1
1
1
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
formed by [a] parts of size 𝑏.
formed by [a] parts of size 𝑏.
formed by [a] parts of size 𝑏.
1
1
1
1
1
1
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
Understand a fraction 𝑏 as the
quantity formed by 1 part
when a whole is partitioned in
b equal parts; understand a
𝑎
fraction 𝑏 as the quantity
formed by [a] parts of size 𝑏.
formed by [a] parts of size 𝑏.
formed by [a] parts of size 𝑏.
1
1
1
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
Explain why a fraction 𝑎⁄𝑏 is
equivalent to a fraction
(𝑛 × 𝑎)⁄(𝑛 × 𝑏) by using
visual fraction models, with
attention to how the number
and size of the parts differ
even though the two fractions
themselves are the same size.
Use this principle to recognize
and generate equivalent
fractions.
𝑎
𝑎
𝑎
Understand a fraction 𝑏 with
Understand a fraction 𝑏 with
Understand a fraction 𝑏 with
𝑎 > 1 as a sum of fractions .
𝑎 > 1 as a sum of fractions .
𝑎 > 1 as a sum of fractions .
1
𝑏
𝑎
1
𝑏
𝑎
1
𝑏
𝑎
Understand a fraction 𝑏 with
Understand a fraction 𝑏 with
Understand a fraction 𝑏 with
𝑎 > 1 as a sum of fractions .
𝑎 > 1 as a sum of fractions .
𝑎 > 1 as a sum of fractions .
1
𝑏
1
𝑏
1
𝑏
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Apply and extend previous
understandings of
multiplication to multiply a
fraction by a whole number.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Interpret a fraction as division
of the numerator by the
denominator (𝑎𝑏=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 equation.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Apply and extend previous
understandings of
multiplication to multiply a
fraction or a whole number by
a fraction.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Solve real world problems
involving multiplication of
fractions and mixed numbers,
e.g., by using visual fraction
models or equations to
represent the problem.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Note: 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.
Note: 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.
Note: 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.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Apply and extend previous
understandings of division to
divide unit fractions by whole
numbers and whole numbers
by unit fractions.
Note: 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.
Note: 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.
Note: 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.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Interpret and compute
quotients of fractions, and
solve word problems involving
division of fractions, e.g., by
using visual fraction models
and equations to represent the
problem.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Solve real-world and
mathematical problems
involving the four operations
with rational numbers.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.
Note: Computations with rational numbers
extend the rules for manipulating fractions to
complex fractions.