Grade 5 Mathematics, Quarter 1, Unit 1.1 Decimals in the · PDF file · 2012-08-03• Add and subtract decimals to hundredths using ... Grade 5 Mathematics, Quarter 1, Unit 1.2 Adding

Southern Rhode Island Regional Collaborative with process support from The Charles A. Dana Center at the University of Texas at Austin

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Grade 5 Mathematics, Quarter 1, Unit 1.1

Decimals in the Number System

Overview Number of Instructional Days: 15 (1 day = 45 minutes)

Content to Be Learned Mathematical Practices to Be Integrated • Recognize place value relationships. In a multi-

digit number, each place represents:

o 10 times as much as the place to the right;

o 1/10 of the place to its left.

• When multiplying/dividing a number by powers of 10, explain patterns based on the number of zeros and placement of decimal point.

• Use whole number exponents to denote powers of ten.

• Read, write, and compare decimals to the thousandths place in multiple ways.

Attend to precision.

• Communicate precisely to others.

• Calculate accurately.

Look for and make use of structure.

• Look closely to understand a pattern.

• Generalize a pattern.

Look for and express regularity in repeated reasoning.

Essential Questions • What effect does multiplying or dividing by ten

or a power of ten have on a number?

• How do you determine which decimal is greater than, less than, or equal to another?

• How does expanded form help you understand the value of each digit in a number?

• When comparing numbers with decimals to the thousandths place, how does expanded form help you?

Grade 5 Mathematics, Quarter 1, Unit 1.1 Decimals in the Number System (15 days)


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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 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.

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 whole-number exponents to denote powers of 10.

5.NBT.3 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., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (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.

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 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. 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.



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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)(x2 + x + 1), and (x – 1)(x3 + x2 + 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 fourth grade, students solidified an understanding of whole number place value a digit in one place represents 10 times what it represents to its right. Students generalized their understanding of whole number place value, wrote whole numbers in expanded form, and compared multidigit whole numbers using <, >, or =. Fourth grade students used decimal notation for fractions and compared decimal fractions. Fractions were limited to denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.

Current Learning

This unit is a critical area for 5th grade.

Students expand their knowledge of whole number place value to include decimal place value to the thousandths place; they will understand that a digit in one place represents 10 times what it represents to its right and 1/10 of what is to its left. The extension of the place value system from whole numbers to decimals is a major intellectual accomplishment involving understanding and skill with base-ten units and fractions. Further, they will write multidigit numbers in expanded form to thousandths and use >, <, and = to compare numbers including decimals.

Future Learning

In sixth grade, students use standard algorithms to fluently add, subtract, multiply, and divide multidigit decimals. Students order rational numbers using symbols and the number line.



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Additional Findings

According to the Progressions K–5 Number and Operations in Base Ten, “The power of the base-ten system is in repeated bundling by ten: 10 tens make a unit called a hundred. Repeating this process of creating new units by bundling in groups of ten creates units called thousand, ten thousand, hundred thousand … In learning about decimals, children partition a one into 10 equal-sized smaller units, each of which is a tenth. Each base-ten unit can be understood in terms of any other base-ten unit. For example, one hundred can be viewed as a tenth of a thousand, 10 tens, 100 ones, or 1,000 tenths. Algorithms for operations in base ten draw on such relationships” (pp. 2–3).


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Adding and Subtracting Whole Numbers and Decimals


Content to Be Learned Mathematical Practices to Be Integrated • Use place value understanding to round

decimals to any place.

• Add and subtract decimals to hundredths using concrete models or drawings.

• Add and subtract decimals to hundredths using written method.

• Apply properties of operations to add and subtract decimals.

• Use understanding of the relationship between addition and subtraction when calculating quantities involving decimals.

• Explain the reasoning used to connect the concrete model to the written method.

Model with mathematics.

• Reflect on whether results make sense.

• Interpret results in the context of a situation.

• Show relationships using place value charts, number lines, and manipulatives.

Attend to precision.

• Communicate precisely to others.

• Calculate accurately.

Reason abstractly and quantitatively.

• Make sense of quantities and their relationships.

• Attend to the meaning of quantities.

• Create coherent representations of the problem.

Essential Questions • In what way does your understanding of place

value help you to round to any place?

• How do you use the relationship between addition and subtraction to add and subtract decimals?

• How would you show the relationship between the concrete model and the written method for addition and subtraction of decimals?

• What written methods can you use to show addition and subtraction of decimals to the hundredths?

• How do properties of operations help you solve addition and subtraction problems involving decimals?

Grade 5 Mathematics, Quarter 1, Unit 1.2 Adding and Subtracting Whole Numbers and Decimals (10 days)


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Written Curriculum


Number and Operations in Base Ten 5.NBT

Understand the place value system.

5.NBT.4 Use place value understanding to round decimals to any place.

Perform operations with multi-digit 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.


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.



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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.


Prior Learning

In fourth grade, students fluently added and subtracted multidigit whole numbers using the standard algorithm. Additionally in fourth grade, students used place value understanding to round multidigit whole numbers to any place. Finally, students cited fraction work with decimals.

Current Learning

In this unit, students build on their understanding of decimal place value, which they first experienced in unit 1.1. This unit is a critical area and a major cluster for fifth grade. Students round decimals to any place and add and subtract decimals to the hundredths place. Be sure to allow plenty of time for students to work concretely with models, drawings, etc. so their strategies are based on place value before moving on to algorithms.

Students apply written methods and properties as well as strategies that they have previously used with whole numbers to their work with decimals. Refer and incorporate Common Core Standards for Mathematics glossary table 1 (p. 88) and table 3 (p. 90) into instruction.

Later this year, students will use knowledge from this unit in the next unit (1.3), Multiplication with Whole Numbers and Decimals.

Future Learning

In sixth grade, students will use standard algorithms to fluently add, subtract, multiply, and divide multi-digit decimals. Students will also order rational numbers using symbols and the number line.



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Additional Findings

According to Principles and Standards for School Mathematics, “Through teacher-orchestrated discussions of problems in context, students can develop useful methods to compute with fractions, decimals, percents, and integers in ways that make sense. Students’ understanding of computation can be enhanced by developing their own methods and sharing them with one another, explaining why their methods work and are reasonable to use, and then comparing their methods with the algorithms traditionally taught in school. In this way, students can appreciate the power and efficiency of the traditional algorithms and also connect them to student-invented methods that may sometimes be less powerful or efficient but are often easier to understand” (p.220).

The book also states, “Representing numbers with various physical materials should be a major part of mathematical instruction in the elementary grades” (p. 33).


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Multiplication With Whole Numbers and Decimals


Content to Be Learned Mathematical Practices to Be Integrated • Multiply multidigit whole numbers using the

standard algorithm.

• Multiply decimals to hundredths, using concrete models or drawings and strategies based on place value and properties of operations.

• Multiply decimals to hundredths, relate the strategy to the written methods, and explain the reasoning used.

Model with mathematics.

• Reflect on whether results make sense.

• Interpret results in the context of a situation.

• Show relationships using symbols, visual models, and manipulatives.

Make sense of problems and persevere in solving them.

• Check answer to problems and determine if the answers make sense.

Look for and make use of structure.

• Look closely to understand a pattern.

• Generalize a pattern.

Essential Questions • How would you demonstrate and explain the

process of multiplying multidigit whole numbers using the standard algorithm?

• How can the methods and strategies you used to learn whole number multiplication help you to multiply decimals?

• Why does multiplying by a decimal result in a product less than one or both of the factors?

• How can you use what you know about place value to explain the relationship between expressions such as 3 x 5, 3 x 0.5, and 0.3 x 0.5?

• How can you model the product of ____x____?

Grade 5 Mathematics, Quarter 1, Unit 1.3 Multiplication With Whole Numbers and Decimals (15 days)


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Written Curriculum


Perform operations with multi-digit whole numbers and with decimals to hundredths.

5.NBT.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

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.


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.

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.



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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 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. 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.


Prior Learning

In third grade, students learned how to multiply one-digit whole numbers by multiples of 10 in the range 10–90. Additionally, they learned to multiply one-digit numbers within 100 and the properties of multiplication.

In fourth grade, students continued to develop their understanding of and fluency with multiplication of whole numbers (four digits by one digit and two digits by two digits) and applied multiplication to real-world problems. Students also focused on developing various strategies for multiplying whole numbers and were exposed to the standard algorithm. Next, they examined how different strategies are related and developed efficient strategies for solving multiplication and division problems, including base ten strategies. Finally, students used place value understanding, the properties of operations, and decimal notation for fractions.

Current Learning

In fifth grade, students build on their conceptual understanding of multiplication by solving problems involving larger numbers and multiplying using strategies and the standard algorithm. Students develop fluency with the multiplication algorithm throughout the year. It is important for students to connect strategies they used in fourth grade to the standard algorithm. Connecting the partial product algorithm to the standard algorithm is particularly helpful. Refer to the Common Core State Standards glossary table 2 (pg. 89) for problem types students have solved in prior grade levels. Students continue to use these problem types to strengthen problem-solving skills as they work with larger numbers. Finally, this is the last year that students focus on multiplication of whole numbers; multiplication of whole numbers using the standard algorithm needs to be mastered by the end of fifth grade.

Students also build on and expand their conceptual understanding of multiplication to multiplying decimals to the hundredths. Students will connect to their prior learning of multiplication of whole numbers by using concrete models, drawings, the partial product algorithm, etc. It is not necessary for students to master the standard algorithm for multiplying decimals.

In future fifth grade units, students will apply their understanding of multiplication to division and fractions.



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Future Learning

In sixth grade, students will apply and extend their previous understanding of multiplication to fractions and the system of rational numbers. Students will also compute fluently with multidigit numbers and find common factors and multiples. They will extend their thinking to solve real-world and mathematical problems involving area, surface area, and volume. Moreover, students will use ratio reasoning to convert measurement units and to manipulate and transform units appropriately when multiplying or dividing quantities. Finally, students will master the standard algorithm for multiplication of decimals.

Additional Findings

Students use understandings of multiplication to develop quick recall of the basic multiplication facts and related division facts. They apply their understanding of models (equal-sized groups, arrays, area models, equal intervals on the number line) place value, properties of operations (in particular the distributive property) as they develop, discuss, and use efficient, accurate, and generalizable methods to multiply multidigit whole numbers. They select appropriate methods and apply them accurately to estimate products or calculate them mentally, depending on the context and numbers involved. They develop fluency with efficient procedures, including the standard algorithm for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems (Curriculum Focal Points, p. 16).

Documents

Grade 5 Mathematics, Quarter 1, Unit 1.1 Decimals in the · PDF file · 2012-08-03• Add and subtract decimals to hundredths using ... Grade 5 Mathematics, Quarter 1, Unit 1.2 Adding