# Understanding Place Value of Whole Numbers and Decimals Including Rounding

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1 Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = minutes) Content to be learned Understand how the value of a digit in one place compares to the value of a digit to its right or left. Explain the difference between the value of two digits through thousandths in a multidigit number. Explain patterns in the number of zeros in a product when multiplying a number by powers of ten. Explain patterns in the placements of a decimal point when a decimal is multiplied by a power of ten. Explain patterns in the placements of a decimal point when a decimal is divided by a power of ten. Use whole number exponents to express powers of ten. Mathematical practices to be integrated Attend to precision. Use clear, precise language, explanations, definitions, and vocabulary. Explain the meaning of symbols. Calculate accurately. Be able to explain understanding to others. Look for and make use of structure. Look for patterns to simplify. Break apart numbers. Look for and express regularity in repeated reasoning. Look for repeated calculations/patterns. Accurate calculations. Essential questions How could you to write a multiple of 10 as a power of 10 using exponents? What can I learn about patterns by looking at the placement of the decimal point after multiplying or dividing by a power of ten? What strategies can I use to round decimals to any place? How do you represent decimals in expanded notation and how is this related to place value? How can you use place value to round any decimal to any place value? 1

2 Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding (14 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT Understand the place value system. 5.NBT.1 5.NBT.2 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left. 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 whole-number exponents to denote powers of 10. Common Core Standards for Mathematical Practice 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. 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 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding (14 days) 4

5 Grade 5 Mathematics, Quarter 1, Unit 1.2 Comparing, Ordering, and Rounding Decimals Overview Number of instructional days: 6 (1 day = minutes) Content to be learned Read decimals to thousandths place value in base-ten numeral (standard), number names or expanded form. Write decimals to thousandths place value in base-ten numeral (standard), number names, or expanded form. Compare decimals to the thousandths using <, >, and =. Use place value understandings to round decimals to any place. Essential questions What do the symbols <, =, and > mean and how do you determine which symbol to use to record your comparison? How would you compare and order whole numbers and decimals through the thousandths? How are decimals and fractions alike? How are they different? Mathematical practices to be integrated Attend to precision. Calculate accurately by rechecking for precision. Define mathematical symbols consistently and appropriately. Look for and make use of structure. Look for patterns to simplify. Break numbers apart. Use a variety of strategies and properties to verify answers. Look for and express regularity in repeated reasoning. Look for repeated calculations and patterns. Make accurate calculations. How can decimals be rounded? How do I explain the meaning of a fraction and its numerator and denominator in relationship to decimals? 5

6 Grade 5 Mathematics, Quarter 1, Unit 1.2 Comparing, Ordering, and Rounding Decimals (6 days) Written Curriculum Common Core State Standards for Mathematical Content Number and Operations in Base Ten 5.NBT Understand the place value system. 5.NBT.3 5.NBT.4 Read, write, and compare decimals to thousandths. a. Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., = (1/10) + 9 (1/100) + 2 (1/1000). b. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. Use place value understanding to round decimals to any place. Common Core Standards for Mathematical Practice 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. 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. 6

8 Grade 5 Mathematics, Quarter 1, Unit 1.2 Comparing, Ordering, and Rounding Decimals (6 days) can learn common fraction-decimal equivalents. They can also learn that some fractions can be expressed as terminating decimals but others cannot (p. 150). 8

9 Grade 5 Mathematics, Quarter 1, Unit 1.3 Addition and Multiplication of Whole Numbers and Decimals Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Use standard algorithm to fluently multiply multidigit whole numbers. Connect the addition strategies to a written method. Connect the multiplication strategies to a written method. Use concrete models or drawings and strategies based on place value to add and multiply decimals to hundredths. Explain the reasoning used and relate the strategy to a written method when adding and multiplying decimals to hundredths. Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Prove two ways. State justification. Model with mathematics. Simplify the situation to a simpler rule allowing for revisions. Draw conclusions, interpret results, revise models if needed. Attend to precision Label for clarification. Calculate accurately by rechecking for precision. Communicate formulated explanations with precision. Essential questions Use standard algorithm to fluently multiply multi-digit whole numbers. Use concrete models or drawings and strategies based on place value to add and multiply decimals to hundredths. Explain the reasoning used and relate the strategy to a written method when adding and multiplying decimals to hundredths. 9

12 Grade 5 Mathematics, Quarter 1, Unit 1.3 Addition and Multiplication of Whole Numbers and Decimals (10 days) By the end of this grade band, students should be computing fluently with whole numbers The computational method that the student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication, and number relationships. (p. 152) Research suggests that by solving problems that require calculation, students develop methods for computing and also learn more about operations and properties. As students develop methods to solve multidigit computation problems, they should be encouraged to record and share their methods. As they do so, they can learn from one another, analyze the efficiency and generalizability of various approaches, and try one another s methods. (p. 153) When students leave grade 5, they should be able to solve problems involving whole number computation and should recognize that each operation will help them solve many different types of problems With these understandings and skills, they should be able to develop strategies for computing with decimals. (p. 149) 12

13 Grade 5 Mathematics, Quarter 1, Unit 1.4 Subtraction and Division of Whole Numbers and Decimals Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Find whole number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain how to find quotients using equations. Relate the subtraction strategy to a written method and explain the reasoning used. Relate the division strategy to a written method and explain the reasoning used. Subtract 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. Essential questions How can I use place value to decompose numbers to find quotients or differences? How can models help us understand the subtraction and division of decimals? How do I recognize what strategy to use for a specific problem? Mathematical practices to be integrated Construct viable arguments and critique the reasoning of others. Prove two ways. State justification. Model with mathematics. Simplify the situation to a simpler rule allowing for revisions. Draw conclusions, interpret results, revise models if needed. Attend to precision. Label for clarification. Calculate accurately by rechecking for precision. Communicate formulated explanations with precision. What is expanded notation and how can I use it to assist in finding the difference or quotient of multi-digit whole numbers? How are repeated subtraction and division related? 13

16 Grade 5 Mathematics, Quarter 1, Unit 1.4 Subtraction and Division of Whole Numbers and Decimals (10 days) By the end of this grade band, students should be computing fluently with whole numbers The computational method that the student uses should be based on mathematical ideas that the student understands well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships. (p. 152) Research suggests that by solving problems that require calculation, students develop methods for computing and also learn more about operations and properties. As students develop methods to solve multidigit computation problems, they should be encouraged to record and share their methods. As they do so, they can learn from one another, analyze the efficiency and generalizability of various approaches, and try one another s methods. (p. 153) When students leave grade 5, they should be able to solve problems involving whole number computation and should recognize that each operation will help them solve many different types of problems With these understandings and skills, they should be able to develop strategies for computing with decimals. (p. 149) 16

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