Kindergarten 
Grade 1 
Grade 2 
Grade 3 
Grade 4 
Grade 5 
Understand addition as putting together and adding to, and understand subtraction as taking apart and taking
from.
1.0 Represent addition and subtraction with objects, fingers, mental images, drawings,2 sounds (e.g., claps), acting out
situations, verbal explanations, expressions, or equations.

Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting
together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with
a symbol for the unknown number to represent the problem.

Represent and solve problems involving addition and subtraction.
1. Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking
from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations
with a symbol for the unknown number to represent the problem.

Represent and solve problems involving multiplication and division.
Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.
For example, describe a context in which a total number of objects can be expressed as 5 × 7.

1.0 Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as
7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations .

Write and interpret numerical expressions.
1.0 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these s

2.0 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to
represent the problem.

Represent and solve problems involving addition and subtraction.
2. 0 Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using
objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Add and subtract within 20.
2. Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two
onedigit numbers.

2.0 Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56
objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8
objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56÷8.

2.0 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a
symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1

2.0 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.
2.1 Express a whole number in the range 2–50 as a product of its prime factors. For example, find the prime factors of 24 and
express 24 as 2 × 2 × 2 × 3. CA
Analyze patterns and relationships.

3.o Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and
record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

Understand and apply properties of operations and the relationship between addition and subtraction.
3. Apply properties of operations as strategies to add and subtract.3 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also
known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

Work with equal groups of objects to gain foundations for multiplication.
3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or
counting them by 2s; write an equation to express an even number as a sum of two equal addends.

3. Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measure
ment quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1

3. Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations,
including problems in which remainders must be interpreted. Represent these problems using equations with a letter
standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation
strategies including rounding.

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 corresponding terms
in the other sequence. Explain informally why this is so.

4.o For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or
drawings, and record the answer with a drawing or equation.

Understand and apply properties of operations and the relationship between addition and subtraction.
4. Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10
when added to 8.

Work with equal groups of objects to gain foundations for multiplication.
4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns;
write an equation to express the total as a sum of equal addends.

4. Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example,
determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 =÷ 3, 6 × 6 = ?.

4. Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its
factors. Determine whether a given whole number in the range 1–100 is a multiple of a given onedigit number. Determine
whether a given whole number in the range 1–100 is prime or composite.

5.NBT.4: Use place value understanding to round
decimals to any place.

5.0 Fluently add and subtract within 5.

Add and subtract within 20.
5. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).


5. Apply properties of operations as strategies to multiply and divide.2 Examples:
If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by
3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that
8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.)

5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit
in the rule itself. 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. Explain informally why the numbers will
continue to alternate in this way.



Add and subtract within 20.
6. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 – 4 =
13 – 3 – 1 = 10 – 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one
knows 12 – 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent
6 + 6 + 1 = 12 + 1 = 13).


6. Understand division as an unknownfactor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when
multiplied by 8 



Work with addition and subtraction equations.
7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false.For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 – 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.


Multiply and divide within 100.
7. Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division
(e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from
memory all products of two onedigit numbers.
Division Facts

4.NF.7: Compare two decimals to
hundredths by reasoning about their size.
Recognize that comparisons are valid
only when the two decimals refer to the
same whole. Record the results of
comparisons with the symbols >, =, <,
and justify the conclusions, e.g., by
using a visual model.



Work with addition and subtraction equations.
8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example,
determine the unknown number that makes the equation true in each of the equations 8 += 11, 5 =– 3, 6 + 6 =.


Solve problems involving the four operations, and identify and explain patterns in arithmetic.
Solve twostep word problems using the four operations. Represent these problems using equations with a letter standing
for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies
including rounding.32






9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using
properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number
can be decomposed into two equal addends.


