# Decimal Fractions

Students expand their conception of what a “number” is as they are introduced to an entirely new category of number, decimals, which they learn to convert, compare, and add in simple cases.

## Unit Summary

Unit 7 introduces students to an entirely new category of number—decimals. Students will explore decimals and their relationship to fractions, seeing that tenths and hundredths are particularly important fractional units because they represent an extension of the place value system into a new kind of number called decimals. Thus, students expand their conception of what a “number” is to encompass this entirely new category, which they will rely on for the remainder of their mathematical education.

Students have previously encountered an example of needing to change their understanding of what a number is in Grade 3, when the term came to include fractions. Their Grade 3 understanding of fractions (3.NF.A), as well as their work with fractions so far this year (4.NF.A, 4.NF.B), will provide the foundation upon which decimal numbers, their equivalence to fractions, their comparison, and their addition will be built. Students also developed an understanding of money in Grade 2, working with quantities either less than one dollar or whole dollar amounts (2.MD.8). But with the knowledge acquired in this unit, students will be able to work with money represented as decimals, as it so often is.

Thus, students rely on their work with fractions to see the importance of a tenth as a fractional unit as an extension of the place value system in Topic A, then expand that understanding to hundredths in Topic B. Throughout Topics A and B, students convert between fraction, decimal, unit, and expanded forms to encourage these connections (4.NF.6). Then students learn to compare decimals in Topic C (4.NF.7) and add decimal fractions in Topic D (4.NF.5). Finally, students apply this decimal understanding to solve word problems, including those particularly related to money, at the end of the unit. Thus, the work with money (4.MD.2) supports the major work and main focus of the unit on decimals.

While students will have ample opportunities to engage with the standards for mathematical practice, they’ll rely heavily on looking for and making use of structure (MP.7), particularly the structure of the place value system. They will also construct viable arguments and critique the reasoning of others (MP.3) using various decimal fraction models to support their reasoning.

In Grade 5, students will build on this solid foundation of decimal fractions to develop an even deeper understanding of decimals' relationship to place value and to perform decimal operations with similar models (5.NBT.1—4, 5.NBT.7). By the end of 6th grade, students will be fluent with the use of the standard algorithm to compute with decimals (6.NS.3). From that point forward, students will use their understanding of decimals as a specific kind of number in their mathematical work, including ratios, functions, and many others.

Pacing: 16 instructional days (14 lessons, 1 flex day, 1 assessment day)

For guidance on adjusting the pacing for the 2020-2021 school year due to school closures, see our 4th Grade Scope and Sequence Recommended Adjustments.

• Expanded Assessment Package
• Problem Sets for Each Lesson
• Student Handout Editor
• Vocabulary Package

## Assessment

This assessment accompanies Unit 7 and should be given on the suggested assessment day or after completing the unit.

## Unit Prep

### Intellectual Prep

?

#### Intellectual Prep for All Units

• Read and annotate “Unit Summary” and “Essential Understandings” portion of the unit plan.
• Do all the Target Tasks and annotate them with the “Unit Summary” and “Essential Understandings” in mind.
• Take the unit assessment.

### Essential Understandings

?

• The fractional units tenths and hundredths are particularly important, as they represent the extension of the place value system to places smaller than one whole.
• “There are several ways to read decimals aloud. For example, 0.15 can be read aloud as ‘1 tenth and 5 hundredths’ or ‘15 hundredths,’ just as 1,500 is sometimes read ‘15 hundred’ or ‘1 thousand, 5 hundred’” (NF Progression, p. 15).
• “The number of digits to the right of the decimal point indicates the number of zeros in the denominator, so that $2.70=\frac{270}{100}$ and $2.7=\frac{27}{10}$” (NF Progression, p. 15).
• Unlike with whole numbers, adding zeroes to the end of a decimal number does not change its value. Fraction conversion can be used to show $2.70=\frac{270}{100}=\frac{10\times27}{10\times10}=\frac{27}{10}=2.7$.
• Place-value units are not symmetric about the decimal point; rather, they are symmetric around the ones place.
• Comparing numbers written in standard form uses the understanding that one of any unit is greater than any amount of a smaller unit. Thus, the largest place values in each number contains the most relevant information when comparing numbers. If both numbers have the same number of largest units, the next largest place value should be attended to next, iteratively, until one digit is larger than another in the same unit. This may mean that a decimal with fewer digits is larger than a decimal with more digits in cases where the larger decimal place values have a larger value (e.g., 0.4 > 0.13).
• The approach to adding decimals fractions is based in the idea of needing to add like units together and, thus, replacing fractions with equivalent fractions that allow for addition of like units (i.e., common denominators).

### Vocabulary

?

decimal expanded form

decimal number

decimal point

fraction expanded form

hundredth

decimal fraction

tenth

### Unit Materials, Representations and Tools

?

• Area model
• Number line
• Place-value chart with decimals to hundredths
• Tape diagram

## Common Core Standards

Key: Major Cluster Supporting Cluster Additional Cluster

### Core Standards

?

##### Measurement and Data
• 4.MD.A.2 — 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.

##### Number and Operations—Fractions
• 4.NF.C.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. Students who can generate equivalent fractions can develop strategies for adding fractions with unlike denominators in general. But addition and subtraction with unlike denominators in general is not a requirement at this grade. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.

• 4.NF.C.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.

• 4.NF.C.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.

?

• 2.MD.C.8

• 4.NBT.A.2

• 3.NF.A.2

• 4.NF.A.1

• 4.NF.A.2

• 4.NF.B.3

• 4.NF.B.4

?

• 5.NBT.A.1

• 5.NBT.A.2

• 5.NBT.A.3

• 5.NBT.A.4

• 5.NBT.B.7

• 5.NF.A.1

### Standards for Mathematical Practice

• CCSS.MATH.PRACTICE.MP1 — Make sense of problems and persevere in solving them.

• CCSS.MATH.PRACTICE.MP2 — Reason abstractly and quantitatively.

• CCSS.MATH.PRACTICE.MP3 — Construct viable arguments and critique the reasoning of others.

• CCSS.MATH.PRACTICE.MP4 — Model with mathematics.

• CCSS.MATH.PRACTICE.MP5 — Use appropriate tools strategically.

• CCSS.MATH.PRACTICE.MP6 — Attend to precision.

• CCSS.MATH.PRACTICE.MP7 — Look for and make use of structure.

• CCSS.MATH.PRACTICE.MP8 — Look for and express regularity in repeated reasoning.