Mississippi College and Career Readiness Standards for
Mathematics Scaffolding Document
Grade 5
September 2016 Page 1 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Operations and Algebraic Thinking (OA)
Write and interpret numerical expressions
5.OA.1
Use parentheses,
brackets, or braces in
numerical expressions,
and evaluate
expressions with these
symbols.
Desired Student Performance
A student should know
● The mathematics symbols for
operations of addition,
subtraction, multiplication,
and division.
● There are numerous ways to
write the different operations
and some situations require
different mathematical
symbols.
● Parentheses are often used
when working with
multiplication and can be
used to illustrate the
Associative Property of
Multiplication and the
A student should understand
● Mathematic symbols help
keep numeric expressions
organized.
● Parentheses group a set of
numbers and operation
symbols together and can also
represent the operation of
multiplication.
● How to attend to precision.
A student should be able to do
● Evaluate expressions by
solving within parentheses
first, within brackets second,
and finally within the braces.
● Recognize that not all
problems will contain all the
mathematical symbols, but
when they are present, an
order of operations must be
followed to complete the
problem.
● Use mathematical symbols
appropriately to organize
numerical expressions.
September 2016 Page 2 of 65
College- and Career-Readiness Standards for Mathematics
Distributive Property of
Multiplication.
● The difference between an
expression and an equation.
● Interpret numerical
expressions and evaluate
them.
● Evaluate, create, and write
numerical expressions.
September 2016 Page 3 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Operations and Algebraic Thinking (OA)
Write and interpret numerical expressions
5.OA.2
Write simple
expressions that record
calculations with
numbers, and interpret
numerical expressions
without evaluating
them. For example,
express the calculation
“add 8 and 7, then
multiply by 2” as 2 × (8
+ 7). Recognize that 3 ×
(18932 + 921) is three
times as large as 18932
+ 921, without having to
calculate the indicated
sum or product.
Desired Student Performance
A student should know
● Parentheses are used to
group expressions together.
● It is possible to multiply any
given expression by another
quantity.
● How the Distributive Property
of Multiplication can be
written as an expression.
A student should understand
● Word problems are real-world
situations and can be
represented using numerical
expressions.
● The expression 14 x 3 is the
same as (14)3, (10 + 4) x 3, or
(10 + 4) + (10 + 4) + (10 + 4).
(There are many other ways to
write the expression as well.)
● Decontextualizing a problem
and organizing the information
into a numeric expression is a
necessary part of
mathematics.
● How to attend to precision.
A student should be able to do
● Represent a word problem or
real-world situation as a
numeric expression.
● Write a problem in various
equivalent expressions.
● Use parentheses and other
mathematical symbols
appropriately.
● Use these mathematical
symbols appropriately to
organize numerical expressions.
● Interpret numerical expressions.
● Evaluate, create, and write
numerical expressions.
September 2016 Page 4 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Operations and Algebraic Thinking (OA)
Analyze patterns and relationships
5.OA.3
Generate two numerical
patterns using two
given rules. Identify
apparent relationships
between corresponding
terms. Form ordered
pairs consisting of
corresponding terms
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 number
0, generate terms in the
resulting sequences,
and observe that the
terms in one sequence
are twice the
Desired Student Performance
A student should know
● How to generate a number
pattern that follows a given
rule. For example: given the
rule “Add 3” and the starting
number 1, generate terms in
the resulting sequence and
observe that the terms appear
to alternate between odd and
even numbers.
A student should understand
● What an ordered pair is and
the relationship between the
coordinates and the
coordinate plane.
● Patterns and finding
relationships between
numbers.
● How to look for and express
regularity in repeated
reasoning.
● How to look for and make use
of structure.
A student should be able to do
● Create real-world and
mathematical problems that
require graphing points in
Quadrant I of a coordinate
plane.
● Interpret coordinate values of
points in the context of the
situation.
● Calculate terms of an ordered
pair given a rule that must be
followed.
● Explain the relationship
between two sets of patterns,
i.e., Given the rule “Add 2” and
a starting number 0, and given
September 2016 Page 5 of 65
College- and Career-Readiness Standards for Mathematics
corresponding terms in
the other sequence.
Explain informally why
this is so.
the rule “Add 6” and a starting
number 0, explain why the
terms in the second sequence
are three times greater than
the numbers in the first
sequence.
September 2016 Page 6 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Understand the Place Value System
5.NBT.1
Recognize that in a
multi-digit number, a
digit in one place
represents 10 times as
much as it represents in
the place to its right and
1/10 of what it
represents in the place
to its left (e.g., “In the
number 3.33, the
underlined digit
represents 3/10, which
is 10 times the amount
represented by the digit
to its right (3/100) and is
1/10 the amount
represented by the digit
to its left (3)).
Desired Student Performance
A student should know
● The names of the place value
columns for whole numbers.
● Ten ones compose a ten, ten
different tens compose a
hundred, and ten different
hundreds compose a
thousand.
● The value of a digit located in
the tenths or hundredths
place.
● The Base Ten System has
place value because it is a
positional notation system.
The numerals 0, 1, 2, 3, 4, 5,
6, 7, 8, and 9 can represent
A student should understand
● A nine in the tens position has
a different value than a nine in
the hundred’s position.
● Columns located to the left of
a given column have a greater
value than columns located to
the right of that column.
● Multiples of 10.
● A fraction bar represents
division.
● How to find the decimal
equivalents for fractions of
1/10, 1/100, 1/1000, etc.
● Multiplying by the fraction 1/10
is the same as dividing by 10,
A student should be able to do
● For a multi-digit number, tell
what value each digit holds.
For example, in 245, the 2 is in
the hundreds place and has a
value of 200.
● Explain the patterns of the
Base Ten System (each
position is 10 times the
position to its right and 1/10 of
the position to its left).
● Write an expression for a
multi-digit number to show the
quantity of each digit. For
example: 345.67 is equivalent
September 2016 Page 7 of 65
College- and Career-Readiness Standards for Mathematics
different values depending
upon their position within a
group of numerals. This is an
efficient way to represent
many quantities with few
numeric symbols.
multiplying by 1/100 is the
same as dividing by 100, etc.
to (3 x 100) + (4 x 10) + (5 x 1)
+ (6 x 1/10) + (7 x 1/100).
● Explain why dividing by 10 is
equivalent to multiplying by
1/10.
September 2016 Page 8 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Understand the Place Value System
5.NBT.2
Explain patterns in the
number of zeros of the
product when
multiplying a number by
powers of 10, and
explain patterns in the
placement of the
decimal point when a
decimal is multiplied or
divided by a power of
10. Use whole-number
exponents to denote
powers of 10.
Desired Student Performance
A student should know
● A conceptual understanding
of the multiplication of whole
numbers.
● A conceptual understanding
of the Distributive Property of
Multiplication.
● Division is the inverse of
multiplication.
● Place value has many
patterns.
● In place value, each column
has a value 10 times that of
the column to the right of it.
Each column has a value
A student should understand
● Exponents are related to the
operation of multiplication.
● The base is the number that is
being multiplied, while the
exponent is the number of
times the base is multiplied.
● Exponents can also be
referred to as powers.
● Patterns are a way of making
meaning without actually
evaluating. For example:
10
2
= 10 x 10 = 100
10
3
= 10 x 10 x 10 = 1,000
10
4
= 10 x 10 x 10 x 10 =
10,000
A student should be able to do
● Explain how the patterns of the
powers of ten relate to
numbers being multiplied by
them.
● Explain 10
2
is the same as
multiplying by 10 x 10, and the
product of this is 100.
● Explain why the problem
6.2 x 10
2
is the same as
6.2 x 100.
● Use patterns and reasoning to
place a decimal in a product or
quotient. For example: The
product of 3.1 x 10
2
must be
close to 300 because 3.1 is
September 2016 Page 9 of 65
College- and Career-Readiness Standards for Mathematics
1/10 of the column to the left
of it.
● How to multiply multi-digit
numbers by a single digit
number as well as multi-digit
numbers by a two-digit
number.
close to 3 and 3 x 100 = 300,
therefore the logical placement
of the decimal is between the
ones place and the tenths
place.
September 2016 Page 10 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Understand the Place Value System
5.NBT.3a
Read, write, and
compare decimals to
thousandths.
Read and write decimals
to thousandths using
base-ten numerals,
number names, and
expanded form, e.g.,
347.392 = 3 × 100 + 4 ×
10 + 7 × 1 + 3 × (1/10) + 9
× (1/100) + 2 × (1/1000).
Desired Student Performance
A student should know
● How to read and write whole
numbers using base-ten
numerals, number names,
and expanded form.
● The relationship between
fractions and their base ten
decimal equivalents.
● Equivalent decimal values.
For example: 0.6 is equivalent
to 0.60.
● How to compare decimals to
the hundredths.
● How to represent tenths and
hundredths using modeling.
A student should understand
● The patterns in the place
value system can be extended
beyond hundredths.
● Thousandths are 1/10 the
value of a hundredth, 1/100
the value of a tenth, and
1/1000 the value of one whole.
● There are multiple ways to
represent any given amount.
● There is no comma to
separate hundredths and
thousandths.
A student should be able to do
● Read and write decimals to
thousandths using base-ten
numerals, number names, and
expanded form.
● Convert numbers to word form
and expanded form.
● Compare the decimal amount
in the various forms and with
varying decimal place values.
September 2016 Page 11 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Understand the Place Value System
5.NBT.3b
Read, write, and
compare decimals to
thousandths.
Compare two decimals
to thousandths based
on meanings of the
digits in each place,
using >, =, and <
symbols to record the
results of comparisons.
Desired Student Performance
A student should know
● How to read and write whole
numbers using base-ten
numerals, number names,
and expanded form.
● How to compare whole
numbers based on the
meanings of the digits in each
place.
● The relationship between
fractions and their base ten
decimal equivalents.
● Equivalent decimal values.
For example: 0.6 is equivalent
to 0.60
A student should understand
● The patterns in the place
value system can be extended
beyond hundredths.
● Thousandths are 1/10 the
value of a hundredth, 1/100
the value of a tenth, and
1/1000 the value of one whole.
● There are multiple ways to
represent any given amount.
● There is no comma to
separate hundredths and
thousandths.
● The number of digits in a
base-ten decimal number
does not determine its value.
A student should be able to do
● Compare decimals to the
thousandths place by using
the symbols >, =, and <.
● Use visual models to show the
value of each digit in a base-
ten decimal number.
● Explain decimal equivalence
by using visual models and/or
fractional equivalence.
● Place decimals on a number
line to demonstrate an
understanding of value. Use
number lines that show tenths,
hundredths, and thousandths.
September 2016 Page 12 of 65
College- and Career-Readiness Standards for Mathematics
● How to compare decimals to
the hundredths place.
● How to represent tenths and
hundredths using modeling.
● The meanings of the symbols
>, =, and <.
For example: 0.7 > 0.299
because 0.7 is closer to one
whole than 0.299.
● Explain that tenths are placed
on a number line between
whole numbers, hundredths
are placed between tenths,
and thousandths are placed
between hundredths.
September 2016 Page 13 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Understand the Place Value System
5.NBT.4
Use place value
understanding to round
decimals to any place.
Desired Student Performance
A student should know
● How to use a number line to
round whole numbers.
● Decimal numbers can be
placed on a number line.
● How to round whole numbers.
For example: If rounding 48 to
the nearest tens place, it
rounds to 50.
A student should understand
● Tenths are placed on a
number line between whole
numbers, hundredths are
placed between tenths, and
thousandths are placed
between hundredths.
● Rounding decimal values is
very similar to rounding whole-
number values.
A student should be able to do
● Place decimals on a number
line.
● Use the number line to
determine what benchmark
number the original number is
closest to on the line.
● Given a base-ten decimal
number, students should be
able to explain what
benchmark two numbers the
given decimal is located
between.
● Round a decimal number to
any given place using place
value understanding.
September 2016 Page 14 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Perform operations with multi-digit whole numbers and decimals to the hundredths
5.NBT.5
Fluently multiply multi-
digit whole numbers
using the standard
algorithm.
Desired Student Performance
A student should know
● How to fluently recall basic
multiplication facts.
● Multiplication using strategies
based on place value. These
strategies could include
partial products algorithms,
distributive property,
rectangular arrays, and area
models.
A student should understand
● The standard algorithm of
multiplication is a “short cut”
for other visual and written
models.
● The standard algorithm
applies the same concepts of
the Distributive Property of
Multiplication. Every digit of
the multiplicand must be
multiplied by every digit in the
multiplier.
● The partial products in the
standard algorithm are the
A student should be able to do
● Explain each of the steps in
the standard multiplication
algorithm and how place value
plays an important role in each
step.
● Explain how the partial
products in the standard
algorithm relate to the place
value of the digits being
multiplied.
● Complete all of steps in the
standard algorithm with the
September 2016 Page 15 of 65
College- and Career-Readiness Standards for Mathematics
results of multiplying by each
digit in the multiplier. A two-
digit multiplier results in two
partial products. A three-digit
multiplier results in three
partial products, and so on.
corresponding place values
lined up appropriately.
● Adhere to precision and
determine the reasonableness
of the final product based on
the numbers multiplied.
● Complete the standard
algorithm fluently to multiply
multi-digit numbers.
September 2016 Page 16 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Perform operations with multi-digit whole numbers and decimals to the hundredths
5.NBT.6
Find whole-number
quotients of whole
numbers with up to
four-digit dividends and
two-digit divisors, using
strategies based on
place value, the
properties of
operations, and/or the
relationship between
multiplication and
division. Illustrate and
explain the calculation
by using equations,
rectangular arrays,
and/or area models.
Desired Student Performance
A student should know
● A conceptual knowledge of
division and division models.
● How to find whole-number
quotients and remainders of
up to four-digit dividends and
one-digit divisors.
● Division is the inverse of
multiplication.
● How to use visual models to
divide whole numbers.
● How to make sense of
problems and persevere in
solving them.
A student should understand
● Dividing with two-digit divisors
is conceptually the same as
dividing with a single-digit
divisor.
● One visual model may be
more appropriate than another
depending on the problem
context.
● The relationship between
multiplication and division.
A student should be able to do
● Divide a whole number
dividend with up to four digits
by a two-digit divisor using any
appropriate strategy.
● Use multiple strategies for
multi-digit division. Area
models illustrate a connection
to multiplication, partial
quotients make a connection
to place value, and concrete
models (base- ten blocks)
demonstrate the
decomposition needed in the
standard algorithm.
September 2016 Page 17 of 65
College- and Career-Readiness Standards for Mathematics
● Illustrate and explain the
solution strategy using
equations, rectangular arrays,
and/or area models.
● Reason with the value of the
dividend and the value of the
divisor to determine if a
quotient is reasonable.
September 2016 Page 18 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations in Base Ten (NBT)
Perform operations with multi-digit whole numbers and decimals to the hundredths
5.NBT.7
Add, subtract, multiply,
and divide decimals to
hundredths, using
concrete models (to
include, but not limited
to: base ten blocks,
decimal tiles, etc.) or
drawings and strategies
based on place value,
properties of
operations, and/or the
relationship between
addition and
subtraction; relate the
strategy to a written
method and explain the
reasoning used.
Desired Student Performance
A student should know
● How to add, subtract,
multiply, and divide whole
numbers using strategies
based on place value and the
properties of operations.
● Addition and subtraction are
inverse operations.
● Multiplication and division are
inverse operations.
● Place value is extremely
important when performing
operations.
A student should understand
● The concept of adding and
subtracting decimals is
conceptually the same as it is
for whole numbers.
● Number lines, concrete
models, and algorithms can all
be used to solve addition,
subtraction, multiplication, and
division problems with decimal
numbers as well as with whole
numbers.
● The relationship between
performing operations with
fractions and with decimal
numbers.
A student should be able to do
● Use number lines, concrete
models (base-ten blocks or
decimal grids) or visual
models to illustrate addition,
subtraction, multiplication, or
division of decimal numbers.
● Apply knowledge of fraction
multiplication and division to
perform decimal operations.
● Use reasoning to place the
decimal in a sum, difference,
product, or quotient.
● Explain how the placement of
the decimal in an answer is
September 2016 Page 19 of 65
College- and Career-Readiness Standards for Mathematics
related to the value of the
numbers calculated.
● Determine which method or
strategy is appropriate for the
given problem.
September 2016 Page 20 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Use equivalent fractions as a strategy to add and subtract fractions
5.NF.1
Add and subtract
fractions with unlike
denominators (including
mixed numbers) by
replacing given
fractions with
equivalent fractions in
such a way as to
produce an equivalent
sum or difference of
fractions with like
denominators. For
example, 2/3 + 5/4 = 8/12
+ 15/12 = 23/12. (In
general, a/b + c/d = (ad +
bc)/bd.)
Desired Student Performance
A student should know
● Adding fractions is joining
separate parts referring to the
same whole.
● How to create an equivalent
fraction for a given fraction using
visual fraction models.
● How to find common
denominators and create
equivalent fractions to compare
fractions.
● A unit fraction has a numerator of
1 and can be combined with
other unit fractions with the same
denominator.
A student should understand
● Equivalent fractions
represent the same part of
a whole. They make it
easier to perform operations
with fractions.
● Multiples and factors are
important and help in
finding equivalent fractions.
● Fractions can be estimated
to the nearest benchmark 0,
½, or 1 whole.
● Mixed numbers can also be
estimated to benchmarks.
● Fractions with different size
denominators can be
A student should be able to do
● Find a common denominator
and create equivalent
fractions for given fractions or
mixed numbers.
● Place a fraction or mixed
number on a number line and
then increase or decrease it
in value (move on the number
line) from this position to
perform an operation (adding
or subtracting).
● Use bar models or visual
models to represent the
adding or subtracting of
September 2016 Page 21 of 65
College- and Career-Readiness Standards for Mathematics
● How to add or subtract mixed
numbers with like denominators.
● Solve word problems involving
addition and subtraction of
fractions with like denominators
by using visual fraction models,
equations, and a number line.
placed on the same number
line.
● Improper fractions are
fractions that represent an
amount greater than one
whole.
fractions or mixed numbers
with unlike denominators.
September 2016 Page 22 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Use equivalent fractions as a strategy to add and subtract fractions
5.NF.2
Solve word problems
involving addition and
subtraction of fractions
referring to the same
whole, including cases
of unlike denominators,
e.g., by using visual
fraction models or
equations to represent
the problem. Use
benchmark fractions
and number sense of
fractions to estimate
mentally and assess the
reasonableness of
answers.
For example, recognize
an incorrect result 2/5 +
1/2 = 3/7, by observing
that 3/7 < 1/2.
Desired Student Performance
A student should know
● Adding fractions is joining
separate parts referring to the
same whole.
● How to use bar models, visual
models, a number line, and
equations to solve addition
and subtraction problems
involving fractions with like
denominators and fractions
with unlike denominators.
● How to compare fractions
with like and unlike
denominators.
● Equivalent fractions represent
the same part of a whole. They
make it easier to perform
operations with fractions.
● Multiples and factors are
important and help in finding
equivalent fractions.
● Fractions can be estimated to
the nearest benchmark 0, ½, or
1 whole.
● Mixed numbers also can be
estimated to benchmarks.
● Fractions with different size
denominators can be placed on
the same number line.
A student should be able to do
● Create equivalent fractions
for given fractions or mixed
numbers.
● Find a common denominator
for given fractions or mixed
numbers.
● Solve word problems
involving addition and
subtraction of fractions with
like or unlike denominators.
● Use bar models, equations,
or a number line to represent
adding or subtracting of
fractions with unlike
denominators.
September 2016 Page 23 of 65
College- and Career-Readiness Standards for Mathematics
● How to estimate a fraction to
the nearest benchmark 0, ½,
and 1.
that represent an amount
greater than one whole.
● Relate fractions to
benchmark fractions (0, ½, 1)
to determine if a solution is
reasonable.
September 2016 Page 24 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.3
Interpret a fraction as
division of the numerator
by the 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
Desired Student Performance
A student should know
● How to divide whole numbers
and what it means to divide
using the partitioning and
repeated subtraction models.
● Division is the inverse of
multiplication.
● How to use visual models to
divide whole numbers with
and without remainders.
● Equivalencies such as
2 tens = 20 ones, 1 = 3/3, 2
=8/4, 1 = 10/10, etc.
● How to make sense of
problems and persevere in
solving them.
● Quotients can be represented
with fractions.
● It is possible to share an
amount such as 3 with a
greater number like 4. The
process will require that the 3
be decomposed into smaller
parts.
● Contexts in word problems help
to determine what operation to
perform and what strategies
might be useful.
● Remainders can be interpreted
in multiple ways and may be
A student should be able to do
● Contextualize and
decontextualize word
problems involving division.
● Produce visual models (bar/
circle) to justify a division
such as 7/8 (i.e., draw 7
wholes and 8 groups.
Partition each whole into 8
pieces and then share the
parts with the 8 groups).
Each group will have seven
pieces, and each piece will
have a size of 1/8, thus each
group will receive 7/8.
September 2016 Page 25 of 65
College- and Career-Readiness Standards for Mathematics
of rice equally by weight,
how many pounds of rice
should each person get?
Between what two whole
numbers does your
answer lie?
number.
● Write an equation to
represent the division shown
in a visual model.
● Estimate the size of the
quotient (part) before
dividing. i.e., ¾ is less than 1
whole.
September 2016 Page 26 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.4a
Apply and extend
previous
understandings of
multiplication to
multiply a fraction or
whole number by a
fraction.
Interpret the product
(a/b) × q as a parts of a
partition of q into b
equal parts;
equivalently, as the
result of a sequence of
operations a × q ÷ b.
For example, use a
visual fraction model to
show (2/3) × 4 = 8/3, and
create a story context
for this equation. Do the
same with (2/3) × (4/5) =
8/15. (In general, (a/b) ×
(c/d) = ac/bd.)
Desired Student Performance
A student should know
● A strong conceptual
understanding of
multiplication as an operation.
● Multiplication can be viewed
as repeated addition, equal-
sized groups, or using an
area model.
● Multiplication by a number
greater than 1 yields a
product greater than the
factors.
● Multiplication by 1 yields a
product that is equal to one of
the factors.
● There are many different
multiplication models that can
be used. The model used
depends on the context of the
problem.
● Multiplying by a number that is
less than one whole will yield a
product that is less than one of
the factors.
A student should be able to do
● Multiply a fraction or whole
number by a fraction and
interpret the product.
● Use visual fraction models
and number lines to show the
steps used in solving a
problem involving
multiplication by a fraction.
● Use benchmarks to estimate
the product and determine if
the solution is reasonable.
● Contextualize and
decontextualize problems by
creating word problems
and/or equations that
September 2016 Page 27 of 65
College- and Career-Readiness Standards for Mathematics
● Multiplication is a
commutative operation.
represent different
multiplication situations and
models.
September 2016 Page 28 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.4b
Apply and extend
previous
understandings of
multiplication to
multiply a fraction or
whole number by a
fraction.
Find the area of a
rectangle with fractional
side lengths by tiling it
with unit squares of the
appropriate unit fraction
side lengths, and show
that the area is the same
as would be found by
multiplying the side
lengths. Multiply
fractional side lengths
to find areas of
rectangles, and
represent fraction
Desired Student Performance
A student should know
● How to find the area of a
given rectangle with whole-
number side lengths using
square units (tiling square
units and finding the total
number of square units).
● How to find the area of a
given rectangle with whole-
number side lengths using
multiplication and addition.
A student should understand
● Rectangles can have
fractional side lengths.
● Finding the area of a rectangle
with fractional side lengths is
similar to finding the area of a
rectangle with whole number
side lengths (the same
process is used for both).
● The total number of square
units used to tile a rectangle
represents the area of that
rectangle.
● Multiplication is a more
efficient process for finding the
area of a rectangle.
A student should be able to do
● Find the area of a rectangle
with fractional side lengths
using unit squares of the
appropriate unit fraction side
lengths.
● Find and explain the
relationship between the
fractional side lengths of the
square unit and the fractional
side lengths of the rectangle.
● Show that counting the square
units used to tile the rectangle
and multiplying the side
lengths of the rectangle
produce the same answer
September 2016 Page 29 of 65
College- and Career-Readiness Standards for Mathematics
products as rectangular
areas.
(similar to finding the area of a
rectangle with whole number
side lengths).
September 2016 Page 30 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.5a
Interpret multiplication
as scaling (resizing), by:
Comparing the size of a
product to the size of
one factor on the basis
of the size of the other
factor, without
performing the
indicated multiplication.
Desired Student Performance
A student should know
● The Identity Property of
Multiplication means that any
number multiplied by 1 equals
the original number. For
example: 6 x 1 = 6 or 126 x 1
= 126.
● How to compare fractions to
benchmarks 0, ½, and 1.
● Multiplication is used for
resizing (scaling).
● Multiplication can produce an
answer less than one or both
of the factors.
A student should understand
● Multiplying by a fraction less
than 1 will yield a product less
than one of the factors.
● When multiplying by
1
2
, the
product is half the value of that
factor times 1. For example, 6
×
1
2
= 3.
● Multiplying by a fraction less
than 1 will result in an answer
less than 6.
● When multiplying two proper
fractions, the product is a part
of a part of a whole. This
A student should be able to do
● Compare the size of a product
of two fractions to the size of
one of the factors, without
performing the indicated
multiplication,
● Make use of the structure of
multiplication with whole
numbers, and apply this
knowledge to predict an
outcome for multiplication of
fractions. (For example, 4x2=8
and 4x1=4; therefore,
multiplying 4 by a fraction less
than 1 will produce an answer
less than 4)
September 2016 Page 31 of 65
College- and Career-Readiness Standards for Mathematics
yields a product that is less
than both factors.
● Use benchmark fractions to
determine if a solution is
reasonable.
September 2016 Page 32 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.5b
Interpret multiplication
as scaling (resizing), by:
Explaining why
multiplying a given
number by a fraction
greater than 1 results in
a product greater than
the given number
(recognizing
multiplication by whole
numbers greater than 1
as a familiar case);
explaining 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
Desired Student Performance
A student should know
● Multiplication can be shown
using repeated addition,
equal-sized groups, or using
an area model.
● Multiplication is commutative.
● Multiplying a number by a
second number greater than
1 equals a product greater
than the original number.
● The Identity Property of
Multiplication means that any
number multiplied by 1 equals
the original number. i.e., 6 x 1
= 6 or b x 1 = b.
A student should understand
● There is pattern in multiplying
whole numbers.
For example:
6 x 3 = 18
6 x 2 = 12
6 x 1 = 6 (Identity Prop.)
6 x 0 = 0 (Zero Prop.)
● A whole number multiplied by
1 will always result in a
product equal to the original
whole number.
● A whole number multiplied by
zero will always result in a
product of zero.
A student should be able to do
● Predict the relative size of the
product for a given
multiplication problem based on
the two factors in the problem.
● Use patterns to reason/ justify
about the size of the product
when multiplying a whole
number by a fraction. For
example: 6 x a/b = must be less
than 6 but greater than 0,
because a/b < 1 and a/b > 0.
● Use patterns to reason/ justify
about the size of the product
when multiplying a fraction by a
fraction.
September 2016 Page 33 of 65
College- and Career-Readiness Standards for Mathematics
effect of multiplying a/b
by 1.
● The Zero Property of
Multiplication. a x 0 = 0
● How to look for and make use
of structure.
● How to look for and express
regularity in repeated
reasoning.
● A whole number can be
multiplied by a fraction.
For example:
3/4 x 3 = 9/4 = 2 1/4
3/4 x 2 = 6/4 = 1 1/2
3/4 x 1 = 3/4
3/4 x 3/3 = 9/12 = 3/4
3/4 x 2/3 = 6/12 = 1/2
3/4 x 0 = 0
Therefore 3/4 multiplied by a
number greater than 1 will
result in a product greater than
3/4. If 3/4 is multiplied by any
number less than one but
greater than 0 the product will
be less than ¾.
September 2016 Page 34 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.6
Solve real world
problems involving
multiplication of
fractions and mixed
numbers, e.g., by using
visual fraction models
or equations to
represent the problem.
Desired Student Performance
A student should know
● Multiplication can be shown
using repeated addition,
equal-sized groups, or using
an area model.
● Multiplying a number by a
second number greater than
1 equals a product greater
than the original number.
● The Identity Property of
Multiplication means that any
number multiplied by 1 equals
the original number. i.e., 6 x 1
= 6 or b x 1 = b.
● How to look for and make use
of structure.
A student should understand
● Mixed numbers can be written
as improper fractions.
● The concepts learned for
multiplying whole numbers
and fractions can be applied
when multiplying by mixed
numbers.
● Mixed numbers represent a
value greater than 1;
therefore, multiplying a
number by a mixed number
will yield an answer that is
greater than the given
number.
A student should be able to do
● Solve real-world multiplication
problems involving fractions
and mixed numbers by creating
a visual model or equation to
solve.
● Make use of patterns to solve
problems.
● Use prior knowledge of
multiplying by fractions (proper
or improper) to solve problems
such as the following:
6 x 4 ½ =
4 ½ = 9/2
So, 6 x 9/2 = 54/2 = 27
September 2016 Page 35 of 65
College- and Career-Readiness Standards for Mathematics
● How to look for and express
regularity in repeated
reasoning.
● Apply an understanding of the
Distributive Property of
Multiplication to solve
problems:
6 x 4 = 24
6 x ½ = 3
So, (6 x 4) + (6 x ½) = 27
September 2016 Page 36 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.7a
Apply and extend
previous
understandings of
division to divide unit
fractions by whole
numbers and whole
numbers by unit
fractions.
1
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
Desired Student Performance
A student should know
● How to divide whole numbers
and what it means to divide
using the partitioning and
repeated subtraction models.
● Division is the inverse of
multiplication.
● How to use visual models to
divide whole numbers with
and without remainders.
● How to contextualize division
problems using whole
numbers.
● Division is not a commutative
operation.
A student should understand
● Creating a visual model to
represent problems helps give
meaning to the problem and
what is happening in the
problem.
● The division model used to
solve a problem depends on
the context of the problem.
● The role of the dividend,
divisor, and quotient.
● A fraction can be divided by a
whole number and the result
will be less than the original
fraction because it was
partitioned into pieces.
A student should be able to do
● Create visual models and
divide unit fractions by whole
numbers.
● Reason through a division
problem (i.e., For ¼ ÷ 3, ask,
“Can ¼ be shared with three
groups?” Explain that if ¼ is
shared with three groups the
quotient will be smaller in size
than ¼).
● Interpret division of a unit
fraction by a non-zero whole
number and compute
quotients. Create a word
problem to represent division
September 2016 Page 37 of 65
College- and Career-Readiness Standards for Mathematics
(1/3) ÷ 4 = 1/12 because
(1/12) × 4 = 1/3.
● Unit fractions have a
numerator of one and can be
combined to create non-unit
fractions.
of a unit fraction by a non-zero
whole number.
September 2016 Page 38 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.7b
Apply and extend
previous
understandings of
division to divide unit
fractions by whole
numbers and whole
numbers by unit
fractions.
1
Interpret division of a
whole number by a unit
fraction, and compute
such quotients. For
example, create a story
context for 4 ÷ (1/5), and
use a visual fraction
model to show the
quotient. Use the
relationship between
multiplication and
division to explain that 4
÷ (1/5) = 20 because 20 ×
(1/5) = 4.
Desired Student Performance
A student should know
● How to divide whole numbers
and what it means to divide
using the partitioning and
repeated subtraction models.
● Division is the inverse of
multiplication.
● How to use visual models to
divide whole numbers with
and without remainders.
● Unit fractions have a
numerator of one.
● Division is not a commutative
operation.
A student should understand
● Creating a visual model to
represent problems helps to
give meaning to the problem
and what is happening in the
problem.
● The division model used to
solve a problem depends on
the context of the problem.
● The role of the dividend,
divisor, and quotient.
● A whole number can be
divided by a fraction and the
result will be greater than the
original whole number.
A student should be able to do
● Create visual models to divide
a whole number by a unit
fraction. Make meaning of a
problem, such as 6 ÷ ½ by
asking, “How many ½ are in
6?” (The quotient will be
greater than 6 because each
whole is composed of two
halves.)
● Create word problems to
represent division problems.
● Draw visual fraction models
(bar/circles) using the
appropriate number of wholes
to find out how many of the
September 2016 Page 39 of 65
College- and Career-Readiness Standards for Mathematics
given unit fraction are found in
the wholes.
September 2016 Page 40 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Number and Operations – Fractions (NF)
Apply and extend previous understandings of multiplication and division to multiply and divide fractions
5.NF.7c
Apply and extend
previous understandings
of division to divide unit
fractions by whole
numbers and whole
numbers by unit
fractions.
1
Solve real world
problems involving
division of unit fractions
by non-zero whole
numbers and division of
whole numbers by unit
fractions, e.g., by using
visual fraction models
and equations to
represent the problem.
For example, how much
chocolate will each
person get if 3 people
share 1/2 lb of chocolate
equally? How many 1/3-
Desired Student Performance
A student should know
● How to divide whole numbers
and what it means to divide
using the partitioning and
repeated subtraction models.
● Division is the inverse of
multiplication.
● How to use visual models to
divide whole numbers with
and without remainders.
● How to contextualize division
problems using whole
numbers.
● Division is not a commutative
operation.
A student should understand
● Creating a visual model to
represent problems helps give
meaning to the problem and
what is happening in the
problem.
● The division model used to
solve a problem depends on
the context of the problem.
● The role of the dividend,
divisor, and quotient.
● Various models can be used
to illustrate given problems.
A student should be able to do
● Solve real-world word
problems involving division of
unit fractions by non-zero
whole numbers.
● Solve real-world problems
involving division of whole
numbers by unit fractions.
● Use visual fraction models and
equations to represent word
problems and solve them.
● Use prior knowledge of
patterns in dividing fractions
and whole numbers to reason
through problems.
September 2016 Page 41 of 65
College- and Career-Readiness Standards for Mathematics
cup servings are in 2
cups of raisins?
● Unit fractions have a
numerator of one.
● How to make sense of
problems and persevere in
solving them.
● Use benchmark fractions to
estimate quotients and
determine the reasonableness
of solutions.
September 2016 Page 42 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Convert like measurement units within a given measurement system
5.MD.1
Convert among
different-sized standard
measurement units
within a given
measurement system
(e.g., convert 5 cm to
0.05 m), and use these
conversions in solving
multi-step, real world
problems.
Desired Student Performance
A student should know
● There are two systems of
measurement (metric and
customary).
● Relative sizes of the different
units in each of the two
different systems.
● There are multiple ways to
represent measurements and
equivalent measurements can
be expressed by using
different units.
● There are different units for
different types of measuring.
For example: There are
different units for mass,
A student should understand
● Units of measurement can be
expressed in terms of a larger
unit or a smaller unit. (For
example: 6 in. = 0.5 ft.)
● Conversions of measurement
units are sometimes
necessary when applying
measurement to the real
world.
● The metric system is a base-
ten system, and the customary
system works in various
bases.
A student should be able to do
● Solve multiple-step, real- world
problems using various units
of measurement (within the
same system).
● Explain equivalents within a
given measurement system.
● Use knowledge of whole
numbers, fractions, and
decimals to compare/convert
units of measurement within a
system.
● Use visual models for
conversions and solve
measurement problems.
September 2016 Page 43 of 65
College- and Career-Readiness Standards for Mathematics
height, capacity, length and
so on.
● Basic concepts of whole
numbers, fractions, and
decimals.
● Apply knowledge of base-ten
place value to conceptually
understand the conversion of
metric units.
● Use measurement tools
appropriately.
September 2016 Page 44 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Represent and interpret data
5.MD.2
Make a line plot to
display a data set of
measurements in
fractions of a unit
(1/2, 1/4, 1/8). Use
operations on fractions
for this grade to solve
problems involving
information presented
in line plots.
For example, given
different measurements
of liquid in identical
beakers, find the
amount of liquid each
beaker would contain if
the total amount in all
the beakers were
redistributed equally.
Desired Student Performance
A student should know
● How to partition a line into
halves, fourths, and eighths.
● How to use a line or line
segment to make a line plot.
● How to interpret and solve
problems with a line plot
using whole numbers and the
unit fractions of 1/2, 1/4, and
1/8.
● How to make sense of
problems and persevere in
solving them.
● How to add and subtract unit
fractions.
A student should understand
● A line plot is used to organize
data.
● Every piece of data in a data
set is displayed on the line plot
with a symbol. Intervals on
the line plot that do not have a
symbol do not contain data.
● Real-world problems can be
represented using a line plot.
A student should be able to do
● Collect real-world data using
fractions 1/2, 1/4, and 1/8, and
create a line plot to display the
results visually.
● Use the results of the line plot
to make observations and/or
inferences about the data.
● Answer questions using a line
plot that has already been
created.
● Use fraction operations of
addition, subtraction,
multiplication, and division to
solve real-world problems
using line plots.
September 2016 Page 45 of 65
College- and Career-Readiness Standards for Mathematics
● How to multiply unit fractions
by whole numbers.
● Find the mean (average) of a
set of data by leveling off the
line plot and redistributing the
data equally.
September 2016 Page 46 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.3a
Recognize volume as an
attribute of solid figures
and understand
concepts of volume
measurement.
A cube with side length
1 unit, called a “unit
cube,” is said to have
“one cubic unit” of
volume, and can be
used to measure
volume.
Desired Student Performance
A student should know
● The names and attributes of
two-dimensional shapes, in
particular, squares and
rectangles.
● The names and attributes of
three-dimensional shapes, in
particular, rectangular prisms
and cubes.
● Two dimensional figures can
be measured using area.
● How to find the area of a
figure using square units or
the standard algorithm.
A student should understand
● Volume is the space that can
be filled in a three-dimensional
figure similar to the way that
area is the space that can be
filled in a two-dimensional
figure.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents can be used to
describe square units and
cubic units.
● Volume can be “packed” or
“filled.” These two different
A student should be able to do
● Explain the concept of volume.
Provide examples in the real
world that represent a
measure of volume.
● Describe the difference
between square units and
cubic units.
● Make connections between
exponents and the relationship
they have with square units
and cubic units.
● Explain how the unit cube is
used to find the volume of an
object.
September 2016 Page 47 of 65
College- and Career-Readiness Standards for Mathematics
● How the formula for the area
of rectangles and squares is
derived.
ideas both represent volume.
For example: packing with unit
cubes versus filling with
liquid/gas.
● Use differing units such as
inches, centimeters, feet, etc.,
to construct a unit cube.
● Select the appropriate unit
cube to use to measure a
three-dimensional space.
September 2016 Page 48 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.3b
Recognize volume as an
attribute of solid figures
and understand
concepts of volume
measurement.
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.
Desired Student Performance
A student should know
● The names and attributes of
two-dimensional shapes
especially squares and
rectangles.
● The names and attributes of
three-dimensional shapes
especially rectangular prisms
and cubes.
● Two-dimensional figures can
be measured using area.
● How to find the area of a
figure using square units or
the standard algorithm.
A student should understand
● Volume is the space in a
three-dimensional figure.
● What it means to find the area
of a two-dimensional figure.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents are used to
describe square units and
cubic units.
● Volume can be “packed” or
“filled.” These two different
ideas both represent volume.
A student should be able to do
● Explain that when finding
volume, unit cubes must be
packed without gaps or
overlays inside a three-
dimensional space.
● The total number of unit cubes
(n) packed into a three-
dimensional figure equals the
volume of the figure.
● Look at examples of different
sized prisms packed with unit
cubes, some packed with no
gaps or overlays and others
packed in an unorganized
manner, and explain which
September 2016 Page 49 of 65
College- and Career-Readiness Standards for Mathematics
● How the formula for the area
of rectangles and squares is
derived.
Packing with unit cubes versus
filling with liquid/gas.
examples accurately represent
the volume of the prism.
September 2016 Page 50 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.4
Measure volumes by
counting unit cubes,
using cubic cm, cubic
in, cubic ft, and
improvised units.
Desired Student Performance
A student should know
● The names and attributes of
two-dimensional shapes,
especially squares and
rectangles.
● The names and attributes of
three-dimensional shapes,
especially rectangular prisms
and cubes.
● Two-dimensional figures can
be measured using area.
● How to find the area of a
figure using square units or
the standard algorithm.
A student should understand
● Volume is the space in a
three-dimensional figure.
● Finding the area for a two-
dimensional figure is similar to
finding the volume of a three-
dimensional figure.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents are used to
describe square units and
cubic units.
● A cubic unit has a length,
width, and height of 1 unit.
A student should be able to do
● Determine the volume of a
rectangular prism using a
concrete or pictorial example,
by counting unit cubes. The
unit cubes may be cubic
centimeters, cubic inches,
cubic feet, or other improvised
units. (These examples should
already have visible unit cubes
associated with them. For
example: a cube or
rectangular prism built from
snap cubes or inch cubes or a
drawing or picture of a
September 2016 Page 51 of 65
College- and Career-Readiness Standards for Mathematics
● How the formulas for the area
of rectangles and squares are
derived.
● Volume can be “packed” or
“filled.” These two different
ideas both represent volume.
Packing with unit cubes versus
filling with liquid/gas.
cube/rectangular prism with
individual unit cubes visible.)
September 2016 Page 52 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.5a
Relate volume to the
operations of
multiplication and
addition and solve real
world and mathematical
problems involving
volume.
Find the volume of a right
rectangular prism with
whole-number side
lengths by packing it with
unit cubes, and show
that the volume is the
same as would be found
by multiplying the edge
lengths, equivalently by
multiplying the height by
the area of the base.
Represent threefold
whole-number products
as volumes, e.g., to
Desired Student Performance
A student should know
● Volume is the space in a
three-dimensional figure.
● Volume can be “packed” or
“filled.” These two different
ideas can be problematic for
students. Packing with unit
cubes versus filling with
liquid/gas.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents are used to
describe square units and
cubic units.
A student should understand
● Unit cubes must be packed
into a prism or cube with no
gaps or overlays to accurately
measure volume.
● Volume can be measured with
cubic units that are improvised
or standardized. The
improvised unit will be referred
to solely as a cubic unit and
has a length, width, and height
of 1 unit.
● A unit cube with 1in. side
lengths is referred to as a
cubic inch, a unit cube with 1
cm. side lengths is referred to
A student should be able to do
● Pack real-world prisms/cubes
with unit cubes such as inch
cubes, centimeter cubes, and
improvised cubes. State the
volume of a given prism/cube
based on how many unit
cubes it holds.
● Calculate the volume of real-
world rectangular prisms by
counting the unit cubes used
for the length, width, and
height and multiplying them to
get the total number of unit
cubes in the volume.
September 2016 Page 53 of 65
College- and Career-Readiness Standards for Mathematics
represent the associative
property of
multiplication.
as a cubic centimeter, and a
unit cube with 1 ft. side lengths
is referred to as a cubic foot.
● It is possible to calculate the
volume of prisms and cubes
that have no unit cubes visible.
● Use addition to determine the
number of unit cubes or
volume in a three-dimensional
shape.
● Solve real-world problems
using the concepts related to
volume.
September 2016 Page 54 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.5b
Relate volume to the
operations of
multiplication and
addition and solve real
world and mathematical
problems involving
volume.
Apply the formulas
V = l × w × h and V = b ×
h for rectangular prisms
to find volumes of right
rectangular prisms with
whole-number edge
lengths in the context of
solving real world and
mathematical problems.
Desired Student Performance
A student should know
● Volume space inside a three-
dimensional figure.
● Volume can be “packed” or
“filled.” These two different
ideas can be problematic for
students. Packing with unit
cubes versus filling with
liquid/gas.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents are used to
describe cubic units.
A student should understand
● Unit cubes must be packed
into a prism or cube with no
gaps or overlays to accurately
measure volume.
● It is possible to calculate the
volume of rectangular
prisms/cubes without counting
every unit cube.
● Volume of rectangular prisms
can be found by multiplying
the total number of unit cubes
needed to form the length of
the prism by the total number
of unit cubes needed to form
the width of the prism by the
A student should be able to do
● Discover the formulas for
volume (l x w x h and b x h)
based on their knowledge of
packing unit cubes into three-
dimensional figures and
counting the cubes.
● Explain the different formulas
V=l x w x h and
V = b x h (l represents length, w
represents width, h represents
height, and b represents the
area of the base).
● Find the volume for real- world
problems using rectangular
September 2016 Page 55 of 65
College- and Career-Readiness Standards for Mathematics
number of unit cubes needed
for the height of the prism.
● The total number of unit cubes
in each layer is equivalent to
the area of the base.
prisms with whole number side
lengths.
September 2016 Page 56 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Measurement and Data (MD)
Geometric measurement: understand concepts of volume and relate volume to multiplication and to addition
5.MD.5c
Relate volume to the
operations of
multiplication and
addition and solve real
world and mathematical
problems involving
volume.
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.
Desired Student Performance
A student should know
● Volume is the space that can
be filled in a three-
dimensional figure.
● Volume can be “packed” or
“filled.” These two different
ideas can be problematic for
students. Packing with unit
cubes vs. filling with
liquid/gas.
● A cubic unit is similar to a
square unit. The difference is
that it has a third dimension,
height.
● Exponents are used to
describe cubic units.
A student should understand
● Unit cubes must be packed
into a prism or cube with no
gaps or overlays to accurately
measure volume.
● It is possible to calculate the
volume of rectangular
prisms/cubes without counting
every unit cube by applying
the formulas V= l x w x h or
V = b x h.
● The total number of unit cubes
in each layer of a rectangular
prism is equivalent to the area
of the base.
A student should be able to do
● Find the volume of different
rectangular prism/cubes by
counting unit cubes and
applying the formulas for
volume.
● Combine two different
rectangular prisms/cubes and
determine the total volume of
the combined prisms. Explain
that if two prisms are
combined, the total volume of
one prism is added to the
volume of the second prism.
● Find the volume of combined
rectangular prisms by
September 2016 Page 57 of 65
College- and Career-Readiness Standards for Mathematics
decomposing them into
separate figures, finding the
volume of each, and then
composing the figures back
together.
September 2016 Page 58 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Geometry (G)
Graph points on the coordinate plane to solve real-world and mathematical problems
5.G.1
Use a pair of
perpendicular 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 line and a given
point in the plane
located by using an
ordered pair of
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
Desired Student Performance
A student should know
● How to use a number line.
● Basic geometric concepts of
points, lines, line segments,
rays, perpendicular lines, and
parallel lines.
● Two lines that cross at a 90-
degree angle are
perpendicular lines.
● The meaning of the words
vertical and horizontal.
A student should understand
● When two lines cross they
form an intersection.
● When perpendicular lines
exist, a plane has been
partitioned by those lines into
fourths. The fourths are also
referred to as quarters. In the
case of coordinate planes,
each quarter is referred to as
a quadrant.
● Each line forming the
perpendicular line set is
labeled for identification. The
horizontal line is known as the
A student should be able to do
● Identify the different parts of the
coordinate grid. Know and
understand the following:
Origin
x-axis
y-axis
Ordered Pair
Quadrant I
Point/Coordinate
● Given an ordered pair, place a
point on the correct coordinate.
● Given a point in Quadrant I,
identify the correct ordered pair.
September 2016 Page 59 of 65
College- and Career-Readiness Standards for Mathematics
the second axis, with
the convention that the
names of the two axes
and the coordinates
correspond (e.g., x-axis
and x-coordinate, y-axis
and y-coordinate).
x-axis and the vertical line is
known as the y-axis.
● When two number lines form
perpendicular lines, a
coordinate grid is created.
● The point where the two lines
cross is known as the origin
and is a starting point.
● A point can be located and
identified by using the x- axis
and the y-axis.
● Explain how to correctly move
and locate points within
Quadrant I.
September 2016 Page 60 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Geometry (G)
Graph points on the coordinate plane to solve real-world and mathematical problems
5.G.2
Represent real world
and mathematical
problems by graphing
points in the first
quadrant of the
coordinate plane, and
interpret coordinate
values of points in the
context of the situation.
Desired Student Performance
A student should know
● Coordinate planes are
created when two
perpendicular lines cross and
a mathematical grid is placed
upon them.
● These perpendicular lines are
labeled as the x-axis and the
y-axis.
● Points within a plane can be
located using an ordered pair
which consists of an x-
coordinate and a y-
coordinate.
A student should understand
● Coordinate grids are a
mathematical concept that can
be applied to the real world.
● Coordinates are used in the
real world to help with locating
and direction. Lines of latitude
and longitude are an example
of how mathematical structure
is applied to the real world.
● Quadrant I can also be useful
when representing real-world
data. This quadrant can allow
us to look for trends in data or
changes in data over time.
A student should be able to do
● Locate points (coordinates)
and follow directions on a
coordinate grid that has been
contextualized using a real-
world example.
● Use maps, pictures, or
drawings with a coordinate
grid imposed upon it to create
real-world math problems that
involve locating and graphing
points within Quadrant I.
● Create Quadrant I using an x-
axis and y-axis and graph
points within Quadrant I that
relate to real-world data.
September 2016 Page 61 of 65
College- and Career-Readiness Standards for Mathematics
● Movement begins at the
origin, follows the x-axis first,
and the y-axis second.
Connect the points to look for
structure/patterns in the data.
This leads to the creation and
interpretation of line graphs.
September 2016 Page 62 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Geometry (G)
Classify two-dimensional figures into categories based on their properties
5.G.3
Understand that
attributes belonging to a
category of two-
dimensional figures
also belong to all
subcategories of that
category.
For example, all
rectangles have four
right angles and
squares are rectangles,
so all squares have four
right angles.
Desired Student Performance
A student should know
● Basic geometric concepts
such as points, lines, line
segments, rays, and angles.
● Special types of lines
including parallel and
perpendicular lines.
● Angles are obtuse, acute, or
right.
● Polygons are closed figures
with straight sides. There are
many different types of
polygons and each is named
based its number of sides and
angles.
A student should understand
● Polygons may appear different
in shape and size, but they
can still be classified together
based on their attributes.
● Regular polygons contain
equal length sides and are a
special type of polygon.
Irregular polygons do not have
to have the same length sides
but still contain attributes that
can be classified.
● Polygons must be classified
based on their attributes and
not solely on their
appearance.
A student should be able to do
● Given the attributes without a
visual picture, a student should
be able to classify and name
the polygon.
● Sort polygons, especially
quadrilaterals, into different
subcategories by explaining the
criterion by which they used to
sort the polygons.
● Compare and contrast the
different polygons.
● Justify, explain, and debate the
categorizing of different types
of polygons.
September 2016 Page 63 of 65
College- and Career-Readiness Standards for Mathematics
● Polygons can be classified
based on attributes. There
can also be a hierarchy for
certain polygons.
● The attributes of these
quadrilaterals: square,
rectangle, trapezoid, and
rhombus. For example, a
trapezoid is a quadrilateral
with at least 1 pair of parallel
sides.
● Different polygons may or may
not contain some of the same
attributes thus creating
subcategories.
For example: Are all
parallelograms squares?
When is a rhombus a square?
Are all squares rectangles?
September 2016 Page 64 of 65
College- and Career-Readiness Standards for Mathematics
GRADE 5
Geometry (G)
Classify two-dimensional figures into categories based on their properties
5.G.4
Classify two-
dimensional figures in a
hierarchy based on
properties.
Desired Student Performance
A student should know
● Basic geometric concepts
such as points, lines, line
segments, rays, and angles.
● Special types of lines
including parallel and
perpendicular lines.
● Angles are obtuse, acute, or
right.
● Polygons are closed figures
with straight sides. There are
many different types of
polygons, and each is named
based on the number of sides
and angles it has.
A student should understand
● Polygons may appear different
in shape and size but they can
still be classified together
based on their attributes.
● Regular polygons contain
equal length sides and are a
special type of polygon.
Irregular polygons do not have
to have the same length sides
but still contain attributes that
can be classified.
● Polygons must be classified
based on their attributes and
not solely on their
appearance.
A student should be able to do
● Given the attributes without a
visual picture, a student
should be able to classify,
draw, and name the polygon.
● Explain why squares are
unique among quadrilaterals.
● Create a hierarchy of
polygons, such as
quadrilaterals, sorted with
those with the most attributes
and narrowing down to those
with the fewest attributes.
September 2016 Page 65 of 65
College- and Career-Readiness Standards for Mathematics
● Polygons can be classified
based on attributes. There
can also be a hierarchy for
certain polygons.
● The attributes of these
quadrilaterals: square,
rectangle, trapezoid, and
rhombus. For example, a
trapezoid is a quadrilateral
with at least 1 pair of parallel
sides.
● Different polygons may or may
not contain some of the same
attributes thus creating
subcategories.
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 is not a requirement at this grade.