All the resources on this page come from the GADOE Framework 5th Grade Unit 2 Overview. Reviewing this information before or during planning can help ensure your instruction is aligned with the standards and is at the depth expected.

Unit 2 Overview

In this unit students will:

• Solve problems by understanding that like whole numbers, the location of a digit in a decimal number determines the value of the digit.

• Understand that rounding decimals should be “sensible” for the context of the problem.

• Understand that decimal numbers can be represented with models.

• Understand that addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values.

UNDERSTAND THE PLACE VALUE SYSTEM

Mathematically proficient students communicate precisely by engaging in discussion about their reasoning using appropriate mathematical language. The terms students should learn to use with increasing precision with this cluster are: place value, decimal, decimal point, patterns, multiply, divide, tenths, thousands, greater than, less than, equal to, ‹, ›, =, compare/ comparison, round.

MGSE5NBT.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.

Students will work with place values from thousandths to one million.

This standard calls for students to reason about the magnitude of numbers. Students should work with the idea that the tens place is ten times as much as the ones place, and the ones place is 1/10 the size of the tens place. In 4th grade, students examined the relationships of the digits in numbers for whole numbers only. This standard extends this understanding to the relationship of decimal fractions. Students use base ten blocks, pictures of base ten blocks, and interactive images of base ten blocks to manipulate and investigate the place value relationships. They use their understanding of unit fractions to compare decimal places and fractional language to describe those comparisons. Before considering the relationship of decimal fractions, students express their understanding that in multi-digit whole numbers, a digit in one place represents 10 times what it represents in the place to its right and 1/10 of what it represents in the place to its left.

Example: A student thinks, “I know that in the number 5555, the 5 in the tens place (5555) represents 50 and the 5 in the hundreds place (5555) represents 500. So, a 5 in the hundreds place is ten times as much as a 5 in the tens place or a 5 in the tens place is 1 10 of the value of a 5 in the hundreds place. Based on the base-10 number system, digits to the left are times as great as digits to the right; likewise, digits to the right are 1 10 of digits to the left. For example, the 8 in 845 has a value of 800 which is ten times as much as the 8 in the number 782. In the same spirit, the 8 in 782 is 1/10 the value of the 8 in 845. To extend this understanding of place value to their work with decimals, students use a model of one unit; they cut it into 10 equal pieces, shade in, or describe 1/10 of that model using fractional language. (“This is 1 out of 10 equal parts. So, it is 1/10 . I can write this using 1/10 or 0.1.”) They repeat the process by finding 1/10 of a 1/10 (e.g., dividing 1/10 into 10 equal parts to arrive at 1/100 or 0.01) and can explain their reasoning: “0.01 is 1/10 of 1 10 thus is 1/100 of the whole unit.” In the number 55.55, each digit is 5, but the value of the digits is different because of the placement.

The 5 that the arrow points to is 1/10 of the 5 to the left and 10 times the 5 to the right. The 5 in the ones place is 1/10 of 50 and 10 times five tenths.

The 5 that the arrow points to is 1/10 of the 5 to the left and 10 times the 5 to the right. The 5 in the tenths place is 10 times five hundredths.

This standard references expanded form of decimals with fractions included. Students should build on their work from 4th grade, where they worked with both decimals and fractions interchangeably. Expanded form is included to build upon work in MGSE.5.NBT.2 and deepen students’ understanding of place value. Students build on the understanding they developed in fourth grade to read, write, and compare decimals to thousandths. They connect their prior experiences with using decimal notation for fractions and addition of fractions with denominators of 10 and 100. They use concrete models and number lines to extend this understanding to decimals to the thousandths. Models may include base ten blocks, place value charts, grids, pictures, drawings, manipulatives, technology-based, etc. They read decimals using fractional language and write decimals in fractional form, as well as in expanded notation. This investigation leads them to understanding equivalence of decimals (0.8 = 0.80 = 0.800).

Comparing decimals builds on work from 4th grade.

Example:

Some equivalent forms of 0.72 are:

Students need to understand the size of decimal numbers and relate them to common benchmarks such as 0, 0.5 (0.50 and 0.500), and 1. Comparing tenths to tenths, hundredths to hundredths, and thousandths to thousandths is simplified if students use their understanding of fractions to compare decimals.

Examples:

Comparing 0.25 and 0.17, a student might think, “25 hundredths is more than 17 hundredths”. They may also think that it is 8 hundredths more. They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is another way to express this comparison.

Comparing 0.207 to 0.26, a student might think, “Both numbers have 2 tenths, so I need to compare the hundredths. The second number has 6 hundredths and the first number has no hundredths so the second number must be larger. Another student might think while writing fractions, “I know that 0.207 is 207 thousandths (and may write 207/1000 ). 0.26 is 26 hundredths (and may write 26/100 ) but I can also think of it as 260 thousandths ( 260/1000 ). So, 260 thousandths is more than 207 thousandths.

MGSE5.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 x (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.

This standard references expanded form of decimals with fractions included. Students should build on their work from 4th grade, where they worked with both decimals and fractions interchangeably. Expanded form is included to build upon work in MGSE.5.NBT.2 and deepen students’ understanding of place value. Students build on the understanding they developed in fourth grade to read, write, and compare decimals to thousandths. They connect their prior experiences with using decimal notation for fractions and addition of fractions with denominators of 10 and 100. They use concrete models and number lines to extend this understanding to decimals to the thousandths. Models may include base ten blocks, place value charts, grids, pictures, drawings, manipulatives, technology-based, etc. They read decimals using fractional language and write decimals in fractional form, as well as in expanded notation. This investigation leads them to understanding equivalence of decimals (0.8 = 0.80 = 0.800).

Examples: Comparing 0.25 and 0.17, a student might think, “25 hundredths is more than 17 hundredths”. They may also think that it is 8 hundredths more. They may write this comparison as 0.25 > 0.17 and recognize that 0.17 < 0.25 is another way to express this comparison.

Comparing 0.207 to 0.26, a student might think, “Both numbers have 2 tenths, so I need to compare the hundredths. The second number has 6 hundredths and the first number has no hundredths so the second number must be larger. Another student might think while writing fractions, “I know that 0.207 is 207 thousandths (and may write 207/1000). 0.26 is 26 hundredths (and may write 26/100) but I can also think of it as 260 thousandths (260/1000). So, 260 thousandths is more than 207 thousandths.

MGSE5.NBT.4 Use place value understanding to round decimals up to the hundredths place.

Rounding

Students should go beyond simply applying an algorithm or procedure for rounding. The expectation is that students have a deep understanding of place value and number sense and can explain and reason about the answers they get when they round. Students should have numerous experiences using a number line to support their work with rounding.

Example:

Round 14.235 to the nearest tenth.

Students recognize that the possible answer must be in tenths thus, it is either 14.2 or 14.3. They then identify that 14.235 is closer to 14.2 (14.20) than to 14.3 (14.30).

Students should use benchmark numbers to support this work. Benchmarks are convenient numbers for comparing and rounding numbers. 0, 0.5, 1, 1.5 are examples of benchmark numbers.

Example:

Which benchmark number is the best estimate of the shaded amount in the model below? Explain your thinking.

MGSE5.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.

This standard builds on the work from 4th grade where students are introduced to decimals and compare them. In5th grade, students begin adding, subtracting, multiplying, and dividing decimals. This work should focus on concrete models and pictorial representations, rather than relying solely on the algorithm. The use of symbolic notations involves having students record the answers to computations (2.25 3= 6.75), but this work should not be done without models or pictures. This standard includes students’ reasoning and explanations of how they use models, pictures, and strategies.

This standard requires students to extend the models and strategies they developed for whole numbers in grades 1-4 to decimal values. Before students are asked to give exact answers, they should estimate answers based on their understanding of operations and the value of the numbers. In this unit, students will only add and subtract decimals. Multiplication and division are addressed in Unit 3.

Examples:

• 3.6 + 1.7

A student might estimate the sum to be larger than 5 because 3.6 is more than 3½ and 1.7 is more than 1½.

• 5.4 – 0.8

A student might estimate the answer to be a little more than 4.4 because a number less than 1 is being subtracted.

Students should be able to express that when they add decimals, they add tenths to tenths and hundredths to hundredths. So, when they are adding in a vertical format (numbers beneath each other), it is important that they write numbers with the same place value beneath each other. This understanding can be reinforced by connecting addition of decimals to their understanding of addition of fractions. Adding fractions with denominators of 10 and 100 is a standard in fourth grade. Example: 4 - 0.3

3 tenths subtracted from 4 wholes. One of the wholes must be divided into tenths.

Example: A recipe for a cake requires 1.25 cups of milk, 0.40 cups of oil, and 0.75 cups of water. How much liquid is in the mixing bowl?

For more detailed information about unpacking the content standards, unpacking a task, math routines and rituals, maintenance activities and more, please refer to the Grade Level Overview.

STANDARDS FOR MATHEMATICAL PRACTICE

This section provides examples of learning experiences for this unit that support the development of the proficiencies described in the Standards for Mathematical Practice. These proficiencies correspond to those developed through the Literacy Standards. The statements provided offer a few examples of connections between the Standards for Mathematical Practice and the Content Standards of this unit. The list is not exhaustive and will hopefully prompt further reflection and discussion.

1. Make sense of problems and persevere in solving them. Students solve problems by applying and extending their understanding of addition and subtraction to decimals. Students seek the meaning of a problem and look for efficient ways to solve it. They determine situations when decimal numbers should be rounded and when they need to be exact.

2. Reason abstractly and quantitatively. Students demonstrate abstract reasoning to connect decimal quantities to fractions, and to compare relative values of decimal numbers. Students round decimal numbers using place value concepts.

3. Construct viable arguments and critique the reasoning of others. Students construct arguments using concrete referents, such as objects, pictures, and drawings. They explain calculations with decimals based upon models and rules that generate patterns. They explain their thinking to others and respond to others’ thinking.

4. Model with mathematics. Students use base ten blocks, drawings, number lines, and equations to represent decimal place value, addition, and subtraction. They determine which models are most efficient for solving problems.

5. Use appropriate tools strategically. Students select and use tools such as graph paper, base ten blocks, and number lines to accurately solve problems with decimals.

6. Attend to precision. Students use clear and precise language, (math talk) in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to decimal place value and use decimal points correctly

7. Look for and make use of structure. Students use properties of operations as strategies to add and subtract with decimals. Students utilize patterns in place value and powers of ten and relate them to rules and graphical representations. Students also use structure to read, write, and compare decimals.

8. Look for and express regularity in repeated reasoning. Students use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and properties of operations to fluently add and subtract decimals.

***Mathematical Practices 1 and 6 should be evident in EVERY lesson***

STANDARDS FOR MATHEMATICAL CONTENT

MGSE5.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.

MGSE5.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.

MGSE5.NBT.4 Use place value understanding to round decimals up to the hundredths place.

MGSE5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models, 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

(NOTE: Addition and subtraction are taught in this unit, but the standard is continued in Unit 3: Multiplication and Division with Decimals.)

BIG IDEAS

• Students will understand that like whole numbers, the location of a digit in decimal numbers determines the value of the digit.

• Students will understand that rounding decimals should be “sensible/reasonable” for the context of the problem.

• Students will understand that decimal numbers can be represented with models including fractions.

• Students will understand that addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values.

• Addition and subtraction with decimals are based on the fundamental concept of adding and subtracting the numbers in like position values---a simple extension from whole numbers.

ESSENTIAL QUESTIONS

• What is the relationship between decimals and fractions?

• How can we read, write, and represent decimal values?

• How are decimal numbers placed on a number line?

• How can rounding decimal numbers be helpful?

• How can you decide if your answer is reasonable?

• How do we compare decimals?

• How are decimals used in batting averages?

• How can estimation help me get closer to 1?

• How can I keep from going over 1?

• Why is place value important when adding whole numbers and decimal numbers?

• How does the placement of a digit affect the value of a decimal number?

• Why is place value important when subtracting whole numbers and decimal numbers?

• What strategies can I use to add and subtract decimals?

• How do you round decimals?

• How does context help me round decimals?

CONCEPTS AND SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

1. Number sense to the tenths place

2. Place value of whole numbers through the millions place

3. Addition and subtraction of whole numbers

4. Representations of fractions as tenths

5. Expressing fractions as decimal numbers

6. Using a number line with decimals

Fluency: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Fluent problem solving does not necessarily mean solving problems within a certain time limit, though there are reasonable limits on how long computation should take. Fluency is based on a deep understanding of quantity and number.

Deep Understanding: Teachers teach more than simply “how to get the answer” and instead support students’ ability to access concepts from a number of perspectives. Therefore, students are able to see math as more than a set of mnemonics or discrete procedures. Students demonstrate deep conceptual understanding of foundational mathematics concepts by applying them to new situations, as well as writing and speaking about their understanding.

Memorization: The rapid recall of arithmetic facts or mathematical procedures. Memorization is often confused with fluency and automaticity. Fluency implies a much richer kind of mathematical knowledge and experience.

Number Sense: Students consider the context of a problem, look at the numbers in a problem, and make a decision about which strategy would be most efficient in each particular problem. Number sense is not a deep understanding of a single strategy, but rather the ability to think flexibly between a variety of strategies in context.

Fluent students:

• flexibly use a combination of deep understanding, number sense, and memorization.

• are fluent in the necessary baseline functions in mathematics so that they are able to spend their thinking and processing time unpacking problems and making meaning from them.

• are able to articulate their reasoning.

• find solutions through a number of different paths.

For more about fluency, see: http://www.youcubed.org/wpcontent/uploads/2015/03/FluencyWithoutFear-2015.pdf and: https://bhi61nm2cr3mkdgk1dtaov18-wpengine.netdna-ssl.com/wp-content/uploads/nctm-timedtests.pdf

STRATEGIES FOR TEACHING AND LEARNING

• Students should be actively engaged by developing their own understanding.

• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.

• Appropriate manipulatives and technology should be used to enhance student learning.

• Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition, which includes self-assessment and reflection.

• Students need to write in mathematics class to explain their thinking, talk about how they perceive topics, and justify their work to others.

Instructional Strategies (Place Value)

In Grade 5, the concept of place value is extended to include decimal values to thousandths. The strategies for Grades 3 and 4 should be drawn upon and extended for whole numbers and decimal numbers. For example, students need to continue to represent, write and state the value of numbers including decimal numbers. For students who are not able to read, write and represent multi-digit numbers, working with decimals will be challenging. Money is a good medium to compare decimals. Present contextual situations that require the comparison of the cost of two items to determine the lower or higher priced item. Students should also be able to identify how many pennies, dimes, dollars and ten dollars, etc., are in a given value. Help students make connections between the number of each type of coin and the value of each coin, and the expanded form of the number. A dime is worth 10 times as much as a penny, but only 1/10 as much as a dollar. Build on the understanding that it always takes ten of the number to the right to make the number to the left. The place value to the right is always 1/10 of the place to its left. Number cards, number cubes, spinners and other manipulatives can be used to generate decimal numbers. For example, have students roll three number cubes, then use those digits to create the largest and smallest numbers to the thousandths place. Ask students to represent the number using numerals, words, and expanded form.

Instructional Resources/Tools

National Library of Virtual Manipulatives; Base Block Decimals, Students use a Ten Frame to demonstrate decimal relationships. http://nlvm.usu.edu/en/nav/frames_asid_264_g_2_t_1.html?from=grade_g_2.html

Instructional Strategies (Decimal Addition and Subtraction)

Students have used various models and strategies to solve problems involving addition and subtraction with whole numbers, such as use of the properties, base ten blocks and number lines. They should apply these strategies and models to decimals before using standard algorithms. With guidance from the teacher, they should understand the connection between the standard algorithm and their strategies. Students should be able to see the connections between the algorithm for adding and subtracting multi-digit whole numbers and adding and subtracting decimal numbers. As students developed efficient strategies for whole number operations, they should also develop efficient strategies with decimal operations. Students should learn to estimate decimal computations before they compute with pencil and paper. The focus on estimation should be on the meaning of the numbers and the operations, not on how many decimal places are involved. For example, to estimate the sum of 32.84 + 4.1, the estimate would be about 37. Students should consider that 32.84 is closer to 33 and 4.1 is closer to 4. The sum of 33 and 4 is 37. Therefore, the sum of 32.84 + 4.1 should be close to 37. Estimates should be used to check answers to determine whether they’re reasonable.

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The terms below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real-life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

• decimal

• fraction

• decimal point

• hundredths

• ones

• place value

• rounding

• tenths

• thousandths

Mathematics Glossary http://www.corestandards.org/Math/Content/mathematics-glossary/glossary