4
th
Grade Mathematics ● Unpacked Contents
For the new Standard Course of Study that will be effective in all North Carolina schools in the 2017-18 School Year.
This document is designed to help North Carolina educators teach the 4
th
Grade Mathematics Standard Course of Study. NCDPI staff are
continually updating and improving these tools to better serve teachers and districts.
What is the purpose of this document?
The purpose of this document is to increase student achievement by ensuring educators understand the expectations of the new standards. This
document may also be used to facilitate discussion among teachers and curriculum staff and to encourage coherence in the sequence, pacing,
and units of study for grade-level curricula. This document, along with on-going professional development, is one of many resources used to
understand and teach the NC SCOS.
What is in the document?
This document includes a detailed clarification of each standard in the grade level along with a sample of questions or directions that may be
used during the instructional sequence to determine whether students are meeting the learning objective outlined by the standard. These items
are included to support classroom instruction and are not intended to reflect summative assessment items. The examples included may not fully
address the scope of the standard. The document also includes a table of contents of the standards organized by domain with hyperlinks to assist
in navigating the electronic version of this instructional support tool.
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Link for: Feedback for NC’s Math Unpacking Documents
We will use your input to refine our unpacking of the standards. Thank You!
Just want the standards alone?
Link for: NC Mathematics Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
2
North Carolina Course of Study – 4
th
Grade Standards
Standards for Mathematical Practice
Operations & Algebraic
Thinking
Number & Operations in
Base Ten
Number & Operations-
Fraction
Measurement & Data Geometry
Represent and solve
problems involving
multiplication and division.
NC.4.OA.1
Use the four operations with
whole numbers to solve
problems.
NC.4.OA.3
Gain familiarity with factors
and multiples.
NC.4.OA.4
Generate and analyze
patterns.
NC.4.OA.5
Generalize place value
understanding for multi-digit
whole numbers.
NC.4.NBT.1
NC.4.NBT.2
NC.4.NBT.7
Use place value
understanding and
properties of operations to
perform multi-digit
arithmetic.
NC.4.NBT.4
NC.4.NBT.5
NC.4.NBT.6
Extend understanding of
fractions.
NC.4.NF.1
NC.4.NF.2
Build fractions from unit
fractions by applying and
extending previous
understandings of
operations on whole
numbers.
NC.4.NF.3
Use unit fractions to
understand operations of
fractions.
NC.4.NF.4
Understand decimal notation
for fractions, and compare
decimal fractions.
NC.4.NF.6
NC.4.NF.7
Solve problems involving
measurement.
NC.4.MD.1
NC.4.MD.2
NC.4.MD.8
Solve problems involving
area and perimeter.
NC.4.MD.3
Represent and interpret
data.
NC.4.MD.4
Understand concepts of
angle and measure angles.
NC.4.MD.6
Classify shapes based on
lines and angles in two-
dimensional figures.
NC.4.G.1
NC.4.G.2
NC.4.G.3
NC 4
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Grade Math Unpacking - Revised June 2022
3
Standards for Mathematical Practice
Explanation and Example
and persevere in solving
them.
Mathematically proficient students in grade 4 know that doing mathematics involves solving problems and discussing how they
solved them. Students explain to themselves the meaning of a problem and look for ways to solve it. Fourth graders may use
concrete objects or pictures to help them conceptualize and solve problems. They may check their thinking by asking
themselves, “Does this make sense?” They listen to the strategies of others and will try different approaches. They often will
use another method to check their answers.
quantitatively.
Mathematically proficient fourth grade students should recognize that a number represents a specific quantity. They connect
the quantity to written symbols and create a logical representation of the problem at hand, considering both the appropriate
units involved and the meaning of quantities. They extend this understanding from whole numbers to their work with fractions
and decimals. Students write simple expressions, record calculations with numbers, and represent or round numbers using
place value concepts.
arguments and critique
the reasoning of others.
In fourth grade mathematically proficient students may construct arguments using concrete referents, such as objects, pictures,
and drawings. They explain their thinking and make connections between models and equations. They refine their
mathematical communication skills as they participate in mathematical discussions involving questions like “How did you get
that?” and “Why is that true?” They explain their thinking to others and respond to others’ thinking.
Mathematically proficient fourth grade students experiment with representing problem situations in multiple ways including
numbers, words (mathematical language), drawing pictures, using objects, making a chart, list, or graph, creating equations,
etc. Students need opportunities to connect the different representations and explain the connections. They should be able to
use all of these representations as needed. Fourth graders should evaluate their results in the context of the situation and
reflect on whether the results make sense.
strategically.
Mathematically proficient fourth grader students consider the available tools (including estimation) when solving a mathematical
problem and decide when certain tools might be helpful. For instance, they may use graph paper or a number line to represent
and compare decimals and protractors to measure angles. They use other measurement tools to understand the relative size of
units within a system and express measurements given in larger units in terms of smaller units.
As fourth grader students develop their mathematical communication skills, they try to use clear and precise language in their
discussions with others and in their own reasoning. They are careful about specifying units of measure and state the meaning
of the symbols they choose. For instance, they use appropriate labels when creating a line plot.
structure.
In fourth grade mathematically proficient students look closely to discover a pattern or structure. For instance, students use
properties of operations to explain calculations (partial products model). They relate representations of counting problems such
as tree diagrams and arrays to the multiplication principal of counting. They generate number or shape patterns that follow a
given rule.
regularity in repeated
Students in fourth grade should notice repetitive actions in computation to make generalizations Students use models to
explain calculations and understand how algorithms work. They also use models to examine patterns and generate their own
algorithms. For example, students use visual fraction models to write equivalent fractions.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
4
Operations and Algebraic Thinking
Represent and solve problems involving multiplication and division.
NC.4.OA.1 Interpret a multiplication equation as a comparison. Multiply or divide to solve word problems involving multiplicative comparisons using models and
equations with a symbol for the unknown number. Distinguish multiplicative comparison from additive comparison.
Clarification
Checking for Understanding
A multiplicative comparison is a situation in which one quantity is multiplied by
a specified number to get another quantity (e.g., “a is n times as much as b”).
In a multiplicative comparison, the underlying question is what factor would
multiply one quantity in order to result in the other. Students should be able to
identify and verbalize which quantity is being multiplied and which number
tells how many times.
Students should be able to translate comparative situations into equations
with a variable and then find the value of the variable. Many opportunities to
solve contextual problems and write and identify equations and statements
for multiplicative comparison should be provided. Likewise, when given an
equation students should be able to create a model and a word problem that
matches the equation.
Students are expected to distinguish between additive and multiplicative
comparisons. Additive comparisons focus on the difference between two
quantities. Multiplicative comparisons focus on one quantity being some
number times larger than another.
For example:
Additive comparison:
Jane has 8 apples and Sam has 5 apples. How many more apples does
Jane have than Sam.
Multiplicative comparison:
Jane has 8 apples and Sam has 5 times as many apples as Jane. How
many apples does Sam have?
In this standard the referent, which is the number of times a quantity is larger
than or smaller than another quantity, should be limited to 10 or less. Further,
while students multiply a fraction by a whole number in Grade 4 (NC.4.NF.4),
multiplicative comparison situations are limited to only whole numbers.
Sally is five years old. Her mom is eight times older. How many years older is
Sally’s mom compared to Sally?
Possible responses:
Student A:
First, I need to find the age of Sally’s mom.
5 x 8 = 40.
The difference between the ages of Sally’s mom and Sally is 40 – 5 = 35.
Student B:
Sally is 5. Sally’s mom is 8 times older which is 8 x 5 or 40. The
difference between their ages is 40 – 5 = 35.
A book costs $18. That is 3 times more than a DVD. How much does a DVD
cost?
Possible response:
Student A:
18 ÷ p = 3
18 ÷ p = 3
or 3 x p = 18
Student B:
The book is $18 and 3 times more than a DVD.
I know that 6 x 3 = 18 so the DVD is $6.
5
5
5
5
5
5
5
5
40
NC 4
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Grade Math Unpacking - Revised June 2022
5
Represent and solve problems involving multiplication and division.
NC.4.OA.1 Interpret a multiplication equation as a comparison. Multiply or divide to solve word problems involving multiplicative comparisons using models and
equations with a symbol for the unknown number. Distinguish multiplicative comparison from additive comparison.
Clarification
Checking for Understanding
Brandi walks for 72 minutes on Saturday. Terrence walks for 8 minutes.
a. Write a question about an additive comparison between the amount of
time that Brandi and Terrence walked.
b. Solve the additive comparison question that you wrote using a picture
or equation.
c. Write a question about a multiplicative comparison between the amount
of time that Brandi and Terrence walked.
d. Solve the multiplicative comparison question that you wrote using a
picture or equation.
Possible response:
Additive comparison questions: What is the difference between the
amount of time that Brandi walked compared to the time that Terrence
walked? How many more minutes did Brandi walk compared to
Terrence? How many fewer minutes did Terrence walk than Brandi?
Multiplicative comparison questions: How many times more minutes did
Brandi walk compared to Terrence? How many times fewer minutes did
Terrence walk compared to Brandi.
NC 4
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Grade Math Unpacking - Revised June 2022
6
Represent and solve problems involving multiplication and division.
NC.4.OA.1 Interpret a multiplication equation as a comparison. Multiply or divide to solve word problems involving multiplicative comparisons using models and
equations with a symbol for the unknown number. Distinguish multiplicative comparison from additive comparison.
Multiplication & Division Situations
Unknown Product
3 x 6 = ?
Group Size Unknown
(How many in each group?”
Division)
3 x ? = 18 and 18 ÷ 3 = ?
Number of Groups Unknown
(“How many groups?” Division)
? x 6 = 18 and 18 ÷ 6 = ?
Equal Groups
There are 3 bags with 6 plums in each
bag. How many plums are there in all?
Measurement example. You need 3
lengths of string, each 6 inches long.
How much string will you need
altogether?
(Grade 3)
If 18 plums are shared equally into 3
bags, then how many plums will be in
each bag?
Measurement example. You have 18
inches of string, which you will cut into
3 equal pieces. How long will each
piece of string be?
(Grade 3)
If 18 plums are to be packed 6 to a bag,
then how many bags are needed?
Measurement example. You have 18
inches of string, which you will cut into
pieces that are 6 inches long. How
many pieces of string will you have?
(Grade 3)
Arrays & Area
There are 3 rows of apples with 6
apples in each row. How many apples
are there?
Area example. What is the area of a 3
cm by 6 cm rectangle?
(Grade 3)
If 18 apples are arranged into 3 equal
rows, how many apples will be in each
row?
Area example. A rectangle has area 18
square centimeters. If one side is 3 cm
long, how long is a side next to it?
(Grade 3)
If 18 apples are arranged into equal
rows of 6 apples, how many rows will
there be?
Area example. A rectangle has area 18
square centimeters. If one side is 6 cm
long, how long is a side next to it?
(Grade 3)
Compare
Compare- Larger Unknown
*New in Grade 4. All three numbers
should be whole numbers in Grade 4.
A blue hat costs $6. A red hat costs 3
times as much as the blue hat. How
much does the red hat cost?
Measurement example. A rubber band
is 6 cm long. How long will the rubber
band be when it is stretched to be 3
times as long?
Compare- Smaller Unknown
*New in Grade 4. All three numbers
should be whole numbers in Grade 4.
A red hat costs $18 and that is 3 times
as much as a blue hat costs. How
much does a blue hat cost?
Measurement example. A rubber band
is stretched to be 18 cm long and that
is 3 times as long as it was at first. How
long was the rubber band at first?
Compare- Difference Unknown
*New in Grade 4. All three numbers
should be whole numbers in Grade 4.
A red hat costs $18 and a blue hat
costs $6. How many times as much
does the red hat cost compared to the
blue hat?
Measurement example. A rubber band
was 6 cm long at first. Now it is
stretched to be 18 cm long. How many
times as long is the rubber band now
as it was at first?
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
7
Use the four operations with whole numbers to solve problems.
NC.4.OA.3 Solve two-step word problems involving the four operations with whole numbers.
• Use estimation strategies to assess reasonableness of answers.
• Interpret remainders in word problems.
• Represent problems using equations with a letter standing for the unknown quantity.
Clarification
Checking for Understanding
The focus in this standard is to have students use and discuss various
strategies for solving word problems using all four operations. Students
should build on the problem solving strategies they developed in earlier
grades and apply those strategies to multi-step problems.
Students should be introduced to a variety of estimation strategies.
Estimation strategies include identifying when estimation is appropriate,
determining the level of accuracy needed, selecting the appropriate method
of estimation, and verifying solutions or determining the reasonableness of
situations using various estimation strategies. Estimation strategies include,
but are not limited to:
• front-end estimation with adjusting (using the highest place value and
estimating from the front end, making adjustments to the estimate by
taking into account the remaining amounts),
• rounding and adjusting (students round down or round up and then
adjust their estimate depending on how much the rounding affected
the original values),
• using friendly or compatible numbers such as factors (students seek
to fit numbers together - e.g., rounding to factors and grouping
numbers together that have round sums like 100 or 1000),
• using benchmark numbers that are easy to compute (students select
close whole numbers for fractions or decimals to determine an
estimate).
Problems should be structured so that all acceptable estimation strategies will
arrive at a reasonable answer. The assessment of estimation strategies
should only have one reasonable answer (500 or 530), or a range (between
500 and 550).
In this standard, students are required to solve division tasks and interpret
remainders. All problems involving remainders should be in a real-world
context that influences how the remainder should be interpreted.
In both Grades 4 and 5 here are ways that students are expected to interpret
remainders:
Two-step word problem with addition and estimation:
On a vacation, your family travels 267 miles on the first day, 194 miles on the
second day and 34 miles on the third day. How many miles did they travel
total? How do you know your answer is reasonable?
Possible responses:
Student 1
I first thought about 267
and 34. I noticed that
their sum is about 300.
Then I knew that 194 is
close to 200. When I
put 300 and 200
together, I get 500.
Student 2
I first thought about
194. It is really close to
200. I also have 2
hundreds in 267. That
gives me a total of 4
hundreds. Then I have
67 in 267 and the 34.
When I put 67 and 34
together that is really
close to 100. When I
add that hundred to the
4 hundreds that I
already had, I end up
with 500.
Student 3
I rounded 267 to 300. I
rounded 194 to 200. I
rounded 34 to 30.
When I added 300, 200
and 30, I know my
answer will be about
530.
Two-step word problem with multiplication and subtraction and estimation:
Your class is collecting bottled water for a service project. The goal is to collect
300 bottles of water. On the first day, Max brings in 6 packs with 6 bottles in
each container. About how many bottles of water still need to be collected?
Possible responses:
Student 1
First, I multiplied 6 and 6 which
equals 36. I’m trying to get to 300. 36
is close to 40, and 40 plus 60 is 100.
Then I need 2 more hundreds. So,
we still need about 260 bottles.
Student 2
First, I multiplied 6 and 6 which
equals 36. I know 36 is about 40 and
300-40 = 260, so we need about 260
more bottles.
NC 4
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Grade Math Unpacking - Revised June 2022
8
Use the four operations with whole numbers to solve problems.
NC.4.OA.3 Solve two-step word problems involving the four operations with whole numbers.
• Use estimation strategies to assess reasonableness of answers.
• Interpret remainders in word problems.
• Represent problems using equations with a letter standing for the unknown quantity.
Clarification
Checking for Understanding
There are 19 pens that need to be shared between 3 friends and myself
(19 4)
Give quotient and
remainder
If we leave the leftover
pens on the table how
many pens does each
person get? (4)
How many pens are
left on the table? (3)
Put remainder in 1
group
If we give all of the
leftovers to only 1
person how many
pens will people
receive?
3 people receive 4
pens
1 person receives 7
pens.
Share remainder
among groups
If we give the leftovers to
different people until we
run out how many pens
will people receive?
3 people receive 5 pens
1 person receives 4 pens
There are 130 children going on the field trip. Twenty-four children can fit
on a bus.
Adding 1 to the
quotient
How many busses are
needed in order to
take all of the
children? (6)
Give quotient and
remainder
How many of the
busses have 24
children? (5)
How many children
are on the bus that
does not have 24
children? (10)
Multiplication and division with estimation and a remainder
Ana bakes 13 dozen cookies. She then puts the into bags with 5 cookies in
each bag. She keeps the leftover cookies for herself. How many bags of
cookies does she have? How many cookies does she keep for herself?
Estimate the number of bags before solving.
Possible responses:
Student A:
For my estimate, 13 and 12 are close to 10 so Ana makes about 100
cookies. If she puts them in 5 bags, she will have about 20 bags since
100 ÷ 5 = 20.
In order to solve this problem, first I multiplied 13 and 12.
In order to determine how many bags of 5 I can
make with my 156 cookies I solved 156 ÷ 5 = __.
The answer using partial quotients is 20 + 10 + 1
which is 31. Ana can make 31 bags of cookies.
There is a remainder of 1 so Ana can have 1 cookie
for herself.
NC 4
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Grade Math Unpacking - Revised June 2022
9
Use the four operations with whole numbers to solve problems.
NC.4.OA.3 Solve two-step word problems involving the four operations with whole numbers.
• Use estimation strategies to assess reasonableness of answers.
• Interpret remainders in word problems.
• Represent problems using equations with a letter standing for the unknown quantity.
Clarification
Checking for Understanding
Interpret remainders in word problems
Write different word problems involving 44 ÷ 6 = ? where the answers are best
represented as:
Problem A: 7
Problem B: 7 r 2
Problem C: 8
Problem D: 7 or 8
Problem E: 9
Possible responses:
Problem A: 7. Mary had 44 pencils. Six pencils fit into each of her
pencil pouches. How many pouches did she fill? 44 ÷ 6 = p; p = 7 r 2.
Mary can fill 7 pouches completely.
Problem B: 7 r 2. Mary had 44 pencils. Six pencils fit into each of her
pencil pouches. How many pouches could she fill and how many
pencils would she have left? 44 ÷ 6 = p; p = 7 r 2; Mary can fill 7
pouches and have 2 left over.
Problem C: 8. Mary had 44 pencils. Six pencils fit into each of her
pencil pouches. What would the fewest number of pouches she would
need in order to hold all of her pencils? 44 ÷ 6 = p; p = 7 r 2; Mary
needs 8 pouches to hold all of the pencils.
Problem D: 7 or 8. Mary had 44 pencils. She divided them equally
among her friends before giving one of the leftovers to each of her
friends. How many pencils could her friends have received? 44 ÷ 6 = p;
p = 7 r 2; Some of her friends received 7 pencils. Two friends received
8 pencils.
Problem E: 9. Mary had 44 pencils. She put them into 6 different bags.
All of the remaining pencils were put into one bag. How many pencils
were in the bat that had the most pencils?
There are 156 students going on a roller coaster. If each car of the roller
coaster holds 8 students, how many roller coaster cars are needed?
156 ÷ 8 = b; b = 19 R 4; They will need 20 cars because 19 cars would
not hold all of the students.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
10
Gain familiarity with factors and multiples.
NC.4.OA.4 Find all factor pairs for whole numbers up to and including 50 to:
• Recognize that a whole number is a multiple of each of its factors.
• Determine whether a given whole number is a multiple of a given one-digit number.
• Determine if the number is prime or composite.
Clarification
Checking for Understanding
This standard requires students to
demonstrate understanding of factors and
multiples of whole numbers up to and
including 50. Factor pairs include two
numbers that when multiplied result in a
particular product. Students should be
given opportunities to explore factor pairs
with concrete objects and drawings to
represent arrays.
Multiples are the result of multiplying two
whole numbers. Multiples can be related
to factors, and this relationship can be
discovered through exploration with
arrays. Students can build on their
understanding of skip counting by a given
number to determine the multiples of the
given number.
As students explore and discover
patterns, they build a conceptual
understanding of prime and composite
numbers. Prime numbers have exactly
two factors, the number one and their own
number. For example, the number 17 has
the factors of 1 and 17. Composite
numbers have more than two factors. For
example, 8 has the factors 1, 2, 4, and 8.
A common misconception is that the
number 1 is prime, when it is neither
prime nor composite. Another common
misconception is that all prime numbers
are odd numbers. This is not true, since
the number 2 has only 2 factors, 1 and 2,
and is also an even number.
Recognize that a whole number is a multiple of each of its factors
Part 1:
There are 24 chairs in the art room. What are the different ways that the chairs can be arranged into equal groups if
you want at least 2 groups and want at least 2 chairs in each group?
• How do you know that you have found every arrangement? Write division equations to show your answers.
Explain how you know that you have found every arrangement.
Part 2:
There are 48 chairs in the multi-purpose room. What are the different ways that the chairs can be arranged into
equal groups if you want at least 2 groups and want at least 2 chairs in each group?
• How do you know that you have found every arrangement? Write division equations to show your answers.
• What relationship do you notice about the size of the groups if the chairs were arranged in 4 groups in both
Part 1 and Part 2?
• What about if the chairs were arranged in 8 groups? Explain why you think this relationship exists.
Possible response:
Part 1: 2 groups of 12, 3 groups of 8, 4 groups of 6, 6 groups of 4, 12 groups of 2. I know I have found
every group because the number of groups and group sizes should be all of the factors of 24 except for the
numbers 1 and 24.
Part 2: 2 groups of 24, 3 groups of 16, 4 groups of 12, 6 groups of 8, 8 groups of 6, 12 groups of 4, 16
groups of 3, 24 groups of 2.
I noticed that in Part 2 the number of chairs in a group is double or twice as large for the same number of
groups. For example, Part 1 had 4 groups of 6 and Part 2 had 4 groups of 12.
In 8 groups we have 8 groups of 3 in Part 1 and in Part 2 we have 8 groups of 6.
A landscaping company visits the school to talk about the possible ways to tile a patio and picnic area near the
playground. The school can afford between 24 and 30 square tiles.
• For each of the proposed number of tiles (24-30), determine all of the possible dimensions of rectangles
you could make.
• The space for the patio is configured so that there cannot be any more than 10 tiles in a row. For the
proposed number of tiles (24-30), determine which numbers would work as the total number of tiles.
• Which number of tiles provides the most flexibility in terms of the possible ways that the tiles could be
arranged? Explain your reasoning.
• Look at the number 29. How many different rectangles can you make? Explain whether 29 is a prime or
composite number.
NC 4
th
Grade Math Unpacking - Revised June 2022
11
Gain familiarity with factors and multiples.
NC.4.OA.4 Find all factor pairs for whole numbers up to and including 50 to:
• Recognize that a whole number is a multiple of each of its factors.
• Determine whether a given whole number is a multiple of a given one-digit number.
• Determine if the number is prime or composite.
Clarification
Checking for Understanding
Determine whether a given whole number is a multiple of a given-one digit number
Katrina and Nico are talking about the number 6 and 2. Katrina says that 6 is a factor of 2, while Nico says that 6 is
a multiple of 2. Who is correct? Use multiplication and division equations as well as the words factor and multiple to
explain your answer.
Possible Response:
Student A:
I put the counters into 2 rows with 3 counters in each row. Since 2 and 3
are the dimensions of my array I know that 2 and 3 are factors and 6 is a
multiple.
Student B:
Nico is correct. Multiples are the products when you multiply two factors. We know that 6 ÷ 2 = 3 and 2 x 3
= 6 so 6 is a multiple and 2 is a factor of 6.
Recognize if a number is prime or composite
Each number below is the area of a rectangle.
11, 12, 13, 14
1. Use square tiles and make all of the rectangles that you can that have that area.
2. Complete the table below. Based on the number of rectangles that you were able to make state whether
each number is prime or composite.
Area
Dimensions of Rectangles
Prime or Composite
11
12
13
14
Based on the number of rectangles that you were able to make state whether each number is prime or composite.
NC 4
th
Grade Math Unpacking - Revised June 2022
12
Gain familiarity with factors and multiples.
NC.4.OA.4 Find all factor pairs for whole numbers up to and including 50 to:
• Recognize that a whole number is a multiple of each of its factors.
• Determine whether a given whole number is a multiple of a given one-digit number.
• Determine if the number is prime or composite.
Clarification
Checking for Understanding
Possible Response:
Area
Dimensions of Rectangles
Prime or Composite
11
1x11, 11x1
Prime
12
1x12, 2x6, 3x4, 4x3, 6x2, 12x1
Composite
13
1x13, 13x1
Prime
14
1x14, 2x7, 7x2, 14x1
Composite
Vikas says that 37 is a prime number since it ends in a 7. Is Vikas correct that all numbers that end in a 7 are
prime? Look at the numbers that are less than 50 that have a 7 in them. Is each one prime or composite?
Use multiplication or division equations to support your answer.
Possible Response:
Student A:
Vikas is not correct. 27 is not prime since 9 x 3 = 27. The other numbers 7, 17, 37, and 47 are prime since
the only multiplication equation that equals those numbers include the factors 1 and that number.
Student B:
Vikas is not correct. 27 has the factors 1, 3, 9, and 27. The other numbers 7, 17, 37, and 47 are prime since
those numbers only have 2 factors which are 1 and itself.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
13
Generate and analyze patterns.
NC.4.OA.5 Generate and analyze a number or shape pattern that follows a given rule.
Clarification
Checking for Understanding
In this standard, students must generate or create a number or shape pattern
when they are given 1 rule. Students are also expected to analyze a pattern
in order to determine or generate the rule that was used to create the pattern.
The standard is a building block for alter grades since the ability to recognize
and explain patterns in mathematics leads students to developing the ability
to make generalizations, a foundational concept in algebraic thinking.
Students need multiple opportunities creating, extending, and analyzing
number and shape patterns.
In terms of vocabulary, patterns and rules are related. A pattern is a
sequence that repeats the same process over and over. A rule dictates the
process.
In Grade 4 students are expected to generate, analyze, and describe patterns
that are growing patterns. Growing patterns are generated by either following
a repeating rule (add 2 to the previous term) or a rule in which the rule
changes each time (add 1 more than we did to generate the previous term).
Pattern
Example
Rule
Repeating
pattern
The core of the pattern is
triangle, triangle, square and
it repeats.
Growing
pattern with a
repeating
rule
The pattern is a growing
pattern since the number
increases. The rule repeats
since the next term is always
generated by adding 2 to the
previous term.
Growing
pattern with a
changing rule
The pattern is a growing
pattern since the number we
add to each term changes.
The rule is to add 2 more than
we added to create the
previous term.
Generating a pattern and describing a rule:
Ted and Nancy both mow lawns during the summer to earn money.
Ted charges $6 per hour.
Nancy charges $12 per hour.
Complete the table to show how much Ted and Nancy would each earn based
on the amount of time that it took to mow a lawn.
Based on the data in the table, what is the rule for Ted? What is the rule for
Nancy?
Possible response:
Ted
Nancy
½ hour
3
6
1 hour
6
12
1 and ½ hours
9
18
2 hours
12
24
2 and ½ hours
15
30
3 hours
18
36
3 and ½ hours
21
42
4 hours
24
48
There are 4 beans in the jar. Each day 3 beans are added. How many beans
are in the jar for each of the first 5 days? Describe the rule for the pattern.
Possible response:
Day
Beans
0
4
1
7
2
10
3
13
4
16
5
19
The rule is that 3 more beans are added each day, which means we
add 3 to the previous term in the pattern.
NC 4
th
Grade Math Unpacking - Revised June 2022
14
Generate and analyze patterns.
NC.4.OA.5 Generate and analyze a number or shape pattern that follows a given rule.
Clarification
Checking for Understanding
A banquet company provides options for table arrangements: triangular tables,
square tables, and hexagonal tables. For each type of table, you can fit one
person on each side of the table. For parties, they want to put all of the tables
together so that every table shares at least one side with another table.
Based on this proposed arrangement, how many people could you sit at 1
triangular table? 2 connected triangular tables? 3 connected triangular tables?
4 connected triangular tables?
Solutions:
1 table: 3 people
2 tables: 4 people
3 tables: 5 people
4 tables: 6 people
Analyzing a pattern and describing the rule
Unique decides to walk each day after school with her friend. Below is the
number of miles she walks each day.
Day
1
2
3
4
5
Miles
1
2
4
7
10
Based on the information in the table is this a growing pattern or a repeating
pattern? What is the rule for the number of miles that Unique walks?
How many miles will Unique walk on Day 6? How many miles will Unique walk
on Day 7? Explain how you know.
Possible response:
The pattern is a growing pattern since the number we add to the
number before increases or grows.
The rule is that we find the next number by adding 1 more than we did
to find the previous number.
NC 4
th
Grade Math Unpacking - Revised June 2022
15
Generate and analyze patterns.
NC.4.OA.5 Generate and analyze a number or shape pattern that follows a given rule.
Clarification
Checking for Understanding
Nina looks at the pattern below.
Complete the table below based on the pattern.
How many squares will be in the 5th, 6th, and 7th terms?
What is the rule that is used to generate the pattern? Explain how you know.
Possible Response:
Term
1
2
3
4
5
6
7
Number
2
6
12
20
30
42
56
Difference between number
and previous number
+2
+4
+6
+8
+10
+12
+14
The rule that was used to generate the pattern was to add one row and
one column to the previous picture. By adding a row and a column you
are adding 2 more squares than the previous term. Specifically, the fifth
term has 30 squares which is 10 more than the term before. The pattern
was generated by adding 2, then adding 4, then adding 6 and so on.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
16
Number and Operations in Base Ten
Generalize place value understanding for multi-digit whole numbers.
NC.4.NBT.1 Explain that in a multi-digit whole number, a digit in one place represents 10 times as much as it represents in the place to its right, up to 100,000.
Clarification
Checking for Understanding
This standard calls for students to extend their understanding of place value
by exploring and explaining the relationship between the magnitude of a digit
and the value of that same digit if it were one place to the left of the original
digit.
Students are expected to reason about the magnitude of digits in a number
and make connections to the idea that a digit to the left of a number has a
value that is 10 times greater than the value of a digit to the immediate right.
What is the value of the 3 in hundreds place compared to the value of the 3 in
the ones place in the number 353? Use pictures, equations, or words to support
your answer.
Possible responses:
Student A:
353 = 300 + 50 + 3. The 3 in the hundreds place has a value of 300. The
3 in the ones place has a value of 3. The 3 in the hundreds place is 100
times greater than the 3 in the ones place.
Student B:
With base ten blocks I see that the 3 in the hundreds place is 300 and
the 3 in the ones place is 3. There are 100 ones in every group of 100 so
300 is 100 times more than the 3 in 353.
Student C:
3 x 10 is 30 and 3 x 10 x 10 is 300. So the value of the 3 in the hundreds
place is 10 x 10 or 100 times greater than the 3 in the ones place.
Brandi said, “In my pocket I have 25 of the same amount of dollar bills.
Part 1: For each of the scenarios below, write an equation and determine the
value of Brandi’s money.
a) 25 one dollar bills
b) 25 ten dollar bills
c) 25 hundred dollar bills
d) 25 one thousand dollar bills
Part 2: Brandi is trying to determine the relationship between the value of the 5
in 250 and the value of the 5 in 25,000. Use pictures, equations, or words to
explain the relationship between the 5’s in the two numbers.
NC 4
th
Grade Math Unpacking - Revised June 2022
17
Generalize place value understanding for multi-digit whole numbers.
NC.4.NBT.1 Explain that in a multi-digit whole number, a digit in one place represents 10 times as much as it represents in the place to its right, up to 100,000.
Clarification
Checking for Understanding
Possible Answers:
Part 1: 25 x 1 = 25; 25 x 10 = 250; 25 x 100 = 2,500; 25 x 1,000 =
25,000
Part 2: In 250 the 5 has a value of 50. In 25,000 the 5 has a value of
5,000. In order to move the 5 one place to the left we have to multiply by
10 so
50 x 10 = 500
50 x 10 x 10 = 5,000.
That means that the 5 in 250 is 10 x 10 or 100 times less than the 5 in
25,000.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
18
Generalize place value understanding for multi-digit whole numbers.
NC.4.NBT.2 Read and write multi-digit whole numbers up to and including 100,000 using numerals, number names, and expanded form.
Clarification
Checking for Understanding
This standard asks for students to write numbers in
various forms. Students should have flexibility with the
different forms of a number, including numbers that
are in nontraditional forms, where there is a number
greater than 9 in a given place.
The written form or number name of a number
requires students to write out a number in words such
as 285 = two hundred eighty-five. Traditional
expanded form is 285 = 200 + 80 + 5. However,
students should have opportunities to explore the idea
that 285 in nontraditional forms could also be written
as 28 tens plus 5 ones or 1 hundred, 18 tens, and 5
ones. They should also be comfortable with
expanding a number by place value using
multiplication in the notation, such as 285 = (2 x 100)
+ (8 x 10) + (5 x 1).
In this standard, students need to understand the role
of commas. Each group of 3 digits between commas
is read as hundreds, tens, and ones followed by the
appropriate unit. For example, the number 97,345
would be read ninety-seven thousand, three hundred
forty-five.
Place value with numerals and expanded form
Juice pouches are packaged in different ways. A box holds 10 pouches. A case holds 10 boxes. A crate
holds 10 cases.
Some students bring in juice boxes for Field Day. The information is below.
Miguel- 1 crate, 12 cases, 3 boxes and 6 pouches.
Aaron- 1 crate, 13 cases, 17 boxes, and 2 pouches.
Sarah- 1 crate, 12 cases, 2 boxes and 17 pouches.
Vicky- 1 crate, 14 cases, 6 boxes, and 9 pouches.
1) If each person were going to reorganize their drink pouches to use as many of the larger
containers as possible, how many of each container would each of them need?
2) How many total drink pouches does each student have?
Place value with number names and expanded form (integrated comparisons NC.4.NBT.7)
Which of the following is greater than 4,050? Use a place value chart or equations to support your
answer.
A. thirty-nine hundreds, 14 tens, 12 ones
B. thirty-eight hundreds, 24 tens, 9 ones
C. two hundred more than thirty-seven hundreds, 14 tens, 8 ones
D. forty hundreds, 4 tens, 19 ones
Possible answers:
Th
Hu
Tens
Ones
Number
A
4
39 + 1
40 = 4 Th + 0 H
0
14 + 1
15 = 1 H + 5 T
5
12 = 1 T + 2 O
2
4,052
B
4
38 + 2 H
40 = 4 Th + 0 H
0
24 = 2 H + 4 T
4
9
4,049
C
3 + 1
4
37 + 1
38 = 3 Th + 8 H
8 + 2 H = 10
10 = 1 Th + 0 H
0
14 = 1 H + 4 T
4
8
4,048
D
4
40 = 4 th + 0 H
0
4 + 1
5
19 = 1 T + 9 O
9
4,059
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
19
Generalize place value understanding for multi-digit whole numbers.
NC.4.NBT.7 Compare two multi-digit numbers up to and including 100,000 based on the values of the digits in each place, using >, =, and < symbols to record
the results of comparisons.
Clarification
Checking for Understanding
In this standard, students use their understanding of groups and
value of digits to compare two numbers by examining the value of
the digits. Students are expected to be able to compare numbers
presented in various forms, including nontraditional forms here the
value is greater than 9 for a given place.
Students should have ample experiences communicating their
comparisons in words before using symbols. Students were
introduced to the symbols greater than (>), less than (<) and equal
to (=) in Grade 1 and continued to use them in Grade 2 to
compare whole numbers, and in Grade 3 to compare fractions.
While students may have the skills to order more than 2 numbers,
this standard focuses on comparing two numbers and using
reasoning about place value to support the use of the various
symbols. This standard may be assessed by having students
order up to 4 numbers based on the understanding that ordering
those numbers includes comparing two numbers at a time.
Compare these two numbers. 75,452 __ 75,455
Possible responses:
Student A
Place Value
75,452 has 75 thousands, 4 hundreds, 5
tens, and 2 ones. 75,455 has 75
thousands, 4 hundreds, 5 tens, and 5
ones. They have the same number of
thousands, hundreds and the same
number of tens, but 455 has 5 ones and
75,452 only has 2 ones. 75,452 is less
than 455.
75,452 < 75,455
Student B
Counting
75,452 is less than 75,455. I know this
because they have the same thousands.
So, I’m going to compare 452 and 455.
When I count up I say 452 before I say
455. 75,452 is less than 75,455.
75,452 < 75,455
.
Find the population of these cities in number form. Then put them in order from least to
greatest:
Thomasville- 17 thousands, 98 tens, 14 ones
Henderson- 17 thousands, 2 hundreds, 15 ones
Elizabeth City- 17 thousands, thirty-two tens, five ones
Davidson- 1 ten thousand, 34 hundreds, 27 ones
Possible answer:
Ten Th
Th
H
T
O
Number
Thom
1
17
7
9
98
99
9
14
4
17,994
Hend
1
17
7
2
1
15
5
17,215
Eliz
1
17
7
3
32
2
5
17,325
Dav
1
1
3
34
4
2
27
7
13,427
Return to Standards
Least to Greatest:
Davidson: 13,427
Henderson: 17,215
Elizabeth City: 17,325
Thomasville: 17,994
NC 4
th
Grade Math Unpacking - Revised June 2022
20
Use place value understanding and properties of operations to perform multi-digit arithmetic.
NC.4.NBT.4 Add and subtract multi-digit whole numbers up to and including 100,000 using the standard algorithm with place value understanding.
Clarification
Checking for Understanding
In this standard, students build on their conceptual understanding of addition
and subtraction, their use of place value and their flexibility with multiple
strategies to make sense of the standard algorithm. They continue to use
place value in describing and justifying the processes they use to add and
subtract. Students are expected to explain their thinking to show
understanding of the algorithm.
This is the first grade level in which students are expected to be proficient at
using the standard algorithm to add and subtract. However, other previously
learned strategies are still appropriate for students to use prior to exploring
and developing proficiency with the algorithm. In Grade 3 students use
expanded form and drawings of base 10 blocks to solve addition and
subtraction problems.
In mathematics, an algorithm is defined by its steps and not by the way those
steps are recorded in writing. With this in mind, minor variations in methods of
recording standard algorithms are acceptable.
Students may ask if it is possible to subtract a larger number from a smaller
number. While it is not the focus or expectation of this standard in this grade,
students should know that it is mathematically possible, and they will be
learning more about that concept in later grades. If the misconception that
larger numbers cannot be subtracted from smaller numbers is confirmed or
reinforced, students may struggle to make the transition to negative numbers
in later grades.
The following amounts of juice were in separate containers after the school’s
parent breakfast.
• Container 1: 750 mL
• Container 2: 1,450 mL
• Container 3: 2,087 mL
Part 1: If all of the liquid was put into one large container how much liquid would
be in the large container?
Part 2: If the container holds 5,000 mL how much more liquid can still be added
until the container is full?
Possible Answer:
Part 1: Part 2:
9 9
1 1 4 10 10 10
1,450 5, 0 0 0
2,087 -4, 2 8 7
+750 7 1 3. There is still room for 713 mL of liquid.
4,287
On a field trip, three different schools send their fourth graders across town to
the high school for a math competition. Each school sends between 120 and
170 students each. There are 417 students total.
1. How many students could have come from each school? Show your
thinking.
2. Find another possible solution to this task. Show your thinking.
Possible answer:
The three schools send a total of 417 so the three addends must total
417. Each should be greater than 120 and smaller than 170.
Examples: 140 + 140 + 137 = 417; 120 + 127 + 170 = 417.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
21
Use place value understanding and properties of operations to perform multi-digit arithmetic.
NC.4.NBT.5 Multiply a whole number of up to three digits by a one-digit whole number, and multiply up to two two-digit numbers with place value understanding
using area models, partial products, and the properties of operations. Use models to make connections and develop the algorithm.
Clarification
Checking for Understanding
In this standard, students extend their understanding of multiplying a single-
digit factor times a multiple of ten (NC.3.NBT.3) to multiplying a single-digit
factor times two- or three-digit factors and two two-digit factors.
Students are expected to apply their understanding of place value and
various forms of a number to compute products. Students will also use area
models, partial products and properties of operations to solve multiplication
problems. Parentheses are not expected until grade 5, so students should
record multiplication using partial products without parentheses.
Connections should be made between models and written equations (as
shown below), but it is not necessary for fourth grade students to use the
standard algorithm. The standard algorithm for multiplication is not an
expectation until fifth grade.
There are 25 dozen cookies in the bakery. What is the total number of cookies
at the bakery?
Possible responses:
Student A
25 x12
I broke 12 up into 10 +
2
25 x 10 = 250
25 x 2 = 50
250 + 50 = 300
Student B
25 x 12
I broke 25 up into 5
groups of 5
5 x 12 = 60
There are 5 groups of 5
in 25
60 x 5 = 300
Student C
25 x 12
I doubled 25 and cut 12
in half to get 50 x 6
50 x 6 = 300
In the cafeteria, there are 14 long tables. Each table seats 16 students. How
many students can eat in the cafeteria at one time?
Possible responses:
Student A:
Using base ten blocks to model this problem, I broke 14 x 16 into this
equation:
10 x 10 + 4 x 10 + 6 x 10 + 6 x 4
1
hundred
4 tens
6 tens
24 ones
10
x
10
4 x
10
6 x
10
6 x
4
100
40
60
24
14 x 16 = 224
NC 4
th
Grade Math Unpacking - Revised June 2022
22
Use place value understanding and properties of operations to perform multi-digit arithmetic.
NC.4.NBT.5 Multiply a whole number of up to three digits by a one-digit whole number, and multiply up to two two-digit numbers with place value understanding
using area models, partial products, and the properties of operations. Use models to make connections and develop the algorithm.
Clarification
Checking for Understanding
Student B:
Using an open array, I broke 16 up
into 10 and 6. I knew 14 x 10 is 140.
For 14 x 6, I broke 6 up into 5 and 1
and did 14 x 6 is 14 x 5 + 14 which is
84. Then I added 140 + 84 = 224
Student C:
1 4
x 1 6
2 4
6 0
4 0
+1 0 0
2 2 4
There are 38 buses in the parking lot, and each bus holds 74 people. How
many people are able to ride the buses?
Possible Response:
I drew an open array and
broke 38 into 30+8 and
broke 74 into 70+4. I then
multiplied the partial products
and added them together.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
23
Use place value understanding and properties of operations to perform multi-digit arithmetic.
NC.4.NBT.6 Find whole-number quotients and remainders with up to three-digit dividends and one-digit divisors with place value understanding using
rectangular arrays, area models, repeated subtraction, partial quotients, properties of operations, and/or the relationship between multiplication and division.
Clarification
Checking for Understanding
In this standard, students build on their understanding of the meaning of
division and the relationship to multiplication by solving division problems in
and out of context that have a three-digit dividend and a one-digit divisor.
The focus of this standard is to build conceptual understanding of division.
Students are expected to use various strategies and explain their thinking.
Students are not expected to master the traditional algorithm until middle
school.
This standard calls for students to explore division through various strategies.
Students should be able to apply their understanding of place value and
various forms of a number to compute quotients. Students will also use arrays
and area models, repeated subtraction, partial quotients and properties of
operations to solve division problems
This standard also intersects division situations that have remainders. Refer
to NC.4.OA.3 for examples and information about how students are expected
to interpret and make sense of remainders in division situations.
The focus of this standard is to build conceptual understanding of division.
Students are expected to use various strategies and explain their thinking.
Students are not expected to master the traditional algorithm until middle
school.
A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She
wants to put the pencils in the boxes so that each box has the same number of
pencils. How many pencils will there be in each box?
Possible responses:
• Using Base 10 Blocks: Students build 260 with base 10 blocks and
distribute them into 4 equal groups. Some students may need to trade
the 2 hundreds for tens but others may easily recognize that 200
divided by 4 is 50.
• Using Place Value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4) = 50 + 15 = 65
• Using Multiplication: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5
= 65; so, 260 ÷ 4 = 65
There are 592 students participating in Field Day. They are put into teams of 8
for the competition. How many teams get created?
Student 1
592 divided by 8
There are 70 8’s in
560
592 - 560 = 32
There are 4 8’s in
32
70 + 4 = 74
Student 2
592 divided by 8
I know that 10 8’s
is 80
If I take out 50 8’s
that is 400
592 - 400 = 192
I can take out 20
more 8’s which is
160
192 - 160 = 32
4 groups of 8 is 32
I have none left
I took out 50, then 20 more, then 4
more. That’s 74
Student 3
I want to get to
592
8 x 25 = 200
8 x 25 = 200
8 x 25 = 200
200 + 200 + 200
= 600
600 - 8 = 592
I had 75 groups of
8 and took one
away, so there
are 74 teams
NC 4
th
Grade Math Unpacking - Revised June 2022
24
Use place value understanding and properties of operations to perform multi-digit arithmetic.
NC.4.NBT.6 Find whole-number quotients and remainders with up to three-digit dividends and one-digit divisors with place value understanding using
rectangular arrays, area models, repeated subtraction, partial quotients, properties of operations, and/or the relationship between multiplication and division.
Clarification
Checking for Understanding
Journey has 150 hair bows. She puts them into bags with 6 hair bows in each
bag. How many bags of hair bows will she have?
Possible Responses:
Open Array/Area Model
150 - 60 = 90
90 - 60 = 30
30 - 30 = 0
I started thinking about how many groups of 6 are in 250. I knew that
10x6 = 60 so I did that 2 times which meant that I had 20 groups of 6
which is 120. I knew that 120 is 30 from 150 and I knew that 5 x 6 = 30
so I could make 5 more bags of hair bows with the 30 that is left. My total
number of bags is 10 + 10 + 5 which is 25.
Partial Quotients
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
25
Number and Operations—Fractions
Extend understanding of fractions.
NC.4.NF.1 Explain why a fraction is equivalent to another fraction by using area and length fraction models, with attention to how the number and size of the
parts differ even though the two fractions themselves are the same size.
Clarification
Checking for Understanding
In this standard, students are expected to use area and length fraction models
to explain how fractions are equivalent to each other. Area models include
circles and rectangles while length models typically focus on number lines.
Students should not do any work on this standard without the use of a model.
Students only work with the denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100 in
this standard.
Kennedy rode her bike down a straight road and stopped at the halfway point
for water. Courtney also biked the same distance but broke her bike ride into 4
equal parts to get water. Did Courtney and Kennedy ever stop at the same
place to get water? How do you know? Draw and label a number line to
support your conclusions.
Possible student response:
Courtney broke her ride into fourths since she had 4 equal parts.
Kennedy stopped at the halfway point. Based on the number line
Courtney stopped
2
4
of the way down the road which is the same point as
one half.
Mr. Gomez and Mr. Lopez each have vegetable gardens that are the same
size. Mr. Gomez plants carrots in 6/8 of his garden. If Mr. Lopez has 4 regions
and wants to plant carrots in the same sized space as Mr. Gomez how many
of the regions will he plant carrots in? Draw a picture and write a sentence to
explain your answer.
Possible response:
Mr. Gomez:
6
8
Mr. Lopez:
I know that
2
8
=
1
4
and
6
8
=
2
8
+
2
8
+
2
8
. That means that Mr. Lopez will have
carrots in
3
4
of his garden since
1
4
+
1
4
+
1
4
=
3
4
.
NC 4
th
Grade Math Unpacking - Revised June 2022
26
Extend understanding of fractions.
NC.4.NF.1 Explain why a fraction is equivalent to another fraction by using area and length fraction models, with attention to how the number and size of the
parts differ even though the two fractions themselves are the same size.
Clarification
Checking for Understanding
Lauren is trying to think about fractions are equivalent to
1
2
.
Part 1: Using the denominators 4, 6, 8, and 12 use models to show all of the
fractions that are equivalent to
1
2
.
Part 2: Pick one of the fractions that is equivalent to
1
2
. Explain how you know
that fraction is equivalent to
1
2
.
Possible Response:
Part 1: Number lines or area models for any of the following fractions.
Part 2: The explanation clearly refers to a model and explains why the
fraction is equivalent to
1
2
.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
27
Extend understanding of fractions.
NC.4.NF.2 Compare two fractions with different numerators and different denominators, using the denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Recognize that
comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the
conclusions by:
• Reasoning about their size and using area and length models.
• Using benchmark fractions 0, ½, and a whole.
• Comparing common numerator or common denominators.
Clarification
Checking for Understanding
In this standard, students are expected to compare two
fractions with the denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.
In Grade 3 students used reasoning and models to compare
fractions that either had the same numerator or the same
denominator. In Grade 4, students reason about their size and
justify their comparison using area and length models, including
circles, rectangles, and number lines. Students are also
expected to use the benchmark fractions 0, ½ and 1 whole to
compare fractions.
Students should be able to put a set of up to 4 fractions in order
based on their size by comparing pairs of fractions.
Students should not use a procedure such as cross
multiplication, for comparing fractions. A student’s justification
that relies solely on explaining the steps of an algorithm would
not demonstrate mastery of this standard.
Crystal and Katie are each running a mile. Crystal ran
3
4
of a mile before stopping for water,
while Katie ran
2
3
of a mile before stopping. Who ran the farthest before stopping? Draw a
picture or write a sentence to support your answer.
Possible responses:
Student 1: Using length
models
Crystal ran more since
3
4
is farther from 0 than
2
3
.
Student 2: Comparing common numerators or denominators
I noticed that Crystal ran
1
4
less than a whole and Katie ran
1
3
less than a whole. Since
1
4
is smaller than
1
3
I know Crystal ran the farthest.
Tammy, Joe, and Lisa went to the movies. Each of them bought a small box of popcorn.
Tammy ate
3
6
of her popcorn. Joe ate
3
8
of his popcorn and Lisa ate
2
3
of her popcorn. Who ate
more?
Possible responses:
• I can compare
3
6
and
3
8
and I know that sixths are larger than eighths, so
3
6
>
3
8
.
(comparing common numerator or denominators)
• When I compare
2
3
to
3
6
, I know that
2
3
is equivalent to 4/6, so
2
3
>
3
6
. (reasoning
about their size)
• I know that
3
8
is less than half and
2
3
is more than half so
3
8
<
2
3
. (using benchmark
fractions)
• If I put the fractions in order from least to greatest based on my comparisons:
3
8
,
3
6
,
2
3
Which of the following fractions is smaller than or equal to
3
4
? For each fraction explain your
reasoning.
2
3
,
5
8
,
9
12
,
5
6
,
4
5
NC 4
th
Grade Math Unpacking - Revised June 2022
28
Extend understanding of fractions.
NC.4.NF.2 Compare two fractions with different numerators and different denominators, using the denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100. Recognize that
comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the
conclusions by:
• Reasoning about their size and using area and length models.
• Using benchmark fractions 0, ½, and a whole.
• Comparing common numerator or common denominators.
Clarification
Checking for Understanding
Possible responses:
Reasoning about their size using models:
Using benchmark fractions:
4
5
.
4
5
is
1
5
away from 1.
3
4
is
1
4
from 1. Since
1
5
is less than
1
4
, that means that
4
5
is closer to
1 than
3
4
is and is the larger fraction.
Comparing common numerator or denominator:
5
8
. I know that
3
4
is equal to
6
8
so
5
8
is less than
3
4
. The same process can be used for
9
12
since
3
4
is equal to
9
12
.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
29
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
NC.4.NF.3 Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• Decompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length
models, and equations.
• Add and subtract fractions, including mixed numbers with like denominators, by replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and subtraction.
• Solve word problems involving addition and subtraction of fractions, including mixed numbers by writing equations from a visual representation of the
problem.
Clarification
Checking for Understanding
NC.4.NF.3 calls for students to explore the meaning of addition and
subtraction with fractional amounts using both areas and length models. This
work is limited to joining, separating, and comparing fractions with like
denominators, and the only denominators that should be used are 2, 3, 4, 5, 6,
8, 10, 12, and 100.
The second bullet focuses on using area and length models to decompose a
fraction or mixed number into smaller fractional amounts, including unit
fractions. A unit fraction is a term that identifies the size of 1 fractional piece in
a whole and has a 1 in the numerator. For example,
1
3
is the unit fraction that
identifies a whole being divided into 3 equal pieces. Just as there are 3, one-
inch units in the length of 3 inches, there are 2,
1
3
units in the fraction
2
3
.
Models should also be used to support students’ work when they add and
subtract fractions in the latter two bullets of this standard. The work
decomposing fractions, including mixed numbers, in the first two bullets of this
standard, is foundational for the last two bullets of this Standard.
While equivalent fractions are not explicitly mentioned in this standard,
students are expected to identify the equivalent fractions of answers to
addition and subtraction problems.
Decompose a fraction into a sum of unit fractions and a sum of fractions
After a pizza party there is
7
8
of a pizza left. Some pieces of the pizza are
cheese, some are pepperoni, and some are vegetable. What are the possible
fractions that could be used to represent the amount of pizza that could be
cheese, pepperoni, and vegetable? Draw a picture and write an equation to
represent the amounts of each type of pizza that are remaining. Find at least
two combinations.
Possible response:
4
8
+
2
8
+
1
8
=
7
8
2
8
+
3
8
+
2
8
=
7
8
Add and subtraction fractions using the properties of operations and the
relationship between addition and subtraction.
The picture shows the amount of crackers that Mrs. Nickel has. She has
7
8
more of a pack of crackers
than Mrs. Fazio. If they
combine their crackers how
many packs of crackers do
they have?
NC 4
th
Grade Math Unpacking - Revised June 2022
30
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
NC.4.NF.3 Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• Decompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length
models, and equations.
• Add and subtract fractions, including mixed numbers with like denominators, by replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and subtraction.
• Solve word problems involving addition and subtraction of fractions, including mixed numbers by writing equations from a visual representation of the
problem.
Clarification
Checking for Understanding
Possible Response:
Mrs. Nickel has 2
5
8
packs. I made a part-part-whole picture to help me.
Since she has more than Mrs. Fazio, we
need to subtract the difference to find Mrs.
Fazio.
2
5
8
= Fazio +
7
8
or Fazio = 2
5
8
-
7
8
I crossed out the
5
8
then I partitioned one
of the other wholes into eighths then crossed out
2
8
from that whole since
I had to cross out a total
of
7
8
. Mrs. Fazio had 1
6
8
.
The total is 1
6
8
+ 2
5
8
. I
solved that by adding 1 + 2 +
6
8
+
5
8
= 3
11
8
.
I know that
11
8
is 1
3
8
so that means my total is 3 + 1 +
3
8
= 4
3
8
.
Which of the following fractions make this statement true?
____ -
5
8
>
a)
11
8
, b) 1
1
6
, c) 1
1
4
, d) 1
1
3
, f) 1
2
10
Possible response:
Since addition and subtraction are opposites, I need to find fractions
greater than
5
8
+
5
8
which is
10
8
or 1
2
8
.
*Comparisons can be made using length or area models, or reasoning
about the size of each fraction.
a) yes, b) no, c) no, d) yes, f) no
NC 4
th
Grade Math Unpacking - Revised June 2022
31
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
NC.4.NF.3 Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• Decompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length
models, and equations.
• Add and subtract fractions, including mixed numbers with like denominators, by replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and subtraction.
• Solve word problems involving addition and subtraction of fractions, including mixed numbers by writing equations from a visual representation of the
problem.
Clarification
Checking for Understanding
Solve word problems involving addition and subtraction of fractions
There is some firewood on the pile. Mr. Mickelson adds
7
8
pounds of firewood.
If there is now 2
1
8
of firewood on the pile how much firewood was first there?
Possible student responses:
Student 1
I wrote the equation 2
1
8
–
7
8
to
find out how much firewood
was first there. I then drew a
picture of 2
1
8
and crossed out
7
8
. I had 1
2
8
or 1
1
4
left.
Student 2
I drew
7
8
and then added on
until I reached 2
1
8
. I then went
back and counted. I added
1
8
+ 1 +
1
8
which is 1
2
8
or 1
1
4
pounds.
Student 3
I renamed 2
1
8
into an equivalent fraction
17
8
. I then took
7
8
away
from
17
8
which got me an answer of
10
8
. When I drew the
picture, I realized
10
8
is 1 whole and
2
8
, which is 1
2
8
pounds.
NC 4
th
Grade Math Unpacking - Revised June 2022
32
Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
NC.4.NF.3 Understand and justify decompositions of fractions with denominators of 2, 3, 4, 5, 6, 8, 10, 12, and 100.
• Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.
• Decompose a fraction into a sum of unit fractions and a sum of fractions with the same denominator in more than one way using area models, length
models, and equations.
• Add and subtract fractions, including mixed numbers with like denominators, by replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and subtraction.
• Solve word problems involving addition and subtraction of fractions, including mixed numbers by writing equations from a visual representation of the
problem.
Clarification
Checking for Understanding
Brielle ran 1
2
3
miles less than Kim. Brielle ran 2
2
3
miles. How far did Kim run?
Draw a number line and an equation to support your answer.
Possible student responses:
Student 1:
I started at 2
2
3
since that was how far Brielle ran. Since Brielle ran less
than Kim I knew I had to add 1
2
3
. I broke the
2
3
up into 2 jumps of a
1
3
so I
could land on 3, then jump to 4, then landed on 4 and
1
3
. An equation is 2
2
3
+ 1
2
3
= 4
1
3
.
Student 2:
I started at 2
2
3
since that was how far Brielle ran. Since Brielle ran less
than Kim I knew I had to add 1
2
3
. I jumped 1 to land on 3
2
3
. I then made 2
jumps of
1
3
and landed on 4
1
3
. An equation is 2
2
3
+ 1
2
3
= 4
1
3
.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
33
Use unit fractions to understand operations of fractions.
NC.4.NF.4 Apply and extend previous understandings of multiplication to:
• Model and explain how fractions can be represented by multiplying a whole number by a unit fraction, using this understanding to multiply a whole
number by any fraction less than one.
• Solve word problems involving multiplication of a fraction by a whole number.
Clarification
Checking for Understanding
This standard calls for students to understand a fraction as a whole number
of groups of a unit fraction (e.g.
3
8
is 3 groups of
1
8
). A unit fraction is a term
that identifies the size of 1 fractional piece in a whole. For example,
1
3
is the
unit fraction that identifies a whole being divided into 3 equal pieces. Just as
there are 3, one inch units in the length of 3 inches, there are 2,
1
3
units in the
fraction
2
3
.
Students also use multiplication of a fraction by a whole number to determine
a fraction of a set.
For example:
There are 12 pizzas.
1
4
of the total number of pizzas is 3.
There are 12 people and each person takes
1
4
of a pizza. The total number
of pizzas eaten by 12 people is 3.
Students use a unit fraction as well as repeated addition to establish a
foundation for the process of multiplying a whole number by a fraction (4 x
2
3
=
2
3
+
2
3
+
2
3
+
2
3
=
(42)
3
.
Students use both area and length models to explore and solve word
problems. All fractions are limited to the denominators of 2, 3, 4, 5, 6, 8, 10,
12.
Express the fraction
3
6
as the product of a whole number and a unit fraction.
Draw a model which supports your answer.
Possible response:
3
6
= 3 x
1
6
.
Tomas and Hector are running at P.E. Tomas runs
3
4
. of a mile. Hector runs 3
times as far as Tomas. How far did Hector run?
Possible response:
Hector ran
3
4
times 3 =
9
4
miles.
Michelle was making bracelets from ribbon. She wanted to make 4 bracelets
and each bracelet needed
2
3
yards of ribbon. How much ribbon does Michelle
need?
Possible response:
Michelle needs
8
3
yards of ribbon.
1
4
x 12 =3
1
4
of 12 =3
(
1
4
+
1
4
+
1
4
+
1
4
) + (
1
4
+
1
4
+
1
4
+
1
4
) + (
1
4
+
1
4
+
1
4
+
1
4
) = 3
Or 12 x
1
4
= 3
NC 4
th
Grade Math Unpacking - Revised June 2022
34
Use unit fractions to understand operations of fractions.
NC.4.NF.4 Apply and extend previous understandings of multiplication to:
• Model and explain how fractions can be represented by multiplying a whole number by a unit fraction, using this understanding to multiply a whole
number by any fraction less than one.
• Solve word problems involving multiplication of a fraction by a whole number.
Clarification
Checking for Understanding
This standard also includes situations involving finding the fractional amount
of a set of objects. In this standard the number in a set is limited to whole
numbers within 20. This work should use set models, arrays, or reasoning
about fractions as strategies to explore fractions of a set situations.
Fractions of a Set
There are 20 apples in the basket.
1
5
of them are red,
3
5
of them are yellow, and
the rest are green. How many apples are there of each color?
Possible response:
Student A:
I drew the 20 apples in the shape of an
array with 5 in each row. I then thought
about the red apples. Since
1
5
means 1
out of 5 I shaded the first circle red in
each row so there were 4 red. I then
shaded the next 3 in each row yellow
since
3
5
means 3 out of 5. There were 15
yellow. I shaded the rest green which
were 5 circles.
Student B:
I started with
1
5
since it is a unit fraction. I know that 5 x 4 = 20 and 20 ÷
5 = 4, so
1
5
of 20 is 4. I then found the yellow apples. Since
1
5
of 20 = 4
then
3
5
of 20 is going to be 3 x
1
5
of 20 or 3 x 4 which is 12. Then to find
the green I subtracted the red and yellow from 20 so 20 - 4 - 12 = 4.
There were 4 green apples.
While trying to find the answer to
4
6
of 18 a group of students shared their ideas.
For each student is their strategy correct? Explain why or why not.
• Max knew that 18 divided into 6 equal groups means that there are 3 in
each group so the answer is 3.
• Oprah drew 18 circles in rows of 6. He shaded the first 4 circles in each
row and counted 12 shaded circles.
• Asher said, “I know that 6 x 3 = 18 so
1
6
of 18 is 3. I then need to find
4
6
of 18 so I added 3 up four times to get 12.
• Ezekiel skip counted by 6 until he reached 18. He realized that 18 was
the 3rd multiple he said so the answer is 3.
NC 4
th
Grade Math Unpacking - Revised June 2022
35
Use unit fractions to understand operations of fractions.
NC.4.NF.4 Apply and extend previous understandings of multiplication to:
• Model and explain how fractions can be represented by multiplying a whole number by a unit fraction, using this understanding to multiply a whole
number by any fraction less than one.
• Solve word problems involving multiplication of a fraction by a whole number.
Clarification
Checking for Understanding
Possible response:
The correct answer is 12. Explanations should use arrays or other
models or reasoning about the relationship between the fraction and the
number of objects in the set.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
36
Understand decimal notation for fractions and compare decimal fractions.
NC.4.NF.6 Use decimal notation to represent fractions.
• Express, model and explain the equivalence between fractions with denominators of 10 and 100.
• Use equivalent fractions to add two fractions with denominators of 10 or 100.
• Represent tenths and hundredths with models, making connections between fractions and decimals.
Clarification
Checking for Understanding
This standard is the first time that students explore decimals. They are
introduced to decimals through area models for fractions that are partitioned
into 10 equal parts (tenths) and decimal grids which are 10x10 grids with 100
squares (hundredths). Students should have ample opportunities to explore
and reason about the idea that a number can be represented as both a
fraction and a decimal. This standard establishes the connection that a
fraction that has been equally partitioned into 10 or 100 equal parts (10
th
and
100ths) can also be written as a decimal.
With the second bullet of this standard students explore how to find an
equivalent fraction for both tenths and hundredths to add two fractions.
Students are expected to find the sum of two fractions using area models or
physical manipulatives such as base-ten blocks. Money should not be used
as a context since it is a non-proportional representation, meaning that a
dime is not ten times larger than a penny.
In the third bullet, models should focus on area models partitioned in tenths or
hundredths, and number lines. Students can also make connections between
fractions with denominators of 10 and 100 and the place value chart.
Express, model, and explain the equivalence between fractions and represent
tenths and hundredths with models
Rosita has
8
10
of a meter of ribbon. However, the directions for her craft product
have directions written about hundredths of a meter. What is an equivalent
decimal to
8
10
to the hundredths place?
Possible response:
I shaded in 8 columns on the decimal grid. That is the
same as
80
100
which can also be written as 0.8 or 0.80.
Shade
4
10
on a decimal grid. On a different decimal grid shade in
4
100
. Explain
how the two fractions are different on the decimal grid. Explain how they are
different when written as decimals using a place value chart.
Possible response:
I shaded both fractions on separate
decimal grids. The grid on the left
shows
4
100
and the grid on the right shows
4
10
.
The fraction
4
100
means 4 out of 100 and
since there are 100 small squares I only shaded in 4 of them. On the
place value chart 4 out of 100 means that I have 4 hundredths so there
is a 4 in the hundredths place and a 0 in the tenths place.
The fraction
4
10
has 4 columns shaded since
that fraction means 4 out of 10 and there are
10 columns. On the place value chart 4 out of
10 or
4
10
is 4 tenths so there is a 4 in the
tenths place.
By reading fraction names, students say 32/100 as thirty-two
hundredths and rewrite this as 0.32 or represent it on a place value
model as shown below.
Hundreds
Tens
Ones
•
Tenths
Hundredths
•
3
2
NC 4
th
Grade Math Unpacking - Revised June 2022
37
Understand decimal notation for fractions and compare decimal fractions.
NC.4.NF.6 Use decimal notation to represent fractions.
• Express, model and explain the equivalence between fractions with denominators of 10 and 100.
• Use equivalent fractions to add two fractions with denominators of 10 or 100.
• Represent tenths and hundredths with models, making connections between fractions and decimals.
Clarification
Checking for Understanding
Use equivalent fractions to add two fractions with denominators of 10 and 100
Mitch swam
5
10
of a mile on Saturday and
39
100
a mile on Sunday. How much did
Mitch swim on the two days? Use a decimal grid to show your answer and write
your answer as a decimal.
Possible response:
89 of the one hundred squares are shaded so Mitch swam
89
100
of a mile. I
can also write
89
100
as 0.89.
NC 4
th
Grade Math Unpacking - Revised June 2022
38
Understand decimal notation for fractions and compare decimal fractions.
NC.4.NF.7 Compare two decimals to hundredths by reasoning about their size using area and length models, and recording the results of comparisons with the
symbols >, =, or <. Recognize that comparisons are valid only when the two decimals refer to the same whole.
Clarification
Checking for Understanding
Students should reason that comparisons
are only valid when they refer to the same
whole. Comparisons should only be done
in Grade 4 with area and length models,
which include decimal grids, decimal
circles, number lines, and meter sticks.
Students should be able to construct their
own models.
Sarah drinks 0.37 Liters of juice. Rochelle drinks 0.4 Liters of juice. Who drank more juice? Draw a picture and
explain your reasoning.
Possible response:
Sarah Rochelle
Sarah drank 37 hundredths of a Liter, while Rochelle drank 4 tenths or 40 hundredths of a Liter. Rochelle
drank more.
Theresa has
4
100
of a Liter of milk in one container and a brand new container. Her recipe has the number 0.4 of
milk in it. Can she make the recipe with the milk in one container or does she need to open a new container?
Possible response:
She needs to open a new container since 0.4 is equal to
4
10
and she has only
4
100
of a Liter in her container.
Denard lives 0.67 km from school and Calvin lives 0.76 km from school. Draw a number line to show who lives the
farthest from school.
Possible student response:
Denard lives 67 hundredths of a kilometer away which is less
than 7 tenths of a kilometer. Calvin lives 76 hundredths of a
kilometer away which is greater than 7 tenths. Calvin lives
farther from school than Denard.
The decimal grid shows the amount of kilograms of turkey that Mrs. Burrell has to make sandwiches. Mrs.
Rigamarole has more than 0.5 kilograms of turkey but less than Mrs. Burrell. How much turkey
could Mrs. Rigamarole have?
Possible response:
Mrs. Rigamarole has more than 0.5 kilograms and less than 0.6 kilograms. She could have as
few as 0.51 or as many as 0.59.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
39
Measurement and Data
Solve problems involving measurement.
NC.4.MD.1 Know relative sizes of measurement units. Solve problems involving metric measurement.
• Measure to solve problems involving metric units: centimeter, meter, gram, kilogram, Liter, milliliter.
• Add, subtract, multiply, and divide to solve one-step word problems involving whole-number measurements of length, mass, and capacity that are given
in metric units.
Clarification
Checking for Understanding
In this standard, students reason about the units of length, capacity and weight
using metric units. Students need to develop a basic understanding of the size
and weight of metric units and apply this understanding when estimating and
measuring. Students should understand how to express larger measurements
in smaller units within the metric system to reinforce notions of place value.
In this standard, word problems should only be one-step and include the same
units. In Grade 4, students are not expected to do conversions between units
before solving problems.
One can of soda holds 376 mL. A large container holds 8 times more punch
than the can of soda. How much soda does the large container hold?
Possible response:
The large container holds 376 x 8 mL.
The container holds 3,008 mL.
I have 4,327 mL of juice. I want to divide the juice equally into 8 containers.
The remaining juice is all poured into one of the containers. How much juice
will be in the container that has the most juice? How much juice will be in the
other containers?
Possible responses:
Student A
I used partial quotients. The answer was 540
with a remainder of 7. That means that one
container had 547 mL and the other 7
containers had 540 mL of juice.
Student B
I multiplied up using 8s. I knew
5x8 = 40
50 x 8 = 400
500 x 8 = 4,000
4x8 = 32
40 x 8 = 320 so 540 x 8 = 4,320 with a remainder of 7.
One container received 547 mL and the rest of them had 540 mL.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
40
Solve problems involving measurement.
NC.4.MD.2 Use multiplicative reasoning to convert metric measurements from a larger unit to a smaller unit using place value understanding, two-column
tables, and length models.
Clarification
Checking for Understanding
In this standard, students should understand how to express larger
measurements in smaller units within the metric system to reinforce notions of
place value. Students will make metric conversions from larger units to smaller
units exploring the relationship between the units. Students may use a two-
column chart to convert from larger to smaller units and record equivalent
measurements.
Through exploration with place value and conversions, students may explore
with various metric prefixes. However, students are only responsible for
knowing conversions between centimeter/meter, gram/kilogram, and
Liter/milliliter.
Students are also expected to use their place value understanding to support
conversions. For example, since 1 meter is 100 centimeters, the number of
centimeters is 100 times the number of meters. Therefore, if a piece of rope is
3 meters, then the rope is 300 cm long.
This standard is focused on whole number conversions of metric units since
students do not multiply decimals until Grade 5.
Crystal has 8 Liters of soda. How many milliliters does she have?
Possible response:
There are 1,000 mL in 1 Liter. So, in 8 Liters the number of mL is 8 x
1,000 which is 8,000.
Complete the table below:
Meters
Centimeters
3
4
5
12
120
Answers: 3 m = 300 cm, 4 m = 400 cm, 5 m = 500 cm, 12 m = 1,200 cm,
120 m = 12,000 cm
Charlotte tells her teacher that a container that holds 2 Liters holds 200 mL of
liquid. Is she correct or incorrect? Explain why.
Possible Response:
Since 1 Liter equals 1000 mL, 2 Liters is equal to 2,000 mL. Charlotte is
incorrect since she said 2 Liters equals 200 mL.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
41
Solve problems involving measurement.
NC.4.MD.8 Solve word problems involving addition and subtraction of time intervals that cross the hour.
Clarification
Checking for Understanding
In this standard, students apply addition and subtraction strategies
to find an end time, amount of time passed, or a start time. In third
grade, students determined elapsed time within an hour. This
standard calls for students to be able to cross over the hour.
Students should use tools such as clocks, timelines, and tables to
solve problems.
This standard could include situations with two activities that
include time that passed (or duration), which may result in a multi-
step problem.
One-step word problem
The movie started at 2:30 pm and lasted for 1 hour and 35 minutes. What time did the
movie end?
Possible response:
Student A:
I started at 2:30 and added 1 hour and moved to 3:30. I then added 30 minutes and
moved to 4:00 p.m. I
then had to move 5
more minutes which
was 4:05 p.m.
Student B:
I decomposed 1 hour 35 minutes into 1 hour + 30 minutes + 5 minutes.
2:30 + 1 hour + 35 minutes.
2:30 + 30 minutes = 3:00
3:00 + 5 minutes = 3:05
3:05 + 1 hour = 4: 05 p.m.
Multi-step word problem
I wake up at 6:47 a.m. I get ready for school
which takes 15 minutes and then eat
breakfast before leaving. If I leave for school
at 7:21 a.m. how long do I have to eat
breakfast?
Possible response:
I added 15 minutes to 6:47 by moving 13 minutes until 7:00 and then moving 2
minutes until 7:02. I then knew I had to eat breakfast until 7:21. In order to find this
amount, I added up 8 until 7:10 and then added 11 more to get to 7:21 My answer is
the distance between 7:02 to 7:21, which is 8 + 11 or 19 minutes.
Return to Standards
NC 4
th
Grade Math Unpacking - Revised June 2022
42
Solve problems involving area and perimeter.
NC.4.MD.3 Solve problems with area and perimeter.
• Find areas of rectilinear figures with known side lengths.
• Solve problems involving a fixed area and varying perimeters and a fixed perimeter and varying areas.
• Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Clarification
Checking for Understanding
In this standard, students will apply their previous understanding of perimeter
and area to problem situations.
Students will be able to
determine the area of a
rectilinear figure. A rectilinear
figure is a polygon that has all
right angles. Recognizing that
area is additive, students will
be able to decompose the
rectilinear figure into
rectangles, determine the area
of the rectangles, and use the
areas of the rectangles to
determine the area of the
rectilinear figure.
Students will solve problems
that involve exploration of the
relationship between perimeter
and area in a rectangle. When
given a fixed area, students will be able to determine all of the possible
dimensions of the rectangle. When given a fixed perimeter, students will be
able to determine all possible areas.
Students learn to apply these understandings and formulas to the solution of
real-world and mathematical problems. Note that “apply the formula” does not
mean write down a memorized formula and put in known values. In fourth
grade, working with perimeter and area of rectangles is still based in models
and strategies.
Find areas of rectilinear figures with known side lengths
Mr. Rutherford is covering the miniature golf course with an artificial grass. How
many 1-foot squares of carpet will he need to cover the entire course?
Possible Response:
Student A
I decided to break the shape into 3 rectangles. The one on the left was 5
feet long and 2 feet wide which is 5x2 or 10 square feet. The middle
rectangle was 3 feet long and 2 feet wide which is 3x2 or 6 square feet.
The right rectangle was 4 feet long and 2 feet wide which is 4x2 or 8
square feet. I add them together to get a total of 10+6+8 or 24 square
feet.
Student B
I counted 24 squares, so the area is 24 square feet.
A storage shed is pictured to the right. What is
the total area?
How could the figure be decomposed to help find
the area?
Possible Response:
I decomposed the shape into three
rectangles to find the area of each
rectangle. The top rectangle is 10 wide x
5 long or 50 square meters. The middle
rectangle has a width of 10 - 6 = 4 and a length of 5 so the area is 5x4 or
20 square meters. The bottom rectangle is 10 x 5 or 50 square meters.
The total area is 50 + 20 + 50 which is 120 square meters.
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Solve problems involving area and perimeter.
NC.4.MD.3 Solve problems with area and perimeter.
• Find areas of rectilinear figures with known side lengths.
• Solve problems involving a fixed area and varying perimeters and a fixed perimeter and varying areas.
• Apply the area and perimeter formulas for rectangles in real world and mathematical problems.
Clarification
Checking for Understanding
Solve problems involving a fixed perimeter and varying areas
At Miguel’s apartment complex they are building a rectangular outdoor eating
space with a perimeter of 32 meters. Side lengths will be whole numbers and
each dimension is less than 13 meters. What are the possible areas? Which
dimensions have the most space for people to eat?
Possible Response:
I made a table of rectangle dimensions with the perimeter of 32 and
different areas. The dimensions of 8x8, which is a square, gives people
the most space to eat.
Dimensions
Perimeter
Area
8x8
32
64
9x7
32
63
10x6
32
60
11x5
32
55
12x4
32
48
Solve problems with a fixed area and varying perimeters
You want to build a region that has an area of 12 square meters. What are the
possible dimensions? Which dimensions require the least amount of fencing?
Possible solution:
Area
Length
Width
Perimeter
12 sq. m
1 m
12 m
26 m
12 sq. m.
2 m
6 m
16 m
12 sq. m
3 m
4 m
14 m
12 sq. m
4 m
3 m
14 m
12 sq. m
6 m
2 m.
16 m
12 sq. m
12 m
1 m
26 m
Return to Standards
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Represent and interpret data.
NC.4.MD.4 Represent and interpret data using whole numbers.
• Collect data by asking a question that yields numerical data.
• Make a representation of data and interpret data in a frequency table, scaled bar graph, and/or line plot.
• Determine whether a survey question will yield categorical or numerical data.
Clarification
Checking for Understanding
In this standard, students will interact with data
by posing a question, interacting with data,
analyzing data, and interpreting data.
Students need ample experiences with creating
and discussing survey questions, data collection,
creating scaled bar graphs and line plots, and
interpretating data. In third grade, students
collected data by asking a question that yielded
categorical data, which is data that can be
grouped into categories. Students in fourth grade
will build on that concept and begin to also ask
questions that provide numerical data, which is
data that is measurable such as time, height,
weight, temperature, etc.
Once data is collected, students should be able
to choose an appropriate representation of
categorical or numerical data and create the
representation. Students will create frequency
tables, scaled bar graphs or line plots based on
the data collected. Graphs should include a title,
categories, category label, key, and data. Once
graphs are created, students should be able to
solve simple one and two-step problems using
the information in the graphs.
In Grade 3, students learned that scaled bar
graphs have a scale on the y-axis in which the
labels do not include every number.
Make a representation of data and interpret data in a line plot
Mrs. Smith’s class tracked the daily high temperatures for 20 days in July. The chart below shows the data
that the class collected.
90
92
93
92
89
93
91
95
88
90
95
94
97
97
94
94
91
90
89
94
a. Create a line plot that shows the frequency of the July high temperatures. Make sure you label the
scale.
b. If the normal daily high temperature for July is 90°, how many days was the high temperature less
than normal?
c. What was the most frequent daily high temperature recorded during the 20 days?
Possible Response:
There were 3 days where the temperature was below 90 degrees or less than normal.
The most frequent daily high temperature was 94 degrees.
Students were given the choice of eating a hot dog, a cheeseburger, or a chicken sandwich at the class
picnic. The data is in the table to the right.
There were some students who did not vote. The number of
students who voted for cheeseburger is 3 times the number of
students who did not vote.
Make a scaled bar graph that shows the data. Include the non-
voters in the graph.
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Represent and interpret data.
NC.4.MD.4 Represent and interpret data using whole numbers.
• Collect data by asking a question that yields numerical data.
• Make a representation of data and interpret data in a frequency table, scaled bar graph, and/or line plot.
• Determine whether a survey question will yield categorical or numerical data.
Clarification
Checking for Understanding
Possible response:
There were 2 non-voters since 6 is 3 times more than 2.
Determine whether a survey question will yield categorical or numerical data
For each question determine whether the responses will be categorical data or numerical data:
How many hours did people sleep last night?
Would you rather hike, swim, or play soccer today?
How many times can you clap your hands in 10 seconds?
At a picnic would you want to eat a hot dog, a cheeseburger, or a salad?
Answers:
Hours of sleep- Numerical
Outdoor activity- Categorical
Clapping hands- Numerical
Picnic- Categorical
Return to Standards
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Understand concepts of angles and measure angles.
NC.4.MD.6 Develop an understanding of angles and angle measurement.
• Understand angles as geometric shapes that are formed wherever two rays share a common endpoint and are measured in degrees.
• Measure and sketch angles in whole-number degrees using a protractor.
• Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.
Clarification
Checking for Understanding
In this standard, students explore angles and their properties. An angle is
formed by two rays that share an endpoint and is measured with reference to
the degrees of a circle.
Students will be able to identify three types of angles: right angle, acute angle,
and obtuse angle
Measure and sketch angles in whole-number degrees using a protractor
Draw Angle ABC that is 30 degrees.
Explain whether Angle ABC is an acute, an obtuse, or a right angle.
Now draw Angle DEF that is 3 times larger than Angle ABC.
Explain whether Angle DEF is an acute, an obtuse, or a right angle.
Now draw Angle GHI that is 5 times larger than Angle ABC.
Explain whether Angle GHI is an acute, an obtuse, or a right angle.
Answers:
ABC is an acute angle that is 30 degrees
DEF is a right angle that is 90 degrees
GHI is an obtuse angle that is 150 degrees.
Solve addition and subtraction problems to find unknown angles on a diagram
in real-world and mathematical problems
If the two rays are perpendicular, what is the value of m?
Possible Response:
The large angle has a measure of 90 degrees so Angle M + 25 + 20 =
90. I can subtract both 25 and 20 from 90 which means Angle M is 45
degrees.
A lawn water sprinkler rotates 85 degress and then pauses. It then rotates an
additional 19 degrees. If it does this 3 times, what is the total degree of the
water sprinkler rotation?
Possible Response:
The water sprinkler goes 68 degrees and then another 19 degrees. That
is a total of 87 degrees. Since it does this 3 times that is a total of 87x3 =
261 degrees.
For example: If the common endpoint of two rays is the center of a
circle, the angle can be measured by considering the fraction of the
circular arc between the points where the rays intersect the circle.
An angle that turns
1
360
of a circle is a “one-degree angle”.
right angle:
An angle that equals one quarter of a full
rotation of a circle, or 90º
acute angle:
An angle that is less than a right angle, or
less than 90º
obtuse angle:
An angle that is more than a right angle, or
more than 90º.
Straight angle:
An angle that is 180º
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Understand concepts of angles and measure angles.
NC.4.MD.6 Develop an understanding of angles and angle measurement.
• Understand angles as geometric shapes that are formed wherever two rays share a common endpoint and are measured in degrees.
• Measure and sketch angles in whole-number degrees using a protractor.
• Solve addition and subtraction problems to find unknown angles on a diagram in real-world and mathematical problems.
Clarification
Checking for Understanding
Students will learn to measure and sketch angles using a protractor. Students
should also have an understanding of benchmark angles such as 45º, 90º and
180º.
Students will explore the additive nature of angle measurements and solve
one- and two-step addition and subtraction problems where an angle measure
is missing. When an angle is decomposed into non-overlapping parts, the
angle measure of the whole is the sum of the angle measures of the parts.
In terms of vocabulary, students explore the concepts of complementary and
supplementary angles, but are not responsible for these terms. Two angles are
called complementary if their measurements have the sum of 90º. Two angles
are called supplementary if their measurements have the sum of 180º. Two
angles with the same vertex that share a side are called adjacent angles.
These terms may come up in classroom discussion, but students are not
responsible for knowing these terms. These concepts are developed
thoroughly in middle school (7
th
grade).
Joey knows that when a clock’s hands are exactly on 12 and 1, the angle
formed by the clock’s hands measures 30º. What is the measure of the angle
formed when a clock’s hands are exactly on the 12 and 4?
Possible Response:
Student A
12 to 1 is 30 degrees. 1 to 2 is another 30 degrees. 2 to 3 is another 30
degrees. 3 to 4 is another 30 degrees.
30 + 30 + 30 + 30 = 120 degrees
Student B
12 to 1 is a rotation of 30 degrees. 12 to 4 is 4 times larger. 30 x 4 is 120
so the angle is 120 degrees when the hands are at 12 and 4.
Return to Standards
For example:
When measuring angles with a
protractor, students will first identify
if the angle is acute or obtuse to
determine which numbers to use.
Acute angles would measure 0º to
89º, and obtuse angles would
measure 91º to 179º. In this
example, the angle is obtuse, so it
would be read as 120º.
In the following example, the angle
measured is obtuse, but facing the
opposite direction of the angle
pictured to the left. Students who
understand that obtuse angles
measure 91º to 179º would know
that this angle measures 135º
rather than 45º.
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Geometry
Classify shapes based on lines and angles in two-dimensional figures.
NC.4.G.1 Draw and identify points, lines, line segments, rays, angles, and perpendicular and parallel lines.
Clarification
Checking for Understanding
This standard asks students to draw two-dimensional geometric objects and
to also identify them in two-dimensional figures.
Students should be able to draw and identify the following figures:
point
angle
line segment
right angle
line
acute angle
ray
obtuse angle
parallel lines
straight angle
perpendicular lines
Students should understand the concept of parallel and perpendicular lines.
Two lines are parallel if they never intersect and are always equidistant. Two
lines are perpendicular if they intersect in right angles (90º).
Draw the following shapes:
Angle ABC is a straight angle.
Ray BD intersects Angle ABC and Angle ABD into a right angle.
Ray BE intersects Angle ABC and Angle EBC is a 30 degree angle.
What is the measure of Angle DBE?
Possible Response:
Draw two different types of quadrilaterals that have two pairs of parallel sides.
Describe what the two shapes have in common. Describe differences between
the two shapes.
Possible responses:
Shapes may include a square, rectangle, rhombus, parallelograms that
are not squares. Descriptions discuss the similarity of 4 sides, 2 pairs of
parallel sides. Differences may include the types of angles and the
length of the sides.
Is it possible to have a right triangle with an obtuse angle? Justify your
reasoning using pictures and words.
Possible response:
A right triangle has a 90-degree angle and the 2 other angles must be
acute. Therefore, any triangle with a right angle can never have an
obtuse angle.
Draw and list the properties of a parallelogram. Draw and list the properties of a
rectangle. How are your drawings and lists alike? How are they different? Be
ready to share your thinking with the class.
Lines AB and CD are parallel. Line FG is perpendicular to lines AB and
CD forming right angles.
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Classify shapes based on lines and angles in two-dimensional figures.
NC.4.G.1 Draw and identify points, lines, line segments, rays, angles, and perpendicular and parallel lines.
Clarification
Checking for Understanding
Possible response:
A parallelogram is a quadrilateral with 2 pairs of parallel sides. The
opposite sides are the same length. A rectangle has all of those
characteristics. In addition, a rectangle always has 4 right angles.
How many acute, obtuse and right angles are in this shape? Explain how you
know. Now draw a shape that has the same name that has 2 right angles.
Possible response:
This shape has 2 acute and 2 obtuse angles. The trapezoid that has 2
right angles is called a right trapezoid. I drew it here.
Return to Standards
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Classify shapes based on lines and angles in two-dimensional figures.
NC.4.G.2 Classify quadrilaterals and triangles based on angle measure, side lengths, and the presence or absence of parallel or perpendicular lines.
Clarification
Checking for Understanding
This standard calls for students to sort and classify quadrilaterals and
triangles based on parallelism, perpendicularity, the lengths of sides, and
angle types.
Students should be able to use side length to classify triangles as equilateral,
isosceles, or scalene; and can use angle size to classify them as acute, right,
or obtuse. They then learn to cross-classify, for example, naming a shape as
a right isosceles triangle.
Students should be able to use the relationships between sides (parallel and
perpendicular) and side lengths to classify quadrilaterals. Students are
expected to know the characteristics of: quadrilateral, trapezoid,
parallelogram, rectangle, square, and rhombus. While working with
quadrilaterals, students’ experiences with drawing and identifying right, acute,
and obtuse angles support them in classifying two-dimensional figures based
on specific angle measurements. They use the benchmark angles of 90°,
180°, and 360° to approximate the measurement of angles. Students are not
expected to know the sum of the interior angles of a triangle or a quadrilateral
in Grade 4.
The notion of congruence (“same size and same shape”) may be part of
classroom conversation, but the concepts of congruence and similarity do not
appear in standards until middle school.
Note: North Carolina has adopted the exclusive definition for a trapezoid. A
trapezoid is a quadrilateral with exactly one pair of parallel sides.
Do you agree with the label on each of the circles in the Venn diagram above?
Describe why some shapes fall in the overlapping sections of the circles.
Add one more figure to the diagram. Explain where it should go and why.
Possible response:
I agree with the labels on the Venn Diagram.
The shapes in the center section have at least one set of parallel sides
and at least one right angle.
Adding two more figures:
A trapezoid with 2 right angles could be added to the center section
since it has 2 right angles and has at least one set of parallel sides.
For each of the following, sketch an example if it is possible. If it is impossible,
say so, and explain why or show a counter example.
• A parallelogram with exactly one right angle. (impossible)
• An isosceles right triangle.
• A rectangle that is not a parallelogram. (impossible)
• A square that is a rhombus
• A trapezoid that is a parallelogram.
• A parallelogram that has 4 right angles.
Possible responses:
Isosceles right triangle:
Every square is a rhombus since it has 4 congruent sides.
A parallelogram that has 4 right angles can be a rectangle or a square.
For example: The square has perpendicular lines because the sides
meet at a corner, forming right angles. It also has parallel sides that are
opposite from each other. I know this because if I changed the sides to
lines that never end, the lines would never intersect and be the same
distance apart. Segments are just parts of lines. All of the line segments
are equal.
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Classify shapes based on lines and angles in two-dimensional figures.
NC.4.G.2 Classify quadrilaterals and triangles based on angle measure, side lengths, and the presence or absence of parallel or perpendicular lines.
Clarification
Checking for Understanding
Identify which of these shapes have perpendicular or parallel sides and justify
your selection.
Square: 2 pairs of parallel sides. 4 pairs of perpendicular sides that form
4 right angles.
Triangle: No parallel and no perpendicular sides.
Pentagon: No parallel and no perpendicular sides.
Trapezoid: Exactly 1 pair of parallel sides and no pairs of perpendicular
sides.
Return to Standards
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Classify shapes based on lines and angles in two-dimensional figures.
NC.4.G.3 Recognize symmetry in a two-dimensional figure, and identify and draw lines of symmetry.
Clarification
Checking for Understanding
In this standard, students determine if figures are symmetrical, and if they are
symmetrical how many lines of symmetry are in a figure. Students explore the
concept that a line of symmetry is a line that divides a figure into two parts
that are identical in shape and size.
Students are expected to identify and draw lines of symmetry in regular and
non-regular polygons, circles, and letters of the alphabet. Circles have an
infinite number of lines of symmetry while all regular and non-regular
polygons have a specific number of lines of symmetry. Students are expected
to understand that some figures may have more than one line of symmetry. In
Grade 4, students only explore line symmetry not rotational symmetry.
Do these figures have lines of symmetry? If so, identify the lines of symmetry.
T H S D
Possible response:
T has 1 vertical line of symmetry through the center.
H has 1 vertical line of symmetry through the center.
S has 0 lines of symmetry.
D has 1 horizontal line of symmetry through the center.
Explain how the number of lines of symmetry differ in a square and a rectangle
that is not a square.
Possible response:
A square has 4 lines of symmetry- a vertical line through the center, a
horizontal line through the center, and 2 diagonal lines from upper left to
bottom right and upper right to bottom left through the center. A
rectangle that is not a square has 2 lines of symmetry- a vertical line
through the center, a horizontal line through the center.
Explain what has more lines of symmetry: a circle or a square.
Possible response:
A circle has more lines of symmetry than a square. A square has 4 lines
of symmetry. A circle has an infinite number of lines of symmetry through
the center of the circle.
Return to Standards
The triangle has one line of symmetry,
and the square has four lines of
symmetry.