4th Grade Mathematics - Investigations
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4th Grade Mathematics - Investigations-15875-991235Unit 4: Multi-Digit Number DivisionTeacher Resource Guide2012-2013In Grade 4, instructional time should focus on three critical areas:Developing understanding and fluency with multi-digit multiplication, and developing understanding of dividing to find quotients involving multi-digit dividends; Students apply their understanding of models for multiplication (equal-sized groups, arrays, area models) and models for division, place value, and the distributive property as they discuss and use efficient methods to estimate and compute products and quotients. They develop fluency with efficient procedures for multiplying whole numbers, understand and explain why the procedures work based on place value and properties, and use them to solve problems. Students interpret remainders based upon the context of the problem. Developing an understanding of fraction equivalence, addition and subtraction of fractions with like denominators, and multiplication of fractions by whole numbers;Students recognize that two different fractions can be equal (12/4 = 6/2), and they develop methods for generating and recognizing equivalent fractions. Students extend understandings about how fractions are built from unit fractions (3/4 = ? + ? + ?), and use the meaning of multiplication to multiply a fraction by a whole number.Understanding that geometric figures can be analyzed and classified based on their properties, such as having parallel sides, perpendicular sides, particular angle measures, and symmetry.Students describe, analyze, compare, and classify two-dimensional shapes. Through building, drawing, and analyzing two-dimensional shapes, students understand properties of two-dimensional objects and solve problems involving symmetry. 4th Grade Mathematics 2012-2013 UnitTime FrameTest ByTRIMESTER 11: Multiplication and Division Concepts6 weeks8/27 – 10/5October 52: Multi-Digit Multiplication6 weeks10/8 – 11/16November 16TRIMESTER 23: Measurement/Geometry4 weeks11/19 – 12/21December 214: Multi-Digit Division7 weeks1/2- 2/22February 22TRIMESTER 35: Fractions7 weeks2/25-4/19April 196: Decimal Fractions6 weeks4/22 – 5/29May 29Math Wiki: IdeasEssential QuestionsEstimation is helpful in understanding if an answer is reasonable.Why do we estimate?When you know what the question is asking you know how to use the remainder. Sometimes the remainder is part of the answer, sometimes it changes the answer and sometimes the remainder can be ignored. How does the remainder affect the answer?IdentifierStandardsMathematical PracticesSTANDARDS4.NBT.64.NBT.1Find 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.Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.1) Make sense of problems and persevere in solving them.2) Reason abstractly and quantitatively.3) Construct viable arguments and critique the reasoning of others.4) Model with mathematics.5) Use appropriate tools strategically.6) Attend to precision.7) Look for and make use of structure. 8) Look for and express regularity in repeated reasoning.4.OA.34.OA.2Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.IdentifierStandardsBloom’sSkillsConceptsSTANDARDS4.NBT.64.NBT.1Find 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.Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right.Remember (1)Understand (2)Find (whole number quotients & remainders) Illustrate and explain (the calculation using equations, arrays, or area models)remaindersdividend divisorquotientequationsrectangular arraysarea models4.OA.34.OA.2Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.Apply (3)Understand (2)Evaluate (5)Solve (multi-step word problems w/ whole numbers & interpret remainders)Represent (problems w/ equations using a letter for the unknown quantity)Assess (reasonableness of answer)equationsunknown quantity mental computationestimation roundingInstructional Strategies for ALL StudentsCritical Reading Prior to Instruction:Teaching Student-Centered Mathematics Grades 3-5, Van de Walle & Lovin, Pearson, 2006, p. 121-128 (building resource, SPED)Real-world context – For students to reach the level of rigor intended for the operations of multiplication and division in the Iowa Core, they must develop understanding of the operations within real-world contexts. The tendency in the United States is to have students solve a lot of problems in a single class period. The focus of these lessons seems to be on how to get answers. In Japan, however, a complete lesson will often revolve around one or two problems and the related discussion (Reys & Reys, 1995). A lesson built around word problems focusing on how students solve the problem is ideal. When the teacher focuses the discussion of the word problems on students’ strategies, attention is turned toward the reasoning and knowledge of the number system necessary to solve the problems. Division Problem Types – There are two types of division problems: Measurement and Partitive. In Measurement problems, the group size is known and the number of groups is unknown. For example: If 18 plums are to be packed 6 to a bag (group size known), then how many bags are needed (number of groups unknown)? In Partitive problems, the number of groups is known and the group size is unknown. For example: If 18 plums are shared equally into 6 bags (number of groups known), then how many plums will be in each bag (group size unknown)? Students need to recognize both types of problems as division situations, should be able to model and solve each type of problem, and should know the units of the result: Is it 3 bags or 3 plums per bag? (See p. 7 in Unit One’s guide for further explanation of the problem types.)Use of models to solve division problems – It is essential for students to learn methods for computation that are meaningful. The primary methods for multi-digit division used in 4th grade are the Rectangle Sections Method, the Partial Quotients Method, the Expanded Notations Method, and the Digit-by-Digit Method. In all four methods, students find partial answers that are combined for the final quotient. This unit is the beginning of multi-digit division for students. The focus is one-digit divisors. Work with these models will continue in 5th grade as students work to become more proficient with division up to two-digit divisors. Keep in mind that the Rectangular Sections Method, the Partial Quotients Method and the Expanded Notation Method are based upon place value much like the Area Model for multi-digit multiplication. The Digit-by-Digit Method (traditional algorithm) will be harder for students because it is based on digits instead of place value. The Partial Quotients Method and Digit-by-Digit Methods are efficient ways to divide multi-digit numbers. With the new Iowa Core, 6th graders are expected to be fluent in whole number operations. Therefore, multi-digit multiplication and multi-digit division are critical priorities for 4th and 5th grades.Interpreting a remainder – More often than not, division does not result in a simple whole number. If there is no context given for a division problem, a remainder can only be dealt with in two ways: It can either remain a quantity left over or be partitioned into fractions. In real contexts, remainders can have three additional effects on answers: The remainder is discarded leaving a smaller whole-number answer, or the remainder can “force” the answer to the next highest whole number, or the answer is rounded to the nearest whole number for an approximate result. Students should not be led to think of remainders as “R3” or “left over.” Remainders should be put in context and dealt with accordingly. (See Real-world context above.)Estimation (to assess reasonableness of an answer) – Approximate numbers and rounding are estimation strategies for multi-digit division. In our number system, some numbers are “nice.” They are easy to think about and work with. Multiples of 10, 100, 5, and 25 are often referred to as nice numbers. Numbers such as 100, 700, and 50 are easier to use than 94, 517, and 762. Multiples of 25 are nice because they are easily combined into multiples of 50 and 100. Flexible thought with numbers and many estimation skills are related to the ability to substitute a nice number for one that is not so nice. This substitution can be helpful in mentally computing an estimate for a multi-digit multiplication problem. Students have primarily been taught to round as an estimation strategy. A context for why it might be necessary to round is typically not emphasized. It is ideal for students to know strategies for nice numbers and rounding and have many opportunities to use the strategies in real-world contexts (adapted from Van de Walle, 2006).Routines/Meaningful Distributed PracticeDistributed Practice that is Meaningful and PurposefulPractice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed.Meaningful: Builds on and extends understandingPurposeful: Links to curriculum goals and targets an identified need based on multiple data sourcesDistributed: Consists of short periods of systematic practice distributed over a long period of timeRoutines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum.. The skills presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.SkillStandardResourceA digit in one place represents ten times what it represents in the place to its right. 4.NBT.1Multiplication and Division fact fluency3.OA.3Draw points, lines, line segments, rays, angles, parallel and perpendicular lines(May be taught in whole group lesson and reinforced during MDP time)4.G.1Solve word problems involving distance, intervals of time, liquid volumes, masses of objects and money 4.MD.2Whole number multiplication4.NBT.5Round multi-digit whole numbers to any place4.NBT.3Multi-digit Addition & Subtraction4.NBT.4Other skills students need to develop based on teacher observations and formative assessments.Investigations Resources for Unit 4 - Multi-digit Division Instructional PlanResourceStandardsAddressedWeek 1 and Week 2Unit 3: Multiple Towers and Division StoriesInvestigation 2 DivisionWeek 3 and Week 4Unit 8: How Many Packages? How Many Groups?Investigation 3 Solving Division ProblemsDo 3.5A before 3.5 (CC90-CC94, C61-C64)Week 5 and Week 6Introduce “Using Groups of the Divisors” strategy.Solve CGI problems including problems in which the remainder must be interpreted.Solve multi-step CGI problems.InvestigationsCGI Problems4.NBT.64.OA.3Additional Focus needed on:Assess the reasonableness of answersSolve equations with variables for the unknown quantityEstimate before solving and explain reasoningCGI ProblemsThese lessons are intended to provide additional resources based on student need.Lesson ProgressionLesson FocusLessons are listed in intended orderTime FrameResourceTeacher DirectionsStandards AddressedMulti-digit subtractionThroughout the unit (as warm up or MDP for whole group review) Wiki: Strategies for multi-digit subtraction These strategies may be reviewed during warm up or MDP. They may also be taught during a small group lesson. 4.NBT.4Multi-step word problems Throughout the unit(a few problem per week)Wiki: Multi-step word problems: without multi-digit divisionTeachers may need to write some problems as well. Teacher-made problems can be shared via a Discussion on the Wiki. Include only addition, subtraction, multiplication and simple division (division facts). Include multi-digit division after the alternative algorithms are taught.4.OA.3Dividing multiples of 103-5 daysThen continue throughout the unit as warm up or MDPWiki: Tack on Trailing Zeros (multiplication) Lesson 22Understanding how to multiply and divide with multiples of 10 is critical when using partial quotients to solve multi-digit division problems. 4.NBT.1Wiki: Tack on Trailing Zeros (multiplication) Lesson 23Wiki: Tack on Trailing Zeros (multiplication) Lesson 24Wiki: Divide by MultiplyingWiki: Tack on Trailing ZerosWiki: Canceling Common ZerosWiki: Breaking up the DividendWiki: Think MoneyLesson ProgressionLesson FocusLessons are listed inintended orderResourceTeacher DirectionsStandards AddressedPartial Quotients Wiki: Partial Quotients Examples Wiki: Division Problem BankTeacher made problemsTeacher-made problems can be shared via a Discussion on the Wiki.Explanation of Partial Quotients Method4.NBT.6Present word problems and allow students to solve them using the Partial Quotients Method. RemaindersWiki: Interpreting the Remainder Problem BankStudents learn what to do with the remainder by solving word problems. Read the paragraph about remainders in the Instructional Strategies.4.NBT.6Multi-digit division word problems Wiki: Division Problem BankTeacher made problemsTeacher-made problems can be shared via a Discussion on the Wiki.Present lots of problems in context. Include multiplication and division. Students use an algorithm/method of choice4.NBT.64.OA.24.NBT.5Multi-step word problems Wiki: Multi-step word problemsTeachers may need to write some problems as well. Teacher-made problems can be shared via a Discussion on the Wiki.Present single and multi-step word problems daily. Include all four operations and interpreting remainders. Students use an algorithm/method of choice. Whole group discussions on the various ways students solved problems is critical.4.OA.3Reasonableness of AnswerThroughout the unitIt is important to discuss the reasonableness of answers. Read the Instructional Strategies for more details. 4.OA.3 ................
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