Overview. Essential Questions. Grade 5 Mathematics, Quarter 3, Unit 3.2 Multiplying and Dividing With Decimals


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1 Multiplying and Dividing With Decimals Overview Number of instruction days: 9 11 (1 day = 90 minutes) Content to Be Learned Multiply decimals to hundredths. Divide decimals to hundredths. Use models, drawings, place value strategies, properties of operations, and/or the relationship between multiplication and division to multiply and divide decimals to hundredths. Use place value understanding to reason about and explain patterns in the placement of the decimal point in the product or quotient when multiplying or dividing decimals. Identify and describe patterns when multiplying decimals by a power of 10. Use wholenumber exponents to indicate powers of 10. Solve multistep, realworld problems that require converting like measurement units within the same measurement system. Mathematical Practices to Be Integrated 1 Make sense of problems and persevere in solving them. Use knowledge of place value to solve problems and perform operations with decimals. 3 Construct viable arguments and critique the reasoning of others. Identify patterns involved in using exponents. Explain calculations based on models and properties of operations. Evaluate calculations for reasonableness. 8 Look for and express regularity in repeated reasoning. Explain patterns in the number of zeros of the product when multiplying a number by a power of 10. Look for and express regularity in repeated reasoning. Essential Questions How can you multiply decimals to hundredths (multiply tenths by tenths and tenths by hundredths)? How can you divide decimals to hundredths? How is multiplying and dividing decimals similar to multiplying and dividing whole numbers? How is it different? What are some different ways you can explain your solution path using visual models? How can you explain patterns in the placement of the decimal point when multiplying or dividing decimals? How can you describe patterns when multiplying decimals by a power of 10? How can you convert like measurement units within the same measurement system when solving multistep, realworld problems? Providence Public Schools D61
2 Multiplying and Dividing With Decimals (9 11 days) Standards Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT Understand the place value system. 5.NBT.2 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use wholenumber exponents to denote powers of 10. Perform operations with multidigit 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. Measurement and Data 5.MD Convert like measurement units within a given measurement system. 5.MD.1 Convert among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multistep, real world problems. 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. 3 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them D62 Providence Public Schools
3 Multiplying and Dividing With Decimals (9 11 days) Grade 5 Mathematics, Quarter 3, Unit 3.2 into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 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 4, students solved problems involving measurement and conversion of measurements from a larger unit to a smaller unit. They read and wrote multidigit whole numbers using baseten numerals, number names, and expanded form. They used place value understanding to round multidigit whole numbers to any place. They compared decimals to the hundredths using comparison symbols. Current Learning In Grade 5, multiplying and dividing with decimals is a major cluster. Students understand why multiplying by a power of 10 shifts the digits of a whole number or decimal that many places to the left. Students perform all four operations on decimals to the hundredths. They use concrete models or drawings and strategies based on place value, properties of operations, the relationship between addition and subtraction, and/or the relationship between multiplication and division. In Grade 5, students convert like measurement units within a given measurement system to support computation with decimals. They also solve multistep, realworld problems involving conversions with differentsized standard units. Future Learning In Grade 6, students will extend their fluency with the standard algorithms using all four operations with decimals. They will extend the baseten number system to negative numbers. Seventhgrade students will, by reasoning about the standard division algorithm, learn that every fraction can be represented with a decimal that either repeats or terminates. Providence Public Schools D63
4 Multiplying and Dividing With Decimals (9 11 days) Additional Findings Students should focus on the meanings of and the relationship between multiplication and division. It is important that students understand what each number in a multiplication or division expression represents. For example, in multiplication, unlike addition, the factors in the problem can refer to different units. (Principles and Standards for School Mathematics, p. 151) Students should continue to refine their understandings of addition, subtraction, multiplication and division as they use these operations with fractions.... Teachers need to be attentive to conceptual obstacles that many students encounter as they make the transition from operations with whole numbers. (Principles and Standards for School Mathematics, p. 218) Assessment When constructing an endofunit 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. Standardsbased 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 performancebased 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. Multiply and divide decimals to hundredths using models, drawings, place value strategies, properties of operations, and/or the relationship between multiplication and division. Explain patterns in the placement of the decimal point in the product or quotient when multiplying or dividing decimals. Multiply decimals to hundredths. Divide decimals to hundredths. Describe patterns when multiplying decimals by a power of 10. Use wholenumber exponents to indicate powers of 10. Solve multistep, realworld problems that require converting like measurement units within the same measurement system. D64 Providence Public Schools
5 Multiplying and Dividing With Decimals (9 11 days) Grade 5 Mathematics, Quarter 3, Unit 3.2 Learning Objectives Students will be able to: Instruction Use models, drawings, place value strategies, properties of operations, and/or the relationship between multiplication and division to multiply and divide decimals to hundredths. Relate the strategies to the written method and explain the reasoning. Use place value understanding to reason about and explain patterns in the placement of the decimal point in the product or quotient when multiplying or dividing decimals. Multiply decimals to hundredths. Divide decimals to hundredths. Identify and describe patterns when multiplying decimals by a power of 10. Use wholenumber exponents to indicate powers of 10. Solve multistep, realworld problems that require converting like measurement units within the same measurement system. Demonstrate understanding of the concepts and skills in this unit. Resources envision Math Grade 5, Pearson Education, Inc., 2009 Topic 7, Multiplying and Dividing Decimals, Teacher Edition Also see Section I, Supplemental Materials Lesson 7.4A, Number Sense: Decimal Multiplication Lesson 7.4B, Models for Multiplying Decimals Lesson 7.6A, Number Sense: Decimal Division 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 Also see Section I, Supplemental Materials Investigation 3A: Adding It Up Investigation 3A: More Decimal Operations Lesson 3A.1, Multiplying Powers of 10 Lesson 3A.2, Multiplying by Small Numbers Providence Public Schools D65
6 Multiplying and Dividing With Decimals (9 11 days) Lesson 3A.3, Multiplying Decimals Lesson 3A.4, Multiplying Decimals, continued Lesson 3A.5, Dividing Powers of 10 Lesson 3A.6, Dividing Decimals Lesson 3A.7, Assessment: Multiplying and Dividing Decimals Lesson 3A.8, Converting Measurements Lesson 3A.9, Converting Measurements, continued Teacher Resources Binder Pearson Online Success Net: Implementing Investigations Site: 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 Money, decimal grids, color pencils, base ten blocks Instructional Considerations Key Vocabulary tenths hundredths divisor dividend quotient product meter centimeter kilometer Planning for Effective Instructional Design and Delivery In this unit, students use many strategies to multiply and divide decimals to hundredths. They also relate the strategies to the written method and explain their reasoning. Included in this is multiplying by multiples of 10 and powers of 10, including 10² which is = 100, and 10³ which is = 1,000. Students should have experiences working with connecting the pattern of the number of zeros in the product when you multiply by powers of 10. Example: ³ = 4.5 ( ) = 4.5 1,000 = 4,500 Students should reason that the exponent above the 10 indicates how many places the decimal point is moving (not just that the decimal point is moving but that you are multiplying or making the number 10 times greater three times) when you multiply by a power of 10. Since we are multiplying by a power of 10 the decimal point moves to the right. Students also need to see examples of division with powers of ten and recognize the pattern that occurs is moving the decimal point to the left. This concept should be explored by the students, not just taught procedurally. D66 Providence Public Schools
7 Multiplying and Dividing With Decimals (9 11 days) Grade 5 Mathematics, Quarter 3, Unit 3.2 Understanding the patterns involved in multiplying and dividing by powers of ten will be helpful when students are converting among differentsized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m). Performing these conversions will reinforce the concept. The work in this unit should focus on concrete models and pictorial representations, rather than relying solely on the algorithm. This is an extension of the strategies that students have used in earlier grades with whole numbers. Students record the answers to computations using equations ( = 6.75), but this work should not be done without models or pictures. This standard includes students reasoning and explanations of how they use models, pictures, and strategies. 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 D67
8 Multiplying and Dividing With Decimals (9 11 days) Notes D68 Providence Public Schools
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