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


 Carol Holt
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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 baseten 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 baseten 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 multidigit whole numbers. 4.NBT.1 4.NBT.2 4.NBT.3 Recognize that in a multidigit 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 multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit 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 multidigit 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
3 Grade 4 Mathematics, Quarter 1, Unit 1.1 Extending Place Value with Whole Numbers to 1,000,000 (10 days) Clarifying the Standards Prior Learning In grade 1, students began to use the symbols <, >, and = as they compared twodigit numbers. In grade 2, students used expanded notation to read and write numbers to 1,000. In grade 3, students rounded numbers to 10 and 100, and multiplied by multiples of 10. Current Learning This is a critical area of study. By the end of this unit students must have a solid understanding of place value of whole numbers to 1,000,000. They generalize this knowledge through reading, writing, comparing, expanding, and rounding whole numbers to 1,000,000. For numbers over 1,000, start with the developmental level of instruction and move towards drillandpractice by the end of this unit. In subsequent fourthgrade units, students will use their understanding of place value to solve problems using the four operations. Future Learning In fifth grade, students will generalize this knowledge to decimals and exponents. In middle school, they will use their understanding of place value to work with rational and irrational numbers. Additional Findings According to Principles and Standards for School Mathematics, what can be most challenging for students is to develop strategies for judging the relative sizes of numbers. They should understand more deeply the multiplicative nature of a number system, including the structure of 786 as 7 x 100, plus 8 x 10, plus 6 x1 (p. 149). According to Progressions for the Common Core State Standards in Mathematics, K 5, Number and Operations in Base Ten, to read numbers between 1,000 and 1,000,000 students need to understand the role of commas. Each sequence of three digits made by commas is read as hundreds, tens, and ones, followed by the name of the appropriate basethousand unit (thousand, million, billion, trillion, etc.) (p. 12). According to A Research Companion to Principles and Standards for School Mathematics, the written place value system is very abstract and can be misleading because the digits in every place look the same. To understand the meaning of the digits in various places, children need to experience with some kind of sizequantity supports (e.g., objects or drawings) that shows tens to be a collection of 10 ones and show hundreds to be simultaneously 10 tens and 100 ones, and so on (p. 78). 3
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 realworld 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 wholenumber 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 multidigit arithmetic. 4.NBT.4 Fluently add and subtract multidigit 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
7 Grade 4 Mathematics, Quarter 1, Unit 1.2 Problem Solving Involving Multidigit Numbers (10 days) 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, twoway 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. 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. Clarifying the Standards Prior Learning In third grade, students solved problems involving perimeter of polygons, given known and unknown side lengths. They started using a variable to represent an unknown quantity. Students fluently added and subtracted within 1,000 using strategies and a range of algorithms. They solved one and twostep problems. Current Learning Through the use of multistep word problems with whole numbers, students apply perimeter formulas and addition/subtraction algorithms. This is the first unit where students are introduced to the standard algorithm for addition/subtraction. It is important to connect the standard algorithm to previously learned strategies and choose the most efficient method based on the problem. Students solve multistep word problems using whole numbers and letters for unknown quantities that result in whole numbers. The concept of perimeter is mastered. Students are fluent using place value strategies including the standard algorithm in addition and subtraction by the end of fourth grade. Therefore, it is essential to continue to practice this skill using wholedigit numbers throughout the year. 7
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 semiinformal 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 readymade 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 onedigit 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 numberline 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 onedigit 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 twocolumn 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 onedigit 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 fourthgrade units, students will multiply multidigit numbers up to four digits by a onedigit whole and two twodigit 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. Fifthgraders with convert differentsize standard measurement units within a given measurement system and use these conversions in solving multistep realworld 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 overpracticed 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, twoway 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
15 Grade 4 Mathematics, Quarter 1, Unit 1.4 Comparing Fractions (10 days) Clarifying the Standards Prior Learning In first and second grade, students developed understanding of fraction concepts as they described equal shares using shapes as halves, thirds, and fourths. In third grade, students developed understanding of fractions as numbers. Students used number lines and visual models to understand and compare fractions with like numerators or denominators. They learned that equivalent means the same size. Note that, in grade 3, fractions were limited to denominators of 2, 3, 4, 6, and 8. Current Learning This is a critical focus in fourth grade. By using visual models, students extend their understanding of equivalent fractions. They are able to recognize and generate equivalent fractions. Students are able to explain why two fractions are equivalent, and why the number and size of parts differ, but the fractions are the same size. Student also compare two fractions with unlike numerators and unlike denominators by using different methods such as visual models, creating common numerators or denominators or comparing them to benchmarks (0, 1/2, 1) Students also record these comparisons using the symbols (<, >, =). In later units, students will continue to extend their understanding of fractions by adding and subtracting fractions with like denominators, and multiply a fraction by a whole number. Note that fractions are limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100 for this grade level. Future Learning Fifthgrade students will build on their understanding of fractions to add and subtract fractions with unlike denominators, multiply fraction by a fraction and divide unit fractions by whole numbers and whole numbers by unit fractions. Sixthgrade students will extend their understanding of multiplication and division to include operations with all fraction quantities. Additional Findings According to Principles and Standards for School Mathematics, During grades 3 5, students should build their understanding of fractions as parts of a whole and as division. They will need to see and explore a variety of models of fractions, focusing primarily on familiar fractions such as halves, thirds, fourths, fifths, sixths, eights, and tenths. By using an area model in which parts of a region are shaded, students can see how fractions are related to a unit whole, compare fractional parts of a whole, and find equivalent fractions. They should develop strategies for ordering and comparing fractions, often using benchmarks such as 1/2 and 1 (p. 150). 15
16 Grade 4 Mathematics, Quarter 1, Unit 1.4 Comparing Fractions (10 days) 16
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