Common Core Unpacked Grade 5

5thGrade Mathematics ● Unpacked Content
For the new Common Core State Standards that will be effective in all North Carolina schools in the 2012-13.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually
updating and improving these tools to better serve teachers.

What is the purpose of this document?
To increase student achievement by ensuring educators understand specifically what the new standards mean a student must know,
understand and be able to do.

What is in the document?
Descriptions of what each standard means a student will know, understand and be able to do. The “unpacking” of the standards done in this
document is an effort to answer a simple question “What does this standard mean that a student must know and be able to do?” and to
ensure the description is helpful, specific and comprehensive for educators.

How do I send Feedback?
We intend the explanations and examples in this document to be helpful and specific. That said, we believe that as this document is used,
teachers and educators will find ways in which the unpacking can be improved and made ever more useful. Please send feedback to us at
feedback@dpi.state.nc.us and we will use your input to refine our unpacking of the standards. Thank You!

Just want the standards alone?
You can find the standards alone at http://corestandards.org/the-standards

Mathematical Vocabulary is identified in bold print. These are words that student should know and be able to use in context.

5th Grade Mathematics ● Unpacked Content

Operations and Algebraic Thinking 5.0A
Common Core Cluster
Write and interpret numerical expressions.

Common Core Standard Unpacking
What do these standards mean a child will know and be able to do?
5.OA.1 Use parentheses, brackets, or 5.OA.1 calls for students to evaluate expressions with parentheses ( ), brackets [ ] and braces { }. In upper levels
braces in numerical expressions, and of mathematics, evaluate means to substitute for a variable and simplify the expression. However at this level
evaluate expressions with these students are to only simplify the expressions because there are no variables.
symbols.
Example:
Evaluate the expression 2{ 5[12 + 5(500 - 100) + 399]}
Students should have experiences working with the order of first evaluating terms in parentheses, then brackets,
and then braces.
The first step would be to subtract 500 – 100 = 400.
Then multiply 400 by 5 = 2,000.
Inside the bracket, there is now [12 + 2,000 + 399]. That equals 2,411.
Next multiply by the 5 outside of the bracket. 2,411 x 5 = 12,055.
Next multiply by the 2 outside of the braces. 12,055 x 2= 24,110.

Mathematically, there cannot be brackets or braces in a problem that does not have parentheses. Likewise, there
cannot be braces in a problem that does not have both parentheses and brackets.
5.OA.2 Write simple expressions that 5.OA.2 refers to expressions. Expressions are a series of numbers and symbols (+, -, x, ÷) without an equals sign.
record calculations with numbers, and Equations result when two expressions are set equal to each other (2 + 3 = 4 + 1).
interpret numerical expressions without
evaluating them. Example:
For example, express the calculation 4(5 + 3) is an expression.
“add 8 and 7, then multiply by 2” as 2 When we compute 4(5 + 3) we are evaluating the expression. The expression equals 32.
× (8 + 7). Recognize that 3 × (18932 + 4(5 + 3) = 32 is an equation.
921) is three times as large as 18932 +
921, without having to calculate the 5.OA.2 calls for students to verbally describe the relationship between expressions without actually calculating
indicated sum or product. them. This standard calls for students to apply their reasoning of the four operations as well as place value while
describing the relationship between numbers. The standard does not include the use of variables, only numbers
and signs for operations.

5th Grade Mathematics ● Unpacked Content page 2

Example:
Write an expression for the steps “double five and then add 26.”
Student
(2 x 5) + 26

Describe how the expression 5(10 x 10) relates to 10 x 10.
Student
The expression 5(10 x 10) is 5 times larger than the expression 10 x 10 since I know that I that 5(10
x 10) means that I have 5 groups of (10 x 10).

Common Core Cluster
Analyze patterns and relationships.

5.OA.3 Generate two numerical 5.OA.3 extends the work from Fourth Grade, where students generate numerical patterns when they are given
patterns using two given rules. one rule. In Fifth Grade, students are given two rules and generate two numerical patterns. The graphs that are
Identify apparent relationships between created should be line graphs to represent the pattern. This is a linear function which is why we get the straight
corresponding terms. Form ordered lines. The Days are the independent variable, Fish are the dependent variables, and the constant rate is what the
pairs consisting of corresponding terms rule identifies in the table.
from the two patterns, and graph the
ordered pairs on a coordinate plane.
For example, given the rule “Add 3”
and the starting number 0, and given
the rule “Add 6” and the starting Examples below:
number 0, generate terms in the
resulting sequences, and observe that
the terms in one sequence are twice the
corresponding terms in the other
sequence. Explain informally why this is
so.


Student
Makes a chart (table) to represent the number of fish that Sam and Terri catch.

Days Sam’s Total Terri’s Total
Number of Fish Number of Fish
0 0 0
1 2 4
2 4 8
3 6 12
4 8 16
5 10 20

Student
Describe the pattern:
Since Terri catches 4 fish each day, and Sam catches 2 fish, the amount of Terri’s fish is
always greater. Terri’s fish is also always twice as much as Sam’s fish. Today, both
Sam and Terri have no fish. They both go fishing each day. Sam catches 2 fish each
day. Terri catches 4 fish each day. How many fish do they have after each of the five
days? Make a graph of the number of fish.

Student
Plot the points on a coordinate plane and make a line graph, and then interpret the
graph.
My graph shows that Terri always has more fish than Sam. Terri’s fish increases at a
higher rate since she catches 4 fish every day. Sam only catches 2 fish every day, so his
number of fish increases at a smaller rate than Terri.
Important to note as well that the lines become increasingly further apart. Identify
apparent relationships between corresponding terms. Additional relationships: The two
lines will never intersect; there will not be a day in which boys have the same total of
fish, explain the relationship between the number of days that has passed and the
number of fish a boy has (2n or 4n, n being the number of days).


Number and Operations in Base Ten 5.NBT
Common Core Cluster
Understand the place value system.

5.NBT.1 Recognize that in a multi-digit 5.NBT.1 calls for students to reason about the magnitude of numbers. Students should work with the idea that the
number, a digit in one place represents tens place is ten times as much as the ones place, and the ones place is 1/10th the size of the tens place.
10 times as much as it represents in the
place to its right and 1/10 of what it Example:
represents in the place to its left. The 2 in the number 542 is different from the value of the 2 in 324. The 2 in 542 represents 2 ones or 2, while the
2 in 324 represents 2 tens or 20. Since the 2 in 324 is one place to the left of the 2 in 542 the value of the 2 is 10
times greater. Meanwhile, the 4 in 542 represents 4 tens or 40 and the 4 in 324 represents 4 ones or 4. Since the 4
in 324 is one place to the right of the 4 in 542 the value of the 4 in the number 324 is 1/10th of its value in the
number 542.

Note the pattern in our base ten number system; all places to the right continue to be divided by ten and that
places to the left of a digit are multiplied by ten.
5.NBT.2 Explain patterns in the number 5.NBT.2 includes multiplying by multiples of 10 and powers of 10, including 102 which is 10 x 10=100, and 103
of zeros of the product when which is 10 x 10 x 10=1,000. Students should have experiences working with connecting the pattern of the
multiplying a number by powers of 10, number of zeros in the product when you multiply by powers of 10.
and explain patterns in the placement of
the decimal point when a decimal is Example:
multiplied or divided by a power of 10. 2.5 x 103 = 2.5 x (10 x 10 x 10) = 2.5 x 1,000 = 2,500 Students should reason that the exponent above the 10
Use whole-number exponents to denote indicates how many places the decimal point is moving (not just that the decimal point is moving but that you are
multiplying or making the number 10 times greater three times) when you multiply by a power of 10. Since we
powers of 10.
are multiplying by a power of 10 the decimal point moves to the right.

350 ÷ 103 = 350 ÷ 1,000 = 0.350 = 0.35 350/10 = 35, 35 /10 = 3.5 3.5 /10 =.0.35, or 350 x 1/10, 35 x 1/10,
3.5 x 1/10 this will relate well to subsequent work with operating with fractions. This example shows that when
we divide by powers of 10, the exponent above the 10 indicates how many places the decimal point is moving
(how many times we are dividing by 10 , the number becomes ten times smaller). Since we are dividing by
powers of 10, the decimal point moves to the left.

Students need to be provided with opportunities to explore this concept and come to this understanding; this
should not just be taught procedurally.


5.NBT.3 Read, write, and compare
decimals to thousandths.
a. Read and write decimals to 5.NBT.3a references expanded form of decimals with fractions included. Students should build on their work
thousandths using base-ten from Fourth Grade, where they worked with both decimals and fractions interchangeably. Expanded form is
numerals, number names, and included to build upon work in 5.NBT.2 and deepen students’ understanding of place value.
expanded form, e.g., 347.392 = 3 ×
100 + 4 × 10 + 7 × 1 + 3 × (1/10) +
9 x (1/100) + 2 x (1/1000)
b. Compare two decimals to 5.NBT.3b comparing decimals builds on work from fourth grade.
thousandths based on meanings of
the digits in each place, using >, =,
and < symbols to record the results
of comparisons.
5.NBT.4 Use place value understanding 5.NBT.4 refers to rounding. Students should go beyond simply applying an algorithm or procedure for rounding.
to round decimals to any place. The expectation is that students have a deep understanding of place value and number sense and can explain and
reason about the answers they get when they round. Students should have numerous experiences using a number
line to support their work with rounding.
5.NBT.4 references rounding. Students should use benchmark numbers to support this work. Benchmarks are
convenient numbers for comparing and rounding numbers. 0., 0.5, 1, 1.5 are examples of benchmark numbers.
Which benchmark number is the best estimate of the shaded amount in the model below? Explain your thinking.


Common Core Cluster
Perform operations with multi-digit whole numbers and with decimals to hundredths.
Students develop understanding of why division procedures work based on the meaning of base-ten numerals and properties of operations. They finalize
fluency with multi-digit addition, subtraction, multiplication, and division. They apply their understandings of models for decimals, decimal notation, and
properties of operations to add and subtract decimals to hundredths. They develop fluency in these computations, and make reasonable estimates of their
results. Students use the relationship between decimals and fractions, as well as the relationship between finite decimals and whole numbers (i.e., a finite
decimal multiplied by an appropriate power of 10 is a whole number), to understand and explain why the procedures for multiplying and dividing finite
decimals make sense. They compute products and quotients of decimals to hundredths efficiently and accurately.
5.NBT.5 Fluently multiply 5.NBT.5 refers to fluency which means accuracy (correct answer), efficiency (a reasonable amount of steps), and flexibility
multi-digit whole numbers (using strategies such as the distributive property or breaking numbers apart also using strategies according to the numbers
using the standard algorithm. in the problem, 26 x 4 may lend itself to (25 x 4 ) + 4 where as another problem might lend itself to making an equivalent
problem 32 x 4 = 64 x 8)). This standard builds upon students’ work with multiplying numbers in Third and Fourth Grade.
In Fourth Grade, students developed understanding of multiplication through using various strategies. While the standard
algorithm is mentioned, alternative strategies are also appropriate to help students develop conceptual understanding. The
size of the numbers should NOT exceed a three-digit factor by a two-digit factor.

Examples of alternative strategies:
There are 225 dozen cookies in the bakery. How many cookies are there?

Student 1 Student 2 Student 3
225 x 12 225x12 I doubled 225 and cut
I broke 12 up into I broke up 225 into 200 and 25. 12 in half to get 450 x
10 and 2. 200 x 12 = 2,400 6. I then doubled 450
225 x 10 = 2,250 I broke 25 up into 5 x 5, so I had 5 again and cut 6 in half
225 x 2 = 450 x 5 x12 or 5 x 12 x 5. to get 900 x 3.
2,250 + 450 = 5 x12= 60. 60 x 5 = 300 900 x 3 = 2,700.
2,700 I then added 2,400 and 300
2,400 + 300 = 2,700.


Draw a array model for 225 x 12…. 200 x 10, 200 x 2, 20 x 10, 20 x 2, 5 x 10, 5 x 2

225 x 12

200 20 5
2,000
400
10 2,000 200 50 200
40
50
+ 10
2 400 40 10 2,700

5.NBT.6 Find whole-number 5.NBT.6 references various strategies for division. Division problems can include remainders. Even though this standard
quotients of whole numbers leads more towards computation, the connection to story contexts is critical. Make sure students are exposed to problems
with up to four-digit dividends where the divisor is the number of groups and where the divisor is the size of the groups.
and two-digit divisors, using
strategies based on place value, Properties – rules about how numbers work
the properties of operations,
and/or the relationship between There are 1,716 students participating in Field Day. They are put into teams of 16 for the competition. How many teams get
created? If you have left over students, what do you do with them?
multiplication and division.
Illustrate and explain the
calculation by using equations,
rectangular arrays, and/or Student 1 Student 2
area models. 1,716 divided by 16 1,716 divided by 16.
There are 100 16’s in 1,716. There are 100 16’s in 1716
1,716 – 1,600 = 116 1,716. -1600 100
I know there are at least 6 16’s. Ten groups of 16 is 160. 116
116 - 96 = 20 That’s too big. -80
I can take out at least 1 more 16. Half of that is 80, which is 5
20 - 16 = 4 5 groups. 36
There were 107 teams with 4 students I know that 2 groups of -32 2
left over. If we put the extra students 16’s is 32. 4
on different team, 4 teams will have 17 I have 4 students left over.
students.


Student 3 Student 4
1,716 ÷ 16 = How many 16’s are in 1,716?
I want to get to 1,716 We have an area of 1,716. I know that one side
I know that 100 16’s equals 1,600 of my array is 16 units long. I used 16 as the
I know that 5 16’s equals 80 height. I am trying to answer the question what
1,600 + 80 = 1,680 is the width of my rectangle if the area is 1,716
Two more groups of 16’s equals 32, and the height is 16. 100 + 7 = 107 R 4
which gets us to 1,712
I am 4 away from 1,716
So we had 100 + 6 + 1 = 107 teams 100 7
Those other 4 students can just hang
out 16 100 x 16 = 1,600 7 x 16
=112

1,716 - 1,600 = 116 116 -
112 =
4

5.NBT.7 Add, subtract, 5.NBT.7 builds on the work from Fourth Grade where students are introduced to decimals and compare them. In Fifth
multiply, and divide decimals Grade, students begin adding, subtracting, multiplying and dividing decimals. This work should focus on concrete models
to hundredths, using concrete and pictorial representations, rather than relying solely on the algorithm. The use of symbolic notations involves having
models or drawings and students record the answers to computations (2.25 x 3= 6.75), but this work should not be done without models or pictures.
strategies based on place value, This standard includes students’ reasoning and explanations of how they use models, pictures, and strategies.
properties of operations, and/or
the relationship between
addition and subtraction; relate
the strategy to a written method
and explain the reasoning used.


Example:
A recipe for a cake requires 1.25 cups of milk, 0.40 cups of oil, and 0.75 cups of water. How much liquid is in the mixing
bowl?
Student 1
1.25 + 0.40 + 0.75
First, I broke the numbers apart:
I broke 1.25 into 1.00 + 0.20 + 0.05
I left 0.40 like it was.
I broke 0.75 into 0.70 + 0.05
I combined my two 0.05s to get 0.10.
I combined 0.40 and 0.20 to get 0.60.
I added the 1 whole from 1.25.
I ended up with 1 whole, 6
tenths, 7 more tenths and 1
1.00 0.20 0.05 more + 0.05 = 0.10
0.05 tenth which equals 2.40
1.00 1.00 cups.
0.40
.00 .00
1.00 0.70 0.05
1.00 1.00
.00 1.00
1.00 .00
1.00

0.40 + 0.20 = 0.60


Student 2
I saw that the 0.25 in 1.25 and the 0.75 for water would combine to equal 1 whole.
I then added the 2 wholes and the 0.40 to get 2.40.
.25 + .75 + 1 + .40 = 2.40

0 50 1 1.5 2 2.5 3

Example of Multiplication:
A gumball costs $0.22. How much do 5 gumballs cost? Estimate the total, and then calculate. Was your estimate close?

I estimate that the total cost will be a little more than a dollar. I know that 5 20’s equal 100 and we have 5 22’s.

I have 10 whole columns shaded and 10 individual boxes shaded. The 10 columns equal 1 whole. The 10 individual boxes equal 10 hundredths or 1
tenth. My answer is $1.10.

My estimate was a little more than a dollar, and my answer was $1.10. I was really close.


Example of Division:
A relay race lasts 4.65 miles. The relay team has 3 runners. If each runner goes the same distance, how far does each team member run? Make an
estimate, find your actual answer, and then compare them.

Runner 1 Runner 3
Runner 1 Runner 2 Runner 3

Runner 2

My estimate is that each runner runs between 1 and 2 miles. If each runner went 2 miles, that would be a total of 6 miles which is too high. If each
runner ran 1 mile, that would be 3 miles, which is too low.

I used the 5 grids above to represent the 4.65 miles. I am going to use all of the first 4 grids and 65 of the squares in the 5th grid. I have to divide the 4
whole grids and the 65 squares into 3 equal groups. I labeled each of the first 3 grids for each runner, so I know that each team member ran at least 1
mile. I then have 1 whole grid and 65 squares to divide up. Each column represents one-tenth. If I give 5 columns to each runner, that means that each
runner has run 1 whole mile and 5 tenths of a mile. Now, I have 15 squares left to divide up. Each runner gets 5 of those squares. So each runner ran 1
mile, 5 tenths and 5 hundredths of a mile. I can write that as 1.55 miles.

My answer is 1.55 and my estimate was between 1 and 2 miles. I was pretty close.


Number and Operation – Fractions 5.NF
Common Core Cluster
Use equivalent fractions as a strategy to add and subtract fractions.
Students apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as
equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of
them.
5.NF.1 Add and subtract fractions with 5.NF.1 builds on the work in Fourth Grade where students add fractions with like denominators. In Fifth Grade,
unlike denominators (including mixed the example provided in the standard has students find a common denominator by finding the product of both
numbers) by replacing given fractions denominators. For 1/3 + 1/6, a common denominator is 18, which is the product of 3 and 6. This process should
with equivalent fractions in such a be introduced using visual fraction models (area models, number lines, etc.) to build understanding before moving
way as to produce an equivalent sum into the standard algorithm.
or difference of fractions with like
denominators. Example:
Present students with the problem 1/3 + 1/6. Encourage students to use the clock face as a model for solving the
For example, 2/3 + 5/4 = 8/12 + 15/12
problem. Have students share their approaches with the class and demonstrate their thinking using the clock
= 23/12. (In general, a/b + c/d = (ad +
model.
bc)/bd.)

5.NF.2 Solve word problems involving 5.NF.2 refers to number sense, which means students’ understanding of fractions as numbers that lie between
addition and subtraction of fractions whole numbers on a number line. Number sense in fractions also includes moving between decimals and fractions
referring to the same whole, including to find equivalents, also being able to use reasoning such as 7/8 is greater than ¾ because 7/8 is missing only 1/8
cases of unlike denominators, e.g., by and ¾ is missing ¼ so 7/8 is closer to a whole Also, students should use benchmark fractions to estimate and
using visual fraction models or examine the reasonableness of their answers. Example here such as 5/8 is greater than 6/10 because 5/8 is 1/8


equations to represent the problem. Use larger than ½(4/8) and 6/10 is only 1/10 larger than ½ (5/10)
benchmark fractions and number Example:
sense of fractions to estimate mentally Your teacher gave you 1/7 of the bag of candy. She also gave your friend 1/3 of the bag of candy. If you and your
and assess the reasonableness of friend combined your candy, what fraction of the bag would you have? Estimate your answer and then calculate.
answers. For example, recognize an How reasonable was your estimate?
incorrect result 2/5 + 1/2 = 3/7, by
observing that 3/7 < 1/2. Student 1
1/7 is really close to 0. 1/3 is larger than 1/7, but still less than 1/2. If we put them together we
might get close to 1/2.
1/7 + 1/3= 3/21 + 7/21 = 10/21. The fraction does not simplify. I know that 10 is half of 20, so
10/21 is a little less than ½.
Another example: 1/7 is close to 1/6 but less than 1/6, and 1/3 is equivalent to 2/6, so I have a
little less than 3/6 or ½.

Common Core Cluster
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
Students also use the meaning of fractions, of multiplication and division, and the relationship between multiplication and division to understand and explain
why the procedures for multiplying and dividing fractions make sense. (Note: this is limited to the case of dividing unit fractions by whole numbers and
whole numbers by unit fractions.)
What does this standards mean a child will know and be able to do?
5.NF.3 Interpret a fraction as 5.NF.3 calls for students to extend their work of partitioning a number line from Third and Fourth Grade. Students need
division of the numerator by the ample experiences to explore the concept that a fraction is a way to represent the division of two quantities.
denominator (a/b = a ÷ b). Solve
word problems involving division
of whole numbers leading to
answers in the form of fractions
or mixed numbers, e.g., by using
visual fraction models or
equations to represent the
problem.
For example, interpret 3/4 as the
result of dividing 3 by 4, noting


that 3/4 multiplied by 4 equals 3,
and that when 3 wholes are
shared equally among 4 people
each person has a share of size
3/4. If 9 people want to share a
50-pound sack of rice equally by
weight, how many pounds of rice
should each person get? Between
what two whole numbers does
your answer lie?
Example:
Your teacher gives 7 packs of paper to your group of 4 students. If you share the paper equally, how much paper does each student get?

Student 1 Student 2 Student 3 Student 4 1 2 3 4 1 2 3 4 1 2 3 4

Pack 1 pack 2 pack 3 pack 4 pack 5 pack 6 pack 7

Each student receives 1 whole pack of paper and ¼ of the each of the 3 packs of paper. So each student gets 1 ¾ packs of paper.

5.NF.4 Apply and extend Students need to develop a fundamental understanding that the multiplication of a fraction by a whole number could be
previous understandings of represented as repeated addition of a unit fraction (e.g., 2 x (1/4) = 1/4 + ¼
multiplication to multiply a
fraction or whole number by a 5.NF.4 extends student’s work of multiplication from earlier grades. In Fourth Grade, students worked with recognizing
fraction. that a fraction such as 3/5 actually could be represented as 3 pieces that are each one-fifth (3 x (1/5)). In Fifth Grade,
a. Interpret the product (a/b) × q students are only multiplying fractions less than one. They are not multiplying mixed numbers until Sixth Grade.
as a parts of a partition of q
into b equal parts; 5.NF.4a references both the multiplication of a fraction by a whole number and the multiplication of two fractions.
equivalently, as the result of a Visual fraction models (area models, tape diagrams, number lines) should be used and created by students during their
sequence of operations a × q work with this standard.
÷ b. Example:
For example, use a visual Three-fourths of the class is boys. Two-thirds of the boys are wearing tennis shoes. What fraction of the class are boys
with tennis shoes?
fraction model to show (2/3)
This question is asking what 2/3 of ¾ is, or what is 2/3 x ¾. What is 2/3 x ¾, in this case you have 2/3 groups of size ¾
× 4 = 8/3, and create a story
( a way to think about it in terms of the language for whole numbers is 4 x 5 you have 4 groups of size 5.
context for this equation. Do The array model is very transferable from whole number work and then to binomials.
the same with (2/3) × (4/5) =
8/15. (In general, (a/b) ×


(c/d) = ac/bd.) Student 1 Student 3
I drew a rectangle to represent the whole Fraction circle could be used to model
class. The four columns represent the fourths student thinking.
of a class. I shaded 3 columns to represent First I shade the fraction circle to show the
the fraction that are boys. I then split the ¾ and then overlay with 2/3 of that?
rectangle with horizontal lines into thirds.
The dark area represents the fraction of the
boys in the class wearing tennis shoes,
which is 6 out of 12. That is 6/12, which
equals 1/2.
¼ ¼ ¼ ¼
1/3

1/3

1/3

Student 2

0 ¼ ½ ¾ 1

b. Find the area of a rectangle 5.NF.4b extends students’ work with area. In Third Grade students determine the area of rectangles and composite
with fractional side lengths rectangles. In Fourth Grade students continue this work. The Fifth Grade standard calls students to continue the process
by tiling it with unit squares of covering (with tiles). Grids (see picture) below can be used to support this work.
of the appropriate unit Example:
fraction side lengths, and The home builder needs to cover a small storage room floor with carpet. The storage room is 4 meters long and half of a
show that the area is the same meter wide. How much carpet do you need to cover the floor of the storage room? Use a grid to show your work and
as would be found by explain your answer.
In the grid below I shaded the top half of 4 boxes. When I added them together, I added ½ four times, which equals 2. I
multiplying the side lengths.


Multiply fractional side could also think about this with multiplication ½ x 4 is equal to 4/2 which is equal to 2.
lengths to find areas of
rectangles, and represent
fraction products as
rectangular areas.

5.NF.5 Interpret multiplication as
scaling (resizing), by:
a. Comparing the size of a
product to the size of one 5.NF.5a calls for students to examine the magnitude of products in terms of the relationship between two types of
factor on the basis of the size problems. This extends the work with 5.OA.1.
of the other factor, without
performing the indicated Example 1: Example 2:
multiplication. Mrs. Jones teaches in a room that is 60 How does the product of 225 x 60
feet wide and 40 feet long. Mr. Thomas compare to the product of 225 x 30?
teaches in a room that is half as wide, but How do you know?
has the same length. How do the Since 30 is half of 60, the product of 22
dimensions and area of Mr. Thomas’ 5x 60 will be double or twice as large as
classroom compare to Mrs. Jones’ room? the product of 225 x 30.
Draw a picture to prove your answer.

b. Explaining why multiplying 5.NF.5b asks students to examine how numbers change when we multiply by fractions. Students should have ample
a given number by a fraction opportunities to examine both cases in the standard: a) when multiplying by a fraction greater than 1, the number
greater than 1 results in a increases and b) when multiplying by a fraction less the one, the number decreases. This standard should be explored
product greater than the given and discussed while students are working with 5.NF.4, and should not be taught in isolation.
number (recognizing
multiplication by whole Example:
numbers greater than 1 as a Mrs. Bennett is planting two flower beds. The first flower bed is 5 meters long and 6/5 meters wide. The second flower
bed is 5 meters long and 5/6 meters wide. How do the areas of these two flower beds compare? Is the value of the area
familiar case); explaining
larger or smaller than 5 square meters? Draw pictures to prove your answer.
why multiplying a given


number by a fraction less than
1 results in a product smaller
than the given number; and
relating the principle of
fraction equivalence a/b = (n
× a)/(n × b) to the effect of
multiplying a/b by 1.
5.NF.6 Solve real world problems 5.NF.6 builds on all of the work done in this cluster. Students should be given ample opportunities to use various
involving multiplication of strategies to solve word problems involving the multiplication of a fraction by a mixed number. This standard could
fractions and mixed numbers, include fraction by a fraction, fraction by a mixed number or mixed number by a mixed number.
e.g., by using visual fraction Example:
models or equations to represent There are 2 ½ bus loads of students standing in the parking lot. The students are getting ready to go on a field trip. 2/5
the problem. of the students on each bus are girls. How many busses would it take to carry only the girls?
Student 1 Student 2
I drew 3 grids and 1 grid represents 1 bus. I cut the third grid in half and I 2 ½ x 2/5=
marked out the right half of the third grid, leaving 2 ½ grids. I then cut each grid I split the 2 ½ into 2
into fifths, and shaded two-fifths of each grid to represent the number of girls. and ½
When I added up the shaded pieces, 2/5 of the 1st and 2nd bus were both shaded, 2 x 2/5 = 4/5
and 1/5 of the last bus was shaded. ½ x 2/5 = 2/10
2/5 + 2/5 + 1/5 = 5/5 = 1 whole bus. I then added 4/5 and
2/10. That equals 1
whole bus load.

5.NF.7 Apply and extend 5.NF.7 is the first time that students are dividing with fractions. In Fourth Grade students divided whole numbers, and
previous understandings of multiplied a whole number by a fraction. The concept unit fraction is a fraction that has a one in the denominator. For
division to divide unit fractions example, the fraction 3/5 is 3 copies of the unit fraction 1/5. 1/5 + 1/5 + 1/5 = 3/5 = 1/5 x 3 or 3 x 1/5
by whole numbers and whole
numbers by unit fractions.1
a. Interpret division of a unit
fraction by a non-zero whole
number, and compute such
quotients.


For example, create a story
context for (1/3) ÷ 4, and use
a visual fraction model to
show the quotient. Use the
relationship between
multiplication and division to
explain that (1/3) ÷ 4 = 1/12
because (1/12) × 4 = 1/3.
1
Students able to multiply
fractions in general can develop
strategies to divide fractions in
general, by reasoning about the
relationship between
multiplication and division. But
division of a fraction by a fraction
is not a requirement at this grade.

5.NF.7a asks students to work with story contexts where a unit fraction is divided by a non-zero whole number. Students should use various fraction models
and reasoning about fractions.

Example:
You have 1/8 of a bag of pens and you need to share them among 3 people. How much of the bag does each person get?
Student 1
Expression 1/ 8 ÷ 3

0 3/24 8/24 16/24 24/24
1/8


Student 2
I drew a rectangle and divided it into 8 columns to represent my 1/8. I shaded the first column. I
then needed to divide the shaded region into 3 parts to represent sharing among 3 people. I
shaded one-third of the first column even darker. The dark shade is 1/24 of the grid or 1/24 of
the bag of pens.

1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
1/3

1/3

1/3

Student 3
1/8 of a bag of pens divided by 3 people. I know that my answer will be less than 1/8 since I’m
sharing 1/8 into 3 groups. I multiplied 8 by 3 and got 24, so my answer is 1/24 of the bag of
pens. I know that my answer is correct because (1/24) x 3 = 3/24 which equals 1/8.

b. Interpret division of a whole 5.NF.7b calls for students to create story contexts and visual fraction models for division situations where a whole
number by a unit fraction, number is being divided by a unit fraction.
and compute such quotients. Example:
For example, create a story Create a story context for 5 ÷ 1/6. Find your answer and then draw a picture to prove your answer and use multiplication
context for 4 ÷ (1/5), and use to reason about whether your answer makes sense. How many 1/6 are there in 5?
a visual fraction model to Student
show the quotient. Use the The bowl holds 5 Liters of water. If we use a scoop that holds 1/6 of a Liter, how many scoops will we need
relationship between in order to fill the entire bowl?
multiplication and division to
I created 5 boxes. Each box represents 1 Liter of water. I then divided each box into sixths to represent the
explain that 4 ÷ (1/5) = 20
size of the scoop. My answer is the number of small boxes, which is 30. That makes sense since 6 x 5 = 30.
because 20 × (1/5) = 4.

1 = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 a whole has 6/6 so five wholes would be 6/6 + 6/6 + 6/6 + 6/6 + 6/6 =30/6


c. Solve real world problems 5.NF.7c extends students’ work from other standards in 5.NF.7. Student should continue to use visual fraction models
involving division of unit and reasoning to solve these real-world problems.
fractions by non-zero whole
numbers and division of Example:
whole numbers by unit How many 1/3-cup servings are in 2 cups of raisins?
fractions, e.g., by using visual
fraction models and Student
equations to represent the I know that there are three 1/3 cup servings in 1 cup of raisins. Therefore, there are 6 servings in
2 cups of raisins. I can also show this since 2 divided by 1/3 = 2 x 3 = 6 servings of raisins.
problem.
For example, how much
chocolate will each person
get if 3 people share ½ lb of
chocolate equally? How
many 1/3-cup servings are 2
cups of raisins?

Measurement and Data 5.MD
Common Core Cluster
Convert like measurement units within a given measurement system.

5.MD.1 Convert among different-sized 5.MD.1 calls for students to convert measurements within the same system of measurement in the context of
standard measurement units within a multi-step, real-world problems. Both customary and standard measurement systems are included; students
given measurement system (e.g., worked with both metric and customary units of length in Second Grade. In Third Grade, students work with
convert 5 cm to 0.05 m), and use these metric units of mass and liquid volume. In Fourth Grade, students work with both systems and begin conversions
conversions in solving multi-step, real within systems in length, mass and volume.
world problems. Students should explore how the base-ten system supports conversions within the metric system.
Example: 100 cm = 1 meter.
From previous grades: relative size, liquid volume, mass, length, kilometer (km), meter (m), centimeter (cm),
kilogram (kg), gram (g), liter (L), milliliter (mL), inch (in), foot (ft), yard (yd), mile (mi), ounce (oz), pound
(lb), cup (c), pint (pt), quart (qt), gallon (gal), hour, minute, second


Common Core Cluster
Represent and interpret data.

5. MD.2 Make a line plot to display a 5.MD.2 This standard provides a context for students to work with fractions by measuring objects to one-eighth of
data set of measurements in fractions a unit. This includes length, mass, and liquid volume. Students are making a line plot of this data and then adding
of a unit (1/2, 1/4, 1/8). Use and subtracting fractions based on data in the line plot.
operations on fractions for this grade
to solve problems involving
information presented in line plots.
For example, given different Example:
Students measured objects in their desk to the nearest ½, ¼, or 1/8 of an inch then displayed data collected on a line
measurements of liquid in identical
plot. How many object measured ¼? ½? If you put all the objects together end to end what would be the total
beakers, find the amount of liquid
length of all the objects?
each beaker would contain if the total
amount in all the beakers were
redistributed equally.

Common Core Cluster
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition.
Students recognize volume as an attribute of three-dimensional space. They understand that volume can be measured by finding the total number of same-
size units of volume required to fill the space without gaps or overlaps. They understand that a 1-unit by 1-unit by 1-unit cube is the standard unit for
measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating and measuring volume. They decompose
three-dimensional shapes and find volumes of right rectangular prisms by viewing them as decomposed into layers of arrays of cubes. They measure
necessary attributes of shapes in order to determine volumes to solve real world and mathematical problems.


5. MD.3 Recognize volume as an 5. MD.3, 5.MD.4, and 5. MD.5 represents the first time that students begin exploring the concept of volume. In
attribute of solid figures and Third Grade, students begin working with area and covering spaces. The concept of volume should be extended
understand concepts of volume from area with the idea that students are covering an area (the bottom of cube) with a layer of unit cubes and then
measurement. adding layers of unit cubes on top of bottom layer (see picture below). Students should have ample experiences
a. A cube with side length 1 unit, with concrete manipulatives before moving to pictorial representations.
called a “unit cube,” is said to have
“one cubic unit” of volume, and
can be used to measure volume.
b. A solid figure which can be packed
without gaps or overlaps using n
unit cubes is said to have a volume
of n cubic units.
(3 x 2) represented by first layer
5. MD.4 Measure volumes by counting (3 x 2) x 5 represented by number of
unit cubes, using cubic cm, cubic in, 3 x 2 layers
cubic ft, and improvised units. (3 x 2) + (3 x 2) + (3 x 2) + (3 x 2)+ (3 x
5. MD.5 Relate volume to the 2) = 6 + 6 + 6 + 6 + 6 + 6 = 30
6 representing the size/area of one layer
operations of multiplication and
addition and solve real world and
mathematical problems involving
volume.
a. Find the volume of a right
rectangular prism with whole-
number side lengths by packing it
with unit cubes, and show that the
5. MD.5a & b involves finding the volume of right rectangular prisms (see picture above). Students should have
volume is the same as would be experiences to describe and reason about why the formula is true. Specifically, that they are covering the bottom
found by multiplying the edge of a right rectangular prism (length x width) with multiple layers (height). Therefore, the formula (length x width
lengths, equivalently by x height) is an extension of the formula for the area of a rectangle.
multiplying the height by the area
of the base. Represent threefold
whole-number products as volumes, 5.MD.5c calls for students to extend their work with the area of composite figures into the context of volume.
e.g., to represent the associative Students should be given concrete experiences of breaking apart (decomposing) 3-dimensional figures into right
property of multiplication. rectangular prisms in order to find the volume of the entire 3-dimensional figure.
b. Apply the formulas V = l × w × h
and V = b × h for rectangular prisms


to find volumes of right rectangular Example:
prisms with whole-number edge
lengths in the context of solving
decomposed figure
real world and mathematical
problems.
c. Recognize volume as additive. Find
volumes of solid figures composed
of two non-overlapping right
rectangular prisms by adding the
volumes of the non-overlapping
parts, applying this technique to
solve real world problems. Example:


Example:

Geometry 5.G
Common Core Cluster
Graph points on the coordinate plane to solve real-world and mathematical problems.

5.G.1 Use a pair of perpendicular 5.G.1 and 5.G.2 deal with only the first quadrant (positive numbers) in the coordinate plane.
number lines, called axes, to define a
coordinate system, with the
intersection of the lines (the origin)
arranged to coincide with the 0 on each Example:
line and a given point in the plane Connect these points in order on the coordinate grid below:
located by using an ordered pair of (2, 2) (2, 4) (2, 6) (2, 8) (4, 5) (6, 8) (6, 6) (6, 4) and (6, 2).
numbers, called its coordinates.
Understand that the first number
indicates how far to travel from the


origin in the direction of one axis, and
the second number indicates how far to
travel in the direction of the second
axis, with the convention that the names
of the two axes and the coordinates What letter is formed on the grid?
correspond (e.g., x-axis and x-
coordinate, y-axis and y-coordinate). Solution: “M” is formed.

Example:
Plot these points on a coordinate grid.
Point A: (2,6)
Point B: (4,6)
Point C: (6,3)
Point D: (2,3)
Connect the points in order. Make sure to connect Point D back to Point A.
1. What geometric figure is formed? What attributes did you use to
identify it?
2. What line segments in this figure are parallel?
3. What line segments in this figure are perpendicular?
solutions: trapezoid, line segments AB and DC are parallel, segments AD and
DC are perpendicular

Example:
Emanuel draws a line segment from (1, 3) to (8, 10). He then draws a line
segment from (0, 2) to (7, 9). If he wants to draw another line segment that is
parallel to those two segments what points will he use?
5.G.2 Represent real world and 5.G.2 references real-world and mathematical problems, including the traveling from one point to another and
mathematical problems by graphing identifying the coordinates of missing points in geometric figures, such as squares, rectangles, and
points in the first quadrant of the parallelograms.
coordinate plane, and interpret
coordinate values of points in the
context of the situation.


Example:
Using the coordinate grid, which
ordered pair represents the
location of the School?
Explain a possible path from the
school to the library.

Common Core Cluster
Classify two-dimensional figures into categories based on their properties.

5.G.3 Understand that attributes 5.G.3 calls for students to reason about the attributes (properties) of shapes. Student should have experiences
belonging to a category of two- discussing the property of shapes and reasoning.
dimensional figures also belong to
all subcategories of that category. Example:
For example, all rectangles have Examine whether all quadrilaterals have right angles. Give examples and non-examples.
four right angles and squares are
rectangles, so all squares have four
right angles.
5.G.4 Classify two-dimensional 5.G.4 this stand build on what was done in 4th grade.
figures in a hierarchy based on Figures from previous grades: polygon, rhombus/rhombi, rectangle, square, triangle, quadrilateral, pentagon,
properties. hexagon, cube, trapezoid, half/quarter circle, circle


Example:
Create a Hierarchy Diagram using the following terms:
polygons – a closed plane figure formed Possible student solution:
from line segments that meet only at their
endpoints.
quadrilaterals - a four-sided polygon.
Polygons
rectangles - a quadrilateral with two pairs of
congruent parallel sides and four right Quadrilaterals
angles.
rhombi – a parallelogram with all four sides Rectangles Rhombi
equal in length.
square – a parallelogram with four congruent Square
sides and four right angles.

quadrilateral – a four-sided Possible student solution:
polygon.
parallelogram – a quadrilateral
with two pairs of parallel and
congruent sides.
rectangle – a quadrilateral with
two pairs of congruent, parallel
sides and four right angles.
rhombus – a parallelogram with
all four sides equal in length.
square – a parallelogram with four
congruent sides and four right
angles.

Student should be able to reason about the attributes of shapes by examining: What are ways to classify triangles?
Why can’t trapezoids and kites be classified as parallelograms? Which quadrilaterals have opposite angles congruent
and why is this true of certain quadrilaterals?, and How many lines of symmetry does a regular polygon have?


Common Core Unpacked Grade 5

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