# Extending Place Value with Whole Numbers to 1,000,000

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1 Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Recognize that a digit in one place represents ten times that of the place to the right. Read and write multidigit numbers to 1,000,000 using numerals, number names, and expanded form. Compare two multidigit whole numbers using <, >, or =. Round multidigit whole numbers to any place. Essential Questions How would you explain the base-ten place value system to another student? What are different ways to represent multidigit whole numbers up to one million? Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Make sense of quantities. Represent numbers using manipulatives and numerals. Provide a context for the number. Look for and make use of structure. Look for patterns. Shift perspective from number form to expanded form. Recognize the structure of the base-ten system. How can you use place value to compare two multidigit whole numbers and why? How is rounding useful in our everyday lives? 1

2 Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 (10 days) Written Curriculum 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, NBT Generalize place value understanding for multi-digit whole numbers. 4.NBT.1 4.NBT.2 4.NBT.3 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. For example, recognize that = 10 by applying concepts of place value and division. Read and write multi-digit whole numbers using base-ten numerals, number names, and expanded form. Compare two multi-digit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round multi-digit whole numbers to any place. Common Core Standards for Mathematical Practice 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. 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. 2

4 Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 (10 days) 4

5 Grade 4 Mathematics, Quarter 1, Unit 1.2 Problem Solving Involving Multidigit Numbers Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Solve multistep whole number addition/ subtraction problems. Apply the perimeter formulas of rectangles in real-world and mathematical situations. Represent problems using equations where a letter stands for an unknown quantity. Mentally compute and estimate answers to assess reasonableness. Use the standard algorithm to add and subtract multidigit whole numbers up to 1,000,000. Essential Questions Why is it important to know the standard algorithm for addition and subtraction? How can you use what you know about numbers to decide on an efficient strategy for adding/subtracting? Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Make sense of quantities and their relationships in problem situations. Create a coherent representation of the problem at hand. Attend to the meaning of quantities. Model with mathematics. Apply the mathematics they know to solve problems arising in everyday life. Routinely interpret their mathematical results in the context of the situation. Reflect on whether results make sense. Attend to precision. Communicate precisely to others. Calculate accurately and efficiently. Give carefully formulated explanations. How would you determine if an answer is reasonable? How is your strategy for solving perimeter connected to a formula? 5

6 Grade 4 Mathematics, Quarter 1, Unit 1.2 Problem Solving Involving Multidigit Numbers (10 days) Written Curriculum Common Core State Standards for Mathematical Content Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor. Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems. 4.OA.3 Solve 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. 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, NBT Use place value understanding and properties of operations to perform multi-digit arithmetic. 4.NBT.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm. Common Core Standards for Mathematical Practice 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. 6

8 Grade 4 Mathematics, Quarter 1, Unit 1.2 Problem Solving Involving Multidigit Numbers (10 days) Future Learning In fifth grade, students extend their place value knowledge from whole numbers to decimals and fractions. They will also continue to solve multistep problems with whole numbers and decimals. Additional Findings According to A Research Companion to Principles and Standards for School Mathematics, current approaches in mathematics focus on understanding, problem solving, and applying knowledge. By teaching with manipulatives and models, we build an understanding of the standard algorithms. In such an approach, conventional algorithms might be the end point of the instructional sequence, but more often a sufficient aim for teachers is to develop semi-informal algorithms. (p. 114) The book also states, one is tempted to teach students these products of the work of the mathematicians of the past in ready-made form, especially if the goal is a more or less routine proficiency in mathematical procedures Teaching student algorithms that they do not understand, however, has limited potential at best and, more important, leads to isolated skills that do not contribute to students general mathematical knowledge. (p. 120) 8

9 Grade 4 Mathematics, Quarter 1, Unit 1.3 Exploring Multiplication Through Problem Solving Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Find factor pairs and multiples in the range Determine prime or composite numbers in the range Interpret multiplication as a comparison. Solve word problems involving multiplicative comparison. Express measurements in a larger unit in terms of a smaller unit. Solve problems involving intervals of time and distance. Represent measurement quantities using diagrams and number lines. Essential Questions What is the relationship between factors and their multiples? How would you identify the multiples of any one-digit number? How do you classify numbers as prime or composite? How is knowing multiples and factors of a number related to conversion of measurement? Mathematical Practices to Be Integrated Make sense of problems and persevere in solving them. Use picture to help conceptualize and solve problems. Make a plan to solve the problem while considering similar problems. Students continually ask themselves, Does this make sense? Reason abstractly and quantitatively. Alternate between abstract and concrete thinking. Make sense of quantities and their relationships. Go beyond just computation, attending to the meaning of quantities. Look for and express regularity in repeated reasoning. Notice if calculations are repeated. Look for shortcuts, explaining their value and efficiency. Continually evaluate the reasonableness of intermediate results. What is the difference between two times a number and two more than a number? What is your strategy for expressing meters to centimeters (e.g., yards to inches, hours to seconds, etc.)? How would you represent measurement quantities using a scaled number-line diagram? 9

10 Grade 4 Mathematics, Quarter 1, Unit 1.3 Exploring Multiplication Through Problem Solving (10 days) Written Curriculum Common Core State Standards for Mathematical Content Operations and Algebraic Thinking 4.OA Use the four operations with whole numbers to solve problems. 4.OA.1 4.OA.2 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 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. 1 1 See Glossary, Table 2. Gain familiarity with factors and multiples. 4.OA.4 Find all factor pairs for a whole number in the range Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range is a multiple of a given one-digit number. Determine whether a given whole number in the range is prime or composite. Measurement and Data 4.MD Solve problems involving measurement and conversion of measurements from a larger unit to a smaller unit. 4.MD.1 4.MD.2 Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. (no strikethrough in highlighted section) Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),... Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including problems involving simple fractions or decimals, and problems that require expressing measurements given in a larger unit in terms of a smaller unit. Represent measurement quantities using diagrams such as number line diagrams that feature a measurement scale. 10

11 Grade 4 Mathematics, Quarter 1, Unit 1.3 Exploring Multiplication Through Problem Solving (10 days) Common Core 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. 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. 11

12 Grade 4 Mathematics, Quarter 1, Unit 1.3 Exploring Multiplication Through Problem Solving (10 days) Clarifying the Standards Prior Learning Students come to fourth grade fluent in all products of two one-digit whole numbers from memory. They use multiplication and division within 100 to solve word problems. They solved problems using all four operations including measurement problems. They studied multiplication in terms of equal groups, arrays, and area. Current Learning Using multiplication with whole numbers to solve problems is a major cluster supported by measurement and conversion of measurement from a larger unit to a smaller unit. This standard appears again in quarter 2 when students multiply multidigit numbers. Students find all the factors pairs and multiples of a numbers in the range of and determine whether it is prime or composite. Students distinguish the difference between multiplicative and additive comparisons (i.e., the difference between two times more and two more ). In future fourth-grade units, students will multiply multidigit numbers up to four digits by a one-digit whole and two two-digit whole numbers. Future Learning In fifth grade, students will develop an understanding of the multiplication of fractions. They will finalize fluency with multidigit multiplication using the standard algorithm. They will also compute products of decimals to the hundredths. Fifth-graders with convert different-size standard measurement units within a given measurement system and use these conversions in solving multi-step real-world problems. Additional Findings As stated in Principles and Standards for School Mathematics, Students who understand the structure of numbers and the relationships among numbers can work with them flexibly (p. 149). Computational fluency should develop in tandem with understanding the role and meaning of arithmetic operation in number systems (Heibert et al., 1997; Thornton 1990, p. 35). On the one hand, computational methods that are over-practiced without understanding are often forgotten or remembered incorrectly (Hiebert 1999; Kamii, Lewis, and Linvington, 1993; Hiebert and Lindquist 1990, p. 35). Part of being able to compute fluency means making smart choices about which tools to use and when (Principles and Standards for School Mathematics, p. 36). 12

13 Grade 4 Mathematics, Quarter 1, Unit 1.4 Comparing Fractions Overview Number of Instructional Days: 10 (1 day = 45 minutes) Content to Be Learned Explain how to generate equivalent fractions by using visual models. Explain how the number and size of parts differ but the two fractions are the same size. Recognize and generate equivalent fractions. Generate equivalent fractions in order to use same numerator or same denominator strategies when comparing fractions with different numerators and denominators. Using symbols (<, >, =) to record comparisons, and use visual models to justify conclusions. Use benchmarks to compare fractions. Recognize that the whole needs to be the same when comparing fractions. Essential Questions How can benchmark numbers be used to compare fractions? What is your strategy for comparing these two fractions? What other strategy can you use to check your reasoning? How are two equivalent fractions the same? How do they differ? Mathematical Practices to Be Integrated Reason abstractly and quantitatively. Make sense of quantities. Move from models to numbers and back again. Model with mathematics. Apply mathematics they know to solve problems arising in everyday life. Identify important quantities in a practical situation and explain their relationships using various tools. Reflect on whether the results make sense. Given two equivalent fractions, what is the relationship between the number and size of the pieces? When comparing fractions, why do both fractions need to refer to the same whole? Why does (doubling, tripling) both the numerator and the denominator of a fraction produce an equivalent fraction? 13

14 Grade 4 Mathematics, Quarter 1, Unit 1.4 Comparing Fractions (10 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations Fractions 3 4.NF 3 Grade 4 expectations in this domain are limited to fractions with denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Extend understanding of fraction equivalence and ordering. 4.NF.1 4.NF.2 Explain why a fraction a/b is equivalent to a fraction (n a)/(n b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Common Core Standards for Mathematical Practice 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. 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 14

16 Grade 4 Mathematics, Quarter 1, Unit 1.4 Comparing Fractions (10 days) 16

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