework. Students will divide multidigit whole numbers with up to four-digit dividends and two-digit divisors. They will also divide decimals to the hundredths and relate their strategy to a written method. In sixth grade students will be expected to divide fluently. Additional Findings In grades 3 5, students should focus on the meanings of, and relationship between, multiplication and division. It is important that students understand what each number in a multiplication or division expression represents. Students in these grades will also encounter situations where the result of division includes a remainder. They should learn the meaning of a remainder by modeling division problems and exploring the size of remainders given a particular divisor. (Principles and Standards for School Mathematics, p. 151) To compute and interpret remainders in word problems, students must reason abstractly and quantitatively, make sense of problems, and look for and express regularity in repeated reasoning as they search for structure in problems with similar interpretations of remainders. (PARCC Model Content Frameworks, p. 21). 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. Providence Public Schools D-95
Dividing Whole Numbers With Remainders (7 9 days) 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. Solve division situations using strategies based on place value, the properties of operations and relationship between multiplication and division. Solve for whole number quotients and remainders with up to four-digit dividends and one-digit divisors. Represent and explain calculations using equations, rectangular arrays, and/or area models. Solve problems using the relationship between multiplication and division. Learning Objectives Students will be able to: Instruction Use strategies based on place value and the properties of operations to solve division situations. Use strategies based on the relationship between multiplication and division to solve division situations. Solve for whole number quotients and remainders with up to four-digit dividends and one-digit divisors. Represent and explain calculations using equations, rectangular arrays, and/or area models. Solve problems using the relationship between multiplication and division. Use pictures and equations to represent remainders in a problem. Demonstrate understanding of the concepts and skills in this unit. Resources envision Math Grade 4, Pearson Education, Inc., 2009 Topic 8, Dividing by 1-Digit Divisors, Teacher Edition Also see Section I, Supplemental Materials Lesson 8.3A, Estimating Quotients for Greater Dividends Lesson 8.3B, Using Objects to Divide: Division as Repeated Subtraction Lesson 8.3C, Division as Repeated Subtraction Lesson 8.8A, Dividing 4-Digit by 1-Digit Numbers Investigations in Numbers, Data, and Space, Grade 4, Pearson Education, Inc., 2008 Implementing Investigations in Grade 4- Implementation Guide Unit 3; Multiple Towers and Division Stories, Teacher Edition Investigation 2: Division D-96 Providence Public Schools
Dividing Whole Numbers With Remainders (7 9 days) Grade 4 Mathematics, Quarter 4, Unit 4.1 Unit 8; How Many Packages? How Many Groups?, Teacher Edition Investigation 3: Solving Division Problems Also see Section I, Supplemental Materials Lesson 3.5A, Dividing 4-Digit Numbers Teacher Resources Binder 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 ample resources to refer to when planning your unit and individual lessons. Materials Number cubes, centimeter grid paper or colored tiles Instructional Considerations Key Vocabulary divisibility dividend divisor quotient remainder Planning for Effective Instructional Design and Delivery In this unit, students build on their third grade work with division within 100. Students will explore division further this year through various strategies. They use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. They illustrate and explain their calculations using equations and visual models. Students need opportunities to develop their understandings by using problems in and out of context. In Lesson 8-5 (TE, page 177), use identifying similarities and differences to review and practice concepts. Students will compare factors and multiples using a Venn diagram. 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. Providence Public Schools D-97
Dividing Whole Numbers With Remainders (7 9 days) Notes D-98 Providence Public Schools