 # Standard 3 : Understand decimal notation for fractions, and compare decimal fractions. (Major Cluster)

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Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

### General Information

Number: MAFS.4.NF.3
Title: Understand decimal notation for fractions, and compare decimal fractions. (Major Cluster)
Type: Cluster
Subject: Mathematics
Grade: 4
Domain-Subdomain: Number and Operations - Fractions

#### Related Standards

This cluster includes the following benchmarks
 Code Description MAFS.4.NF.3.5: Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100. MAFS.4.NF.3.6: Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram. MAFS.4.NF.3.7: Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

#### Related Access Points

This cluster includes the following access points.

#### Access Points

 Access Point Number Access Point Title MAFS.4.NF.3.AP.5a: Find the equivalent fraction with denominators that are multiples of 10. MAFS.4.NF.3.AP.6a: Identify the equivalent decimal form for a benchmark fraction. MAFS.4.NF.3.AP.6b: Match a fraction (with a denominator of 10 or 100) with its decimal equivalent (5/10 = 0.5). MAFS.4.NF.3.AP.6c: Read, write, or select decimals to the tenths place. MAFS.4.NF.3.AP.6d: Read, write, or select decimals to the hundredths place. MAFS.4.NF.3.AP.7a: Use =, <, or > to compare two decimals (decimals in multiples of .10). MAFS.4.NF.3.AP.7b: Compare two decimals expressed to the tenths place with a value of less than 1 using a visual model. MAFS.4.NF.3.AP.7c: Compare two decimals expressed to the hundredths place with a value of less than 1 using a visual model.

#### Related Resources

Vetted resources educators can use to teach the concepts and skills in this topic.

#### Original Student Tutorials

 Name Description Fractions at the Fair: Equivalent Tenths and Hundredths: Learn about equivalent 10ths and 100ths and how to calculate these equivalent fractions at the fair in this interactive tutorial. Return to Deciland: Locating Decimals on a Number Line: Learn about decimals on a number line and comparing decimals to save the Decis from a wizard's spell in this interactive tutorial. Deci Land Escape: Help Rich escape Deci Land by learning how to write decimals that are related to fractions with denominators of 10 and 100 in this interactive tutorial.

#### Assessments

 Name Description Sample 4 - Fourth Grade Math State Interim Assessment: This is a State Interim Assessment for fourth grade. Sample 3 - Fourth Grade Math State Interim Assessment: This is a State Interim Assessment for fourth grade. Sample 1 - Fourth Grade Math State Interim Assessment: This is a State Interim Assessment for fourth grade.

#### Educational Games

 Name Description Fraction Quiz: Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit. Decimal and Fraction: This interactive Flash applet has students match fractions with their equivalent one- or two-place decimals. Students have a chance to correct errors until all matches are made.

#### Formative Assessments

 Name Description Using Models to Compare Decimals: Students are asked to compare decimals by drawing a visual model and record the comparison using the less than, greater than, or equal to symbol. Tenths and Hundredths: Students are asked if an equation involving the sum of two fractions is true or false.  Then students are asked to find the sum of two fractions. Comparing Decimals in Context: Students are asked to compare two pairs of decimals in the context of word problems and to record a comparison using an inequality symbol. Adding Five Tenths: Students express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and are then asked to add the fraction to another fraction with a denominator of 100. Comparing Four Tenths: Students are asked to consider two grids with different sized wholes and determine if both models show four-tenths. Compare Decimals: Students are asked to compare four pairs of decimals using the less than, greater than, or equal to symbols. Using Benchmark Decimals on a Number Line: Students are asked to use benchmark decimals to place four fractions on a number line. Hundredths and Tenths: Students are asked if an equation is true or false. Then students are asked to find the sum of two fractions. Fractions to Decimals: Students are given four fractions and asked to write each in decimal form. Seven Tenths: Students express a fraction with a denominator of 10 as an equivalent fraction with a denominator of 100 and are then asked to add the fraction to another fraction with a denominator of 100. Using Benchmark Fractions on a Number Line: Students are asked to use benchmark fractions to place four decimals on a number line. Decimals to Fractions: Students are given four decimals and asked to write each as a fraction.

#### Lesson Plans

 Name Description What Sort of Decimal Do You Have?: This is a thought-provoking fraction/decimal comparison task. The fractions and decimals have been selected for this task to promote the use the following strategies when suitable: • comparing to benchmark fractions, • finding a common denominator, • finding a common numerator. Field Day Fractions: This is a Model Eliciting Activity (MEA) activity which requires the students to convert fractions to decimals, order the decimals, and then design a process for ranking the classrooms from quantitative and qualitative data and then re-test their procedure on a new set of data. Ultimately, the students have to write a letter explaining and supporting the step-by-step process they used. Cookies, Fractions and Decimals, Oh My!: This lesson asks students to recommend which cookie the owners of The Cookie Jar should add to their menu. Before they make their decision, the students have to convert decimal notation and fractions with denominators 10 and 100 to fractions with like denominators. Then they will be able to see exactly how many people voted for each cookie and they can factor in that information along with additional cookie facts to make their final recommendation. We All Scream for ICE CREAM - MEA: In this MEA, students will work in collaborative groups to solve multi-step problems with whole numbers, fractions, decimals and percent by using different mathematical operations. The students will be asked to assist an ice cream shop owner, who is planning a promotional program "Flavor of the Month," to rank the ice cream flavors based on the data provided. Students will need to read a data table, rank the flavors, convert the fraction amount to a percent and decimal and per serving costs to a decimal as well. A twist is added to the problem when one of the flavors is too expensive to make because of seasonal availability but two new flavors are added to be calculated. An additional twist is given by adding an adult survey to the second data table. The students will need to recalculate the new percent and decimals for the additional flavors. Donuts and Decimals: In this MEA, students will convert fractions into decimals and then compare the decimals to decide which donut a donut shop should add to their menu. Cell Phone Inquiry: Students will determine what cell phone would be the best phone for their teacher to purchase. Factors to consider are price, touch screen, camera, voice command, weight and size. Farming Fractions: This lesson begins with a whole-group flower garden anchor chart activity. Independent practice follows with students using a 100"s grid to plant a ten-row vegetable garden with four kinds of vegetables. They write equivalent fractions and decimals for the part of the garden in which each vegetable is planted. Decimals are then written in expanded form. Comparing and Ordering Decimals: In this cooperative learning activity, students will have five sets of decimal cards to sort and put in order - least to greatest. The lesson starts with a short whole group activity and then breaks off in to structured groups. The teacher is free to interact with each of the groups and monitor progress, participation, and understanding. Amazing Alice Cookies: Students will help Amazing Alice Cookies choose the perfect chocolate chip brand to use for their cookies. Students will be given data in the form of fractions and decimals. Fourth grade students will compare decimals and order and compare fractions. Students will write a letter describing their procedure to the client. Cookies and Treats: Fourth graders will help Cookies and Treats find eco-friendly packaging for its cookies. Students will work with decimals and data in order to develop a procedure for ranking and choosing packaging for cookies. Equivalency Detectives: Fractions and Decimals!: This is a lesson intended to reinforce students' ability to find equivalent fractions and decimals. The lesson requires prior essential vocabulary knowledge, and a basic understanding of converting fractions to decimals and decimals to fractions (specifically tenths and hundredths). Fractions Undercover!: Students will correctly model and discover fractions and their decimal equivalents through the use of decimal grids and base ten blocks. Dynamic Decimals, Fractions and Money!: In this lesson, students will realize the connection between fractions, decimals and money through the use of a 100 grid. Happy Hundredths (Lesson 2 of 3): In this lesson, students will work with math manipulatives to understand that it takes 100 hundredths of something to make one whole. They will use manipulatives with money (pennies and dollars), fractions (one hundredth pieces and one whole pieces), and base ten blocks (units and wholes) to show different values. They will express values with combinations of the given manipulatives and draw their solutions. This lesson 2 of 3 in a unit on fraction and decimal concepts Playground Picks: In this open-ended real world problem, students will work in groups to determine a procedure for ranking playground equipment to help a school purchase new equipment for their playground. Students will need to find like denominators, make decisions based on a data table, and write a letter to the school providing evidence for their decisions. Students will need to trade off between the cost of the equipment, its safety rating and student opinions. Shopping with Tenths and Hundredths (Lesson 3 of 3 in a Unit): This is the third lesson of a three-lesson unit about tenths and hundredths. Please see "happy hundredths" and "terrific tenths" before attempting this lesson. Students will work with math manipulatives to understand that it takes 100 hundredths or 10 tenths of something to make one whole. They will understand that it takes 10 hundredths to make 1 tenth. They will use manipulatives with money (pennies, dimes, dollars), fractions (one hundredth pieces, one tenth pieces, one whole pieces), and base ten blocks (units, rods, wholes) to show different values. They will express values with combinations of the given manipulatives and draw their solutions. Terrific Tenths (Lesson 1 of 3): In this lesson, students will work with math manipulatives to understand that it takes 10 tenths of something to make one whole. They will use manipulatives with money (dimes), fractions (one tenth pieces), and base ten blocks (rods) to show different values. They will express values with combinations of the given manipulatives and draw their solutions. This is lesson one in a three lesson unit.  Lesson two (HAPPY HUNDREDTHS) deals with hundredths and lesson three deals with combining and solving problems with tenths and hundredths (SHOPPING WITH TENTHS AND HUNDREDTHS).

#### Problem-Solving Tasks

 Name Description Adding Tenths and Hundredths: The purpose of this task is adding fractions being with a focus on tenths and hundredths. Each part of this task emphasizes a unique aspect of 4.NF.5. How Many Tenths and Hundredths?: The purpose of this task is for students to finish the equations to make true statements. Parts (a) and (b) have the same solution, which emphasizes that the order in which we add doesn't matter (because addition is commutative), while parts (c) and (d) emphasize that the position of a digit in a decimal number is critical. The student must really think to encode the quantity in positional notation. In parts (e), (f), and (g), the base-ten units in 14 hundredths are bundled in different ways. In part (e), "hundredths" are thought of as units: 14 things = 10 things + 4 things. Part (h) addresses the notion of equivalence between hundredths and tenths. Fraction Equivalence: Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. 6/8=9/12), but it can always be framed as subdividing the same quantity in different ways. Expanded Fractions and Decimals: The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit. Dimes and Pennies: The purpose of this task is to help students gain a better understanding of fractions through the use of dimes and pennies. Using Place Value: Each part of this task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

#### Tutorials

 Name Description Adding Two Fractions with Denominators 10 and 100: The Khan Academy tutorial video presents a visual fraction model for adding 3/10 + 7/100 . Comparing Two Decimals with a Visual Model: In this Khan Academy tutorial video two decimals are compared using grid diagrams. Decimals as Words: In this Khan Academy video decimals are written and spoken in words. Decimals and Fractions from Grid and Number-Line Representations: The Khan Academy video uses grid diagrams and number-line representations to say and write equivalent decimals and fractions. Grid Representations of Decimals: The Khan Academy video illustrates how to determine and write the decimal represented by shaded grids. Visually Converting from Tenths to Hundredths: In this Khan Academy video a fraction is converted from tenths to hundredths using grid diagrams. Introduction to Decimals: This tutorial for student audiences will provide a basic introduction to decimals. The tutorial presents a decimal as another way to represent a fraction. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer gaining an orange circle and a wrong answer graying out. Some "Try This" sections will read the decimal to the students as well.

#### Virtual Manipulative

 Name Description Fraction Models: An interactive tool to represent a fraction circle, rectangle, or set model with numerators and denominators ranging from 1 to 100. The decimal and percent equivalents of the created fraction are also displayed.

#### Student Resources

Vetted resources students can use to learn the concepts and skills in this topic.

#### Original Student Tutorials

 Title Description Fractions at the Fair: Equivalent Tenths and Hundredths: Learn about equivalent 10ths and 100ths and how to calculate these equivalent fractions at the fair in this interactive tutorial. Return to Deciland: Locating Decimals on a Number Line: Learn about decimals on a number line and comparing decimals to save the Decis from a wizard's spell in this interactive tutorial. Deci Land Escape: Help Rich escape Deci Land by learning how to write decimals that are related to fractions with denominators of 10 and 100 in this interactive tutorial.

#### Educational Games

 Title Description Fraction Quiz: Test your fraction skills by answering questions on this site. This quiz asks you to simplify fractions, convert fractions to decimals and percentages, and answer algebra questions involving fractions. You can even choose difficulty level, question types, and time limit. Decimal and Fraction: This interactive Flash applet has students match fractions with their equivalent one- or two-place decimals. Students have a chance to correct errors until all matches are made.

#### Problem-Solving Tasks

 Title Description Adding Tenths and Hundredths: The purpose of this task is adding fractions being with a focus on tenths and hundredths. Each part of this task emphasizes a unique aspect of 4.NF.5. How Many Tenths and Hundredths?: The purpose of this task is for students to finish the equations to make true statements. Parts (a) and (b) have the same solution, which emphasizes that the order in which we add doesn't matter (because addition is commutative), while parts (c) and (d) emphasize that the position of a digit in a decimal number is critical. The student must really think to encode the quantity in positional notation. In parts (e), (f), and (g), the base-ten units in 14 hundredths are bundled in different ways. In part (e), "hundredths" are thought of as units: 14 things = 10 things + 4 things. Part (h) addresses the notion of equivalence between hundredths and tenths. Fraction Equivalence: Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. 6/8=9/12), but it can always be framed as subdividing the same quantity in different ways. Expanded Fractions and Decimals: The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit. Dimes and Pennies: The purpose of this task is to help students gain a better understanding of fractions through the use of dimes and pennies. Using Place Value: Each part of this task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

#### Tutorials

 Title Description Adding Two Fractions with Denominators 10 and 100: The Khan Academy tutorial video presents a visual fraction model for adding 3/10 + 7/100 . Comparing Two Decimals with a Visual Model: In this Khan Academy tutorial video two decimals are compared using grid diagrams. Decimals as Words: In this Khan Academy video decimals are written and spoken in words. Decimals and Fractions from Grid and Number-Line Representations: The Khan Academy video uses grid diagrams and number-line representations to say and write equivalent decimals and fractions. Grid Representations of Decimals: The Khan Academy video illustrates how to determine and write the decimal represented by shaded grids. Visually Converting from Tenths to Hundredths: In this Khan Academy video a fraction is converted from tenths to hundredths using grid diagrams. Introduction to Decimals: This tutorial for student audiences will provide a basic introduction to decimals. The tutorial presents a decimal as another way to represent a fraction. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer gaining an orange circle and a wrong answer graying out. Some "Try This" sections will read the decimal to the students as well.

#### Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this topic.

#### Problem-Solving Tasks

 Title Description Adding Tenths and Hundredths: The purpose of this task is adding fractions being with a focus on tenths and hundredths. Each part of this task emphasizes a unique aspect of 4.NF.5. How Many Tenths and Hundredths?: The purpose of this task is for students to finish the equations to make true statements. Parts (a) and (b) have the same solution, which emphasizes that the order in which we add doesn't matter (because addition is commutative), while parts (c) and (d) emphasize that the position of a digit in a decimal number is critical. The student must really think to encode the quantity in positional notation. In parts (e), (f), and (g), the base-ten units in 14 hundredths are bundled in different ways. In part (e), "hundredths" are thought of as units: 14 things = 10 things + 4 things. Part (h) addresses the notion of equivalence between hundredths and tenths. Fraction Equivalence: Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. 6/8=9/12), but it can always be framed as subdividing the same quantity in different ways. Expanded Fractions and Decimals: The purpose of this task is for students to show they understand the connection between fraction and decimal notation by writing the same numbers both ways. Comparing and contrasting the two solutions shown below shows why decimal notation can be confusing. The first solution shows the briefest way to represent each number, and the second solution makes all the zeros explicit. Dimes and Pennies: The purpose of this task is to help students gain a better understanding of fractions through the use of dimes and pennies. Using Place Value: Each part of this task highlights a slightly different aspect of place value as it relates to decimal notation. More than simply being comfortable with decimal notation, the point is for students to be able to move fluidly between and among the different ways that a single value can be represented and to understand the relative size of the numbers in each place.

#### Tutorial

 Title Description Introduction to Decimals: This tutorial for student audiences will provide a basic introduction to decimals. The tutorial presents a decimal as another way to represent a fraction. Students will be able to navigate the teaching portion of the tutorial at their own pace and test their understanding after each step of the lesson with a "Try This" section. The "Try This" section will monitor students answers and self-check by a right answer gaining an orange circle and a wrong answer graying out. Some "Try This" sections will read the decimal to the students as well.