Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Maryland College and Career-Ready Standards
In this unit, 4th graders are formally introduced to decimals for the first time. Students build upon their knowledge of fractions and the base-ten system to
develop an understanding of decimals to the hundredths place. The relationship between fractions and decimals is formed using visuals including area
models, grids, and number lines and is used to develop students’ understanding that decimals (like fractions) can be used to name values less than 1 or
values between whole numbers. Additionally, students utilize the whole number patterns in the base-ten system and extend them to the right of the ones
place with decimals.
Students will apply their understanding of equivalent fractions and decimals in order to understand relative size and to compare/order and add/subtract
decimals. Students will make connections between adding/subtracting fractions as they apply these concepts to word problems and line plots involving
measurement data.
They will add and subtract fractions with the same denominator and convert an improper fraction to a mixed number by decomposing the fraction into a
sum of a whole number and a number less than 1. Throughout this unit, students will build a deep number sense of fraction and decimal numbers.
Research
When teaching fraction operation, remember that students built their understandings of operations with whole numbers. We can use their understanding
of what operations mean to give meaning to fraction computations. However students must have a strong understanding of fractions as numbers to be
successful with fraction computation. Without a strong foundation of fractions, fraction computation will be rules without reason, which is an
unacceptable goal.
It is important to give students ample opportunities to develop fraction sense and not rush to rules of fraction computation. Some guidelines to keep in
mind:
1. Begin with contextual tasks and allow students to develop their own methods and meaning of adding and subtracting fractions.
2. Connect the meaning of fraction computation to whole number computation.
3. Encourage students to estimate and use informal methods to add and subtract. (Will the answer be over or under 1, 2…?)
4. Explore operations of fractions with models. Use models to prove answers and/or solve problems.
Teaching Student-Centered Mathematics Grades 3-5, John A. Van de Walle, LouAnn H. Lovin
In grade 4, students use place value and decimal notation to represent fractional amounts. A student’s understanding of decimal notation requires the
use of basic meanings of fractions with denominators of 10, 100, and 1,000, as well as the idea that the same fractional amount can be represented by
many equivalent symbols and the understanding of the relationships between the positions in place-value notation. (Focus in Grade 4: Teaching with
Curriculum Focal Points)
Students can observe that when moving to the right across the places, each place has the value of the one before it divided by 10. Students use this
pattern of dividing by 10 as they develop decimal place-value concepts by observing that this pattern continues to the right of the ones place indefinitely
and results in the decimal place values. . . .the decimal place values continue to get infinitely smaller to the right just as the whole-number places
continue to get infinitely larger to the left. (Focus in Grade 4: Teaching with Curriculum Focal Points)
To help students see the connection between fractions and decimals, we can do three things. First, we can use familiar fraction concepts and models to
explore rational numbers that are easily represented by decimals: tenths, hundredths, and thousandths. Second, we can help students see how the baseten system can be extended to include numbers less than 1 as well as large numbers. Third, we can help children use models to make meaningful
translations between fractions and decimals. (Teaching Student-Centered Mathematics, Grades 3-5)
1
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
The chart below highlights the key understandings of Unit 4 along with important questions that teachers should pose to
promote these understandings. The chart also includes key vocabulary that should be modeled by teachers and used by
Enduring Understandings
Students will understand that:
Essential Questions

How can we determine the operation(s)
needed to solve multi-step word
problems?

Problems can be represented
by an equation with an
unknown quantity.
What strategy can I use to determine the
reasonableness of my answer?

How can we use math properties,
patterns, and rules to solve problems?

Estimation strategies can be
used to determine if an answer
is reasonable.

How are unit fractions used to compose
other fractions?

Problems can be represented
and solved accurately using a
variety of strategies.

How can whole numbers be expressed as
fractions?

Fractions can be represented
in multiple ways.

How can we use measurement to solve
problems?

Fractions can be composed
and decomposed.

How can we create line plots to represent
data?

Measurement can be used to
solve problems.

What is a decimal?

Line plots can be constructed
to represent data.

How are fractions and decimals related?

How can decimals be represented?

How can we use reasoning to compare
decimals?


Multistep word-problems can
be solved by determining the
operation(s) needed to solve
the problem.
Key Vocabulary
Addend
Compare
Data
Decimal
Decimal Fraction
Denominator
Difference
Dividend
Divisor
Equivalent
Equivalent decimals
Estimation
Factor
Greater than
Hundredths
Improper fraction
Interval
Less than
Line plot
Mixed number
Multiple
Multiplication
Numerator
Product
Quotient
Reasonableness
Remainder
Rounding
Sum
Tenths
Unit fraction
Unknown
Value
Background Reading
Teaching StudentCentered Mathematics
– Grades 3-5
Focus in Grade 4:
Teaching with
Curriculum Focal Points
pp. 30 - 36
Putting the Practices
into Action:
Implementing the
Common Core
Standards for
Mathematical Practice
K-8
2
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
students to show precision of language when communicating mathematically.
Throughout this cluster, students will develop their use of the 8 Mathematical Practices while learning the instructional standards. The
mathematical practices in the shaded boxes should be emphasized during instruction this unit due the how well they connect with the
content standards in this unit.
Standards for Mathematical
Practice
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
3. Construct viable arguments and
critique the reasoning of others
4. Model with mathematics
Connections to this Cluster
Solve problems in which the solution is not immediately evident.
To determine and articulate what the problem is asking:

Ask students to restate the problem in their own words.

Have students turn to a partner to state the problem.

Discuss familiar problems (When have we seen something like this before? What did we do?)
To self-monitor progress and change directions when necessary:

Have students talk or write about how they got “stuck” and then “unstuck” when solving a problem.

Think aloud to show students how to change course when needed.
To demonstrate perseverance in problem-solving and identify different ways to solve a problem:

Make a classroom list of possible strategies.

Acknowledge those who modify their thinking and persevere to get to the solution and have students show
and talk about how they solved problems.

Encourage students to show at least two ways to solve a problem.
Write an equation for a situation and be able to explain how the equation relates to the situation presented. Solve
the equation outside of the context of the problem, and then connect the solution back to the situation presented.
To make sense of quantities and their relationships in problem situations:

Represent a given situations problem with the equation using a variable for the unknown.

Write a situation problem that matches a specific equation.

Represent multistep computation with an accurate string of reasoning using symbols appropriately.
Explain why a problem with an estimated answer may have more than one solution, based on an understanding of
place value.

Provide opportunities for students to reason through, explore, and explain student-invented strategies.

Provide alternate approaches/solutions for students to analyze and critique.

Find errors in flawed work samples. Allow students to explain using words and concrete examples.
Write equations for various real world problem situations and solve problems about the situations. Represent the
unknown with a variable.
 Represent problem situations in multiple ways including numbers, words (mathematical language), drawing
pictures, using objects, and creating equations
 Connect the different representations and explain the connections.
 Evaluate their results in the context of the situation and reflect on whether the results make sense.
3
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in
repeated reasoning
Use manipulatives, bar models (tape diagrams), and drawings that represent mathematical situations.

Consider the available tools (including estimation) when solving a mathematical problem and decide when
certain tools might be helpful.

Explain how tools can assist them in seeing patterns and relationships with numbers.
Create accurate drawings and representations of mathematical situations.
Use specific math vocabulary to communicate mathematical ideas.
Compute accurately.

Develop and display anchor charts with precise math vocabulary.

Orally rephrase student explanations using appropriate vocabulary.
Look closely to discover a pattern or structure and apply the properties of operations to solve problems.

Notice patterns when finding equivalent fractions. Use a 100 chart to find fractions equivalent to 2/3.

Make observations about the relationship between mixed numbers and equivalent improper fractions.
Notice repetitive actions in computation and make generalizations about rules and “short-cuts” to get to answers
more quickly.

Use patterns in measurement conversions to solve multiplicative comparison problems.

Use models to examine patterns and generate alternative algorithms.
4
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Maryland College and CareerReady Standards
Build fractions from unit fractions by applying and extending previous
understandings of operations on whole numbers.
4.NF.A.3
c. Add and subtract
mixed numbers with like
denominators, e.g., by
replacing each mixed
number with an
equivalent fraction,
and/or by using properties
of operations and the
relationship between
addition and subtraction.
SMP
2. Reason abstractly and
quantitatively.
3. Construct viable
arguments and critique the
reasoning of others.
6. Attend to precision.
7. Look for and make use of
structure.
Formative
Assessments
Instructional Strategies and Resource Support
FA.4.NF.A3.ca
A separate algorithm for mixed numbers in addition and subtraction is not
needed. Students will apply their knowledge of adding or subtracting the whole
numbers first. Then, they will work with the fractions using the same strategies they
have applied to problems that contained only fractions.
Note: Mixed numbers are introduced for the first time in Fourth Grade. Students
should have ample experiences of adding and subtracting mixed numbers where
they work with mixed numbers or convert mixed numbers so that the numerator is
equal to or greater than the denominator.
Students may compose and decompose mixed numbers in multiple ways to add
or subtract.
Example : 2 3/5 + 3 2/5 = _____
2
Solution A
Solution B
Solution C
2 + 3 =
2 + 3 =
2 + 3 =
+
2+3=5
3

+ = = 1
5 + 1 = 6
3
2 +

3 = 5 
2 =

3 = 
+== 6
5 += 5 = 6
SMP:
Students show work to multistep algorithms in the form of valid chains of reasoning using
symbols such as equal signs appropriately.
Students will not get full credit for nonsense statements (such as 3 ¾ =15/4 + ¾ = 18/4) even
if the solution is correct.
(PARCConline.org – evidence tables)
Fill in the blanks with a
number that makes the
equation true.
7 =+ 3+____

2++____= 4

4+ ___ + 2= 8
FA.4.NF.A3.cb
Show two ways to solve the
problem below.
2 + 4


FA.4.NF.A3.cc
Solve:
6 -1

Place the difference from the
above problem on the
number line between the two
whole numbers it lies
between.
FA.4.NF.A3.cd
Write two equations that
could represent the work on
the number line below.
1
1 223
5
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Build fractions from unit fractions by applying and extending previous understandings
of operations on whole numbers.
Maryland College and CareerReady Standards
Instructional Strategies and Resource Support
4.NF.A.3
Use a visual fraction model to solve a problem like this:
d. Solve word problems
involving addition and
subtraction of fractions
referring to the same
whole and having like
denominators, e.g., by
using visual fraction
models and equations to
represent the problem
Example A:
SMP
1. Make sense of
problems and persevere
in solving them
4. Model with
mathematics
5. Use appropriate tools
strategically.
Formative
Assessments
FA.4.NF.A3.da
Paul has 4 1/6 pizzas left over from his Super Bowl party. After giving some pizza to his
friend, he has 2 4/6 of a pizza left. How much pizza did Paul give to his friend?
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Mr. Foot recorded that
8inches of snow fell in
January. In February we
had 11inches of snow.
How much snow did we
get in the first 2 months
of the year?
Paul has 4 1/6 pizzas to start. This is the same as 25/6. I shaded in 25/6 to show what Paul
has to start. I placed Xs to show the 2 4/6 Paul gave to his friend. Paul has 1 3/6 pizzas
left.
Example B:
Kim and Amanda need 6 3/8 feet of ribbon for gift wrapping. Amanda has 2 1/8 feet of
ribbon and Kim has 4 3/8 feet of ribbon. How much ribbon do they have altogether? Will
it be enough to complete the project? Explain why or why not.
Possible Reasoning:
FA.4.NF.A3.db
Mrs. Greenthumb’s
dogwood tree was 14
feet tall this morning.
She clipped back some
branches, and it is now 9
feet tall. How much did
Mrs. Greenthumb trim off
the tree?
If I add 2 1/8 feet and 4 3/8 feet, I will find the total amount of ribbon Kim and Amanda
have. I can add the whole numbers (2 + 4) and the fractions (1/8 + 3/8). Together, the
girls have 6 4/8 feet of ribbon. This is greater than the 6 3/8 feet of ribbon they need (1/8
foot more).
Students should also use number lines to represent fraction word problems.
Example C:
Bill started art class with 2 1/4 pounds of clay. When he finished making his dog
sculpture, he had 3/4 of a pound of clay left. How much does Bill’s sculpture weigh?
1
1 1lbs.
0
1
2
3
6
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Solve problems involving measurement and conversion of measurements from a larger
unit to a smaller unit.
Maryland College and CareerReady Standards
4.MD.B.4
Make a line plot to display a
data set of measurements in
fractions of a unit (1/2, 1/4,
1/8). Solve problems
involving addition and
subtraction of fractions by
using information presented
in line plots.
For example, from a line plot
find and interpret the difference
in length between the longest
and shortest specimens in an
insect collection.
Instructional Strategies and Resource Support
Teaching Student-Centered Mathematics – Grades 3-5
pp. 333
Creating line plots using measurement data requires application of
equivalent fractions from unit 3 and can give context for students to add and
subtract fractions with like denominators in this unit.
Example: Complete the line plot using the data below:
1 lbs
1
SMP
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
4. Model with mathematics
6. Attend to precision
1 lbs
1 lbs
2 lbs
 lbs
Tina started to create a line plot to
show the length of her fingers.
But she made a mistake on the number
line below.
Correct her mistake and complete the
line plot.
2 , 3 , 4  , 3, 2 , 3,
4 , 4 , 3, 2
1  lbs
1lbs
FA.4.MD.B4a
Length of Tina’s fingers in inches:
Weight of packages to be mailed:
1lbs
Formative Assessments
1lbs
1 lbs
2 lbs
1lbs
FA.4.MD.B4b
Tyler works at the science museum
organizing the rare insect collection.
He records the lengths of different
bugs including a variety of beetles and
cockroaches. Below is a line plot
containing the lengths of locusts.
11111112
Weight of Packages in lbs
What is the weight of the heaviest and lightest package combined?
How many packages weighed 1lbs?
What is the difference in weight between all of the 2 pound packages
combined and all of the 1packages combined?
Comparing two line plots with different intervals like in the illustrative
mathematics task linked below (1/4 of an inch compared to 1/8 of an inch)
can allow for a variety of questions and a deeper understanding of
measurement.
FA.4.MD.B4c
Below is a line plot showing how long
Wally’s spent working on homework
each day last month.
Hours Spent on Homework
illustrative mathematics 4.MD.B.4 button illustration
7
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Solve problems involving measurement and conversion of measurements from a larger
unit to a smaller unit.
Maryland College and CareerReady Standards
4.MD.B.4
Make a line plot to display a
data set of measurements in
fractions of a unit (1/2, 1/4,
1/8). Solve problems
involving addition and
subtraction of fractions by
using information presented
in line plots.
For example, from a line plot
find and interpret the difference
in length between the longest
and shortest specimens in an
insect collection.
SMP
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
4. Model with mathematics
6. Attend to precision
Instructional Strategies and Resource Support
Formative Assessments
http://www.k-5mathteachingresources.com/supportfiles/lengthofantslineplot.pdf

After students create the line plot from the data in the link above,
have them round each fraction to the nearest ¼ inch and have
them explain how the data looks different.

Once they have rounded the data to the nearest ¼ inch, ask the
students if the length of the objects changed since the data looks
different.
(Example on the k-5 task linked above, the shortest ant is 1/8 of an
inch, which would round to ¼ . The data looks different, but the
length of the actual ant didn’t change.)
Note: students can get confused about what the x’s or dots stand
for (the number with the most x’s means it has the most data not
that it is the largest object)

Have students look at a line plot and write questions that could be
answered using addition and/or subtraction.

Line plots can be counting by 1/4’s and have all denominators of
4 or it could have halves on there as well. This is another way to
practice equivalent fractions and relate to a ruler.
Note: measuring to the nearest 1/8 of an inch is not a standard, but it
this unit would be a good place to review ruler use.
8
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Extend understanding of fraction equivalence and ordering.
Maryland College and Career-Ready
Standards
4.NF.1
Explain why a fraction a/b is
equivalent to a fraction (n ×
a)/(n × b) by using visual
fraction models, with attention
to how the number and size of
the parts differ even though
the two fractions themselves
are the same size. Use this
principle to recognize and
generate equivalent fractions.
Instructional Strategies and Resource Support
Formative Assessments
This standard was previously introduced in unit 3 as students
worked with fraction equivalence and comparison. In unit 3,
students built equivalence understanding using fraction models
and number lines. Additionally, they began to make
connections between model representations and multiplying the
numerator and denominator by the same number. In this unit,
students will continue to strengthen their equivalence knowledge
primarily in conjunction with 2 other standards:
4.MD.B.4 – In creating, interpreting, and manipulating line plots
with fractional data, students will utilize fraction equivalence and
comparison specifically with measurements in halves, fourths,
and eighths. Students begin to recognize and use the
and that comparisons can be
made using this knowledge. For example, if then > or
<. Refer to pp. 7-8 of the instructional guide for more
understanding that
SMP
1. Make sense of problems and
persevere in solving them
7. Look for and make use of
structure.
information.
4.NF.C.5 - In this standard, students explore decimal fractions
which are fractions with a denominator that is a power of ten (ex.
10 and 100). They begin to develop and then use equivalency
understanding when converting tenths to hundredths with
standards 4.NF.6 and 4.NF.7. Students will develop their
understanding of decimal equivalency initially with grid models
and number lines and incorporate this understanding in using
equations. Refer to pp. 10-11 of the instructional guide for more
information.
9
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Maryland College and CareerReady Standards
Understand decimal notation for fractions, and compare decimal
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.
For example, express 3/10
as 30/100 and add 3/10 +
4/100 = 34/100
NOTE: 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.
SMP
2. Reason abstractly and
quantitatively
3. Construct viable arguments
and critique the reasoning of
others.
5. Use appropriate tools
strategically.
6. Attend to precision
Instructional Strategies and Resource Support
In this standard, students explore decimal fractions which are fractions with
a denominator that is a power of ten (ex. 10 and 100). They begin to
develop equivalency understanding when converting tenths to
hundredths.
Because it involves partitioning into 10 equal parts and treating the parts
as numbers called one tenth and one hundredth, work with these fractions
can be used as preparation to extend the base-ten system to non-whole
numbers.
Money concepts provide context to help foster equivalency reasoning
with decimal fractions. For example, since there are 10 dimes in a dollar, 4
dimes is 4/10 of a dollar. It is also equal to 40/100 of a dollar, because it is
40 cents, and there are 100 cents in a dollar.
Formative Assessments
FA.4.NF.C.5.a
Write an addition sentence
in which the sum equals
. Make one addend have
a denominator of 10 and
the other addend have a
denominator of 100.
FA.4.NF.C.5.b
The tile floor of Eli’s Pizzeria is
shown below. Diners eat in the
shaded section, and the kitchen
is in the unshaded section.
4
4 x 10
40
=
=
10
10 x 10
100
Students can represent this understanding using grids. They can partition
one whole into ten equal parts (tenths). Each tenth is partitioned into ten
equal parts (hundredths).
FA.4.NF.C.5.c
There are 10 dimes in a dollar,
so one dime = of a dollar.
There are 100 pennies in a
dollar, so one penny = of a
dollar.
SMP
Students base explanations/reasoning on diagrams (whether provided in the prompt or
constructed by the student in her response), connecting the diagrams to a written
(symbolic) method.
Megan has 4 dimes and 20
pennies. Shade in the model to
show the fraction of a dollar
that Megan has.
10
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and CareerReady Standards
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.
For example, express 3/10 as
30/100 and add 3/10 + 4/100 =
34/100
NOTE: 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.
Instructional Strategies and Resource Support
Grade 4 students learn to add decimal fractions by converting
them to fractions with the same denominator, in preparation for
general fraction addition in Grade 5:
Formative Assessments
FA.4.NF.C.5.d
Find the sum of each.
 


Each part of this Illustrative task highlights different aspects of
4.NF.C.5 including the commutative property, attention to the
denominator, sums less than and greater than one, and
changing hundredths to tenths.
http://www.illustrativemathematics.org/illustrations/153

FA.4.NF.C.5.e
Sam solved the following
problems and is sure he has all
problems correct. Draw a box
around the equations that are
true.



SMP
2. Reason abstractly and
quantitatively
3. Construct viable arguments and
critique the reasoning of others.
5. Use appropriate tools
strategically.
6. Attend to precision
11
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and CareerReady Standards
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.
SMP
1. Make sense of problems
and persevere in solving them
2. Reason abstractly and
quantitatively
4. Model with mathematics
6. Attend to precision
Instructional Strategies and Resource Support
Formative Assessments
This standard is 4th grade’s introduction to decimals.
The decimal point is used to signify the location of the ones place, but its location
may suggest there should be a “oneths" place to its right in order to create
symmetry with respect to the decimal point. However, because one is the basic
unit from which the other base-ten units are derived, the symmetry occurs instead
with the ones place.
FA.4.NF.C.6.a
Plot the fraction or decimal
on each number line.
Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, April
2011, page 12
Students apply and build upon the work with decimal fractions in order to
understand that a number can be represented as both a fraction and a decimal.
Students make connections between fractions with denominators of 10 and 100,
visual models, decimal digi blocks, and a place value chart. By reading fraction
names, students say 54/100 as fifty-four hundredths and rewrite this as 0.54 or
represent it on a place value model.
Hundreds
Tens
Ones
.
.
0
Tenths
Hundredths
5
4
Using number lines, students begin to explore the size of a decimal relative to
whole numbers as well as other decimals.
0.54
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
FA.4.NF.C.6.b
Partition and label the
number line to show the
location of 3.65 on the line.
FA.4.NF.C.6.c
Look at the number line
below.


















0
0.1
0.4 0.5 0.6
1.0
 0.2 0.3
 0.7



 0.8
 0.9







 (or 5 tenths) and less
Students can see that 0.54 is greater than 50 hundredths








than 60 hundredths (or 6 tenths). It is also less than 1 whole. 
12
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and CareerReady Standards
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.
SMP
1. Make sense of problems and
persevere in solving them
2. Reason abstractly and
quantitatively
4. Model with mathematics
6. Attend to precision
Instructional Strategies and Resource Support
In exploring place value, students need to understand decimal values
in a variety of ways:
Fraction:
1 
Decimal:
1.73
Model:
Formative Assessments
FA.4.NF.C.6d
Scott spent 7 dimes and 6
pennies to buy a bottle of
water. Represent this
fractional amount in each
form.
 + 
Expanded Fraction Form:
1+
Expanded Decimal Form:
1 + 0.7 + 0.03
Having students read decimal names aloud will help them make the
connection to fractions. When reading 1.73, we say “one and seventythree hundredths” which is how we would say the value as a mixed
number.
It takes time for students to build fluency with decimal representations.
Layered cards help students explore place value and decimal
equivalency such as six tenths is sixty hundredths. Additionally, students
strengthen their understanding of decimal parts by decomposing
decimals (example: 17 hundredths = 1 tenth + 4 hundredths + 3
hundredths).
http://www.illustrativemathematics.org/illustrations/103
13
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and Career-Ready
Standards
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.
Instructional Strategies and Resource Support
Formative Assessments
Focus in Grade 4 Teaching with Curriculum Focal Points , p. 46-48
As with fractions, students should understand that decimal
comparisons are only valid when referring to the same size whole.
Area models, decimal grids, decimal digi blocks, and number lines
are valuable tools to help students reason about the relative size of
decimals.
When comparing decimals, students use their understanding of the
relationship among one, tenths, and hundredths. They recognize that
10 tenths = 1 whole and 10 hundredths = 1 tenth. Students can apply
this understanding to compare 3.4 and 3.39, for example: 3.39 is 1
hundredth less than 3.4, because 3.4 is equivalent to 3.40.
FA.4.NF.C.7.a
Patty covered 0.4 of her family
room floor with carpeting. The
amount of carpeting in Mark’s
family room covers more than
0.4 but less than 0.5.
Shade in how Mark’s floor may
look.
FA.4.NF.C.7.b
Represent each decimal using
the decimal grids.
SMP
3. Construct viable arguments and
critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision
Decimal-Fraction Connections
Students use their knowledge of comparing fractions with like
denominators to compare decimals. For example, to compare 0.3
and 0.4, this can be thought of as 3/10 and 4/10. Students recognize
that there are fewer tenths in 3/10 than 4/10. So, 3/10 < 4/10 and 0.3 <
0.4.
Additionally, students can use their understanding of equivalent
fractions to compare decimals. Often students compare decimals
incorrectly, such as with 0.6 and 0.25 and state that 0.6 < 0.25,
because 6 < 25. When using equivalencies, students can reason that
0.6 = 6/10 = 60/100. Therefore, 60/100 > 25/100 or 0.6 > 0.25.
0.13
0.3
0.09
FA.4.NF.C.7.c
Place a checkmark to indicate if
each inequality is true or false. If
it false, explain your reasoning.
True
False
Reason
0.6 > 0.40
0.17 < 0.34
0.8 < 0.69
14
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and Career-Ready
Standards
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.
SMP
3. Construct viable arguments and
critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision
Instructional Strategies and Resource Support
Using Decimal Place Value
In grade 3, students compared whole numbers by considering the
greatest place in each number first. They can apply this same
reasoning when comparing decimals. For example:
Comparison
2.4 > 0.99
Reasoning
Whole numbers are greater than tenths;
2 is greater than 0, so 2.4 is greater than
0.99
0.09 < 0.8
Hundredths are less than tenths; so, 0.09 is
less than 0.8
231.5 > 97.16
Hundreds are greater than tens; 200 > 90,
so 231.5 is greater than 97.16
Students also apply this understanding when comparing decimals
that have the same value in the greatest place. They then
compare values in each place from left to right.
Comparison
Reasoning

4.85 > 4.82


The greatest place is the ones place. 4
ones = 4 ones
The next largest place is the tenths place.
8 tenths = 8 tenths
The next greatest place is the hundredths
place. 5 hundredths > 2 hundredths, so
4.85 is greater than 4.82
Formative Assessments
FA.4.NF.C.7d
At the track meet, Sean and
Wesley competed in the
hundred-yard dash. Spectators
could not tell who won, because
the boys ran side-by-side for the
entire race.
The times were posted. Sean
yelled, “Yes, I won! My time
was 10.3 seconds.” Wesley said,
“Wait a minute. I won, because
my time was 10.18 seconds.”
FA.4.NF.C.7e
Name three decimals between
1.85 and 2.1.
_____ ______ _____
Use a visual model to show
your thinking.
15
Carroll County Public Schools Elementary Mathematics Instructional Guide (Grade 4)
Unit 4: Addition/Subtraction with Fractions and Decimal Fractions – 20 Days
Understand decimal notation for fractions, and compare decimal
fractions.
Maryland College and Career-Ready
Standards
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.
Instructional Strategies and Resource Support
Formative Assessments
Monetary Connections
A strong understanding of money will help students build fraction
and decimal concepts.
Since 100 pennies are in a dollar:
Coins
1 penny
6 dimes
2 pennies
Fraction

Value
Decimal
1
0.01
62 
0.62
40
0.40 or 0.4



4 dimes
or
SMP
3. Construct viable arguments and
critique the reasoning of others.
5. Use appropriate tools strategically.
6. Attend to precision
16